From a9fe2c3dd30071ccb48baf3291b65d46960b14a6 Mon Sep 17 00:00:00 2001 From: Rahul Shah Date: Wed, 17 Feb 2016 15:28:00 -0500 Subject: [PATCH 001/402] KLSadd insertion first attempt --- KLSadd_insertion/FileChangerLatex.py | 64 + KLSadd_insertion/KLSadd.pdf | Bin 0 -> 414643 bytes KLSadd_insertion/KLSadd.tex | 2637 ++++++++++ KLSadd_insertion/KoLeSw.pdf | Bin 0 -> 4597173 bytes KLSadd_insertion/chap01.tex | 1542 ++++++ KLSadd_insertion/chap09.tex | 3037 ++++++++++++ KLSadd_insertion/chap14.tex | 5734 ++++++++++++++++++++++ KLSadd_insertion/linetest.py | 165 + KLSadd_insertion/newtempchap09.pdf | Bin 0 -> 319561 bytes KLSadd_insertion/newtempchap09.tex | 3934 +++++++++++++++ KLSadd_insertion/newtempchap14.tex | 6695 ++++++++++++++++++++++++++ KLSadd_insertion/svmono.cls | 1690 +++++++ KLSadd_insertion/tempchap09.tex | 3926 +++++++++++++++ KLSadd_insertion/tempchap14.tex | 5734 ++++++++++++++++++++++ KLSadd_insertion/updateChapters.py | 117 + KLSadd_insertion/updated14.tex | 4162 ++++++++++++++++ KLSadd_insertion/updated9.tex | 4162 ++++++++++++++++ 17 files changed, 43599 insertions(+) create mode 100644 KLSadd_insertion/FileChangerLatex.py create mode 100644 KLSadd_insertion/KLSadd.pdf create mode 100644 KLSadd_insertion/KLSadd.tex create mode 100644 KLSadd_insertion/KoLeSw.pdf create mode 100644 KLSadd_insertion/chap01.tex create mode 100644 KLSadd_insertion/chap09.tex create mode 100644 KLSadd_insertion/chap14.tex create mode 100644 KLSadd_insertion/linetest.py create mode 100644 KLSadd_insertion/newtempchap09.pdf create mode 100644 KLSadd_insertion/newtempchap09.tex create mode 100644 KLSadd_insertion/newtempchap14.tex create mode 100644 KLSadd_insertion/svmono.cls create mode 100644 KLSadd_insertion/tempchap09.tex create mode 100644 KLSadd_insertion/tempchap14.tex create mode 100644 KLSadd_insertion/updateChapters.py create mode 100644 KLSadd_insertion/updated14.tex create mode 100644 KLSadd_insertion/updated9.tex diff --git a/KLSadd_insertion/FileChangerLatex.py b/KLSadd_insertion/FileChangerLatex.py new file mode 100644 index 0000000..1ef75ff --- /dev/null +++ b/KLSadd_insertion/FileChangerLatex.py @@ -0,0 +1,64 @@ +""" +Rahul Shah +11/20/15 +FileChangerLatex.py +Reads in from KLSadd.tex, finds places to be changed and adds +to end of section +""" + +#Open the document to be read +#Figure out which chapter to read + +subsection = "" +a = -1 +chapter ="" +list = [] +with open("KLSadd.tex", "rb") as file: + for line in file: + #finds all mentions of subsections in KLSadd + #used to determine chapter number + #tests every subsection to find if it has a number + #eliminates Introduction, etc + chapter = "" + subsection = "" + toCopy = False + if("\\subsection*{" in line): + #finds the subsection and subsubsections + for word in line: + try: + chapter+= str(int(word)) + except: + a = -1 + + if(word =="."): + chapter += "." + + if(chapter is not ""): + list.append(chapter) + #END OF LOOP why can't I have nice brackets like java :( + """ + list contains all of the subsections. Begin chapter file searching + """ + words = "" + for line2 in file: + words += line2 + + print(words) + numFile = "" + fileList = [] + for s in list: + numFile = "" + if("." in s[0:2]): + numFile = s[0] + else: + numFile = s[0:2] + fileList.append(int(numFile)) + + #numFile now contains the right number of the file to access + #stored in fileList + + + + + file.close() +#end of FileChangerLatex.py diff --git a/KLSadd_insertion/KLSadd.pdf b/KLSadd_insertion/KLSadd.pdf new file mode 100644 index 0000000000000000000000000000000000000000..c0562e08dde4eec14d3e67b54f20dbaa5f694ed5 GIT binary patch literal 414643 zcma&NW6Urxx32lvwr$(CZQHhO+t&Nowr$(CjrsODGntd@NoF@qm8Snzd);@sNEJlH zXc_5Pp-AVKN7kVj83`B&?2W9TczB@bWlZhNT`UNgnAw>K{#Svb7qhf+F?AxK7qc;R zF%>a2wl^__;^TvIc5yN_w1x86jMb1%K4OFGovU-Wk*u|95{>dQl~2BQOOkADWL?^j zCIjCnvn?~N5SF}nZ|vg*0K7bKyOUE@j|LGy0^#F1_i)u2sj}Z2vG?io^Kt(pXiA+9 zO{&o2#Zwcdn-r>JQjHKnts48eOh5nev-jR&+wn>srn=Pr&4Yo3tKqIwMT2z>1;Jdo zijA>Zb7yIRo$yVI{Jnkb-VLaB6kp1dy`!1Dm8f_9so-suRju>){jQ%BNvdI!ic)GM zYi_Nfm-TK{oqArFexH#j#F<{JK6o<9?s+iNm_Sl1b7)1g%$tY94;!6l z(%fC(qe`Qk>;oGA30A98O39q;#zY3pSU=Z5p%gaWda2!fv9G2xi)Lq1CK8UH6gswS zpPMK`pJH(W>(M5FNJtPB{~E(O4f{Y<)CJXZlX2399_xZ`aHH_Ex|v1j;zh{4*CgQU zN)#f>KVmW4n#!!DS}KZYqVAry5B_!$fwm!*_ZSYnWhnF(2PbM=1GqQtdWZugbdZFq zB@AEUgYc=Zjao-y&LMsQZ41C1FeWd0M;I1lo&Sw1Gv9K>P&bQrhNQNgN#c$;D!jFI zF8N^pciF)n7f<2YWhM_?aS=_}&EX&&LCl-~oqKp{Vkw1_B)R`yHL}8#VQ%7r-#s|U+N4H`Bc&rFMKu9x=wW|F zi3CULagI~~K*5;zbE>UI2{6YYg-RU{7J(myAV60Ub(^rfP&5ytvfzNONm4h^z6=Q} zRkD5t<^C}OW&tc1c#=L*osfip<4oF} z7)6#24XSZWSJY2j%)C4#Y-96juv>RHr>=pI(K9Pt5{O)sg|%K7U|_A7*<0E5&}y)u zGN-sNF8)2$9?sU;EISleN0-`f=X;^1G-pbfx?SNybdgnU-?$KVeNB|V zx;e#YKJ(q3yyA^@ z!@X;VZU7@E{SIE*Y_HR*)v+lf7mT#(&(I(RjI5uRh1RanB+VOO?$Y^dpFts`IXrz& zFj9a%OB2ssC5xf+v>H!YWRmZ*HaCZ3ADuf3{?nDRqgcY%Ia>M)9&v%d;Vcz0R$SK@H++_CkKCdSmzJVXc`* zjWoy6zYgrmPP{1up?{E?v@_|H$r8V<_oPGN$I;ExTSyvBIB`LG{22A~szzX=aMjDS z?q}|s8A#nuk_@;IE<+C%JP=BQsEgMdhs>|4t8(JfGeFAJYOr8u7pn}=r@@8lGw(+& zr|d~&^8!$U^T)^*0SnE%4OG#y0qRupo`w__umd_(5C6b$jO{9{y(BGXvF~%6bt~aV zI|IQj9E3NAIr<(A(43tF%2!C9bku)wdnHvTXUG(CGxJ;tdfpLfk4~w)RapWK9pkV= z^pEeWAJ97jnt48BgD~2?<+u_`N1x73{e3y*EL7S}S?Yc^9xt*J(*;trVjO@nfd z7JaCISW36kziu9hF29Y3pN|$eM{s@eaQSQD`67jyUxy21^5oNleFsm@jVa;JOjzE9 z=ysJ4@0Gnw8^$TG+pMp${6LC$?T1`5snA_zDQS&rF2oCh{AUuOGyVB`zo1d8o!OyG z?M(h}ochn`zfg>s<-ao-BOA;An8|WAq@B0e5PHAV9lR)6kqL)`>&ntuYPNXbBh8&0 zn2;laNLEn}5>#-`*4)vdl}cpD<>-k5KwlpW9}EjA*1;6--!zZ={B?48`P4o8Ij4>^ zK+v-ZP-)~A2?FWmpZ1M#b;9OsdJ=fPgZRO+h_0oy+oI#q>GAT-0pm6BPH@!42b4r; zW>IOo*E2_yPKmw|Mi$iFpz-B^JN4vAqGdv_w-k42MbLKj=+o{WxHg9>SwXeR{9={i zyuWo9y32fNr z72UFg`ge68NGzvES(dM9eCZbkF~PAF=WT$kx_`c4V0wIBW+A_PLkr_$)H-PU!-9hd z9JKB}*#h_f?OeH)qTjO8+C-Oi-;TCZ_uOgxKCN6mahF-9MvF+H+D*}86jWIHoU~?&)PT28xQY^YT2u!=x$QA=&E~|< zG0SVvC0Bgeo6Yj7(?l}gPs}((0_YT0VXM^c#Jw6n7Y;YsZg~1zw!7@Qm|0^5u%v3f zyU2-e;Q^du`h(WXBaQSBzvNWqKo)TE51l}k>%f8^oEzI-ZQ8u73p2hZm~DV*62=WV zF~=K>dJKDe!vn!Pn^mX`g5(SN?_{hd0LNQ^FZgesFYLpt%y}>s3pL2Vvb_VV(0C?=^&OBG%5VDT&F^P-Xp|W7$a~Xn;g1B3DO2tb zj^ed)d!PG!D&uiZn+=_`vx;-~9eqp;&eaD=pYOC#aV8tB+ss-}i%x2S?aI79`Xow9 z;4PJE89_A3zgvdpC-bWF!eU>M!9?2Bn->E|VLk z0dmm*CfvqyfjMHukR7PG1y;=a%)xNDn1CT@65#av2e$c1KW6`K4z~`$`tGb( zHd1sDM#4u90lIX(j;siY*kcepz#urLL{n1m5i{jCt+%hng4O8~3AIsa^8-`Qat5gf ziHbvi zSJlM6$ww9!AmRi#B!P$aQ?)Z+qDP}qy%_TKDunK$IxBc81gP&{0j*Ri9BKycaTsjK`1rpzh(tXfNdHSDb8Q$tC*E!1&>H z>HsU^hTP$=Df4gM*C*vvc~4T|p}pTrt=c?1D`UE<+4%a?^84nWvi5R3sx@3`bY{?P z>5UGJl^FD=u_Nj!%VLe?ZR(0LL!4%4UWyBy@{$e$0%G#}^fLjK8zu=A*w#YYfQ{N# zI13;3XJ1boAuO?*F-nS|Tk`kt0Aoddic%ug0MiyR~^Bp&0x~Pk8`U7z4ulo7# zFBb>re}B0+nOOg~nLzOW?(O~j1ll1e+PkW@UQUqg@oHiDPROOy0VR1R61mmnH0s! 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+\documentclass[twoside,11pt]{article} +\usepackage[pdftex]{hyperref} +\hypersetup{ + colorlinks = true, + allcolors = {red}, +} +\usepackage{color} +\usepackage{amsmath} +\usepackage{amssymb} +\usepackage{xparse} +\usepackage{cite} +\renewcommand\citeform[1]{K#1} +\setlength{\textwidth}{16cm} +\setlength{\textheight}{21cm} +\setlength{\topmargin}{0cm} +\setlength{\oddsidemargin}{0.2mm} +\setlength{\evensidemargin}{0.2mm} + +\newcommand\sa{\smallskipamount} +\newcommand\sLP{\\[\sa]} +\newcommand\sPP{\\[\sa]\indent} +\newcommand\ba{\bigskipamount} +\newcommand\bLP{\\[\ba]} +\newcommand\CC{\mathbb{C}} +\newcommand\RR{\mathbb{R}} +\newcommand\ZZ{\mathbb{Z}} +\newcommand\al\alpha +\newcommand\be\beta +\newcommand\ga\gamma +\newcommand\de\delta +\newcommand\tha\theta +\newcommand\la\lambda +\newcommand\om\omega +\newcommand\Ga{\Gamma} +\newcommand\half{\frac12} +\newcommand\thalf{\tfrac12} +\newcommand\iy\infty +\newcommand\wt{\widetilde} +\newcommand\const{{\rm const.}\,} +\newcommand\Zpos{\ZZ_{>0}} +\newcommand\Znonneg{\ZZ_{\ge0}} +\newcommand{\hyp}[5]{\,\mbox{}_{#1}F_{#2}\!\left( + \genfrac{}{}{0pt}{}{#3}{#4};#5\right)} +\newcommand{\qhyp}[5]{\,\mbox{}_{#1}\phi_{#2}\!\left( + \genfrac{}{}{0pt}{}{#3}{#4};#5\right)} +\newcommand\LHS{left-hand side} +\newcommand\RHS{right-hand side} +\renewcommand\Re{{\rm Re}\,} +\renewcommand\Im{{\rm Im}\,} + +\NewDocumentCommand\mycite{m g}{% + \IfNoValueTF{#2} + {[\hyperlink{#1}{#1}]} + {[\hyperlink{#1}{#1}, #2]}% +} +\newcommand\mybibitem[1]{\bibitem[#1]{#1}\hypertarget{#1}{}} + +%\NewDocumentCommand{\myciteKLS}{m g}{% +% \IfNoValueTF{#2} +% {\hyperlink{KLS#1}{[#1]}} +% {\hyperlink{KLS#1}{[#1, #2]}}% +%} +\NewDocumentCommand{\myciteKLS}{m g}{% + \IfNoValueTF{#2} + {[\hyperlink{KLS#1}{#1}]} + {[\hyperlink{KLS#1}{#1}, #2]}% +} +\newcommand\mybibitemKLS[1]{\bibitem[#1]{#1}\hypertarget{KLS#1}{}} + +\begin{document} + +\title{Additions to the formula lists in +``Hypergeometric orthogonal polynomials and their $q$-analogues'' +by Koekoek, Lesky and Swarttouw} +\author{Tom H. Koornwinder} +\date{June 19, 2015} +\maketitle +\begin{abstract} +This report gives a rather arbitrary choice of formulas for +($q$-)hypergeometric orthogonal polynomials which the author missed +while consulting Chapters 9 and 14 in the book +``Hypergeometric orthogonal polynomials and their $q$-analogues'' +by Koekoek, Lesky and Swarttouw. The systematics of these chapters will be followed +here, in particular for the numbering of subsections and of references. +\end{abstract} +% +\subsection*{Introduction} +\label{sec_intro} +This report contains some formulas about ($q$-)hypergeometric +orthogonal polynomials which I missed but wanted to use +while consulting Chapters 9 and 14 in the book \mycite{KLS}: +\sLP +R. Koekoek, P.~A. Lesky and R.~F. Swarttouw, +{\em Hypergeometric orthogonal polynomials and their $q$-analogues}, +Springer-Verlag, 2010. +\sLP +These chapters form together the (slightly extended) successor of the report +\sLP +R.~Koekoek and R.~F. Swarttouw, +{\em The Askey-scheme of hypergeometric orthogonal +polynomials and its $q$-analogue}, +Report 98-17, Faculty of Technical Mathematics and Informatics, +Delft University of Technology, 1998; +\url{http://aw.twi.tudelft.nl/~koekoek/askey/}. +\sPP +Certainly these chapters give complete lists of formulas of special type, for instance +orthogonality relations and three-term recurrence relations. But outside these narrow +categories there are many other +formulas for ($q$-)orthogonal polynomials which one wants to have available. +Often one can find the desired formula in one of the +\hyperref[sec_ref1]{standard references} listed at the end of this report. +Sometimes it is only available in a journal or a less common monograph. +Just for my own comfort, I have brought together some of these formulas. +This will possibly also be helpful for some other users. + +Usually, any type of formula I give for a special class of polynomials, will suggest +a similar formula for many other classes, but I have not aimed at completeness +by filling in a formula of such type at all places. The resulting choice of formulas is +rather arbitrary, just depending on the formulas which I happened to need or which raised my interest. +For each formula I give a suitable reference or I sketch a +proof. +It is my intention to gradually extend this collection of formulas. +% +\subsection*{Conventions} +\label{sec_conv} +The (x.y) and (x.y.z) type subsection numbers, the +(x.y.z) type formula numbers, and the [x] type citation numbers +refer to \mycite{KLS}. +The (x) type formula numbers refer to this manuscript and the [Kx] type citation numbers refer to citations which are not in \mycite{KLS}. +Some standard references like \mycite{DLMF} +are given by special acronyms. + +$N$ is always a positive integer. Always assume $n$ to be a nonnegative +integer or, if $N$ is present, to be in $\{0,1,\ldots,N\}$. +Throughout assume $00$ and $0\le\frac{4xy}{(1-x-y)^2}<1$. +% +\paragraph{$q$-Hypergeometric series of base $q^{-1}$} +By \mycite{GR}{Exercise 1.4(i)}: +\begin{equation} +\qhyp rs{a_1,\ldots,a_r}{b_1,\ldots b_s}{q^{-1},z} +=\qhyp{s+1}s{a_1^{-1},\ldots a_r^{-1},0,\ldots,0} +{b_1^{-1},\ldots,b_s^{-1}}{q,\frac{qa_1\ldots a_rz}{b_1\ldots b_s}} +\label{154} +\end{equation} +for $r\le s+1$, $a_1,\ldots,a_r,b_1,\ldots,b_s\ne0$. +In the non-terminating case, for $00$ and $(c,d)=(\overline a,\overline b)$ or $(\overline b,\overline a)$.\\ +Thus, under these assumptions, the continuous Hahn polynomial +$p_n(x;a,b,c,d)$ +is symmetric in $a,b$ and in $c,d$. +This follows from the orthogonality relation (9.4.2) +together with the value of its coefficient of $x^n$ given in (9.4.4b).\\ +As a consequence, it is sufficient to give generating function (9.4.11). Then the generating +function (9.4.12) will follow by symmetry in the parameters. +% +\paragraph{Uniqueness of orthogonality measure} +The coefficient of $p_{n-1}(x)$ in (9.4.4) behaves as $O(n^2)$ as $n\to\iy$. +Hence \eqref{93} holds, by which the orthogonality measure is unique. +% +\paragraph{Special cases} +In the following special case there is a reduction to +Meixner-Pollaczek: +\begin{equation} +p_n(x;a,a+\thalf,a,a+\thalf)= +\frac{(2a)_n (2a+\thalf)_n}{(4a)_n}\,P_n^{(2a)}(2x;\thalf\pi). +\end{equation} +See \myciteKLS{342}{(2.6)} (note that in \myciteKLS{342}{(2.3)} the +Meixner-Pollaczek polynonmials are defined different from (9.7.1), +without a constant factor in front). + +For $0\al>-1,\quad m,n<-\thalf(\al+\be+1),\\ +&\frac{h_n}{h_0}= +\frac{n+\al+\be+1}{2n+\al+\be+1}\, +\frac{(\al+1)_n(\be+1)_n}{(\al+\be+2)_n\,n!}\,,\quad +h_0=\frac{2^{\al+\be+1}\Ga(\al+1)\Ga(-\al-\be-1)}{\Ga(-\be)}\,. +\end{split} +\label{122} +\end{equation} + +% +\paragraph{Symmetry} +\begin{equation} +P_n^{(\al,\be)}(-x)=(-1)^n\,P_n^{(\be,\al)}(x). +\label{48} +\end{equation} +Use (9.8.2) and (9.8.5b) or see \mycite{DLMF}{Table 18.6.1}. +% +\paragraph{Special values} +\begin{equation} +P_n^{(\al,\be)}(1)=\frac{(\al+1)_n}{n!}\,,\quad +P_n^{(\al,\be)}(-1)=\frac{(-1)^n(\be+1)_n}{n!}\,,\quad +\frac{P_n^{(\al,\be)}(-1)}{P_n^{(\al,\be)}(1)}=\frac{(-1)^n(\be+1)_n}{(\al+1)_n}\,. +\label{50} +\end{equation} +Use (9.8.1) and \eqref{48} or see \mycite{DLMF}{Table 18.6.1}. +% +\paragraph{Generating functions} +Formula (9.8.15) was first obtained by Brafman \myciteKLS{109}. +% +\paragraph{Bilateral generating functions} +For $0\le r<1$ and $x,y\in[-1,1]$ we have in terms of $F_4$ (see~\eqref{62}): +\begin{align} +&\sum_{n=0}^\iy\frac{(\al+\be+1)_n\,n!}{(\al+1)_n(\be+1)_n}\,r^n\, +P_n^{(\al,\be)}(x)\,P_n^{(\al,\be)}(y) +=\frac1{(1+r)^{\al+\be+1}} +\nonumber\\ +&\qquad\quad\times F_4\Big(\thalf(\al+\be+1),\thalf(\al+\be+2);\al+1,\be+1; +\frac{r(1-x)(1-y)}{(1+r)^2},\frac{r(1+x)(1+y)}{(1+r)^2}\Big), +\label{58}\sLP +&\sum_{n=0}^\iy\frac{2n+\al+\be+1}{n+\al+\be+1} +\frac{(\al+\be+2)_n\,n!}{(\al+1)_n(\be+1)_n}\,r^n\, +P_n^{(\al,\be)}(x)\,P_n^{(\al,\be)}(y) +=\frac{1-r}{(1+r)^{\al+\be+2}}\nonumber\\ +&\qquad\quad\times F_4\Big(\thalf(\al+\be+2),\thalf(\al+\be+3);\al+1,\be+1; +\frac{r(1-x)(1-y)}{(1+r)^2},\frac{r(1+x)(1+y)}{(1+r)^2}\Big). +\label{59} +\end{align} +Formulas \eqref{58} and \eqref{59} were first +given by Bailey \myciteKLS{91}{(2.1), (2.3)}. +See Stanton \myciteKLS{485} for a shorter proof. +(However, in the second line of +\myciteKLS{485}{(1)} $z$ and $Z$ should be interchanged.)$\;$ +As observed in Bailey \myciteKLS{91}{p.10}, \eqref{59} follows +from \eqref{58} +by applying the operator $r^{-\half(\al+\be-1)}\,\frac d{dr}\circ r^{\half(\al+\be+1)}$ +to both sides of \eqref{58}. +In view of \eqref{60}, formula \eqref{59} is the Poisson kernel for Jacobi +polynomials. The \RHS\ of \eqref{59} makes clear that this kernel is positive. +See also the discussion in Askey \myciteKLS{46}{following (2.32)}. +% +\paragraph{Quadratic transformations} +\begin{align} +\frac{C_{2n}^{(\al+\half)}(x)}{C_{2n}^{(\al+\half)}(1)} +=\frac{P_{2n}^{(\al,\al)}(x)}{P_{2n}^{(\al,\al)}(1)} +&=\frac{P_n^{(\al,-\half)}(2x^2-1)}{P_n^{(\al,-\half)}(1)}\,, +\label{51}\\ +\frac{C_{2n+1}^{(\al+\half)}(x)}{C_{2n+1}^{(\al+\half)}(1)} +=\frac{P_{2n+1}^{(\al,\al)}(x)}{P_{2n+1}^{(\al,\al)}(1)} +&=\frac{x\,P_n^{(\al,\half)}(2x^2-1)}{P_n^{(\al,\half)}(1)}\,. +\label{52} +\end{align} +See p.221, Remarks, last two formulas together with \eqref{50} and \eqref{49}. +Or see \mycite{DLMF}{(18.7.13), (18.7.14)}. +% +\paragraph{Differentiation formulas} +Each differentiation formula is given in two equivalent forms. +\begin{equation} +\begin{split} +\frac d{dx}\left((1-x)^\al P_n^{(\al,\be)}(x)\right)&= +-(n+\al)\,(1-x)^{\al-1} P_n^{(\al-1,\be+1)}(x),\\ +\left((1-x)\frac d{dx}-\al\right)P_n^{(\al,\be)}(x)&= +-(n+\al)\,P_n^{(\al-1,\be+1)}(x). +\end{split} +\label{68} +\end{equation} +% +\begin{equation} +\begin{split} +\frac d{dx}\left((1+x)^\be P_n^{(\al,\be)}(x)\right)&= +(n+\be)\,(1+x)^{\be-1} P_n^{(\al+1,\be-1)}(x),\\ +\left((1+x)\frac d{dx}+\be\right)P_n^{(\al,\be)}(x)&= +(n+\be)\,P_n^{(\al+1,\be-1)}(x). +\end{split} +\label{69} +\end{equation} +Formulas \eqref{68} and \eqref{69} follow from +\mycite{DLMF}{(15.5.4), (15.5.6)} +together with (9.8.1). They also follow from each other by \eqref{48}. +% +\paragraph{Generalized Gegenbauer polynomials} +These are defined by +\begin{equation} +S_{2m}^{(\al,\be)}(x):=\const P_m^{(\al,\be)}(2x^2-1),\qquad +S_{2m+1}^{(\al,\be)}(x):=\const x\,P_m^{(\al,\be+1)}(2x^2-1) +\label{70} +\end{equation} +in the notation of \myciteKLS{146}{p.156} +(see also \cite{K27}), while \cite[Section 1.5.2]{K26} +has $C_n^{(\la,\mu)}(x)=\const\allowbreak\times S_n^{(\la-\half,\mu-\half)}(x)$. +For $\al,\be>-1$ we have the orthogonality relation +\begin{equation} +\int_{-1}^1 S_m^{(\al,\be)}(x)\,S_n^{(\al,\be)}(x)\,|x|^{2\be+1}(1-x^2)^\al\,dx +=0\qquad(m\ne n). +\label{71} +\end{equation} +For $\be=\al-1$ generalized Gegenbauer polynomials are limit cases of +continuous $q$-ultraspherical polynomials, see \eqref{176}. + +If we define the {\em Dunkl operator} $T_\mu$ by +\begin{equation} +(T_\mu f)(x):=f'(x)+\mu\,\frac{f(x)-f(-x)}x +\label{72} +\end{equation} +and if we choose the constants in \eqref{70} as +\begin{equation} +S_{2m}^{(\al,\be)}(x)=\frac{(\al+\be+1)_m}{(\be+1)_m}\, P_m^{(\al,\be)}(2x^2-1),\quad +S_{2m+1}^{(\al,\be)}(x)=\frac{(\al+\be+1)_{m+1}}{(\be+1)_{m+1}}\, +x\,P_m^{(\al,\be+1)}(2x^2-1) +\label{73} +\end{equation} +then (see \cite[(1.6)]{K5}) +\begin{equation} +T_{\be+\half}S_n^{(\al,\be)}=2(\al+\be+1)\,S_{n-1}^{(\al+1,\be)}. +\label{74} +\end{equation} +Formula \eqref{74} with \eqref{73} substituted gives rise to two +differentiation formulas involving Jacobi polynomials which are equivalent to +(9.8.7) and \eqref{69}. + +Composition of \eqref{74} with itself gives +\[ +T_{\be+\half}^2S_n^{(\al,\be)}=4(\al+\be+1)(\al+\be+2)\,S_{n-2}^{(\al+2,\be)}, +\] +which is equivalent to the composition of (9.8.7) and \eqref{69}: +\begin{equation} +\left(\frac{d^2}{dx^2}+\frac{2\be+1}x\,\frac d{dx}\right)P_n^{(\al,\be)}(2x^2-1) +=4(n+\al+\be+1)(n+\be)\,P_{n-1}^{(\al+2,\be)}(2x^2-1). +\label{75} +\end{equation} +Formula \eqref{75} was also given in \myciteKLS{322}{(2.4)}. +% +\subsection*{9.8.1 Gegenbauer / Ultraspherical} +\label{sec9.8.1} +% +\paragraph{Notation} +Here the Gegenbauer polynomial is denoted by $C_n^\la$ instead of $C_n^{(\la)}$. +% +\paragraph{Orthogonality relation} +Write the \RHS\ of (9.8.20) as $h_n\,\de_{m,n}$. Then +\begin{equation} +\frac{h_n}{h_0}= +\frac\la{\la+n}\,\frac{(2\la)_n}{n!}\,,\quad +h_0=\frac{\pi^\half\,\Ga(\la+\thalf)}{\Ga(\la+1)},\quad +\frac{h_n}{h_0\,(C_n^\la(1))^2}= +\frac\la{\la+n}\,\frac{n!}{(2\la)_n}\,. +\label{61} +\end{equation} +% +\paragraph{Hypergeometric representation} +Beside (9.8.19) we have also +\begin{equation} +C_n^\lambda(x)=\sum_{\ell=0}^{\lfloor n/2\rfloor}\frac{(-1)^{\ell}(\lambda)_{n-\ell}} +{\ell!\;(n-2\ell)!}\,(2x)^{n-2\ell} +=(2x)^{n}\,\frac{(\lambda)_{n}}{n!}\, +\hyp21{-\thalf n,-\thalf n+\thalf}{1-\la-n}{\frac1{x^2}}. +\label{57} +\end{equation} +See \mycite{DLMF}{(18.5.10)}. +% +\paragraph{Special value} +\begin{equation} +C_n^{\la}(1)=\frac{(2\la)_n}{n!}\,. +\label{49} +\end{equation} +Use (9.8.19) or see \mycite{DLMF}{Table 18.6.1}. +% +\paragraph{Expression in terms of Jacobi} +% +\begin{equation} +\frac{C_n^\la(x)}{C_n^\la(1)}= +\frac{P_n^{(\la-\half,\la-\half)}(x)}{P_n^{(\la-\half,\la-\half)}(1)}\,,\qquad +C_n^\la(x)=\frac{(2\la)_n}{(\la+\thalf)_n}\,P_n^{(\la-\half,\la-\half)}(x). +\label{65} +\end{equation} +% +\paragraph{Re: (9.8.21)} +By iteration of recurrence relation (9.8.21): +\begin{multline} +x^2 C_n^\la(x)= +\frac{(n+1)(n+2)}{4(n+\la)(n+\la+1)}\,C_{n+2}^\la(x)+ +\frac{n^2+2n\la+\la-1}{2(n+\la-1)(n+\la+1)}\,C_n^\la(x)\\ ++\frac{(n+2\la-1)(n+2\la-2)}{4(n+\la)(n+\la-1)}\,C_{n-2}^\la(x). +\label{6} +\end{multline} +% +\paragraph{Bilateral generating functions} +\begin{multline} +\sum_{n=0}^\iy\frac{n!}{(2\la)_n}\,r^n\,C_n^\la(x)\,C_n^\la(y) +=\frac1{(1-2rxy+r^2)^\la}\,\hyp21{\thalf\la,\thalf(\la+1)}{\la+\thalf} +{\frac{4r^2(1-x^2)(1-y^2)}{(1-2rxy+r^2)^2}}\\ +(r\in(-1,1),\;x,y\in[-1,1]). +\label{66} +\end{multline} +For the proof put $\be:=\al$ in \eqref{58}, then use \eqref{63} and \eqref{65}. +The Poisson kernel for Gegenbauer polynomials can be derived in a similar way +from \eqref{59}, or alternatively by applying the operator +$r^{-\la+1}\frac d{dr}\circ r^\la$ to both sides of \eqref{66}: +\begin{multline} +\sum_{n=0}^\iy\frac{\la+n}\la\,\frac{n!}{(2\la)_n}\,r^n\,C_n^\la(x)\,C_n^\la(y) +=\frac{1-r^2}{(1-2rxy+r^2)^{\la+1}}\\ +\times\hyp21{\thalf(\la+1),\thalf(\la+2)}{\la+\thalf} +{\frac{4r^2(1-x^2)(1-y^2)}{(1-2rxy+r^2)^2}}\qquad +(r\in(-1,1),\;x,y\in[-1,1]). +\label{67} +\end{multline} +Formula \eqref{67} was obtained by Gasper \& Rahman \myciteKLS{234}{(4.4)} +as a limit case of their formula for the Poisson kernel for continuous +$q$-ultraspherical polynomials. +% +\paragraph{Trigonometric expansions} +By \mycite{DLMF}{(18.5.11), (15.8.1)}: +\begin{align} +C_n^{\la}(\cos\tha) +&=\sum_{k=0}^n\frac{(\la)_k(\la)_{n-k}}{k!\,(n-k)!}\,e^{i(n-2k)\tha} +=e^{in\tha}\frac{(\la)_n}{n!}\, +\hyp21{-n,\la}{1-\la-n}{e^{-2i\tha}}\label{103}\\ +&=\frac{(\la)_n}{2^\la n!}\, +e^{-\half i\la\pi}e^{i(n+\la)\tha}\,(\sin\tha)^{-\la}\, +\hyp21{\la,1-\la}{1-\la-n}{\frac{i e^{-i\tha}}{2\sin\tha}}\label{104}\\ +&=\frac{(\la)_n}{n!}\,\sum_{k=0}^\iy\frac{(\la)_k(1-\la)_k}{(1-\la-n)_k k!}\, +\frac{\cos((n-k+\la)\tha+\thalf(k-\la)\pi)}{(2\sin\tha)^{k+\la}}\,.\label{105} +\end{align} +In \eqref{104} and \eqref{105} we require that +$\tfrac16\pi<\tha<\tfrac56\pi$. Then the convergence is absolute for $\la>\thalf$ +and conditional for $0<\la\le\thalf$. + +By \mycite{DLMF}{(14.13.1), (14.3.21), (15.8.1)]}: +\begin{align} +C_n^\la(\cos\tha)&=\frac{2\Ga(\la+\thalf)}{\pi^\half\Ga(\la+1)}\, +\frac{(2\la)_n}{(\la+1)_n}\,(\sin\tha)^{1-2\la}\, +\sum_{k=0}^\iy\frac{(1-\la)_k(n+1)_k}{(n+\la+1)_k k!}\, +\sin\big((2k+n+1)\tha\big) +\label{7}\\ +&=\frac{2\Ga(\la+\thalf)}{\pi^\half\Ga(\la+1)}\, +\frac{(2\la)_n}{(\la+1)_n}\,(\sin\tha)^{1-2\la}\, +\Im\!\!\left(e^{i(n+1)\tha}\,\hyp21{1-\la,n+1}{n+\la+1}{e^{2i\tha}}\right)\nonumber\\ +&=\frac{2^\la\Ga(\la+\thalf)}{\pi^\half\Ga(\la+1)}\, +\frac{(2\la)_n}{(\la+1)_n}\,(\sin\tha)^{-\la}\, +\Re\!\!\left(e^{-\thalf i\la\pi}e^{i(n+\la)\tha}\, +\hyp21{\la,1-\la}{1+\la+n}{\frac{e^{i\tha}}{2i\sin\tha}}\right)\nonumber\\ +&=\frac{2^{2\la}\Ga(\la+\thalf)}{\pi^\half\Ga(\la+1)}\,\frac{(2\la)_n}{(\la+1)_n}\, +\sum_{k=0}^\iy\frac{(\la)_k(1-\la)_k}{(1+\la+n)_k k!}\, +\frac{\cos((n+k+\la)\tha-\thalf(k+\la)\pi)}{(2\sin\tha)^{k+\la}}\,. +\label{106} +\end{align} +We require that $0<\tha<\pi$ in \eqref{7} and $\tfrac16\pi<\tha<\tfrac56\pi$ in +\eqref{106} The convergence is absolute for $\la>\thalf$ and conditional for +$0<\la\le\thalf$. +For $\la\in\Zpos$ the above series terminate after the term with +$k=\la-1$. +Formulas \eqref{7} and \eqref{106} are also given in +\mycite{Sz}{(4.9.22), (4.9.25)}. +% +\paragraph{Fourier transform} +\begin{equation} +\frac{\Ga(\la+1)}{\Ga(\la+\thalf)\,\Ga(\thalf)}\, +\int_{-1}^1 \frac{C_n^\la(y)}{C_n^\la(1)}\,(1-y^2)^{\la-\half}\, +e^{ixy}\,dy +=i^n\,2^\la\,\Ga(\la+1)\,x^{-\la}\,J_{\la+n}(x). +\label{8} +\end{equation} +See \mycite{DLMF}{(18.17.17) and (18.17.18)}. +% +\paragraph{Laplace transforms} +\begin{equation} +\frac2{n!\,\Ga(\la)}\, +\int_0^\iy H_n(tx)\,t^{n+2\la-1}\,e^{-t^2}\,dt=C_n^\la(x). +\label{56} +\end{equation} +See Nielsen \cite[p.48, (4) with p.47, (1) and p.28, (10)]{K4} (1918) +or Feldheim \cite[(28)]{K3} (1942). +\begin{equation} +\frac2{\Ga(\la+\thalf)}\,\int_0^1 \frac{C_n^\la(t)}{C_n^\la(1)}\, +(1-t^2)^{\la-\half}\,t^{-1}\,(x/t)^{n+2\la+1}\,e^{-x^2/t^2}\,dt +=2^{-n}\,H_n(x)\,e^{-x^2}\quad(\la>-\thalf). +\label{46} +\end{equation} +Use Askey \& Fitch \cite[(3.29)]{K2} for $\al=\pm\thalf$ together with +\eqref{48}, \eqref{51}, \eqref{52}, \eqref{54} and \eqref{55}. +\paragraph{Addition formula} (see \mycite{AAR}{(9.8.5$'$)]}) +\begin{multline} +R_n^{(\al,\al)}\big(xy+(1-x^2)^\half(1-y^2)^\half t\big) +=\sum_{k=0}^n \frac{(-1)^k(-n)_k\,(n+2\al+1)_k}{2^{2k}((\al+1)_k)^2}\\ +\times(1-x^2)^{k/2} R_{n-k}^{(\al+k,\al+k)}(x)\,(1-y^2)^{k/2} R_{n-k}^{(\al+k,\al+k)}(y)\, +\om_k^{(\al-\half,\al-\half)}\,R_k^{(\al-\half,\al-\half)}(t), +\label{108} +\end{multline} +where +\[ +R_n^{(\al,\be)}(x):=P_n^{(\al,\be)}(x)/P_n^{(\al,\be)}(1),\quad +\om_n^{(\al,\be)}:=\frac{\int_{-1}^1 (1-x)^\al(1+x)^\be\,dx} +{\int_{-1}^1 (R_n^{(\al,\be)}(x))^2\,(1-x)^\al(1+x)^\be\,dx}\,. +\] +% +\subsection*{9.8.2 Chebyshev} +\label{sec9.8.2} +In addition to the Chebyshev polynomials $T_n$ of the first kind (9.8.35) +and $U_n$ of the second kind (9.8.36), +\begin{align} +T_n(x)&:=\frac{P_n^{(-\half,-\half)}(x)}{P_n^{(-\half,-\half)}(1)} +=\cos(n\tha),\quad x=\cos\tha,\\ +U_n(x)&:=(n+1)\,\frac{P_n^{(\half,\half)}(x)}{P_n^{(\half,\half)}(1)} +=\frac{\sin((n+1)\tha)}{\sin\tha}\,,\quad x=\cos\tha, +\end{align} +we have Chebyshev polynomials $V_n$ {\em of the third kind} +and $W_n$ {\em of the fourth kind}, +\begin{align} +V_n(x)&:=\frac{P_n^{(-\half,\half)}(x)}{P_n^{(-\half,\half)}(1)} +=\frac{\cos((n+\thalf)\tha)}{\cos(\thalf\tha)}\,,\quad x=\cos\tha,\\ +W_n(x)&:=(2n+1)\,\frac{P_n^{(\half,-\half)}(x)}{P_n^{(\half,-\half)}(1)} +=\frac{\sin((n+\thalf)\tha)}{\sin(\thalf\tha)}\,,\quad x=\cos\tha, +\end{align} +see \cite[Section 1.2.3]{K20}. Then there is the symmetry +\begin{equation} +V_n(-x)=(-1)^n W_n(x). +\label{140} +\end{equation} + +The names of Chebyshev polynomials of the third and fourth kind +and the notation $V_n(x)$ are due to Gautschi \cite{K21}. +The notation $W_n(x)$ was first used by Mason \cite{K22}. +Names and notations for Chebyshev polynomials of the third and fourth +kind are interchanged in \mycite{AAR}{Remark 2.5.3} and +\mycite{DLMF}{Table 18.3.1}. +% +\subsection*{9.9 Pseudo Jacobi (or Routh-Romanovski)} +\label{sec9.9} +In this section in \mycite{KLS} the pseudo Jacobi polynomial $P_n(x;\nu,N)$ in (9.9.1) +is considered +for $N\in\ZZ_{\ge0}$ and $n=0,1,\ldots,n$. However, we can more generally take +$-\thalf\be>0\;\;{\rm or}\;\;\al<\be<0). +\] +Then $P_n$ can be expressed as a Meixner polynomial: +\[ +P_n(x)=(-k_2(\al\be)^{-1})_n\,\be^n\, +M_n\left(-\,\frac{x+k_2\al^{-1}}{\al-\be},-k_2(\al\be)^{-1},\be\al^{-1}\right). +\] + +In 1938 Gottlieb \cite[\S2]{K1} introduces polynomials $l_n$ ``of Laguerre type'' +which turn out to be special Meixner polynomials: +$l_n(x)=e^{-n\la} M_n(x;1,e^{-\la})$. +% +\paragraph{Uniqueness of orthogonality measure} +The coefficient of $p_{n-1}(x)$ in (9.10.4) behaves as $O(n^2)$ as $n\to\iy$. +Hence \eqref{93} holds, by which the orthogonality measure is unique. +% +\subsection*{9.11 Krawtchouk} +\label{sec9.11} +% +\paragraph{Special values} +By (9.11.1) and the binomial formula: +\begin{equation} +K_n(0;p,N)=1,\qquad +K_n(N;p,N)=(1-p^{-1})^n. +\label{9} +\end{equation} +The self-duality (p.240, Remarks, first formula) +\begin{equation} +K_n(x;p,N)=K_x(n;p,N)\qquad (n,x\in \{0,1,\ldots,N\}) +\label{147} +\end{equation} +combined with \eqref{9} yields: +\begin{equation} +K_N(x;p,N)=(1-p^{-1})^x\qquad(x\in\{0,1,\ldots,N\}). +\label{148} +\end{equation} +% +\paragraph{Symmetry} +By the orthogonality relation (9.11.2): +\begin{equation} +\frac{K_n(N-x;p,N)}{K_n(N;p,N)}=K_n(x;1-p,N). +\label{10} +\end{equation} +By \eqref{10} and \eqref{147} we have also +\begin{equation} +\frac{K_{N-n}(x;p,N)}{K_N(x;p,N)}=K_n(x;1-p,N) +\qquad(n,x\in\{0,1,\ldots,N\}), +\label{149} +\end{equation} +and, by \eqref{149}, \eqref{10} and \eqref{9}, +\begin{equation} +K_{N-n}(N-x;p,N)=\left(\frac p{p-1}\right)^{n+x-N}K_n(x;p,N) +\qquad(n,x\in\{0,1,\ldots,N\}). +\label{150} +\end{equation} +A particular case of \eqref{10} is: +\begin{equation} +K_n(N-x;\thalf,N)=(-1)^n K_n(x;\thalf,N). +\label{11} +\end{equation} +Hence +\begin{equation} +K_{2m+1}(N;\thalf,2N)=0. +\label{12} +\end{equation} +From (9.11.11): +\begin{equation} +K_{2m}(N;\thalf,2N)=\frac{(\thalf)_m}{(-N+\thalf)_m}\,. +\label{13} +\end{equation} +% +\paragraph{Quadratic transformations} +\begin{align} +K_{2m}(x+N;\thalf,2N)&=\frac{(\thalf)_m}{(-N+\thalf)_m}\, +R_m(x^2;-\thalf,-\thalf,N), +\label{31}\\ +K_{2m+1}(x+N;\thalf,2N)&=-\,\frac{(\tfrac32)_m}{N\,(-N+\thalf)_m}\, +x\,R_m(x^2-1;\thalf,\thalf,N-1), +\label{33}\\ +K_{2m}(x+N+1;\thalf,2N+1)&=\frac{(\tfrac12)_m}{(-N-\thalf)_m}\, +R_m(x(x+1);-\thalf,\thalf,N), +\label{32}\\ +K_{2m+1}(x+N+1;\thalf,2N+1)&=\frac{(\tfrac32)_m}{(-N-\thalf)_{m+1}}\, +(x+\thalf)\,R_m(x(x+1);\thalf,-\thalf,N), +\label{34} +\end{align} +where $R_m$ is a dual Hahn polynomial (9.6.1). For the proofs use +(9.6.2), (9.11.2), (9.6.4) and (9.11.4). +% +\paragraph{Generating functions} +\begin{multline} +\sum_{x=0}^N\binom Nx K_m(x;p,N)K_n(x;q,N)z^x\\ +=\left(\frac{p-z+pz}p\right)^m +\left(\frac{q-z+qz}q\right)^n +(1+z)^{N-m-n} +K_m\left(n;-\,\frac{(p-z+pz)(q-z+qz)}z,N\right). +\label{107} +\end{multline} +This follows immediately from Rosengren \cite[(3.5)]{K8}, which goes back +to Meixner \cite{K9}. +% +\subsection*{9.12 Laguerre} +\label{sec9.12} +\paragraph{Notation} +Here the Laguerre polynomial is denoted by $L_n^\al$ instead of +$L_n^{(\al)}$. +% +\paragraph{Hypergeometric representation} +\begin{align} +L_n^\al(x)&= +\frac{(\al+1)_n}{n!}\,\hyp11{-n}{\al+1}x +\label{182}\\ +&=\frac{(-x)^n}{n!} \hyp20{-n,-n-\al}-{-\,\frac1x} +\label{183}\\ +&=\frac{(-x)^n}{n!}\,C_n(n+\al;x), +\label{184} +\end{align} +where $C_n$ in \eqref{184} is a +\hyperref[sec9.14]{Charlier polynomial}. +Formula \eqref{182} is (9.12.1). Then \eqref{183} follows by reversal +of summation. Finally \eqref{184} follows by \eqref{183} and \eqref{179}. +It is also the remark on top of p.244 in \mycite{KLS}, and it is essentially +\myciteKLS{416}{(2.7.10)}. +% +\paragraph{Uniqueness of orthogonality measure} +The coefficient of $p_{n-1}(x)$ in (9.12.4) behaves as $O(n^2)$ as $n\to\iy$. +Hence \eqref{93} holds, by which the orthogonality measure is unique. +% +\paragraph{Special value} +\begin{equation} +L_n^{\al}(0)=\frac{(\al+1)_n}{n!}\,. +\label{53} +\end{equation} +Use (9.12.1) or see \mycite{DLMF}{18.6.1)}. +% +\paragraph{Quadratic transformations} +\begin{align} +H_{2n}(x)&=(-1)^n\,2^{2n}\,n!\,L_n^{-1/2}(x^2), +\label{54}\\ +H_{2n+1}(x)&=(-1)^n\,2^{2n+1}\,n!\,x\,L_n^{1/2}(x^2). +\label{55} +\end{align} +See p.244, Remarks, last two formulas. +Or see \mycite{DLMF}{(18.7.19), (18.7.20)}. +% +\paragraph{Fourier transform} +\begin{equation} +\frac1{\Ga(\al+1)}\,\int_0^\iy \frac{L_n^\al(y)}{L_n^\al(0)}\, +e^{-y}\,y^\al\,e^{ixy}\,dy= +i^n\,\frac{y^n}{(iy+1)^{n+\al+1}}\,, +\label{14} +\end{equation} +see \mycite{DLMF}{(18.17.34)}. +% +\paragraph{Differentiation formulas} +Each differentiation formula is given in two equivalent forms. +\begin{equation} +\frac d{dx}\left(x^\al L_n^\al(x)\right)= +(n+\al)\,x^{\al-1} L_n^{\al-1}(x),\qquad +\left(x\frac d{dx}+\al\right)L_n^\al(x)= +(n+\al)\,L_n^{\al-1}(x). +\label{76} +\end{equation} +% +\begin{equation} +\frac d{dx}\left(e^{-x} L_n^\al(x)\right)= +-e^{-x} L_n^{\al+1}(x),\qquad +\left(\frac d{dx}-1\right)L_n^\al(x)= +-L_n^{\al+1}(x). +\label{77} +\end{equation} +% +Formulas \eqref{76} and \eqref{77} follow from +\mycite{DLMF}{(13.3.18), (13.3.20)} +together with (9.12.1). +% +\paragraph{Generalized Hermite polynomials} +See \myciteKLS{146}{p.156}, \cite[Section 1.5.1]{K26}. +These are defined by +\begin{equation} +H_{2m}^\mu(x):=\const L_m^{\mu-\half}(x^2),\qquad +H_{2m+1}^\mu(x):=\const x\,L_m^{\mu+\half}(x^2). +\label{78} +\end{equation} +Then for $\mu>-\thalf$ we have orthogonality relation +\begin{equation} +\int_{-\iy}^{\iy} H_m^\mu(x)\,H_n^\mu(x)\,|x|^{2\mu}e^{-x^2}\,dx +=0\qquad(m\ne n). +\label{79} +\end{equation} +Let the Dunkl operator $T_\mu$ be defined by \eqref{72}. +If we choose the constants in \eqref{78} as +\begin{equation} +H_{2m}^\mu(x)=\frac{(-1)^m(2m)!}{(\mu+\thalf)_m}\,L_m^{\mu-\half}(x^2),\qquad +H_{2m+1}^\mu(x)=\frac{(-1)^m(2m+1)!}{(\mu+\thalf)_{m+1}}\, + x\,L_m^{\mu+\half}(x^2) + \label{80} +\end{equation} +then (see \cite[(1.6)]{K5}) +\begin{equation} +T_\mu H_n^\mu=2n\,H_{n-1}^\mu. +\label{81} +\end{equation} +Formula \eqref{81} with \eqref{80} substituted gives rise to two +differentiation formulas involving Laguerre polynomials which are equivalent to +(9.12.6) and \eqref{76}. + +Composition of \eqref{81} with itself gives +\[ +T_\mu^2 H_n^\mu=4n(n-1)\,H_{n-2}^\mu, +\] +which is equivalent to the composition of (9.12.6) and \eqref{76}: +\begin{equation} +\left(\frac{d^2}{dx^2}+\frac{2\al+1}x\,\frac d{dx}\right)L_n^\al(x^2) +=-4(n+\al)\,L_{n-1}^\al(x^2). +\label{82} +\end{equation} +% +\subsection*{9.14 Charlier} +\label{sec9.14} +% +\paragraph{Hypergeometric representation} +\begin{align} +C_n(x;a)&=\hyp20{-n,-x}-{-\,\frac1a} +\label{179}\\ +&=\frac{(-x)_n}{a^n} \hyp11{-n}{x-n+1}a +\label{180}\\ +&=\frac{n!}{(-a)^n}\,L_n^{x-n}(a), +\label{181} +\end{align} +where $L_n^\al(x)$ is a +\hyperref[sec9.12]{Laguerre polynomial}. +Formula \eqref{179} is (9.14.1). Then \eqref{180} follows by reversal +of the summation. Finally \eqref{181} follows by \eqref{180} and +(9.12.1). It is also the Remark on p.249 of \mycite{KLS}, and it +was earlier given in \myciteKLS{416}{(2.7.10)}. +% +\paragraph{Uniqueness of orthogonality measure} +The coefficient of $p_{n-1}(x)$ in (9.14.4) behaves as $O(n)$ as $n\to\iy$. +Hence \eqref{93} holds, by which the orthogonality measure is unique. +% +\subsection*{9.15 Hermite} +\label{sec9.15} +% +\paragraph{Uniqueness of orthogonality measure} +The coefficient of $p_{n-1}(x)$ in (9.15.4) behaves as $O(n)$ as $n\to\iy$. +Hence \eqref{93} holds, by which the orthogonality measure is unique. +% +\paragraph{Fourier transforms} +\begin{equation} +\frac1{\sqrt{2\pi}}\,\int_{-\iy}^\iy H_n(y)\,e^{-\half y^2}\,e^{ixy}\,dy= +i^n\,H_n(x)\,e^{-\half x^2}, +\label{15} +\end{equation} +see \mycite{AAR}{(6.1.15) and Exercise 6.11}. +\begin{equation} +\frac1{\sqrt\pi}\,\int_{-\iy}^\iy H_n(y)\,e^{-y^2}\,e^{ixy}\,dy= +i^n\,x^n\,e^{-\frac14 x^2}, +\label{16} +\end{equation} +see \mycite{DLMF}{(18.17.35)}. +\begin{equation} +\frac{i^n}{2\sqrt\pi}\,\int_{-\iy}^\iy y^n\,e^{-\frac14 y^2}\,e^{-ixy}\,dy= +H_n(x)\,e^{-x^2}, +\label{17} +\end{equation} +see \mycite{AAR}{(6.1.4)}. +% +\subsection*{14.1 Askey-Wilson} +\label{sec14.1} +% +\paragraph{Symmetry} +The Askey-Wilson polynomials $p_n(x;a,b,c,d\,|\,q)$ are \,|\,symmetric +in $a,b,c,d$. +\sLP +This follows from the orthogonality relation (14.1.2) +together with the value of its coefficient of $x^n$ given in (14.1.5b). +Alternatively, combine (14.1.1) with \mycite{GR}{(III.15)}.\\ +As a consequence, it is sufficient to give generating function (14.1.13). Then the generating +functions (14.1.14), (14.1.15) will follow by symmetry in the parameters. +% +\paragraph{Basic hypergeometric representation} +In addition to (14.1.1) we have (in notation \eqref{111}): +\begin{multline} +p_n(\cos\tha;a,b,c,d\,|\, q) +=\frac{(ae^{-i\tha},be^{-i\tha},ce^{-i\tha},de^{-i\tha};q)_n} +{(e^{-2i\tha};q)_n}\,e^{in\tha}\\ +\times {}_8W_7\big(q^{-n}e^{2i\tha};ae^{i\tha},be^{i\tha}, +ce^{i\tha},de^{i\tha},q^{-n};q,q^{2-n}/(abcd)\big). +\label{113} +\end{multline} +This follows from (14.1.1) by combining (III.15) and (III.19) in +\mycite{GR}. +It is also given in \myciteKLS{513}{(4.2)}, but be aware for some slight errors. +The symmetry in $a,b,c,d$ is evident from \eqref{113}. +% +\paragraph{Special value} +\begin{equation} +p_n\big(\thalf(a+a^{-1});a,b,c,d\,|\, q\big)=a^{-n}\,(ab,ac,ad;q)_n\,, +\label{40} +\end{equation} +and similarly for arguments $\thalf(b+b^{-1})$, $\thalf(c+c^{-1})$ and +$\thalf(d+d^{-1})$ by symmetry of $p_n$ in $a,b,c,d$. +% +\paragraph{Trivial symmetry} +\begin{equation} +p_n(-x;a,b,c,d\,|\, q)=(-1)^n p_n(x;-a,-b,-c,-d\,|\, q). +\label{41} +\end{equation} +Both \eqref{40} and \eqref{41} are obtained from (14.1.1). +% +\paragraph{Re: (14.1.5)} +Let +\begin{equation} +p_n(x):=\frac{p_n(x;a,b,c,d\,|\, q)}{2^n(abcdq^{n-1};q)_n}=x^n+\wt k_n x^{n-1} ++\cdots\;. +\label{18} +\end{equation} +Then +\begin{equation} +\wt k_n=-\frac{(1-q^n)(a+b+c+d-(abc+abd+acd+bcd)q^{n-1})} +{2(1-q)(1-abcdq^{2n-2})}\,. +\label{19} +\end{equation} +This follows because $\tilde k_n-\tilde k_{n+1}$ equals the coefficient +$\thalf\bigl(a+a^{-1}-(A_n+C_n)\bigr)$ of $p_n(x)$ in (14.1.5). +% +\paragraph{Generating functions} +Rahman \myciteKLS{449}{(4.1), (4.9)} gives: +\begin{align} +&\sum_{n=0}^\iy \frac{(abcdq^{-1};q)_n a^n}{(ab,ac,ad,q;q)_n}\,t^n\, +p_n(\cos\tha;a,b,c,d\,|\,q) +\nonumber\\ +&=\frac{(abcdtq^{-1};q)_\iy}{(t;q)_\iy}\, +\qhyp65{(abcdq^{-1})^\half,-(abcdq^{-1})^\half,(abcd)^\half, +-(abcd)^\half,a e^{i\tha},a e^{-i\tha}} +{ab,ac,ad,abcdtq^{-1},qt^{-1}}{q,q} +\nonumber\\ +&+\frac{(abcdq^{-1},abt,act,adt,ae^{i\tha},ae^{-i\tha};q)_\iy} +{(ab,ac,ad,t^{-1},ate^{i\tha},ate^{-i\tha};q)_\iy} +\nonumber\\ +&\times\qhyp65{t(abcdq^{-1})^\half,-t(abcdq^{-1})^\half,t(abcd)^\half, +-t(abcd)^\half,at e^{i\tha},at e^{-i\tha}} +{abt,act,adt,abcdt^2q^{-1},qt}{q,q}\quad(|t|<1). +\label{185} +\end{align} +In the limit \eqref{109} the first term on the \RHS\ of \eqref{185} +tends to the \LHS\ of (9.1.15), while the second term tends formally +to 0. The special case $ad=bc$ of \eqref{185} was earlier given in +\myciteKLS{236}{(4.1), (4.6)}. +% +\paragraph{Limit relations}\quad\sLP +{\bf Askey-Wilson $\longrightarrow$ Wilson}\\ +Instead of (14.1.21) we can keep a polynomial of degree $n$ while the limit is approached: +\begin{equation} +\lim_{q\to1}\frac{p_n(1-\thalf x(1-q)^2;q^a,q^b,q^c,q^d\,|\, q)}{(1-q)^{3n}} +=W_n(x;a,b,c,d). +\label{109} +\end{equation} +For the proof first derive the corresponding limit for the monic polynomials by comparing +(14.1.5) with (9.4.4). +\bLP +{\bf Askey-Wilson $\longrightarrow$ Continuous Hahn}\\ +Instead of (14.4.15) we can keep a polynomial of degree $n$ while the limit is approached: +\begin{multline} +\lim_{q\uparrow1} +\frac{p_n\big(\cos\phi-x(1-q)\sin\phi;q^a e^{i\phi},q^b e^{i\phi},q^{\overline a} e^{-i\phi}, +q^{\overline b} e^{-i\phi}\,|\, q\big)} +{(1-q)^{2n}}\\ +=(-2\sin\phi)^n\,n!\,p_n(x;a,b,\overline a,\overline b)\qquad +(0<\phi<\pi). +\label{177} +\end{multline} +Here the \RHS\ has a continuous Hahn polynomial (9.4.1). +For the proof first derive the corresponding limit for the monic polynomials by comparing +(14.1.5) with (9.1.5). +In fact, define the monic polynomial +\[ +\wt p_n(x):= +\frac{p_n\big(\cos\phi-x(1-q)\sin\phi;q^a e^{i\phi},q^b e^{i\phi},q^{\overline a} e^{-i\phi}, +q^{\overline b} e^{-i\phi}\,|\, q\big)} +{(-2(1-q)\sin\phi)^n\,(abcdq^{n-1};q)_n}\,. +\] +Then it follows from (14.1.5) that +\begin{equation*} +x\,\wt p_n(x)=\wt p_{n+1}(x) ++\frac{(1-q^a)e^{i\phi}+(1-q^{-a})e^{-i\phi}+\wt A_n+\wt C_n}{2(1-q)\sin\phi}\,\wt p_n(x)\\ ++\frac{\wt A_{n-1} \wt C_n}{(1-q)^2 \sin^2\phi}\,\wt p_{n-1}(x), +\end{equation*} +where $\wt A_n$ and $\wt C_n$ are as given after (14.1.3) with $a,b,c,d$ replaced by +$q^a e^{i\phi},q^b e^{i\phi},q^{\overline a} e^{-i\phi},q^{\overline b} e^{-i\phi}$. +Then the recurrence equation for $\wt p_n(x)$ tends for $q\uparrow 1$ to +the recurrence equation (9.4.4) with $c=\overline a$, $d=\overline b$. +\bLP +{\bf Askey-Wilson $\longrightarrow$ Meixner-Pollaczek}\\ +Instead of (14.9.15) we can keep a polynomial of degree $n$ while the limit is approached: +\begin{equation} +\lim_{q\uparrow1} +\frac{p_n\big(\cos\phi-x(1-q)\sin\phi; +q^\la e^{i\phi},0,q^\la e^{-i\phi},0\,|\, q\big)}{(1-q)^n} +=n!\,P_n^{(\la)}(x;\pi-\phi)\quad +(0<\phi<\pi). +\label{178} +\end{equation} +Here the \RHS\ has a Meixner-Pollaczek polynomial (9.7.1). +For the proof first derive the corresponding limit for the monic polynomials by comparing +(14.1.5) with (9.7.4). +In fact, define the monic polynomial +\[ +\wt p_n(x):= +\frac{p_n\big(\cos\phi-x(1-q)\sin\phi; +q^\la e^{i\phi},0,q^\la e^{-i\phi},0\,|\, q\big)}{(-2(1-q)\sin\phi)^n}\,. +\] +Then it follows from (14.1.5) that +\begin{equation*} +x\,\wt p_n(x)=\wt p_{n+1}(x) ++\frac{(1-q^\la)e^{i\phi}+(1-q^{-\la})e^{-i\phi}+\wt A_n+\wt C_n} +{2(1-q)\sin\phi}\,\wt p_n(x)\\ ++\frac{\wt A_{n-1} \wt C_n}{(1-q)^2 \sin^2\phi}\,\wt p_{n-1}(x), +\end{equation*} +where $\wt A_n$ and $\wt C_n$ are as given after (14.1.3) with $a,b,c,d$ replaced by +$q^\la e^{i\phi},0,q^\la e^{-i\phi},0$. +Then the recurrence equation for $\wt p_n(x)$ tends for $q\uparrow 1$ to +the recurrence equation (9.7.4). +% +\paragraph{References} +See also Koornwinder \cite{K7}. +% +\subsection*{14.2 $q$-Racah} +\label{sec14.2} +\paragraph{Symmetry} +\begin{equation} +R_n(x;\al,\be,q^{-N-1},\de\,|\, q) +=\frac{(\be q,\al\de^{-1}q;q)_n}{(\al q,\be\de q;q)_n}\,\de^n\, +R_n(\de^{-1}x;\be,\al,q^{-N-1},\de^{-1}\,|\, q). +\label{84} +\end{equation} +This follows from (14.2.1) combined with \mycite{GR}{(III.15)}. +\sLP +In particular, +\begin{equation} +R_n(x;\al,\be,q^{-N-1},-1\,|\, q) +=\frac{(\be q,-\al q;q)_n}{(\al q,-\be q;q)_n}\,(-1)^n\, +R_n(-x;\be,\al,q^{-N-1},-1\,|\, q), +\label{85} +\end{equation} +and +\begin{equation} +R_n(x;\al,\al,q^{-N-1},-1\,|\, q) +=(-1)^n\,R_n(-x;\al,\al,q^{-N-1},-1\,|\, q), +\label{86} +\end{equation} + +\paragraph{Trivial symmetry} +Clearly from (14.2.1): +\begin{equation} +R_n(x;\al,\be,\ga,\de\,|\, q)=R_n(x;\be\de,\al\de^{-1},\ga,\de\,|\, q) +=R_n(x;\ga,\al\be\ga^{-1},\al,\ga\de\al^{-1}\,|\, q). +\label{83} +\end{equation} +For $\al=q^{-N-1}$ this shows that the three cases +$\al q=q^{-N}$ or $\be\de q=q^{-N}$ or $\ga q=q^{-N}$ of (14.2.1) +are not essentially different. +% +\paragraph{Duality} +It follows from (14.2.1) that +\begin{equation} +R_n(q^{-y}+\ga\de q^{y+1};q^{-N-1},\be,\ga,\de\,|\, q) +=R_y(q^{-n}+\be q^{n-N};\ga,\de,q^{-N-1},\be\,|\, q)\quad +(n,y=0,1,\ldots,N). +\end{equation} +% +\subsection*{14.3 Continuous dual $q$-Hahn} +\label{sec14.3} +The continuous dual $q$-Hahn polynomials are the special case $d=0$ of the +Askey-Wilson polynomials: +\[ +p_n(x;a,b,c\,|\, q):=p_n(x;a,b,c,0\,|\, q). +\] +Hence all formulas in \S14.3 are specializations for $d=0$ of formulas in \S14.1. +% +\subsection*{14.4 Continuous $q$-Hahn} +\label{sec14.4} +The continuous $q$-Hahn polynomials are the special case +of Askey-Wilson polynomials with parameters +$a e^{i\phi},b e^{i\phi},a e^{-i\phi},b e^{-i\phi}$: +\[ +p_n(x;a,b,\phi\,|\, q):= +p_n(x;a e^{i\phi},b e^{i\phi},a e^{-i\phi},b e^{-i\phi}\,|\, q). +\] +In \myciteKLS{72}{(4.29)} and \mycite{GR}{(7.5.43)} +(who write $p_n(x;a,b\,|\,q)$, $x=\cos(\tha+\phi)$) +and in \mycite{KLS}{\S14.4} (who writes $p_n(x;a,b,c,d;q)$, +$x=\cos(\tha+\phi)$) +the parameter +dependence on $\phi$ is incorrectly omitted. + +Since all formulas in \S14.4 are specializations of formulas in \S14.1, +there is no real need to give these specializations explicitly. +In particular, the limit (14.4.15) is in fact a limit from Askey-Wilson to +continuous $q$-Hahn. See also \eqref{177}. +% +\subsection*{14.5 Big $q$-Jacobi} +\label{sec14.5} +% +\paragraph{Different notation} +See p.442, Remarks: +\begin{equation} +P_n(x;a,b,c,d;q):=P_n(qac^{-1}x;a,b,-ac^{-1}d;q) +=\qhyp32{q^{-n},q^{n+1}ab,qac^{-1}x}{qa,-qac^{-1}d}{q,q}. +\label{123} +\end{equation} +Furthermore, +\begin{equation} +P_n(x;a,b,c,d;q)=P_n(\la x;a,b,\la c,\la d;q), +\label{141} +\end{equation} +\begin{equation} +P_n(x;a,b,c;q)=P_n(-q^{-1}c^{-1}x;a,b,-ac^{-1},1;q) +\label{142} +\end{equation} +% +\paragraph{Orthogonality relation} +(equivalent to (14.5.2), see also \cite[(2.42), (2.41), (2.36), (2.35)]{K17}). +Let $c,d>0$ and either $a\in (-c/(qd),1/q)$, $b\in(-d/(cq),1/q)$ or +$a/c=-\overline b/d\notin\RR$. Then +\begin{equation} +\int_{-d}^c P_m(x;a,b,c,d;q) P_n(x;a,b,c,d;q)\, +\frac{(qx/c,-qx/d;q)_\iy}{(qax/c,-qbx/d;q)_\iy}\,d_qx=h_n\,\de_{m,n}\,, +\label{124} +\end{equation} +where +\begin{equation} +\frac{h_n}{h_0}=q^{\half n(n-1)}\left(\frac{q^2a^2d}c\right)^n\, +\frac{1-qab}{1-q^{2n+1}ab}\, +\frac{(q,qb,-qbc/d;q)_n}{(qa,qab,-qad/c;q)_n} +\label{125} +\end{equation} +and +\begin{equation} +h_0=(1-q)c\,\frac{(q,-d/c,-qc/d,q^2ab;q)_\iy} +{(qa,qb,-qbc/d,-qad/c;q)_\iy}\,. +\label{126} +\end{equation} +% +\paragraph{Other hypergeometric representation and asymptotics} +\begin{align} +&P_n(x;a,b,c,d;q) +=\frac{(-qbd^{-1}x;q)_n}{(-q^{-n}a^{-1}cd^{-1};q)_n}\, +\qhyp32{q^{-n},q^{-n}b^{-1},cx^{-1}}{qa,-q^{-n}b^{-1}dx^{-1}}{q,q} +\label{138}\\ +&\qquad=(qac^{-1}x)^n\,\frac{(qb,cx^{-1};q)_n}{(qa,-qac^{-1}d;q)_n}\, +\qhyp32{q^{-n},q^{-n}a^{-1},-qbd^{-1}x}{qb,q^{1-n}c^{-1}x} +{q,-q^{n+1}ac^{-1}d} +\label{132}\\ +&\qquad=(qac^{-1}x)^n\,\frac{(qb,q;q)_n}{(-qac^{-1}d;q)_n}\, +\sum_{k=0}^n\frac{(cx^{-1};q)_{n-k}}{(q,qa;q)_{n-k}}\, +\frac{(-qbd^{-1}x;q)_k}{(qb,q;q)_k}\,(-1)^k q^{\half k(k-1)}(-dx^{-1})^k. +\label{133} +\end{align} +Formula \eqref{138} follows from \eqref{123} by +\mycite{GR}{(III.11)} and next \eqref{132} follows by series inversion +\mycite{GR}{Exercise 1.4(ii)}. +Formulas \eqref{138} and \eqref{133} are also given in +\mycite{Ism}{(18.4.28), (18.4.29)}. +It follows from \eqref{132} or \eqref{133} that +(see \myciteKLS{298}{(1.17)} or \mycite{Ism}{(18.4.31)}) +\begin{equation} +\lim_{n\to\iy}(qac^{-1}x)^{-n} P_n(x;a,b,c,d;q) +=\frac{(cx^{-1},-dx^{-1};q)_\iy}{(-qac^{-1}d,qa;q)_\iy}\,, +\label{134} +\end{equation} +uniformly for $x$ in compact subsets of $\CC\backslash\{0\}$. +(Exclusion of the spectral points $x=cq^m,dq^m$ ($m=0,1,2,\ldots$), +as was done in \myciteKLS{298} and \mycite{Ism}, is not necessary. However, +while \eqref{134} yields 0 at these points, a more refined asymptotics +at these points is given in \myciteKLS{298} and \mycite{Ism}.)$\;$ +For the proof of \eqref{134} use that +\begin{equation} +\lim_{n\to\iy}(qac^{-1}x)^{-n} P_n(x;a,b,c,d;q) +=\frac{(qb,cx^{-1};q)_n}{(qa,-qac^{-1}d;q)_n}\, +\qhyp11{-qbd^{-1}x}{qb}{q,-dx^{-1}}, +\label{135} +\end{equation} +which can be evaluated by \mycite{GR}{(II.5)}. +Formula \eqref{135} follows formally from \eqref{132}, and it follows rigorously, by +dominated convergence, from \eqref{133}. +% +\paragraph{Symmetry} +(see \cite[\S2.5]{K17}). +\begin{equation} +\frac{P_n(-x;a,b,c,d;q)}{P_n(-d/(qb);a,b,c,d;q)} +=P_n(x;b,a,d,c;q). +\end{equation} +% +\paragraph{Special values} +\begin{align} +P_n(c/(qa);a,b,c,d;q)&=1,\\ +P_n(-d/(qb);a,b,c,d;q)&=\left(-\,\frac{ad}{bc}\right)^n\, +\frac{(qb,-qbc/d;q)_n}{(qa,-qad/c;q)_n}\,,\\ +P_n(c;a,b,c,d;q)&= +q^{\half n(n+1)}\left(\frac{ad}c\right)^n +\frac{(-qbc/d;q)_n}{(-qad/c;q)_n}\,,\\ +P_n(-d;a,b,c,d;q)&=q^{\half n(n+1)} (-a)^n\,\frac{(qb;q)_n}{(qa;q)_n}\,. +\end{align} +% +\paragraph{Quadratic transformations} +(see \cite[(2.48), (2.49)]{K17} and \eqref{128}).\\ +These express big $q$-Jacobi polynomials $P_m(x;a,a,1,1;q)$ in terms of little +$q$-Jacobi polynomials (see \S14.12). +\begin{align} +P_{2n}(x;a,a,1,1;q)&=\frac{p_n(x^2;q^{-1},a^2;q^2)}{p_n((qa)^{-2};q^{-1},a^2;q^2)}\,, +\label{130}\\ +P_{2n+1}(x;a,a,1,1;q)&=\frac{qax\,p_n(x^2;q,a^2;q^2)}{p_n((qa)^{-2};q,a^2;q^2)}\,. +\label{131} +\end{align} +Hence, by (14.12.1), \mycite{GR}{Exercise 1.4(ii)} and \eqref{128}, +\begin{align} +P_n(x;a,a,1,1;q)&=\frac{(qa^2;q^2)_n}{(qa^2;q)_n}\,(qax)^n\, +\qhyp21{q^{-n},q^{-n+1}}{q^{-2n+1}a^{-2}}{q^2,(ax)^{-2}} +\label{136}\\ +&=\frac{(q;q)_n}{(qa^2;q)_n}\,(qa)^n\, +\sum_{k=0}^{[\half n]}(-1)^k q^{k(k-1)} +\frac{(qa^2;q^2)_{n-k}}{(q^2;q^2)_k\,(q;q)_{n-2k}}\,x^{n-2k}. +\label{137} +\end{align} +% +\paragraph{$q$-Chebyshev polynomials} +In \eqref{123}, with $c=d=1$, the cases $a=b=q^{-\half}$ and $a=b=q^\half$ can be considered +as $q$-analogues of the Chebyshev polynomials of the first and second kind, respectively +(\S9.8.2) because of the limit (14.5.17). The quadratic relations \eqref{130}, \eqref{131} +can also be specialized to these cases. The definition of the $q$-Chebyshev polynomials +may vary by normalization and by dilation of argument. They were considered in +\cite{K18}. +By \myciteKLS{24}{p.279} and \eqref{130}, \eqref{131}, the {\em Al-Salam-Ismail polynomials} +$U_n(x;a,b)$ ($q$-dependence suppressed) in the case $a=q$ can be expressed as +$q$-Chebyshev polynomials of the second kind: +\begin{equation*} +U_n(x,q,b)=(q^{-3} b)^{\half n}\,\frac{1-q^{n+1}}{1-q}\, +P_n(b^{-\half}x;q^\half,q^\half,1,1;q). +\end{equation*} +Similarly, by \cite[(5.4), (5.1), (5.3)]{K19} and \eqref{130}, \eqref{131}, Cigler's $q$-Chebyshev +polynomials $T_n(x,s,q)$ and $U_n(x,s,q)$ +can be expressed in terms of the $q$-Chebyshev cases of \eqref{123}: +\begin{align*} +T_n(x,s,q)&=(-s)^{\half n}\,P_n((-qs)^{-\half} x;q^{-\half},q^{-\half},1,1;q),\\ +U_n(x,s,q)&=(-q^{-2}s)^{\half n}\,\frac{1-q^{n+1}}{1-q}\, +P_n((-qs)^{-\half} x;q^{\half},q^{\half},1,1;q). +\end{align*} +% +\paragraph{Limit to Discrete $q$-Hermite I} +\begin{equation} +\lim_{a\to0} a^{-n}\,P_n(x;a,a,1,1;q)=q^n\,h_n(x;q). +\label{139} +\end{equation} +Here $h_n(x;q)$ is given by (14.28.1). +For the proof of \eqref{139} use \eqref{138}. +% +\paragraph{Pseudo big $q$-Jacobi polynomials} +Let $a,b,c,d\in\CC$, $z_+>0$, $z_-<0$ such that +$\tfrac{(ax,bx;q)_\iy}{(cx,dx;q)_\iy}>0$ for $x\in z_- q^\ZZ\cup z_+ q^\ZZ$. +Then $(ab)/(qcd)>0$. Assume that $(ab)/(qcd)<1$. +Let $N$ be the largest nonnegative integer such that $q^{2N}>(ab)/(qcd)$. +Then +\begin{multline} +\int_{z_- q^\ZZ\cup z_+ q^\ZZ}P_m(cx;c/b,d/a,c/a;q)\,P_n(cx;c/b,d/a,c/a;q)\, +\frac{(ax,bx;q)_\iy}{(cx,dx;q)_\iy}\,d_qx=h_n\de_{m,n}\\ +(m,n=0,1,\ldots,N), +\label{114} +\end{multline} +where +\begin{equation} +\frac{h_n}{h_0}=(-1)^n\left(\frac{c^2}{ab}\right)^n q^{\half n(n-1)} q^{2n}\, +\frac{(q,qd/a,qd/b;q)_n}{(qcd/(ab),qc/a,qc/b;q)_n}\, +\frac{1-qcd/(ab)}{1-q^{2n+1}cd/(ab)} +\label{115} +\end{equation} +and +\begin{equation} +h_0=\int_{z_- q^\ZZ\cup z_+ q^\ZZ}\frac{(ax,bx;q)_\iy}{(cx,dx;q)_\iy}\,d_qx +=(1-q)z_+\, +\frac{(q,a/c,a/d,b/c,b/d;q)_\iy}{(ab/(qcd);q)_\iy}\, +\frac{\tha(z_-/z_+,cdz_-z_+;q)}{\tha(cz_-,dz_-,cz_+,dz_+;q)}\,. +\label{116} +\end{equation} +See Groenevelt \& Koelink \cite[Prop.~2.2]{K14}. +Formula \eqref{116} was first given by Slater \cite[(5)]{K15} as an evaluation +of a sum of two ${}_2\psi_2$ series. +The same formula is given in Slater \myciteKLS{471}{(7.2.6)} and in +\mycite{GR}{Exercise 5.10}, but in both cases with the same slight error, +see \cite[2nd paragraph after Lemma 2.1]{K14} for correction. +The theta function is given by \eqref{117}. +Note that +\begin{equation} +P_n(cx;c/b,d/a,c/a;q)=P_n(-q^{-1}ax;c/b,d/a,-a/b,1;q). +\label{145} +\end{equation} + +In \cite{K29} the weights of the pseudo big $q$-Jacobi polynomials +occur in certain measures on the space of $N$-point configurations +on the so-called extended Gelfand-Tsetlin graph. +% +\subsubsection*{Limit relations} +\paragraph{Pseudo big $q$-Jacobi $\longrightarrow$ Discrete Hermite II} +\begin{equation} +\lim_{a\to\iy}i^n q^{\half n(n-1)} P_n(q^{-1}a^{-1}ix;a,a,1,1;q)= +\wt h_n(x;q). +\label{144} +\end{equation} +For the proof use \eqref{137} and \eqref{143}. +Note that $P_n(q^{-1}a^{-1}ix;a,a,1,1;q)$ is obtained from the +\RHS\ of \eqref{144} by replacing $a,b,c,d$ by $-ia^{-1},ia^{-1},i,-i$. +% +\paragraph{Pseudo big $q$-Jacobi $\longrightarrow$ Pseudo Jacobi} +\begin{equation} +\lim_{q\uparrow1}P_n(iq^{\half(-N-1+i\nu)}x;-q^{-N-1},-q^{-N-1},q^{-N+i\nu-1};q) +=\frac{P_n(x;\nu,N)}{P_n(-i;\nu,N)}\,. +\label{118} +\end{equation} +Here the big $q$-Jacobi polynomial on the \LHS\ equals +$P_n(cx;c/b,d/a,c/a;q)$ with\\ +$a=iq^{\half(N+1-i\nu)}$, $b=-iq^{\half(N+1+i\nu)}$, +$c=iq^{\half(-N-1+i\nu)}$, $d=-iq^{\half(-N-1-i\nu)}$. +% +\subsection*{14.7 Dual $q$-Hahn} +\label{sec14.7} +\paragraph{Orthogonality relation} +More generally we have (14.7.2) with positive weights in any of the following +cases: +(i) $0<\ga q<1$, $0<\de q<1$;\quad +(ii) $0<\ga q<1$, $\de<0$;\quad +(iii) $\ga<0$, $\de>q^{-N}$;\quad +(iv) $\ga>q^{-N}$, $\de>q^{-N}$;\quad +(v) $01,\;qb1$ +if $p>q^{-1}$. +% +\paragraph{History} +The origin of the name of the quantum $q$-Krawtchouk polynomials +is by their interpretation +as matrix elements of irreducible corepresentations of (the quantized +function algebra of) the quantum group $SU_q(2)$ considered +with respect to its quantum subgroup $U(1)$. The orthogonality +relation and dual orthogonality relation of these polynomials +are an expression of the unitarity of these corepresentations. +See for instance \myciteKLS{343}{Section 6}. +% +\subsection*{14.16 Affine $q$-Krawtchouk} +\label{sec14.16} +% +\paragraph{$q$-Hypergeometric representation} +For $n=0,1,\ldots,N$ +(see (14.16.1)): +\begin{align} +K_n^{\rm Aff}(y;p,N;q) +&=\frac1{(p^{-1}q^{-1};q^{-1})_n}\,\qhyp21{q^{-n},q^{-N}y^{-1}}{q^{-N}}{q,p^{-1}y} +\label{156}\\ +&=\qhyp32{q^{-n},y,0}{q^{-N},pq}{q,q}. +\label{157} +\end{align} +% +\paragraph{Self-duality} +By \eqref{157}: +\begin{equation} +K_n^{\rm Aff}(q^{-x};p,N;q)=K_x^{\rm Aff}(q^{-n};p,N;q)\qquad +(n,x\in\{0,1,\ldots,N\}). +\label{168} +\end{equation} +% +\paragraph{Special values} +By \eqref{156} and \mycite{GR}{(II.4)}: +\begin{equation} +K_n^{\rm Aff}(1;p,N;q)=1,\qquad +K_n^{\rm Aff}(q^{-N};p,N;q)=\frac1{((pq)^{-1};q^{-1})_n}\,. +\label{165} +\end{equation} +By \eqref{165} and \eqref{168} we have also +\begin{equation} +K_N^{\rm Aff}(q^{-x};p,N;q)=\frac1{((pq)^{-1};q^{-1})_x}\,. +\label{170} +\end{equation} +% +\paragraph{Limit for $q\to1$ to Krawtchouk} (see (14.16.14) and Section \hyperref[sec9.11]{9.11}): +\begin{align} +\lim_{q\to1} K_n^{\rm Aff}(1+(1-q)x;p,N;q)&=K_n(x;1-p,N), +\label{162}\\ +\lim_{q\to1} K_n^{\rm Aff}(q^{-x};p,N;q)&=K_n(x;1-p,N). +\label{166} +\end{align} +% +\paragraph{A relation between quantum and affine $q$-Krawtchouk}\quad\\ +By \eqref{152}, \eqref{156}, \eqref{165} and \eqref{168} +we have for $x\in\{0,1,\ldots,N\}$: +\begin{align} +K_{N-n}^{\rm qtm}(q^{-x};p^{-1}q^{-N-1},N;q) +&=\frac{K_x^{\rm Aff}(q^{-n};p,N;q)}{K_x^{\rm Aff}(q^{-N};p,N;q)} +\label{171}\\ +&=\frac{K_n^{\rm Aff}(q^{-x};p,N;q)}{K_N^{\rm Aff}(q^{-x};p,N;q)}\,. +\label{172} +\end{align} +Formula \eqref{171} is given in \cite[formula after (12)]{K24} +and \cite[(59)]{K25}. +In view of \eqref{164} and \eqref{166} +formula \eqref{172} has \eqref{149} as a limit case for +$q\to 1$. +% +\paragraph{Affine $q^{-1}$-Krawtchouk} +By \eqref{156}, \eqref{165}, +\eqref{154} and \eqref{152} (see also p.505, first formula): +\begin{align} +\frac{K_n^{\rm Aff}(y;p,N;q^{-1})}{K_n^{\rm Aff}(q^N;p,N;q^{-1})} +&=\qhyp21{q^{-n},q^{-N}y}{q^{-N}}{q,p^{-1}q^{n+1}} +\label{158}\\ +&=K_n^{\rm qtm}(q^{-N}y;p^{-1},N;q). +\label{159} +\end{align} +Formula \eqref{159} is equivalent to \eqref{160}. +Just as for \eqref{160}, it tends after suitable substitutions to +\eqref{10} as $q\to1$. + +The orthogonality relation (14.16.2) holds with positive weights for $q>1$ +if $0-1) +\label{119} +\end{equation} +with +\begin{equation} +\frac{h_n}{h_0}=\frac{(q^{\al+1};q)_n}{(q;q)_n q^n},\qquad +h_0=-\,\frac{(q^{-\al};q)_\iy}{(q;q)_\iy}\,\frac\pi{\sin(\pi\al)}\,. +\label{120} +\end{equation} +The expression for $h_0$ (which is Askey's $q$-gamma evaluation +\cite[(4.2)]{K16}) +should be interpreted by continuity in $\al$ for +$\al\in\Znonneg$. +Explicitly we can write +\begin{equation} +h_n=q^{-\half\al(\al+1)}\,(q;q)_\al\,\log(q^{-1})\qquad(\al\in\Znonneg). +\label{121} +\end{equation} +% +\subsubsection*{Expansion of $x^n$} +\begin{equation} +x^n=q^{-\half n(n+2\al+1)}\,(q^{\al+1};q)_n\, +\sum_{k=0}^n\frac{(q^{-n};q)_k}{(q^{\al+1};q)_k}\,q^k\,L_k^\al(x;q). +\label{37} +\end{equation} +This follows from \eqref{36} by the equality given in the Remark at the end +of \S14.20. Alternatively, it can be derived in the same way as \eqref{36} +from the generating function (14.21.14). +% +\subsubsection*{Quadratic transformations} +$q$-Laguerre polynomials $L_n^\al(x;q)$ with $\al=\pm\half$ are +related to discrete $q$-Hermite II polynomials $\wt h_n(x;q)$: +\begin{align} +L_n^{-1/2}(x^2;q^2)&= +\frac{(-1)^n q^{2n^2-n}}{(q^2;q^2)_n}\,\wt h_{2n}(x;q), +\label{38}\\ +xL_n^{1/2}(x^2;q^2)&= +\frac{(-1)^n q^{2n^2+n}}{(q^2;q^2)_n}\,\wt h_{2n+1}(x;q). +\label{39} +\end{align} +These follows from \eqref{28} and \eqref{29}, respectively, by applying +the equalities given in the Remarks at the end of \S14.20 and \S14.28. +% +\subsection*{14.27 Stieltjes-Wigert} +\label{sec14.27} +% +\subsubsection*{An alternative weight function} +The formula on top of p.547 should be corrected as +\begin{equation} +w(x)=\frac\ga{\sqrt\pi}\,x^{-\half}\exp(-\ga^2\ln^2 x),\quad x>0,\quad +{\rm with}\quad\ga^2=-\,\frac1{2\ln q}\,. +\label{94} +\end{equation} +For $w$ the weight function given in \mycite{Sz}{\S2.7} the \RHS\ of \eqref{94} +equals $\const w(q^{-\half}x)$. See also +\mycite{DLMF}{\S18.27(vi)}. +% +\subsection*{14.28 Discrete $q$-Hermite I} +\label{sec14.28} +% +\paragraph{History} +Discrete $q$ Hermite I polynomials (not yet with this name) first occurred in +Hahn \myciteKLS{261}, see there p.29, case V and the $q$-weight $\pi(x)$ given by +the second expression on line 4 of p.30. However note that on the line on p.29 dealing with +case V, one should read $k^2=q^{-n}$ instead of $k^2=-q^n$. Then, with the indicated +substitutions, \myciteKLS{261}{(4.11), (4.12)} yield constant multiples of +$h_{2n}(q^{-1}x;q)$ and $h_{2n+1}(q^{-1}x;q)$, respectively, + due to the quadratic transformations \eqref{28}, \eqref{29} together with (4.20.1). +% +\subsection*{14.29 Discrete $q$-Hermite II} +\label{sec14.29} +% +\paragraph{Basic hypergeometric representation}(see (14.29.1)) +\begin{equation} +\wt h_n(x;q)=x^n\,\qhyp21{q^{-n},q^{-n+1}}0{q^2,-q^2 x^{-2}}. +\label{143} +\end{equation} +% +\renewcommand{\refname}{Standard references} +\begin{thebibliography}{999999} +\label{sec_ref1} +% +\mybibitem{AAR} +G. E. Andrews, R. Askey and R. 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Koornwinder, +{$q$-Special functions, a tutorial}, +{\tt arXiv:math/9403216v2 [math.CA]}, 2013. + % + \bibitem{K10} + N. N. Leonenko and N. \v{S}uvak, +{\em Statistical inference for student diffusion process}, +Stoch. Anal. Appl. 28 (2010), 972--1002. +% +\bibitem{K22} +J. C. Mason, +{\em Chebyshev polynomials of the second, third and fourth kinds in +approximation, indefinite integration, and integral transforms}, +J. Comput. Appl. Math. 49 (1993), 169--178. +% +\bibitem{K20} +J. C. Mason and D. Handscomb, +{\em Chebyshev polynomials}, +Chapman \& Hall / CRC, 2002. +% +\bibitem{K9} +J. Meixner, +{\em Umformung gewisser Reihen, deren Glieder Produkte hypergeometrischer +Funktionen sind}, +Deutsche Math. 6 (1942), 341--349. +% +\bibitem{K4} +N. Nielsen, +{\em Recherches sur les polyn\^omes d' Hermite}, +Kgl. Danske Vidensk. Selsk. Math.-Fys. Medd. I.6, K\o benhavn, 1918. +% +\bibitem{K12} +J. Peetre, +{\em Correspondence principle for the quantized annulus, Romanovski polynomials, +and Morse potential}, +J. Funct. Anal. 117 (1993), 377--400. +% +\bibitem{K8} +H. Rosengren, +{\em Multivariable orthogonal polynomials and coupling coefficients for +discrete series representations}, +SIAM J. Math. Anal. 30 (1999), 233--272. +% +\bibitem{K13} +E. J. Routh, +{\em On some properties of certain solutions of a differential equation of the second order}, +Proc. London Math. Soc. 16 (1885), 245--261. +% +\bibitem{K6} +J. A. Shohat and J. D. Tamarkin, +{\em The problem of moments}, +American Mathematical Society, 1943. +% +\bibitem{K15} +L. J. Slater, +{\em General transformations of bilateral series}, +Quart. J. Math., Oxford Ser. (2) 3 (1952), 73--80. +% +%until K29 +\end{thebibliography} +\quad\\ +\begin{footnotesize} +\begin{quote} +{T. H. Koornwinder, Korteweg-de Vries Institute, University of Amsterdam,\\ +P.O.\ Box 94248, 1090 GE Amsterdam, The Netherlands; + +\vspace{\smallskipamount} +email: }{\tt T.H.Koornwinder@uva.nl} +\end{quote} +\end{footnotesize} +\end{document} diff --git a/KLSadd_insertion/KoLeSw.pdf b/KLSadd_insertion/KoLeSw.pdf new file mode 100644 index 0000000000000000000000000000000000000000..ac31ef684d42c0ab1dc310c02b1ea9dd16a21937 GIT binary patch literal 4597173 zcmc$`1yohf*D$Pr(jAJZbR*o$ZMr)J1PN&gDG6z$RZ>c%rMr}p5=2D=B&0+@LK*=@ zN(A0>P=0@ZUZ3Z`zHfc+`dDk$%$c+Io*grL&m3l7b`2T1OHh6oDf{$H&pT2gC`1q? z$OK_>G_z$A5fMn02Md5CDH9VDDUpY><@Mu{peQ6M5#-k&fN<1e za`@E;g#&c4`cP;%Rv#LUg^Pe-(?dXEShz?S6!j-PBwP@R)rW*Zu=+59SoF{k3>MF5 z7zEHN`1c>#FmO=QfAk?ySaM*{aLgZYFgOzWCw~YCQV{(I91+Nw4m2EVZXnSJIF>EZC}8&d zd%hsiXc#se28lH;1`5NH10#sS!o|ShSYu!iC~SHtYblmi6;{Lmk|LO~%gEMB01H^YL1L9lV*Sb9c55$HeXF5n#D zf9eVaMI*5Ig93i^&v6C+q8C9J3X5k!gdi5rf=CpW{sb{ta~B1JAh2u%6U3T-C>R`T zZJ}TY)Sq($1w&!m9EL&s>90_5D7H@F*z*qsha<6Zk=VM2qp|!r3h?(>Iz>RS)*A}& ze^~ZK0FLrcdmw-njMev-ago5}{Syu;h&@jMF|cd|I8>}LkVp(RJv3H~LLo6&F%N}; zV69aYN)USuqF~rM1r&%iuTV%Ve}Dr1u=qn^AXxfCL$LJ>>HmP1m36127xbUI3VpSb9N2 zF~~plj247KvE&fMj)Q1HI0B1jK_nDQHUJeH4lu}{I)Fj3d>9%A!{!AB>|n8Qky!f@ zAkxCI;DFGL-S-zg!?9xuKn+``zy=#j4mb*ir2{w`iKPQL27_fc1YqDlWdmFtHXK~= z&vl7L0GmB5T)@F#^{zklAQ~lzwXZ`1`$Q~zpb$tb*#KvZWh2yIY=pvK>mCh-V%ZY?7rg)z5KCV4 zUwi~GESAmD;9>0Fdq*?|0>OdPU|@oO;zEuNcmL3HbvCuJvvejU0#1xtz+n+M z_QBz30S#x%TR)G4NQ}T`M@QGAQzPIXm@W$_I$U==dIZ!9KFSz~$U-F1au`XNjI^|z z3<4u5BZn3ggbPZ`O3FeJNExV@2u4~Gj**i^2}02_lECRi3N433pu@sTi;fD{u92oD(GFbhLt z&?o^p8#`A^X8}1oQ&&qFOLIpHO94d?1P%vIz(5TF&%XdD3M!z*1Z;`~!ILixEeL^# ziMhBsTbkMfN8HgaA7T&f!NCR371&ymN^Hjr$=NsxX6AC{B(^ww2sI9y2pf(~P?p?9 zD1q&FW$!{W;hCUNifNi%ZRb+e?reQpt-Y)PzqgNW)5@GLe^cEwAUEY)`E^@8VW@r3 z&wb7Jt;Jlb@7q-B*~4aG#%Ro5$&+ob^6KGhR$O6b+&rJ>*j(|1j%`rWaz__hRLT+6Y16QQCm%t$OQ#JUC!TtOV~~MX2;}W?g~0FzaYIY z^@^(|^-nKq$85^x#1$7FtJvW*zolHe6H zCcI#1t0VkJn^RRlUKdH56vV{jgyDqsG!yK6>OE7g)6fADf5HmB7KjBmp;qxWHIO z-@8B6M14J-zxm}l`@2tvN$=jG`}I=ge)SC(G_%8o8fwi*APnUvzl&t=L<8622S8eX+jy40%E69|Fnf zM5vPyPlX?B_-De__59Y7-d`z0(mc}V=&ymP_swmQQeS0cSj}5Y{kr|dulGzrO5Fir zBDv4SI`2C6*iF$27LE947*q;WtxvXTxK`>)WtEM`?3qRHx-yOQjnMQl@pE&h>F~=N zUf*``2xutAA0BcqnaIIlE?|a;}V^G*yr8?dsH( zQ$-y645gec8-1@P!_8B!@4`pP-Kih7d?N;qhu_q7E%8V%Uyw>SSVcYys ztG@F)|DHybjjesNFX>+Ug7KYY^C82E2U`a}Le)BnOIYhoc4{SOBsJ812dZzHt8s1w zALhTMmG4S<_--I%Qabu9`IFU_N(JS(&l!*CM9`_j4trFecAm}9KZK-SHzpmqa~qXX zWIlX)mCyk*Bi6bfw54vxlUz;gBR=qca3a0o)%}YQm3Z~sCbH4OT27vt5x?#gk}B_| zYnAhX%Pjf%BFkyeTNgGnDhJhbN-Y`5B5DM@xb0aNzGb8@?GCt|zeTHU5< zLe+duC>b1^Xqgy!6JBOMxvJ6jDfDfGuc7<5EAuGdRjZbemM2CjQ5Nnlh=7P662@=w zJP-qCukK!Xdtq#8A+K@pL@$QMb~@7j90+& z^VEa>K`q@oPd;vLMnw2&2sSl2a&o?un6hiPmTG##HhL~%j zo21WX&>ooNq}w0f>2^(vk&9d`9!=|e(S2_<+>NZRCg5YQFsJbI)#axs9AhEJ?ryOT zi%=pKgY>uiw6*EqxrR18ns2I2B_s(Nz1jaK-)<%-dP_&h$#(7Xhz5%TK615fdHj9lBb zJ~11D8~967-n*w7C`A-eo7Ym-&^;H)4@Af6g=eU53Ahz?5|4SVOiU7Mw`91np3R4` z^6ohwnt4E%?!}8N)mdAU+T9iPZv8^#&D+E`vhrl)p>VeD=1_3A!Fxyf;vjFeWb(%^ z2ja9g(iZVaNm~^Tegs8Xnlv2lICW6YC$7v7#3*Q%=TgU?Y8f~mJ?mw%m62j6WYZAZ z1oP;7KXjwv%n7fCc45=2FQR0mTkDPWl_aheWbTieZHSg|J6Tq&Tynv9E5jY@AlKc3 z2+}CuZs#6Gq^R%b)fVNO@WO<@>eaoRDY^cJ<%~!*^&I)bZJbkay+wC-N$oF z)c85Qyz4$2zIAqxoOI|x#FS`i(9|c@+gQ!1o)&YxVKJWL$?xhVAGB5WxO(4yBg{S0 zZe;ZQ^3FnAJ9+m7!<)Aj=AH(9kPG#iU`RIdS;(0m#!c7SR2MNx5)#o6SrBfRe?2hM zlX&sML)eJ1$iN!r);FJ5eJ*#r-;xhTlkbY5uZWA)`T7uCb6{W$jB62RjK4Tx;A7Rc z+a_vt$Ljz&urZaM(!PCO^tQ=ijJ;b9cHz2xCnaZ!lt=KzCx#v3W_CNJsGdpnu2H+OssHb*F{zVBQ*L$>WoJ~->U zHYdM*d(^jOdm@1K1Nqto@dyazK?K=_eW&%>TMhHtIs%Y}20jc+~2U z`Pqow$(nw1Qbw{ewD`_4 zO})o<3KKp^V#iV3XLdS?zk&3iy^Xh_yW6D%lIlKN@&l)~&0%Kh)Oo2y&k$LM279v8 zTs(Ll1&z*rHuT0*+*dV1R=@0UGzU4v2gMehN)Ky!{xbZ*01CO==thd@u0}Q(kljfy zxAtdpYY}bY`2MPL+MS-~UOq*<9kR;sgeqHm=pFMf^cNP2CHR_?>gY_qnBSB1A*AY@ zMx39f@;9H-RrY0Vf6x|?SN|eQf~oZB(0plY&pMgHZNiUuQ%h`oCn(BJw@QkcC~%RJ zw^R8)3dVDdZ$a4}uh3X=3x2yQ6RtYNvU ze5YNDnF5EGk|T=Nu#3w7kQatK)m8bB^&veGZy5ELvlq+s!mcMsNe)obgib;a$8*n~ z3~0fl37phZfpBOTZLT0U>4WD(+}?N~=n${2=0p$YHd0+Bypnf~ztx`hKt=jyNrj

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0 HcmV?d00001 diff --git a/KLSadd_insertion/chap01.tex b/KLSadd_insertion/chap01.tex new file mode 100644 index 0000000..d2b4c10 --- /dev/null +++ b/KLSadd_insertion/chap01.tex @@ -0,0 +1,1542 @@ +\documentclass[envcountchap,graybox]{svmono} + +\addtolength{\textwidth}{1mm} + +\usepackage{amsmath,amssymb} + +\usepackage{amsfonts} +%\usepackage{breqn} +\usepackage{DLMFmath} +\usepackage{DRMFfcns} + +\usepackage{mathptmx} +\usepackage{helvet} +\usepackage{courier} + +\usepackage{makeidx} +\usepackage{graphicx} + +\usepackage{multicol} +\usepackage[bottom]{footmisc} + +\makeindex + +\def\bibname{Bibliography} +\def\refname{Bibliography} + +\def\theequation{\thesection.\arabic{equation}} + +\smartqed + +\let\corollary=\undefined +\spnewtheorem{corollary}[theorem]{Corollary}{\bfseries}{\itshape} + +\newcounter{rom} + +\newcommand{\hyp}[5]{\mbox{}_{#1}{F}_{#2} +\left(\genfrac{}{}{0pt}{}{#3}{#4}\,;\,#5\right)} + +\newcommand{\qhyp}[5]{\mbox{}_{#1}{\phi}_{#2} +\left(\genfrac{}{}{0pt}{}{#3}{#4}\,;\,q,\,#5\right)} + +\newcommand{\mathindent}{\hspace{7.5mm}} + +\newcommand{\e}{\textrm{e}} + +\renewcommand{\E}{\textrm{E}} + +\renewcommand{\textfraction}{-1} + +\renewcommand{\Gamma}{\varGamma} + +\renewcommand{\leftlegendglue}{\hfil} + +\settowidth{\tocchpnum}{14\enspace} +\settowidth{\tocsecnum}{14.30\enspace} +\settowidth{\tocsubsecnum}{14.12.1\enspace} + +\makeatletter +\def\cleartoversopage{\clearpage\if@twoside\ifodd\c@page + \hbox{}\newpage\if@twocolumn\hbox{}\newpage\fi + \else\fi\fi} + +\newcommand{\clearemptyversopage}{ + \clearpage{\pagestyle{empty}\cleartoversopage}} +\makeatother + +\oddsidemargin -1.5cm +\topmargin -2.0cm +\textwidth 16.3cm +\textheight 25cm + +\begin{document} + +\author{Roelof Koekoek\\[2.5mm]Peter A. Lesky\\[2.5mm]Ren\'e F. Swarttouw} +\title{Hypergeometric orthogonal polynomials and their\\$q$-analogues} +\subtitle{-- Monograph --} +\maketitle + +\frontmatter + +\pagenumbering{roman} + +\large + +\chapter{Definitions and miscellaneous formulas} +\label{Definitions} + + +\section{Orthogonal polynomials}\index{Orthogonal polynomials} +\par\setcounter{equation}{0} +\label{orthogonal polynomials} + +A system of polynomials $\left\{p_n(x)\right\}_{n=0}^{\infty}$ with degree$[p_n(x)]=n$ for all +$n\in\{0,1,2,\ldots\}$ is called orthogonal on an interval $(a,b)$ with respect to the weight +function $w(x)\geq 0$ if +\begin{equation} +\label{orth1} +\int_a^bp_m(x)p_n(x)w(x)\,dx=0,\quad m\neq n,\quad m,n\in\{0,1,2,\ldots\}. +\end{equation} +Here $w(x)$ is continuous or piecewise continuous or integrable, and such that +\begin{equation} +0<\int_a^bx^{2n}w(x)\,dx<\infty +% \constraint{ $n\in\{0,1,2,\ldots\}$ } +\end{equation} +More generally, $w(x)\,dx$ may be replaced in this definition by a positive measure +$d\alpha(x)$, where $\alpha(x)$ is a bounded nondecreasing function on $[a,b]\cap\mathbb{R}$ +with an infinite number of points of increase, and such that +\begin{equation} +0<\int_a^bx^{2n}\,d\alpha(x)<\infty +% \constraint{ $n\in\{0,1,2,\ldots\}$ } +\end{equation} +If this function $\alpha(x)$ is constant between its (countably many) jump points then we have +the situation of positive weights $w_x$ on a countable subset $X$ of $\mathbb{R}$. Then the +system $\{p_n(x)\}_{n=0}^{\infty}$ is orthogonal on $X$ with respect to these weights as follows: +\begin{equation} +\label{orth2} +\sum_{x\in X}p_m(x)p_n(x)w_x=0,\quad m\neq n,\quad m,n\in\{0,1,2,\ldots\}. +\end{equation} +The case of weights $w_x$ ($x\in X$) on a finite set $X$ of $N+1$ points yields orthogonality +for a finite system of polynomials $\{p_n(x)\}_{n=0}^N$: +\begin{equation} +\label{orth3} +\sum_{x\in X}p_m(x)p_n(x)w_x=0,\quad m\neq n,\quad m,n\in\{0,1,2,\ldots,N\}. +\end{equation} +The orthogonality relations (\ref{orth1}), (\ref{orth2}) or (\ref{orth3}) determine the +polynomials $\{p_n(x)\}_{n=0}^{\infty}$ up to constant factors, which may be fixed by a +suitable normalization. We set: +\begin{equation} +\label{norm1} +\sigma_n=\int_a^b\left\{p_n(x)\right\}^2w(x)\,dx,\quad n=0,1,2,\ldots, +\end{equation} +\begin{equation} +\label{norm2} +\sigma_n=\sum_{x\in X}\left\{p_n(x)\right\}^2w_x,\quad n=0,1,2,\ldots +\end{equation} +or +\begin{equation} +\label{norm3} +\sigma_n=\sum_{x\in X}\left\{p_n(x)\right\}^2w_x,\quad n=0,1,2,\ldots,N, +\end{equation} +respectively and +\begin{equation} +\label{polynomial} +p_n(x)=k_nx^n+\,\textrm{lower order terms},\quad n=0,1,2,\ldots. +\end{equation} +Then we have the following normalizations: + +\begin{enumerate}\itemsep2.5mm +\item $\sigma_n=1$ for all $n=0,1,2,\ldots$. In that case the system of polynomials is called +orthonormal. If moreover $k_n>0$, for instance, then the polynomials are uniquely determined. +\item $k_n=1$ for all $n=0,1,2,\ldots$. In that case the system of polynomials is called +monic. Also in this case the polynomials are uniquely determined. +\end{enumerate} +Polynomials $\{p_n(x)\}_{n=0}^{\infty}$ with degree$[p_n(x)]=n$ for all $n\in\{0,1,2,\ldots\}$ +which are orthogonal with respect to a (piecewise) continuous or integrable weight function +$w(x)$ as in (\ref{orth1}) are called continuous orthogonal polynomials.\\ +Polynomials $\{p_n(x)\}_{n=0}^N$ with degree$[p_n(x)]=n$ for all $n\in\{0,1,2,\ldots,N\}$ and +possibly $N\rightarrow\infty$ which are orthogonal with respect to a countable subset $X$ of +$\mathbb{R}$ as in (\ref{orth2}) or (\ref{orth3}) are called discrete orthogonal polynomials.\\ +By using the Kronecker delta, defined by +\begin{equation} +\label{Kronecker} +\,\delta_{mn}:=\left\{\begin{array}{ll}0, &m\neq n,\\[5mm] +1, & m=n,\end{array}\right.\quad m,n\in\{0,1,2,\ldots\}, +\end{equation} +orthogonality relations can be written in the form +\begin{equation} +\label{orth4} +\int_a^bp_m(x)p_n(x)w(x)\,dx=\sigma_n\,\delta_{mn},\quad m,n\in\{0,1,2,\ldots\} +\end{equation} +or (with possibly $N\rightarrow\infty$) +\begin{equation} +\label{orth5} +\sum_{x\in X}p_m(x)p_n(x)w_x=\sigma_n\,\delta_{mn},\quad m,n\in\{0,1,2,\ldots,N\}, +\end{equation} +respectively. + +\section{The gamma and beta function} +\par\setcounter{equation}{0} +\label{gamma function} + +For $\textrm{Re}\,z>0$ the gamma function can be defined by the +gamma integral $\EulerGamma@{z}$ +\begin{equation} +\label{DefGamma}\index{Gamma function} +\Gamma(z):=\int_0^{\infty}t^{z-1}\e^{-t}\,dt,\quad\textrm{Re}\,z>0. +\end{equation} +This gamma function satisfies the well-known functional equation +\begin{equation} +\label{GammaFunctional} +\Gamma(z+1)=z\Gamma(z) +% \constraint{ $\Gamma(1)=1$ } +\end{equation} +which shows that $\Gamma(n+1)=n!$ for $n\in\{0,1,2,\ldots\}$. +For non-integral values of $z$, the gamma function also satisfies the reflection formula +\begin{equation} +\label{GammaReflection} +\Gamma(z)\Gamma(1-z)=\frac{\pi}{\sin(\pi z)},\quad z\notin\mathbb{Z}. +\end{equation} +Hence we have $\Gamma(1/2)=\sqrt{\pi}$, which implies that +\begin{equation} +\label{IntExp} +\int_{-\infty}^{\infty}\e^{-x^2}\,dx=2\int_0^{\infty}\e^{-x^2}\,dx +=\int_0^{\infty}t^{-1/2}\e^{-t}\,dt=\Gamma(1/2)=\sqrt{\pi}. +\end{equation} +More general we have +\begin{equation} +\label{IntHermite} +\int_{-\infty}^{\infty}\e^{-\alpha^2x^2-2\beta x}\,dx +=\sqrt{\frac{\pi}{\alpha^2}}\,\e^{\beta^2/\alpha^2}, +% \constraint{ +% $\alpha,\beta\in\Real$ & +% $\alpha\neq 0$ } +\end{equation} +Further we have Legendre's duplication formula +\begin{equation} +\label{GammaDuplication} +\Gamma(z)\Gamma(z+1/2)=2^{1-2z}\sqrt{\pi}\,\Gamma(2z) +% \constraint{ +% $z\in\Complex$ & +% $2z\neq 0,-1,-2,\ldots$ } +\end{equation} +and Stirling's asymptotic formula +\begin{equation} +\label{GammaAsymptotic} +\Gamma(z)\sim\sqrt{2\pi}\,z^{z-1/2}\e^{-z},\quad\textrm{Re}\,z\rightarrow\infty. +\end{equation} +For $z=x+iy$ with $x,y\in\mathbb{R}$ we also have +\begin{equation} +\label{GammaAsymptotic2} +\Gamma(x+iy)\sim\sqrt{2\pi}\,|y|^{x-1/2}\e^{-|y|\pi/2},\quad|y|\rightarrow\infty +\end{equation} +and for the ratio of two gamma functions we have the asymptotic +formula +\begin{equation} +\label{GammaAsymptotic3} +\frac{\Gamma(z+a)}{\Gamma(z+b)}\sim z^{a-b},\quad +% \constraint{ +% $a,b\in\Complex$ & +% $|z|\rightarrow\infty$ } +\end{equation} + +For $\textrm{Re}\,x>0$ and $\textrm{Re}\,y>0$ the beta function can be defined +by the integral +\begin{equation} +\label{DefBeta}\index{Beta function} +\textrm{B}(x,y):=\int_0^1t^{x-1}(1-t)^{y-1}\,dt +% \constraint{ +% $\textrm{Re}\,x>0$ & +% $\textrm{Re}\,y>0$ } +\end{equation} +The connection between the beta function and the gamma function is +given by the relation +\begin{equation} +\label{BetaGamma} +\textrm{B}(x,y)=\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)} +% \constraint{ +% $\textrm{Re}\,x>0$ & +% $\textrm{Re}\,y>0$ } +\end{equation} +There is another beta integral due to Cauchy, which can be written as +\begin{equation} +\label{Cauchy} +\frac{1}{2\pi}\int_{-\infty}^{\infty}\frac{dt}{(r+it)^{\rho}(s-it)^{\sigma}} +=\frac{(r+s)^{1-\rho-\sigma}\Gamma(\rho+\sigma-1)}{\Gamma(\rho)\Gamma(\sigma)} +\end{equation} +for $\textrm{Re}\,r>0$, $\textrm{Re}\,s>0$ and $\textrm{Re}(\rho+\sigma)>1$. + +\section{The shifted factorial and binomial coefficients} +\par\setcounter{equation}{0} +\label{pochhammer symbol} + +The shifted factorial -- or Pochhammer symbol -- is defined by +\begin{equation} +\label{Pochhammer}\index{Shifted factorial} +(a)_0:=1\quad\textrm{and}\quad (a)_k:=\prod_{n=1}^k(a+n-1) +% \constraint{ $k=1,2,3,\ldots$ } +\end{equation} +This can be seen as a generalization of the factorial since +$$(1)_n=n!,\quad n=0,1,2,\ldots.$$ + +The binomial coefficient can be defined by +\begin{equation} +\label{binomialcoeff}\index{Binomial coefficient} +\binom{\alpha}{\beta}:=\frac{\Gamma(\alpha+1)}{\Gamma(\beta+1)\Gamma(\alpha-\beta+1)}. +\end{equation} +For integer values of the parameter $\beta$ we have +$$\binom{\alpha}{k}:=\frac{(-\alpha)_k}{k!}(-1)^k,\quad k=0,1,2,\ldots$$ +and when the parameter $\alpha$ is an integer too, we have +\begin{equation} +\binom{n}{k}:=\frac{n!}{k!\,(n-k)!} +% \constraint{ +% $k=0,1,2,\ldots,n$ & +% $n=0,1,2,\ldots$ } +\end{equation} +The latter formula can be used to show that +$$\binom{2n}{n}=\frac{\left(\frac{1}{2}\right)_n}{n!}4^n,\quad n=0,1,2,\ldots.$$ + +\section{Hypergeometric functions} +\par\setcounter{equation}{0} +\label{hypergeometric functions} + +The hypergeometric function $\mbox{}_rF_s$ is defined by the series +\begin{equation} +\label{DefHyp}\index{Hypergeometric function} +\hyp{r}{s}{a_1,\ldots,a_r}{b_1,\ldots,b_s}{z}:= +\sum\limits_{k=0}^{\infty}\frac{(a_1,\ldots,a_r)_k}{(b_1,\ldots,b_s)_k} +\frac{z^k}{k!}, +\end{equation} +where +$$(a_1,\ldots,a_r)_k:=(a_1)_k\cdots(a_r)_k.$$ +Of course, the parameters must be such that the denominator factors in the +terms of the series are never zero. When one of the numerator parameters +$a_j$ equals $-n$, where $n$ is a nonnegative integer, this hypergeometric +function is a polynomial in $z$. Otherwise the radius of convergence $\rho$ of +the hypergeometric series is given by +$$\rho=\left\{\begin{array}{ll} +\displaystyle \infty & \quad\textrm{if}\quad r < s+1\\[5mm] +\displaystyle 1 & \quad\textrm{if}\quad r = s+1\\[5mm] +\displaystyle 0 & \quad\textrm{if}\quad r > s+1.\end{array}\right.$$ + +A hypergeometric series of the form (\ref{DefHyp}) is called +balanced\index{Hypergeometric function!Balanced} or +Saalsch\"{u}tzian\index{Hypergeometric function!Saalsch\"{u}tzian} +if $r=s+1$, $z=1$ and $a_1+a_2+ \ldots +a_{s+1}+1=b_1+b_2+ \ldots +b_s$. + +Many limit relations between hypergeometric orthogonal polynomials are based on the observations that +\begin{equation} +\label{hypsub} +\hyp{r}{s}{a_1,\ldots,a_{r-1},\mu}{b_1,\ldots,b_{s-1},\mu}{z} +=\hyp{r-1}{s-1}{a_1,\ldots,a_{r-1}}{b_1,\ldots,b_{s-1}}{z}, +\end{equation} +\begin{equation} +\label{hyplim1} +\lim\limits_{\lambda\rightarrow\infty} +\hyp{r}{s}{a_1,\ldots,a_{r-1},\lambda a_r}{b_1,\ldots,b_s}{\frac{z}{\lambda}} +=\hyp{r-1}{s}{a_1,\ldots,a_{r-1}}{b_1,\ldots,b_s}{a_rz}, +\end{equation} +\begin{equation} +\label{hyplim2} +\lim\limits_{\lambda\rightarrow\infty} +\hyp{r}{s}{a_1,\ldots,a_r}{b_1,\ldots,b_{s-1},\lambda b_s}{\lambda z}= +\hyp{r}{s-1}{a_1,\ldots,a_r}{b_1,\ldots,b_{s-1}}{\frac{z}{b_s}} +\end{equation} +and +\begin{equation} +\label{hyplim3} +\lim\limits_{\lambda\rightarrow\infty} +\hyp{r}{s}{a_1,\ldots,a_{r-1},\lambda a_r}{b_1,\ldots,b_{s-1},\lambda b_s}{z} +=\hyp{r-1}{s-1}{a_1,\ldots,a_{r-1}}{b_1,\ldots,b_{s-1}}{\frac{a_rz}{b_s}}. +\end{equation} + +Mostly, the left-hand side of (\ref{hypsub}) occurs as a limit case where some numerator +parameter and some denominator parameter tend to the same value. + +All families of discrete orthogonal polynomials $\{P_n(x)\}_{n=0}^N$ are +defined for $n=0,1,2,\ldots,N$, where $N$ is a positive integer. In these +cases something like (\ref{hypsub}) occurs in the hypergeometric representation when $n=N$. +In these cases we have to be aware of the fact that we still have a polynomial (in that case of +degree $N$). For instance, if we take $n=N$ in the hypergeometric representation (\ref{DefHahn}) +of the Hahn polynomials, we have +$$Q_N(x;\alpha,\beta,N)=\sum_{k=0}^N\frac{(N+\alpha+\beta+1)_k(-x)_k}{(\alpha+1)_kk!}.$$ +So these cases must be understood by continuity. + +In cases of discrete orthogonal polynomials, we need a special notation for +some of the generating functions. We define +$$\left[f(t)\right]_N:=\sum_{k=0}^N\frac{f^{(k)}(0)}{k!}t^k,$$ +for every function $f$ for which $f^{(k)}(0)$, $k=0,1,2,\ldots,N$ exists. As +an example of the use of this $N$th partial sum of a power series in $t$ we +remark that the generating function (\ref{GenKrawtchouk2}) for the +Krawtchouk polynomials must be understood as follows: the $N$th partial sum +of +$$\e^t\,\hyp{1}{1}{-x}{-N}{-\frac{t}{p}}= +\sum_{k=0}^{\infty}\frac{t^k}{k!}\sum_{m=0}^x\frac{(-x)_m}{(-N)_mm!} +\left(-\frac{t}{p}\right)^m$$ +equals +$$\sum_{n=0}^N\frac{K_n(x;p,N)}{n!}t^n$$ +for $x=0,1,2,\ldots,N$. + +The classical exponential function $\e^z$ and the trigonometric functions +$\cos z$ and $\sin z$ can be expressed in terms of hypergeometric functions +as +\begin{equation} +\label{exp} +\e^z=\hyp{0}{0}{-}{-}{z}, +\end{equation} +\begin{equation} +\label{cos} +\cos z=\hyp{0}{1}{-}{\frac{1}{2}}{-\frac{z^2}{4}} +\end{equation} +and +\begin{equation} +\label{sin} +\sin z=z\,\hyp{0}{1}{-}{\frac{3}{2}}{-\frac{z^2}{4}}. +\end{equation} + +Further we have the well-known Bessel function of the first kind $J_{\nu}(z)$, which can be +defined by +\begin{equation} +\label{Bessel}\index{Bessel function} +J_{\nu}(z):=\frac{\left(\frac{1}{2}z\right)^{\nu}}{\Gamma(\nu+1)}\, +\hyp{0}{1}{-}{\nu+1}{-\frac{z^2}{4}}. +\end{equation} + +\section{The binomial theorem and other summation formulas} +\par\setcounter{equation}{0} +\label{summation formulas} + +One of the most important summation formulas for hypergeometric series is +given by the binomial theorem: +\begin{equation} +\label{binomial}\index{Binomial theorem}\index{Summation formula!Binomial theorem} +\hyp{1}{0}{a}{-}{z}=\sum_{n=0}^{\infty}\frac{(a)_n}{n!}z^n=(1-z)^{-a}, +% \constraint{ $\quad |z|<1$ } +\end{equation} +which is a generalization of Newton's binomium +\begin{equation} +\label{Newton} +\hyp{1}{0}{-n}{-}{z}=\sum_{k=0}^n\frac{(-n)_k}{k!}z^k +=\sum_{k=0}^n\binom{n}{k}(-z)^k=(1-z)^n,\quad n=0,1,2,\ldots. +\end{equation} + +We also have Gauss's summation formula +\begin{equation} +\label{Gauss}\index{Gauss summation formula}\index{Summation formula!Gauss} +\hyp{2}{1}{a,b}{c}{1}= +\frac{\Gamma(c)\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)} +% \constraint{ $\textrm{Re}(c-a-b)>0$ } +\end{equation} +and the Vandermonde or Chu-Vandermonde summation formula +\begin{equation} +\label{ChuVandermonde} +\index{Vandermonde summation formula}\index{Summation formula!Vandermonde} +\index{Chu-Vandermonde summation formula}\index{Summation formula!Chu-Vandermonde} +\hyp{2}{1}{-n,b}{c}{1}=\frac{(c-b)_n}{(c)_n},\quad n=0,1,2,\ldots. +\end{equation} + +On the next level we have the summation formula +\begin{equation} +\label{PfaffSaalschutz} +\index{Saalsch\"{u}tz summation formula}\index{Summation formula!Saalsch\"{u}tz} +\index{Pfaff-Saalsch\"{u}tz summation formula}\index{Summation formula!Pfaff-Saalsch\"{u}tz} +\hyp{3}{2}{-n,a,b}{c,1+a+b-c-n}{1}=\frac{(c-a)_n(c-b)_n}{(c)_n(c-a-b)_n}, +\quad n=0,1,2,\ldots, +\end{equation} +which is called the Saalsch\"{u}tz or Pfaff-Saalsch\"{u}tz summation formula. + +For a very-well-poised ${}_5F_4$ we have the summation formula +\begin{eqnarray} +\label{wellpoised}\index{Summation formula!for a very-well-poised ${}_5F_4$} +& &\hyp{5}{4}{1+a/2,a,b,c,d}{a/2,1+a-b,1+a-c,1+a-d}{1}\nonumber\\ +& &{}=\frac{\Gamma(1+a-b)\Gamma(1+a-c)\Gamma(1+a-d)\Gamma(1+a-b-c-d)} +{\Gamma(1+a)\Gamma(1+a-b-c)\Gamma(1+a-b-d)\Gamma(1+a-c-d)}. +\end{eqnarray} +The limit case $d\rightarrow -\infty$ leads to +\begin{equation} +\label{wellpoisedlimit} +\hyp{4}{3}{1+a/2,a,b,c}{a/2,1+a-b,1+a-c}{-1} +=\frac{\Gamma(1+a-b)\Gamma(1+a-c)}{\Gamma(1+a)\Gamma(1+a-b-c)}. +\end{equation} + +Finally, we mention Dougall's bilateral sum +\begin{equation} +\label{Dougall1}\index{Dougall's bilateral sum}\index{Summation formula!Dougall} +\sum_{n=-\infty}^{\infty}\frac{\Gamma(n+a)\Gamma(n+b)}{\Gamma(n+c)\Gamma(n+d)} +=\frac{\Gamma(a)\Gamma(1-a)\Gamma(b)\Gamma(1-b)\Gamma(c+d-a-b-1)} +{\Gamma(c-a)\Gamma(d-a)\Gamma(c-b)\Gamma(d-b)}, +\end{equation} +which holds for $\textrm{Re}(a+b)+1<\textrm{Re}(c+d)$.\\ +By using +$$\Gamma(n+a)\Gamma(n+b)=\frac{\Gamma(a)\Gamma(1-a)\Gamma(b)\Gamma(1-b)} +{\Gamma(1-a-n)\Gamma(1-b-n)},\quad n\in\mathbb{Z},$$ +Dougall's bilateral sum can also be written in the form +\begin{eqnarray} +\label{Dougall2} +& &\sum_{n=-\infty}^{\infty}\frac{1}{\Gamma(n+c)\Gamma(n+d)\Gamma(1-a-n)\Gamma(1-b-n)}\nonumber\\ +& &{}=\frac{\Gamma(c+d-a-b-1)}{\Gamma(c-a)\Gamma(d-a)\Gamma(c-b)\Gamma(d-b)} +\end{eqnarray} +for $\textrm{Re}(a+b)+1<\textrm{Re}(c+d)$. + +\section{Some integrals} +\par\setcounter{equation}{0} +\label{integals1} + +For the ${}_2F_1$ hypergeometric function we have Euler's integral representation +\begin{equation} +\label{Eulerint}\index{Euler's integral representation for a ${}_2F_1$} +\index{Hypergeometric function!Euler's integral representation} +\hyp{2}{1}{a,b}{c}{z}=\frac{\Gamma(c)}{\Gamma(b)\Gamma(c-b)} +\int_0^1t^{b-1}(1-t)^{c-b-1}(1-zt)^{-a}\,dt, +\end{equation} +where $\textrm{Re}\,c>\textrm{Re}\,b>0$, which holds for $z\in\mathbb{C}\setminus(1,\infty)$. +Here it is understood that $\arg t=\arg(1-t)=0$ and that $(1-zt)^{-a}$ has its +principal value. + +We also have Barnes' integral representation +\begin{equation} +\label{Barnes}\index{Barnes' integral representation} +\index{Hypergeometric function!Barnes' integral representation} +\frac{\Gamma(a)\Gamma(b)}{\Gamma(c)}\,\hyp{2}{1}{a,b}{c}{z}=\frac{1}{2\pi i} +\int_{-i\infty}^{i\infty}\frac{\Gamma(a+s)\Gamma(b+s)\Gamma(-s)}{\Gamma(c+s)}(-z)^s\,ds +\end{equation} +for $|z|<1$ with $\arg(-z)<\pi$, where the path of integration is deformed +if necessary, to separate the decreasing poles $s=-a-n$ and $s=-b-n$ from the +increasing poles $s=n$ for $n\in\{0,1,2,\ldots\}$. Such a path always exists if +$a,b\notin\{\ldots,-3,-2,-1\}$. + +Secondly, we have the Mellin-Barnes integral or Barnes' first lemma +\begin{eqnarray} +\label{Mellin-Barnes1}\index{Mellin-Barnes integral}\index{Barnes' first lemma} +& &\frac{1}{2\pi i}\int_{-i\infty}^{i\infty}\Gamma(a+s)\Gamma(b+s)\Gamma(c-s)\Gamma(d-s)\,ds\nonumber\\ +& &{}=\frac{\Gamma(a+c)\Gamma(a+d)\Gamma(b+c)\Gamma(b+d)}{\Gamma(a+b+c+d)} +\end{eqnarray} +for $\textrm{Re}(a+b+c+d)<1$, where the contour must also be taken in a such a +way that the increasing poles and the decreasing poles remain separate. By using +analytic continuation, one can avoid the condition $\textrm{Re}(a+b+c+d)<1$. The +Barnes integral (\ref{Mellin-Barnes1}) can be seen as a continuous analogue of +Gauss' summation formula (\ref{Gauss}). + +We also have a continuous analogue of the Pfaff-Saalsch\"utz summation +formula (\ref{PfaffSaalschutz}) given by the integral, which is also +called Barnes' second lemma\index{Barnes' second lemma}, +\begin{eqnarray} +\label{Mellin-Barnes2} +& &\frac{1}{2\pi i}\int_{-i\infty}^{i\infty} +\frac{\Gamma(a+s)\Gamma(b+s)\Gamma(c+s)\Gamma(1-d-s)\Gamma(-s)}{\Gamma(e+s)}\,ds\nonumber\\ +& &{}=\frac{\Gamma(a)\Gamma(b)\Gamma(c)\Gamma(1+a-d)\Gamma(1+b-d)\Gamma(1+c-d)} +{\Gamma(e-a)\Gamma(e-b)\Gamma(e-c)} +\end{eqnarray} +with $d+e=a+b+c+1$, where the contour must be taken in a such a way that +the increasing poles and the decreasing poles stay separated. + +A continuous analogue of the summation formula (\ref{wellpoised}) for a very-well-poised +${}_5F_4$ is given by Bailey's integral (see also \cite{Bailey35}) +\begin{eqnarray} +\label{Bailey1}\index{Bailey's integral} +& &\frac{1}{2\pi i}\int_{-i\infty}^{i\infty} +\frac{\Gamma(1+a/2+s)\Gamma(a+s)\Gamma(b+s)\Gamma(c+s)\Gamma(d+s)\Gamma(b-a-s)\Gamma(-s)} +{\Gamma(a/2+s)\Gamma(1+a-c+s)\Gamma(1+a-d+s)}\,ds\nonumber\\ +& &{}=\frac{\Gamma(b)\Gamma(c)\Gamma(d)\Gamma(b+c-a)\Gamma(b+d-a)} +{2\,\Gamma(1+a-c-d)\Gamma(b+c+d-a)} +\end{eqnarray} +which can also be written in a more symmetric form +\begin{eqnarray} +\label{Bailey2} +& &\frac{1}{2\pi i}\int_{-i\infty}^{i\infty} +\frac{\Gamma(a+s)\Gamma(b+s)\Gamma(c+s)\Gamma(d+s)}{\Gamma(a+2s)}\nonumber\\ +& &{}\mathindent\times\frac{\Gamma(-s)\Gamma(b-a-s)\Gamma(c-a-s)\Gamma(d-a-s)}{\Gamma(-a-2s)}\,ds\nonumber\\ +& &{}=\frac{2\,\Gamma(b)\Gamma(c)\Gamma(d)\Gamma(b+c-a)\Gamma(b+d-a)\Gamma(c+d-a)}{\Gamma(b+c+d-a)} +\end{eqnarray} +and also in the following form due to Wilson (see also \cite{Wilson80}) +\begin{eqnarray} +\label{Bailey3} +& &\frac{1}{2\pi i}\int_{-i\infty}^{i\infty} +\frac{\Gamma(a+s)\Gamma(b+s)\Gamma(c+s)\Gamma(d+s)\Gamma(a-s)\Gamma(b-s)\Gamma(c-s)\Gamma(d-s)} +{\Gamma(2s)\Gamma(-2s)}\,ds\nonumber\\ +& &{}=\frac{2\,\Gamma(a+b)\Gamma(a+c)\Gamma(a+d)\Gamma(b+c)\Gamma(b+d)\Gamma(c+d)}{\Gamma(a+b+c+d)}, +\end{eqnarray} +where the contours must be taken in a such a way that the increasing poles and the +decreasing poles remain separate. + +We also mention the integral +\begin{equation} +\label{Slater} +\frac{1}{2\pi i}\int_{-i\infty}^{i\infty}\Gamma(a+s)\Gamma(b-s)z^{-s}\,ds +=\Gamma(a+b)\frac{z^a}{(1+z)^{a+b}}. +\end{equation} +Again the contour must be taken in such a way that the increasing poles of $\Gamma(b-s)$ and +the decreasing poles of $\Gamma(a+s)$ remain separate. + +The integrals (\ref{Mellin-Barnes1}) through (\ref{Bailey3}) are all special cases of a more +general formula (id est formula (4.5.1.2) in \cite{Slater}), which has more interesting and +useful special cases such as +\begin{equation} +\label{Slater1} +\frac{1}{2\pi i}\int_{-i\infty}^{i\infty}\frac{\Gamma(a+s)\Gamma(b-s)}{\Gamma(c+s)\Gamma(d-s)}\,ds +=\frac{\Gamma(a+b)\Gamma(c+d-a-b-1)}{\Gamma(c+d-1)\Gamma(c-a)\Gamma(d-b)}, +\end{equation} +where the contour must be taken in a such a way that the increasing poles and the decreasing poles +remain separate and +\begin{eqnarray} +\label{Slater2} +& &\frac{1}{2\pi i}\int_{-i\infty}^{i\infty} +\frac{\Gamma(a+s)\Gamma(b+s)\Gamma(c+s)\Gamma(-s)\Gamma(b-a-s)\Gamma(c-a-s)} +{\Gamma(a+2s)\Gamma(-a-2s)}\,ds\nonumber\\ +& &{}=\frac{1}{2}\Gamma(b)\Gamma(c)\Gamma(b+c-a), +\end{eqnarray} +where the contour must be taken in a such a way that the increasing poles and the decreasing poles +remain separate. + +If $a,b,c,d$ are positive or $b=\overline{a}$ and/or $d=\overline{c}$ and the real parts are +positive, Wilson's integral (\ref{Bailey3}) can be written in the form +\begin{eqnarray} +\label{Wilson1}\index{Wilson's integral} +& &\frac{1}{2\pi}\int_0^{\infty} +\left|\frac{\Gamma(a+ix)\Gamma(b+ix)\Gamma(c+ix)\Gamma(d+ix)}{\Gamma(2ix)}\right|^2\,dx\nonumber\\ +& &{}=\frac{\Gamma(a+b)\Gamma(a+c)\Gamma(a+d)\Gamma(b+c)\Gamma(b+d)\Gamma(c+d)}{\Gamma(a+b+c+d)}. +\end{eqnarray} +The limit case $d\rightarrow\infty$ leads to +\begin{equation} +\label{Wilson2} +\frac{1}{2\pi}\int_0^{\infty} +\left|\frac{\Gamma(a+ix)\Gamma(b+ix)\Gamma(c+ix)}{\Gamma(2ix)}\right|^2\,dx +=\Gamma(a+b)\Gamma(a+c)\Gamma(b+c). +\end{equation} + +\section{Transformation formulas for generalized hypergeometric functions} +\par\setcounter{equation}{0} +\label{transformation formulas} + +In this section we list a number of transformation formulas which can be used to +transform hypergeometric representations and other formulas into equivalent +but different forms. + +First of all we have Euler's transformation formula: +\begin{equation} +\label{Eulertrans}\index{Euler's transformation formula} +\index{Transformation formula!Euler} +\hyp{2}{1}{a,b}{c}{z}=(1-z)^{c-a-b}\,\hyp{2}{1}{c-a,c-b}{c}{z}. +\end{equation} + +Another transformation formula for the ${}_2F_1$ series, which is also due to Euler, is +\begin{equation} +\label{PfaffKummer}\index{Pfaff-Kummer transformation formula} +\index{Transformation formula!Pfaff-Kummer} +\hyp{2}{1}{a,b}{c}{z}=(1-z)^{-a}\,\hyp{2}{1}{a,c-b}{c}{\frac{z}{z-1}}. +\end{equation} +This transformation formula is also known as the Pfaff or Pfaff-Kummer transformation formula. + +As a limit case of the Pfaff-Kummer transformation formula we have Kummer's transformation +formula for the confluent hypergeometric series: +\begin{equation} +\label{Kummer}\index{Kummer's transformation formula}\index{Transformation formula!Kummer} +\hyp{1}{1}{a}{c}{z}=\e^z\,\hyp{1}{1}{c-a}{c}{-z}. +\end{equation} + +If we reverse the order of summation in a terminating ${}_1F_1$ series, we +obtain a ${}_2F_0$ series; in fact we have +\begin{equation} +\label{reverse1} +\hyp{1}{1}{-n}{a}{x}=\frac{(-x)^n}{(a)_n}\,\hyp{2}{0}{-n,-a-n+1}{-}{-\frac{1}{x}}, +\quad n=0,1,2,\ldots. +\end{equation} +If we apply this technique to a terminating ${}_2F_1$ series, we find +\begin{equation} +\label{reverse2} +\hyp{2}{1}{-n,b}{c}{x}=\frac{(b)_n}{(c)_n}(-x)^n\,\hyp{2}{1}{-n,-c-n+1}{-b-n+1}{\frac{1}{x}}, +\quad n=0,1,2,\ldots. +\end{equation} + +On the next level we have Whipple's transformation formula for a terminating balanced +${}_4F_3$ series: +\begin{eqnarray} +\label{Whipple}\index{Whipple's transformation formula}\index{Transformation formula!Whipple} +& &\hyp{4}{3}{-n,a,b,c}{d,e,f}{1}\nonumber\\ +& &{}=\frac{(e-a)_n(f-a)_n}{(e)_n(f)_n}\hyp{4}{3}{-n,a,d-b,d-c}{d,a-e-n+1,a-f-n+1}{1} +\end{eqnarray} +provided that $a+b+c+1=d+e+f+n$. + +\section{The $q$-shifted factorial} +\par\setcounter{equation}{0} +\label{qshifted factorial} + +The theory of $q$-analogues or $q$-extensions of classical formulas and +functions is based on the observation that +$$\lim\limits_{q\rightarrow 1}\frac{1-q^{\alpha}}{1-q}=\alpha.$$ +Therefore the number $(1-q^{\alpha})/(1-q)$ is sometimes called the +\emph{basic number} $[\alpha]$. For $q\neq 0$ and $q\neq 1$ we define +\begin{equation} +\label{basic} +[\alpha]:=\frac{1-q^{\alpha}}{1-q}, +\end{equation} +which implies that +\begin{equation} +\label{basicdef} +[0]=0,\quad [n]=\frac{1-q^n}{1-q}=\sum_{k=0}^{n-1}q^k,\quad n=1,2,3,\ldots. +\end{equation} + +Now we can give a $q$-analogue of the Pochhammer symbol $(a)_k$ defined by +(\ref{Pochhammer}): +\begin{equation} +\label{qsh}\index{q-Shifted factorial@$q$-Shifted factorial} +(a;q)_0:=1\quad\textrm{and}\quad (a;q)_k:=\prod_{n=1}^k(1-aq^{n-1}) +% \constraint{ $k=1,2,3,\ldots$ } +\end{equation} +It is clear that +\begin{equation} +\label{limitqsh} +\lim\limits_{q\rightarrow 1}\frac{(q^{\alpha};q)_k}{(1-q)^k}=(\alpha)_k. +\end{equation} +The symbols $(a;q)_k$ are called $q$-shifted factorials. For negative subscripts we define +\begin{equation} +\label{qsh1} +(a;q)_{-k}:=\frac{1}{\displaystyle\prod_{n=1}^{k}(1-aq^{-n})},\quad a\neq q,q^2,q^3,\ldots,q^k, +\quad k=1,2,3,\ldots. +\end{equation} +Now we have +\begin{equation} +\label{qsh2} +(a;q)_{-n}=\frac{1}{(aq^{-n};q)_n}=\frac{(-qa^{-1})^n}{(qa^{-1};q)_n} +q^{\binom{n}{2}},\quad n=0,1,2,\ldots. +\end{equation} + +If we replace $q$ by $q^{-1}$, we obtain +\begin{equation} +\label{qsh3} +(a;q^{-1})_n=(a^{-1};q)_n(-a)^nq^{-\binom{n}{2}},\quad a\neq 0. +\end{equation} + +We can also define +$$(a;q)_{\infty}=\prod_{k=0}^{\infty}(1-aq^k),\quad 0<|q|<1.$$ +This implies that +\begin{equation} +\label{qsh4} +(a;q)_n=\frac{(a;q)_{\infty}}{(aq^n;q)_{\infty}},\quad 0<|q|<1, +\end{equation} +and, for any complex number $\lambda$, +\begin{equation} +\label{qsh5} +(a;q)_{\lambda}=\frac{(a;q)_{\infty}}{(aq^{\lambda};q)_{\infty}},\quad 0<|q|<1, +\end{equation} +where the principal value of $q^{\lambda}$ is taken. + +Finally, we list a number of transformation formulas for the $q$-shifted +factorials, where $k$ and $n$ are nonnegative integers: +\begin{equation} +\label{qsh6} +(a;q)_{n+k}=(a;q)_n(aq^n;q)_k, +\end{equation} +\begin{equation} +\label{qsh7} +\frac{(aq^n;q)_k}{(aq^k;q)_n}=\frac{(a;q)_k}{(a;q)_n}, +\end{equation} +\begin{equation} +\label{qsh8} +(aq^k;q)_{n-k}=\frac{(a;q)_n}{(a;q)_k},\quad k=0,1,2,\ldots,n, +\end{equation} +\begin{equation} +\label{qsh9} +(a;q)_n=(a^{-1}q^{1-n};q)_n(-a)^nq^{\binom{n}{2}},\quad a\neq 0, +\end{equation} +\begin{equation} +\label{qsh10} +(aq^{-n};q)_n=(a^{-1}q;q)_n(-a)^nq^{-n-\binom{n}{2}},\quad a\neq 0, +\end{equation} +\begin{equation} +\label{qsh11} +\frac{(aq^{-n};q)_n}{(bq^{-n};q)_n}=\frac{(a^{-1}q;q)_n}{(b^{-1}q;q)_n} +\left(\frac{a}{b}\right)^n +% \constraint{ +% $a\neq 0$ & +% $b\neq 0$ } +\end{equation} +\begin{equation} +\label{qsh12} +(a;q)_{n-k}=\frac{(a;q)_n}{(a^{-1}q^{1-n};q)_k}\left(-\frac{q}{a}\right)^k +q^{\binom{k}{2}-nk},\quad a\neq 0,\quad k=0,1,2,\ldots,n, +\end{equation} +\begin{eqnarray} +\label{qsh13} +& &\frac{(a;q)_{n-k}}{(b;q)_{n-k}}=\frac{(a;q)_n}{(b;q)_n} +\frac{(b^{-1}q^{1-n};q)_k}{(a^{-1}q^{1-n};q)_k}\left(\frac{b}{a}\right)^k,\nonumber\\ +& &{}\mathindent\mathindent\mathindent a\neq 0,\quad b\neq 0,\quad k=0,1,2,\ldots,n, +\end{eqnarray} +\begin{equation} +\label{qsh14} +(q^{-n};q)_k=\frac{(q;q)_n}{(q;q)_{n-k}}(-1)^kq^{\binom{k}{2}-nk}, +\quad k=0,1,2,\ldots,n, +\end{equation} +\begin{equation} +\label{qsh15} +(aq^{-n};q)_k=\frac{(a^{-1}q;q)_n}{(a^{-1}q^{1-k};q)_n}(a;q)_kq^{-nk}, +\quad a\neq 0, +\end{equation} +\begin{eqnarray} +\label{qsh16} +& &(aq^{-n};q)_{n-k}=\frac{(a^{-1}q;q)_n}{(a^{-1}q;q)_k} +\left(-\frac{a}{q}\right)^{n-k}q^{\binom{k}{2}-\binom{n}{2}},\nonumber\\ +& &{}\mathindent\mathindent\mathindent a\neq 0,\quad k=0,1,2,\ldots,n, +\end{eqnarray} +\begin{equation} +\label{qsh17} +(a;q)_{2n}=(a;q^2)_n(aq;q^2)_n, +\end{equation} +\begin{equation} +\label{qsh18} +(a^2;q^2)_n=(a;q)_n(-a;q)_n, +\end{equation} +\begin{equation} +\label{qsh19} +(a;q)_{\infty}=(a;q^2)_{\infty}(aq;q^2)_{\infty},\quad 0<|q|<1, +\end{equation} +\begin{equation} +\label{qsh20} +(a^2;q^2)_{\infty}=(a;q)_{\infty}(-a;q)_{\infty},\quad 0<|q|<1. +\end{equation} + +We remark that by using (\ref{qsh18}) we have +\begin{equation} +\label{qsh21} +\frac{1-a^2q^{2n}}{1-a^2}=\frac{(a^2q^2;q^2)_n}{(a^2;q^2)_n} +=\frac{(aq;q)_n(-aq;q)_n}{(a;q)_n(-a;q)_n}. +\end{equation} + +\section{The $q$-gamma function and $q$-binomial coefficients} +\par\setcounter{equation}{0} +\label{qgamma function} + +The $q$-gamma function is defined by +\begin{equation} +\label{DefqGamma}\index{q-Gamma function@$q$-Gamma function} +\Gamma_q(x):=\frac{(q;q)_{\infty}}{(q^x;q)_{\infty}}(1-q)^{1-x},\quad 01$ the $q$-gamma function can be defined by +\begin{equation} +\label{DefqGamma2} +\Gamma_q(x)=\frac{(q^{-1};q^{-1})_{\infty}}{(q^{-x};q^{-1})_{\infty}} +q^{\binom{x}{2}}(q-1)^{1-x},\quad q>1. +\end{equation} + +The $q$-binomial coefficient is defined by +\begin{equation} +\label{DefqBinomial1}\index{q-Binomial coefficient@$q$-Binomial coefficient} +\genfrac{[}{]}{0pt}{}{n}{k}_q:=\frac{(q;q)_n}{(q;q)_k(q;q)_{n-k}}=\genfrac{[}{]}{0pt}{}{n}{n-k}_q, +\quad k=0,1,2,\ldots,n, +\end{equation} +where $n$ denotes a nonnegative integer. + +This definition can be generalized in the following way. For arbitrary +complex $\alpha$ we have +\begin{equation} +\label{DefqBinomial2} +\genfrac{[}{]}{0pt}{}{\alpha}{k}_q:=\frac{(q^{-\alpha};q)_k}{(q;q)_k} +(-1)^kq^{k\alpha-\binom{k}{2}}. +\end{equation} +Or more general, for all complex $\alpha$ and $\beta$ and $0<|q|<1$ we have +\begin{equation} +\label{DefqBinomial3} +\genfrac{[}{]}{0pt}{}{\alpha}{\beta}_q:=\frac{\Gamma_q(\alpha+1)}{\Gamma_q(\beta+1)\Gamma_q(\alpha-\beta+1)} +=\frac{(q^{\beta+1};q)_{\infty}(q^{\alpha-\beta+1};q)_{\infty}} +{(q;q)_{\infty}(q^{\alpha+1};q)_{\infty}}. +\end{equation} + +For instance this implies that +$$\frac{(q^{\alpha+1};q)_n}{(q;q)_n}=\genfrac{[}{]}{0pt}{}{n+\alpha}{n}_q.$$ + +Note that +$$\lim\limits_{q\rightarrow 1}\genfrac{[}{]}{0pt}{}{\alpha}{\beta}_q +=\frac{\Gamma(\alpha+1)}{\Gamma(\beta+1)\Gamma(\alpha-\beta+1)}=\binom{\alpha}{\beta}.$$ + +Finally, we remark that +\begin{equation} +\label{qqn} +\frac{1}{(q;q)_n}=\sum_{k=0}^n\frac{q^k}{(q;q)_k},\quad n=0,1,2,\ldots, +\end{equation} +which can easily be shown by induction, and that +\begin{equation} +\label{aqn} +(a;q)_n=\sum_{k=0}^n\genfrac{[}{]}{0pt}{}{n}{k}_qq^{\binom{k}{2}}(-a)^k,\quad n=0,1,2,\ldots, +\end{equation} +which is a special case of (\ref{qbinomialn}). + +\section{Basic hypergeometric functions} +\par\setcounter{equation}{0}\label{hypergeometric} +\label{qhypergeometric functions} + +The basic hypergeometric or $q$-hypergeometric function $\mbox{}_r\phi_s$ is defined by the series +\begin{eqnarray} +\label{DefBasHyp}\index{Basic hypergeometric function} +& &\qhyp{r}{s}{a_1,\ldots,a_r}{b_1,\ldots,b_s}{z}\nonumber\\ +& &{}:=\sum\limits_{k=0}^{\infty}\frac{(a_1,\ldots,a_r;q)_k}{(b_1,\ldots,b_s;q)_k} +(-1)^{(1+s-r)k}q^{(1+s-r)\binom{k}{2}}\frac{z^k}{(q;q)_k}, +\end{eqnarray} +where +$$(a_1,\ldots,a_r;q)_k:=(a_1;q)_k\cdots(a_r;q)_k.$$ +Again we assume that the parameters are such that the denominator factors +in the terms of the series are never zero. If one of the numerator +parameters $a_j$ equals $q^{-n}$, where $n$ is a nonnegative integer, this +basic hypergeometric function is a polynomial in $z$. Otherwise the radius of +convergence $\rho$ of the basic hypergeometric series is given by +$$\rho=\left\{\begin{array}{ll} +\displaystyle \infty & \quad\textrm{if}\quad r < s+1\\[5mm] +\displaystyle 1 & \quad\textrm{if}\quad r = s+1\\[5mm] +\displaystyle 0 & \quad\textrm{if}\quad r > s+1.\end{array}\right.$$ + +The special case $r=s+1$ reads +$$\qhyp{s+1}{s}{a_1,\ldots,a_{s+1}}{b_1,\ldots,b_s}{z}= +\sum\limits_{k=0}^{\infty}\frac{(a_1,\ldots,a_{s+1};q)_k}{(b_1,\ldots,b_s;q)_k} +\frac{z^k}{(q;q)_k}.$$ +This basic hypergeometric series was first introduced by Heine in 1846; therefore it is +sometimes called Heine's\index{Heine's series} series. A basic hypergeometric series of +this form is called balanced\index{Basic hypergeometric function!Balanced} +or Saalsch\"{u}tzian\index{Basic hypergeometric function!Saalsch\"{u}tzian} if $z=q$ and +$a_1a_2 \cdots a_{s+1}q=b_1b_2 \cdots b_s$. + +The $q$-hypergeometric function is a $q$-analogue of the hypergeometric function +defined by (\ref{DefHyp}) since +$$\lim\limits_{q\rightarrow 1} +\qhyp{r}{s}{q^{a_1},\ldots,q^{a_r}}{q^{b_1},\ldots,q^{b_s}}{(q-1)^{1+s-r}z} +=\hyp{r}{s}{a_1,\ldots,a_r}{b_1,\ldots,b_s}{z}.$$ +This limit will be used frequently in chapter~\ref{BasicHyperOrtPol}. In all +cases the hypergeometric series involved is in fact a polynomial so that +convergence is guaranteed. + +In the sequel of this paragraph we also assume that each (basic) hypergeometric +function is in fact a polynomial. We remark that +$$\lim\limits_{a_r\rightarrow\infty} +\qhyp{r}{s}{a_1,\ldots,a_r}{b_1,\ldots,b_s}{\frac{z}{a_r}}= +\qhyp{r-1}{s}{a_1,\ldots,a_{r-1}}{b_1,\ldots,b_s}{z}.$$ +In fact this is the reason for the factors +$(-1)^{(1+s-r)k}q^{(1+s-r)\binom{k}{2}}$ in the definition (\ref{DefBasHyp}) +of the basic hypergeometric function. + +Many limit relations between basic hypergeometric orthogonal polynomials +are based on the observations that +\begin{equation} +\label{qhypsub} +\qhyp{r}{s}{a_1,\ldots,a_{r-1},\mu}{b_1,\ldots,b_{s-1},\mu}{z}= +\qhyp{r-1}{s-1}{a_1,\ldots,a_{r-1}}{b_1,\ldots,b_{s-1}}{z}, +\end{equation} +\begin{equation} +\label{qhyplim1} +\lim\limits_{\lambda\rightarrow\infty} +\qhyp{r}{s}{a_1,\ldots,a_{r-1},\lambda a_r}{b_1,\ldots,b_s}{\frac{z}{\lambda}} +=\qhyp{r-1}{s}{a_1,\ldots,a_{r-1}}{b_1,\ldots,b_s}{a_rz}, +\end{equation} +\begin{equation} +\label{qhyplim2} +\lim\limits_{\lambda\rightarrow\infty} +\qhyp{r}{s}{a_1,\ldots,a_r}{b_1,\ldots,b_{s-1},\lambda b_s}{\lambda z}= +\qhyp{r}{s-1}{a_1,\ldots,a_r}{b_1,\ldots,b_{s-1}}{\frac{z}{b_s}}, +\end{equation} +and +\begin{equation} +\label{qhyplim3} +\lim\limits_{\lambda\rightarrow\infty} +\qhyp{r}{s}{a_1,\ldots,a_{r-1},\lambda a_r}{b_1,\ldots,b_{s-1},\lambda b_s}{z} +=\qhyp{r-1}{s-1}{a_1,\ldots,a_{r-1}}{b_1,\ldots,b_{s-1}}{\frac{a_rz}{b_s}}. +\end{equation} + +Mostly, the left-hand side of (\ref{qhypsub}) occurs as a limit case when some numerator +parameter and some denominator parameter tend to the same value. + +All families of discrete orthogonal polynomials $\{P_n(x)\}_{n=0}^N$ are +defined for $n=0,1,2,\ldots,N$, where $N$ is a positive integer. In these +cases something like (\ref{qhypsub}) occurs in the basic hypergeometric representation +when $n=N$. In these cases we have to be aware of the fact that we still have a polynomial +(in that case of degree $N$). For instance, if we take $n=N$ in the basic hypergeometric +representation (\ref{DefqHahn}) of the $q$-Hahn polynomials, we have +$$Q_N(q^{-x};\alpha,\beta,N|q)=\sum_{k=0}^N +\frac{(\alpha\beta q^{N+1};q)_k(q^{-x};q)_k}{(\alpha q;q)_k(q;q)_k}q^k.$$ +So these cases must be understood by continuity. + +\section{The $q$-binomial theorem and other summation formulas} +\par\setcounter{equation}{0} +\label{qsummation formulas} + +A $q$-analogue of the binomial theorem (\ref{binomial}) is called the $q$-binomial theorem: +\begin{equation} +\label{qbinomial}\index{q-Binomial theorem@$q$-Binomial theorem} +\index{Summation formula!q-Binomial theorem@$q$-Binomial theorem} +\qhyp{1}{0}{a}{-}{z}=\sum_{n=0}^{\infty}\frac{(a;q)_n}{(q;q)_n}z^n= +\frac{(az;q)_{\infty}}{(z;q)_{\infty}},\quad 0<|q|<1,\quad |z|<1. +\end{equation} +For $a=q^{-n}$ with $n$ a nonnegative integer we find +\begin{equation} +\label{qbinomialn} +\qhyp{1}{0}{q^{-n}}{-}{z}=(zq^{-n};q)_n,\quad n=0,1,2,\ldots. +\end{equation} +In fact this is a $q$-analogue of Newton's binomium (\ref{Newton}). +Note that the case $a=0$ of (\ref{qbinomial}) is the limit case ($n\rightarrow\infty$) of (\ref{qqn}). + +Gauss's summation formula (\ref{Gauss}) and the Vandermonde or Chu-Vandermonde +summation formula (\ref{ChuVandermonde}) have the following $q$-analogues: +\begin{equation} +\label{qChuVandermonde1}\index{q-Gauss summation formula@$q$-Gauss summation formula} +\index{Summation formula!q-Gauss@$q$-Gauss} +\qhyp{2}{1}{a,b}{c}{\frac{c}{ab}}=\frac{(a^{-1}c,b^{-1}c;q)_{\infty}} +{(c,a^{-1}b^{-1}c;q)_{\infty}},\quad 0<|q|<1,\quad\left|\frac{c}{ab}\right|<1, +\end{equation} +\begin{equation} +\label{qChuVandermonde2}\index{q-Vandermonde summation formula@$q$-Vandermonde summation formula} +\index{Summation formula!q-Vandermonde@$q$-Vandermonde} +\index{q-Chu-Vandermonde summation formula@$q$-Chu-Vandermonde summation formula} +\index{Summation formula!q-Chu-Vandermonde@$q$-Chu-Vandermonde} +\qhyp{2}{1}{q^{-n},b}{c}{\frac{cq^n}{b}}=\frac{(b^{-1}c;q)_n}{(c;q)_n},\quad n=0,1,2,\ldots +\end{equation} +and +\begin{equation} +\label{qChuVandermonde3} +\qhyp{2}{1}{q^{-n},b}{c}{q}=\frac{(b^{-1}c;q)_n}{(c;q)_n}b^n,\quad n=0,1,2,\ldots. +\end{equation} +By taking the limit $b\rightarrow\infty$ in (\ref{qChuVandermonde1}) we also obtain a summation +formula for the ${}_1\phi_1$ series: +\begin{equation} +\label{sum1phi1}\index{Summation formula!for a ${}_1\phi_1$} +\qhyp{1}{1}{a}{c}{\frac{c}{a}}=\frac{(a^{-1}c;q)_{\infty}}{(c;q)_{\infty}},\quad 0<|q|<1. +\end{equation} +By taking the limit $c\rightarrow\infty$ in (\ref{qChuVandermonde2}) we obtain a summation +formula for a terminating ${}_2\phi_0$ series: +\begin{equation} +\label{sum2phi0}\index{Summation formula!for a terminating ${}_2\phi_0$} +\qhyp{2}{0}{q^{-n},b}{-}{\frac{q^n}{b}}=\frac{1}{b^n},\quad n=0,1,2,\ldots. +\end{equation} +By taking the limit $a\rightarrow\infty$ in (\ref{sum1phi1}) we obtain +\begin{equation} +\label{sum0phi1}\index{Summation formula!for a ${}_0\phi_1$} +\qhyp{0}{1}{-}{c}{c}=\frac{1}{(c;q)_{\infty}},\quad 0<|q|<1. +\end{equation} + +On the ${}_3\phi_2$ level we have the summation formula +\begin{equation} +\label{qPfaffSaalschutz} +\index{q-Saalschutz summation formula@$q$-Saalsch\"{u}tz summation formula} +\index{Summation formula!q-Saalschutz@$q$-Saalsch\"{u}tz} +\index{q-Pfaff-Saalschutz summation formula@$q$-Pfaff-Saalsch\"{u}tz summation formula} +\index{Summation formula!q-Pfaff-Saalschutz@$q$-Pfaff-Saalsch\"{u}tz} +\qhyp{3}{2}{q^{-n},a,b}{c,abc^{-1}q^{1-n}}{q}=\frac{(a^{-1}c,b^{-1}c;q)_n} +{(c,a^{-1}b^{-1}c;q)_n},\quad n=0,1,2,\ldots, +\end{equation} +which is a $q$-analogue of the Saalsch\"utz or Pfaff-Saalsch\"utz summation formula +(\ref{PfaffSaalschutz}). + +On the ${}_6\phi_5$ level we have the Jackson summation formula +\begin{eqnarray} +\label{qJackson1}\index{Jackson's summation formula}\index{Summation formula!Jackson} +\index{Summation formula!for a very-well-poised ${}_6\phi_5$} +& &\qhyp{6}{5}{q\sqrt{a},-q\sqrt{a},a,b,c,d}{\sqrt{a},-\sqrt{a},ab^{-1}q,ac^{-1}q,ad^{-1}q}{\frac{aq}{bcd}}\nonumber\\ +& &{}=\frac{(aq,ab^{-1}c^{-1}q,ab^{-1}d^{-1}q,ac^{-1}d^{-1}q;q)_{\infty}} +{(ab^{-1}q,ac^{-1}q,ad^{-1}q,ab^{-1}c^{-1}d^{-1}q;q)_{\infty}},\quad\left|\frac{aq}{bcd}\right|<1 +\end{eqnarray} +for a non-terminating very-well-poised ${}_6\phi_5$. We also have the summation formula +\begin{eqnarray} +\label{qJackson2} +& &\qhyp{6}{5}{q\sqrt{a},-q\sqrt{a},a,b,c,q^{-n}}{\sqrt{a},-\sqrt{a},ab^{-1}q,ac^{-1}q,aq^{n+1}}{\frac{aq^{n+1}}{bc}}\nonumber\\ +& &{}=\frac{(aq,ab^{-1}c^{-1}q;q)_n}{(ab^{-1}q,ac^{-1}q;q)_n},\quad n=0,1,2,\ldots +\end{eqnarray} +for a terminating very-well-poised ${}_6\phi_5$, which is also due to Jackson. +By taking the limit $c\rightarrow 0$ we obtain +\begin{eqnarray} +\label{qJackson3} +& &\qhyp{6}{4}{q\sqrt{a},-q\sqrt{a},a,b,0,q^{-n}}{\sqrt{a},-\sqrt{a},ab^{-1}q,aq^{n+1}}{\frac{q^n}{b}}\nonumber\\ +& &{}=\frac{(aq;q)_n}{(ab^{-1}q;q)_n}b^{-n},\quad n=0,1,2,\ldots. +\end{eqnarray} +If we take the limit $b\rightarrow 0$ we obtain +\begin{eqnarray} +\label{qJackson4} +& &\qhyp{6}{3}{q\sqrt{a},-q\sqrt{a},a,0,0,q^{-n}}{\sqrt{a},-\sqrt{a},aq^{n+1}}{\frac{q^{n-1}}{a}}\nonumber\\ +& &{}=(-1)^na^{-n}q^{-\binom{n+1}{2}}(aq;q)_n,\quad n=0,1,2,\ldots. +\end{eqnarray} + +\section{More integrals} +\par\setcounter{equation}{0} +\label{integrals2} + +In this section we list some $q$-extensions of the beta integral. +For instance, we will need the following integral: +\begin{equation} +\label{int1} +\int_0^{\infty}x^{c-1}\frac{(-ax,-bq/x;q)_{\infty}}{(-x,-q/x;q)_{\infty}}\,dx +=\frac{\pi}{\sin(\pi c)}\,\frac{(ab,q^c,q^{1-c};q)_{\infty}}{(bq^c,aq^{-c},q;q)_{\infty}}, +\end{equation} +which holds for $00$ and $\left|aq^{-c}\right|<1$. +If we set $b=1$ in (\ref{int1}), we find that +\begin{equation} +\label{int2} +\int_0^{\infty}x^{c-1}\frac{(-ax;q)_{\infty}}{(-x;q)_{\infty}}\,dx +=\frac{\pi}{\sin(\pi c)}\,\frac{(a,q^{1-c};q)_{\infty}}{(aq^{-c},q;q)_{\infty}} +\end{equation} +for $00$ and $\left|aq^{-c}\right|<1$, +and by taking the limit $c\rightarrow 1$ in (\ref{int1}), we obtain +\begin{equation} +\label{int3} +\int_0^{\infty}\frac{(-ax,-bq/x;q)_{\infty}}{(-x,-q/x;q)_{\infty}}\,dx +=-\ln q\,\frac{(ab,q;q)_{\infty}}{(bq,a/q;q)_{\infty}} +\end{equation} +for $00$ and non-real parameters occur in conjugate pairs, then +\begin{eqnarray} +\label{OrthIWilson} +& &\frac{1}{2\pi}\int_0^{\infty} +\left|\frac{\Gamma(a+ix)\Gamma(b+ix)\Gamma(c+ix)\Gamma(d+ix)}{\Gamma(2ix)}\right|^2\nonumber\\ +& &{}\mathindent\times W_m(x^2;a,b,c,d)W_n(x^2;a,b,c,d)\,dx\nonumber\\ +& &{}=\frac{\Gamma(n+a+b)\cdots\Gamma(n+c+d)}{\Gamma(2n+a+b+c+d)}(n+a+b+c+d-1)_nn!\,\delta_{mn}, +\end{eqnarray} +where +\begin{eqnarray*} +& &\Gamma(n+a+b)\cdots\Gamma(n+c+d)\\ +& &{}=\Gamma(n+a+b)\Gamma(n+a+c)\Gamma(n+a+d)\Gamma(n+b+c)\Gamma(n+b+d)\Gamma(n+c+d). +\end{eqnarray*} +If $a<0$ and $a+b$, $a+c$, $a+d$ are positive or a pair of complex conjugates +occur with positive real parts, then +\begin{eqnarray} +\label{OrtIIWilson} +& &\frac{1}{2\pi}\int_0^{\infty}\left|\frac{\Gamma(a+ix)\Gamma(b+ix)\Gamma(c+ix)\Gamma(d+ix)}{\Gamma(2ix)}\right|^2\nonumber\\ +& &{}\mathindent\mathindent\times W_m(x^2;a,b,c,d)W_n(x^2;a,b,c,d)\,dx\nonumber\\ +& &{}\mathindent{}+\frac{\Gamma(a+b)\Gamma(a+c)\Gamma(a+d)\Gamma(b-a)\Gamma(c-a)\Gamma(d-a)}{\Gamma(-2a)}\nonumber\\ +& &{}\mathindent\mathindent{}\times\sum_{\begin{array}{c} +{\scriptstyle k=0,1,2\ldots}\\{\scriptstyle a+k<0}\end{array}} +\frac{(2a)_k(a+1)_k(a+b)_k(a+c)_k(a+d)_k}{(a)_k(a-b+1)_k(a-c+1)_k(a-d+1)_kk!}\nonumber\\ +& &{}\mathindent\mathindent\mathindent\mathindent{}\times W_m(-(a+k)^2;a,b,c,d)W_n(-(a+k)^2;a,b,c,d)\nonumber\\ +& &{}=\frac{\Gamma(n+a+b)\cdots\Gamma(n+c+d)}{\Gamma(2n+a+b+c+d)}(n+a+b+c+d-1)_nn!\,\delta_{mn}. +\end{eqnarray} + +\subsection*{Recurrence relation} +\begin{equation} +\label{RecWilson} +-\left(a^2+x^2\right){\tilde{W}}_n(x^2) +=A_n{\tilde{W}}_{n+1}(x^2)-\left(A_n+C_n\right){\tilde{W}}_n(x^2)+C_n{\tilde{W}}_{n-1}(x^2), +\end{equation} +where +$${\tilde{W}}_n(x^2):={\tilde{W}}_n(x^2;a,b,c,d)=\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}$$ +and +$$\left\{\begin{array}{l} +\displaystyle A_n=\frac{(n+a+b+c+d-1)(n+a+b)(n+a+c)(n+a+d)}{(2n+a+b+c+d-1)(2n+a+b+c+d)}\\[5mm] +\displaystyle C_n=\frac{n(n+b+c-1)(n+b+d-1)(n+c+d-1)}{(2n+a+b+c+d-2)(2n+a+b+c+d-1)}. +\end{array}\right.$$ + +\subsection*{Normalized recurrence relation} +\begin{equation} +\label{NormRecWilson} +xp_n(x)=p_{n+1}(x)+(A_n+C_n-a^2)p_n(x)+A_{n-1}C_np_{n-1}(x), +\end{equation} +where +$$W_n(x^2;a,b,c,d)=(-1)^n(n+a+b+c+d-1)_np_n(x^2).$$ + +\subsection*{Difference equation} +\begin{eqnarray} +\label{dvWilson} +& &n(n+a+b+c+d-1)y(x)\nonumber\\ +& &{}=B(x)y(x+i)-\left[B(x)+D(x)\right]y(x)+D(x)y(x-i), +\end{eqnarray} +where +$$y(x)=W_n(x^2;a,b,c,d)$$ +and +$$\left\{\begin{array}{l} +\displaystyle B(x)=\frac{(a-ix)(b-ix)(c-ix)(d-ix)}{2ix(2ix-1)}\\[5mm] +\displaystyle D(x)=\frac{(a+ix)(b+ix)(c+ix)(d+ix)}{2ix(2ix+1)}. +\end{array}\right.$$ + +\subsection*{Forward shift operator} +\begin{eqnarray} +\label{shift1WilsonI} +& &W_n((x+\textstyle\frac{1}{2}i)^2;a,b,c,d) +-W_n((x-\textstyle\frac{1}{2}i)^2;a,b,c,d)\nonumber\\ +& &{}=-2inx(n+a+b+c+d-1)W_{n-1}(x^2;a+\textstyle\frac{1}{2},b+\textstyle\frac{1}{2}, +c+\textstyle\frac{1}{2},d+\textstyle\frac{1}{2}) +\end{eqnarray} +or equivalently +\begin{eqnarray} +\label{shift1WilsonII} +& &\frac{\delta W_n(x^2;a,b,c,d)}{\delta x^2}\nonumber\\ +& &{}=-n(n+a+b+c+d-1)W_{n-1}(x^2;a+\textstyle\frac{1}{2},b+\textstyle\frac{1}{2}, +c+\textstyle\frac{1}{2},d+\textstyle\frac{1}{2}). +\end{eqnarray} + +\newpage + +\subsection*{Backward shift operator} +\begin{eqnarray} +\label{shift2WilsonI} +& &(a-\textstyle\frac{1}{2}-ix)(b-\textstyle\frac{1}{2}-ix) +(c-\textstyle\frac{1}{2}-ix)(d-\textstyle\frac{1}{2}-ix) +W_n((x+\textstyle\frac{1}{2}i)^2;a,b,c,d)\nonumber\\ +& &{}\mathindent{}-(a-\textstyle\frac{1}{2}+ix)(b-\textstyle\frac{1}{2}+ix) +(c-\textstyle\frac{1}{2}+ix)(d-\textstyle\frac{1}{2}+ix) +W_n((x-\textstyle\frac{1}{2}i)^2;a,b,c,d)\nonumber\\ +& &{}=-2ix W_{n+1}(x^2;a-\textstyle\frac{1}{2},b-\textstyle\frac{1}{2}, +c-\textstyle\frac{1}{2},d-\textstyle\frac{1}{2}) +\end{eqnarray} +or equivalently +\begin{eqnarray} +\label{shift2WilsonII} +& &\frac{\delta\left[\omega(x;a,b,c,d)W_n(x^2;a,b,c,d)\right]}{\delta x^2}\nonumber\\ +& &{}=\omega(x;a-\textstyle\frac{1}{2},b-\textstyle\frac{1}{2}, +c-\textstyle\frac{1}{2},d-\textstyle\frac{1}{2}) +W_{n+1}(x^2;a-\textstyle\frac{1}{2},b-\textstyle\frac{1}{2},c-\textstyle\frac{1}{2},d-\textstyle\frac{1}{2}), +\end{eqnarray} +where +$$\omega(x;a,b,c,d):=\frac{1}{2ix}\left|\frac{\Gamma(a+ix)\Gamma(b+ix)\Gamma(c+ix)\Gamma(d+ix)}{\Gamma(2ix)}\right|^2.$$ + +\subsection*{Rodrigues-type formula} +\begin{eqnarray} +\label{RodWilson} +& &\omega(x;a,b,c,d)W_n(x^2;a,b,c,d)\nonumber\\ +& &{}=\left(\frac{\delta}{\delta x^2}\right)^n +\left[\omega(x;a+\textstyle\frac{1}{2}n,b+\textstyle\frac{1}{2}n, +c+\textstyle\frac{1}{2}n,d+\textstyle\frac{1}{2}n)\right]. +\end{eqnarray} + +\subsection*{Generating functions} +\begin{equation} +\label{GenWilson1} +\hyp{2}{1}{a+ix,b+ix}{a+b}{t}\,\hyp{2}{1}{c-ix,d-ix}{c+d}{t}= +\sum_{n=0}^{\infty}\frac{W_n(x^2;a,b,c,d)t^n}{(a+b)_n(c+d)_nn!}. +\end{equation} + +\begin{equation} +\label{GenWilson2} +\hyp{2}{1}{a+ix,c+ix}{a+c}{t}\,\hyp{2}{1}{b-ix,d-ix}{b+d}{t}= +\sum_{n=0}^{\infty}\frac{W_n(x^2;a,b,c,d)t^n}{(a+c)_n(b+d)_nn!}. +\end{equation} + +\begin{equation} +\label{GenWilson3} +\hyp{2}{1}{a+ix,d+ix}{a+d}{t}\,\hyp{2}{1}{b-ix,c-ix}{b+c}{t}= +\sum_{n=0}^{\infty}\frac{W_n(x^2;a,b,c,d)t^n}{(a+d)_n(b+c)_nn!}. +\end{equation} + +\begin{eqnarray} +\label{GenWilson4} +& &(1-t)^{1-a-b-c-d}\nonumber\\ +& &{}\mathindent\times\hyp{4}{3}{\frac{1}{2}(a+b+c+d-1),\frac{1}{2}(a+b+c+d),a+ix,a-ix} +{a+b,a+c,a+d}{-\frac{4t}{(1-t)^2}}\nonumber\\ +& &{}=\sum_{n=0}^{\infty}\frac{(a+b+c+d-1)_n}{(a+b)_n(a+c)_n(a+d)_nn!}W_n(x^2;a,b,c,d)t^n. +\end{eqnarray} + +\subsection*{Limit relations} + +\subsubsection*{Wilson $\rightarrow$ Continuous dual Hahn} +The continuous dual Hahn polynomials given by (\ref{DefContDualHahn}) can be found from the +Wilson polynomials by dividing by $(a+d)_n$ and letting $d\rightarrow\infty$: +\begin{equation} +\lim_{d\rightarrow\infty}\frac{W_n(x^2;a,b,c,d)}{(a+d)_n}=S_n(x^2;a,b,c). +\end{equation} + +\subsubsection*{Wilson $\rightarrow$ Continuous Hahn} +The continuous Hahn polynomials given by (\ref{DefContHahn}) are obtained from the Wilson +polynomials by the substitutions $a\rightarrow a-it$, $b\rightarrow b-it$, $c\rightarrow c+it$, +$d\rightarrow d+it$ and $x\rightarrow x+t$ and the limit $t\rightarrow\infty$ in the following +way: +\begin{equation} +\lim_{t\rightarrow\infty} +\frac{W_n((x+t)^2;a-it,b-it,c+it,d+it)}{(-2t)^nn!}=p_n(x;a,b,c,d). +\end{equation} + +\subsubsection*{Wilson $\rightarrow$ Jacobi} +The Jacobi polynomials given by (\ref{DefJacobi}) can be found from the Wilson +polynomials by substituting $a=b=\frac{1}{2}(\alpha+1)$, $c=\frac{1}{2}(\beta+1)+it$, +$d=\frac{1}{2}(\beta+1)-it$ and $x\rightarrow t\sqrt{\frac{1}{2}(1-x)}$ in the +definition (\ref{DefWilson}) of the Wilson polynomials and taking the limit +$t\rightarrow\infty$. In fact we have +\begin{eqnarray} +& &\lim_{t\rightarrow\infty}\frac{W_n(\frac{1}{2}(1-x)t^2;\frac{1}{2}(\alpha+1), +\frac{1}{2}(\alpha+1),\frac{1}{2}(\beta+1)+it,\frac{1}{2}(\beta+1)-it)}{t^{2n}n!}\nonumber\\ +& &{}=P_n^{(\alpha,\beta)}(x). +\end{eqnarray} + +\subsection*{Remarks} +Note that for $k0$, $c=\bar{a}$ and $d=\bar{b}$, then +\begin{eqnarray} +\label{OrtContHahn} +& &\frac{1}{2\pi}\int_{-\infty}^{\infty}\Gamma(a+ix)\Gamma(b+ix)\Gamma(c-ix)\Gamma(d-ix) +p_m(x;a,b,c,d)p_n(x;a,b,c,d)\,dx\nonumber\\ +& &{}=\frac{\Gamma(n+a+c)\Gamma(n+a+d)\Gamma(n+b+c)\Gamma(n+b+d)} +{(2n+a+b+c+d-1)\Gamma(n+a+b+c+d-1)n!}\,\delta_{mn}. +\end{eqnarray} + +\subsection*{Recurrence relation} +\begin{equation} +\label{RecContHahn} +(a+ix)\tilde{p}_n(x)=A_n\tilde{p}_{n+1}(x)-\left(A_n+C_n\right)\tilde{p}_n(x)+C_n\tilde{p}_{n-1}(x), +\end{equation} +where +$$\tilde{p}_n(x):=\tilde{p}_n(x;a,b,c,d)=\frac{n!}{i^n(a+c)_n(a+d)_n}p_n(x;a,b,c,d)$$ +and +$$\left\{\begin{array}{l} +\displaystyle A_n=-\frac{(n+a+b+c+d-1)(n+a+c)(n+a+d)}{(2n+a+b+c+d-1)(2n+a+b+c+d)}\\[5mm] +\displaystyle C_n=\frac{n(n+b+c-1)(n+b+d-1)}{(2n+a+b+c+d-2)(2n+a+b+c+d-1)}. +\end{array}\right.$$ + +\subsection*{Normalized recurrence relation} +\begin{equation} +\label{NormRecContHahn} +xp_n(x)=p_{n+1}(x)+i(A_n+C_n+a)p_n(x)-A_{n-1}C_np_{n-1}(x), +\end{equation} +where +$$p_n(x;a,b,c,d)=\frac{(n+a+b+c+d-1)_n}{n!}p_n(x).$$ + +\subsection*{Difference equation} +\begin{eqnarray} +\label{dvContHahn} +& &n(n+a+b+c+d-1)y(x)\nonumber\\ +& &{}=B(x)y(x+i)-\left[B(x)+D(x)\right]y(x)+D(x)y(x-i), +\end{eqnarray} +where +$$y(x)=p_n(x;a,b,c,d)$$ +and +$$\left\{\begin{array}{l} +\displaystyle B(x)=(c-ix)(d-ix)\\[5mm] +\displaystyle D(x)=(a+ix)(b+ix). +\end{array}\right.$$ + +\subsection*{Forward shift operator} +\begin{eqnarray} +\label{shift1ContHahnI} +& &p_n(x+\textstyle\frac{1}{2}i;a,b,c,d)-p_n(x-\textstyle\frac{1}{2}i;a,b,c,d)\nonumber\\ +& &{}=i(n+a+b+c+d-1)p_{n-1}(x;a+\textstyle\frac{1}{2}, +b+\textstyle\frac{1}{2},c+\textstyle\frac{1}{2},d+\textstyle\frac{1}{2}) +\end{eqnarray} +or equivalently +\begin{equation} +\label{shift1ContHahnII} +\frac{\delta p_n(x;a,b,c,d)}{\delta x}=(n+a+b+c+d-1) +p_{n-1}(x;a+\textstyle\frac{1}{2},b+\textstyle\frac{1}{2}, +c+\textstyle\frac{1}{2},d+\textstyle\frac{1}{2}). +\end{equation} + +\subsection*{Backward shift operator} +\begin{eqnarray} +\label{shift2ContHahnI} +& &(c-\textstyle\frac{1}{2}-ix)(d-\textstyle\frac{1}{2}-ix) +p_n(x+\textstyle\frac{1}{2}i;a,b,c,d)\nonumber\\ +& &{}\mathindent{}-(a-\textstyle\frac{1}{2}+ix)(b-\textstyle\frac{1}{2}+ix) +p_n(x-\textstyle\frac{1}{2}i;a,b,c,d)\nonumber\\ +& &{}=\frac{n+1}{i}p_{n+1}(x;a-\textstyle\frac{1}{2}, +b-\textstyle\frac{1}{2},c-\textstyle\frac{1}{2},d-\textstyle\frac{1}{2}) +\end{eqnarray} +or equivalently +\begin{eqnarray} +\label{shift2ContHahnII} +& &\frac{\delta\left[\omega(x;a,b,c,d)p_n(x;a,b,c,d)\right]}{\delta x}\nonumber\\ +& &{}=-(n+1)\omega(x;a-\textstyle\frac{1}{2},b-\textstyle\frac{1}{2}, +c-\textstyle\frac{1}{2},d-\textstyle\frac{1}{2})\nonumber\\ +& &{}\mathindent\times p_{n+1}(x;a-\textstyle\frac{1}{2},b-\textstyle\frac{1}{2},c-\textstyle\frac{1}{2},d-\textstyle\frac{1}{2}), +\end{eqnarray} +where +$$\omega(x;a,b,c,d)=\Gamma(a+ix)\Gamma(b+ix)\Gamma(c-ix)\Gamma(d-ix).$$ + +\subsection*{Rodrigues-type formula} +\begin{eqnarray} +\label{RodContHahn} +& &\omega(x;a,b,c,d)p_n(x;a,b,c,d)\nonumber\\ +& &{}=\frac{(-1)^n}{n!}\left(\frac{\delta}{\delta x}\right)^n +\left[\omega(x;a+\textstyle\frac{1}{2}n,b+\textstyle\frac{1}{2}n, +c+\textstyle\frac{1}{2}n,d+\textstyle\frac{1}{2}n)\right]. +\end{eqnarray} + +\subsection*{Generating functions} +\begin{equation} +\label{GenContHahn1} +\hyp{1}{1}{a+ix}{a+c}{-it}\,\hyp{1}{1}{d-ix}{b+d}{it}= +\sum_{n=0}^{\infty}\frac{p_n(x;a,b,c,d)}{(a+c)_n(b+d)_n}t^n. +\end{equation} + +\begin{equation} +\label{GenContHahn2} +\hyp{1}{1}{a+ix}{a+d}{-it}\,\hyp{1}{1}{c-ix}{b+c}{it}= +\sum_{n=0}^{\infty}\frac{p_n(x;a,b,c,d)}{(a+d)_n(b+c)_n}t^n. +\end{equation} + +\begin{eqnarray} +\label{GenContHahn3} +& &(1-t)^{1-a-b-c-d}\,\hyp{3}{2}{\frac{1}{2}(a+b+c+d-1),\frac{1}{2}(a+b+c+d),a+ix} +{a+c,a+d}{-\frac{4t}{(1-t)^2}}\nonumber\\ +& &{}=\sum_{n=0}^{\infty} +\frac{(a+b+c+d-1)_n}{(a+c)_n(a+d)_ni^n}p_n(x;a,b,c,d)t^n. +\end{eqnarray} + +\subsection*{Limit relations} + +\subsubsection*{Wilson $\rightarrow$ Continuous Hahn} +The continuous Hahn polynomials are obtained from the Wilson polynomials given by +(\ref{DefWilson}) by the substitution $a\rightarrow a-it$, $b\rightarrow b-it$, +$c\rightarrow c+it$, $d\rightarrow d+it$ and $x\rightarrow x+t$ and the limit +$t\rightarrow\infty$ in the following way: +$$\lim_{t\rightarrow\infty} +\frac{W_n((x+t)^2;a-it,b-it,c+it,d+it)}{(-2t)^nn!}=p_n(x;a,b,c,d).$$ + +\subsubsection*{Continuous Hahn $\rightarrow$ Meixner-Pollaczek} +The Meixner-Pollaczek polynomials given by (\ref{DefMP}) can be obtained from the +continuous Hahn polynomials by setting $x\rightarrow x+t$, $a=\lambda-it$, $c=\lambda+it$ +and $b=d=t\tan\phi$ and taking the limit $t\rightarrow\infty$: +\begin{equation} +\lim_{t\rightarrow\infty}\frac{p_n(x+t;\lambda-it,t\tan\phi,\lambda+it,t\tan\phi)}{t^n} +=\frac{P_n^{(\lambda)}(x;\phi)}{(\cos\phi)^n}. +\end{equation} + +\subsubsection*{Continuous Hahn $\rightarrow$ Jacobi} +The Jacobi polynomials given by (\ref{DefJacobi}) follow from the continuous Hahn polynomials +by the substitution $x\rightarrow \frac{1}{2}xt$, $a=\frac{1}{2}(\alpha+1-it)$, +$b=\frac{1}{2}(\beta+1+it)$, $c=\frac{1}{2}(\alpha+1+it)$ and $d=\frac{1}{2}(\beta+1-it)$, +division by $t^n$ and the limit $t\rightarrow\infty$: +\begin{eqnarray} +& &\lim_{t\rightarrow\infty} +\frac{p_n(\frac{1}{2}xt;\frac{1}{2}(\alpha+1-it),\frac{1}{2}(\beta+1+it), +\frac{1}{2}(\alpha+1+it),\frac{1}{2}(\beta+1-it))}{t^n}\nonumber\\ +& &{}=P_n^{(\alpha,\beta)}(x). +\end{eqnarray} + +\subsubsection*{Continuous Hahn $\rightarrow$ Pseudo Jacobi} +The pseudo Jacobi polynomials given by (\ref{DefPseudoJacobi}) follow from the continuous +Hahn polynomials by the substitution $x\rightarrow xt$, $a=\frac{1}{2}(-N+i\nu-2t)$, +$b=\frac{1}{2}(-N-i\nu+2t)$, $c=\frac{1}{2}(-N-i\nu-2t)$ and $d=\frac{1}{2}(-N+i\nu+2t)$, +division by $t^n$ and the limit $t\rightarrow\infty$: +\begin{eqnarray} +& &\lim_{t\rightarrow\infty}\frac{p_n(xt;\frac{1}{2}(-N+i\nu-2t),\frac{1}{2}(-N-i\nu+2t), +\frac{1}{2}(-N+i\nu-2t),\frac{1}{2}(-N-i\nu+2t))}{t^n}\nonumber\\ +& &{}=\frac{(n-2N-1)_n}{n!}P_n(x;\nu,N). +\end{eqnarray} + +\subsection*{Remark} +Since we have for $k-1$ and $\beta>-1$, or for $\alpha<-N$ and $\beta<-N$, we have +\begin{eqnarray} +\label{OrtHahn} +& &\sum_{x=0}^N\binom{\alpha +x}{x}\binom{\beta+N-x}{N-x}Q_m(x;\alpha,\beta,N)Q_n(x;\alpha,\beta,N)\nonumber\\ +& &{}=\frac{(-1)^n(n+\alpha+\beta+1)_{N+1}(\beta+1)_nn!}{(2n+\alpha+\beta+1)(\alpha+1)_n(-N)_nN!}\,\delta_{mn}. +\end{eqnarray} + +\subsection*{Recurrence relation} +\begin{equation} +\label{RecHahn} +-xQ_n(x)=A_nQ_{n+1}(x)-\left(A_n+C_n\right)Q_n(x)+C_nQ_{n-1}(x), +\end{equation} +where +$$Q_n(x):=Q_n(x;\alpha,\beta,N)$$ +and +$$\left\{\begin{array}{l} +\displaystyle A_n=\frac{(n+\alpha+\beta+1)(n+\alpha+1)(N-n)}{(2n+\alpha+\beta+1)(2n+\alpha+\beta+2)}\\[5mm] +\displaystyle C_n=\frac{n(n+\alpha+\beta+N+1)(n+\beta)}{(2n+\alpha+\beta)(2n+\alpha+\beta+1)}. +\end{array}\right.$$ + +\subsection*{Normalized recurrence relation} +\begin{equation} +\label{NormRecHahn} +xp_n(x)=p_{n+1}(x)+\left(A_n+C_n\right)p_n(x)+A_{n-1}C_np_{n-1}(x), +\end{equation} +where +$$Q_n(x;\alpha,\beta,N)=\frac{(n+\alpha+\beta+1)_n}{(\alpha+1)_n(-N)_n}p_n(x).$$ + +\subsection*{Difference equation} +\begin{equation} +\label{dvHahn} +n(n+\alpha+\beta+1)y(x)=B(x)y(x+1)-\left[B(x)+D(x)\right]y(x)+D(x)y(x-1), +\end{equation} +where +$$y(x)=Q_n(x;\alpha,\beta,N)$$ +and +$$\left\{\begin{array}{l} +\displaystyle B(x)=(x+\alpha+1)(x-N)\\[5mm] +\displaystyle D(x)=x(x-\beta-N-1). +\end{array}\right.$$ + +\subsection*{Forward shift operator} +\begin{eqnarray} +\label{shift1HahnI} +& &Q_n(x+1;\alpha,\beta,N)-Q_n(x;\alpha,\beta,N)\nonumber\\ +& &{}=-\frac{n(n+\alpha+\beta+1)}{(\alpha+1)N}Q_{n-1}(x;\alpha+1,\beta+1,N-1) +\end{eqnarray} +or equivalently +\begin{equation} +\label{shift1HahnII} +\Delta Q_n(x;\alpha,\beta,N)=-\frac{n(n+\alpha+\beta+1)}{(\alpha+1)N}Q_{n-1}(x;\alpha+1,\beta+1,N-1). +\end{equation} + +\subsection*{Backward shift operator} +\begin{eqnarray} +\label{shift2HahnI} +& &(x+\alpha)(N+1-x)Q_n(x;\alpha,\beta,N)-x(\beta+N+1-x)Q_n(x-1;\alpha,\beta,N)\nonumber\\ +& &{}=\alpha(N+1)Q_{n+1}(x;\alpha-1,\beta-1,N+1) +\end{eqnarray} +or equivalently +\begin{eqnarray} +\label{shift2HahnII} +& &\nabla\left[\omega(x;\alpha,\beta,N)Q_n(x;\alpha,\beta,N)\right]\nonumber\\ +& &{}=\frac{N+1}{\beta}\omega(x;\alpha-1,\beta-1,N+1)Q_{n+1}(x;\alpha-1,\beta-1,N+1), +\end{eqnarray} +where +$$\omega(x;\alpha,\beta,N)=\binom{\alpha+x}{x}\binom{\beta+N-x}{N-x}.$$ + +\subsection*{Rodrigues-type formula} +\begin{eqnarray} +\label{RodHahn} +& &\omega(x;\alpha,\beta,N)Q_n(x;\alpha,\beta,N)\nonumber\\ +& &{}=\frac{(-1)^n(\beta+1)_n}{(-N)_n}\nabla^n\left[\omega(x;\alpha+n,\beta+n,N-n)\right]. +\end{eqnarray} + +\subsection*{Generating functions} +For $x=0,1,2,\ldots,N$ we have +\begin{equation} +\label{GenHahn1} +\hyp{1}{1}{-x}{\alpha+1}{-t}\,\hyp{1}{1}{x-N}{\beta+1}{t} +=\sum_{n=0}^N\frac{(-N)_n}{(\beta+1)_nn!}Q_n(x;\alpha,\beta,N)t^n. +\end{equation} + +\begin{eqnarray} +\label{GenHahn2} +& &\hyp{2}{0}{-x,-x+\beta+N+1}{-}{-t}\,\hyp{2}{0}{x-N,x+\alpha+1}{-}{t}\nonumber\\ +& &{}=\sum_{n=0}^N\frac{(-N)_n(\alpha+1)_n}{n!}Q_n(x;\alpha,\beta,N)t^n. +\end{eqnarray} + +\begin{eqnarray} +\label{GenHahn3} +& &\left[(1-t)^{-\alpha-\beta-1}\,\hyp{3}{2}{\frac{1}{2}(\alpha+\beta+1),\frac{1}{2}(\alpha+\beta+2),-x} +{\alpha+1,-N}{-\frac{4t}{(1-t)^2}}\right]_N\nonumber\\ +& &{}=\sum_{n=0}^N\frac{(\alpha+\beta+1)_n}{n!}Q_n(x;\alpha,\beta,N)t^n. +\end{eqnarray} + +\subsection*{Limit relations} + +\subsubsection*{Racah $\rightarrow$ Hahn} +If we take $\gamma+1=-N$ and let $\delta\rightarrow\infty$ in the definition (\ref{DefRacah}) +of the Racah polynomials, we obtain the Hahn polynomials. Hence +$$\lim_{\delta\rightarrow\infty} +R_n(\lambda(x);\alpha,\beta,-N-1,\delta)=Q_n(x;\alpha,\beta,N).$$ +And if we take $\delta=-\beta-N-1$ and let $\gamma\rightarrow\infty$ in the definition (\ref{DefRacah}) +of the Racah polynomials, we also obtain the Hahn polynomials: +$$\lim_{\gamma\rightarrow\infty} +R_n(\lambda(x);\alpha,\beta,\gamma,-\beta-N-1)=Q_n(x;\alpha,\beta,N).$$ +Another way to do this is to take $\alpha+1=-N$ and $\beta\rightarrow\beta+\gamma+N+1$ in +the definition (\ref{DefRacah}) of the Racah polynomials and then take the limit +$\delta\rightarrow\infty$. In that case we obtain the Hahn polynomials in the following way: +$$\lim_{\delta\rightarrow\infty} +R_n(\lambda(x);-N-1,\beta+\gamma+N+1,\gamma,\delta)=Q_n(x;\gamma,\beta,N).$$ + +\subsubsection*{Hahn $\rightarrow$ Jacobi} +To find the Jacobi polynomials given by (\ref{DefJacobi}) from the Hahn polynomials we take +$x\rightarrow Nx$ and let $N\rightarrow\infty$. In fact we have +\begin{equation} +\lim_{N\rightarrow\infty} +Q_n(Nx;\alpha,\beta,N)=\frac{P_n^{(\alpha,\beta)}(1-2x)}{P_n^{(\alpha,\beta)}(1)}. +\end{equation} + +\subsubsection*{Hahn $\rightarrow$ Meixner} +The Meixner polynomials given by (\ref{DefMeixner}) can be obtained from the Hahn polynomials +by taking $\alpha=b-1$, $\beta=N(1-c)c^{-1}$ and letting $N\rightarrow\infty$: +\begin{equation} +\lim_{N\rightarrow\infty} +Q_n(x;b-1,N(1-c)c^{-1},N)=M_n(x;b,c). +\end{equation} + +\subsubsection*{Hahn $\rightarrow$ Krawtchouk} +The Krawtchouk polynomials given by (\ref{DefKrawtchouk}) are obtained from the Hahn polynomials +if we take $\alpha=pt$ and $\beta=(1-p)t$ and let $t\rightarrow\infty$: +\begin{equation} +\lim_{t\rightarrow\infty}Q_n(x;pt,(1-p)t,N)=K_n(x;p,N). +\end{equation} + +\subsection*{Remark} +If we interchange the role of $x$ and $n$ in (\ref{DefHahn}) we obtain the +dual Hahn polynomials given by (\ref{DefDualHahn}). + +\noindent +Since +$$Q_n(x;\alpha,\beta,N)=R_x(\lambda(n);\alpha,\beta,N)$$ +we obtain the dual orthogonality relation for the Hahn polynomials from the +orthogonality relation (\ref{OrtDualHahn}) of the dual Hahn polynomials: +\begin{eqnarray*} +& &\sum_{n=0}^N\frac{(2n+\alpha+\beta+1)(\alpha+1)_n(-N)_nN!}{(-1)^n(n+\alpha+\beta+1)_{N+1}(\beta+1)_nn!} +Q_n(x;\alpha,\beta,N)Q_n(y;\alpha,\beta,N)\\ +& &{}=\frac{\delta_{xy}}{\dbinom{\alpha+x}{x}\dbinom{\beta+N-x}{N-x}},\quad x,y \in \{0,1,2,\ldots,N\}. +\end{eqnarray*} + +\subsection*{References} +\cite{AlSalam90}, \cite{AndrewsAskey85}, \cite{Area+II}, \cite{Askey75}, +\cite{Askey89I}, \cite{Askey2005}, \cite{AskeyGasper77}, \cite{AskeyWilson85}, +\cite{AtakRahmanSuslov}, \cite{AtakSuslov88}, \cite{Chihara78}, +\cite{Ciesielski}, \cite{Cooper+}, \cite{Dette95}, \cite{Dunkl76}, +\cite{Dunkl78I}, \cite{Gasper73I}, \cite{Gasper74}, \cite{HoareRahman}, +\cite{Ismail77}, \cite{Ismail2005II}, \cite{Karlin61}, \cite{Koorn81}, \cite{Koorn88}, +\cite{LabelleYehI}, \cite{LabelleYehII}, \cite{Laine}, \cite{Lesky62}, +\cite{Lesky88}, \cite{Lesky89}, \cite{Lesky94I}, \cite{Lesky95II}, +\cite{LewanowiczII}, \cite{Neuman}, \cite{Nikiforov+}, \cite{NikiforovUvarov}, +\cite{Rahman76III}, \cite{Rahman78I}, \cite{Rahman78II}, \cite{Rahman81III}, +\cite{Sablonniere}, \cite{Stanton84}, \cite{Stanton90}, \cite{Wilson80}, \cite{Wilson70II}, +\cite{Zarzo+}. + + +\section{Dual Hahn}\index{Dual Hahn polynomials}\index{Hahn polynomials!Dual} + +\par\setcounter{equation}{0} + +\subsection*{Hypergeometric representation} +\begin{equation} +\label{DefDualHahn} +R_n(\lambda(x);\gamma,\delta,N)= +\hyp{3}{2}{-n,-x,x+\gamma+\delta+1}{\gamma+1,-N}{1},\quad n=0,1,2,\ldots,N, +\end{equation} +where +$$\lambda(x)=x(x+\gamma+\delta+1).$$ + +\subsection*{Orthogonality relation} +For $\gamma>-1$ and $\delta>-1$, or for $\gamma<-N$ and $\delta<-N$, we have +\begin{eqnarray} +\label{OrtDualHahn} +& &\sum_{x=0}^N\frac{(2x+\gamma+\delta+1)(\gamma+1)_x(-N)_xN!}{(-1)^x(x+\gamma+\delta+1)_{N+1}(\delta+1)_xx!} +R_m(\lambda(x);\gamma,\delta,N)R_n(\lambda(x);\gamma,\delta,N)\nonumber\\ +& &{}=\frac{\,\delta_{mn}}{\dbinom{\gamma+n}{n}\dbinom{\delta+N-n}{N-n}}. +\end{eqnarray} + +\subsection*{Recurrence relation} +\begin{equation} +\label{RecDualHahn} +\lambda(x)R_n(\lambda(x)) +=A_nR_{n+1}(\lambda(x))-\left(A_n+C_n\right)R_n(\lambda(x))+C_nR_{n-1}(\lambda(x)), +\end{equation} +where +$$R_n(\lambda(x)):=R_n(\lambda(x);\gamma,\delta,N)$$ +and +$$\left\{\begin{array}{l} +\displaystyle A_n=(n+\gamma+1)(n-N)\\[5mm] +\displaystyle C_n=n(n-\delta-N-1). +\end{array}\right.$$ + +\subsection*{Normalized recurrence relation} +\begin{equation} +\label{NormRecDualHahn} +xp_n(x)=p_{n+1}(x)-(A_n+C_n)p_n(x)+A_{n-1}C_np_{n-1}(x), +\end{equation} +where +$$R_n(\lambda(x);\gamma,\delta,N)=\frac{1}{(\gamma+1)_n(-N)_n}p_n(\lambda(x)).$$ + +\subsection*{Difference equation} +\begin{equation} +\label{dvDualHahn} +-ny(x)=B(x)y(x+1)-\left[B(x)+D(x)\right]y(x)+D(x)y(x-1), +\end{equation} +where +$$y(x)=R_n(\lambda(x);\gamma,\delta,N)$$ +and +$$\left\{\begin{array}{l} +\displaystyle B(x)=\frac{(x+\gamma+1)(x+\gamma+\delta+1)(N-x)}{(2x+\gamma+\delta+1)(2x+\gamma+\delta+2)}\\[5mm] +\displaystyle D(x)=\frac{x(x+\gamma+\delta+N+1)(x+\delta)}{(2x+\gamma+\delta)(2x+\gamma+\delta+1)}. +\end{array}\right.$$ + +\subsection*{Forward shift operator} +\begin{eqnarray} +\label{shift1DualHahnI} +& &R_n(\lambda(x+1);\gamma,\delta,N)-R_n(\lambda(x);\gamma,\delta,N)\nonumber\\ +& &{}=-\frac{n(2x+\gamma+\delta+2)}{(\gamma+1)N}R_{n-1}(\lambda(x);\gamma+1,\delta,N-1) +\end{eqnarray} +or equivalently +\begin{equation} +\label{shift1DualHahnII} +\frac{\Delta R_n(\lambda(x);\gamma,\delta,N)}{\Delta\lambda(x)}= +-\frac{n}{(\gamma+1)N}R_{n-1}(\lambda(x);\gamma+1,\delta,N-1). +\end{equation} + +\subsection*{Backward shift operator} +\begin{eqnarray} +\label{shift2DualHahnI} +& &(x+\gamma)(x+\gamma+\delta)(N+1-x)R_n(\lambda(x);\gamma,\delta,N)\nonumber\\ +& &{}\mathindent{}-x(x+\gamma+\delta+N+1)(x+\delta)R_n(\lambda(x-1);\gamma,\delta,N)\nonumber\\ +& &{}=\gamma(N+1)(2x+\gamma+\delta)R_{n+1}(\lambda(x);\gamma-1,\delta,N+1) +\end{eqnarray} +or equivalently +\begin{eqnarray} +\label{shift2DualHahnII} +& &\frac{\nabla\left[\omega(x;\gamma,\delta,N)R_n(\lambda(x);\gamma,\delta,N)\right]}{\nabla\lambda(x)}\nonumber\\ +& &{}=\frac{1}{\gamma+\delta}\omega(x;\gamma-1,\delta,N+1)R_{n+1}(\lambda(x);\gamma-1,\delta,N+1), +\end{eqnarray} +where +$$\omega(x;\gamma,\delta,N)=\frac{(-1)^x(\gamma+1)_x(\gamma+\delta+1)_x(-N)_x} +{(\gamma+\delta+N+2)_x(\delta+1)_xx!}.$$ + +\newpage + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodDualHahn} +\omega(x;\gamma,\delta,N)R_n(\lambda(x);\gamma,\delta,N) +=(\gamma+\delta+1)_n\left(\nabla_{\lambda}\right)^n\left[\omega(x;\gamma+n,\delta,N-n)\right], +\end{equation} +where +$$\nabla_{\lambda}:=\frac{\nabla}{\nabla\lambda(x)}.$$ + +\subsection*{Generating functions} +For $x=0,1,2,\ldots,N$ we have +\begin{equation} +\label{GenDualHahn1} +(1-t)^{N-x}\,\hyp{2}{1}{-x,-x-\delta}{\gamma+1}{t} +=\sum_{n=0}^N\frac{(-N)_n}{n!}R_n(\lambda(x);\gamma,\delta,N)t^n. +\end{equation} + +\begin{eqnarray} +\label{GenDualHahn2} +& &(1-t)^x\,\hyp{2}{1}{x-N,x+\gamma+1}{-\delta-N}{t}\nonumber\\ +& &=\sum_{n=0}^N\frac{(\gamma+1)_n(-N)_n}{(-\delta-N)_nn!}R_n(\lambda(x);\gamma,\delta,N)t^n. +\end{eqnarray} + +\begin{equation} +\label{GenDualHahn3} +\left[\expe^t\,\hyp{2}{2}{-x,x+\gamma+\delta+1}{\gamma+1,-N}{-t}\right]_N=\sum_{n=0}^N +\frac{R_n(\lambda(x);\gamma,\delta,N)}{n!}t^n. +\end{equation} + +\begin{eqnarray} +\label{GenDualHahn4} +& &\left[(1-t)^{-\epsilon}\,\hyp{3}{2}{\epsilon,-x,x+\gamma+\delta+1}{\gamma+1,-N}{\frac{t}{t-1}}\right]_N\nonumber\\ +& &{}=\sum_{n=0}^N\frac{(\epsilon)_n}{n!}R_n(\lambda(x);\gamma,\delta,N)t^n, +\quad\textrm{$\epsilon$ arbitrary}. +\end{eqnarray} + +\subsection*{Limit relations} + +\subsubsection*{Racah $\rightarrow$ Dual Hahn} +If we take $\alpha+1=-N$ and let $\beta\rightarrow\infty$ in the definition (\ref{DefRacah}) +of the Racah polynomials, then we obtain the dual Hahn polynomials: +$$\lim_{\beta\rightarrow\infty} +R_n(\lambda(x);-N-1,\beta,\gamma,\delta)=R_n(\lambda(x);\gamma,\delta,N).$$ +And if we take $\beta=-\delta-N-1$ and let $\alpha\rightarrow\infty$ in the definition (\ref{DefRacah}) +of the Racah polynomials, then we also obtain the dual Hahn polynomials: +$$\lim_{\alpha\rightarrow\infty} +R_n(\lambda(x);\alpha,-\delta-N-1,\gamma,\delta)=R_n(\lambda(x);\gamma,\delta,N).$$ +Finally, if we take $\gamma+1=-N$ and $\delta\rightarrow\alpha+\delta+N+1$ in the definition +(\ref{DefRacah}) of the Racah polynomials and take the limit +$\beta\rightarrow\infty$ we find the dual Hahn polynomials in the following way: +$$\lim_{\beta\rightarrow\infty} +R_n(\lambda(x);\alpha,\beta,-N-1,\alpha+\delta+N+1)=R_n(\lambda(x);\alpha,\delta,N).$$ + +\subsubsection*{Dual Hahn $\rightarrow$ Meixner} +The Meixner polynomials given by (\ref{DefMeixner}) are obtained from the dual Hahn polynomials +if we take $\gamma=\beta-1$ and $\delta=N(1-c)c^{-1}$ and let $N\rightarrow\infty$: +\begin{equation} +\lim_{N\rightarrow\infty} +R_n(\lambda(x);\beta-1,N(1-c)c^{-1},N)=M_n(x;\beta,c). +\end{equation} + +\subsubsection*{Dual Hahn $\rightarrow$ Krawtchouk} +The Krawtchouk polynomials given by (\ref{DefKrawtchouk}) can be obtained from the dual +Hahn polynomials by setting $\gamma=pt$, $\delta=(1-p)t$ and letting $t\rightarrow\infty$: +\begin{equation} +\lim_{t\rightarrow\infty}R_n(\lambda(x);pt,(1-p)t,N)=K_n(x;p,N). +\end{equation} + +\subsection*{Remark} +If we interchange the role of $x$ and $n$ in the definition (\ref{DefDualHahn}) +of the dual Hahn polynomials we obtain the Hahn polynomials given by (\ref{DefHahn}). + +\noindent +Since +$$R_n(\lambda(x);\gamma,\delta,N)=Q_x(n;\gamma,\delta,N)$$ +we obtain the dual orthogonality relation for the dual Hahn polynomials +from the orthogonality relation (\ref{OrtHahn}) for the Hahn polynomials: +\begin{eqnarray*} +& &\sum_{n=0}^N\binom{\gamma+n}{n}\binom{\delta+N-n}{N-n} R_n(\lambda(x);\gamma,\delta,N)R_n(\lambda(y);\gamma,\delta,N)\\ +& &{}=\frac{(-1)^x(x+\gamma+\delta+1)_{N+1}(\delta+1)_xx!} +{(2x+\gamma+\delta+1)(\gamma+1)_x(-N)_xN!}\delta_{xy},\quad x,y \in \{0,1,2,\ldots,N\}. +\end{eqnarray*} + +\subsection*{References} +\cite{Askey2005}, \cite{AskeyWilson85}, \cite{AtakRahmanSuslov}, \cite{AtakSuslov88}, +\cite{Ismail2005II}, \cite{Karlin61}, \cite{Koorn81}, \cite{Koorn88}, \cite{Lesky93}, +\cite{Lesky94I}, \cite{Lesky95I}, \cite{Lesky95II}, \cite{Nikiforov+}, \cite{NikiforovUvarov}, +\cite{Rahman81II}, \cite{Stanton84}, \cite{Wilson80}. + + +\section{Meixner-Pollaczek}\index{Meixner-Pollaczek polynomials} + +\par\setcounter{equation}{0} + +\subsection*{Hypergeometric representation} +\begin{equation} +\label{DefMP} +P_n^{(\lambda)}(x;\phi)=\frac{(2\lambda)_n}{n!}\expe^{in\phi}\, +\hyp{2}{1}{-n,\lambda+ix}{2\lambda}{1-\expe^{-2i\phi}}. +\end{equation} + +\subsection*{Orthogonality relation} +\begin{equation} +\label{OrtMP} +\frac{1}{2\pi}\int_{-\infty}^{\infty} +\e^{(2\phi-\pi)x}\left|\Gamma(\lambda+ix)\right|^2 +P_m^{(\lambda)}(x;\phi)P_n^{(\lambda)}(x;\phi)\,dx +{}=\frac{\Gamma(n+2\lambda)}{(2\sin\phi)^{2\lambda}n!}\,\delta_{mn}, +% \constraint{ +% $\lambda > 0$ & +% $0 < \phi < \pi$ } +\end{equation} + +\subsection*{Recurrence relation} +\begin{eqnarray} +\label{RecMP} +& &(n+1)P_{n+1}^{(\lambda)}(x;\phi)-2\left[x\sin\phi+(n+\lambda)\cos\phi\right] +P_n^{(\lambda)}(x;\phi)\nonumber\\ +& &{}\mathindent{}+(n+2\lambda-1)P_{n-1}^{(\lambda)}(x;\phi)=0. +\end{eqnarray} + +\subsection*{Normalized recurrence relation} +\begin{equation} +\label{NormRecMP} +xp_n(x)=p_{n+1}(x)-\left(\frac{n+\lambda}{\tan\phi}\right)p_n(x)+ +\frac{n(n+2\lambda-1)}{4\sin^2\phi}p_{n-1}(x), +\end{equation} +where +$$P_n^{(\lambda)}(x;\phi)=\frac{(2\sin\phi)^n}{n!}p_n(x).$$ + +\subsection*{Difference equation} +\begin{eqnarray} +\label{dvMP} +& &\e^{i\phi}(\lambda-ix)y(x+i)+2i\left[x\cos\phi-(n+\lambda)\sin\phi\right]y(x)\nonumber\\ +& &{}\mathindent{}-\e^{-i\phi}(\lambda+ix)y(x-i)=0,\quad y(x)=P_n^{(\lambda)}(x;\phi). +\end{eqnarray} + +\subsection*{Forward shift operator} +\begin{equation} +\label{shift1MPI} +P_n^{(\lambda)}(x+\textstyle\frac{1}{2}i;\phi)- +P_n^{(\lambda)}(x-\textstyle\frac{1}{2}i;\phi)=(\expe^{i\phi}-\expe^{-i\phi}) +P_{n-1}^{(\lambda+\frac{1}{2})}(x;\phi) +\end{equation} +or equivalently +\begin{equation} +\label{shift1MPII} +\frac{\delta P_n^{(\lambda)}(x;\phi)}{\delta x} +=2\sin\phi\,P_{n-1}^{(\lambda+\frac{1}{2})}(x;\phi). +\end{equation} + +\subsection*{Backward shift operator} +\begin{eqnarray} +\label{shift2MPI} +& &\e^{i\phi}(\lambda-\textstyle\frac{1}{2}-ix) +P_n^{(\lambda)}(x+\textstyle\frac{1}{2}i;\phi)+ +\e^{-i\phi}(\lambda-\textstyle\frac{1}{2}+ix) +P_n^{(\lambda)}(x-\textstyle\frac{1}{2}i;\phi)\nonumber\\ +& &{}=(n+1)P_{n+1}^{(\lambda-\frac{1}{2})}(x;\phi) +\end{eqnarray} +or equivalently +\begin{equation} +\label{shift2MPII} +\frac{\delta\left[\omega(x;\lambda,\phi)P_n^{(\lambda)}(x;\phi)\right]}{\delta x}= +-(n+1)\omega(x;\lambda-\textstyle\frac{1}{2},\phi) +P_{n+1}^{(\lambda-\frac{1}{2})}(x;\phi), +\end{equation} +where +$$\omega(x;\lambda,\phi)=\Gamma(\lambda+ix)\Gamma(\lambda-ix)\e^{(2\phi-\pi)x}.$$ + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodMP} +\omega(x;\lambda,\phi)P_n^{(\lambda)}(x;\phi)=\frac{(-1)^n}{n!} +\left(\frac{\delta}{\delta x}\right)^n\left[\omega(x;\lambda+\textstyle\frac{1}{2}n,\phi)\right]. +\end{equation} + +\newpage + +\subsection*{Generating functions} +\begin{equation} +\label{GenMP1} +(1-\expe^{i\phi}t)^{-\lambda+ix}(1-\expe^{-i\phi}t)^{-\lambda-ix}= +\sum_{n=0}^{\infty}P_n^{(\lambda)}(x;\phi)t^n. +\end{equation} + +\begin{equation} +\label{GenMP2} +\e^t\,\hyp{1}{1}{\lambda+ix}{2\lambda}{(\expe^{-2i\phi}-1)t}= +\sum_{n=0}^{\infty}\frac{P_n^{(\lambda)}(x;\phi)}{(2\lambda)_n\expe^{in\phi}}t^n. +\end{equation} + +\begin{eqnarray} +\label{GenMP3} +& &(1-t)^{-\gamma}\,\hyp{2}{1}{\gamma,\lambda+ix}{2\lambda}{\frac{(1-\e^{-2i\phi})t}{t-1}}\nonumber\\ +& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n}{(2\lambda)_n}\frac{P_n^{(\lambda)}(x;\phi)}{\e^{in\phi}}t^n, +\quad\textrm{$\gamma$ arbitrary}. +\end{eqnarray} + +\subsection*{Limit relations} + +\subsubsection*{Continuous dual Hahn $\rightarrow$ Meixner-Pollaczek} +The Meixner-Pollaczek polynomials can be obtained from the continuous dual Hahn polynomials +given by (\ref{DefContDualHahn}) by the substitutions $x\rightarrow x-t$, $a=\lambda+it$, +$b=\lambda-it$ and $c=t\cot\phi$ and the limit $t\rightarrow\infty$: +$$\lim_{t\rightarrow\infty}\frac{S_n((x-t)^2;\lambda+it,\lambda-it,t\cot\phi)}{t^nn!} +=\frac{P_n^{(\lambda)}(x;\phi)}{(\sin\phi)^n}.$$ + +\subsubsection*{Continuous Hahn $\rightarrow$ Meixner-Pollaczek} +By setting $x\rightarrow x+t$, $a=\lambda-it$, $c=\lambda+it$ and $b=d=t\tan\phi$ in the +definition (\ref{DefContHahn}) of the continuous Hahn polynomials and taking the limit +$t\rightarrow\infty$ we obtain the Meixner-Pollaczek polynomials: +$$\lim_{t\rightarrow\infty}\frac{p_n(x+t;\lambda-it,t\tan\phi,\lambda+it,t\tan\phi)}{t^nn!} +=\frac{P_n^{(\lambda)}(x;\phi)}{(\cos\phi)^n}.$$ + +\newpage + +\subsubsection*{Meixner-Pollaczek $\rightarrow$ Laguerre} +The Laguerre polynomials given by (\ref{DefLaguerre}) can be obtained from the +Meixner-Pollaczek polynomials by the substitution $\lambda=\frac{1}{2}(\alpha+1)$, +$x\rightarrow -\frac{1}{2}\phi^{-1}x$ and the limit $\phi\rightarrow 0$: +\begin{equation} +\lim_{\phi\rightarrow 0} +P_n^{(\frac{1}{2}\alpha+\frac{1}{2})}(-\textstyle\frac{1}{2}\phi^{-1}x;\phi)=L_n^{(\alpha)}(x). +\end{equation} + +\subsubsection*{Meixner-Pollaczek $\rightarrow$ Hermite} +The Hermite polynomials given by (\ref{DefHermite}) are obtained from the Meixner-Pollaczek +polynomials if we substitute $x\rightarrow (\sin\phi)^{-1}(x\sqrt{\lambda}-\lambda\cos\phi)$ +and then let $\lambda\rightarrow\infty$: +\begin{equation} +\lim_{\lambda\rightarrow\infty} +\lambda^{-\frac{1}{2}n}P_n^{(\lambda)} +((\sin\phi)^{-1}(x\sqrt{\lambda}-\lambda\cos\phi);\phi)=\frac{H_n(x)}{n!}. +\end{equation} + +\subsection*{Remark} +Since we have for $k-1$ and $\beta>-1$ we have +\begin{eqnarray} +\label{OrtJacobi1} +& &\int_{-1}^1(1-x)^{\alpha}(1+x)^{\beta}P_m^{(\alpha,\beta)}(x)P_n^{(\alpha,\beta)}(x)\,dx\nonumber\\ +& &{}=\frac{2^{\alpha+\beta+1}}{2n+\alpha+\beta+1}\frac{\Gamma(n+\alpha+1)\Gamma(n+\beta+1)}{\Gamma(n+\alpha+\beta+1)n!}\,\delta_{mn}. +\end{eqnarray} +For $\alpha+\beta<-2N-1$, $\beta>-1$ and $m,n\in\{0,1,2,\ldots,N\}$ we also have +\begin{eqnarray} +\label{OrtJacobi2} +& &\int_1^{\infty}(x+1)^{\alpha}(x-1)^{\beta}P_m^{(\alpha,\beta)}(-x)P_n^{(\alpha,\beta)}(-x)\,dx\nonumber\\ +& &{}=-\frac{2^{\alpha+\beta+1}}{2n+\alpha+\beta+1}\frac{\Gamma(-n-\alpha-\beta)\Gamma(n+\alpha+\beta+1)}{\Gamma(-n-\alpha)n!}\,\delta_{mn}. +\end{eqnarray} + +\subsection*{Recurrence relation} +\begin{eqnarray} +\label{RecJacobi} +xP_n^{(\alpha,\beta)}(x)&=&\frac{2(n+1)(n+\alpha+\beta+1)}{(2n+\alpha+\beta+1)(2n+\alpha+\beta+2)}P_{n+1}^{(\alpha,\beta)}(x)\nonumber\\ +& &{}\mathindent{}+\frac{\beta^2-\alpha^2}{(2n+\alpha+\beta)(2n+\alpha+\beta+2)}P_n^{(\alpha,\beta)}(x)\nonumber\\ +& &{}\mathindent\mathindent{}+\frac{2(n+\alpha)(n+\beta)}{(2n+\alpha+\beta)(2n+\alpha+\beta+1)}P_{n-1}^{(\alpha,\beta)}(x). +\end{eqnarray} + +\subsection*{Normalized recurrence relation} +\begin{eqnarray} +\label{NormRecJacobi} +xp_n(x)&=&p_{n+1}(x)+\frac{\beta^2-\alpha^2}{(2n+\alpha+\beta)(2n+\alpha+\beta+2)}p_n(x)\nonumber\\ +& &{}\mathindent{}+\frac{4n(n+\alpha)(n+\beta)(n+\alpha+\beta)} +{(2n+\alpha+\beta-1)(2n+\alpha+\beta)^2(2n+\alpha+\beta+1)}p_{n-1}(x) +\end{eqnarray} +where +$$P_n^{(\alpha,\beta)}(x)=\frac{(n+\alpha+\beta+1)_n}{2^nn!}p_n(x).$$ + +\subsection*{Differential equation} +\begin{eqnarray} +\label{dvJacobi} +& &(1-x^2)y''(x)+\left[\beta-\alpha-(\alpha+\beta+2)x\right]y'(x)\nonumber\\ +& &{}\mathindent{}+n(n+\alpha+\beta+1)y(x)=0,\quad y(x)=P_n^{(\alpha,\beta)}(x). +\end{eqnarray} + +\subsection*{Forward shift operator} +\begin{equation} +\label{shift1Jacobi} +\frac{d}{dx}P_n^{(\alpha,\beta)}(x)=\frac{n+\alpha+\beta+1}{2}P_{n-1}^{(\alpha+1,\beta+1)}(x). +\end{equation} + +\subsection*{Backward shift operator} +\begin{eqnarray} +\label{shift2JacobiI} +& &(1-x^2)\frac{d}{dx}P_n^{(\alpha,\beta)}(x)+ +\left[(\beta-\alpha)-(\alpha+\beta)x\right]P_n^{(\alpha,\beta)}(x)\nonumber\\ +& &{}=-2(n+1)P_{n+1}^{(\alpha-1,\beta-1)}(x) +\end{eqnarray} +or equivalently +\begin{eqnarray} +\label{shift2JacobiII} +& &\frac{d}{dx}\left[(1-x)^\alpha(1+x)^\beta P_n^{(\alpha,\beta)}(x)\right]\nonumber\\ +& &{}=-2(n+1)(1-x)^{\alpha-1}(1+x)^{\beta-1}P_{n+1}^{(\alpha-1,\beta-1)}(x). +\end{eqnarray} + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodJacobi} +(1-x)^{\alpha}(1+x)^{\beta}P_n^{(\alpha,\beta)}(x)= +\frac{(-1)^n}{2^nn!}\left(\frac{d}{dx}\right)^n +\left[(1-x)^{n+\alpha}(1+x)^{n+\beta}\right]. +\end{equation} + +\subsection*{Generating functions} +\begin{equation} +\label{GenJacobi1} +\frac{2^{\alpha+\beta}}{R(1+R-t)^{\alpha}(1+R+t)^{\beta}}= +\sum_{n=0}^{\infty}P_n^{(\alpha,\beta)}(x)t^n,\quad R=\sqrt{1-2xt+t^2}. +\end{equation} + +\begin{eqnarray} +\label{GenJacobi2} +& &\hyp{0}{1}{-}{\alpha+1}{\frac{(x-1)t}{2}}\,\hyp{0}{1}{-}{\beta+1}{\frac{(x+1)t}{2}}\nonumber\\ +& &{}=\sum_{n=0}^{\infty}\frac{P_n^{(\alpha,\beta)}(x)}{(\alpha+1)_n(\beta+1)_n}t^n. +\end{eqnarray} + +\begin{eqnarray} +\label{GenJacobi3} +& &(1-t)^{-\alpha-\beta-1}\,\hyp{2}{1}{\frac{1}{2}(\alpha+\beta+1),\frac{1}{2}(\alpha+\beta+2)} +{\alpha+1}{\frac{2(x-1)t}{(1-t)^2}}\nonumber\\ +& &{}=\sum_{n=0}^{\infty}\frac{(\alpha+\beta+1)_n}{(\alpha+1)_n}P_n^{(\alpha,\beta)}(x)t^n. +\end{eqnarray} + +\begin{eqnarray} +\label{GenJacobi4} +& &(1+t)^{-\alpha-\beta-1}\,\hyp{2}{1}{\frac{1}{2}(\alpha+\beta+1),\frac{1}{2}(\alpha+\beta+2)} +{\beta+1}{\frac{2(x+1)t}{(1+t)^2}}\nonumber\\ +& &{}=\sum_{n=0}^{\infty}\frac{(\alpha+\beta+1)_n}{(\beta+1)_n}P_n^{(\alpha,\beta)}(x)t^n. +\end{eqnarray} + +\begin{eqnarray} +\label{GenJacobi5} +& &\hyp{2}{1}{\gamma,\alpha+\beta+1-\gamma}{\alpha+1}{\frac{1-R-t}{2}}\, +\hyp{2}{1}{\gamma,\alpha+\beta+1-\gamma}{\beta+1}{\frac{1-R+t}{2}}\nonumber\\ +& &{}=\sum_{n=0}^{\infty} +\frac{(\gamma)_n(\alpha+\beta+1-\gamma)_n}{(\alpha+1)_n(\beta+1)_n}P_n^{(\alpha,\beta)}(x)t^n, +\quad R=\sqrt{1-2xt+t^2} +\end{eqnarray} +with $\gamma$ arbitrary. + +\subsection*{Limit relations} + +\subsubsection*{Wilson $\rightarrow$ Jacobi} +The Jacobi polynomials can be found from the Wilson polynomials given by (\ref{DefWilson}) by +substituting $a=b=\frac{1}{2}(\alpha+1)$, $c=\frac{1}{2}(\beta+1)+it$, +$d=\frac{1}{2}(\beta+1)-it$ and $x\rightarrow t\sqrt{\frac{1}{2}(1-x)}$ in the +definition (\ref{DefWilson}) of the Wilson polynomials and taking the limit +$t\rightarrow\infty$. In fact we have +$$\lim_{t\rightarrow\infty} +\frac{W_n(\frac{1}{2}(1-x)t^2;\frac{1}{2}(\alpha+1), +\frac{1}{2}(\alpha+1),\frac{1}{2}(\beta+1)+it,\frac{1}{2}(\beta+1)-it)} +{t^{2n}n!}=P_n^{(\alpha,\beta)}(x).$$ + +\subsubsection*{Continuous Hahn $\rightarrow$ Jacobi} +The Jacobi polynomials follow from the continuous Hahn polynomials given by (\ref{DefContHahn}) +by using the substitution $x\rightarrow \frac{1}{2}xt$, $a=\frac{1}{2}(\alpha+1-it)$, +$b=\frac{1}{2}(\beta+1+it)$, $c=\frac{1}{2}(\alpha+1+it)$ and $d=\frac{1}{2}(\beta+1-it)$ +in (\ref{DefContHahn}), division by $t^n$ and the limit $t\rightarrow\infty$: +$$\lim_{t\rightarrow\infty} +\frac{p_n(\frac{1}{2}xt;\frac{1}{2}(\alpha+1-it),\frac{1}{2}(\beta+1+it), +\frac{1}{2}(\alpha+1+it),\frac{1}{2}(\beta+1-it))}{t^n}=P_n^{(\alpha,\beta)}(x).$$ + +\subsubsection*{Hahn $\rightarrow$ Jacobi} +To find the Jacobi polynomials from the Hahn polynomials given by (\ref{DefHahn}) we take +$x\rightarrow Nx$ in (\ref{DefHahn}) and let $N\rightarrow\infty$. In fact we have +$$\lim_{N\rightarrow\infty} +Q_n(Nx;\alpha,\beta,N)=\frac{P_n^{(\alpha,\beta)}(1-2x)}{P_n^{(\alpha,\beta)}(1)}.$$ + +\subsubsection*{Jacobi $\rightarrow$ Laguerre} +The Laguerre polynomials given by (\ref{DefLaguerre}) can be obtained from the Jacobi +polynomials by setting $x\rightarrow 1-2\beta^{-1}x$ and then the limit $\beta\rightarrow\infty$: +\begin{equation} +\lim_{\beta\rightarrow\infty} +P_n^{(\alpha,\beta)}(1-2\beta^{-1}x)=L_n^{(\alpha)}(x). +\end{equation} + +\subsubsection*{Jacobi $\rightarrow$ Bessel} +The Bessel polynomials given by (\ref{DefBessel}) are obtained from the Jacobi polynomials +if we take $\beta=a-\alpha$ and let $\alpha\rightarrow-\infty$: +\begin{equation} +\lim_{\alpha\rightarrow-\infty} +\frac{P_n^{(\alpha,a-\alpha)}(1+\alpha x)}{P_n^{(\alpha,a-\alpha)}(1)}=y_n(x;a). +\end{equation} + +\subsubsection*{Jacobi $\rightarrow$ Hermite} +The Hermite polynomials given by (\ref{DefHermite}) follow from the Jacobi polynomials +by taking $\beta=\alpha$ and letting $\alpha\rightarrow\infty$ in the following way: +\begin{equation} +\lim_{\alpha\rightarrow\infty} +\alpha^{-\frac{1}{2}n}P_n^{(\alpha,\alpha)}(\alpha^{-\frac{1}{2}}x)=\frac{H_n(x)}{2^nn!}. +\end{equation} + +\subsection*{Remarks} +The definition (\ref{DefJacobi}) of the Jacobi polynomials can also be written as: +$$P_n^{(\alpha,\beta)}(x)=\frac{1}{n!}\sum_{k=0}^n\frac{(-n)_k}{k!} +(n+\alpha+\beta+1)_k(\alpha+k+1)_{n-k}\left(\frac{1-x}{2}\right)^k.$$ +In this way the Jacobi polynomials can also be seen as polynomials in the parameters $\alpha$ +and $\beta$. Therefore they can be defined for all $\alpha$ and $\beta$. Then we have the +following connection with the Meixner polynomials given by (\ref{DefMeixner}): +$$\frac{(\beta)_n}{n!}M_n(x;\beta,c)=P_n^{(\beta-1,-n-\beta-x)}((2-c)c^{-1}).$$ + +\noindent +The Jacobi polynomials are related to the pseudo Jacobi polynomials defined by +(\ref{DefPseudoJacobi}) in the following way: +$$P_n(x;\nu,N)=\frac{(-2i)^nn!}{(n-2N-1)_n}P_n^{(-N-1+i\nu,-N-1-i\nu)}(ix).$$ + +\noindent +The Jacobi polynomials are also related to the Gegenbauer (or ultraspherical) polynomials +given by (\ref{DefGegenbauer}) by the quadratic transformations: +$$C_{2n}^{(\lambda)}(x)=\frac{(\lambda)_n}{(\frac{1}{2})_n} +P_n^{(\lambda-\frac{1}{2},-\frac{1}{2})}(2x^2-1)$$ +and +$$C_{2n+1}^{(\lambda)}(x)=\frac{(\lambda)_{n+1}}{(\frac{1}{2})_{n+1}} +xP_n^{(\lambda-\frac{1}{2},\frac{1}{2})}(2x^2-1).$$ + +\subsection*{References} +\cite{Abram}, \cite{Ahmed+82}, \cite{Allaway89}, \cite{NAlSalam66}, \cite{AlSalam64}, +\cite{AlSalamChihara72}, \cite{AndrewsAskey85}, \cite{AndrewsAskeyRoy}, \cite{Askey68}, +\cite{Askey70}, \cite{Askey72}, \cite{Askey73}, \cite{Askey74}, \cite{Askey75}, \cite{Askey78}, +\cite{Askey89I}, \cite{Askey2005}, \cite{AskeyFitch}, \cite{AskeyGasper71I}, \cite{AskeyGasper71II}, +\cite{AskeyGasper76}, \cite{AskeyWainger}, \cite{AskeyWilson85}, \cite{Bailey38}, \cite{Brafman51}, +\cite{Brown}, \cite{Carlitz61II}, \cite{Carlitz67}, \cite{ChenIsmail91}, \cite{Chihara78}, +\cite{Chow+}, \cite{Ciesielski}, \cite{Cooper+}, \cite{DetteStudden92}, \cite{DetteStudden95}, +\cite{DijksmaKoorn}, \cite{DimitrovRafaeli}, \cite{DimitrovRodrigues}, \cite{Doha2002I}, +\cite{Doha2003I}, \cite{Doha2004II}, \cite{Dunkl84}, \cite{ElbertLaforgia87II}, +\cite{ElbertLaforgiaRodono}, \cite{Erdelyi+}, \cite{Faldey}, \cite{FoataLeroux}, +\cite{Gasper69}, \cite{Gasper70I}, \cite{Gasper70II}, \cite{Gasper71I}, \cite{Gasper71II}, +\cite{Gasper72I}, \cite{Gasper72II}, \cite{Gasper73I}, \cite{Gasper73II}, \cite{Gasper74}, +\cite{Gasper77}, \cite{Gatteschi87}, \cite{Gautschi2008}, \cite{Gautschi2009I}, +\cite{Gautschi2009II}, \cite{GautschiLeopardi}, \cite{GawronskiShawyer}, \cite{Godoy+}, +\cite{Grad}, \cite{Hajmirzaahmad94}, \cite{HartmannStephan}, \cite{Horton}, \cite{Ismail74}, +\cite{Ismail77}, \cite{Ismail96}, \cite{Ismail2005II}, \cite{IsmailLi}, +\cite{IsmailMassonRahman}, \cite{Kochneff97I}, \cite{Koekoek99}, \cite{Koekoek2000}, +\cite{Koelink96II}, \cite{Koorn73}, \cite{Koorn74}, \cite{Koorn75}, \cite{Koorn77II}, +\cite{Koorn78}, \cite{Koorn72}, \cite{Koorn85}, \cite{Koorn88}, \cite{KuijlaarsMartinez}, +\cite{Kuijlaars+}, \cite{LabelleYehI}, \cite{LabelleYehII}, \cite{Laine}, \cite{Lesky95II}, +\cite{Lesky96}, \cite{Li96}, \cite{Li97}, \cite{LopezTemme2004}, \cite{Luke}, \cite{Martinez}, +\cite{Mathai}, \cite{Meijer}, \cite{Miller89}, \cite{MoakSaffVarga}, \cite{Nikiforov+}, +\cite{NikiforovUvarov}, \cite{Olver}, \cite{Prasad}, \cite{Rahman76I}, \cite{Rahman76II}, +\cite{Rahman77}, \cite{Rahman81I}, \cite{RahmanShah}, \cite{Rainville}, \cite{Rusev}, +\cite{Shi}, \cite{Srivastava69II}, \cite{Srivastava71}, \cite{Srivastava82}, +\cite{SrivastavaSinghal}, \cite{Stanton80I}, \cite{Stanton90}, \cite{Szego75}, \cite{Temme}, +\cite{Vertesi}, \cite{Wimp87}, \cite{WongZhang94}, \cite{WongZhang2006}, \cite{Zarzo+}, +\cite{Zayed}. + +\section*{Special cases} + +\subsection{Gegenbauer / Ultraspherical}\index{Gegenbauer polynomials} +\index{Ultraspherical polynomials} + +\par + +\subsection*{Hypergeometric representation} +The Gegenbauer (or ultraspherical) polynomials are Jacobi polynomials with +$\alpha=\beta=\lambda-\frac{1}{2}$ and another normalization: +\begin{eqnarray} +\label{DefGegenbauer} +C_n^{(\lambda)}(x)&=&\frac{(2\lambda)_n}{(\lambda+\frac{1}{2})_n} +P_n^{(\lambda-\frac{1}{2},\lambda-\frac{1}{2})}(x)\nonumber\\ +&=&\frac{(2\lambda)_n}{n!}\,\hyp{2}{1}{-n,n+2\lambda} +{\lambda+\frac{1}{2}}{\frac{1-x}{2}},\quad\lambda\neq 0. +\end{eqnarray} + +\subsection*{Orthogonality relation} +\begin{eqnarray} +\label{OrtGegenbauer} +& &\int_{-1}^1(1-x^2)^{\lambda-\frac{1}{2}}C_m^{(\lambda)}(x)C_n^{(\lambda)}(x)\,dx\nonumber\\ +& &{}=\frac{\pi\Gamma(n+2\lambda)2^{1-2\lambda}}{\left\{\Gamma(\lambda)\right\}^2(n+\lambda)n!}\,\delta_{mn}, +\quad\lambda>-\frac{1}{2}\quad\lambda\neq 0. +\end{eqnarray} + +\subsection*{Recurrence relation} +\begin{equation} +\label{RecGegenbauer} +2(n+\lambda)xC_n^{(\lambda)}(x)=(n+1)C_{n+1}^{(\lambda)}(x)+(n+2\lambda-1)C_{n-1}^{(\lambda)}(x). +\end{equation} + +\subsection*{Normalized recurrence relation} +\begin{equation} +\label{NormRecGegenbauer} +xp_n(x)=p_{n+1}(x)+\frac{n(n+2\lambda-1)}{4(n+\lambda-1)(n+\lambda)}p_{n-1}(x), +\end{equation} +where +$$C_n^{(\lambda)}(x)=\frac{2^n(\lambda)_n}{n!}p_n(x).$$ + +\subsection*{Differential equation} +\begin{equation} +\label{dvGegenbauer} +(1-x^2)y''(x)-(2\lambda+1)xy'(x)+n(n+2\lambda)y(x)=0,\quad y(x)=C_n^{(\lambda)}(x). +\end{equation} + +\subsection*{Forward shift operator} +\begin{equation} +\label{shift1Gegenbauer} +\frac{d}{dx}C_n^{(\lambda)}(x)=2\lambda C_{n-1}^{(\lambda+1)}(x). +\end{equation} + +\subsection*{Backward shift operator} +\begin{equation} +\label{shift2GegenbauerI} +(1-x^2)\frac{d}{dx}C_n^{(\lambda)}(x)+(1-2\lambda)xC_n^{(\lambda)}(x)= +-\frac{(n+1)(2\lambda+n-1)}{2(\lambda-1)} C_{n+1}^{(\lambda-1)}(x) +\end{equation} +or equivalently +\begin{eqnarray} +\label{shift2GegenbauerII} +& &\frac{d}{dx}\left[(1-x^2)^{\lambda-\textstyle\frac{1}{2}}C_n^{(\lambda)}(x)\right]\nonumber\\ +& &{}=-\frac{(n+1)(2\lambda+n-1)}{2(\lambda-1)}(1-x^2)^{\lambda-\textstyle\frac{3}{2}}C_{n+1}^{(\lambda-1)}(x). +\end{eqnarray} + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodGegenbauer} +(1-x^2)^{\lambda-\frac{1}{2}}C_n^{(\lambda)}(x)= +\frac{(2\lambda)_n(-1)^n}{(\lambda+\frac{1}{2})_n2^nn!}\left(\frac{d}{dx}\right)^n +\left[(1-x^2)^{\lambda+n-\frac{1}{2}}\right]. +\end{equation} + +\subsection*{Generating functions} +\begin{equation} +\label{GenGegenbauer1} +(1-2xt+t^2)^{-\lambda}=\sum_{n=0}^{\infty}C_n^{(\lambda)}(x)t^n. +\end{equation} + +\begin{equation} +\label{GenGegenbauer2} +R^{-1}\left(\frac{1+R-xt}{2}\right)^{\frac{1}{2}-\lambda}=\sum_{n=0}^{\infty} +\frac{(\lambda+\frac{1}{2})_n}{(2\lambda)_n}C_n^{(\lambda)}(x)t^n, +\quad R=\sqrt{1-2xt+t^2}. +\end{equation} + +\begin{equation} +\label{GenGegenbauer3} +\hyp{0}{1}{-}{\lambda+\frac{1}{2}}{\frac{(x-1)t}{2}}\, +\hyp{0}{1}{-}{\lambda+\frac{1}{2}}{\frac{(x+1)t}{2}} +=\sum_{n=0}^{\infty}\frac{C_n^{(\lambda)}(x)} +{(2\lambda)_n(\lambda+\frac{1}{2})_n}t^n. +\end{equation} + +\begin{equation} +\label{GenGegenbauer4} +\e^{xt}\,\hyp{0}{1}{-}{\lambda+\frac{1}{2}}{\frac{(x^2-1)t^2}{4}}= +\sum_{n=0}^{\infty}\frac{C_n^{(\lambda)}(x)}{(2\lambda)_n}t^n. +\end{equation} + +\begin{eqnarray} +\label{GenGegenbauer5} +& &\hyp{2}{1}{\gamma,2\lambda-\gamma}{\lambda+\frac{1}{2}}{\frac{1-R-t}{2}}\, +\hyp{2}{1}{\gamma,2\lambda-\gamma}{\lambda+\frac{1}{2}}{\frac{1-R+t}{2}}\nonumber\\ +& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n(2\lambda-\gamma)_n} +{(2\lambda)_n(\lambda+\frac{1}{2})_n}C_n^{(\lambda)}(x)t^n, +\quad R=\sqrt{1-2xt+t^2},\quad\textrm{$\gamma$ arbitrary}. +\end{eqnarray} + +\begin{eqnarray} +\label{GenGegenbauer6} +& &(1-xt)^{-\gamma}\,\hyp{2}{1}{\frac{1}{2}\gamma,\frac{1}{2}\gamma+\frac{1}{2}} +{\lambda+\frac{1}{2}}{\frac{(x^2-1)t^2}{(1-xt)^2}}\nonumber\\ +& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n}{(2\lambda)_n}C_n^{(\lambda)}(x)t^n, +\quad\textrm{$\gamma$ arbitrary}. +\end{eqnarray} + +\subsection*{Limit relation} + +\subsubsection*{Gegenbauer / Ultraspherical $\rightarrow$ Hermite} +The Hermite polynomials given by (\ref{DefHermite}) follow from the Gegenbauer (or +ultraspherical) polynomials by taking $\lambda=\alpha+\frac{1}{2}$ and letting +$\alpha\rightarrow\infty$ in the following way: +\begin{equation} +\lim_{\alpha\rightarrow\infty} +\alpha^{-\frac{1}{2}n}C_n^{(\alpha+\frac{1}{2})}(\alpha^{-\frac{1}{2}}x)=\frac{H_n(x)}{n!}. +\end{equation} + +\subsection*{Remarks} +The case $\lambda=0$ needs another normalization. In that case we have the +Chebyshev polynomials of the first kind described in the next subsection. + +\noindent +The Gegenbauer (or ultraspherical) polynomials are related to the Jacobi polynomials +given by (\ref{DefJacobi}) by the quadratic transformations: +$$C_{2n}^{(\lambda)}(x)=\frac{(\lambda)_n}{(\frac{1}{2})_n} +P_n^{(\lambda-\frac{1}{2},-\frac{1}{2})}(2x^2-1)$$ +and +$$C_{2n+1}^{(\lambda)}(x)=\frac{(\lambda)_{n+1}}{(\frac{1}{2})_{n+1}} +xP_n^{(\lambda-\frac{1}{2},\frac{1}{2})}(2x^2-1).$$ + +\subsection*{References} +\cite{Abram}, \cite{Ahmed+86}, \cite{AndrewsAskeyRoy}, \cite{Area+I}, \cite{Askey67}, \cite{Askey74}, \cite{Askey75}, +\cite{Askey89I}, \cite{AskeyFitch}, \cite{AskeyKoornRahman}, \cite{Berg}, \cite{BilodeauI}, +\cite{BojanovNikolov}, \cite{Brafman51}, \cite{Brafman57I}, \cite{Brown}, +\cite{BustozIsmail82}, \cite{BustozIsmail83}, \cite{BustozIsmail89}, \cite{BustozSavage79}, +\cite{BustozSavage80}, \cite{Carlitz61II}, \cite{Chihara78}, \cite{Common}, \cite{Danese}, +\cite{Dette94}, \cite{DijksmaKoorn}, \cite{DilcherStolarsky}, \cite{Dimitrov96}, +\cite{Dimitrov2003}, \cite{Doha2002II}, \cite{Driver}, \cite{ElbertLaforgia86I}, +\cite{ElbertLaforgia86II}, \cite{ElbertLaforgia90}, \cite{Erdelyi+}, \cite{Exton96}, +\cite{Gasper69}, \cite{Gasper72II}, \cite{Gasper85}, \cite{Grad}, \cite{Ismail74}, +\cite{Ismail2005II}, \cite{Koekoek2000}, \cite{Koorn88}, \cite{Laforgia}, \cite{LewanowiczI}, +\cite{Lorch}, \cite{Mathai}, \cite{Nagel}, \cite{Nikiforov+}, \cite{NikiforovUvarov}, +\cite{RahmanShah}, \cite{Rainville}, \cite{Reimer}, \cite{Sartoretto}, \cite{Srivastava71}, +\cite{Szego75}, \cite{Temme}, \cite{Viswanathan}, \cite{Zayed}. + +\subsection{Chebyshev}\index{Chebyshev polynomials} + +\par + +\subsection*{Hypergeometric representation} +The Chebyshev polynomials of the first kind can be obtained from the Jacobi +polynomials by taking $\alpha=\beta=-\frac{1}{2}$: +\begin{equation} +\label{DefChebyshevI} +T_n(x)=\frac{P_n^{(-\frac{1}{2},-\frac{1}{2})}(x)}{P_n^{(-\frac{1}{2},-\frac{1}{2})}(1)} +=\hyp{2}{1}{-n,n}{\frac{1}{2}}{\frac{1-x}{2}} +\end{equation} +and the Chebyshev polynomials of the second kind can be obtained from the +Jacobi polynomials by taking $\alpha=\beta=\frac{1}{2}$: +\begin{equation} +\label{DefChebyshevII} +U_n(x)=(n+1)\frac{P_n^{(\frac{1}{2},\frac{1}{2})}(x)}{P_n^{(\frac{1}{2},\frac{1}{2})}(1)} +=(n+1)\,\hyp{2}{1}{-n,n+2}{\frac{3}{2}}{\frac{1-x}{2}}. +\end{equation} + +\subsection*{Orthogonality relation} +\begin{equation} +\label{OrtChebyshevI} +\int_{-1}^1(1-x^2)^{-\frac{1}{2}}T_m(x)T_n(x)\,dx= +\left\{\begin{array}{ll} +\displaystyle\frac{\cpi}{2}\,\delta_{mn}, & n\neq 0\\[5mm] +\cpi\,\delta_{mn}, & n=0. +\end{array}\right. +\end{equation} + +\begin{equation} +\label{OrtChebyshevII} +\int_{-1}^1(1-x^2)^{\frac{1}{2}}U_m(x)U_n(x)\,dx=\frac{\cpi}{2}\,\delta_{mn}. +\end{equation} + +\subsection*{Recurrence relations} +\begin{equation} +\label{RecChebyshevI} +2xT_n(x)=T_{n+1}(x)+T_{n-1}(x),\quad T_{0}(x)=1\quad\textrm{and}\quad T_1(x)=x. +\end{equation} + +\begin{equation} +\label{RecChebyshevII} +2xU_n(x)=U_{n+1}(x)+U_{n-1}(x),\quad U_{0}(x)=1\quad\textrm{and}\quad U_1(x)=2x. +\end{equation} + +\subsection*{Normalized recurrence relations} +\begin{equation} +\label{NormRecChebyshevI} +xp_n(x)=p_{n+1}(x)+\frac{1}{4}p_{n-1}(x), +\end{equation} +where +$$T_1(x)=p_1(x)=x\quad\textrm{and}\quad T_n(x)=2^np_n(x),\quad n\neq 1.$$ + +\begin{equation} +\label{NormRecChebyshevII} +xp_n(x)=p_{n+1}(x)+\frac{1}{4}p_{n-1}(x), +\end{equation} +where +$$U_n(x)=2^np_n(x).$$ + +\subsection*{Differential equations} +\begin{equation} +\label{dvChebyshevI} +(1-x^2)y''(x)-xy'(x)+n^2y(x)=0,\quad y(x)=T_n(x). +\end{equation} + +\begin{equation} +\label{dvChebyshevII} +(1-x^2)y''(x)-3xy'(x)+n(n+2)y(x)=0,\quad y(x)=U_n(x). +\end{equation} + +\subsection*{Forward shift operator} +\begin{equation} +\label{shift1Chebyshev} +\frac{d}{dx}T_n(x)=nU_{n-1}(x). +\end{equation} + +\subsection*{Backward shift operator} +\begin{equation} +\label{shift2ChebyshevI} +(1-x^2)\frac{d}{dx}U_n(x)-xU_n(x)=-(n+1)T_{n+1}(x) +\end{equation} +or equivalently +\begin{equation} +\label{shift2ChebyshevII} +\frac{d}{dx}\left[\left(1-x^2\right)^{\frac{1}{2}}U_n(x)\right] +=-(n+1)\left(1-x^2\right)^{-\frac{1}{2}}T_{n+1}(x). +\end{equation} + +\subsection*{Rodrigues-type formulas} +\begin{equation} +\label{RodChebyshevI} +(1-x^2)^{-\frac{1}{2}}T_n(x)=\frac{(-1)^n}{(\frac{1}{2})_n2^n} +\left(\frac{d}{dx}\right)^n\left[(1-x^2)^{n-\frac{1}{2}}\right]. +\end{equation} + +\begin{equation} +\label{RodChebyshevII} +(1-x^2)^{\frac{1}{2}}U_n(x)=\frac{(n+1)(-1)^n}{(\frac{3}{2})_n2^n} +\left(\frac{d}{dx}\right)^n\left[(1-x^2)^{n+\frac{1}{2}}\right]. +\end{equation} + +\subsection*{Generating functions} +\begin{equation} +\label{GenChebyshevI1} +\frac{1-xt}{1-2xt+t^2}=\sum_{n=0}^{\infty}T_n(x)t^n. +\end{equation} + +\begin{equation} +\label{GenChebyshevI2} +R^{-1}\sqrt{\frac{1}{2}(1+R-xt)}=\sum_{n=0}^{\infty} +\frac{\left(\frac{1}{2}\right)_n}{n!}T_n(x)t^n,\quad R=\sqrt{1-2xt+t^2}. +\end{equation} + +\begin{equation} +\label{GenChebyshevI3} +\hyp{0}{1}{-}{\frac{1}{2}}{\frac{(x-1)t}{2}}\, +\hyp{0}{1}{-}{\frac{1}{2}}{\frac{(x+1)t}{2}}= +\sum_{n=0}^{\infty}\frac{T_n(x)}{\left(\frac{1}{2}\right)_nn!}t^n. +\end{equation} + +\begin{equation} +\label{GenChebyshevI4} +\e^{xt}\,\hyp{0}{1}{-}{\frac{1}{2}}{\frac{(x^2-1)t^2}{4}}= +\sum_{n=0}^{\infty}\frac{T_n(x)}{n!}t^n. +\end{equation} + +\begin{eqnarray} +\label{GenChebyshevI5} +& &\hyp{2}{1}{\gamma,-\gamma}{\frac{1}{2}}{\frac{1-R-t}{2}}\, +\hyp{2}{1}{\gamma,-\gamma}{\frac{1}{2}}{\frac{1-R+t}{2}}\nonumber\\ +& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n(-\gamma)_n}{\left(\frac{1}{2}\right)_nn!}T_n(x)t^n, +\quad R=\sqrt{1-2xt+t^2},\quad\textrm{$\gamma$ arbitrary}. +\end{eqnarray} + +\begin{eqnarray} +\label{GenChebyshevI6} +& &(1-xt)^{-\gamma}\,\hyp{2}{1}{\frac{1}{2}\gamma,\frac{1}{2}\gamma+\frac{1}{2}}{\frac{1}{2}} +{\frac{(x^2-1)t^2}{(1-xt)^2}}\nonumber\\ +& &=\sum_{n=0}^{\infty}\frac{(\gamma)_n}{n!}T_n(x)t^n,\quad\textrm{$\gamma$ arbitrary}. +\end{eqnarray} + +\begin{equation} +\label{GenChebyshevII1} +\frac{1}{1-2xt+t^2}=\sum_{n=0}^{\infty}U_n(x)t^n. +\end{equation} + +\begin{equation} +\label{GenChebyshevII2} +\frac{1}{R\sqrt{\frac{1}{2}(1+R-xt)}}=\sum_{n=0}^{\infty} +\frac{\left(\frac{3}{2}\right)_n}{(n+1)!}U_n(x)t^n,\quad R=\sqrt{1-2xt+t^2}. +\end{equation} + +\begin{equation} +\label{GenChebyshevII3} +\hyp{0}{1}{-}{\frac{3}{2}}{\frac{(x-1)t}{2}}\, +\hyp{0}{1}{-}{\frac{3}{2}}{\frac{(x+1)t}{2}}= +\sum_{n=0}^{\infty}\frac{U_n(x)}{\left(\frac{3}{2}\right)_n(n+1)!}t^n. +\end{equation} + +\begin{equation} +\label{GenChebyshevII4} +\e^{xt}\,\hyp{0}{1}{-}{\frac{3}{2}}{\frac{(x^2-1)t^2}{4}}= +\sum_{n=0}^{\infty}\frac{U_n(x)}{(n+1)!}t^n. +\end{equation} + +\begin{eqnarray} +\label{GenChebyshevII5} +& &\hyp{2}{1}{\gamma,2-\gamma}{\frac{3}{2}}{\frac{1-R-t}{2}}\, +\hyp{2}{1}{\gamma,2-\gamma}{\frac{3}{2}}{\frac{1-R+t}{2}}\nonumber\\ +& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n(2-\gamma)_n}{\left(\frac{3}{2}\right)_n(n+1)!}U_n(x)t^n, +\quad R=\sqrt{1-2xt+t^2},\quad\textrm{$\gamma$ arbitrary}. +\end{eqnarray} + +\begin{eqnarray} +\label{GenChebyshevII6} +& &(1-xt)^{-\gamma}\,\hyp{2}{1}{\frac{1}{2}\gamma,\frac{1}{2}\gamma+\frac{1}{2}}{\frac{3}{2}} +{\frac{(x^2-1)t^2}{(1-xt)^2}}\nonumber\\ +& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n}{(n+1)!}U_n(x)t^n,\quad\textrm{$\gamma$ arbitrary}. +\end{eqnarray} + +\subsection*{Remarks} +The Chebyshev polynomials can also be written as: +$$T_n(x)=\cos(n\theta),\quad x=\cos\theta$$ +and +$$U_n(x)=\frac{\sin (n+1)\theta}{\sin\theta},\quad x=\cos\theta.$$ +Further we have +$$U_n(x)=C_n^{(1)}(x)$$ +where $C_n^{(\lambda)}(x)$ denotes the Gegenbauer (or ultraspherical) +polynomial given by (\ref{DefGegenbauer}) in the preceding subsection. + +\subsection*{References} +\cite{Abram}, \cite{AskeyFitch}, \cite{AskeyGasperHarris}, \cite{AskeyIsmail76}, +\cite{Bavinck95}, \cite{Chihara78}, \cite{Danese}, \cite{DilcherStolarsky}, +\cite{Erdelyi+}, \cite{Grad}, \cite{HartmannStephan}, \cite{Ismail2005II}, \cite{Koekoek2000}, +\cite{Luke}, \cite{Mathai}, \cite{Nikiforov+}, \cite{NikiforovUvarov}, \cite{Rainville}, +\cite{Rayes+}, \cite{Rivlin}, \cite{SainteViennot}, \cite{Szego75}, \cite{Temme}, +\cite{Wilson70I}, \cite{Zayed}, \cite{Zhang}, \cite{ZhangWang}. + +\subsection{Legendre / Spherical}\index{Legendre polynomials} +\index{Spherical polynomials} + +\par + +\subsection*{Hypergeometric representation} +The Legendre (or spherical) polynomials are Jacobi polynomials with $\alpha=\beta=0$: +\begin{equation} +\label{DefLegendre} +P_n(x)=P_n^{(0,0)}(x)=\hyp{2}{1}{-n,n+1}{1}{\frac{1-x}{2}}. +\end{equation} + +\subsection*{Orthogonality relation} +\begin{equation} +\label{OrtLegendre} +\int_{-1}^1P_m(x)P_n(x)\,dx=\frac{2}{2n+1}\,\delta_{mn}. +\end{equation} + +\subsection*{Recurrence relation} +\begin{equation} +\label{RecLegendre} +(2n+1)xP_n(x)=(n+1)P_{n+1}(x)+nP_{n-1}(x). +\end{equation} + +\subsection*{Normalized recurrence relation} +\begin{equation} +\label{NormRecLegendre} +xp_n(x)=p_{n+1}(x)+\frac{n^2}{(2n-1)(2n+1)}p_{n-1}(x), +\end{equation} +where +$$P_n(x)=\binom{2n}{n}\frac{1}{2^n}p_n(x).$$ + +\subsection*{Differential equation} +\begin{equation} +\label{dvLegendre} +(1-x^2)y''(x)-2xy'(x)+n(n+1)y(x)=0,\quad y(x)=P_n(x). +\end{equation} + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodLegendre} +P_n(x)=\frac{(-1)^n}{2^nn!}\left(\frac{d}{dx}\right)^n\left[(1-x^2)^n\right]. +\end{equation} + +\subsection*{Generating functions} +\begin{equation} +\label{GenLegendre1} +\frac{1}{\sqrt{1-2xt+t^2}}=\sum_{n=0}^{\infty}P_n(x)t^n. +\end{equation} + +\begin{equation} +\label{GenLegendre2} +\hyp{0}{1}{-}{1}{\frac{(x-1)t}{2}}\,\hyp{0}{1}{-}{1}{\frac{(x+1)t}{2}}= +\sum_{n=0}^{\infty}\frac{P_n(x)}{(n!)^2}t^n. +\end{equation} + +\begin{equation} +\label{GenLegendre3} +\e^{xt}\,\hyp{0}{1}{-}{1}{\frac{(x^2-1)t^2}{4}}= +\sum_{n=0}^{\infty}\frac{P_n(x)}{n!}t^n. +\end{equation} + +\begin{eqnarray} +\label{GenLegendre4} +& &\hyp{2}{1}{\gamma,1-\gamma}{1}{\frac{1-R-t}{2}}\, +\hyp{2}{1}{\gamma,1-\gamma}{1}{\frac{1-R+t}{2}}\nonumber\\ +& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n(1-\gamma)_n}{(n!)^2}P_n(x)t^n, +\quad R=\sqrt{1-2xt+t^2},\quad\textrm{$\gamma$ arbitrary}. +\end{eqnarray} + +\begin{eqnarray} +\label{GenLegendre5} +& &(1-xt)^{-\gamma}\,\hyp{2}{1}{\frac{1}{2}\gamma,\frac{1}{2}\gamma+\frac{1}{2}}{1} +{\frac{(x^2-1)t^2}{(1-xt)^2}}\nonumber\\ +& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n}{n!}P_n(x)t^n,\quad\textrm{$\gamma$ arbitrary}. +\end{eqnarray} + +\subsection*{References} +\cite{Abram}, \cite{Alladi}, \cite{AlSalam90}, \cite{Bhonsle}, \cite{Brafman51}, +\cite{Carlitz57II}, \cite{Chihara78}, \cite{Danese}, \cite{Dattoli2001}, +\cite{DilcherStolarsky}, \cite{ElbertLaforgia94}, \cite{Erdelyi+}, \cite{Grad}, +\cite{Mathai}, \cite{Nikiforov+}, \cite{NikiforovUvarov}, \cite{Olver}, \cite{Rainville}, +\cite{Szego75}, \cite{Temme}, \cite{Zayed}. + +\newpage + +\section{Pseudo Jacobi}\index{Pseudo Jacobi polynomials}\index{Jacobi polynomials!Pseudo} + +\par\setcounter{equation}{0} + +\subsection*{Hypergeometric representation} +\begin{eqnarray} +\label{DefPseudoJacobi} +P_n(x;\nu,N)&=&\frac{(-2i)^n(-N+i\nu)_n}{(n-2N-1)_n}\,\hyp{2}{1}{-n,n-2N-1}{-N+i\nu}{\frac{1-ix}{2}}\\ +&=&(x+i)^n\,\hyp{2}{1}{-n,N+1-n-i\nu}{2N+2-2n}{\frac{2}{1-ix}},\quad n=0,1,2,\ldots,N.\nonumber +\end{eqnarray} + +\subsection*{Orthogonality relation} +\begin{eqnarray} +\label{OrtPseudoJacobi} +& &\frac{1}{2\pi}\int_{-\infty}^{\infty}(1+x^2)^{-N-1}\e^{2\nu\arctan x}P_m(x;\nu,N)P_n(x;\nu,N)\,dx\nonumber\\ +& &{}=\frac{\Gamma(2N+1-2n)\Gamma(2N+2-2n)2^{2n-2N-1}n!}{\Gamma(2N+2-n)\left|\Gamma(N+1-n+i\nu)\right|^2}\,\delta_{mn}. +\end{eqnarray} + +\subsection*{Recurrence relation} +\begin{eqnarray} +\label{RecPseudoJacobi} +xP_n(x;\nu,N)&=&P_{n+1}(x;\nu,N)+\frac{(N+1)\nu}{(n-N-1)(n-N)}P_n(x;\nu,N)\nonumber\\ +& &{}\mathindent{}-\frac{n(n-2N-2)}{(2n-2N-3)(n-N-1)^2(2n-2N-1)}\nonumber\\ +& &{}\mathindent\mathindent\times(n-N-1-i\nu)(n-N-1+i\nu)P_{n-1}(x;\nu,N). +\end{eqnarray} + +\subsection*{Normalized recurrence relation} +\begin{eqnarray} +\label{NormRecPseudoJacobi} +xp_n(x)&=&p_{n+1}(x)+\frac{(N+1)\nu}{(n-N-1)(n-N)}p_n(x)\nonumber\\ +& &{}\mathindent{}-\frac{n(n-2N-2)(n-N-1-i\nu)(n-N-1+i\nu)}{(2n-2N-3)(n-N-1)^2(2n-2N-1)}p_{n-1}(x), +\end{eqnarray} +where +$$P_n(x;\nu,N)=p_n(x).$$ + +\subsection*{Differential equation} +\begin{equation} +\label{dvPseudoJacobi} +(1+x^2)y''(x)+2\left(\nu-Nx\right)y'(x)-n(n-2N-1)y(x)=0, +\end{equation} +where +$$y(x)=P_n(x;\nu,N).$$ + +\subsection*{Forward shift operator} +\begin{equation} +\label{shift1PseudoJacobi} +\frac{d}{dx}P_n(x;\nu,N)=nP_{n-1}(x;\nu,N-1). +\end{equation} + +\subsection*{Backward shift operator} +\begin{eqnarray} +\label{shift2PseudoJacobiI} +& &(1+x^2)\frac{d}{dx}P_n(x;\nu,N)+2\left[\nu-(N+1)x\right]P_n(x;\nu,N)\nonumber\\ +& &{}=(n-2N-2)P_{n+1}(x;\nu,N+1) +\end{eqnarray} +or equivalently +\begin{eqnarray} +\label{shift2PseudoJacobiII} +& &\frac{d}{dx}\left[(1+x^2)^{-N-1}\e^{2\nu\arctan x}P_n(x;\nu,N)\right]\nonumber\\ +& &{}=(n-2N-2)(1+x^2)^{-N-2}\e^{2\nu\arctan x}P_{n+1}(x;\nu,N+1). +\end{eqnarray} + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodPseudoJacobi} +P_n(x;\nu,N)=\frac{(1+x^2)^{N+1}\expe^{-2\nu\arctan x}}{(n-2N-1)_n} +\left(\frac{d}{dx}\right)^n\left[(1+x^2)^{n-N-1}\expe^{2\nu\arctan x}\right]. +\end{equation} + +\subsection*{Generating function} +\begin{eqnarray} +\label{GenPseudoJacobi} +& &\left[\hyp{0}{1}{-}{-N+i\nu}{(x+i)t}\,\hyp{0}{1}{-}{-N-i\nu}{(x-i)t}\right]_N\nonumber\\ +& &{}=\sum_{n=0}^N\frac{(n-2N-1)_n}{(-N+i\nu)_n(-N-i\nu)_nn!}P_n(x;\nu,N)t^n. +\end{eqnarray} + +\subsection*{Limit relation} + +\subsubsection*{Continuous Hahn $\rightarrow$ Pseudo Jacobi} +The pseudo Jacobi polynomials follow from the continuous Hahn polynomials given by +(\ref{DefContHahn}) by the substitutions $x\rightarrow xt$, $a=\frac{1}{2}(-N+i\nu-2t)$, +$b=\frac{1}{2}(-N-i\nu+2t)$, $c=\frac{1}{2}(-N-i\nu-2t)$ and $d=\frac{1}{2}(-N+i\nu+2t)$, +division by $t^n$ and the limit $t\rightarrow\infty$: +\begin{eqnarray*} +& &\lim_{t\rightarrow\infty}\frac{p_n(xt;\frac{1}{2}(-N+i\nu-2t),\frac{1}{2}(-N-i\nu+2t), +\frac{1}{2}(-N+i\nu-2t),\frac{1}{2}(-N-i\nu+2t))}{t^n}\\ +& &{}=\frac{(n-2N-1)_n}{n!}P_n(x;\nu,N). +\end{eqnarray*} + +\subsection*{Remarks} +Since we have for $k 0$ & +% $0 < c < 1$ } +\end{equation} + +\subsection*{Recurrence relation} +\begin{eqnarray} +\label{RecMeixner} +(c-1)xM_n(x;\beta,c)&=&c(n+\beta)M_{n+1}(x;\beta,c)\nonumber\\ +& &{}\mathindent{}-\left[n+(n+\beta)c\right]M_n(x;\beta,c)+nM_{n-1}(x;\beta,c). +\end{eqnarray} + +\subsection*{Normalized recurrence relation} +\begin{equation} +\label{NormRecMeixner} +xp_n(x)=p_{n+1}(x)+\frac{n+(n+\beta)c}{1-c}p_n(x)+ +\frac{n(n+\beta-1)c}{(1-c)^2}p_{n-1}(x), +\end{equation} +where +$$M_n(x;\beta,c)=\frac{1}{(\beta)_n}\left(\frac{c-1}{c}\right)^np_n(x).$$ + +\subsection*{Difference equation} +\begin{equation} +\label{dvMeixner} +n(c-1)y(x)=c(x+\beta)y(x+1)-\left[x+(x+\beta)c\right]y(x)+xy(x-1), +\end{equation} +where +$$y(x)=M_n(x;\beta,c).$$ + +\subsection*{Forward shift operator} +\begin{equation} +\label{shift1MeixnerI} +M_n(x+1;\beta,c)-M_n(x;\beta,c)= +\frac{n}{\beta}\left(\frac{c-1}{c}\right)M_{n-1}(x;\beta+1,c) +\end{equation} +or equivalently +\begin{equation} +\label{shift1MeixnerII} +\Delta M_n(x;\beta,c)=\frac{n}{\beta}\left(\frac{c-1}{c}\right)M_{n-1}(x;\beta+1,c). +\end{equation} + +\subsection*{Backward shift operator} +\begin{equation} +\label{shift2MeixnerI} +c(\beta+x-1)M_n(x;\beta,c)-xM_n(x-1;\beta,c)=c(\beta-1)M_{n+1}(x;\beta-1,c) +\end{equation} +or equivalently +\begin{equation} +\label{shift2MeixnerII} +\nabla\left[\frac{(\beta)_xc^x}{x!}M_n(x;\beta,c)\right]= +\frac{(\beta-1)_xc^x}{x!}M_{n+1}(x;\beta-1,c). +\end{equation} + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodMeixner} +\frac{(\beta)_xc^x}{x!}M_n(x;\beta,c)=\nabla^n\left[\frac{(\beta+n)_xc^x}{x!}\right]. +\end{equation} + +\subsection*{Generating functions} +\begin{equation} +\label{GenMeixner1} +\left(1-\frac{t}{c}\right)^x(1-t)^{-x-\beta}= +\sum_{n=0}^{\infty}\frac{(\beta)_n}{n!}M_n(x;\beta,c)t^n. +\end{equation} + +\begin{equation} +\label{GenMeixner2} +\e^t\,\hyp{1}{1}{-x}{\beta}{\left(\frac{1-c}{c}\right)t}= +\sum_{n=0}^{\infty}\frac{M_n(x;\beta,c)}{n!}t^n. +\end{equation} + +\begin{equation} +\label{GenMeixner3} +(1-t)^{-\gamma}\,\hyp{2}{1}{\gamma,-x}{\beta}{\frac{(1-c)t}{c(1-t)}} +=\sum_{n=0}^{\infty}\frac{(\gamma)_n}{n!}M_n(x;\beta,c)t^n, +\quad\textrm{$\gamma$ arbitrary}. +\end{equation} + +\subsection*{Limit relations} + +\subsubsection*{Hahn $\rightarrow$ Meixner} +If we take $\alpha=b-1$, $\beta=N(1-c)c^{-1}$ in the definition (\ref{DefHahn}) +of the Hahn polynomials and let $N\rightarrow\infty$ we find the Meixner polynomials: +$$\lim_{N\rightarrow\infty} +Q_n(x;b-1,N(1-c)c^{-1},N)=M_n(x;b,c).$$ + +\subsubsection*{Dual Hahn $\rightarrow$ Meixner} +To obtain the Meixner polynomials from the dual Hahn polynomials we have to take +$\gamma=\beta-1$ and $\delta=N(1-c)c^{-1}$ in the definition (\ref{DefDualHahn}) of +the dual Hahn polynomials and let $N\rightarrow\infty$: +$$\lim_{N\rightarrow\infty} +R_n(\lambda(x);\beta-1,N(1-c)c^{-1},N)=M_n(x;\beta,c).$$ + +\subsubsection*{Meixner $\rightarrow$ Laguerre} +The Laguerre polynomials given by (\ref{DefLaguerre}) are obtained from the Meixner polynomials +if we take $\beta=\alpha+1$ and $x\rightarrow (1-c)^{-1}x$ and let $c\rightarrow 1$: +\begin{equation} +\lim_{c\rightarrow 1} +M_n((1-c)^{-1}x;\alpha+1,c)=\frac{L_n^{(\alpha)}(x)}{L_n^{(\alpha)}(0)}. +\end{equation} + +\subsubsection*{Meixner $\rightarrow$ Charlier} +The Charlier polynomials given by (\ref{DefCharlier}) are obtained from the Meixner polynomials +if we take $c=(a+\beta)^{-1}a$ and let $\beta\rightarrow\infty$: +\begin{equation} +\lim_{\beta\rightarrow\infty} +M_n(x;\beta,(a+\beta)^{-1}a)=C_n(x;a). +\end{equation} + +\subsection*{Remarks} +The Meixner polynomials are related to the Jacobi polynomials given by (\ref{DefJacobi}) +in the following way: +$$\frac{(\beta)_n}{n!}M_n(x;\beta,c)=P_n^{(\beta-1,-n-\beta-x)}((2-c)c^{-1}).$$ + +\noindent +The Meixner polynomials are also related to the Krawtchouk polynomials given by +(\ref{DefKrawtchouk}) in the following way: +$$K_n(x;p,N)=M_n(x;-N,(p-1)^{-1}p).$$ + +\subsection*{References} +\cite{Allaway76}, \cite{NAlSalam66}, \cite{AlSalam90}, \cite{AlSalamChihara76}, +\cite{AlSalamIsmail76}, \cite{Alvarez+}, \cite{AndrewsAskey85}, \cite{Area+II}, \cite{Askey75}, +\cite{Askey89I}, \cite{Askey2005}, \cite{AskeyGasper77}, \cite{AskeyIsmail76}, +\cite{AskeyWilson85}, \cite{AtakRahmanSuslov}, \cite{AtakSuslov88}, \cite{Bavinck98}, +\cite{BavinckHaeringen}, \cite{Campigotto+}, \cite{Chihara78}, \cite{Cooper+}, +\cite{Erdelyi+}, \cite{FoataLabelle}, \cite{Gabutti}, \cite{GabuttiMathis}, \cite{Gasper73I}, +\cite{Gasper74}, \cite{HoareRahman}, \cite{Ismail2005II}, \cite{IsmailLetVal88}, +\cite{IsmailLi}, \cite{IsmailMuldoon}, \cite{IsmailStanton97}, \cite{JinWong}, +\cite{Karlin58}, \cite{Koekoek2000}, \cite{Koorn88}, \cite{LabelleYehI}, \cite{LabelleYehII}, +\cite{Lesky89}, \cite{Lesky94I}, \cite{Lesky95II}, \cite{LewanowiczII}, \cite{Meixner}, +\cite{Nikiforov+}, \cite{NikiforovUvarov}, \cite{Rahman78I}, \cite{ValentAssche}, +\cite{Viennot}, \cite{Zarzo+}, \cite{Zeng90}. + + +\section{Krawtchouk}\index{Krawtchouk polynomials} + +\par\setcounter{equation}{0} + +\subsection*{Hypergeometric representation} +\begin{equation} +\label{DefKrawtchouk} +K_n(x;p,N)=\hyp{2}{1}{-n,-x}{-N}{\frac{1}{p}},\quad n=0,1,2,\ldots,N. +\end{equation} + +\subsection*{Orthogonality relation} +\begin{eqnarray} +\label{OrtKrawtchouk} +& &\sum_{x=0}^N\binom{N}{x}p^x(1-p)^{N-x} K_m(x;p,N)K_n(x;p,N)\nonumber\\ +& &{}=\frac{(-1)^nn!}{(-N)_n}\left(\frac{1-p}{p}\right)^n\,\delta_{mn},\quad0 < p < 1. +\end{eqnarray} + +\subsection*{Recurrence relation} +\begin{eqnarray} +\label{RecKrawtchouk} +-xK_n(x;p,N)&=&p(N-n)K_{n+1}(x;p,N)\nonumber\\ +& &{}\mathindent{}-\left[p(N-n)+n(1-p)\right]K_n(x;p,N)\nonumber\\ +& &{}\mathindent\mathindent{}+n(1-p)K_{n-1}(x;p,N). +\end{eqnarray} + +\subsection*{Normalized recurrence relation} +\begin{eqnarray} +\label{NormRecKrawtchouk} +xp_n(x)&=&p_{n+1}(x)+\left[p(N-n)+n(1-p)\right]p_n(x)\nonumber\\ +& &{}\mathindent{}+np(1-p)(N+1-n)p_{n-1}(x), +\end{eqnarray} +where +$$K_n(x;p,N)=\frac{1}{(-N)_np^n}p_n(x).$$ + +\subsection*{Difference equation} +\begin{eqnarray} +\label{dvKrawtchouk} +-ny(x)&=&p(N-x)y(x+1)\nonumber\\ +& &{}\mathindent{}-\left[p(N-x)+x(1-p)\right]y(x)+x(1-p)y(x-1), +\end{eqnarray} +where +$$y(x)=K_n(x;p,N).$$ + +\subsection*{Forward shift operator} +\begin{equation} +\label{shift1KrawtchoukI} +K_n(x+1;p,N)-K_n(x;p,N)=-\frac{n}{Np}K_{n-1}(x;p,N-1) +\end{equation} +or equivalently +\begin{equation} +\label{shift1KrawtchoukII} +\Delta K_n(x;p,N)=-\frac{n}{Np}K_{n-1}(x;p,N-1). +\end{equation} + +\subsection*{Backward shift operator} +\begin{eqnarray} +\label{shift2KrawtchoukI} +& &(N+1-x)K_n(x;p,N)-x\left(\frac{1-p}{p}\right)K_n(x-1;p,N)\nonumber\\ +& &{}=(N+1)K_{n+1}(x;p,N+1) +\end{eqnarray} +or equivalently +\begin{equation} +\label{shift2KrawtchoukII} +\nabla\left[\binom{N}{x}\left(\frac{p}{1-p}\right)^xK_n(x;p,N)\right]= +\binom{N+1}{x}\left(\frac{p}{1-p}\right)^xK_{n+1}(x;p,N+1). +\end{equation} + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodKrawtchouk} +\binom{N}{x}\left(\frac{p}{1-p}\right)^xK_n(x;p,N)= +\nabla^n\left[\binom{N-n}{x}\left(\frac{p}{1-p}\right)^x\right]. +\end{equation} + +\subsection*{Generating functions} +For $x=0,1,2,\ldots,N$ we have +\begin{equation} +\label{GenKrawtchouk1} +\left(1-\frac{(1-p)}{p}t\right)^x(1+t)^{N-x}= +\sum_{n=0}^N\binom{N}{n}K_n(x;p,N)t^n. +\end{equation} + +\begin{equation} +\label{GenKrawtchouk2} +\left[\expe^t\,\hyp{1}{1}{-x}{-N}{-\frac{t}{p}}\right]_N= +\sum_{n=0}^N\frac{K_n(x;p,N)}{n!}t^n. +\end{equation} + +\begin{eqnarray} +\label{GenKrawtchouk3} +& &\left[(1-t)^{-\gamma}\,\hyp{2}{1}{\gamma,-x}{-N}{\frac{t}{p(t-1)}}\right]_N\nonumber\\ +& &{}=\sum_{n=0}^N\frac{(\gamma)_n}{n!}K_n(x;p,N)t^n, +\quad\textrm{$\gamma$ arbitrary}. +\end{eqnarray} + +\subsection*{Limit relations} + +\subsubsection*{Hahn $\rightarrow$ Krawtchouk} +If we take $\alpha=pt$ and $\beta=(1-p)t$ in the definition (\ref{DefHahn}) of the Hahn +polynomials and let $t\rightarrow\infty$ we obtain the Krawtchouk polynomials: +$$\lim_{t\rightarrow\infty}Q_n(x;pt,(1-p)t,N)=K_n(x;p,N).$$ + +\subsubsection*{Dual Hahn $\rightarrow$ Krawtchouk} +The Krawtchouk polynomials follow from the dual Hahn polynomials given by +(\ref{DefDualHahn}) if we set $\gamma=pt$, $\delta=(1-p)t$ and let $t\rightarrow\infty$: +$$\lim_{t\rightarrow\infty}R_n(\lambda(x);pt,(1-p)t,N)=K_n(x;p,N).$$ + +\subsubsection*{Krawtchouk $\rightarrow$ Charlier} +The Charlier polynomials given by (\ref{DefCharlier}) can be found from the Krawtchouk +polynomials by taking $p=N^{-1}a$ and letting $N\rightarrow\infty$: +\begin{equation} +\lim_{N\rightarrow\infty}K_n(x;N^{-1}a,N)=C_n(x;a). +\end{equation} + +\subsubsection*{Krawtchouk $\rightarrow$ Hermite} +The Hermite polynomials given by (\ref{DefHermite}) follow from the Krawtchouk polynomials +by setting $x\rightarrow pN+x\sqrt{2p(1-p)N}$ and then letting $N\rightarrow\infty$: +\begin{equation} +\lim_{N\rightarrow\infty} +\sqrt{\binom{N}{n}}K_n(pN+x\sqrt{2p(1-p)N};p,N) +=\frac{\displaystyle (-1)^nH_n(x)}{\displaystyle\sqrt{2^nn!\left(\frac{p}{1-p}\right)^n}}. +\end{equation} + +\subsection*{Remarks} +The Krawtchouk polynomials are self-dual, which means that +$$K_n(x;p,N)=K_x(n;p,N),\quad n,x\in\{0,1,2,\ldots,N\}.$$ +By using this relation we easily obtain the so-called dual orthogonality +relation from the orthogonality relation (\ref{OrtKrawtchouk}): +$$\sum_{n=0}^N\binom{N}{n}p^n(1-p)^{N-n} K_n(x;p,N)K_n(y;p,N)= +\frac{\displaystyle\left(\frac{1-p}{p}\right)^x}{\dbinom{N}{x}}\delta_{xy},$$ +where $0 < p < 1$ and $x,y\in\{0,1,2,\ldots,N\}$. + +\noindent +The Krawtchouk polynomials are related to the Meixner polynomials given by (\ref{DefMeixner}) +in the following way: +$$K_n(x;p,N)=M_n(x;-N,(p-1)^{-1}p).$$ + +\subsection*{References} +\cite{AlSalam90}, \cite{AndrewsAskey85}, \cite{Area+II}, \cite{Askey75}, \cite{Askey89I}, +\cite{AskeyGasper77}, \cite{AskeyWilson85}, \cite{AtakRahmanSuslov}, \cite{Campigotto+}, +\cite{LChiharaStanton}, \cite{Chihara78}, \cite{Dette95}, \cite{Dominici}, \cite{Dunkl76}, +\cite{Dunkl84}, \cite{DunklRamirez}, \cite{Erdelyi+}, \cite{FeinsilverSchott}, +\cite{Gasper73I}, \cite{Gasper74}, \cite{HoareRahman}, \cite{Ismail2005II}, \cite{Karlin58}, +\cite{Koorn82}, \cite{Koorn88}, \cite{LabelleYehI}, \cite{LabelleYehII}, \cite{Lesky62}, +\cite{Lesky89}, \cite{Lesky94I}, \cite{Lesky95II}, \cite{LewanowiczII}, \cite{Nikiforov+}, +\cite{NikiforovUvarov}, \cite{Qiu}, \cite{Rahman78I}, \cite{Rahman79}, \cite{Stanton84}, +\cite{Stanton90}, \cite{Szego75}, \cite{Zarzo+}, \cite{Zeng90}. + + +\section{Laguerre}\index{Laguerre polynomials} + +\par\setcounter{equation}{0} + +\subsection*{Hypergeometric representation} +\begin{equation} +\label{DefLaguerre} +L_n^{(\alpha)}(x)=\frac{(\alpha+1)_n}{n!}\,\hyp{1}{1}{-n}{\alpha+1}{x}. +\end{equation} + +\subsection*{Orthogonality relation} +\begin{equation} +\label{OrtLaguerre} +\int_0^{\infty}\expe^{-x}x^{\alpha}L_m^{(\alpha)}(x)L_n^{(\alpha)}(x)\,dx= +\frac{\Gamma(n+\alpha+1)}{n!}\,\delta_{mn},\quad\alpha > -1. +\end{equation} + +\subsection*{Recurrence relation} +\begin{equation} +\label{RecLaguerre} +(n+1)L_{n+1}^{(\alpha)}(x)-(2n+\alpha+1-x)L_n^{(\alpha)}(x)+(n+\alpha)L_{n-1}^{(\alpha)}(x)=0. +\end{equation} + +\subsection*{Normalized recurrence relation} +\begin{equation} +\label{NormRecLaguerre} +xp_n(x)=p_{n+1}(x)+(2n+\alpha+1)p_n(x)+n(n+\alpha)p_{n-1}(x), +\end{equation} +where +$$L_n^{(\alpha)}(x)=\frac{(-1)^n}{n!}p_n(x).$$ + +\subsection*{Differential equation} +\begin{equation} +\label{dvLaguerre} +xy''(x)+(\alpha+1-x)y'(x)+ny(x)=0,\quad y(x)=L_n^{(\alpha)}(x). +\end{equation} + +\newpage + +\subsection*{Forward shift operator} +\begin{equation} +\label{shift1Laguerre} +\frac{d}{dx}L_n^{(\alpha)}(x)=-L_{n-1}^{(\alpha+1)}(x). +\end{equation} + +\subsection*{Backward shift operator} +\begin{equation} +\label{shift2LaguerreI} +x\frac{d}{dx}L_n^{(\alpha)}(x)+(\alpha-x)L_n^{(\alpha)}(x)=(n+1)L_{n+1}^{(\alpha-1)}(x) +\end{equation} +or equivalently +\begin{equation} +\label{shift2LaguerreII} +\frac{d}{dx}\left[\expe^{-x}x^{\alpha}L_n^{(\alpha)}(x)\right]=(n+1)\expe^{-x}x^{\alpha-1}L^{(\alpha-1)}_{n+1}(x). +\end{equation} + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodLaguerre} +\e^{-x}x^{\alpha}L_n^{(\alpha)}(x)=\frac{1}{n!}\left(\frac{d}{dx}\right)^n\left[\expe^{-x}x^{n+\alpha}\right]. +\end{equation} + +\subsection*{Generating functions} +\begin{equation} +\label{GenLaguerre1} +(1-t)^{-\alpha-1}\exp\left(\frac{xt}{t-1}\right)= +\sum_{n=0}^{\infty}L_n^{(\alpha)}(x)t^n. +\end{equation} + +\begin{equation} +\label{GenLaguerre2} +\e^t\,\hyp{0}{1}{-}{\alpha+1}{-xt} +=\sum_{n=0}^{\infty}\frac{L_n^{(\alpha)}(x)}{(\alpha+1)_n}t^n. +\end{equation} + +\begin{equation} +\label{GenLaguerre3} +(1-t)^{-\gamma}\,\hyp{1}{1}{\gamma}{\alpha+1}{\frac{xt}{t-1}} +=\sum_{n=0}^{\infty}\frac{(\gamma)_n}{(\alpha+1)_n}L_n^{(\alpha)}(x)t^n, +\quad\textrm{$\gamma$ arbitrary}. +\end{equation} + +\subsection*{Limit relations} + +\subsubsection*{Meixner-Pollaczek $\rightarrow$ Laguerre} +The Laguerre polynomials can be obtained from the Meixner-Pollaczek polynomials given by +(\ref{DefMP}) by the substitution $\lambda=\frac{1}{2}(\alpha+1)$, +$x\rightarrow -\frac{1}{2}\phi^{-1}x$ and the limit $\phi\rightarrow 0$: +$$\lim_{\phi\rightarrow 0} +P_n^{(\frac{1}{2}\alpha+\frac{1}{2})}(-\textstyle\frac{1}{2}\phi^{-1}x;\phi)=L_n^{(\alpha)}(x).$$ + +\subsubsection*{Jacobi $\rightarrow$ Laguerre} +The Laguerre polynomials are obtained from the Jacobi polynomials given by (\ref{DefJacobi}) +if we set $x\rightarrow 1-2\beta^{-1}x$ and then take the limit $\beta\rightarrow\infty$: +$$\lim_{\beta\rightarrow\infty} +P_n^{(\alpha,\beta)}(1-2\beta^{-1}x)=L_n^{(\alpha)}(x).$$ + +\subsubsection*{Meixner $\rightarrow$ Laguerre} +If we take $\beta=\alpha+1$ and $x\rightarrow (1-c)^{-1}x$ in the definition +(\ref{DefMeixner}) of the Meixner polynomials and let $c\rightarrow 1$ we obtain +the Laguerre polynomials: +$$\lim_{c\rightarrow 1} +M_n((1-c)^{-1}x;\alpha+1,c)=\frac{L_n^{(\alpha)}(x)}{L_n^{(\alpha)}(0)}.$$ + +\subsubsection*{Laguerre $\rightarrow$ Hermite} +The Hermite polynomials given by (\ref{DefHermite}) can be obtained from the Laguerre +polynomials by taking the limit $\alpha\rightarrow\infty$ in the following way: +\begin{equation} +\lim_{\alpha\rightarrow\infty} +\left(\frac{2}{\alpha}\right)^{\frac{1}{2}n} +L_n^{(\alpha)}((2\alpha)^{\frac{1}{2}}x+\alpha)=\frac{(-1)^n}{n!}H_n(x). +\end{equation} + +\subsection*{Remarks} +The definition (\ref{DefLaguerre}) of the Laguerre polynomials can also be +written as: +$$L_n^{(\alpha)}(x)=\frac{1}{n!}\sum_{k=0}^n\frac{(-n)_k}{k!}(\alpha+k+1)_{n-k}x^k.$$ +In this way the Laguerre polynomials can also be seen as polynomials in the parameter $\alpha$. +Therefore they can be defined for all $\alpha$. + +\noindent +The Laguerre polynomials are related to the Bessel polynomials given by (\ref{DefBessel}) +in the following way: +$$L_n^{(\alpha)}(x)=\frac{(-x)^n}{n!}y_n(2x^{-1};-2n-\alpha-1).$$ + +\noindent +The Laguerre polynomials are related to the Charlier polynomials given by (\ref{DefCharlier}) +in the following way: +$$\frac{(-a)^n}{n!}C_n(x;a)=L_n^{(x-n)}(a).$$ + +\noindent +The Laguerre polynomials and the Hermite polynomials given by (\ref{DefHermite}) are also +connected by the following quadratic transformations: +$$H_{2n}(x)=(-1)^nn!\,2^{2n}L_n^{(-\frac{1}{2})}(x^2)$$ +and +$$H_{2n+1}(x)=(-1)^nn!\,2^{2n+1}xL_n^{(\frac{1}{2})}(x^2).$$ + +\noindent +In combinatorics the Laguerre polynomials with $\alpha=0$ are often called Rook +polynomials. + +\subsection*{References} +\cite{Abdul}, \cite{Abram}, \cite{Ahmed+82}, \cite{Allaway76}, \cite{Allaway91}, +\cite{NAlSalam66}, \cite{AlSalam64}, \cite{AlSalam90}, \cite{AlSalamChihara72}, +\cite{AlSalamChihara76}, \cite{AndrewsAskey85}, \cite{AndrewsAskeyRoy}, \cite{Askey68}, +\cite{Askey75}, \cite{Askey89I}, \cite{Askey2005}, \cite{AskeyGasper76}, \cite{AskeyGasper77}, +\cite{AskeyIsmail76}, \cite{AskeyIsmailKoorn}, \cite{AskeyWilson85}, \cite{Barrett}, +\cite{Bavinck96}, \cite{Brafman51}, \cite{Brafman57II}, \cite{Brenke}, \cite{Brown}, +\cite{BustozSavage79}, \cite{BustozSavage80}, \cite{Carlitz60}, \cite{Carlitz61I}, +\cite{Carlitz61II}, \cite{Carlitz62}, \cite{Carlitz68}, \cite{ChenSrivastava}, +\cite{ChenIsmail91}, \cite{Chihara68II}, \cite{Chihara78}, \cite{Cohen}, \cite{Cooper+}, +\cite{Dattoli2001}, \cite{DetteStudden92}, \cite{DetteStudden95}, \cite{Dimitrov2003}, +\cite{Doha2003II}, \cite{ElbertLaforgia87I}, \cite{Erdelyi}, \cite{Erdelyi+}, \cite{Exton98}, +\cite{Faldey}, \cite{Gasper73II}, \cite{Gasper77}, \cite{Gatteschi2002}, \cite{Gawronski87}, +\cite{Gawronski93}, \cite{Gillis}, \cite{GillisIsmailOffer}, \cite{Godoy+}, \cite{Grad}, +\cite{Hajmirzaahmad95}, \cite{Ismail74}, \cite{Ismail77}, \cite{Ismail2005II}, +\cite{IsmailLetVal88}, \cite{IsmailLi}, \cite{IsmailMassonRahman}, \cite{IsmailMuldoon}, +\cite{IsmailStanton97}, \cite{IsmailTamhankar}, \cite{Jackson}, \cite{Karlin58}, +\cite{Kochneff95}, \cite{Kochneff97II}, \cite{Koekoek2000}, \cite{Koorn77I}, \cite{Koorn78}, +\cite{Koorn85}, \cite{Koorn88}, \cite{Krasikov2003}, \cite{Krasikov2005}, +\cite{KuijlaarsMcLaughlin}, \cite{KwonLittle}, \cite{LabelleYehI}, \cite{LabelleYehII}, +\cite{LabelleYehIII}, \cite{Lee97I}, \cite{Lee97II}, \cite{Lee97III}, \cite{Lesky95II}, +\cite{Lesky96}, \cite{LewanowiczI}, \cite{LopezTemme2004}, \cite{Mathai}, \cite{Meixner}, +\cite{Nikiforov+}, \cite{NikiforovUvarov}, \cite{Olver}, \cite{PerezPinar}, \cite{Pittaluga}, +\cite{Rainville}, \cite{SainteViennot}, \cite{Srivastava66}, \cite{Srivastava69I}, +\cite{Srivastava69II}, \cite{Srivastava70}, \cite{Srivastava71}, \cite{Szego75}, \cite{Temme}, +\cite{Trickovic}, \cite{Trivedi}, \cite{Viennot}. + + +\section{Bessel}\index{Bessel polynomials} + +\par\setcounter{equation}{0} + +\subsection*{Hypergeometric representation} +\begin{eqnarray} +\label{DefBessel} +y_n(x;a)&=&\hyp{2}{0}{-n,n+a+1}{-}{-\frac{x}{2}}\\ +&=&(n+a+1)_n\left(\frac{x}{2}\right)^n\,\hyp{1}{1}{-n}{-2n-a}{\frac{2}{x}}, +\quad n=0,1,2,\ldots,N.\nonumber +\end{eqnarray} + +\subsection*{Orthogonality relation} +\begin{eqnarray} +\label{OrtBessel} +& &\int_0^{\infty}x^a\e^{-\frac{2}{x}}y_m(x;a)y_n(x;a)\,dx\nonumber\\ +& &=-\frac{2^{a+1}}{2n+a+1}\Gamma(-n-a)n!\,\delta_{mn},\quad a<-2N-1. +\end{eqnarray} + +\subsection*{Recurrence relation} +\begin{eqnarray} +\label{RecBessel} +& &2(n+a+1)(2n+a)y_{n+1}(x;a)\nonumber\\ +& &{}=(2n+a+1)\left[2a+(2n+a)(2n+a+2)x\right]y_n(x;a)\nonumber\\ +& &{}\mathindent{}+2n(2n+a+2)y_{n-1}(x;a). +\end{eqnarray} + +\subsection*{Normalized recurrence relation} +\begin{eqnarray} +\label{NormRecBessel} +xp_n(x)&=&p_{n+1}(x)-\frac{2a}{(2n+a)(2n+a+2)}p_n(x)\nonumber\\ +& &{}\mathindent{}-\frac{4n(n+a)}{(2n+a-1)(2n+a)^2(2n+a+1)}p_{n-1}(x), +\end{eqnarray} +where +$$y_n(x;a)=\frac{(n+a+1)_n}{2^n}p_n(x).$$ + +\subsection*{Differential equation} +\begin{equation} +\label{dvBessel} +x^2y''(x)+\left[(a+2)x+2\right]y'(x)-n(n+a+1)y(x)=0,\quad y(x)=y_n(x;a). +\end{equation} + +\subsection*{Forward shift operator} +\begin{equation} +\label{shift1Bessel} +\frac{d}{dx}y_n(x;a)=\frac{n(n+a+1)}{2}y_{n-1}(x;a+2). +\end{equation} + +\subsection*{Backward shift operator} +\begin{equation} +\label{shift2BesselI} +x^2\frac{d}{dx}y_n(x;a)+(ax+2)y_n(x;a)=2y_{n+1}(x;a-2) +\end{equation} +or equivalently +\begin{equation} +\label{shift2BesselII} +\frac{d}{dx}\left[x^a\expe^{-\frac{2}{x}}y_n(x;a)\right]=2x^{a-2}\expe^{-\frac{2}{x}}y_{n+1}(x;a-2). +\end{equation} + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodBessel} +y_n(x;a)=2^{-n}x^{-a}\expe^{\frac{2}{x}}D^n\left(x^{2n+a}\expe^{-\frac{2}{x}}\right). +\end{equation} + +\subsection*{Generating function} +\begin{equation} +\label{GenBessel} +\left(1-2xt\right)^{-\frac{1}{2}}\left(\frac{2}{1+\sqrt{1-2xt}}\right)^a +\exp\left(\frac{2t}{1+\sqrt{1-2xt}}\right)=\sum_{n=0}^{\infty}y_n(x;a)\frac{t^n}{n!}. +\end{equation} + +\subsection*{Limit relation} + +\subsubsection*{Jacobi $\rightarrow$ Bessel} +If we take $\beta=a-\alpha$ in the definition (\ref{DefJacobi}) of the Jacobi polynomials +and let $\alpha\rightarrow -\infty$ we find the Bessel polynomials: +$$\lim_{\alpha\rightarrow -\infty} +\frac{P_n^{(\alpha,a-\alpha)}(1+\alpha x)}{P_n^{(\alpha,a-\alpha)}(1)}=y_n(x;a).$$ + +\subsection*{Remarks} +The following notations are also used for the Bessel polynomials: +$$y_n(x;a,b)=y_n(2b^{-1}x;a)\quad\textrm{and}\quad\theta_n(x;a,b)=x^ny_n(x^{-1};a,b).$$ +However, the Bessel polynomials essentially depend on only one parameter. + +\noindent +The Bessel polynomials are related to the Laguerre polynomials given by (\ref{DefLaguerre}) +in the following way: +$$L_n^{(\alpha)}(x)=\frac{(-x)^n}{n!}y_n(2x^{-1};-2n-\alpha-1).$$ + +\noindent +The special case $a=-2N-2$ of the Bessel polynomials can be obtained from the pseudo Jacobi +polynomials by setting $x\rightarrow\nu x$ in the definition (\ref{DefPseudoJacobi}) of the +pseudo Jacobi polynomials and taking the limit $\nu\rightarrow\infty$ in the following way: +$$\lim\limits_{\nu\rightarrow\infty}\frac{P_n(\nu x;\nu,N)}{\nu^n} +=\frac{2^n}{(n-2N-1)_n}y_n(x;-2N-2).$$ + +\subsection*{References} +\cite{Andrade}, \cite{BergVignat}, \cite{Carlitz57I}, \cite{Dattoli2003}, +\cite{DohaAhmed2004}, \cite{DohaAhmed2006}, \cite{Grosswald}, \cite{Ismail2005II}, +\cite{KrallFrink}, \cite{Lesky98}, \cite{NikiforovUvarov}. + + +\section{Charlier}\index{Charlier polynomials} + +\par\setcounter{equation}{0} + +\subsection*{Hypergeometric representation} +\begin{equation} +\label{DefCharlier} +C_n(x;a)=\hyp{2}{0}{-n,-x}{-}{-\frac{1}{a}}. +\end{equation} + +\subsection*{Orthogonality relation} +\begin{equation} +\label{OrtCharlier} +\sum_{x=0}^{\infty}\frac{a^x}{x!}C_m(x;a)C_n(x;a)= +a^{-n}\expe^an!\,\delta_{mn},\quad a > 0. +\end{equation} + +\subsection*{Recurrence relation} +\begin{equation} +\label{RecCharlier} +-xC_n(x;a)=aC_{n+1}(x;a)-(n+a)C_n(x;a)+nC_{n-1}(x;a). +\end{equation} + +\subsection*{Normalized recurrence relation} +\begin{equation} +\label{NormRecCharlier} +xp_n(x)=p_{n+1}(x)+(n+a)p_n(x)+nap_{n-1}(x), +\end{equation} +where +$$C_n(x;a)=\left(-\frac{1}{a}\right)^np_n(x).$$ + +\subsection*{Difference equation} +\begin{equation} +\label{dvCharlier} +-ny(x)=ay(x+1)-(x+a)y(x)+xy(x-1),\quad y(x)=C_n(x;a). +\end{equation} + +\subsection*{Forward shift operator} +\begin{equation} +\label{shift1CharlierI} +C_n(x+1;a)-C_n(x;a)=-\frac{n}{a}C_{n-1}(x;a) +\end{equation} +or equivalently +\begin{equation} +\label{shift1CharlierII} +\Delta C_n(x;a)=-\frac{n}{a}C_{n-1}(x;a). +\end{equation} + +\subsection*{Backward shift operator} +\begin{equation} +\label{shift2CharlierI} +C_n(x;a)-\frac{x}{a}C_n(x-1;a)=C_{n+1}(x;a) +\end{equation} +or equivalently +\begin{equation} +\label{shift2CharlierII} +\nabla\left[\frac{a^x}{x!}C_n(x;a)\right]=\frac{a^x}{x!}C_{n+1}(x;a). +\end{equation} + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodCharlier} +\frac{a^x}{x!}C_n(x;a)=\nabla^n\left[\frac{a^x}{x!}\right]. +\end{equation} + +\subsection*{Generating function} +\begin{equation} +\label{GenCharlier} +\e^t\left(1-\frac{t}{a}\right)^x=\sum_{n=0}^{\infty}\frac{C_n(x;a)}{n!}t^n. +\end{equation} + +\subsection*{Limit relations} + +\subsubsection*{Meixner $\rightarrow$ Charlier} +If we take $c=(a+\beta)^{-1}a$ in the definition (\ref{DefMeixner}) of the Meixner polynomials +and let $\beta\rightarrow\infty$ we find the Charlier polynomials: +$$\lim_{\beta\rightarrow\infty}M_n(x;\beta,(a+\beta)^{-1}a)=C_n(x;a).$$ + +\subsubsection*{Krawtchouk $\rightarrow$ Charlier} +The Charlier polynomials can be found from the Krawtchouk polynomials given by +(\ref{DefKrawtchouk}) by taking $p=N^{-1}a$ and letting $N\rightarrow\infty$: +$$\lim_{N\rightarrow\infty}K_n(x;N^{-1}a,N)=C_n(x;a).$$ + +\subsubsection*{Charlier $\rightarrow$ Hermite} +The Hermite polynomials given by (\ref{DefHermite}) are obtained from the Charlier polynomials +if we set $x\rightarrow (2a)^{1/2}x+a$ and let $a\rightarrow\infty$. In fact we have +\begin{equation} +\lim_{a\rightarrow\infty} +(2a)^{\frac{1}{2}n}C_n((2a)^{\frac{1}{2}}x+a;a)=(-1)^nH_n(x). +\end{equation} + +\subsection*{Remark} +The Charlier polynomials are related to the Laguerre polynomials given by (\ref{DefLaguerre}) +in the following way: +$$\frac{(-a)^n}{n!}C_n(x;a)=L_n^{(x-n)}(a).$$ + +\subsection*{References} +\cite{Allaway76}, \cite{NAlSalam66}, \cite{AlSalam90}, \cite{AlSalamChihara76}, +\cite{AlSalamIsmail76}, \cite{AndrewsAskey85}, \cite{Area+II}, \cite{Askey75}, +\cite{AskeyGasper77}, \cite{AskeyWilson85}, \cite{AtakRahmanSuslov}, \cite{Bavinck98}, +\cite{BavinckKoekoek}, \cite{Chihara78}, \cite{Chihara79}, \cite{Dunkl76}, \cite{Erdelyi+}, +\cite{Gasper73I}, \cite{Gasper74}, \cite{Goh}, \cite{HoareRahman}, \cite{IsmailLetVal88}, +\cite{Koekoek2000}, \cite{Koorn88}, \cite{Krasikov2002}, \cite{LabelleYehI}, +\cite{LabelleYehII}, \cite{LabelleYehIII}, \cite{Lesky62}, \cite{Lesky89}, \cite{Lesky94I}, +\cite{Lesky95II}, \cite{LewanowiczII}, \cite{LopezTemme2004}, \cite{Meixner}, \cite{Nikiforov+}, +\cite{NikiforovUvarov}, \cite{Szafraniec}, \cite{Szego75}, \cite{Viennot}, \cite{Zarzo+}, +\cite{Zeng90}. + + +\section{Hermite}\index{Hermite polynomials} + +\par\setcounter{equation}{0} + +\subsection*{Hypergeometric representation} +\begin{equation} +\label{DefHermite} +H_n(x)=(2x)^n\,\hyp{2}{0}{-n/2,-(n-1)/2}{-}{-\frac{1}{x^2}}. +\end{equation} + +\subsection*{Orthogonality relation} +\begin{equation} +\label{OrtHermite} +\frac{1}{\sqrt{\cpi}}\int_{-\infty}^{\infty}\expe^{-x^2}H_m(x)H_n(x)\,dx +=2^nn!\,\delta_{mn}. +\end{equation} + +\subsection*{Recurrence relation} +\begin{equation} +\label{RecHermite} +H_{n+1}(x)-2xH_n(x)+2nH_{n-1}(x)=0. +\end{equation} + +\subsection*{Normalized recurrence relation} +\begin{equation} +\label{NormRecHermite} +xp_n(x)=p_{n+1}(x)+\frac{n}{2}p_{n-1}(x), +\end{equation} +where +$$H_n(x)=2^np_n(x).$$ + +\subsection*{Differential equation} +\begin{equation} +\label{dvHermite} +y''(x)-2xy'(x)+2ny(x)=0,\quad y(x)=H_n(x). +\end{equation} + +\subsection*{Forward shift operator} +\begin{equation} +\label{shift1Hermite} +\frac{d}{dx}H_n(x)=2nH_{n-1}(x). +\end{equation} + +\subsection*{Backward shift operator} +\begin{equation} +\label{shift2HermiteI} +\frac{d}{dx}H_n(x)-2xH_n(x)=-H_{n+1}(x) +\end{equation} +or equivalently +\begin{equation} +\label{shift2HermiteII} +\frac{d}{dx}\left[\expe^{-x^2}H_n(x)\right]=-\expe^{-x^2}H_{n+1}(x). +\end{equation} + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodHermite} +\e^{-x^2}H_n(x)=(-1)^n\left(\frac{d}{dx}\right)^n\left[\expe^{-x^2}\right]. +\end{equation} + +\subsection*{Generating functions} +\begin{equation} +\label{GenHermite1} +\exp\left(2xt-t^2\right)=\sum_{n=0}^{\infty}\frac{H_n(x)}{n!}t^n. +\end{equation} + +\begin{equation} +\label{GenHermite2} +\left\{\begin{array}{l} +\displaystyle\expe^t\cos(2x\sqrt{t})=\sum_{n=0}^{\infty} +\frac{(-1)^n}{(2n)!}H_{2n}(x)t^n\\[5mm] +\displaystyle\frac{\expe^t}{\sqrt{t}}\sin(2x\sqrt{t})=\sum_{n=0}^{\infty} +\frac{(-1)^n}{(2n+1)!}H_{2n+1}(x)t^n. +\end{array}\right. +\end{equation} + +\begin{equation} +\label{GenHermite3} +\left\{\begin{array}{l} +\displaystyle \expe^{-t^2}\cosh(2xt)=\sum_{n=0}^{\infty} +\frac{H_{2n}(x)}{(2n)!}t^{2n}\\[5mm] +\displaystyle\expe^{-t^2}\sinh(2xt)=\sum_{n=0}^{\infty} +\frac{H_{2n+1}(x)}{(2n+1)!}t^{2n+1}. +\end{array}\right. +\end{equation} + +\begin{equation} +\label{GenHermite4} +\left\{\begin{array}{l} +\displaystyle (1+t^2)^{-\gamma}\,\hyp{1}{1}{\gamma}{\frac{1}{2}}{\frac{x^2t^2}{1+t^2}}= +\sum_{n=0}^{\infty}\frac{(\gamma)_n}{(2n)!}H_{2n}(x)t^{2n}\\[5mm] +\displaystyle\frac{xt}{\sqrt{1+t^2}}\, +\hyp{1}{1}{\gamma+\frac{1}{2}}{\frac{3}{2}}{\frac{x^2t^2}{1+t^2}} +=\sum_{n=0}^{\infty}\frac{(\gamma+\frac{1}{2})_n}{(2n+1)!}H_{2n+1}(x)t^{2n+1} +\end{array}\right. +\end{equation} +with $\gamma$ arbitrary. + +\begin{equation} +\label{GenHermite5} +\frac{1+2xt+4t^2}{(1+4t^2)^{\frac{3}{2}}}\exp\left(\frac{4x^2t^2}{1+4t^2}\right) +=\sum_{n=0}^{\infty}\frac{H_n(x)}{\lfloor n/2\rfloor\,!}t^n, +\end{equation} +where $\lfloor n/2\rfloor$ denotes the largest integer smaller than or equal to $n/2$. + +\subsection*{Limit relations} + +\subsubsection*{Meixner-Pollaczek $\rightarrow$ Hermite} +If we take $x\rightarrow (\sin\phi)^{-1}(x\sqrt{\lambda}-\lambda\cos\phi)$ +in the definition (\ref{DefMP}) of the Meixner-Pollaczek polynomials and +then let $\lambda\rightarrow\infty$ we obtain the Hermite polynomials: +$$\lim_{\lambda\rightarrow\infty} +\lambda^{-\frac{1}{2}n}P_n^{(\lambda)} +((\sin\phi)^{-1}(x\sqrt{\lambda}-\lambda\cos\phi);\phi)=\frac{H_n(x)}{n!}.$$ + +\subsubsection*{Jacobi $\rightarrow$ Hermite} +The Hermite polynomials follow from the Jacobi polynomials given by (\ref{DefJacobi}) by +taking $\beta=\alpha$ and letting $\alpha\rightarrow\infty$ in the following way: +$$\lim_{\alpha\rightarrow\infty} +\alpha^{-\frac{1}{2}n}P_n^{(\alpha,\alpha)}(\alpha^{-\frac{1}{2}}x)=\frac{H_n(x)}{2^nn!}.$$ + +\subsubsection*{Gegenbauer / Ultraspherical $\rightarrow$ Hermite} +The Hermite polynomials follow from the Gegenbauer (or ultraspherical) polynomials given by +(\ref{DefGegenbauer}) by taking $\lambda=\alpha+\frac{1}{2}$ and letting +$\alpha\rightarrow\infty$ in the following way: +$$\lim_{\alpha\rightarrow\infty} +\alpha^{-\frac{1}{2}n}C_n^{(\alpha+\frac{1}{2})}(\alpha^{-\frac{1}{2}}x)=\frac{H_n(x)}{n!}.$$ + +\subsubsection*{Krawtchouk $\rightarrow$ Hermite} +The Hermite polynomials follow from the Krawtchouk polynomials given by (\ref{DefKrawtchouk}) +by setting $x\rightarrow pN+x\sqrt{2p(1-p)N}$ and then letting $N\rightarrow\infty$: +$$\lim_{N\rightarrow\infty} +\sqrt{\binom{N}{n}}K_n(pN+x\sqrt{2p(1-p)N};p,N) +=\frac{\displaystyle (-1)^nH_n(x)}{\displaystyle\sqrt{2^nn!\left(\frac{p}{1-p}\right)^n}}.$$ + +\subsubsection*{Laguerre $\rightarrow$ Hermite} +The Hermite polynomials can be obtained from the Laguerre polynomials given by +(\ref{DefLaguerre}) by taking the limit $\alpha\rightarrow\infty$ in the following way: +$$\lim_{\alpha\rightarrow\infty} +\left(\frac{2}{\alpha}\right)^{\frac{1}{2}n} +L_n^{(\alpha)}((2\alpha)^{\frac{1}{2}}x+\alpha)=\frac{(-1)^n}{n!}H_n(x).$$ + +\subsubsection*{Charlier $\rightarrow$ Hermite} +If we set $x\rightarrow (2a)^{1/2}x+a$ in the definition (\ref{DefCharlier}) +of the Charlier polynomials and let $a\rightarrow\infty$ we find the Hermite +polynomials. In fact we have +$$\lim_{a\rightarrow\infty} +(2a)^{\frac{1}{2}n}C_n((2a)^{\frac{1}{2}}x+a;a)=(-1)^nH_n(x).$$ + +\subsection*{Remarks} +The Hermite polynomials can also be written as: +$$\frac{H_n(x)}{n!}=\sum_{k=0}^{\lfloor n/2\rfloor} +\frac{(-1)^k(2x)^{n-2k}}{k!\,(n-2k)!},$$ +where $\lfloor n/2\rfloor$ denotes the largest integer smaller than or equal to $n/2$. + +\noindent +The Hermite polynomials and the Laguerre polynomials given by (\ref{DefLaguerre}) are also +connected by the following quadratic transformations: +$$H_{2n}(x)=(-1)^nn!\,2^{2n}L_n^{(-\frac{1}{2})}(x^2)$$ +and +$$H_{2n+1}(x)=(-1)^nn!\,2^{2n+1}xL_n^{(\frac{1}{2})}(x^2).$$ + +\subsection*{References} +\cite{Abram}, \cite{NAlSalam66}, \cite{AlSalam90}, \cite{AlSalamChihara72}, +\cite{AlSalamChihara76}, \cite{AndrewsAskey85}, \cite{AndrewsAskeyRoy}, \cite{Area+I}, +\cite{Askey68}, \cite{Askey75}, \cite{Askey89I}, \cite{AskeyGasper76}, \cite{AskeyWilson85}, +\cite{Azor}, \cite{Berg}, \cite{BilodeauII}, \cite{Brafman51}, \cite{Brafman57II}, +\cite{Brenke}, \cite{CarlitzSrivastava}, \cite{ChenSrivastava}, \cite{Chihara78}, +\cite{Cohen}, \cite{Danese}, \cite{DetteStudden92}, \cite{DetteStudden95}, \cite{Doha2004I}, +\cite{Erdelyi+}, \cite{Faldey}, \cite{Gawronski87}, \cite{Gawronski93}, \cite{Gillis+}, +\cite{Godoy+}, \cite{Grad}, \cite{Ismail74}, \cite{Ismail2005II}, \cite{IsmailStanton97}, +\cite{Koekoek2000}, \cite{Koorn88}, \cite{Krasikov2004}, \cite{Kwon+}, \cite{LabelleYehIII}, +\cite{Lesky95II}, \cite{Lesky96}, \cite{LewanowiczI}, \cite{LopezTemme99}, \cite{Mathai}, +\cite{Meixner}, \cite{Nikiforov+}, \cite{NikiforovUvarov}, \cite{Olver}, \cite{Pittaluga}, +\cite{Rainville}, \cite{SainteViennot}, \cite{Srivastava71}, \cite{SrivastavaMathur}, +\cite{Szego75}, \cite{Temme}, \cite{TemmeLopez2000}, \cite{Trickovic}, \cite{Viennot}, +\cite{Weisner59}, \cite{Wyman}. + +\end{document} diff --git a/KLSadd_insertion/chap14.tex b/KLSadd_insertion/chap14.tex new file mode 100644 index 0000000..0b5f396 --- /dev/null +++ b/KLSadd_insertion/chap14.tex @@ -0,0 +1,5734 @@ +\documentclass[envcountchap,graybox]{svmono} + +\addtolength{\textwidth}{1mm} + +\usepackage{amsmath,amssymb} + +\usepackage{amsfonts} +%\usepackage{breqn} +\usepackage{DLMFmath} +\usepackage{DRMFfcns} + +\usepackage{mathptmx} +\usepackage{helvet} +\usepackage{courier} + +\usepackage{makeidx} +\usepackage{graphicx} + +\usepackage{multicol} +\usepackage[bottom]{footmisc} + +\makeindex + +\def\bibname{Bibliography} +\def\refname{Bibliography} + +\def\theequation{\thesection.\arabic{equation}} + +\smartqed + +\let\corollary=\undefined +\spnewtheorem{corollary}[theorem]{Corollary}{\bfseries}{\itshape} + +\newcounter{rom} + +\newcommand{\hyp}[5]{\mbox{}_{#1}{F}_{#2} +\left(\genfrac{}{}{0pt}{}{#3}{#4}\,;\,#5\right)} + +\newcommand{\qhyp}[5]{\mbox{}_{#1}{\phi}_{#2} +\left(\genfrac{}{}{0pt}{}{#3}{#4}\,;\,q,\,#5\right)} + +\newcommand{\qhypK}[5]{\,\mbox{}_{#1}\phi_{#2}\!\left( + \genfrac{}{}{0pt}{}{#3}{#4};#5\right)} + +\newcommand{\mathindent}{\hspace{7.5mm}} + +\newcommand{\e}{\textrm{e}} + +\renewcommand{\E}{\textrm{E}} + +\renewcommand{\textfraction}{-1} + +\renewcommand{\Gamma}{\varGamma} + +\renewcommand{\leftlegendglue}{\hfil} + +\settowidth{\tocchpnum}{14\enspace} +\settowidth{\tocsecnum}{14.30\enspace} +\settowidth{\tocsubsecnum}{14.12.1\enspace} + +\makeatletter +\def\cleartoversopage{\clearpage\if@twoside\ifodd\c@page + \hbox{}\newpage\if@twocolumn\hbox{}\newpage\fi + \else\fi\fi} + +\newcommand{\clearemptyversopage}{ + \clearpage{\pagestyle{empty}\cleartoversopage}} +\makeatother + +\oddsidemargin -1.5cm +\topmargin -2.0cm +\textwidth 16.3cm +\textheight 25cm + +\begin{document} + +\author{Roelof Koekoek\\[2.5mm]Peter A. Lesky\\[2.5mm]Ren\'e F. Swarttouw} +\title{Hypergeometric orthogonal polynomials and their\\$q$-analogues} +\subtitle{-- Monograph --} +\maketitle + +\frontmatter + +\large + +\pagenumbering{roman} +\addtocounter{chapter}{13} + +\chapter{Basic hypergeometric orthogonal polynomials} +\label{BasicHyperOrtPol} + + +In this chapter we deal with all families of basic hypergeometric orthogonal polynomials +appearing in the $q$-analogue of the Askey scheme on the pages~\pageref{qscheme1} and +\pageref{qscheme2}. For each family of orthogonal polynomials we state the most important +properties such as a representation as a basic hypergeometric function, orthogonality +relation(s), the three-term recurrence relation, the second-order $q$-difference equation, +the forward shift (or degree lowering) and backward shift (or degree raising) operator, a +Rodrigues-type formula and some generating functions. Throughout this chapter we assume +that $01$ and $b,c,d$ are real or one is real and the other two are complex conjugates,\\ +$\max(|b|,|c|,|d|)<1$ and the pairwise products of $a,b,c$ and $d$ have +absolute value less than $1$, then we have another orthogonality relation +given by: +\begin{eqnarray} +\label{OrtAskeyWilsonII} +& &\frac{1}{2\pi}\int_{-1}^1\frac{w(x)}{\sqrt{1-x^2}}p_m(x;a,b,c,d|q)p_n(x;a,b,c,d|q)\,dx\nonumber\\ +& &{}\mathindent{}+\sum_{\begin{array}{c}\scriptstyle k\\ \scriptstyle 11$ and $b$ and $c$ are real or complex conjugates, +$\max(|b|,|c|)<1$ and the pairwise products of $a,b$ and $c$ have +absolute value less than $1$, then we have another orthogonality relation +given by: +\begin{eqnarray} +\label{OrtContDualqHahnII} +& &\frac{1}{2\pi}\int_{-1}^1\frac{w(x)}{\sqrt{1-x^2}}p_m(x;a,b,c|q)p_n(x;a,b,c|q)\,dx\nonumber\\ +& &{}\mathindent{}+\sum_{\begin{array}{c}\scriptstyle k\\ \scriptstyle 10$) in the definition (\ref{DefBigqJacobi}) +of the big $q$-Jacobi polynomials and take the limit $c\rightarrow -\infty$ we +obtain the $q$-Meixner polynomials given by (\ref{DefqMeixner}): +\begin{equation} +\lim_{c\rightarrow -\infty}P_n(q^{-x};a,-a^{-1}cd^{-1},c;q)=M_n(q^{-x};a,d;q). +\end{equation} + +\subsubsection*{Big $q$-Jacobi $\rightarrow$ Jacobi} +If we set $c=0$, $a=q^{\alpha}$ and $b=q^{\beta}$ in the definition (\ref{DefBigqJacobi}) +of the big $q$-Jacobi polynomials and let $q\rightarrow 1$ we find the Jacobi +polynomials given by (\ref{DefJacobi}): +\begin{equation} +\lim_{q\rightarrow 1}P_n(x;q^{\alpha},q^{\beta},0;q)=\frac{P_n^{(\alpha,\beta)}(2x-1)}{P_n^{(\alpha,\beta)}(1)}. +\end{equation} +If we take $c=-q^{\gamma}$ for arbitrary real $\gamma$ instead of $c=0$ we +find +\begin{equation} +\lim_{q\rightarrow 1}P_n(x;q^{\alpha},q^{\beta},-q^{\gamma};q)=\frac{P_n^{(\alpha,\beta)}(x)}{P_n^{(\alpha,\beta)}(1)}. +\end{equation} + +\subsection*{Remarks} +The big $q$-Jacobi polynomials with $c=0$ and the little +$q$-Jacobi polynomials given by (\ref{DefLittleqJacobi}) are related +in the following way: +$$P_n(x;a,b,0;q)=\frac{(bq;q)_n}{(aq;q)_n}(-1)^na^nq^{n+\binom{n}{2}}p_n(a^{-1}q^{-1}x;b,a|q).$$ + +\noindent +Sometimes the big $q$-Jacobi polynomials are defined in terms of four +parameters instead of three. In fact the polynomials given by the definition +$$P_n(x;a,b,c,d;q)=\qhyp{3}{2}{q^{-n},abq^{n+1},ac^{-1}qx}{aq,-ac^{-1}dq}{q}$$ +are orthogonal on the interval $[-d,c]$ with respect to the weight function +$$\frac{(c^{-1}qx,-d^{-1}qx;q)_{\infty}}{(ac^{-1}qx,-bd^{-1}qx;q)_{\infty}}d_qx.$$ +These polynomials are not really different from those given by +(\ref{DefBigqJacobi}) since we have +$$P_n(x;a,b,c,d;q)=P_n(ac^{-1}qx;a,b,-ac^{-1}d;q)$$ +and +$$P_n(x;a,b,c;q)=P_n(x;a,b,aq,-cq;q).$$ + +\subsection*{References} +\cite{NAlSalam89}, \cite{AlSalam90}, \cite{AndrewsAskey85}, \cite{AtakKlimyk2004}, +\cite{AtakRahmanSuslov}, \cite{DattaGriffin}, \cite{FloreaniniVinetII}, +\cite{GasperRahman90}, \cite{GrunbaumHaine96}, \cite{Gupta92}, \cite{Hahn}, +\cite{Ismail86I}, \cite{IsmailStanton97}, \cite{IsmailWilson}, \cite{KalninsMiller88}, +\cite{KoelinkE}, \cite{Koorn90II}, \cite{Koorn93}, \cite{Koorn2007}, \cite{Miller89}, +\cite{Nikiforov+}, \cite{NoumiMimachi90II}, \cite{NoumiMimachi90III}, +\cite{NoumiMimachi91}, \cite{Spiridonov97}, \cite{SrivastavaJain90}. + + +\section*{Special case} + +\subsection{Big $q$-Legendre} +\index{Big q-Legendre polynomials@Big $q$-Legendre polynomials} +\index{q-Legendre polynomials@$q$-Legendre polynomials!Big} +\par + +\subsection*{Basic hypergeometric representation} The big $q$-Legendre polynomials are big $q$-Jacobi polynomials +with $a=b=1$: +\begin{equation} +\label{DefBigqLegendre} +P_n(x;c;q)=\qhyp{3}{2}{q^{-n},q^{n+1},x}{q,cq}{q}. +\end{equation} + +\subsection*{Orthogonality relation} +\begin{eqnarray} +\label{OrtBigqLegendre} +& &\int_{cq}^{q}P_m(x;c;q)P_n(x;c;q)\,d_qx\nonumber\\ +& &{}=q(1-c)\frac{(1-q)}{(1-q^{2n+1})} +\frac{(c^{-1}q;q)_n}{(cq;q)_n}(-cq^2)^nq^{\binom{n}{2}}\,\delta_{mn},\quad c<0. +\end{eqnarray} + +\subsection*{Recurrence relation} +\begin{eqnarray} +\label{RecBigqLegendre} +(x-1)P_n(x;c;q)&=&A_nP_{n+1}(x;c;q)-\left(A_n+C_n\right)P_n(x;c;q)\nonumber\\ +& &{}\mathindent{}+C_nP_{n-1}(x;c;q), +\end{eqnarray} +where +$$\left\{\begin{array}{l} +\displaystyle A_n=\frac{(1-q^{n+1})(1-cq^{n+1})}{(1+q^{n+1})(1-q^{2n+1})}\\ +\\ +\displaystyle C_n=-cq^{n+1}\frac{(1-q^n)(1-c^{-1}q^n)}{(1+q^n)(1-q^{2n+1})}. +\end{array}\right.$$ + +\subsection*{Normalized recurrence relation} +\begin{equation} +\label{NormRecBigqLegendre} +xp_n(x)=p_{n+1}(x)+\left[1-(A_n+C_n)\right]p_n(x)+A_{n-1}C_np_{n-1}(x), +\end{equation} +where +$$P_n(x;c;q)=\frac{(q^{n+1};q)_n}{(q,cq;q)_n}p_n(x).$$ + +\subsection*{$q$-Difference equation} +\begin{eqnarray} +\label{dvBigqLegendre} +& &q^{-n}(1-q^n)(1-q^{n+1})x^2y(x)\nonumber\\ +& &{}=B(x)y(qx)-\left[B(x)+D(x)\right]y(x)+D(x)y(q^{-1}x), +\end{eqnarray} +where +$$y(x)=P_n(x;c;q)$$ +and +$$\left\{\begin{array}{l}\displaystyle B(x)=q(x-1)(x-c)\\ +\\ +\displaystyle D(x)=(x-q)(x-cq).\end{array}\right.$$ + +\subsection*{Rodrigues-type formula} +\begin{eqnarray} +\label{RodBigqLegendre} +P_n(x;c;q)&=&\frac{c^nq^{n(n+1)}(1-q)^n}{(q,cq;q)_n} +\left(\mathcal{D}_q\right)^n\left[(q^{-n}x,c^{-1}q^{-n}x;q)_n\right]\\ +&=&\frac{(1-q)^n}{(q,cq;q)_n}\left(\mathcal{D}_q\right)^n +\left[(qx^{-1},cqx^{-1};q)_nx^{2n}\right].\nonumber +\end{eqnarray} + +\subsection*{Generating functions} +\begin{equation} +\label{GenBigqLegendre1} +\qhyp{2}{1}{qx^{-1},0}{q}{xt}\,\qhyp{1}{1}{c^{-1}x}{q}{cqt} +=\sum_{n=0}^{\infty}\frac{(cq;q)_n}{(q,q;q)_n}P_n(x;c;q)t^n. +\end{equation} + +\begin{equation} +\label{GenBigqLegendre2} +\qhyp{2}{1}{cqx^{-1},0}{cq}{xt}\,\qhyp{1}{1}{c^{-1}x}{c^{-1}q}{qt} +=\sum_{n=0}^{\infty}\frac{P_n(x;c;q)}{(c^{-1}q;q)_n}t^n. +\end{equation} + +\subsection*{Limit relations} + +\subsubsection*{Big $q$-Legendre $\rightarrow$ Legendre / Spherical} +If we set $c=0$ in the definition (\ref{DefBigqLegendre}) of the big +$q$-Legendre polynomials and let $q\rightarrow 1$ we simply obtain the Legendre +(or spherical) polynomials given by (\ref{DefLegendre}): +\begin{equation} +\lim_{q\rightarrow 1}P_n(x;0;q)=P_n(2x-1). +\end{equation} +If we take $c=-q^{\gamma}$ for arbitrary real $\gamma$ instead of $c=0$ we +find +\begin{equation} +\lim_{q\rightarrow 1}P_n(x;-q^{\gamma};q)=P_n(x). +\end{equation} + +\subsection*{References} +\cite{Koelink95I}, \cite{Koorn90II}. + + +\section{$q$-Hahn}\index{q-Hahn polynomials@$q$-Hahn polynomials} +\par\setcounter{equation}{0} + +\subsection*{Basic hypergeometric representation} +\begin{equation} +\label{DefqHahn} +Q_n(q^{-x};\alpha,\beta,N|q)=\qhyp{3}{2}{q^{-n},\alpha\beta q^{n+1},q^{-x}} +{\alpha q,q^{-N}}{q},\quad n=0,1,2,\ldots,N. +\end{equation} + +\subsection*{Orthogonality relation} +For $0<\alpha q<1$ and $0<\beta q<1$, or for $\alpha>q^{-N}$ and $\beta>q^{-N}$, we have +\begin{eqnarray} +\label{OrtqHahn} +& &\sum_{x=0}^N\frac{(\alpha q,q^{-N};q)_x}{(q,\beta^{-1}q^{-N};q)_x}(\alpha\beta q)^{-x} +Q_m(q^{-x};\alpha,\beta,N|q)Q_n(q^{-x};\alpha,\beta,N|q)\nonumber\\ +& &{}=\frac{(\alpha\beta q^2;q)_N}{(\beta q;q)_N(\alpha q)^N} +\frac{(q,\alpha\beta q^{N+2},\beta q;q)_n}{(\alpha q,\alpha\beta q,q^{-N};q)_n}\, +\frac{(1-\alpha\beta q)(-\alpha q)^n}{(1-\alpha\beta q^{2n+1})} +q^{\binom{n}{2}-Nn}\,\delta_{mn}. +\end{eqnarray} + +\newpage + +\subsection*{Recurrence relation} +\begin{eqnarray} +\label{RecqHahn} +-\left(1-q^{-x}\right)Q_n(q^{-x})&=&A_nQ_{n+1}(q^{-x})-\left(A_n+C_n\right)Q_n(q^{-x})\nonumber\\ +& &{}\mathindent{}+C_nQ_{n-1}(q^{-x}), +\end{eqnarray} +where +$$Q_n(q^{-x}):=Q_n(q^{-x};\alpha,\beta,N|q)$$ +and +$$\left\{\begin{array}{l} +\displaystyle A_n=\frac{(1-q^{n-N})(1-\alpha q^{n+1})(1-\alpha\beta q^{n+1})}{(1-\alpha\beta q^{2n+1})(1-\alpha\beta q^{2n+2})}\\ +\\ +\displaystyle C_n=-\frac{\alpha q^{n-N}(1-q^n)(1-\alpha\beta q^{n+N+1})(1-\beta q^n)}{(1-\alpha\beta q^{2n})(1-\alpha\beta q^{2n+1})}. +\end{array}\right.$$ + +\subsection*{Normalized recurrence relation} +\begin{equation} +\label{NormRecqHahn} +xp_n(x)=p_{n+1}(x)+\left[1-(A_n+C_n)\right]p_n(x)+A_{n-1}C_np_{n-1}(x), +\end{equation} +where +$$Q_n(q^{-x};\alpha,\beta,N|q)= +\frac{(\alpha\beta q^{n+1};q)_n}{(\alpha q,q^{-N};q)_n}p_n(q^{-x}).$$ + +\subsection*{$q$-Difference equation} +\begin{eqnarray} +\label{dvqHahn} +& &q^{-n}(1-q^n)(1-\alpha\beta q^{n+1})y(x)\nonumber\\ +& &{}=B(x)y(x+1)-\left[B(x)+D(x)\right]y(x)+D(x)y(x-1), +\end{eqnarray} +where +$$y(x)=Q_n(q^{-x};\alpha,\beta,N|q)$$ +and +$$\left\{\begin{array}{l}\displaystyle B(x)=(1-q^{x-N})(1-\alpha q^{x+1})\\ +\\ +\displaystyle D(x)=\alpha q(1-q^x)(\beta-q^{x-N-1}).\end{array}\right.$$ + +\newpage + +\subsection*{Forward shift operator} +\begin{eqnarray} +\label{shift1qHahnI} +& &Q_n(q^{-x-1};\alpha,\beta,N|q)-Q_n(q^{-x};\alpha,\beta,N|q)\nonumber\\ +& &{}=\frac{q^{-n-x}(1-q^n)(1-\alpha\beta q^{n+1})}{(1-\alpha q)(1-q^{-N})} +Q_{n-1}(q^{-x};\alpha q,\beta q,N-1|q) +\end{eqnarray} +or equivalently +\begin{eqnarray} +\label{shift1qHahnII} +\frac{\Delta Q_n(q^{-x};\alpha,\beta,N|q)}{\Delta q^{-x}} +&=&\frac{q^{-n+1}(1-q^n)(1-\alpha\beta q^{n+1})}{(1-q)(1-\alpha q)(1-q^{-N})}\nonumber\\ +& &{}\mathindent{}\times Q_{n-1}(q^{-x};\alpha q,\beta q,N-1|q). +\end{eqnarray} + +\subsection*{Backward shift operator} +\begin{eqnarray} +\label{shift2qHahnI} +& &(1-\alpha q^x)(1-q^{x-N-1})Q_n(q^{-x};\alpha,\beta,N|q)\nonumber\\ +& &{}\mathindent{}-\alpha(1-q^x)(\beta-q^{x-N-1})Q_n(q^{-x+1};\alpha,\beta,N|q)\nonumber\\ +& &{}=q^x(1-\alpha)(1-q^{-N-1})Q_{n+1}(q^{-x};\alpha q^{-1},\beta q^{-1},N+1|q) +\end{eqnarray} +or equivalently +\begin{eqnarray} +\label{shift2qHahnII} +& &\frac{\nabla\left[w(x;\alpha,\beta,N|q)Q_n(q^{-x};\alpha,\beta,N|q)\right]}{\nabla q^{-x}}\nonumber\\ +& &{}=\frac{1}{1-q}w(x;\alpha q^{-1},\beta q^{-1},N+1|q)Q_{n+1}(q^{-x};\alpha q^{-1},\beta q^{-1},N+1|q), +\end{eqnarray} +where +$$w(x;\alpha,\beta,N|q)=\frac{(\alpha q,q^{-N};q)_x}{(q,\beta^{-1}q^{-N};q)_x}(\alpha\beta)^{-x}.$$ + +\subsection*{Rodrigues-type formula} +\begin{eqnarray} +\label{RodqHahn} +& &w(x;\alpha,\beta,N|q)Q_n(q^{-x};\alpha,\beta,N|q)\nonumber\\ +& &{}=(1-q)^n\left(\nabla_q\right)^n\left[w(x;\alpha q^n,\beta q^n,N-n|q)\right], +\end{eqnarray} +where +$$\nabla_q:=\frac{\nabla}{\nabla q^{-x}}.$$ + +\subsection*{Generating functions} For $x=0,1,2,\ldots,N$ we have +\begin{eqnarray} +\label{GenqHahn1} +& &\qhyp{1}{1}{q^{-x}}{\alpha q}{\alpha qt}\,\qhyp{2}{1}{q^{x-N},0}{\beta q}{q^{-x}t}\nonumber\\ +& &{}=\sum_{n=0}^N\frac{(q^{-N};q)_n}{(\beta q,q;q)_n}Q_n(q^{-x};\alpha,\beta,N|q)t^n. +\end{eqnarray} + +\begin{eqnarray} +\label{GenqHahn2} +& &\qhyp{2}{1}{q^{-x},\beta q^{N+1-x}}{0}{-\alpha q^{x-N+1}t}\, +\qhyp{2}{0}{q^{x-N},\alpha q^{x+1}}{-}{-q^{-x}t}\nonumber\\ +& &{}=\sum_{n=0}^N\frac{(\alpha q,q^{-N};q)_n}{(q;q)_n}q^{-\binom{n}{2}}Q_n(q^{-x};\alpha,\beta,N|q)t^n. +\end{eqnarray} + +\subsection*{Limit relations} + +\subsubsection*{$q$-Racah $\rightarrow$ $q$-Hahn} +The $q$-Hahn polynomials follow from the $q$-Racah polynomials by the substitution +$\delta=0$ and $\gamma q=q^{-N}$ in the definition (\ref{DefqRacah}) of the +$q$-Racah polynomials: +$$R_n(\mu(x);\alpha,\beta,q^{-N-1},0|q)=Q_n(q^{-x};\alpha,\beta,N|q).$$ +Another way to obtain the $q$-Hahn polynomials from the $q$-Racah +polynomials is by setting $\gamma=0$ and $\delta=\beta^{-1}q^{-N-1}$ in the definition +(\ref{DefqRacah}): +$$R_n(\mu(x);\alpha,\beta,0,\beta^{-1}q^{-N-1}|q)=Q_n(q^{-x};\alpha,\beta,N|q).$$ +And if we take $\alpha q=q^{-N}$, $\beta\rightarrow\beta\gamma q^{N+1}$ and $\delta=0$ in the +definition (\ref{DefqRacah}) of the $q$-Racah polynomials we find the +$q$-Hahn polynomials given by (\ref{DefqHahn}) in the following way: +$$R_n(\mu(x);q^{-N-1},\beta\gamma q^{N+1},\gamma,0|q)=Q_n(q^{-x};\gamma,\beta,N|q).$$ +Note that $\mu(x)=q^{-x}$ in each case. + +\subsubsection*{$q$-Hahn $\rightarrow$ Little $q$-Jacobi} +If we set $x\rightarrow N-x$ in the definition (\ref{DefqHahn}) of the $q$-Hahn +polynomials and take the limit $N\rightarrow\infty$ we find the little $q$-Jacobi polynomials: +\begin{equation} +\lim _{N\rightarrow\infty} Q_n(q^{x-N};\alpha,\beta,N|q)=p_n(q^x;\alpha,\beta|q), +\end{equation} +where $p_n(q^x;\alpha,\beta|q)$ is given by (\ref{DefLittleqJacobi}). + +\subsubsection*{$q$-Hahn $\rightarrow$ $q$-Meixner} +The $q$-Meixner polynomials given by (\ref{DefqMeixner}) can be obtained from the $q$-Hahn polynomials +by setting $\alpha=b$ and $\beta=-b^{-1}c^{-1}q^{-N-1}$ in the definition (\ref{DefqHahn}) of the +$q$-Hahn polynomials and letting $N\rightarrow\infty$: +\begin{equation} +\lim_{N\rightarrow\infty} +Q_n(q^{-x};b,-b^{-1}c^{-1}q^{-N-1},N|q)=M_n(q^{-x};b,c;q). +\end{equation} + +\subsubsection*{$q$-Hahn $\rightarrow$ Quantum $q$-Krawtchouk} +The quantum $q$-Krawtchouk polynomials given by (\ref{DefQuantumqKrawtchouk}) +simply follow from the $q$-Hahn polynomials by setting $\beta=p$ in the definition (\ref{DefqHahn}) of +the $q$-Hahn polynomials and taking the limit $\alpha\rightarrow\infty$: +\begin{equation} +\lim_{\alpha\rightarrow\infty}Q_n(q^{-x};\alpha,p,N|q)=K_n^{qtm}(q^{-x};p,N;q). +\end{equation} + +\subsubsection*{$q$-Hahn $\rightarrow$ $q$-Krawtchouk} +If we set $\beta=-\alpha^{-1}q^{-1}p$ in the definition (\ref{DefqHahn}) of the $q$-Hahn polynomials +and then let $\alpha\rightarrow 0$ we obtain the $q$-Krawtchouk polynomials given by +(\ref{DefqKrawtchouk}): +\begin{equation} +\lim_{\alpha\rightarrow 0} +Q_n(q^{-x};\alpha,-\alpha^{-1}q^{-1}p,N|q)=K_n(q^{-x};p,N;q). +\end{equation} + +\subsubsection*{$q$-Hahn $\rightarrow$ Affine $q$-Krawtchouk} +The affine $q$-Krawtchouk polynomials given by (\ref{DefAffqKrawtchouk}) +can be obtained from the $q$-Hahn polynomials by the substitution $\alpha=p$ and +$\beta=0$ in (\ref{DefqHahn}): +\begin{equation} +Q_n(q^{-x};p,0,N|q)=K_n^{Aff}(q^{-x};p,N;q). +\end{equation} + +\subsubsection*{$q$-Hahn $\rightarrow$ Hahn} +The Hahn polynomials given by (\ref{DefHahn}) simply follow from the $q$-Hahn +polynomials given by (\ref{DefqHahn}), after setting $\alpha\rightarrow q^{\alpha}$ +and $\beta\rightarrow q^{\beta}$, in the following way: +\begin{equation} +\lim_{q\rightarrow 1}Q_n(q^{-x};q^{\alpha},q^{\beta},N|q)=Q_n(x;\alpha,\beta,N). +\end{equation} + +\subsection*{Remark} +The $q$-Hahn polynomials given by (\ref{DefqHahn}) and the dual $q$-Hahn +polynomials given by (\ref{DefDualqHahn}) are related in the following way: +$$Q_n(q^{-x};\alpha,\beta,N|q)=R_x(\mu(n);\alpha,\beta,N|q),$$ +with +$$\mu(n)=q^{-n}+\alpha\beta q^{n+1}$$ +or +$$R_n(\mu(x);\gamma,\delta,N|q)=Q_x(q^{-n};\gamma,\delta,N|q),$$ +where +$$\mu(x)=q^{-x}+\gamma\delta q^{x+1}.$$ + +\subsection*{References} +\cite{AlSalam90}, \cite{AlvarezRonveaux}, \cite{AndrewsAskey85}, +\cite{AskeyWilson79}, \cite{AskeyWilson85}, \cite{AtakRahmanSuslov}, +\cite{Dunkl78II}, \cite{Fischer95}, \cite{GasperRahman84}, +\cite{GasperRahman90}, \cite{Hahn}, \cite{KalninsMiller88}, +\cite{Koelink96I}, \cite{KoelinkKoorn}, \cite{Koorn89III}, \cite{Koorn90II}, +\cite{Nikiforov+}, \cite{NoumiMimachi90II}, \cite{Rahman82}, +\cite{Stanton80II}, \cite{Stanton84}, \cite{Stanton90}. + + +\section{Dual $q$-Hahn}\index{Dual q-Hahn polynomials@Dual $q$-Hahn polynomials} +\index{q-Hahn polynomials@$q$-Hahn polynomials!Dual} +\par\setcounter{equation}{0} + +\subsection*{Basic hypergeometric representation} +\begin{equation} +\label{DefDualqHahn} +R_n(\mu(x);\gamma,\delta,N|q)= +\qhyp{3}{2}{q^{-n},q^{-x},\gamma\delta q^{x+1}}{\gamma q,q^{-N}}{q},\quad n=0,1,2,\ldots,N, +\end{equation} +where +$$\mu(x):=q^{-x}+\gamma\delta q^{x+1}.$$ + +\newpage + +\subsection*{Orthogonality relation} +For $0<\gamma q<1$ and $0<\delta q<1$, or for $\gamma>q^{-N}$ and $\delta>q^{-N}$, we have +\begin{eqnarray} +\label{OrtDualqHahn} +& &\sum_{x=0}^N\frac{(\gamma q,\gamma\delta q,q^{-N};q)_x}{(q,\gamma\delta q^{N+2},\delta q;q)_x} +\frac{(1-\gamma\delta q^{2x+1})}{(1-\gamma\delta q)(-\gamma q)^x}q^{Nx-\binom{x}{2}}\nonumber\\ +& &{}\mathindent{}\times R_m(\mu(x);\gamma,\delta,N|q)R_n(\mu(x);\gamma,\delta,N|q)\nonumber\\ +& &{}=\frac{(\gamma\delta q^2;q)_N}{(\delta q;q)_N}(\gamma q)^{-N} +\frac{(q,\delta^{-1}q^{-N};q)_n}{(\gamma q,q^{-N};q)_n}(\gamma\delta q)^n\,\delta_{mn}. +\end{eqnarray} + +\subsection*{Recurrence relation} +\begin{eqnarray} +\label{RecDualqHahn} +& &-\left(1-q^{-x}\right)\left(1-\gamma\delta q^{x+1}\right)R_n(\mu(x))\nonumber\\ +& &{}=A_nR_{n+1}(\mu(x))-\left(A_n+C_n\right)R_n(\mu(x))+C_nR_{n-1}(\mu(x)), +\end{eqnarray} +where +$$R_n(\mu(x)):=R_n(\mu(x);\gamma,\delta,N|q)$$ +and +$$\left\{\begin{array}{l} +\displaystyle A_n=\left(1-q^{n-N}\right)\left(1-\gamma q^{n+1}\right)\\ +\\ +\displaystyle C_n=\gamma q\left(1-q^n\right)\left(\delta -q^{n-N-1}\right). +\end{array}\right.$$ + +\subsection*{Normalized recurrence relation} +\begin{eqnarray} +\label{NormRecDualqHahn} +xp_n(x)&=&p_{n+1}(x)+\left[1+\gamma\delta q-(A_n+C_n)\right]p_n(x)\nonumber\\ +& &{}\mathindent{}+\gamma q(1-q^n)(1-\gamma q^n)\nonumber\\ +& &{}\mathindent\mathindent{}\times(1-q^{n-N-1})(\delta-q^{n-N-1})p_{n-1}(x), +\end{eqnarray} +where +$$R_n(\mu(x);\gamma,\delta,N|q)=\frac{1}{(\gamma q,q^{-N};q)_n}p_n(\mu(x)).$$ + +\subsection*{$q$-Difference equation} +\begin{equation} +\label{dvDualqHahn} +q^{-n}(1-q^n)y(x)=B(x)y(x+1)-\left[B(x)+D(x)\right]y(x) ++D(x)y(x-1), +\end{equation} +where +$$y(x)=R_n(\mu(x);\gamma,\delta,N|q)$$ +and +$$\left\{\begin{array}{l}\displaystyle B(x)=\frac{(1-q^{x-N})(1-\gamma q^{x+1})(1-\gamma\delta q^{x+1})} +{(1-\gamma\delta q^{2x+1})(1-\gamma\delta q^{2x+2})}\\ +\\ +\displaystyle D(x)=-\frac{\gamma q^{x-N}(1-q^x)(1-\gamma\delta q^{x+N+1})(1-\delta q^x)} +{(1-\gamma\delta q^{2x})(1-\gamma\delta q^{2x+1})}.\end{array}\right.$$ + +\subsection*{Forward shift operator} +\begin{eqnarray} +\label{shift1DualqHahnI} +& &R_n(\mu(x+1);\gamma,\delta,N|q)-R_n(\mu(x);\gamma,\delta,N|q)\nonumber\\ +& &{}=\frac{q^{-n-x}(1-q^n)(1-\gamma\delta q^{2x+2})}{(1-\gamma q)(1-q^{-N})} +R_{n-1}(\mu(x);\gamma q,\delta,N-1|q) +\end{eqnarray} +or equivalently +\begin{eqnarray} +\label{shift1DualqHahnII} +& &\frac{\Delta R_n(\mu(x);\gamma,\delta,N|q)}{\Delta\mu(x)}\nonumber\\ +& &{}=\frac{q^{-n+1}(1-q^n)}{(1-q)(1-\gamma q)(1-q^{-N})}R_{n-1}(\mu(x);\gamma q,\delta,N-1|q). +\end{eqnarray} + +\subsection*{Backward shift operator} +\begin{eqnarray} +\label{shift2DualqHahnI} +& &(1-\gamma q^x)(1-\gamma\delta q^x)(1-q^{x-N-1})R_n(\mu(x);\gamma,\delta,N|q)\nonumber\\ +& &{}\mathindent{}+\gamma q^{x-N-1}(1-q^x)(1-\gamma\delta q^{x+N+1})(1-\delta q^x)R_n(\mu(x-1);\gamma,\delta,N|q)\nonumber\\ +& &{}=q^x(1-\gamma)(1-q^{-N-1})(1-\gamma\delta q^{2x})R_{n+1}(\mu(x);\gamma q^{-1},\delta,N+1|q) +\end{eqnarray} +or equivalently +\begin{eqnarray} +\label{shift2DualqHahnII} +& &\frac{\nabla\left[w(x;\gamma,\delta,N|q)R_n(\mu(x);\gamma,\delta,N|q)\right]}{\nabla\mu(x)}\nonumber\\ +& &{}=\frac{1}{(1-q)(1-\gamma\delta)}w(x;\gamma q^{-1},\delta,N+1|q)\nonumber\\ +& &{}\mathindent{}\times R_{n+1}(\mu(x);\gamma q^{-1},\delta,N+1|q), +\end{eqnarray} +where +$$w(x;\gamma,\delta,N|q)=\frac{(\gamma q,\gamma\delta q,q^{-N};q)_x} +{(q,\gamma\delta q^{N+2},\delta q;q)_x}(-\gamma^{-1})^x q^{Nx-\binom{x}{2}}.$$ + +\subsection*{Rodrigues-type formula} +\begin{eqnarray} +\label{RodDualqHahn} +& &w(x;\gamma,\delta,N|q)R_n(\mu(x);\gamma,\delta,N|q)\nonumber\\ +& &{}=(1-q)^n(\gamma\delta q;q)_n +\left(\nabla_{\mu}\right)^n\left[w(x;\gamma q^n,\delta,N-n|q)\right], +\end{eqnarray} +where +$$\nabla_{\mu}:=\frac{\nabla}{\nabla\mu(x)}.$$ + +\subsection*{Generating functions} For $x=0,1,2,\ldots,N$ we have +\begin{eqnarray} +\label{GenDualqHahn1} +& &(q^{-N}t;q)_{N-x}\cdot\qhyp{2}{1}{q^{-x},\delta^{-1}q^{-x}}{\gamma q}{\gamma\delta q^{x+1}t}\nonumber\\ +& &{}=\sum_{n=0}^N\frac{(q^{-N};q)_n}{(q;q)_n}R_n(\mu(x);\gamma,\delta,N|q)t^n. +\end{eqnarray} + +\begin{eqnarray} +\label{GenDualqHahn2} +& &(\gamma\delta qt;q)_x\cdot\qhyp{2}{1}{q^{x-N},\gamma q^{x+1}}{\delta^{-1}q^{-N}}{q^{-x}t}\nonumber\\ +& &{}=\sum_{n=0}^N +\frac{(q^{-N},\gamma q;q)_n}{(\delta^{-1}q^{-N},q;q)_n}R_n(\mu(x);\gamma,\delta,N|q)t^n. +\end{eqnarray} + +\subsection*{Limit relations} + +\subsubsection*{$q$-Racah $\rightarrow$ Dual $q$-Hahn} +To obtain the dual $q$-Hahn polynomials from the $q$-Racah polynomials we have to +take $\beta=0$ and $\alpha q=q^{-N}$ in (\ref{DefqRacah}): +$$R_n(\mu(x);q^{-N-1},0,\gamma,\delta|q)=R_n(\mu(x);\gamma,\delta,N|q),$$ +with +$$\mu(x)=q^{-x}+\gamma\delta q^{x+1}.$$ +We may also take $\alpha=0$ and $\beta=\delta^{-1}q^{-N-1}$ in (\ref{DefqRacah}) to obtain +the dual $q$-Hahn polynomials from the $q$-Racah polynomials: +$$R_n(\mu(x);0,\delta^{-1}q^{-N-1},\gamma,\delta|q)=R_n(\mu(x);\gamma,\delta,N|q).$$ +And if we take $\gamma q=q^{-N}$, $\delta\rightarrow\alpha\delta q^{N+1}$ and $\beta=0$ in the +definition (\ref{DefqRacah}) of the $q$-Racah polynomials we find the dual +$q$-Hahn polynomials given by (\ref{DefDualqHahn}) in the following way: +$$R_n(\mu(x);\alpha,0,q^{-N-1},\alpha\delta q^{N+1}|q)=R_n({\tilde \mu}(x);\alpha,\delta,N|q),$$ +with +$${\tilde \mu}(x)=q^{-x}+\alpha\delta q^{x+1}.$$ + +\subsubsection*{Dual $q$-Hahn $\rightarrow$ Affine $q$-Krawtchouk} +The affine $q$-Krawtchouk polynomials given by (\ref{DefAffqKrawtchouk}) +can be obtained from the dual $q$-Hahn polynomials by the substitution $\gamma=p$ +and $\delta=0$ in (\ref{DefDualqHahn}): +\begin{equation} +R_n(\mu(x);p,0,N|q)=K_n^{Aff}(q^{-x};p,N;q). +\end{equation} +Note that $\mu(x)=q^{-x}$ in this case. + +\subsubsection*{Dual $q$-Hahn $\rightarrow$ Dual $q$-Krawtchouk} +The dual $q$-Krawtchouk polynomials given by (\ref{DefDualqKrawtchouk}) can +be obtained from the dual $q$-Hahn polynomials by setting $\delta=c\gamma^{-1}q^{-N-1}$ +in (\ref{DefDualqHahn}) and letting $\gamma\rightarrow 0$: +\begin{equation} +\lim_{\gamma\rightarrow 0} +R_n(\mu(x);\gamma,c\gamma^{-1}q^{-N-1},N|q)=K_n(\lambda(x);c,N|q). +\end{equation} + +\subsubsection*{Dual $q$-Hahn $\rightarrow$ Dual Hahn} +The dual Hahn polynomials given by (\ref{DefDualHahn}) follow from the dual $q$-Hahn +polynomials by simply taking the limit $q\rightarrow 1$ in the definition (\ref{DefDualqHahn}) +of the dual $q$-Hahn polynomials after applying the substitution +$\gamma\rightarrow q^{\gamma}$ and $\delta\rightarrow q^{\delta}$: +\begin{equation} +\lim_{q\rightarrow 1}R_n(\mu(x);q^{\gamma},q^{\delta},N|q)=R_n(\lambda(x);\gamma,\delta,N), +\end{equation} +where +$$\left\{\begin{array}{l} +\displaystyle\mu(x)=q^{-x}+q^{x+\gamma+\delta+1}\\[5mm] +\displaystyle\lambda(x)=x(x+\gamma+\delta+1). +\end{array}\right.$$ + +\subsection*{Remark} +The dual $q$-Hahn polynomials given by (\ref{DefDualqHahn}) and the +$q$-Hahn polynomials given by (\ref{DefqHahn}) are related in the following way: +$$Q_n(q^{-x};\alpha,\beta,N|q)=R_x(\mu(n);\alpha,\beta,N|q),$$ +with +$$\mu(n)=q^{-n}+\alpha\beta q^{n+1}$$ +or +$$R_n(\mu(x);\gamma,\delta,N|q)=Q_x(q^{-n};\gamma,\delta,N|q),$$ +where +$$\mu(x)=q^{-x}+\gamma\delta q^{x+1}.$$ + +\subsection*{References} +\cite{AlvarezSmirnov}, \cite{AndrewsAskey85}, \cite{AskeyWilson79}, +\cite{AskeyWilson85}, \cite{AtakRahmanSuslov}, \cite{GasperRahman90}, +\cite{KoelinkKoorn}, \cite{Nikiforov+}, \cite{Stanton84}. + + +\section{Al-Salam-Chihara}\index{Al-Salam-Chihara polynomials} +\par\setcounter{equation}{0} + +\subsection*{Basic hypergeometric representation} +\begin{eqnarray} +\label{DefAlSalamChihara} +Q_n(x;a,b|q)&=&\frac{(ab;q)_n}{a^n}\, +\qhyp{3}{2}{q^{-n},a\e^{i\theta},a\e^{-i\theta}}{ab,0}{q}\\ +&=&(a\e^{i\theta};q)_n\e^{-in\theta}\,\qhyp{2}{1}{q^{-n},b\e^{-i\theta}} +{a^{-1}q^{-n+1}\e^{-i\theta}}{a^{-1}q\e^{i\theta}}\nonumber\\ +&=&(b\e^{-i\theta};q)_n\e^{in\theta}\,\qhyp{2}{1}{q^{-n},a\e^{i\theta}} +{b^{-1}q^{-n+1}\e^{i\theta}}{b^{-1}q\e^{-i\theta}},\quad x=\cos\theta.\nonumber +\end{eqnarray} + +\subsection*{Orthogonality relation} +If $a$ and $b$ are real or complex conjugates and $\max(|a|,|b|)<1$, then we have the following orthogonality relation +\begin{equation} +\label{OrtAlSalamChiharaI} +\frac{1}{2\pi}\int_{-1}^1\frac{w(x)}{\sqrt{1-x^2}}Q_m(x;a,b|q)Q_n(x;a,b|q)\,dx +=\frac{\,\delta_{mn}}{(q^{n+1},abq^n;q)_{\infty}}, +\end{equation} +where +$$w(x):=w(x;a,b|q)=\left|\frac{(\e^{2i\theta};q)_{\infty}} +{(a\e^{i\theta},b\e^{i\theta};q)_{\infty}}\right|^2 +=\frac{h(x,1)h(x,-1)h(x,q^{\frac{1}{2}})h(x,-q^{\frac{1}{2}})}{h(x,a)h(x,b)},$$ +with +$$h(x,\alpha):=\prod_{k=0}^{\infty}\left(1-2\alpha xq^k+\alpha^2q^{2k}\right) +=\left(\alpha\e^{i\theta},\alpha\e^{-i\theta};q\right)_{\infty},\quad x=\cos\theta.$$ +If $a>1$ and $|ab|<1$, then we have another orthogonality relation given by: +\begin{eqnarray} +\label{OrtAlSalamChiharaII} +& &\frac{1}{2\pi}\int_{-1}^1\frac{w(x)}{\sqrt{1-x^2}}Q_m(x;a,b|q)Q_n(x;a,b|q)\,dx\nonumber\\ +& &{}\mathindent{}+\sum_{\begin{array}{c}\scriptstyle k\\ \scriptstyle 10. +\end{eqnarray} + +\subsection*{Recurrence relation} +\begin{eqnarray} +\label{RecqMeixner} +& &q^{2n+1}(1-q^{-x})M_n(q^{-x})\nonumber\\ +& &{}=c(1-bq^{n+1})M_{n+1}(q^{-x})\nonumber\\ +& &{}\mathindent{}-\left[c(1-bq^{n+1})+q(1-q^n)(c+q^n)\right]M_n(q^{-x})\nonumber\\ +& &{}\mathindent\mathindent{}+q(1-q^n)(c+q^n)M_{n-1}(q^{-x}), +\end{eqnarray} +where +$$M_n(q^{-x}):=M_n(q^{-x};b,c;q).$$ + +\subsection*{Normalized recurrence relation} +\begin{eqnarray} +\label{NormRecqMeixner} +xp_n(x)&=&p_{n+1}(x)+ +\left[1+q^{-2n-1}\left\{c(1-bq^{n+1})+q(1-q^n)(c+q^n)\right\}\right]p_n(x)\nonumber\\ +& &{}\mathindent{}+cq^{-4n+1}(1-q^n)(1-bq^n)(c+q^n)p_{n-1}(x), +\end{eqnarray} +where +$$M_n(q^{-x};b,c;q)=\frac{(-1)^nq^{n^2}}{(bq;q)_nc^n}p_n(q^{-x}).$$ + +\subsection*{$q$-Difference equation} +\begin{equation} +\label{dvqMeixner} +-(1-q^n)y(x)=B(x)y(x+1)-\left[B(x)+D(x)\right]y(x)+D(x)y(x-1), +\end{equation} +where +$$y(x)=M_n(q^{-x};b,c;q)$$ +and +$$\left\{\begin{array}{l}\displaystyle B(x)=cq^x(1-bq^{x+1})\\ +\\ +\displaystyle D(x)=(1-q^x)(1+bcq^x).\end{array}\right.$$ + +\subsection*{Forward shift operator} +\begin{eqnarray} +\label{shift1qMeixnerI} +& &M_n(q^{-x-1};b,c;q)-M_n(q^{-x};b,c;q)\nonumber\\ +& &{}=-\frac{q^{-x}(1-q^n)}{c(1-bq)}M_{n-1}(q^{-x};bq,cq^{-1};q) +\end{eqnarray} +or equivalently +\begin{equation} +\label{shift1qMeixnerII} +\frac{\Delta M_n(q^{-x};b,c;q)}{\Delta q^{-x}} +=-\frac{q(1-q^n)}{c(1-q)(1-bq)}M_{n-1}(q^{-x};bq,cq^{-1};q). +\end{equation} + +\subsection*{Backward shift operator} +\begin{eqnarray} +\label{shift2qMeixnerI} +& &cq^x(1-bq^x)M_n(q^{-x};b,c;q)-(1-q^x)(1+bcq^x)M_n(q^{-x+1};b,c;q)\nonumber\\ +& &{}=cq^x(1-b)M_{n+1}(q^{-x};bq^{-1},cq;q) +\end{eqnarray} +or equivalently +\begin{eqnarray} +\label{shift2qMeixnerII} +& &\frac{\nabla\left[w(x;b,c;q)M_n(q^{-x};b,c;q)\right]}{\nabla q^{-x}}\nonumber\\ +& &{}=\frac{1}{1-q}w(x;bq^{-1},cq;q)M_{n+1}(q^{-x};bq^{-1},cq;q), +\end{eqnarray} +where +$$w(x;b,c;q)=\frac{(bq;q)_x}{(q,-bcq;q)_x}c^xq^{\binom{x+1}{2}}.$$ + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodqMeixner} +w(x;b,c;q)M_n(q^{-x};b,c;q)=(1-q)^n\left(\nabla_q\right)^n\left[w(x;bq^n,cq^{-n};q)\right], +\end{equation} +where +$$\nabla_q:=\frac{\nabla}{\nabla q^{-x}}.$$ + +\subsection*{Generating functions} +\begin{equation} +\label{GenqMeixner1} +\frac{1}{(t;q)_{\infty}}\,\qhyp{1}{1}{q^{-x}}{bq}{-c^{-1}qt} +=\sum_{n=0}^{\infty}\frac{M_n(q^{-x};b,c;q)}{(q;q)_n}t^n. +\end{equation} + +\begin{eqnarray} +\label{GenqMeixner2} +& &\frac{1}{(t;q)_{\infty}}\,\qhyp{1}{1}{-b^{-1}c^{-1}q^{-x}}{-c^{-1}q}{bqt}\nonumber\\ +& &{}=\sum_{n=0}^{\infty}\frac{(bq;q)_n}{(-c^{-1}q,q;q)_n}M_n(q^{-x};b,c;q)t^n. +\end{eqnarray} + +\subsection*{Limit relations} + +\subsubsection*{Big $q$-Jacobi $\rightarrow$ $q$-Meixner} +If we set $b=-a^{-1}cd^{-1}$ (with $d>0$) in the definition (\ref{DefBigqJacobi}) +of the big $q$-Jacobi polynomials and take the limit $c\rightarrow -\infty$ we +obtain the $q$-Meixner polynomials given by (\ref{DefqMeixner}): +\begin{equation} +\lim_{c\rightarrow -\infty}P_n(q^{-x};a,-a^{-1}cd^{-1},c;q)=M_n(q^{-x};a,d;q). +\end{equation} + +\subsubsection*{$q$-Hahn $\rightarrow$ $q$-Meixner} +The $q$-Meixner polynomials given by (\ref{DefqMeixner}) can be obtained from the $q$-Hahn polynomials +by setting $\alpha=b$ and $\beta=-b^{-1}c^{-1}q^{-N-1}$ in the definition (\ref{DefqHahn}) of the +$q$-Hahn polynomials and letting $N\rightarrow\infty$: +$$\lim_{N\rightarrow\infty}Q_n(q^{-x};b,-b^{-1}c^{-1}q^{-N-1},N|q)=M_n(q^{-x};b,c;q).$$ + +\subsubsection*{$q$-Meixner $\rightarrow$ $q$-Laguerre} +The $q$-Laguerre polynomials given by (\ref{DefqLaguerre}) can be obtained +from the $q$-Meixner polynomials given by (\ref{DefqMeixner}) by setting +$b=q^{\alpha}$ and $q^{-x}\rightarrow cq^{\alpha}x$ in the definition +(\ref{DefqMeixner}) of the $q$-Meixner polynomials and then taking the limit +$c\rightarrow\infty$: +\begin{equation} +\lim_{c\rightarrow\infty}M_n(cq^{\alpha}x;q^{\alpha},c;q)= +\frac{(q;q)_n}{(q^{\alpha+1};q)_n}L_n^{(\alpha)}(x;q). +\end{equation} + +\subsubsection*{$q$-Meixner $\rightarrow$ $q$-Charlier} +The $q$-Charlier polynomials given by (\ref{DefqCharlier}) can easily be +obtained from the $q$-Meixner given by (\ref{DefqMeixner}) by setting $b=0$ +in the definition (\ref{DefqMeixner}) of the $q$-Meixner polynomials: +\begin{equation} +M_n(x;0,c;q)=C_n(x;c;q). +\end{equation} + +\subsubsection*{$q$-Meixner $\rightarrow$ Al-Salam-Carlitz~II} +The Al-Salam-Carlitz~II polynomials given by (\ref{DefAlSalamCarlitzII}) +can be obtained from the $q$-Meixner polynomials given by (\ref{DefqMeixner}) +by setting $b=-ac^{-1}$ in the definition (\ref{DefqMeixner}) of the +$q$-Meixner polynomials and then taking the limit $c\rightarrow 0$: +\begin{equation} +\lim_{c\rightarrow 0}M_n(x;-ac^{-1},c;q)= +\left(-\frac{1}{a}\right)^nq^{\binom{n}{2}}V_n^{(a)}(x;q). +\end{equation} + +\subsubsection*{$q$-Meixner $\rightarrow$ Meixner} +To find the Meixner polynomials given by (\ref{DefMeixner}) from the $q$-Meixner polynomials given by +(\ref{DefqMeixner}) we set $b=q^{\beta-1}$ and $c\rightarrow (1-c)^{-1}c$ and let $q\rightarrow 1$: +\begin{equation} +\lim_{q\rightarrow 1}M_n(q^{-x};q^{\beta-1},(1-c)^{-1}c;q)=M_n(x;\beta,c). +\end{equation} + +\subsection*{Remarks} +The $q$-Meixner polynomials given by (\ref{DefqMeixner}) and the little +$q$-Jacobi polynomials given by (\ref{DefLittleqJacobi}) are related in the +following way: +$$M_n(q^{-x};b,c;q)=p_n(-c^{-1}q^n;b,b^{-1}q^{-n-x-1}|q).$$ + +\noindent +The $q$-Meixner polynomials and the quantum $q$-Krawtchouk polynomials +given by (\ref{DefQuantumqKrawtchouk}) are related in the following way: +$$K_n^{qtm}(q^{-x};p,N;q)=M_n(q^{-x};q^{-N-1},-p^{-1};q).$$ + +\subsection*{References} +\cite{AlSalam90}, \cite{AlSalamVerma82II}, \cite{AlSalamVerma88}, \cite{AlvarezRonveaux}, +\cite{AtakAtakKlimyk}, \cite{AtakRahmanSuslov}, \cite{Campigotto+}, \cite{GasperRahman90}, +\cite{Hahn}, \cite{Ismail2005I}, \cite{Nikiforov+}, \cite{Smirnov}. + + +\section{Quantum $q$-Krawtchouk} +\index{Quantum q-Krawtchouk polynomials@Quantum $q$-Krawtchouk polynomials} +\index{q-Krawtchouk polynomials@$q$-Krawtchouk polynomials!Quantum} +\par\setcounter{equation}{0} + +\subsection*{Basic hypergeometric representation} +\begin{equation} +\label{DefQuantumqKrawtchouk} +K_n^{qtm}(q^{-x};p,N;q)= +\qhyp{2}{1}{q^{-n},q^{-x}}{q^{-N}}{pq^{n+1}},\quad n=0,1,2,\ldots,N. +\end{equation} + +\subsection*{Orthogonality relation} +\begin{eqnarray} +\label{OrtQuantumqKrawtchouk} +& &\sum_{x=0}^N\frac{(pq;q)_{N-x}}{(q;q)_x(q;q)_{N-x}}(-1)^{N-x}q^{\binom{x}{2}} +K_m^{qtm}(q^{-x};p,N;q)K_n^{qtm}(q^{-x};p,N;q)\nonumber\\ +& &{}=\frac{(-1)^np^N(q;q)_{N-n}(q,pq;q)_n}{(q,q;q)_N} +q^{\binom{N+1}{2}-\binom{n+1}{2}+Nn}\,\delta_{mn},\quad p>q^{-N}. +\end{eqnarray} + +\subsection*{Recurrence relation} +\begin{eqnarray} +\label{RecQuantumqKrawtchouk} +& &-pq^{2n+1}(1-q^{-x})K_n^{qtm}(q^{-x})\nonumber\\ +& &{}=(1-q^{n-N})K_{n+1}^{qtm}(q^{-x})\nonumber\\ +& &{}\mathindent{}-\left[(1-q^{n-N})+q(1-q^n)(1-pq^n)\right]K_n^{qtm}(q^{-x})\nonumber\\ +& &{}\mathindent\mathindent{}+q(1-q^n)(1-pq^n)K_{n-1}^{qtm}(q^{-x}), +\end{eqnarray} +where +$$K_n^{qtm}(q^{-x}):=K_n^{qtm}(q^{-x};p,N;q).$$ + +\subsection*{Normalized recurrence relation} +\begin{eqnarray} +\label{NormRecQuantumqKrawtchouk} +xp_n(x)&=&p_{n+1}(x)+ +\left[1-p^{-1}q^{-2n-1}\left\{(1-q^{n-N})+q(1-q^n)(1-pq^n)\right\}\right]p_n(x)\nonumber\\ +& &{}\mathindent{}+p^{-2}q^{-4n+1}(1-q^n)(1-pq^n)(1-q^{n-N-1})p_{n-1}(x), +\end{eqnarray} +where +$$K_n^{qtm}(q^{-x};p,N;q)=\frac{p^nq^{n^2}}{(q^{-N};q)_n}p_n(q^{-x}).$$ + +\subsection*{$q$-Difference equation} +\begin{equation} +\label{dvQuantumqKrawtchouk} +-p(1-q^n)y(x)=B(x)y(x+1)-\left[B(x)+D(x)\right]y(x)+D(x)y(x-1), +\end{equation} +where +$$y(x)=K_n^{qtm}(q^{-x};p,N;q)$$ +and +$$\left\{\begin{array}{l}\displaystyle B(x)=-q^x(1-q^{x-N})\\ +\\ +\displaystyle D(x)=(1-q^x)(p-q^{x-N-1}).\end{array}\right.$$ + +\subsection*{Forward shift operator} +\begin{eqnarray} +\label{shift1QuantumqKrawtchoukI} +& &K_n^{qtm}(q^{-x-1};p,N;q)-K_n^{qtm}(q^{-x};p,N;q)\nonumber\\ +& &{}=\frac{pq^{-x}(1-q^n)}{1-q^{-N}}K_{n-1}^{qtm}(q^{-x};pq,N-1;q) +\end{eqnarray} +or equivalently +\begin{equation} +\label{shift1QuantumqKrawtchoukII} +\frac{\Delta K_n^{qtm}(q^{-x};p,N;q)}{\Delta q^{-x}}= +\frac{pq(1-q^n)}{(1-q)(1-q^{-N})}K_{n-1}^{qtm}(q^{-x};pq,N-1;q). +\end{equation} + +\subsection*{Backward shift operator} +\begin{eqnarray} +\label{shift2QuantumqKrawtchoukI} +& &(1-q^{x-N-1})K_n^{qtm}(q^{-x};p,N;q)\nonumber\\ +& &{}\mathindent{}+q^{-x}(1-q^x)(p-q^{x-N-1})K_n^{qtm}(q^{-x+1};p,N;q)\nonumber\\ +& &{}=(1-q^{-N-1})K_{n+1}^{qtm}(q^{-x};pq^{-1},N+1;q) +\end{eqnarray} +or equivalently +\begin{eqnarray} +\label{shift2QuantumqKrawtchoukII} +& &\frac{\nabla\left[w(x;p,N;q)K_n^{qtm}(q^{-x};p,N;q)\right]}{\nabla q^{-x}}\nonumber\\ +& &{}=\frac{1}{1-q}w(x;pq^{-1},N+1;q)K_{n+1}^{qtm}(q^{-x};pq^{-1},N+1;q), +\end{eqnarray} +where +$$w(x;p,N;q)=\frac{(q^{-N};q)_x}{(q,p^{-1}q^{-N};q)_x}(-p)^{-x}q^{\binom{x+1}{2}}.$$ + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodQuantumqKrawtchouk} +w(x;p,N;q)K_n^{qtm}(q^{-x};p,N;q)=(1-q)^n\left(\nabla_q\right)^n\left[w(x;pq^n,N-n;q)\right], +\end{equation} +where +$$\nabla_q:=\frac{\nabla}{\nabla q^{-x}}.$$ + +\subsection*{Generating functions} For $x=0,1,2,\ldots,N$ we have +\begin{eqnarray} +\label{GenQuantumqKrawtchouk1} +& &(q^{x-N}t;q)_{N-x}\cdot\qhyp{2}{1}{q^{-x},pq^{N+1-x}}{0}{q^{x-N}t}\nonumber\\ +& &{}=\sum_{n=0}^N\frac{(q^{-N};q)_n}{(q;q)_n}K_n^{qtm}(q^{-x};p,N;q)t^n. +\end{eqnarray} + +\begin{eqnarray} +\label{GenQuantumqKrawtchouk2} +& &(q^{-x}t;q)_x\cdot\qhyp{2}{1}{q^{x-N},0}{pq}{q^{-x}t}\nonumber\\ +& &{}=\sum_{n=0}^N\frac{(q^{-N};q)_n}{(pq,q;q)_n}K_n^{qtm}(q^{-x};p,N;q)t^n. +\end{eqnarray} + +\subsection*{Limit relations} + +\subsubsection*{$q$-Hahn $\rightarrow$ Quantum $q$-Krawtchouk} +The quantum $q$-Krawtchouk polynomials given by (\ref{DefQuantumqKrawtchouk}) +simply follow from the $q$-Hahn polynomials by setting $\beta=p$ in the definition (\ref{DefqHahn}) of +the $q$-Hahn polynomials and taking the limit $\alpha\rightarrow\infty$: +$$\lim_{\alpha\rightarrow\infty}Q_n(q^{-x};\alpha,p,N|q)=K_n^{qtm}(q^{-x};p,N;q).$$ + +\subsubsection*{Quantum $q$-Krawtchouk $\rightarrow$ Al-Salam-Carlitz~II} +If we set $p=a^{-1}q^{-N-1}$ in the definition (\ref{DefQuantumqKrawtchouk}) +of the quantum $q$-Krawtchouk polynomials and let $N\rightarrow\infty$ we +obtain the Al-Salam-Carlitz~II polynomials given by +(\ref{DefAlSalamCarlitzII}). In fact we have +\begin{equation} +\lim_{N\rightarrow\infty}K_n^{qtm}(x;a^{-1}q^{-N-1},N;q)= +\left(-\frac{1}{a}\right)^nq^{\binom{n}{2}}V_n^{(a)}(x;q). +\end{equation} + +\subsubsection*{Quantum $q$-Krawtchouk $\rightarrow$ Krawtchouk} +The Krawtchouk polynomials given by (\ref{DefKrawtchouk}) easily follow from +the quantum $q$-Krawtchouk polynomials given by (\ref{DefQuantumqKrawtchouk}) +in the following way: +\begin{equation} +\lim_{q\rightarrow 1}K_n^{qtm}(q^{-x};p,N;q)=K_n(x;p^{-1},N). +\end{equation} + +\subsection*{Remarks} +The quantum $q$-Krawtchouk polynomials given by +(\ref{DefQuantumqKrawtchouk}) and the $q$-Meixner polynomials given by +(\ref{DefqMeixner}) are related in the following way: +$$K_n^{qtm}(q^{-x};p,N;q)=M_n(q^{-x};q^{-N-1},-p^{-1};q).$$ + +\noindent +The quantum $q$-Krawtchouk polynomials are related to the affine +$q$-Krawtchouk polynomials given by (\ref{DefAffqKrawtchouk}) +by the transformation $q\leftrightarrow q^{-1}$ in the following way: +$$K_n^{qtm}(q^x;p,N;q^{-1})=(p^{-1}q;q)_n\left(-\frac{p}{q}\right)^nq^{-\binom{n}{2}} +K_n^{Aff}(q^{x-N};p^{-1},N;q).$$ + +\subsection*{References} +\cite{GasperRahman90}, \cite{Koorn89III}, \cite{Koorn90II}, \cite{Smirnov}. + + +\section{$q$-Krawtchouk}\index{q-Krawtchouk polynomials@$q$-Krawtchouk polynomials} +\par\setcounter{equation}{0} + +\subsection*{Basic hypergeometric representation} For $n=0,1,2,\ldots,N$ we have +\begin{eqnarray} +\label{DefqKrawtchouk} +K_n(q^{-x};p,N;q)&=&\qhyp{3}{2}{q^{-n},q^{-x},-pq^n}{q^{-N},0}{q}\\ +&=&\frac{(q^{x-N};q)_n}{(q^{-N};q)_nq^{nx}}\, +\qhyp{2}{1}{q^{-n},q^{-x}}{q^{N-x-n+1}}{-pq^{n+N+1}}.\nonumber +\end{eqnarray} + +\subsection*{Orthogonality relation} +\begin{eqnarray} +\label{OrtqKrawtchouk} +& &\sum_{x=0}^N\frac{(q^{-N};q)_x}{(q;q)_x}(-p)^{-x}K_m(q^{-x};p,N;q)K_n(q^{-x};p,N;q)\nonumber\\ +& &{}=\frac{(q,-pq^{N+1};q)_n}{(-p,q^{-N};q)_n}\frac{(1+p)}{(1+pq^{2n})}\nonumber\\ +& &{}\mathindent{}\times (-pq;q)_Np^{-N}q^{-\binom{N+1}{2}} +\left(-pq^{-N}\right)^nq^{n^2}\,\delta_{mn},\quad p>0. +\end{eqnarray} + +\subsection*{Recurrence relation} +\begin{eqnarray} +\label{RecqKrawtchouk} +-\left(1-q^{-x}\right)K_n(q^{-x})&=&A_nK_{n+1}(q^{-x})-\left(A_n+C_n\right)K_n(q^{-x})\nonumber\\ +& &{}\mathindent{}+C_nK_{n-1}(q^{-x}), +\end{eqnarray} +where +$$K_n(q^{-x}):=K_n(q^{-x};p,N;q)$$ +and +$$\left\{\begin{array}{l} +\displaystyle A_n=\frac{(1-q^{n-N})(1+pq^n)}{(1+pq^{2n})(1+pq^{2n+1})}\\ +\\ +\displaystyle C_n=-pq^{2n-N-1}\frac{(1+pq^{n+N})(1-q^n)}{(1+pq^{2n-1})(1+pq^{2n})}. +\end{array}\right.$$ + +\subsection*{Normalized recurrence relation} +\begin{equation} +\label{NormRecqKrawtchouk} +xp_n(x)=p_{n+1}(x)+\left[1-(A_n+C_n)\right]p_n(x)+A_{n-1}C_np_{n-1}(x), +\end{equation} +where +$$K_n(q^{-x};p,N;q)=\frac{(-pq^n;q)_n}{(q^{-N};q)_n}p_n(q^{-x}).$$ + +\subsection*{$q$-Difference equation} +\begin{eqnarray} +\label{dvqKrawtchouk} +& &q^{-n}(1-q^n)(1+pq^n)y(x)\nonumber\\ +& &{}=(1-q^{x-N})y(x+1)-\left[(1-q^{x-N})-p(1-q^x)\right]y(x)\nonumber\\ +& &{}\mathindent{}-p(1-q^x)y(x-1), +\end{eqnarray} +where +$$y(x)=K_n(q^{-x};p,N;q).$$ + +\subsection*{Forward shift operator} +\begin{eqnarray} +\label{shift1qKrawtchoukI} +& &K_n(q^{-x-1};p,N;q)-K_n(q^{-x};p,N;q)\nonumber\\ +& &{}=\frac{q^{-n-x}(1-q^n)(1+pq^n)}{1-q^{-N}}K_{n-1}(q^{-x};pq^2,N-1;q) +\end{eqnarray} +or equivalently +\begin{equation} +\label{shift1qKrawtchoukII} +\frac{\Delta K_n(q^{-x};p,N;q)}{\Delta q^{-x}}= +\frac{q^{-n+1}(1-q^n)(1+pq^n)}{(1-q)(1-q^{-N})}K_{n-1}(q^{-x};pq^2,N-1;q). +\end{equation} + +\subsection*{Backward shift operator} +\begin{eqnarray} +\label{shift2qKrawtchoukI} +& &(1-q^{x-N-1})K_n(q^{-x};p,N;q)+pq^{-1}(1-q^x)K_n(q^{-x+1};p,N;q)\nonumber\\ +& &{}=q^x(1-q^{-N-1})K_{n+1}(q^{-x};pq^{-2},N+1;q) +\end{eqnarray} +or equivalently +\begin{eqnarray} +\label{shift2qKrawtchoukII} +& &\frac{\nabla\left[w(x;p,N;q)K_n(q^{-x};p,N;q)\right]}{\nabla q^{-x}}\nonumber\\ +& &{}=\frac{1}{1-q}w(x;pq^{-2},N+1;q)K_{n+1}(q^{-x};pq^{-2},N+1;q), +\end{eqnarray} +where +$$w(x;p,N;q)=\frac{(q^{-N};q)_x}{(q;q)_x}\left(-\frac{q}{p}\right)^x.$$ + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodqKrawtchouk} +w(x;p,N;q)K_n(q^{-x};p,N;q)=(1-q)^n\left(\nabla_q\right)^n\left[w(x;pq^{2n},N-n;q)\right], +\end{equation} +where +$$\nabla_q:=\frac{\nabla}{\nabla q^{-x}}.$$ + +\subsection*{Generating function} For $x=0,1,2,\ldots,N$ we have +\begin{eqnarray} +\label{GenqKrawtchouk} +& &\qhyp{1}{1}{q^{-x}}{0}{pqt}\,\qhyp{2}{0}{q^{x-N},0}{-}{-q^{-x}t}\nonumber\\ +& &{}=\sum_{n=0}^N\frac{(q^{-N};q)_n}{(q;q)_n}q^{-\binom{n}{2}}K_n(q^{-x};p,N;q)t^n. +\end{eqnarray} + +\subsection*{Limit relations} + +\subsubsection*{$q$-Racah $\rightarrow$ $q$-Krawtchouk} +The $q$-Krawtchouk polynomials given by (\ref{DefqKrawtchouk}) can be obtained from +the $q$-Racah polynomials by setting $\alpha q=q^{-N}$, $\beta=-pq^N$ and +$\gamma=\delta=0$ in the definition (\ref{DefqRacah}) of the $q$-Racah polynomials: +$$R_n(q^{-x};q^{-N-1},-pq^N,0,0|q)=K_n(q^{-x};p,N;q).$$ +Note that $\mu(x)=q^{-x}$ in this case. + +\subsubsection*{$q$-Hahn $\rightarrow$ $q$-Krawtchouk} +If we set $\beta=-\alpha^{-1}q^{-1}p$ in the definition (\ref{DefqHahn}) of the $q$-Hahn polynomials +and then let $\alpha\rightarrow 0$ we obtain the $q$-Krawtchouk polynomials given by +(\ref{DefqKrawtchouk}): +$$\lim_{\alpha\rightarrow 0} +Q_n(q^{-x};\alpha,-\alpha^{-1}q^{-1}p,N|q)=K_n(q^{-x};p,N;q).$$ + +\subsubsection*{$q$-Krawtchouk $\rightarrow$ $q$-Bessel} +If we set $x\rightarrow N-x$ in the definition (\ref{DefqKrawtchouk}) of the +$q$-Krawtchouk polynomials and then take the limit $N\rightarrow\infty$ we +obtain the $q$-Bessel polynomials given by (\ref{DefqBessel}): +\begin{equation} +\lim_{N\rightarrow\infty}K_n(q^{x-N};p,N;q)=y_n(q^x;p;q). +\end{equation} + +\subsubsection*{$q$-Krawtchouk $\rightarrow$ $q$-Charlier} +By setting $p=a^{-1}q^{-N}$ in the definition (\ref{DefqKrawtchouk}) of the +$q$-Krawtchouk polynomials and then taking the limit $N\rightarrow\infty$ we +obtain the $q$-Charlier polynomials given by (\ref{DefqCharlier}): +\begin{equation} +\lim_{N\rightarrow\infty}K_n(q^{-x};a^{-1}q^{-N},N;q)=C_n(q^{-x};a;q). +\end{equation} + +\subsubsection*{$q$-Krawtchouk $\rightarrow$ Krawtchouk} +If we take the limit $q\rightarrow 1$ in the definition (\ref{DefqKrawtchouk}) of the $q$-Krawtchouk +polynomials we simply find the Krawtchouk polynomials given by (\ref{DefKrawtchouk}) in the +following way: +\begin{equation} +\lim_{q\rightarrow 1}K_n(q^{-x};p,N;q)=K_n(x;(p+1)^{-1},N). +\end{equation} + +\subsection*{Remark} +The $q$-Krawtchouk polynomials given by (\ref{DefqKrawtchouk}) and the +dual $q$-Krawtchouk polynomials given by (\ref{DefDualqKrawtchouk}) are +related in the following way: +$$K_n(q^{-x};p,N;q)=K_x(\lambda(n);-pq^N,N|q)$$ +with +$$\lambda(n)=q^{-n}-pq^n$$ +or +$$K_n(\lambda(x);c,N|q)=K_x(q^{-n};-cq^{-N},N;q)$$ +with +$$\lambda(x)=q^{-x}+cq^{x-N}.$$ + +\subsection*{References} +\cite{AlvarezRonveaux}, \cite{AskeyWilson79}, \cite{AtakRahmanSuslov}, +\cite{Campigotto+}, \cite{GasperRahman90}, \cite{Nikiforov+}, +\cite{NoumiMimachi91}, \cite{Stanton80III}, \cite{Stanton84}. + + +\newpage + +\section{Affine $q$-Krawtchouk} +\index{Affine q-Krawtchouk polynomials@Affine $q$-Krawtchouk polynomials} +\index{q-Krawtchouk polynomials@$q$-Krawtchouk polynomials!Affine} +\par\setcounter{equation}{0} + +\subsection*{Basic hypergeometric representation} +\begin{eqnarray} +\label{DefAffqKrawtchouk} +K_n^{Aff}(q^{-x};p,N;q)&=&\qhyp{3}{2}{q^{-n},0,q^{-x}}{pq,q^{-N}}{q}\\ +&=&\frac{(-pq)^nq^{\binom{n}{2}}}{(pq;q)_n}\, +\qhyp{2}{1}{q^{-n},q^{x-N}}{q^{-N}}{\frac{q^{-x}}{p}},\quad n=0,1,2,\ldots,N.\nonumber +\end{eqnarray} + +\subsection*{Orthogonality relation} +\begin{eqnarray} +\label{OrtAffqKrawtchouk} +& &\sum_{x=0}^N\frac{(pq;q)_x(q;q)_N}{(q;q)_x(q;q)_{N-x}}(pq)^{-x}K_m^{Aff}(q^{-x};p,N;q)K_n^{Aff}(q^{-x};p,N;q)\nonumber\\ +& &{}=(pq)^{n-N}\frac{(q;q)_n(q;q)_{N-n}}{(pq;q)_n(q;q)_N}\,\delta_{mn},\quad 01$, then we have another orthogonality relation given by: +\begin{eqnarray} +\label{OrtContBigqHermite2} +& &\frac{1}{2\pi}\int_{-1}^1\frac{w(x)}{\sqrt{1-x^2}}H_m(x;a|q)H_n(x;a|q)\,dx\nonumber\\ +& &{}\mathindent{}+\sum_{\begin{array}{c}\scriptstyle k\\ \scriptstyle 1-1 +\end{eqnarray} +and +\begin{eqnarray} +\label{OrtqLaguerre2} +& &\sum_{k=-\infty}^{\infty}\frac{q^{k\alpha+k}}{(-cq^k;q)_{\infty}}L_m^{(\alpha)}(cq^k;q)L_n^{(\alpha)}(cq^k;q)\nonumber\\ +& &{}=\frac{(q,-cq^{\alpha+1},-c^{-1}q^{-\alpha};q)_{\infty}} +{(q^{\alpha+1},-c,-c^{-1}q;q)_{\infty}}\frac{(q^{\alpha+1};q)_n}{(q;q)_nq^n}\,\delta_{mn}, +\quad\alpha>-1,\quad c>0. +\end{eqnarray} +For $c=1$ the latter orthogonality relation can also be written as +\begin{eqnarray} +\label{OrtqLaguerre3} +& &\int_0^{\infty}\frac{x^{\alpha}}{(-x;q)_{\infty}}L_m^{(\alpha)}(x;q)L_n^{(\alpha)}(x;q)\,d_qx\nonumber\\ +& &{}=\frac{1-q}{2}\,\frac{(q,-q^{\alpha+1},-q^{-\alpha};q)_{\infty}} +{(q^{\alpha+1},-q,-q;q)_{\infty}}\frac{(q^{\alpha+1};q)_n}{(q;q)_nq^n}\,\delta_{mn},\quad\alpha>-1. +\end{eqnarray} + +\newpage + +\subsection*{Recurrence relation} +\begin{eqnarray} +\label{RecqLaguerre} +-q^{2n+\alpha+1}xL_n^{(\alpha)}(x;q)&=&(1-q^{n+1})L_{n+1}^{(\alpha)}(x;q)\nonumber\\ +& &{}\mathindent{}-\left[(1-q^{n+1})+q(1-q^{n+\alpha})\right]L_n^{(\alpha)}(x;q)\nonumber\\ +& &{}\mathindent\mathindent{}+q(1-q^{n+\alpha})L_{n-1}^{(\alpha)}(x;q). +\end{eqnarray} + +\subsection*{Normalized recurrence relation} +\begin{eqnarray} +\label{NormRecqLaguerre} +xp_n(x)&=&p_{n+1}(x)+q^{-2n-\alpha-1}\left[(1-q^{n+1})+q(1-q^{n+\alpha})\right]p_n(x)\nonumber\\ +& &{}\mathindent{}+q^{-4n-2\alpha+1}(1-q^n)(1-q^{n+\alpha})p_{n-1}(x), +\end{eqnarray} +where +$$L_n^{(\alpha)}(x;q)=\frac{(-1)^nq^{n(n+\alpha)}}{(q;q)_n}p_n(x).$$ + +\subsection*{$q$-Difference equation} +\begin{equation} +\label{dvqLaguerre} +-q^{\alpha}(1-q^n)xy(x)=q^{\alpha}(1+x)y(qx)-\left[1+q^{\alpha}(1+x)\right]y(x)+y(q^{-1}x), +\end{equation} +where +$$y(x)=L_n^{(\alpha)}(x;q).$$ + +\subsection*{Forward shift operator} +\begin{equation} +\label{shift1qLaguerreI} +L_n^{(\alpha)}(x;q)-L_n^{(\alpha)}(qx;q)=-q^{\alpha+1}xL_{n-1}^{(\alpha+1)}(qx;q) +\end{equation} +or equivalently +\begin{equation} +\label{shift1qLaguerreII} +\mathcal{D}_qL_n^{(\alpha)}(x;q)=-\frac{q^{\alpha+1}}{1-q}L_{n-1}^{(\alpha+1)}(qx;q). +\end{equation} + +\subsection*{Backward shift operator} +\begin{equation} +\label{shift2qLaguerreI} +L_n^{(\alpha)}(x;q)-q^{\alpha}(1+x)L_n^{(\alpha)}(qx;q)=(1-q^{n+1})L_{n+1}^{(\alpha-1)}(x;q) +\end{equation} +or equivalently +\begin{equation} +\label{shift2qLaguerreII} +\mathcal{D}_q\left[w(x;\alpha;q)L_n^{(\alpha)}(x;q)\right]= +\frac{1-q^{n+1}}{1-q}w(x;\alpha-1;q)L_{n+1}^{(\alpha-1)}(x;q), +\end{equation} +where +$$w(x;\alpha;q)=\frac{x^{\alpha}}{(-x;q)_{\infty}}.$$ + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodqLaguerre} +w(x;\alpha;q)L_n^{(\alpha)}(x;q)= +\frac{(1-q)^n}{(q;q)_n}\left(\mathcal{D}_q\right)^n\left[w(x;\alpha+n;q)\right]. +\end{equation} + +\subsection*{Generating functions} +\begin{equation} +\label{GenqLaguerre1} +\frac{1}{(t;q)_{\infty}}\,\qhyp{1}{1}{-x}{0}{q^{\alpha+1}t} +=\sum_{n=0}^{\infty}L_n^{(\alpha)}(x;q)t^n. +\end{equation} + +\begin{equation} +\label{GenqLaguerre2} +\frac{1}{(t;q)_{\infty}}\,\qhyp{0}{1}{-}{q^{\alpha+1}}{-q^{\alpha+1}xt} +=\sum_{n=0}^{\infty}\frac{L_n^{(\alpha)}(x;q)}{(q^{\alpha+1};q)_n}t^n. +\end{equation} + +\begin{equation} +\label{GenqLaguerre3} +(t;q)_{\infty}\cdot\qhyp{0}{2}{-}{q^{\alpha+1},t}{-q^{\alpha+1}xt} +=\sum_{n=0}^{\infty}\frac{(-1)^nq^{\binom{n}{2}}}{(q^{\alpha+1};q)_n}L_n^{(\alpha)}(x;q)t^n. +\end{equation} + +\begin{eqnarray} +\label{GenqLaguerre4} +& &\frac{(\gamma t;q)_{\infty}}{(t;q)_{\infty}}\,\qhyp{1}{2}{\gamma}{q^{\alpha+1},\gamma t}{-q^{\alpha+1}xt}\nonumber\\ +& &{}=\sum_{n=0}^{\infty}\frac{(\gamma;q)_n}{(q^{\alpha+1};q)_n}L_n^{(\alpha)}(x;q)t^n, +\quad\textrm{$\gamma$ arbitrary}. +\end{eqnarray} + +\subsection*{Limit relations} + +\subsubsection*{Little $q$-Jacobi $\rightarrow$ $q$-Laguerre} +If we substitute $a=q^{\alpha}$ and $x\rightarrow -b^{-1}q^{-1}x$ in the definition +(\ref{DefLittleqJacobi}) of the little $q$-Jacobi polynomials and then take the limit +$b\rightarrow -\infty$ we find the $q$-Laguerre polynomials given by (\ref{DefqLaguerre}): +$$\lim_{b\rightarrow -\infty}p_n(-b^{-1}q^{-1}x;q^{\alpha},b|q)= +\frac{(q;q)_n}{(q^{\alpha+1};q)_n}L_n^{(\alpha)}(x;q).$$ + +\subsubsection*{$q$-Meixner $\rightarrow$ $q$-Laguerre} +The $q$-Laguerre polynomials given by (\ref{DefqLaguerre}) can be obtained +from the $q$-Meixner polynomials given by (\ref{DefqMeixner}) by setting +$b=q^{\alpha}$ and $q^{-x}\rightarrow cq^{\alpha}x$ in the definition +(\ref{DefqMeixner}) of the $q$-Meixner polynomials and then taking the limit +$c\rightarrow\infty$: +$$\lim_{c\rightarrow\infty}M_n(cq^{\alpha}x;q^{\alpha},c;q)= +\frac{(q;q)_n}{(q^{\alpha+1};q)_n}L_n^{(\alpha)}(x;q).$$ + +\subsubsection*{$q$-Laguerre $\rightarrow$ Stieltjes-Wigert} +If we set $x\rightarrow xq^{-\alpha}$ in the definition +(\ref{DefqLaguerre}) of the $q$-Laguerre polynomials and take the limit +$\alpha\rightarrow\infty$ we simply obtain the Stieltjes-Wigert polynomials given +by (\ref{DefStieltjesWigert}): +\begin{equation} +\lim_{\alpha\rightarrow\infty}L_n^{(\alpha)}\left(xq^{-\alpha};q\right)=S_n(x;q). +\end{equation} + +\subsubsection*{$q$-Laguerre $\rightarrow$ Laguerre / Charlier} +If we set $x\rightarrow (1-q)x$ in the definition (\ref{DefqLaguerre}) +of the $q$-Laguerre polynomials and take the limit $q\rightarrow 1$ we obtain +the Laguerre polynomials given by (\ref{DefLaguerre}): +\begin{equation} +\lim_{q\rightarrow 1}L_n^{(\alpha)}((1-q)x;q)=L_n^{(\alpha)}(x). +\end{equation} + +If we set $x\rightarrow -q^{-x}$ and $q^{\alpha}=a^{-1}(q-1)^{-1}$ (or +$\alpha=-(\ln q)^{-1}\ln (q-1)a$) in the definition (\ref{DefqLaguerre}) of the +$q$-Laguerre polynomials, multiply by $(q;q)_n$, and take the limit +$q\rightarrow 1$ we obtain the Charlier polynomials given by (\ref{DefCharlier}): +\begin{equation} +\lim_{q\rightarrow 1}\,(q;q)_nL_n^{(\alpha)}(-q^{-x};q)=C_n(x;a), +\end{equation} +where +$$q^{\alpha}=\frac{1}{a(q-1)}\quad\textrm{or}\quad\alpha=-\frac{\ln (q-1)a}{\ln q}.$$ + +\subsection*{Remarks} +The $q$-Laguerre polynomials are sometimes called the +generalized Stieltjes-Wigert polynomials. + +If we replace $q$ by $q^{-1}$ we obtain the little $q$-Laguerre +(or Wall) polynomials given by (\ref{DefLittleqLaguerre}) in the following +way: +$$L_n^{(\alpha)}(x;q^{-1})=\frac{(q^{\alpha+1};q)_n}{(q;q)_nq^{n\alpha}}p_n(-x;q^{\alpha}|q).$$ + +\noindent +The $q$-Laguerre polynomials given by (\ref{DefqLaguerre}) and the $q$-Bessel polynomials +given by (\ref{DefqBessel}) are related in the following way: +$$\frac{y_n(q^x;a;q)}{(q;q)_n}=L_n^{(x-n)}(aq^n;q).$$ + +\noindent +The $q$-Laguerre polynomials given by (\ref{DefqLaguerre}) and the +$q$-Charlier polynomials given by (\ref{DefqCharlier}) are related in the +following way: +$$\frac{C_n(-x;-q^{-\alpha};q)}{(q;q)_n}=L_n^{(\alpha)}(x;q).$$ + +\noindent +Since the Stieltjes and Hamburger moment problems corresponding to the +$q$-Laguerre polynomials are indeterminate there exist many different weight +functions. + +\subsection*{References} +\cite{NAlSalam89}, \cite{AlSalam90}, \cite{Askey86}, \cite{Askey89I}, \cite{AskeyWilson85}, +\cite{AtakAtakI}, \cite{ChenIsmailMuttalib}, \cite{Chihara68II}, \cite{Chihara78}, +\cite{Chihara79}, \cite{Chris}, \cite{Exton77}, \cite{GasperRahman90}, \cite{GrunbaumHaine96}, +\cite{Ismail2005I}, \cite{IsmailRahman98}, \cite{Jain95}, \cite{Moak}. + + +\section{$q$-Bessel} +\index{q-Bessel polynomials@$q$-Bessel polynomials} +\par\setcounter{equation}{0} + +\subsection*{Basic hypergeometric representation} +\begin{eqnarray} +\label{DefqBessel} +y_n(x;a;q)&=&\qhyp{2}{1}{q^{-n},-aq^n}{0}{qx}\\ +&=&(q^{-n+1}x;q)_n\cdot\qhyp{1}{1}{q^{-n}}{q^{-n+1}x}{-aq^{n+1}x}\nonumber\\ +&=&\left(-aq^nx\right)^n\cdot\qhyp{2}{1}{q^{-n},x^{-1}}{0}{-\frac{q^{-n+1}}{a}}.\nonumber +\end{eqnarray} + +\newpage + +\subsection*{Orthogonality relation} +\begin{eqnarray} +\label{OrtqBessel} +& &\sum_{k=0}^{\infty}\frac{a^k}{(q;q)_k}q^{\binom{k+1}{2}}y_m(q^k;a;q)y_n(q^k;a;q)\nonumber\\ +& &{}=(q;q)_n(-aq^n;q)_{\infty}\frac{a^nq^{\binom{n+1}{2}}}{(1+aq^{2n})}\,\delta_{mn},\quad a>0. +\end{eqnarray} + +\subsection*{Recurrence relation} +\begin{equation} +\label{RecqBessel} +-xy_n(x;a;q)=A_ny_{n+1}(x;a;q)-(A_n+C_n)y_n(x;a;q)+C_ny_{n-1}(x;a;q), +\end{equation} +where +$$\left\{\begin{array}{l}\displaystyle A_n=q^n\frac{(1+aq^n)}{(1+aq^{2n})(1+aq^{2n+1})}\\ +\\ +\displaystyle C_n=aq^{2n-1}\frac{(1-q^n)}{(1+aq^{2n-1})(1+aq^{2n})}.\end{array}\right.$$ + +\subsection*{Normalized recurrence relation} +\begin{equation} +\label{NormRecqBessel} +xp_n(x)=p_{n+1}(x)+(A_n+C_n)p_n(x)+A_{n-1}C_np_{n-1}(x), +\end{equation} +where +$$y_n(x;a;q)=(-1)^nq^{-\binom{n}{2}}(-aq^n;q)_np_n(x).$$ + +\subsection*{$q$-Difference equation} +\begin{eqnarray} +\label{dvqBessel} +& &-q^{-n}(1-q^n)(1+aq^n)xy(x)\nonumber\\ +& &{}=axy(qx)-(ax+1-x)y(x)+(1-x)y(q^{-1}x), +\end{eqnarray} +where +$$y(x)=y_n(x;a;q).$$ + +\newpage + +\subsection*{Forward shift operator} +\begin{equation} +\label{shift1qBesselI} +y_n(x;a;q)-y_n(qx;a;q)=-q^{-n+1}(1-q^n)(1+aq^n)xy_{n-1}(x;aq^2;q) +\end{equation} +or equivalently +\begin{equation} +\label{shift1qBesselII} +\mathcal{D}_qy_n(x;a;q)=-\frac{q^{-n+1}(1-q^n)(1+aq^n)}{1-q}y_{n-1}(x;aq^2;q). +\end{equation} + +\subsection*{Backward shift operator} +\begin{equation} +\label{shift2qBesselI} +aq^{x-1}y_n(q^x;a;q)-(1-q^x)y_n(q^{x-1}x;a;q)=-y_{n+1}(q^x;aq^{-2};q) +\end{equation} +or equivalently +\begin{equation} +\label{shift2qBesselII} +\frac{\nabla\left[w(x;a;q)y_n(q^x;a;q)\right]}{\nabla q^x}= +\frac{q^2}{a(1-q)}w(x;aq^{-2};q)y_{n+1}(q^x;aq^{-2};q), +\end{equation} +where +$$w(x;a;q)=\frac{a^xq^{\binom{x}{2}}}{(q;q)_x}.$$ + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodqBessel} +w(x;a;q)y_n(q^x;a;q)=a^n(1-q)^nq^{n(n-1)}\left(\nabla_q\right)^n\left[w(x;aq^{2n};q)\right], +\end{equation} +where +$$\nabla_q:=\frac{\nabla}{\nabla q^x}.$$ + +\subsection*{Generating functions} +\begin{eqnarray} +\label{GenqBessel1} +& &\qhyp{0}{1}{-}{0}{-aq^{x+1}t}\,\qhyp{2}{0}{q^{-x},0}{-}{q^xt}\nonumber\\ +& &{}=\sum_{n=0}^{\infty}\frac{y_n(q^x;a;q)}{(q;q)_n}t^n,\quad x=0,1,2,\ldots. +\end{eqnarray} + +\begin{equation} +\label{GenqBessel2} +\frac{(t;q)_{\infty}}{(xt;q)_{\infty}}\,\qhyp{1}{3}{xt}{0,0,t}{-aqxt}= +\sum_{n=0}^{\infty}\frac{(-1)^nq^{\binom{n}{2}}}{(q;q)_n}y_n(x;a;q)t^n. +\end{equation} + +\subsection*{Limit relations} + +\subsubsection*{Little $q$-Jacobi $\rightarrow$ $q$-Bessel} +If we set $b\rightarrow -a^{-1}q^{-1}b$ in the definition (\ref{DefLittleqJacobi}) of +the little $q$-Jacobi polynomials and then take the limit $a\rightarrow 0$ we obtain +the $q$-Bessel polynomials given by (\ref{DefqBessel}): +$$\lim_{a\rightarrow 0}p_n(x;a,-a^{-1}q^{-1}b|q)=y_n(x;b;q).$$ + +\subsubsection*{$q$-Krawtchouk $\rightarrow$ $q$-Bessel} +If we set $x\rightarrow N-x$ in the definition (\ref{DefqKrawtchouk}) of the +$q$-Krawtchouk polynomials and then take the limit $N\rightarrow\infty$ we +obtain the $q$-Bessel polynomials given by (\ref{DefqBessel}): +$$\lim_{N\rightarrow\infty}K_n(q^{x-N};p,N;q)=y_n(q^x;p;q).$$ + +\subsubsection*{$q$-Bessel $\rightarrow$ Stieltjes-Wigert} +The Stieltjes-Wigert polynomials given by (\ref{DefStieltjesWigert}) can be obtained +from the $q$-Bessel polynomials by setting $x\rightarrow a^{-1}x$ in the definition +(\ref{DefqBessel}) of the $q$-Bessel polynomials and then taking the limit +$a\rightarrow\infty$. In fact we have +\begin{equation} +\lim_{a\rightarrow\infty}y_n(a^{-1}x;a;q)=(q;q)_nS_n(x;q). +\end{equation} + +\subsubsection*{$q$-Bessel $\rightarrow$ Bessel} +If we set $x\rightarrow -\frac{1}{2}(1-q)^{-1}x$ and $a\rightarrow -q^{a+1}$ in the definition +(\ref{DefqBessel}) of the $q$-Bessel polynomials and take the limit $q\rightarrow 1$ +we find the Bessel polynomials given by (\ref{DefBessel}): +\begin{equation} +\lim_{q\rightarrow 1}y_n(-\textstyle\frac{1}{2}(1-q)^{-1}x;-q^{a+1};q)=y_n(x;a). +\end{equation} + +\subsubsection*{$q$-Bessel $\rightarrow$ Charlier} +If we set $x\rightarrow q^x$ and $a\rightarrow a(1-q)$ in the definition (\ref{DefqBessel}) +of the $q$-Bessel polynomials and take the limit $q\rightarrow 1$ we find the Charlier +polynomials given by (\ref{DefCharlier}): +\begin{equation} +\lim_{q\rightarrow 1}\frac{y_n(q^x;a(1-q);q)}{(q-1)^n}=a^nC_n(x;a). +\end{equation} + +\subsection*{Remark} +In \cite{Koekoek94} and \cite{Koekoek98} these $q$-Bessel polynomials were called +\emph{alternative $q$-Charlier polynomials}. + +\noindent +The $q$-Bessel polynomials given by (\ref{DefqBessel}) and the $q$-Laguerre +polynomials given by (\ref{DefqLaguerre}) are related in the following way: +$$\frac{y_n(q^x;a;q)}{(q;q)_n}=L_n^{(x-n)}(aq^n;q).$$ + +\subsection*{Reference} +\cite{DattaGriffin}. + + +\section{$q$-Charlier}\index{q-Charlier polynomials@$q$-Charlier polynomials} +\par\setcounter{equation}{0} + +\subsection*{Basic hypergeometric representation} +\begin{eqnarray} +\label{DefqCharlier} +C_n(q^{-x};a;q)&=&\qhyp{2}{1}{q^{-n},q^{-x}}{0}{-\frac{q^{n+1}}{a}}\\ +&=&(-a^{-1}q;q)_n\cdot\qhyp{1}{1}{q^{-n}}{-a^{-1}q}{-\frac{q^{n+1-x}}{a}}.\nonumber +\end{eqnarray} + +\subsection*{Orthogonality relation} +\begin{eqnarray} +\label{OrtqCharlier} +& &\sum_{x=0}^{\infty}\frac{a^x}{(q;q)_x}q^{\binom{x}{2}}C_m(q^{-x};a;q)C_n(q^{-x};a;q)\nonumber\\ +& &{}=q^{-n}(-a;q)_{\infty}(-a^{-1}q,q;q)_n\,\delta_{mn},\quad a>0. +\end{eqnarray} + +\newpage + +\subsection*{Recurrence relation} +\begin{eqnarray} +\label{RecqCharlier} +& &q^{2n+1}(1-q^{-x})C_n(q^{-x})\nonumber\\ +& &{}=aC_{n+1}(q^{-x})-\left[a+q(1-q^n)(a+q^n)\right]C_n(q^{-x})\nonumber\\ +& &{}\mathindent{}+q(1-q^n)(a+q^n)C_{n-1}(q^{-x}), +\end{eqnarray} +where +$$C_n(q^{-x}):=C_n(q^{-x};a;q).$$ + +\subsection*{Normalized recurrence relation} +\begin{eqnarray} +\label{NormRecqCharlier} +xp_n(x)&=&p_{n+1}(x)+\left[1+q^{-2n-1}\left\{a+q(1-q^n)(a+q^n)\right\}\right]p_n(x)\nonumber\\ +& &{}\mathindent{}+aq^{-4n+1}(1-q^n)(a+q^n)p_{n-1}(x), +\end{eqnarray} +where +$$C_n(q^{-x};a;q)=\frac{(-1)^nq^{n^2}}{a^n}p_n(q^{-x}).$$ + +\subsection*{$q$-Difference equation} +\begin{equation} +\label{dvqCharlier} +q^ny(x)=aq^xy(x+1)-q^x(a-1)y(x)+(1-q^x)y(x-1), +\end{equation} +where +$$y(x)=C_n(q^{-x};a;q).$$ + +\subsection*{Forward shift operator} +\begin{equation} +\label{shift1qCharlierI} +C_n(q^{-x-1};a;q)-C_n(q^{-x};a;q)=-a^{-1}q^{-x}(1-q^n)C_{n-1}(q^{-x};aq^{-1};q) +\end{equation} +or equivalently +\begin{equation} +\label{shift1qCharlierII} +\frac{\Delta C_n(q^{-x};a;q)}{\Delta q^{-x}}=-\frac{q(1-q^n)}{a(1-q)}C_{n-1}(q^{-x};aq^{-1};q). +\end{equation} + +\newpage + +\subsection*{Backward shift operator} +\begin{equation} +\label{shift2qCharlierI} +C_n(q^{-x};a;q)-a^{-1}q^{-x}(1-q^x)C_n(q^{-x+1};a;q)=C_{n+1}(q^{-x};aq;q) +\end{equation} +or equivalently +\begin{equation} +\label{shift2qCharlierII} +\frac{\nabla\left[w(x;a;q)C_n(q^{-x};a;q)\right]}{\nabla q^{-x}} +=\frac{1}{1-q}w(x;aq;q)C_{n+1}(q^{-x};aq;q), +\end{equation} +where +$$w(x;a;q)=\frac{a^xq^{\binom{x+1}{2}}}{(q;q)_x}.$$ + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodqCharlier} +w(x;a;q)C_n(q^{-x};a;q)=(1-q)^n\left(\nabla_q\right)^n\left[w(x;aq^{-n};q)\right], +\end{equation} +where +$$\nabla_q:=\frac{\nabla}{\nabla q^{-x}}.$$ + +\subsection*{Generating functions} +\begin{equation} +\label{GenqCharlier1} +\frac{1}{(t;q)_{\infty}}\,\qhyp{1}{1}{q^{-x}}{0}{-a^{-1}qt} +=\sum_{n=0}^{\infty}\frac{C_n(q^{-x};a;q)}{(q;q)_n}t^n. +\end{equation} + +\begin{equation} +\label{GenqCharlier2} +\frac{1}{(t;q)_{\infty}}\,\qhyp{0}{1}{-}{-a^{-1}q}{-a^{-1}q^{-x+1}t} +=\sum_{n=0}^{\infty}\frac{C_n(q^{-x};a;q)}{(-a^{-1}q,q;q)_n}t^n. +\end{equation} + +\subsection*{Limit relations} + +\subsubsection*{$q$-Meixner $\rightarrow$ $q$-Charlier} +The $q$-Charlier polynomials given by (\ref{DefqCharlier}) can easily be +obtained from the $q$-Meixner given by (\ref{DefqMeixner}) by setting $b=0$ +in the definition (\ref{DefqMeixner}) of the $q$-Meixner polynomials: +\begin{equation} +M_n(x;0,c;q)=C_n(x;c;q). +\end{equation} + +\subsubsection*{$q$-Krawtchouk $\rightarrow$ $q$-Charlier} +By setting $p=a^{-1}q^{-N}$ in the definition (\ref{DefqKrawtchouk}) of the +$q$-Krawtchouk polynomials and then taking the limit $N\rightarrow\infty$ we +obtain the $q$-Charlier polynomials given by (\ref{DefqCharlier}): +$$\lim_{N\rightarrow\infty}K_n(q^{-x};a^{-1}q^{-N},N;q)=C_n(q^{-x};a;q).$$ + +\subsubsection*{$q$-Charlier $\rightarrow$ Stieltjes-Wigert} +If we set $q^{-x}\rightarrow ax$ in the definition (\ref{DefqCharlier}) of the +$q$-Charlier polynomials and take the limit $a\rightarrow\infty$ we obtain +the Stieltjes-Wigert polynomials given by (\ref{DefStieltjesWigert}) in the +following way: +\begin{equation} +\lim_{a\rightarrow\infty}C_n(ax;a;q)=(q;q)_nS_n(x;q). +\end{equation} + +\subsubsection*{$q$-Charlier $\rightarrow$ Charlier} +If we set $a\rightarrow a(1-q)$ in the definition (\ref{DefqCharlier}) +of the $q$-Charlier polynomials and take the limit $q\rightarrow 1$ we obtain the +Charlier polynomials given by (\ref{DefCharlier}): +\begin{equation} +\lim_{q\rightarrow 1}C_n(q^{-x};a(1-q);q)=C_n(x;a). +\end{equation} + +\subsection*{Remark} +The $q$-Charlier polynomials given by (\ref{DefqCharlier}) and the +$q$-Laguerre polynomials given by (\ref{DefqLaguerre}) are related in the +following way: +$$\frac{C_n(-x;-q^{-\alpha};q)}{(q;q)_n}=L_n^{(\alpha)}(x;q).$$ + +\subsection*{References} +\cite{AlvarezRonveaux}, \cite{AtakRahmanSuslov}, \cite{GasperRahman90}, +\cite{Hahn}, \cite{Koelink96III}, \cite{Nikiforov+}, \cite{Zeng95}. + + +\section{Al-Salam-Carlitz~I}\index{Al-Salam-Carlitz~I polynomials} +\par\setcounter{equation}{0} + +\subsection*{Basic hypergeometric representation} +\begin{equation} +\label{DefAlSalamCarlitzI} +U_n^{(a)}(x;q)=(-a)^nq^{\binom{n}{2}}\,\qhyp{2}{1}{q^{-n},x^{-1}}{0}{\frac{qx}{a}}. +\end{equation} + +\subsection*{Orthogonality relation} +\begin{eqnarray} +\label{OrtAlSalamCarlitzI} +& &\int_a^1(qx,a^{-1}qx;q)_{\infty}U_m^{(a)}(x;q)U_n^{(a)}(x;q)\,d_qx\nonumber\\ +& &{}=(-a)^n(1-q)(q;q)_n(q,a,a^{-1}q;q)_{\infty}q^{\binom{n}{2}}\,\delta_{mn},\quad a<0. +\end{eqnarray} + +\subsection*{Recurrence relation} +\begin{equation} +\label{RecAlSalamCarlitzI} +xU_n^{(a)}(x;q)=U_{n+1}^{(a)}(x;q)+(a+1)q^nU_n^{(a)}(x;q) +-aq^{n-1}(1-q^n)U_{n-1}^{(a)}(x;q). +\end{equation} + +\subsection*{Normalized recurrence relation} +\begin{equation} +\label{NormRecAlSalamCarlitzI} +xp_n(x)=p_{n+1}(x)+(a+1)q^np_n(x)-aq^{n-1}(1-q^n)p_{n-1}(x), +\end{equation} +where +$$U_n^{(a)}(x;q)=p_n(x).$$ + +\subsection*{$q$-Difference equation} +\begin{eqnarray} +\label{dvAlSalamCarlitzI} +(1-q^n)x^2y(x)&=&aq^{n-1}y(qx)-\left[aq^{n-1}+q^n(1-x)(a-x)\right]y(x)\nonumber\\ +& &{}\mathindent{}+q^n(1-x)(a-x)y(q^{-1}x), +\end{eqnarray} +where +$$y(x)=U_n^{(a)}(x;q).$$ + +\subsection*{Forward shift operator} +\begin{equation} +\label{shift1AlSalamCarlitzI-I} +U_n^{(a)}(x;q)-U_n^{(a)}(qx;q)=(1-q^n)xU_{n-1}^{(a)}(x;q) +\end{equation} +or equivalently +\begin{equation} +\label{shift1AlSalamCarlitzI-II} +\mathcal{D}_qU_n^{(a)}(x;q)=\frac{1-q^n}{1-q}U_{n-1}^{(a)}(x;q). +\end{equation} + +\subsection*{Backward shift operator} +\begin{equation} +\label{shift2AlSalamCarlitzI-I} +aU_n^{(a)}(x;q)-(1-x)(a-x)U_n^{(a)}(q^{-1}x;q)=-q^{-n}xU_{n+1}^{(a)}(x;q) +\end{equation} +or equivalently +\begin{equation} +\label{shift2AlSalamCarlitzI-II} +\mathcal{D}_{q^{-1}}\left[w(x;a;q)U_n^{(a)}(x;q)\right]= +\frac{q^{-n+1}}{a(1-q)}w(x;a;q)U_{n+1}^{(a)}(x;q), +\end{equation} +where +$$w(x;a;q)=(qx,a^{-1}qx;q)_{\infty}.$$ + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodAlSalamCarlitzI} +w(x;a;q)U_n^{(a)}(x;q)=a^nq^{\frac{1}{2}n(n-3)}(1-q)^n +\left(\mathcal{D}_{q^{-1}}\right)^n\left[w(x;a;q)\right]. +\end{equation} + +\subsection*{Generating function} +\begin{equation} +\label{GenAlSalamCarlitzI} +\frac{(t,at;q)_{\infty}}{(xt;q)_{\infty}}= +\sum_{n=0}^{\infty}\frac{U_n^{(a)}(x;q)}{(q;q)_n}t^n. +\end{equation} + +\subsection*{Limit relations} + +\subsubsection*{Big $q$-Laguerre $\rightarrow$ Al-Salam-Carlitz~I} +If we set $x\rightarrow aqx$ and $b\rightarrow ab$ in the definition +(\ref{DefBigqLaguerre}) of the big $q$-Laguerre polynomials and take the +limit $a\rightarrow 0$ we obtain the Al-Salam-Carlitz~I polynomials given by +(\ref{DefAlSalamCarlitzI}): +$$\lim_{a\rightarrow 0}\frac{P_n(aqx;a,ab;q)}{a^n}=q^nU_n^{(b)}(x;q).$$ + +\subsubsection*{Dual $q$-Krawtchouk $\rightarrow$ Al-Salam-Carlitz~I} +If we set $c=a^{-1}$ in the definition (\ref{DefDualqKrawtchouk}) +of the dual $q$-Krawtchouk polynomials and take the limit +$N\rightarrow\infty$ we simply obtain the Al-Salam-Carlitz~I polynomials +given by (\ref{DefAlSalamCarlitzI}): +$$\lim_{N\rightarrow\infty}K_n(\lambda(x);a^{-1},N|q)= +\left(-\frac{1}{a}\right)^nq^{-\binom{n}{2}}U_n^{(a)}(q^x;q).$$ +Note that $\lambda(x)=q^{-x}+a^{-1}q^{x-N}$. + +\subsubsection*{Al-Salam-Carlitz~I $\rightarrow$ Discrete $q$-Hermite~I} +The discrete $q$-Hermite~I polynomials given by +(\ref{DefDiscreteqHermiteI}) can easily be obtained from the +Al-Salam-Carlitz~I polynomials given by (\ref{DefAlSalamCarlitzI}) by the +substitution $a=-1$: +\begin{equation} +U_n^{(-1)}(x;q)=h_n(x;q). +\end{equation} + +\subsubsection*{Al-Salam-Carlitz~I $\rightarrow$ Charlier / Hermite} +If we set $a\rightarrow a(q-1)$ and $x\rightarrow q^x$ in the definition +(\ref{DefAlSalamCarlitzI}) of the Al-Salam-Carlitz~I polynomials and take the +limit $q\rightarrow 1$ after dividing by $a^n(1-q)^n$ we obtain the Charlier +polynomials given by (\ref{DefCharlier}): +\begin{equation} +\lim_{q\rightarrow 1}\frac{U_n^{(a(q-1))}(q^x;q)}{(1-q)^n}=a^nC_n(x;a). +\end{equation} + +If we set $x\rightarrow x\sqrt{1-q^2}$ and $a\rightarrow a\sqrt{1-q^2}-1$ +in the definition (\ref{DefAlSalamCarlitzI}) of the Al-Salam-Carlitz~I +polynomials, divide by $(1-q^2)^{\frac{n}{2}}$, and let $q$ tend to $1$ we +obtain the Hermite polynomials given by (\ref{DefHermite}) with shifted +argument. In fact we have +\begin{equation} +\lim_{q\rightarrow 1}\frac{U_n^{(a\sqrt{1-q^2}-1)}(x\sqrt{1-q^2};q)} +{(1-q^2)^{\frac{n}{2}}}=\frac{H_n(x-a)}{2^n}. +\end{equation} + +\subsection*{Remark} +The Al-Salam-Carlitz~I polynomials are related to the Al-Salam-Carlitz~II +polynomials given by (\ref{DefAlSalamCarlitzII}) in the following way: +$$U_n^{(a)}(x;q^{-1})=V_n^{(a)}(x;q).$$ + +\subsection*{References} +\cite{AlSalam90}, \cite{AlSalamCarlitz}, \cite{AlSalamChihara76}, \cite{AskeySuslovII}, +\cite{AtakRahmanSuslov}, \cite{Chihara68II}, \cite{Chihara78}, \cite{DattaGriffin}, +\cite{Dehesa}, \cite{DohaAhmed2005}, \cite{GasperRahman90}, \cite{Ismail85I}, +\cite{IsmailMuldoon}, \cite{Kim}, \cite{Zeng95}. + + +\section{Al-Salam-Carlitz~II}\index{Al-Salam-Carlitz~II polynomials} +\par\setcounter{equation}{0} + +\subsection*{Basic hypergeometric representation} +\begin{equation} +\label{DefAlSalamCarlitzII} +V_n^{(a)}(x;q)= +(-a)^nq^{-\binom{n}{2}}\,\qhyp{2}{0}{q^{-n},x}{-}{\frac{q^n}{a}}. +\end{equation} + +\subsection*{Orthogonality relation} +\begin{eqnarray} +\label{OrtAlSalamCarlitzII} +& &\sum_{k=0}^{\infty}\frac{q^{k^2}a^k}{(q;q)_k(aq;q)_k} +V_m^{(a)}(q^{-k};q)V_n^{(a)}(q^{-k};q)\nonumber\\ +& &{}=\frac{(q;q)_na^n}{(aq;q)_{\infty}q^{n^2}}\,\delta_{mn},\quad 00,\quad\textrm{with}\quad +\gamma^2=-\frac{1}{2\ln q}.$$ + +\subsection*{References} +\cite{Askey86}, \cite{Askey89I}, \cite{AtakAtakIII}, \cite{Chihara70}, \cite{Chihara78}, +\cite{Dehesa}, \cite{Ismail2005I}, \cite{Nikiforov+}, \cite{Stieltjes}, \cite{Szego75}, +\cite{ValentAssche}, \cite{Wigert}. + + +\section{Discrete $q$-Hermite~I} +\index{Discrete q-Hermite~I polynomials@Discrete $q$-Hermite~I polynomials} +\index{q-Hermite~I polynomials@$q$-Hermite~I polynomials!Discrete} +\par\setcounter{equation}{0} + +\subsection*{Basic hypergeometric representation} +The discrete $q$-Hermite~I polynomials are Al-Salam-Carlitz~I polynomials +with $a=-1$: +\begin{eqnarray} +\label{DefDiscreteqHermiteI} +h_n(x;q)=U_n^{(-1)}(x;q)&=&q^{\binom{n}{2}}\,\qhyp{2}{1}{q^{-n},x^{-1}}{0}{-qx}\\ +&=&x^n\qhypK{2}{0}{q^{-n},q^{-n+1}}{0}{q^2,\frac{q^{2n-1}}{x^2}}\nonumber +\end{eqnarray} + +\subsection*{Orthogonality relation} +\begin{eqnarray} +\label{OrtDiscreteqHermiteI} +& &\int_{-1}^1(qx,-qx;q)_{\infty}h_m(x;q)h_n(x;q)\,d_qx\nonumber\\ +& &{}=(1-q)(q;q)_n(q,-1,-q;q)_{\infty}q^{\binom{n}{2}}\,\delta_{mn}. +\end{eqnarray} + +\subsection*{Recurrence relation} +\begin{equation} +\label{RecDiscreteqHermiteI} +xh_n(x;q)=h_{n+1}(x;q)+q^{n-1}(1-q^n)h_{n-1}(x;q). +\end{equation} + +\subsection*{Normalized recurrence relation} +\begin{equation} +\label{NormRecDiscreteqHermiteI} +xp_n(x)=p_{n+1}(x)+q^{n-1}(1-q^n)p_{n-1}(x), +\end{equation} +where +$$h_n(x;q)=p_n(x).$$ + +\subsection*{$q$-Difference equation} +\begin{equation} +\label{dvDiscreteqHermiteI} +-q^{-n+1}x^2y(x)=y(qx)-(1+q)y(x)+q(1-x^2)y(q^{-1}x), +\end{equation} +where +$$y(x)=h_n(x;q).$$ + +\subsection*{Forward shift operator} +\begin{equation} +\label{shift1DiscreteqHermiteI-I} +h_n(x;q)-h_n(qx;q)=(1-q^n)xh_{n-1}(x;q) +\end{equation} +or equivalently +\begin{equation} +\label{shift1DiscreteqHermiteI-II} +\mathcal{D}_qh_n(x;q)=\frac{1-q^n}{1-q}h_{n-1}(x;q). +\end{equation} + +\subsection*{Backward shift operator} +\begin{equation} +\label{shift2DiscreteqHermiteI-I} +h_n(x;q)-(1-x^2)h_n(q^{-1}x;q)=q^{-n}xh_{n+1}(x;q) +\end{equation} +or equivalently +\begin{equation} +\label{shift2DiscreteqHermiteI-II} +\mathcal{D}_{q^{-1}}\left[w(x;q)h_n(x;q)\right] +=-\frac{q^{-n+1}}{1-q}w(x;q)h_{n+1}(x;q), +\end{equation} +where +$$w(x;q)=(qx,-qx;q)_{\infty}.$$ + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodDiscreteqHermiteI} +w(x;q)h_n(x;q)=(q-1)^nq^{\frac{1}{2}n(n-3)} +\left(\mathcal{D}_{q^{-1}}\right)^n\left[w(x;q)\right]. +\end{equation} + +\subsection*{Generating function} +\begin{equation} +\label{GenDiscreteqHermiteI} +\frac{(t^2;q^2)_{\infty}}{(xt;q)_{\infty}}= +\sum_{n=0}^{\infty}\frac{h_n(x;q)}{(q;q)_n}t^n. +\end{equation} + +\subsection*{Limit relations} + +\subsubsection*{Al-Salam-Carlitz~I $\rightarrow$ Discrete $q$-Hermite~I} +The discrete $q$-Hermite~I polynomials given by +(\ref{DefDiscreteqHermiteI}) can easily be obtained from the +Al-Salam-Carlitz~I polynomials given by (\ref{DefAlSalamCarlitzI}) by the +substitution $a=-1$: +$$U_n^{(-1)}(x;q)=h_n(x;q).$$ + +\subsubsection*{Discrete $q$-Hermite~I $\rightarrow$ Hermite} +The Hermite polynomials given by (\ref{DefHermite}) can be found +from the discrete $q$-Hermite~I polynomials given by +(\ref{DefDiscreteqHermiteI}) in the following way: +\begin{equation} +\lim_{q\rightarrow 1}\frac{h_n(x\sqrt{1-q^2};q)} +{(1-q^2)^{\frac{n}{2}}}=\frac{H_n(x)}{2^n}. +\end{equation} + +\subsection*{Remark} The discrete $q$-Hermite~I polynomials are related to the +discrete $q$-Hermite~II polynomials given by (\ref{DefDiscreteqHermiteII}) +in the following way: +$$h_n(ix;q^{-1})=i^n{\tilde h}_n(x;q).$$ + +\subsection*{References} +\cite{AlSalam90}, \cite{AlSalamCarlitz}, \cite{AtakRahmanSuslov}, +\cite{BergIsmail}, \cite{BustozIsmail82}, \cite{GasperRahman90}, \cite{Hahn}, +\cite{Koorn97}. + + +\newpage + +\section{Discrete $q$-Hermite~II} +\index{Discrete q-Hermite~II polynomials@Discrete $q$-Hermite~II polynomials} +\index{q-Hermite~II polynomials@$q$-Hermite~II polynomials!Discrete} +\par\setcounter{equation}{0} + +\subsection*{Basic hypergeometric representation} +The discrete $q$-Hermite~II polynomials are Al-Salam-Carlitz~II polynomials +with $a=-1$: +\begin{eqnarray} +\label{DefDiscreteqHermiteII} +{\tilde h}_n(x;q)=i^{-n}V_n^{(-1)}(ix;q) +&=&i^{-n}q^{-\binom{n}{2}}\,\qhyp{2}{0}{q^{-n},ix}{-}{-q^n}\\ +&=&x^n +\qhypK{2}{1}{q^{-n},q^{-n+1}}{0}{q^2,-\frac{q^2}{x^2}}\nonumber +\end{eqnarray} + +\subsection*{Orthogonality relation} +\begin{eqnarray} +\label{OrtDiscreteqHermiteIIa} +& &\sum_{k=-\infty}^{\infty}\left[{\tilde h}_m(cq^k;q) +{\tilde h}_n(cq^k;q)+{\tilde h}_m(-cq^k;q){\tilde h}_n(-cq^k;q)\right] +w(cq^k;q)q^k\nonumber\\ +& &{}=2\frac{(q^2,-c^2q,-c^{-2}q;q^2)_{\infty}}{(q,-c^2,-c^{-2}q^2;q^2)_{\infty}} +\frac{(q;q)_n}{q^{n^2}}\,\delta_{mn},\quad c>0, +\end{eqnarray} +where +$$w(x;q)=\frac{1}{(ix,-ix;q)_{\infty}}=\frac{1}{(-x^2;q^2)_{\infty}}.$$ +For $c=1$ this orthogonality relation can also be written as +\begin{equation} +\label{OrtDiscreteqHermiteIIb} +\int_{-\infty}^{\infty}\frac{{\tilde h}_m(x;q){\tilde h}_n(x;q)}{(-x^2;q^2)_{\infty}}\,d_qx +=\frac{\left(q^2,-q,-q;q^2\right)_{\infty}}{\left(q^3,-q^2,-q^2;q^2\right)_{\infty}} +\frac{(q;q)_n}{q^{n^2}}\,\delta_{mn}. +\end{equation} + +\subsection*{Recurrence relation} +\begin{equation} +\label{RecDiscreteqHermiteII} +x{\tilde h}_n(x;q)={\tilde h}_{n+1}(x;q)+q^{-2n+1}(1-q^n){\tilde h}_{n-1}(x;q). +\end{equation} + +\subsection*{Normalized recurrence relation} +\begin{equation} +\label{NormRecDiscreteqHermiteII} +xp_n(x)=p_{n+1}(x)+q^{-2n+1}(1-q^n)p_{n-1}(x), +\end{equation} +where +$${\tilde h}_n(x;q)=p_n(x).$$ + +\subsection*{$q$-Difference equation} +\begin{eqnarray} +\label{dvDiscreteqHermiteII} +& &-(1-q^n)x^2{\tilde h}_n(x;q)\nonumber\\ +& &{}=(1+x^2){\tilde h}_n(qx;q)-(1+x^2+q){\tilde h}_n(x;q)+q{\tilde h}_n(q^{-1}x;q). +\end{eqnarray} + +\subsection*{Forward shift operator} +\begin{equation} +\label{shift1DiscreteqHermiteII-I} +\tilde{h}_n(x;q)-\tilde{h}_n(qx;q)=q^{-n+1}(1-q^n)x\tilde{h}_{n-1}(qx;q) +\end{equation} +or equivalently +\begin{equation} +\label{shift1DiscreteqHermiteII-II} +\mathcal{D}_q\tilde{h}_n(x;q)=\frac{q^{-n+1}(1-q^n)}{1-q}\tilde{h}_{n-1}(qx;q). +\end{equation} + +\subsection*{Backward shift operator} +\begin{equation} +\label{shift2DiscreteqHermiteII-I} +\tilde{h}_n(x;q)-(1+x^2)\tilde{h}_n(qx;q)=-q^nx\tilde{h}_{n+1}(x;q) +\end{equation} +or equivalently +\begin{equation} +\label{shift2DiscreteqHermiteII-II} +\mathcal{D}_q\left[w(x;q)\tilde{h}_n(x;q)\right] +=-\frac{q^n}{1-q}w(x;q)\tilde{h}_{n+1}(x;q). +\end{equation} + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodDiscreteqHermiteII} +w(x;q)\tilde{h}_n(x;q)=(q-1)^nq^{-\binom{n}{2}} +\left(\mathcal{D}_q\right)^n\left[w(x;q)\right]. +\end{equation} + +\subsection*{Generating functions} +\begin{equation} +\label{GenDiscreteqHermiteII1} +\frac{(-xt;q)_{\infty}}{(-t^2;q^2)_{\infty}}=\sum_{n=0}^{\infty} +\frac{q^{\binom{n}{2}}}{(q;q)_n}{\tilde h}_n(x;q)t^n. +\end{equation} + +\begin{equation} +\label{GenDiscreteqHermiteII2} +(-it;q)_{\infty}\cdot\qhyp{1}{1}{ix}{-it}{it}=\sum_{n=0}^{\infty} +\frac{(-1)^nq^{n(n-1)}}{(q;q)_n}{\tilde h}_n(x;q)t^n. +\end{equation} + +\subsection*{Limit relations} + +\subsubsection*{Al-Salam-Carlitz~II $\rightarrow$ Discrete $q$-Hermite~II} +The discrete $q$-Hermite~II polynomials given by +(\ref{DefDiscreteqHermiteII}) follow from the Al-Salam-Carlitz~II +polynomials given by (\ref{DefAlSalamCarlitzII}) by the substitution $a=-1$ +in the following way: +$$i^{-n}V_n^{(-1)}(ix;q)={\tilde h}_n(x;q).$$ + +\subsubsection*{Discrete $q$-Hermite~II $\rightarrow$ Hermite} +The Hermite polynomials given by (\ref{DefHermite}) can be found +from the discrete $q$-Hermite~II polynomials given by +(\ref{DefDiscreteqHermiteII}) in the following way: +\begin{equation} +\lim_{q\rightarrow 1}\frac{{\tilde h}_n(x\sqrt{1-q^2};q)} +{(1-q^2)^{\frac{n}{2}}}=\frac{H_n(x)}{2^n}. +\end{equation} + +\subsection*{Remark} The discrete $q$-Hermite~II polynomials are related to the +discrete $q$-Hermite~I polynomials given by (\ref{DefDiscreteqHermiteI}) +in the following way: +$${\tilde h}_n(x;q^{-1})=i^{-n}h_n(ix;q).$$ + +\subsection*{References} +\cite{BergIsmail}, \cite{Koorn97}. + +\end{document} diff --git a/KLSadd_insertion/linetest.py b/KLSadd_insertion/linetest.py new file mode 100644 index 0000000..19de78a --- /dev/null +++ b/KLSadd_insertion/linetest.py @@ -0,0 +1,165 @@ +""" +TODO: Rewrite the whole thing so another human being can read it +jeez this is like my handwriting; it's nonsense to everyone but me +This started out as a test file but I just kept writing it +and writing it +and writing it +and now its the project +""" + +entireFile = "" +subsections = [] +index = 0 +headings = [] +sections = [] +mathPeople = [] +newCommands = [] +#can you tell I like lists? +name = "" +temp = "" +with open("KLSadd.tex", "r") as file: + entireFile = file.readlines() + for word in entireFile: + index+=1 + if("smallskipamount" in word): + newCommands.append(index-1) + if("mybibitem[1]" in word): + newCommands.append(index) + #1/27/16 RS added \large\bf KLSadd: to make KLSadd additions appear like the other paragraphs + if("paragraph{" in word): + temp = word[0:word.find("{")+ 1] + "\large\\bf KLSadd: " + word[word.find("{")+ 1: word.find("}")+1] + entireFile[entireFile.index(word): entireFile.index(word)+1] = temp + if("\\subsection*{" in word and "." in word): + subsections.append(index) + headings.append(word) + name = word[word.find(" ")+1:word.find("}")] + mathPeople.append(name) + name = "" + #1/29/16 get all of the new commands + comms = ''.join(entireFile[newCommands[0]:newCommands[1]]) + paras = [] + for i in range(len(subsections)-1): + str = ''.join(entireFile[subsections[i]: subsections[i+1]-1]) + paras.append(str) + sec = "" + for f in headings: + for i in f: + try: + int(i) + sec += i + except: + pass + if("." in i): + sec+=i + sections.append(sec) + + sec = "" + #sections now holds all sections + + numFile = "" + filenums = [] + for s in sections: + numFile = "" + if("." in s[0:2]): + numFile = s[0] + else: + numFile = s[0:2] + filenums.append(int(numFile)) + #filenums now has the file numbers like 1,9,14,etc + + #SO python is nothing like java, I have to read in the entire chapter + #file and then change it and THEN put it back into the chapter file + + #chapter 9 + with open("tempchap9.tex", "r") as ch9: + entire9 = ch9.readlines() + ch9.close() + #chapter 14 + with open("tempchap14.tex", "r") as ch14: + entire14 = ch14.readlines() + ch14.close() + + references9 = [] + footmiscIndex = 0 + index = 0 + beginIndex9= -1 + #now I have a list of the names, only store indices + #of the paragraph if its in a section with these names + #TODO make a method to do this + canAdd = False + for word in entire9: + index+=1 + if("begin{document}" in word): + beginIndex9 += index + if("footmisc" in word): + footmiscIndex+= index + #need to check subsection and section because + #KLSadd also fixes some subsections + if("section{" in word or "subsection*{" in word): + w = word[word.find("{")+1:word.find("}")] + if(w in mathPeople): + canAdd = True + if("\\subsection*{References}" in word and canAdd == True): + references9.append(index) + canAdd = False + canAdd = False + references14 = [] + index = 0 + footmiscIndex14 = 0 + beginIndex14 = -1 + for word in entire14: + index+=1 + if("begin{document}" in word): + beginIndex14 += index + if("footmisc" in word): + footmiscIndex14+= index + #need to check subsection and section because + #KLSadd also fixes some subsections + if("section{" in word or "subsection*{" in word): + w = word[word.find("{")+1:word.find("}")] + if(w in mathPeople): + canAdd = True + if("\\subsection*{References}" in word and canAdd == True): + references14.append(index) + canAdd = False + #references9 is now full of indexes to put fixed paragraphs + + #TODO make into a method + with open("newtempchap09.tex", "w") as temp9: + count = 0 + + #1/29/16 add hyperref, xparse, cite packages after the footmisc package + entire9[footmiscIndex] += "\\usepackage[pdftex]{hyperref} \n\\usepackage {xparse} \n\\usepackage{cite} \n" + #1/29/16 add in the newcommands from KLSadd.tex + entire9[beginIndex9-1] += comms + #1/27/16 add \large\bf KLSadd: after every paragraph{ + for i in references9: + entire9[i-3] += "% RS: add begin" + entire9[i-3] += paras[count] + entire9[i-3] += "% RS: add end" + count+=1 + entire9[i-1] += paras[count] + + + str = ''.join(entire9) + temp9.write(str) + + + + #works, as far as I know, huzzah + #references14 is now full of indexes to put fixed paragraphs + with open("newtempchap14.tex", "w") as temp14: + entire14[footmiscIndex14] += "\\usepackage[pdftex]{hyperref} \n\\usepackage {xparse} \n\\usepackage{cite} \n" + entire14[beginIndex14 -3] += comms + for i in references14: + entire14[i-3] += " RS: add begin" + entire14[i-3] += paras[count] + #entire14[i-3] += "% RS: add end" + count+=1 + str = ''.join(entire14) + temp14.write(str) + + + + + diff --git a/KLSadd_insertion/newtempchap09.pdf b/KLSadd_insertion/newtempchap09.pdf new file mode 100644 index 0000000000000000000000000000000000000000..bec3d85a50c47d3eae72c71fd38c175ac81a2ab1 GIT binary patch literal 319561 zcma&NW2|UFvn{%8+qSK}Y}>YN+qP}nwry)K+s57Jy_1)hd?&f5Gk?rqo%F1pHEN7e zMJg{WO3O&c3Pn1EUmX^xu|H}<8%VvcZQ+E88qt*gd~X}%{C>pD7-yGQ&hJ@jZNB8QXjuyFp57gbKmV%2%v2=4qg9}Pi!qsI` zjtQE1uOFJX4+qB7_{#3hRGGoXSnU2H1Z7Fq-qXpJA?S&*UjyzBo*c5SYR+f1`AX00 z+)v&39r*rQ3DTKwO~B|73?JV(aJ+qEdfh7&2SymO&|lb?2oAB`7=;TtcF`6~9aIlK 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\genfrac{}{}{0pt}{}{#3}{#4};#5\right)} +\newcommand\LHS{left-hand side} +\newcommand\RHS{right-hand side} +\renewcommand\Re{{\rm Re}\,} +\renewcommand\Im{{\rm Im}\,} + +\NewDocumentCommand\mycite{m g}{% + \IfNoValueTF{#2} + {[\hyperlink{#1}{#1}]} + {[\hyperlink{#1}{#1}, #2]}% +} +\newcommand\mybibitem[1]{\bibitem[#1]{#1}\hypertarget{#1}{}} +\begin{document} + +\author{Roelof Koekoek\\[2.5mm]Peter A. Lesky\\[2.5mm]Ren\'e F. Swarttouw} +\title{Hypergeometric orthogonal polynomials and their\\$q$-analogues} +\subtitle{-- Monograph --} +\maketitle + +\frontmatter + +\large + +\addtocounter{chapter}{8} +\pagenumbering{roman} +\chapter{Hypergeometric orthogonal polynomials} +\label{HyperOrtPol} + + +In this chapter we deal with all families of hypergeometric orthogonal polynomials +appearing in the Askey scheme on page~\pageref{scheme}. For each family of orthogonal +polynomials we state the most important properties such as a representation as a +hypergeometric function, orthogonality relation(s), the three-term recurrence relation, +the second-order differential or difference equation, the forward shift (or degree lowering) +and backward shift (or degree raising) operator, a Rodrigues-type formula and some generating +functions. In each case we use the notation which seems to be most common in the literature. +Moreover, in each case we mention the connection between various families by stating the +appropriate limit relations. See also \cite{Terwilliger2006} for an algebraic approach of +this Askey scheme and \cite{TemmeLopez2001} for a view from asymptotic analysis. +For notations the reader is referred to chapter~\ref{Definitions}. + +\section{Wilson}\index{Wilson polynomials} + +\par + +\subsection*{Hypergeometric representation} +\begin{eqnarray} +\label{DefWilson} +& &\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\nonumber\\ +& &{}=\hyp{4}{3}{-n,n+a+b+c+d-1,a+ix,a-ix}{a+b,a+c,a+d}{1}. +\end{eqnarray} + +\newpage + +\subsection*{Orthogonality relation} +If $Re(a,b,c,d)>0$ and non-real parameters occur in conjugate pairs, then +\begin{eqnarray} +\label{OrthIWilson} +& &\frac{1}{2\pi}\int_0^{\infty} +\left|\frac{\Gamma(a+ix)\Gamma(b+ix)\Gamma(c+ix)\Gamma(d+ix)}{\Gamma(2ix)}\right|^2\nonumber\\ +& &{}\mathindent\times W_m(x^2;a,b,c,d)W_n(x^2;a,b,c,d)\,dx\nonumber\\ +& &{}=\frac{\Gamma(n+a+b)\cdots\Gamma(n+c+d)}{\Gamma(2n+a+b+c+d)}(n+a+b+c+d-1)_nn!\,\delta_{mn}, +\end{eqnarray} +where +\begin{eqnarray*} +& &\Gamma(n+a+b)\cdots\Gamma(n+c+d)\\ +& &{}=\Gamma(n+a+b)\Gamma(n+a+c)\Gamma(n+a+d)\Gamma(n+b+c)\Gamma(n+b+d)\Gamma(n+c+d). +\end{eqnarray*} +If $a<0$ and $a+b$, $a+c$, $a+d$ are positive or a pair of complex conjugates +occur with positive real parts, then +\begin{eqnarray} +\label{OrtIIWilson} +& &\frac{1}{2\pi}\int_0^{\infty}\left|\frac{\Gamma(a+ix)\Gamma(b+ix)\Gamma(c+ix)\Gamma(d+ix)}{\Gamma(2ix)}\right|^2\nonumber\\ +& &{}\mathindent\mathindent\times W_m(x^2;a,b,c,d)W_n(x^2;a,b,c,d)\,dx\nonumber\\ +& &{}\mathindent{}+\frac{\Gamma(a+b)\Gamma(a+c)\Gamma(a+d)\Gamma(b-a)\Gamma(c-a)\Gamma(d-a)}{\Gamma(-2a)}\nonumber\\ +& &{}\mathindent\mathindent{}\times\sum_{\begin{array}{c} +{\scriptstyle k=0,1,2\ldots}\\{\scriptstyle a+k<0}\end{array}} +\frac{(2a)_k(a+1)_k(a+b)_k(a+c)_k(a+d)_k}{(a)_k(a-b+1)_k(a-c+1)_k(a-d+1)_kk!}\nonumber\\ +& &{}\mathindent\mathindent\mathindent\mathindent{}\times W_m(-(a+k)^2;a,b,c,d)W_n(-(a+k)^2;a,b,c,d)\nonumber\\ +& &{}=\frac{\Gamma(n+a+b)\cdots\Gamma(n+c+d)}{\Gamma(2n+a+b+c+d)}(n+a+b+c+d-1)_nn!\,\delta_{mn}. +\end{eqnarray} + +\subsection*{Recurrence relation} +\begin{equation} +\label{RecWilson} +-\left(a^2+x^2\right){\tilde{W}}_n(x^2) +=A_n{\tilde{W}}_{n+1}(x^2)-\left(A_n+C_n\right){\tilde{W}}_n(x^2)+C_n{\tilde{W}}_{n-1}(x^2), +\end{equation} +where +$${\tilde{W}}_n(x^2):={\tilde{W}}_n(x^2;a,b,c,d)=\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}$$ +and +$$\left\{\begin{array}{l} +\displaystyle A_n=\frac{(n+a+b+c+d-1)(n+a+b)(n+a+c)(n+a+d)}{(2n+a+b+c+d-1)(2n+a+b+c+d)}\\[5mm] +\displaystyle C_n=\frac{n(n+b+c-1)(n+b+d-1)(n+c+d-1)}{(2n+a+b+c+d-2)(2n+a+b+c+d-1)}. +\end{array}\right.$$ + +\subsection*{Normalized recurrence relation} +\begin{equation} +\label{NormRecWilson} +xp_n(x)=p_{n+1}(x)+(A_n+C_n-a^2)p_n(x)+A_{n-1}C_np_{n-1}(x), +\end{equation} +where +$$W_n(x^2;a,b,c,d)=(-1)^n(n+a+b+c+d-1)_np_n(x^2).$$ + +\subsection*{Difference equation} +\begin{eqnarray} +\label{dvWilson} +& &n(n+a+b+c+d-1)y(x)\nonumber\\ +& &{}=B(x)y(x+i)-\left[B(x)+D(x)\right]y(x)+D(x)y(x-i), +\end{eqnarray} +where +$$y(x)=W_n(x^2;a,b,c,d)$$ +and +$$\left\{\begin{array}{l} +\displaystyle B(x)=\frac{(a-ix)(b-ix)(c-ix)(d-ix)}{2ix(2ix-1)}\\[5mm] +\displaystyle D(x)=\frac{(a+ix)(b+ix)(c+ix)(d+ix)}{2ix(2ix+1)}. +\end{array}\right.$$ + +\subsection*{Forward shift operator} +\begin{eqnarray} +\label{shift1WilsonI} +& &W_n((x+\textstyle\frac{1}{2}i)^2;a,b,c,d) +-W_n((x-\textstyle\frac{1}{2}i)^2;a,b,c,d)\nonumber\\ +& &{}=-2inx(n+a+b+c+d-1)W_{n-1}(x^2;a+\textstyle\frac{1}{2},b+\textstyle\frac{1}{2}, +c+\textstyle\frac{1}{2},d+\textstyle\frac{1}{2}) +\end{eqnarray} +or equivalently +\begin{eqnarray} +\label{shift1WilsonII} +& &\frac{\delta W_n(x^2;a,b,c,d)}{\delta x^2}\nonumber\\ +& &{}=-n(n+a+b+c+d-1)W_{n-1}(x^2;a+\textstyle\frac{1}{2},b+\textstyle\frac{1}{2}, +c+\textstyle\frac{1}{2},d+\textstyle\frac{1}{2}). +\end{eqnarray} + +\newpage + +\subsection*{Backward shift operator} +\begin{eqnarray} +\label{shift2WilsonI} +& &(a-\textstyle\frac{1}{2}-ix)(b-\textstyle\frac{1}{2}-ix) +(c-\textstyle\frac{1}{2}-ix)(d-\textstyle\frac{1}{2}-ix) +W_n((x+\textstyle\frac{1}{2}i)^2;a,b,c,d)\nonumber\\ +& &{}\mathindent{}-(a-\textstyle\frac{1}{2}+ix)(b-\textstyle\frac{1}{2}+ix) +(c-\textstyle\frac{1}{2}+ix)(d-\textstyle\frac{1}{2}+ix) +W_n((x-\textstyle\frac{1}{2}i)^2;a,b,c,d)\nonumber\\ +& &{}=-2ix W_{n+1}(x^2;a-\textstyle\frac{1}{2},b-\textstyle\frac{1}{2}, +c-\textstyle\frac{1}{2},d-\textstyle\frac{1}{2}) +\end{eqnarray} +or equivalently +\begin{eqnarray} +\label{shift2WilsonII} +& &\frac{\delta\left[\omega(x;a,b,c,d)W_n(x^2;a,b,c,d)\right]}{\delta x^2}\nonumber\\ +& &{}=\omega(x;a-\textstyle\frac{1}{2},b-\textstyle\frac{1}{2}, +c-\textstyle\frac{1}{2},d-\textstyle\frac{1}{2}) +W_{n+1}(x^2;a-\textstyle\frac{1}{2},b-\textstyle\frac{1}{2},c-\textstyle\frac{1}{2},d-\textstyle\frac{1}{2}), +\end{eqnarray} +where +$$\omega(x;a,b,c,d):=\frac{1}{2ix}\left|\frac{\Gamma(a+ix)\Gamma(b+ix)\Gamma(c+ix)\Gamma(d+ix)}{\Gamma(2ix)}\right|^2.$$ + +\subsection*{Rodrigues-type formula} +\begin{eqnarray} +\label{RodWilson} +& &\omega(x;a,b,c,d)W_n(x^2;a,b,c,d)\nonumber\\ +& &{}=\left(\frac{\delta}{\delta x^2}\right)^n +\left[\omega(x;a+\textstyle\frac{1}{2}n,b+\textstyle\frac{1}{2}n, +c+\textstyle\frac{1}{2}n,d+\textstyle\frac{1}{2}n)\right]. +\end{eqnarray} + +\subsection*{Generating functions} +\begin{equation} +\label{GenWilson1} +\hyp{2}{1}{a+ix,b+ix}{a+b}{t}\,\hyp{2}{1}{c-ix,d-ix}{c+d}{t}= +\sum_{n=0}^{\infty}\frac{W_n(x^2;a,b,c,d)t^n}{(a+b)_n(c+d)_nn!}. +\end{equation} + +\begin{equation} +\label{GenWilson2} +\hyp{2}{1}{a+ix,c+ix}{a+c}{t}\,\hyp{2}{1}{b-ix,d-ix}{b+d}{t}= +\sum_{n=0}^{\infty}\frac{W_n(x^2;a,b,c,d)t^n}{(a+c)_n(b+d)_nn!}. +\end{equation} + +\begin{equation} +\label{GenWilson3} +\hyp{2}{1}{a+ix,d+ix}{a+d}{t}\,\hyp{2}{1}{b-ix,c-ix}{b+c}{t}= +\sum_{n=0}^{\infty}\frac{W_n(x^2;a,b,c,d)t^n}{(a+d)_n(b+c)_nn!}. +\end{equation} + +\begin{eqnarray} +\label{GenWilson4} +& &(1-t)^{1-a-b-c-d}\nonumber\\ +& &{}\mathindent\times\hyp{4}{3}{\frac{1}{2}(a+b+c+d-1),\frac{1}{2}(a+b+c+d),a+ix,a-ix} +{a+b,a+c,a+d}{-\frac{4t}{(1-t)^2}}\nonumber\\ +& &{}=\sum_{n=0}^{\infty}\frac{(a+b+c+d-1)_n}{(a+b)_n(a+c)_n(a+d)_nn!}W_n(x^2;a,b,c,d)t^n. +\end{eqnarray} + +\subsection*{Limit relations} + +\subsubsection*{Wilson $\rightarrow$ Continuous dual Hahn} +The continuous dual Hahn polynomials given by (\ref{DefContDualHahn}) can be found from the +Wilson polynomials by dividing by $(a+d)_n$ and letting $d\rightarrow\infty$: +\begin{equation} +\lim_{d\rightarrow\infty}\frac{W_n(x^2;a,b,c,d)}{(a+d)_n}=S_n(x^2;a,b,c). +\end{equation} + +\subsubsection*{Wilson $\rightarrow$ Continuous Hahn} +The continuous Hahn polynomials given by (\ref{DefContHahn}) are obtained from the Wilson +polynomials by the substitutions $a\rightarrow a-it$, $b\rightarrow b-it$, $c\rightarrow c+it$, +$d\rightarrow d+it$ and $x\rightarrow x+t$ and the limit $t\rightarrow\infty$ in the following +way: +\begin{equation} +\lim_{t\rightarrow\infty} +\frac{W_n((x+t)^2;a-it,b-it,c+it,d+it)}{(-2t)^nn!}=p_n(x;a,b,c,d). +\end{equation} + +\subsubsection*{Wilson $\rightarrow$ Jacobi} +The Jacobi polynomials given by (\ref{DefJacobi}) can be found from the Wilson +polynomials by substituting $a=b=\frac{1}{2}(\alpha+1)$, $c=\frac{1}{2}(\beta+1)+it$, +$d=\frac{1}{2}(\beta+1)-it$ and $x\rightarrow t\sqrt{\frac{1}{2}(1-x)}$ in the +definition (\ref{DefWilson}) of the Wilson polynomials and taking the limit +$t\rightarrow\infty$. In fact we have +\begin{eqnarray} +& &\lim_{t\rightarrow\infty}\frac{W_n(\frac{1}{2}(1-x)t^2;\frac{1}{2}(\alpha+1), +\frac{1}{2}(\alpha+1),\frac{1}{2}(\beta+1)+it,\frac{1}{2}(\beta+1)-it)}{t^{2n}n!}\nonumber\\ +& &{}=P_n^{(\alpha,\beta)}(x). +\end{eqnarray} + +\subsection*{Remarks} +Note that for $k0$, $c=\bar{a}$ and $d=\bar{b}$, then +\begin{eqnarray} +\label{OrtContHahn} +& &\frac{1}{2\pi}\int_{-\infty}^{\infty}\Gamma(a+ix)\Gamma(b+ix)\Gamma(c-ix)\Gamma(d-ix) +p_m(x;a,b,c,d)p_n(x;a,b,c,d)\,dx\nonumber\\ +& &{}=\frac{\Gamma(n+a+c)\Gamma(n+a+d)\Gamma(n+b+c)\Gamma(n+b+d)} +{(2n+a+b+c+d-1)\Gamma(n+a+b+c+d-1)n!}\,\delta_{mn}. +\end{eqnarray} + +\subsection*{Recurrence relation} +\begin{equation} +\label{RecContHahn} +(a+ix)\tilde{p}_n(x)=A_n\tilde{p}_{n+1}(x)-\left(A_n+C_n\right)\tilde{p}_n(x)+C_n\tilde{p}_{n-1}(x), +\end{equation} +where +$$\tilde{p}_n(x):=\tilde{p}_n(x;a,b,c,d)=\frac{n!}{i^n(a+c)_n(a+d)_n}p_n(x;a,b,c,d)$$ +and +$$\left\{\begin{array}{l} +\displaystyle A_n=-\frac{(n+a+b+c+d-1)(n+a+c)(n+a+d)}{(2n+a+b+c+d-1)(2n+a+b+c+d)}\\[5mm] +\displaystyle C_n=\frac{n(n+b+c-1)(n+b+d-1)}{(2n+a+b+c+d-2)(2n+a+b+c+d-1)}. +\end{array}\right.$$ + +\subsection*{Normalized recurrence relation} +\begin{equation} +\label{NormRecContHahn} +xp_n(x)=p_{n+1}(x)+i(A_n+C_n+a)p_n(x)-A_{n-1}C_np_{n-1}(x), +\end{equation} +where +$$p_n(x;a,b,c,d)=\frac{(n+a+b+c+d-1)_n}{n!}p_n(x).$$ + +\subsection*{Difference equation} +\begin{eqnarray} +\label{dvContHahn} +& &n(n+a+b+c+d-1)y(x)\nonumber\\ +& &{}=B(x)y(x+i)-\left[B(x)+D(x)\right]y(x)+D(x)y(x-i), +\end{eqnarray} +where +$$y(x)=p_n(x;a,b,c,d)$$ +and +$$\left\{\begin{array}{l} +\displaystyle B(x)=(c-ix)(d-ix)\\[5mm] +\displaystyle D(x)=(a+ix)(b+ix). +\end{array}\right.$$ + +\subsection*{Forward shift operator} +\begin{eqnarray} +\label{shift1ContHahnI} +& &p_n(x+\textstyle\frac{1}{2}i;a,b,c,d)-p_n(x-\textstyle\frac{1}{2}i;a,b,c,d)\nonumber\\ +& &{}=i(n+a+b+c+d-1)p_{n-1}(x;a+\textstyle\frac{1}{2}, +b+\textstyle\frac{1}{2},c+\textstyle\frac{1}{2},d+\textstyle\frac{1}{2}) +\end{eqnarray} +or equivalently +\begin{equation} +\label{shift1ContHahnII} +\frac{\delta p_n(x;a,b,c,d)}{\delta x}=(n+a+b+c+d-1) +p_{n-1}(x;a+\textstyle\frac{1}{2},b+\textstyle\frac{1}{2}, +c+\textstyle\frac{1}{2},d+\textstyle\frac{1}{2}). +\end{equation} + +\subsection*{Backward shift operator} +\begin{eqnarray} +\label{shift2ContHahnI} +& &(c-\textstyle\frac{1}{2}-ix)(d-\textstyle\frac{1}{2}-ix) +p_n(x+\textstyle\frac{1}{2}i;a,b,c,d)\nonumber\\ +& &{}\mathindent{}-(a-\textstyle\frac{1}{2}+ix)(b-\textstyle\frac{1}{2}+ix) +p_n(x-\textstyle\frac{1}{2}i;a,b,c,d)\nonumber\\ +& &{}=\frac{n+1}{i}p_{n+1}(x;a-\textstyle\frac{1}{2}, +b-\textstyle\frac{1}{2},c-\textstyle\frac{1}{2},d-\textstyle\frac{1}{2}) +\end{eqnarray} +or equivalently +\begin{eqnarray} +\label{shift2ContHahnII} +& &\frac{\delta\left[\omega(x;a,b,c,d)p_n(x;a,b,c,d)\right]}{\delta x}\nonumber\\ +& &{}=-(n+1)\omega(x;a-\textstyle\frac{1}{2},b-\textstyle\frac{1}{2}, +c-\textstyle\frac{1}{2},d-\textstyle\frac{1}{2})\nonumber\\ +& &{}\mathindent\times p_{n+1}(x;a-\textstyle\frac{1}{2},b-\textstyle\frac{1}{2},c-\textstyle\frac{1}{2},d-\textstyle\frac{1}{2}), +\end{eqnarray} +where +$$\omega(x;a,b,c,d)=\Gamma(a+ix)\Gamma(b+ix)\Gamma(c-ix)\Gamma(d-ix).$$ + +\subsection*{Rodrigues-type formula} +\begin{eqnarray} +\label{RodContHahn} +& &\omega(x;a,b,c,d)p_n(x;a,b,c,d)\nonumber\\ +& &{}=\frac{(-1)^n}{n!}\left(\frac{\delta}{\delta x}\right)^n +\left[\omega(x;a+\textstyle\frac{1}{2}n,b+\textstyle\frac{1}{2}n, +c+\textstyle\frac{1}{2}n,d+\textstyle\frac{1}{2}n)\right]. +\end{eqnarray} + +\subsection*{Generating functions} +\begin{equation} +\label{GenContHahn1} +\hyp{1}{1}{a+ix}{a+c}{-it}\,\hyp{1}{1}{d-ix}{b+d}{it}= +\sum_{n=0}^{\infty}\frac{p_n(x;a,b,c,d)}{(a+c)_n(b+d)_n}t^n. +\end{equation} + +\begin{equation} +\label{GenContHahn2} +\hyp{1}{1}{a+ix}{a+d}{-it}\,\hyp{1}{1}{c-ix}{b+c}{it}= +\sum_{n=0}^{\infty}\frac{p_n(x;a,b,c,d)}{(a+d)_n(b+c)_n}t^n. +\end{equation} + +\begin{eqnarray} +\label{GenContHahn3} +& &(1-t)^{1-a-b-c-d}\,\hyp{3}{2}{\frac{1}{2}(a+b+c+d-1),\frac{1}{2}(a+b+c+d),a+ix} +{a+c,a+d}{-\frac{4t}{(1-t)^2}}\nonumber\\ +& &{}=\sum_{n=0}^{\infty} +\frac{(a+b+c+d-1)_n}{(a+c)_n(a+d)_ni^n}p_n(x;a,b,c,d)t^n. +\end{eqnarray} + +\subsection*{Limit relations} + +\subsubsection*{Wilson $\rightarrow$ Continuous Hahn} +The continuous Hahn polynomials are obtained from the Wilson polynomials given by +(\ref{DefWilson}) by the substitution $a\rightarrow a-it$, $b\rightarrow b-it$, +$c\rightarrow c+it$, $d\rightarrow d+it$ and $x\rightarrow x+t$ and the limit +$t\rightarrow\infty$ in the following way: +$$\lim_{t\rightarrow\infty} +\frac{W_n((x+t)^2;a-it,b-it,c+it,d+it)}{(-2t)^nn!}=p_n(x;a,b,c,d).$$ + +\subsubsection*{Continuous Hahn $\rightarrow$ Meixner-Pollaczek} +The Meixner-Pollaczek polynomials given by (\ref{DefMP}) can be obtained from the +continuous Hahn polynomials by setting $x\rightarrow x+t$, $a=\lambda-it$, $c=\lambda+it$ +and $b=d=t\tan\phi$ and taking the limit $t\rightarrow\infty$: +\begin{equation} +\lim_{t\rightarrow\infty}\frac{p_n(x+t;\lambda-it,t\tan\phi,\lambda+it,t\tan\phi)}{t^n} +=\frac{P_n^{(\lambda)}(x;\phi)}{(\cos\phi)^n}. +\end{equation} + +\subsubsection*{Continuous Hahn $\rightarrow$ Jacobi} +The Jacobi polynomials given by (\ref{DefJacobi}) follow from the continuous Hahn polynomials +by the substitution $x\rightarrow \frac{1}{2}xt$, $a=\frac{1}{2}(\alpha+1-it)$, +$b=\frac{1}{2}(\beta+1+it)$, $c=\frac{1}{2}(\alpha+1+it)$ and $d=\frac{1}{2}(\beta+1-it)$, +division by $t^n$ and the limit $t\rightarrow\infty$: +\begin{eqnarray} +& &\lim_{t\rightarrow\infty} +\frac{p_n(\frac{1}{2}xt;\frac{1}{2}(\alpha+1-it),\frac{1}{2}(\beta+1+it), +\frac{1}{2}(\alpha+1+it),\frac{1}{2}(\beta+1-it))}{t^n}\nonumber\\ +& &{}=P_n^{(\alpha,\beta)}(x). +\end{eqnarray} + +\subsubsection*{Continuous Hahn $\rightarrow$ Pseudo Jacobi} +The pseudo Jacobi polynomials given by (\ref{DefPseudoJacobi}) follow from the continuous +Hahn polynomials by the substitution $x\rightarrow xt$, $a=\frac{1}{2}(-N+i\nu-2t)$, +$b=\frac{1}{2}(-N-i\nu+2t)$, $c=\frac{1}{2}(-N-i\nu-2t)$ and $d=\frac{1}{2}(-N+i\nu+2t)$, +division by $t^n$ and the limit $t\rightarrow\infty$: +\begin{eqnarray} +& &\lim_{t\rightarrow\infty}\frac{p_n(xt;\frac{1}{2}(-N+i\nu-2t),\frac{1}{2}(-N-i\nu+2t), +\frac{1}{2}(-N+i\nu-2t),\frac{1}{2}(-N-i\nu+2t))}{t^n}\nonumber\\ +& &{}=\frac{(n-2N-1)_n}{n!}P_n(x;\nu,N). +\end{eqnarray} + +\subsection*{Remark} +Since we have for $k0$ and $(c,d)=(\overline a,\overline b)$ or $(\overline b,\overline a)$.\\ +Thus, under these assumptions, the continuous Hahn polynomial +$p_n(x;a,b,c,d)$ +is symmetric in $a,b$ and in $c,d$. +This follows from the orthogonality relation (9.4.2) +together with the value of its coefficient of $x^n$ given in (9.4.4b).\\ +As a consequence, it is sufficient to give generating function (9.4.11). Then the generating +function (9.4.12) will follow by symmetry in the parameters. +% +\paragraph{\large\bf KLSadd: Uniqueness of orthogonality measure}The coefficient of $p_{n-1}(x)$ in (9.4.4) behaves as $O(n^2)$ as $n\to\iy$. +Hence \eqref{93} holds, by which the orthogonality measure is unique. +% +\paragraph{\large\bf KLSadd: Special cases}In the following special case there is a reduction to +Meixner-Pollaczek: +\begin{equation} +p_n(x;a,a+\thalf,a,a+\thalf)= +\frac{(2a)_n (2a+\thalf)_n}{(4a)_n}\,P_n^{(2a)}(2x;\thalf\pi). +\end{equation} +See \myciteKLS{342}{(2.6)} (note that in \myciteKLS{342}{(2.3)} the +Meixner-Pollaczek polynonmials are defined different from (9.7.1), +without a constant factor in front). + +For $0-1$ and $\beta>-1$, or for $\alpha<-N$ and $\beta<-N$, we have +\begin{eqnarray} +\label{OrtHahn} +& &\sum_{x=0}^N\binom{\alpha +x}{x}\binom{\beta+N-x}{N-x}Q_m(x;\alpha,\beta,N)Q_n(x;\alpha,\beta,N)\nonumber\\ +& &{}=\frac{(-1)^n(n+\alpha+\beta+1)_{N+1}(\beta+1)_nn!}{(2n+\alpha+\beta+1)(\alpha+1)_n(-N)_nN!}\,\delta_{mn}. +\end{eqnarray} + +\subsection*{Recurrence relation} +\begin{equation} +\label{RecHahn} +-xQ_n(x)=A_nQ_{n+1}(x)-\left(A_n+C_n\right)Q_n(x)+C_nQ_{n-1}(x), +\end{equation} +where +$$Q_n(x):=Q_n(x;\alpha,\beta,N)$$ +and +$$\left\{\begin{array}{l} +\displaystyle A_n=\frac{(n+\alpha+\beta+1)(n+\alpha+1)(N-n)}{(2n+\alpha+\beta+1)(2n+\alpha+\beta+2)}\\[5mm] +\displaystyle C_n=\frac{n(n+\alpha+\beta+N+1)(n+\beta)}{(2n+\alpha+\beta)(2n+\alpha+\beta+1)}. +\end{array}\right.$$ + +\subsection*{Normalized recurrence relation} +\begin{equation} +\label{NormRecHahn} +xp_n(x)=p_{n+1}(x)+\left(A_n+C_n\right)p_n(x)+A_{n-1}C_np_{n-1}(x), +\end{equation} +where +$$Q_n(x;\alpha,\beta,N)=\frac{(n+\alpha+\beta+1)_n}{(\alpha+1)_n(-N)_n}p_n(x).$$ + +\subsection*{Difference equation} +\begin{equation} +\label{dvHahn} +n(n+\alpha+\beta+1)y(x)=B(x)y(x+1)-\left[B(x)+D(x)\right]y(x)+D(x)y(x-1), +\end{equation} +where +$$y(x)=Q_n(x;\alpha,\beta,N)$$ +and +$$\left\{\begin{array}{l} +\displaystyle B(x)=(x+\alpha+1)(x-N)\\[5mm] +\displaystyle D(x)=x(x-\beta-N-1). +\end{array}\right.$$ + +\subsection*{Forward shift operator} +\begin{eqnarray} +\label{shift1HahnI} +& &Q_n(x+1;\alpha,\beta,N)-Q_n(x;\alpha,\beta,N)\nonumber\\ +& &{}=-\frac{n(n+\alpha+\beta+1)}{(\alpha+1)N}Q_{n-1}(x;\alpha+1,\beta+1,N-1) +\end{eqnarray} +or equivalently +\begin{equation} +\label{shift1HahnII} +\Delta Q_n(x;\alpha,\beta,N)=-\frac{n(n+\alpha+\beta+1)}{(\alpha+1)N}Q_{n-1}(x;\alpha+1,\beta+1,N-1). +\end{equation} + +\subsection*{Backward shift operator} +\begin{eqnarray} +\label{shift2HahnI} +& &(x+\alpha)(N+1-x)Q_n(x;\alpha,\beta,N)-x(\beta+N+1-x)Q_n(x-1;\alpha,\beta,N)\nonumber\\ +& &{}=\alpha(N+1)Q_{n+1}(x;\alpha-1,\beta-1,N+1) +\end{eqnarray} +or equivalently +\begin{eqnarray} +\label{shift2HahnII} +& &\nabla\left[\omega(x;\alpha,\beta,N)Q_n(x;\alpha,\beta,N)\right]\nonumber\\ +& &{}=\frac{N+1}{\beta}\omega(x;\alpha-1,\beta-1,N+1)Q_{n+1}(x;\alpha-1,\beta-1,N+1), +\end{eqnarray} +where +$$\omega(x;\alpha,\beta,N)=\binom{\alpha+x}{x}\binom{\beta+N-x}{N-x}.$$ + +\subsection*{Rodrigues-type formula} +\begin{eqnarray} +\label{RodHahn} +& &\omega(x;\alpha,\beta,N)Q_n(x;\alpha,\beta,N)\nonumber\\ +& &{}=\frac{(-1)^n(\beta+1)_n}{(-N)_n}\nabla^n\left[\omega(x;\alpha+n,\beta+n,N-n)\right]. +\end{eqnarray} + +\subsection*{Generating functions} +For $x=0,1,2,\ldots,N$ we have +\begin{equation} +\label{GenHahn1} +\hyp{1}{1}{-x}{\alpha+1}{-t}\,\hyp{1}{1}{x-N}{\beta+1}{t} +=\sum_{n=0}^N\frac{(-N)_n}{(\beta+1)_nn!}Q_n(x;\alpha,\beta,N)t^n. +\end{equation} + +\begin{eqnarray} +\label{GenHahn2} +& &\hyp{2}{0}{-x,-x+\beta+N+1}{-}{-t}\,\hyp{2}{0}{x-N,x+\alpha+1}{-}{t}\nonumber\\ +& &{}=\sum_{n=0}^N\frac{(-N)_n(\alpha+1)_n}{n!}Q_n(x;\alpha,\beta,N)t^n. +\end{eqnarray} + +\begin{eqnarray} +\label{GenHahn3} +& &\left[(1-t)^{-\alpha-\beta-1}\,\hyp{3}{2}{\frac{1}{2}(\alpha+\beta+1),\frac{1}{2}(\alpha+\beta+2),-x} +{\alpha+1,-N}{-\frac{4t}{(1-t)^2}}\right]_N\nonumber\\ +& &{}=\sum_{n=0}^N\frac{(\alpha+\beta+1)_n}{n!}Q_n(x;\alpha,\beta,N)t^n. +\end{eqnarray} + +\subsection*{Limit relations} + +\subsubsection*{Racah $\rightarrow$ Hahn} +If we take $\gamma+1=-N$ and let $\delta\rightarrow\infty$ in the definition (\ref{DefRacah}) +of the Racah polynomials, we obtain the Hahn polynomials. Hence +$$\lim_{\delta\rightarrow\infty} +R_n(\lambda(x);\alpha,\beta,-N-1,\delta)=Q_n(x;\alpha,\beta,N).$$ +And if we take $\delta=-\beta-N-1$ and let $\gamma\rightarrow\infty$ in the definition (\ref{DefRacah}) +of the Racah polynomials, we also obtain the Hahn polynomials: +$$\lim_{\gamma\rightarrow\infty} +R_n(\lambda(x);\alpha,\beta,\gamma,-\beta-N-1)=Q_n(x;\alpha,\beta,N).$$ +Another way to do this is to take $\alpha+1=-N$ and $\beta\rightarrow\beta+\gamma+N+1$ in +the definition (\ref{DefRacah}) of the Racah polynomials and then take the limit +$\delta\rightarrow\infty$. In that case we obtain the Hahn polynomials in the following way: +$$\lim_{\delta\rightarrow\infty} +R_n(\lambda(x);-N-1,\beta+\gamma+N+1,\gamma,\delta)=Q_n(x;\gamma,\beta,N).$$ + +\subsubsection*{Hahn $\rightarrow$ Jacobi} +To find the Jacobi polynomials given by (\ref{DefJacobi}) from the Hahn polynomials we take +$x\rightarrow Nx$ and let $N\rightarrow\infty$. In fact we have +\begin{equation} +\lim_{N\rightarrow\infty} +Q_n(Nx;\alpha,\beta,N)=\frac{P_n^{(\alpha,\beta)}(1-2x)}{P_n^{(\alpha,\beta)}(1)}. +\end{equation} + +\subsubsection*{Hahn $\rightarrow$ Meixner} +The Meixner polynomials given by (\ref{DefMeixner}) can be obtained from the Hahn polynomials +by taking $\alpha=b-1$, $\beta=N(1-c)c^{-1}$ and letting $N\rightarrow\infty$: +\begin{equation} +\lim_{N\rightarrow\infty} +Q_n(x;b-1,N(1-c)c^{-1},N)=M_n(x;b,c). +\end{equation} + +\subsubsection*{Hahn $\rightarrow$ Krawtchouk} +The Krawtchouk polynomials given by (\ref{DefKrawtchouk}) are obtained from the Hahn polynomials +if we take $\alpha=pt$ and $\beta=(1-p)t$ and let $t\rightarrow\infty$: +\begin{equation} +\lim_{t\rightarrow\infty}Q_n(x;pt,(1-p)t,N)=K_n(x;p,N). +\end{equation} + +\subsection*{Remark} +If we interchange the role of $x$ and $n$ in (\ref{DefHahn}) we obtain the +dual Hahn polynomials given by (\ref{DefDualHahn}). + +\noindent +Since +$$Q_n(x;\alpha,\beta,N)=R_x(\lambda(n);\alpha,\beta,N)$$ +we obtain the dual orthogonality relation for the Hahn polynomials from the +orthogonality relation (\ref{OrtDualHahn}) of the dual Hahn polynomials: +\begin{eqnarray*} +& &\sum_{n=0}^N\frac{(2n+\alpha+\beta+1)(\alpha+1)_n(-N)_nN!}{(-1)^n(n+\alpha+\beta+1)_{N+1}(\beta+1)_nn!} +Q_n(x;\alpha,\beta,N)Q_n(y;\alpha,\beta,N)\\ +& &{}=\frac{\delta_{xy}}{\dbinom{\alpha+x}{x}\dbinom{\beta+N-x}{N-x}},\quad x,y \in \{0,1,2,\ldots,N\}. +\end{eqnarray*} +% RS: add begin\label{sec9.5} +% +\paragraph{\large\bf KLSadd: Special values}\begin{equation} +Q_n(0;\al,\be,N)=1,\quad +Q_n(N;\al,\be,N)=\frac{(-1)^n(\be+1)_n}{(\al+1)_n}\,. +\label{95} +\end{equation} +Use (9.5.1) and compare with (9.8.1) and \eqref{50}. + +From (9.5.3) and \eqref{1} it follows that +\begin{equation} +Q_{2n}(N;\al,\al,2N)=\frac{(\thalf)_n(N+\al+1)_n}{(-N+\thalf)_n(\al+1)_n}\,. +\label{30} +\end{equation} +From (9.5.1) and \mycite{DLMF}{(15.4.24)} it follows that +\begin{equation} +Q_N(x;\al,\be,N)=\frac{(-N-\be)_x}{(\al+1)_x}\qquad(x=0,1,\ldots,N). +\label{44} +\end{equation} +% +\paragraph{\large\bf KLSadd: Symmetries}By the orthogonality relation (9.5.2): +\begin{equation} +\frac{Q_n(N-x;\al,\be,N)}{Q_n(N;\al,\be,N)}=Q_n(x;\be,\al,N), +\label{96} +\end{equation} +It follows from \eqref{97} and \eqref{45} that +\begin{equation} +\frac{Q_{N-n}(x;\al,\be,N)}{Q_N(x;\al,\be,N)} +=Q_n(x;-N-\be-1,-N-\al-1,N) +\qquad(x=0,1,\ldots,N). +\label{100} +\end{equation} +% +\paragraph{\large\bf KLSadd: Duality}The Remark on p.208 gives the duality between Hahn and dual Hahn polynomials: +% +\begin{equation} +Q_n(x;\al,\be,N)=R_x(n(n+\al+\be+1);\al,\be,N)\quad(n,x\in\{0,1,\ldots N\}). +\label{45} +\end{equation} +% +% RS: add end +\subsection*{References} +\cite{AlSalam90}, \cite{AndrewsAskey85}, \cite{Area+II}, \cite{Askey75}, +\cite{Askey89I}, \cite{Askey2005}, \cite{AskeyGasper77}, \cite{AskeyWilson85}, +\cite{AtakRahmanSuslov}, \cite{AtakSuslov88}, \cite{Chihara78}, +\cite{Ciesielski}, \cite{Cooper+}, \cite{Dette95}, \cite{Dunkl76}, +\cite{Dunkl78I}, \cite{Gasper73I}, \cite{Gasper74}, \cite{HoareRahman}, +\cite{Ismail77}, \cite{Ismail2005II}, \cite{Karlin61}, \cite{Koorn81}, \cite{Koorn88}, +\cite{LabelleYehI}, \cite{LabelleYehII}, \cite{Laine}, \cite{Lesky62}, +\cite{Lesky88}, \cite{Lesky89}, \cite{Lesky94I}, \cite{Lesky95II}, +\cite{LewanowiczII}, \cite{Neuman}, \cite{Nikiforov+}, \cite{NikiforovUvarov}, +\cite{Rahman76III}, \cite{Rahman78I}, \cite{Rahman78II}, \cite{Rahman81III}, +\cite{Sablonniere}, \cite{Stanton84}, \cite{Stanton90}, \cite{Wilson80}, \cite{Wilson70II}, +\cite{Zarzo+}. + + +\section{Dual Hahn}\index{Dual Hahn polynomials}\index{Hahn polynomials!Dual} + +\par\setcounter{equation}{0} + +\subsection*{Hypergeometric representation} +\begin{equation} +\label{DefDualHahn} +R_n(\lambda(x);\gamma,\delta,N)= +\hyp{3}{2}{-n,-x,x+\gamma+\delta+1}{\gamma+1,-N}{1},\quad n=0,1,2,\ldots,N, +\end{equation} +where +$$\lambda(x)=x(x+\gamma+\delta+1).$$ + +\subsection*{Orthogonality relation} +For $\gamma>-1$ and $\delta>-1$, or for $\gamma<-N$ and $\delta<-N$, we have +\begin{eqnarray} +\label{OrtDualHahn} +& &\sum_{x=0}^N\frac{(2x+\gamma+\delta+1)(\gamma+1)_x(-N)_xN!}{(-1)^x(x+\gamma+\delta+1)_{N+1}(\delta+1)_xx!} +R_m(\lambda(x);\gamma,\delta,N)R_n(\lambda(x);\gamma,\delta,N)\nonumber\\ +& &{}=\frac{\,\delta_{mn}}{\dbinom{\gamma+n}{n}\dbinom{\delta+N-n}{N-n}}. +\end{eqnarray} + +\subsection*{Recurrence relation} +\begin{equation} +\label{RecDualHahn} +\lambda(x)R_n(\lambda(x)) +=A_nR_{n+1}(\lambda(x))-\left(A_n+C_n\right)R_n(\lambda(x))+C_nR_{n-1}(\lambda(x)), +\end{equation} +where +$$R_n(\lambda(x)):=R_n(\lambda(x);\gamma,\delta,N)$$ +and +$$\left\{\begin{array}{l} +\displaystyle A_n=(n+\gamma+1)(n-N)\\[5mm] +\displaystyle C_n=n(n-\delta-N-1). +\end{array}\right.$$ + +\subsection*{Normalized recurrence relation} +\begin{equation} +\label{NormRecDualHahn} +xp_n(x)=p_{n+1}(x)-(A_n+C_n)p_n(x)+A_{n-1}C_np_{n-1}(x), +\end{equation} +where +$$R_n(\lambda(x);\gamma,\delta,N)=\frac{1}{(\gamma+1)_n(-N)_n}p_n(\lambda(x)).$$ + +\subsection*{Difference equation} +\begin{equation} +\label{dvDualHahn} +-ny(x)=B(x)y(x+1)-\left[B(x)+D(x)\right]y(x)+D(x)y(x-1), +\end{equation} +where +$$y(x)=R_n(\lambda(x);\gamma,\delta,N)$$ +and +$$\left\{\begin{array}{l} +\displaystyle B(x)=\frac{(x+\gamma+1)(x+\gamma+\delta+1)(N-x)}{(2x+\gamma+\delta+1)(2x+\gamma+\delta+2)}\\[5mm] +\displaystyle D(x)=\frac{x(x+\gamma+\delta+N+1)(x+\delta)}{(2x+\gamma+\delta)(2x+\gamma+\delta+1)}. +\end{array}\right.$$ + +\subsection*{Forward shift operator} +\begin{eqnarray} +\label{shift1DualHahnI} +& &R_n(\lambda(x+1);\gamma,\delta,N)-R_n(\lambda(x);\gamma,\delta,N)\nonumber\\ +& &{}=-\frac{n(2x+\gamma+\delta+2)}{(\gamma+1)N}R_{n-1}(\lambda(x);\gamma+1,\delta,N-1) +\end{eqnarray} +or equivalently +\begin{equation} +\label{shift1DualHahnII} +\frac{\Delta R_n(\lambda(x);\gamma,\delta,N)}{\Delta\lambda(x)}= +-\frac{n}{(\gamma+1)N}R_{n-1}(\lambda(x);\gamma+1,\delta,N-1). +\end{equation} + +\subsection*{Backward shift operator} +\begin{eqnarray} +\label{shift2DualHahnI} +& &(x+\gamma)(x+\gamma+\delta)(N+1-x)R_n(\lambda(x);\gamma,\delta,N)\nonumber\\ +& &{}\mathindent{}-x(x+\gamma+\delta+N+1)(x+\delta)R_n(\lambda(x-1);\gamma,\delta,N)\nonumber\\ +& &{}=\gamma(N+1)(2x+\gamma+\delta)R_{n+1}(\lambda(x);\gamma-1,\delta,N+1) +\end{eqnarray} +or equivalently +\begin{eqnarray} +\label{shift2DualHahnII} +& &\frac{\nabla\left[\omega(x;\gamma,\delta,N)R_n(\lambda(x);\gamma,\delta,N)\right]}{\nabla\lambda(x)}\nonumber\\ +& &{}=\frac{1}{\gamma+\delta}\omega(x;\gamma-1,\delta,N+1)R_{n+1}(\lambda(x);\gamma-1,\delta,N+1), +\end{eqnarray} +where +$$\omega(x;\gamma,\delta,N)=\frac{(-1)^x(\gamma+1)_x(\gamma+\delta+1)_x(-N)_x} +{(\gamma+\delta+N+2)_x(\delta+1)_xx!}.$$ + +\newpage + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodDualHahn} +\omega(x;\gamma,\delta,N)R_n(\lambda(x);\gamma,\delta,N) +=(\gamma+\delta+1)_n\left(\nabla_{\lambda}\right)^n\left[\omega(x;\gamma+n,\delta,N-n)\right], +\end{equation} +where +$$\nabla_{\lambda}:=\frac{\nabla}{\nabla\lambda(x)}.$$ + +\subsection*{Generating functions} +For $x=0,1,2,\ldots,N$ we have +\begin{equation} +\label{GenDualHahn1} +(1-t)^{N-x}\,\hyp{2}{1}{-x,-x-\delta}{\gamma+1}{t} +=\sum_{n=0}^N\frac{(-N)_n}{n!}R_n(\lambda(x);\gamma,\delta,N)t^n. +\end{equation} + +\begin{eqnarray} +\label{GenDualHahn2} +& &(1-t)^x\,\hyp{2}{1}{x-N,x+\gamma+1}{-\delta-N}{t}\nonumber\\ +& &=\sum_{n=0}^N\frac{(\gamma+1)_n(-N)_n}{(-\delta-N)_nn!}R_n(\lambda(x);\gamma,\delta,N)t^n. +\end{eqnarray} + +\begin{equation} +\label{GenDualHahn3} +\left[\expe^t\,\hyp{2}{2}{-x,x+\gamma+\delta+1}{\gamma+1,-N}{-t}\right]_N=\sum_{n=0}^N +\frac{R_n(\lambda(x);\gamma,\delta,N)}{n!}t^n. +\end{equation} + +\begin{eqnarray} +\label{GenDualHahn4} +& &\left[(1-t)^{-\epsilon}\,\hyp{3}{2}{\epsilon,-x,x+\gamma+\delta+1}{\gamma+1,-N}{\frac{t}{t-1}}\right]_N\nonumber\\ +& &{}=\sum_{n=0}^N\frac{(\epsilon)_n}{n!}R_n(\lambda(x);\gamma,\delta,N)t^n, +\quad\textrm{$\epsilon$ arbitrary}. +\end{eqnarray} + +\subsection*{Limit relations} + +\subsubsection*{Racah $\rightarrow$ Dual Hahn} +If we take $\alpha+1=-N$ and let $\beta\rightarrow\infty$ in the definition (\ref{DefRacah}) +of the Racah polynomials, then we obtain the dual Hahn polynomials: +$$\lim_{\beta\rightarrow\infty} +R_n(\lambda(x);-N-1,\beta,\gamma,\delta)=R_n(\lambda(x);\gamma,\delta,N).$$ +And if we take $\beta=-\delta-N-1$ and let $\alpha\rightarrow\infty$ in the definition (\ref{DefRacah}) +of the Racah polynomials, then we also obtain the dual Hahn polynomials: +$$\lim_{\alpha\rightarrow\infty} +R_n(\lambda(x);\alpha,-\delta-N-1,\gamma,\delta)=R_n(\lambda(x);\gamma,\delta,N).$$ +Finally, if we take $\gamma+1=-N$ and $\delta\rightarrow\alpha+\delta+N+1$ in the definition +(\ref{DefRacah}) of the Racah polynomials and take the limit +$\beta\rightarrow\infty$ we find the dual Hahn polynomials in the following way: +$$\lim_{\beta\rightarrow\infty} +R_n(\lambda(x);\alpha,\beta,-N-1,\alpha+\delta+N+1)=R_n(\lambda(x);\alpha,\delta,N).$$ + +\subsubsection*{Dual Hahn $\rightarrow$ Meixner} +The Meixner polynomials given by (\ref{DefMeixner}) are obtained from the dual Hahn polynomials +if we take $\gamma=\beta-1$ and $\delta=N(1-c)c^{-1}$ and let $N\rightarrow\infty$: +\begin{equation} +\lim_{N\rightarrow\infty} +R_n(\lambda(x);\beta-1,N(1-c)c^{-1},N)=M_n(x;\beta,c). +\end{equation} + +\subsubsection*{Dual Hahn $\rightarrow$ Krawtchouk} +The Krawtchouk polynomials given by (\ref{DefKrawtchouk}) can be obtained from the dual +Hahn polynomials by setting $\gamma=pt$, $\delta=(1-p)t$ and letting $t\rightarrow\infty$: +\begin{equation} +\lim_{t\rightarrow\infty}R_n(\lambda(x);pt,(1-p)t,N)=K_n(x;p,N). +\end{equation} + +\subsection*{Remark} +If we interchange the role of $x$ and $n$ in the definition (\ref{DefDualHahn}) +of the dual Hahn polynomials we obtain the Hahn polynomials given by (\ref{DefHahn}). + +\noindent +Since +$$R_n(\lambda(x);\gamma,\delta,N)=Q_x(n;\gamma,\delta,N)$$ +we obtain the dual orthogonality relation for the dual Hahn polynomials +from the orthogonality relation (\ref{OrtHahn}) for the Hahn polynomials: +\begin{eqnarray*} +& &\sum_{n=0}^N\binom{\gamma+n}{n}\binom{\delta+N-n}{N-n} R_n(\lambda(x);\gamma,\delta,N)R_n(\lambda(y);\gamma,\delta,N)\\ +& &{}=\frac{(-1)^x(x+\gamma+\delta+1)_{N+1}(\delta+1)_xx!} +{(2x+\gamma+\delta+1)(\gamma+1)_x(-N)_xN!}\delta_{xy},\quad x,y \in \{0,1,2,\ldots,N\}. +\end{eqnarray*} +% RS: add begin\label{sec9.6} +% +\paragraph{\large\bf KLSadd: Special values}By \eqref{44} and \eqref{45} we have +\begin{equation} +R_n(N(N+\ga+\de+1);\ga,\de,N)=\frac{(-N-\de)_n}{(\ga+1)_n}\,. +\label{47} +\end{equation} +It follows from \eqref{95} and \eqref{45} that +\begin{equation} +R_N(x(x+\ga+\de+1);\ga,\de,N) +=\frac{(-1)^x(\de+1)_x}{(\ga+1)_x}\qquad(x=0,1,\ldots,N). +\label{101} +\end{equation} +% +\paragraph{\large\bf KLSadd: Symmetries}Write the weight in (9.6.2) as +\begin{equation} +w_x(\al,\be,N):=N!\,\frac{2x+\ga+\de+1}{(x+\ga+\de+1)_{N+1}}\, +\frac{(\ga+1)_x}{(\de+1)_x}\,\binom Nx. +\label{98} +\end{equation} +Then +\begin{equation} +(\de+1)_N\,w_{N-x}(\ga,\de,N)= +(-\ga-N)_N\,w_x(-\de-N-1,-\ga-N-1,N). +\label{99} +\end{equation} +Hence, by (9.6.2), +\begin{equation} +\frac{R_n((N-x)(N-x+\ga+\de+1);\ga,\de,N)}{R_n(N(N+\ga+\de+1);\ga,\de,N)} +=R_n(x(x-2N-\ga-\de-1);-N-\de-1,-N-\ga-1,N). +\label{97} +\end{equation} +Alternatively, \eqref{97} follows from (9.6.1) and +\mycite{DLMF}{(16.4.11)}. + +It follows from \eqref{96} and \eqref{45} that +\begin{equation} +\frac{R_{N-n}(x(x+\ga+\de+1);\ga,\de,N)} +{R_N(x(x+\ga+\de+1);\ga,\de,N)} +=R_n(x(x+\ga+\de+1);\de,\ga,N)\qquad(x=0,1,\ldots,N). +\label{102} +\end{equation} +% +\paragraph{\large\bf KLSadd: Re: (9.6.11).}The generating function (9.6.11) can be written in a more conceptual way as +\begin{equation} +(1-t)^x\,\hyp21{x-N,x+\ga+1}{-\de-N}t=\frac{N!}{(\de+1)_N}\, +\sum_{n=0}^N \om_n\,R_n(\la(x);\ga,\de,N)\,t^n, +\label{2} +\end{equation} +where +\begin{equation} +\om_n:=\binom{\ga+n}n \binom{\de+N-n}{N-n}, +\label{3} +\end{equation} +i.e., the denominator on the \RHS\ of (9.6.2). +By the duality between Hahn polynomials and dual Hahn polynomials (see \eqref{45}) the above generating function can be rewritten in +terms of Hahn polynomials: +\begin{equation} +(1-t)^n\,\hyp21{n-N,n+\al+1}{-\be-N}t=\frac{N!}{(\be+1)_N}\, +\sum_{x=0}^N w_x\,Q_n(x;\al,\be,N)\,t^x, +\label{4} +\end{equation} +where +\begin{equation} +w_x:=\binom{\al+x}x \binom{\be+N-x}{N-x}, +\label{5} +\end{equation} +i.e., the weight occurring in the orthogonality relation (9.5.2) +for Hahn polynomials. +\paragraph{\large\bf KLSadd: Re: (9.6.15).}There should be a closing bracket before the equality sign. +% +% RS: add end +\subsection*{References} +\cite{Askey2005}, \cite{AskeyWilson85}, \cite{AtakRahmanSuslov}, \cite{AtakSuslov88}, +\cite{Ismail2005II}, \cite{Karlin61}, \cite{Koorn81}, \cite{Koorn88}, \cite{Lesky93}, +\cite{Lesky94I}, \cite{Lesky95I}, \cite{Lesky95II}, \cite{Nikiforov+}, \cite{NikiforovUvarov}, +\cite{Rahman81II}, \cite{Stanton84}, \cite{Wilson80}. + + +\section{Meixner-Pollaczek}\index{Meixner-Pollaczek polynomials} + +\par\setcounter{equation}{0} + +\subsection*{Hypergeometric representation} +\begin{equation} +\label{DefMP} +P_n^{(\lambda)}(x;\phi)=\frac{(2\lambda)_n}{n!}\expe^{in\phi}\, +\hyp{2}{1}{-n,\lambda+ix}{2\lambda}{1-\expe^{-2i\phi}}. +\end{equation} + +\subsection*{Orthogonality relation} +\begin{equation} +\label{OrtMP} +\frac{1}{2\pi}\int_{-\infty}^{\infty} +\e^{(2\phi-\pi)x}\left|\Gamma(\lambda+ix)\right|^2 +P_m^{(\lambda)}(x;\phi)P_n^{(\lambda)}(x;\phi)\,dx +{}=\frac{\Gamma(n+2\lambda)}{(2\sin\phi)^{2\lambda}n!}\,\delta_{mn}, +% \constraint{ +% $\lambda > 0$ & +% $0 < \phi < \pi$ } +\end{equation} + +\subsection*{Recurrence relation} +\begin{eqnarray} +\label{RecMP} +& &(n+1)P_{n+1}^{(\lambda)}(x;\phi)-2\left[x\sin\phi+(n+\lambda)\cos\phi\right] +P_n^{(\lambda)}(x;\phi)\nonumber\\ +& &{}\mathindent{}+(n+2\lambda-1)P_{n-1}^{(\lambda)}(x;\phi)=0. +\end{eqnarray} + +\subsection*{Normalized recurrence relation} +\begin{equation} +\label{NormRecMP} +xp_n(x)=p_{n+1}(x)-\left(\frac{n+\lambda}{\tan\phi}\right)p_n(x)+ +\frac{n(n+2\lambda-1)}{4\sin^2\phi}p_{n-1}(x), +\end{equation} +where +$$P_n^{(\lambda)}(x;\phi)=\frac{(2\sin\phi)^n}{n!}p_n(x).$$ + +\subsection*{Difference equation} +\begin{eqnarray} +\label{dvMP} +& &\e^{i\phi}(\lambda-ix)y(x+i)+2i\left[x\cos\phi-(n+\lambda)\sin\phi\right]y(x)\nonumber\\ +& &{}\mathindent{}-\e^{-i\phi}(\lambda+ix)y(x-i)=0,\quad y(x)=P_n^{(\lambda)}(x;\phi). +\end{eqnarray} + +\subsection*{Forward shift operator} +\begin{equation} +\label{shift1MPI} +P_n^{(\lambda)}(x+\textstyle\frac{1}{2}i;\phi)- +P_n^{(\lambda)}(x-\textstyle\frac{1}{2}i;\phi)=(\expe^{i\phi}-\expe^{-i\phi}) +P_{n-1}^{(\lambda+\frac{1}{2})}(x;\phi) +\end{equation} +or equivalently +\begin{equation} +\label{shift1MPII} +\frac{\delta P_n^{(\lambda)}(x;\phi)}{\delta x} +=2\sin\phi\,P_{n-1}^{(\lambda+\frac{1}{2})}(x;\phi). +\end{equation} + +\subsection*{Backward shift operator} +\begin{eqnarray} +\label{shift2MPI} +& &\e^{i\phi}(\lambda-\textstyle\frac{1}{2}-ix) +P_n^{(\lambda)}(x+\textstyle\frac{1}{2}i;\phi)+ +\e^{-i\phi}(\lambda-\textstyle\frac{1}{2}+ix) +P_n^{(\lambda)}(x-\textstyle\frac{1}{2}i;\phi)\nonumber\\ +& &{}=(n+1)P_{n+1}^{(\lambda-\frac{1}{2})}(x;\phi) +\end{eqnarray} +or equivalently +\begin{equation} +\label{shift2MPII} +\frac{\delta\left[\omega(x;\lambda,\phi)P_n^{(\lambda)}(x;\phi)\right]}{\delta x}= +-(n+1)\omega(x;\lambda-\textstyle\frac{1}{2},\phi) +P_{n+1}^{(\lambda-\frac{1}{2})}(x;\phi), +\end{equation} +where +$$\omega(x;\lambda,\phi)=\Gamma(\lambda+ix)\Gamma(\lambda-ix)\e^{(2\phi-\pi)x}.$$ + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodMP} +\omega(x;\lambda,\phi)P_n^{(\lambda)}(x;\phi)=\frac{(-1)^n}{n!} +\left(\frac{\delta}{\delta x}\right)^n\left[\omega(x;\lambda+\textstyle\frac{1}{2}n,\phi)\right]. +\end{equation} + +\newpage + +\subsection*{Generating functions} +\begin{equation} +\label{GenMP1} +(1-\expe^{i\phi}t)^{-\lambda+ix}(1-\expe^{-i\phi}t)^{-\lambda-ix}= +\sum_{n=0}^{\infty}P_n^{(\lambda)}(x;\phi)t^n. +\end{equation} + +\begin{equation} +\label{GenMP2} +\e^t\,\hyp{1}{1}{\lambda+ix}{2\lambda}{(\expe^{-2i\phi}-1)t}= +\sum_{n=0}^{\infty}\frac{P_n^{(\lambda)}(x;\phi)}{(2\lambda)_n\expe^{in\phi}}t^n. +\end{equation} + +\begin{eqnarray} +\label{GenMP3} +& &(1-t)^{-\gamma}\,\hyp{2}{1}{\gamma,\lambda+ix}{2\lambda}{\frac{(1-\e^{-2i\phi})t}{t-1}}\nonumber\\ +& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n}{(2\lambda)_n}\frac{P_n^{(\lambda)}(x;\phi)}{\e^{in\phi}}t^n, +\quad\textrm{$\gamma$ arbitrary}. +\end{eqnarray} + +\subsection*{Limit relations} + +\subsubsection*{Continuous dual Hahn $\rightarrow$ Meixner-Pollaczek} +The Meixner-Pollaczek polynomials can be obtained from the continuous dual Hahn polynomials +given by (\ref{DefContDualHahn}) by the substitutions $x\rightarrow x-t$, $a=\lambda+it$, +$b=\lambda-it$ and $c=t\cot\phi$ and the limit $t\rightarrow\infty$: +$$\lim_{t\rightarrow\infty}\frac{S_n((x-t)^2;\lambda+it,\lambda-it,t\cot\phi)}{t^nn!} +=\frac{P_n^{(\lambda)}(x;\phi)}{(\sin\phi)^n}.$$ + +\subsubsection*{Continuous Hahn $\rightarrow$ Meixner-Pollaczek} +By setting $x\rightarrow x+t$, $a=\lambda-it$, $c=\lambda+it$ and $b=d=t\tan\phi$ in the +definition (\ref{DefContHahn}) of the continuous Hahn polynomials and taking the limit +$t\rightarrow\infty$ we obtain the Meixner-Pollaczek polynomials: +$$\lim_{t\rightarrow\infty}\frac{p_n(x+t;\lambda-it,t\tan\phi,\lambda+it,t\tan\phi)}{t^nn!} +=\frac{P_n^{(\lambda)}(x;\phi)}{(\cos\phi)^n}.$$ + +\newpage + +\subsubsection*{Meixner-Pollaczek $\rightarrow$ Laguerre} +The Laguerre polynomials given by (\ref{DefLaguerre}) can be obtained from the +Meixner-Pollaczek polynomials by the substitution $\lambda=\frac{1}{2}(\alpha+1)$, +$x\rightarrow -\frac{1}{2}\phi^{-1}x$ and the limit $\phi\rightarrow 0$: +\begin{equation} +\lim_{\phi\rightarrow 0} +P_n^{(\frac{1}{2}\alpha+\frac{1}{2})}(-\textstyle\frac{1}{2}\phi^{-1}x;\phi)=L_n^{(\alpha)}(x). +\end{equation} + +\subsubsection*{Meixner-Pollaczek $\rightarrow$ Hermite} +The Hermite polynomials given by (\ref{DefHermite}) are obtained from the Meixner-Pollaczek +polynomials if we substitute $x\rightarrow (\sin\phi)^{-1}(x\sqrt{\lambda}-\lambda\cos\phi)$ +and then let $\lambda\rightarrow\infty$: +\begin{equation} +\lim_{\lambda\rightarrow\infty} +\lambda^{-\frac{1}{2}n}P_n^{(\lambda)} +((\sin\phi)^{-1}(x\sqrt{\lambda}-\lambda\cos\phi);\phi)=\frac{H_n(x)}{n!}. +\end{equation} + +\subsection*{Remark} +Since we have for $k-1$ and $\beta>-1$ we have +\begin{eqnarray} +\label{OrtJacobi1} +& &\int_{-1}^1(1-x)^{\alpha}(1+x)^{\beta}P_m^{(\alpha,\beta)}(x)P_n^{(\alpha,\beta)}(x)\,dx\nonumber\\ +& &{}=\frac{2^{\alpha+\beta+1}}{2n+\alpha+\beta+1}\frac{\Gamma(n+\alpha+1)\Gamma(n+\beta+1)}{\Gamma(n+\alpha+\beta+1)n!}\,\delta_{mn}. +\end{eqnarray} +For $\alpha+\beta<-2N-1$, $\beta>-1$ and $m,n\in\{0,1,2,\ldots,N\}$ we also have +\begin{eqnarray} +\label{OrtJacobi2} +& &\int_1^{\infty}(x+1)^{\alpha}(x-1)^{\beta}P_m^{(\alpha,\beta)}(-x)P_n^{(\alpha,\beta)}(-x)\,dx\nonumber\\ +& &{}=-\frac{2^{\alpha+\beta+1}}{2n+\alpha+\beta+1}\frac{\Gamma(-n-\alpha-\beta)\Gamma(n+\alpha+\beta+1)}{\Gamma(-n-\alpha)n!}\,\delta_{mn}. +\end{eqnarray} + +\subsection*{Recurrence relation} +\begin{eqnarray} +\label{RecJacobi} +xP_n^{(\alpha,\beta)}(x)&=&\frac{2(n+1)(n+\alpha+\beta+1)}{(2n+\alpha+\beta+1)(2n+\alpha+\beta+2)}P_{n+1}^{(\alpha,\beta)}(x)\nonumber\\ +& &{}\mathindent{}+\frac{\beta^2-\alpha^2}{(2n+\alpha+\beta)(2n+\alpha+\beta+2)}P_n^{(\alpha,\beta)}(x)\nonumber\\ +& &{}\mathindent\mathindent{}+\frac{2(n+\alpha)(n+\beta)}{(2n+\alpha+\beta)(2n+\alpha+\beta+1)}P_{n-1}^{(\alpha,\beta)}(x). +\end{eqnarray} + +\subsection*{Normalized recurrence relation} +\begin{eqnarray} +\label{NormRecJacobi} +xp_n(x)&=&p_{n+1}(x)+\frac{\beta^2-\alpha^2}{(2n+\alpha+\beta)(2n+\alpha+\beta+2)}p_n(x)\nonumber\\ +& &{}\mathindent{}+\frac{4n(n+\alpha)(n+\beta)(n+\alpha+\beta)} +{(2n+\alpha+\beta-1)(2n+\alpha+\beta)^2(2n+\alpha+\beta+1)}p_{n-1}(x) +\end{eqnarray} +where +$$P_n^{(\alpha,\beta)}(x)=\frac{(n+\alpha+\beta+1)_n}{2^nn!}p_n(x).$$ + +\subsection*{Differential equation} +\begin{eqnarray} +\label{dvJacobi} +& &(1-x^2)y''(x)+\left[\beta-\alpha-(\alpha+\beta+2)x\right]y'(x)\nonumber\\ +& &{}\mathindent{}+n(n+\alpha+\beta+1)y(x)=0,\quad y(x)=P_n^{(\alpha,\beta)}(x). +\end{eqnarray} + +\subsection*{Forward shift operator} +\begin{equation} +\label{shift1Jacobi} +\frac{d}{dx}P_n^{(\alpha,\beta)}(x)=\frac{n+\alpha+\beta+1}{2}P_{n-1}^{(\alpha+1,\beta+1)}(x). +\end{equation} + +\subsection*{Backward shift operator} +\begin{eqnarray} +\label{shift2JacobiI} +& &(1-x^2)\frac{d}{dx}P_n^{(\alpha,\beta)}(x)+ +\left[(\beta-\alpha)-(\alpha+\beta)x\right]P_n^{(\alpha,\beta)}(x)\nonumber\\ +& &{}=-2(n+1)P_{n+1}^{(\alpha-1,\beta-1)}(x) +\end{eqnarray} +or equivalently +\begin{eqnarray} +\label{shift2JacobiII} +& &\frac{d}{dx}\left[(1-x)^\alpha(1+x)^\beta P_n^{(\alpha,\beta)}(x)\right]\nonumber\\ +& &{}=-2(n+1)(1-x)^{\alpha-1}(1+x)^{\beta-1}P_{n+1}^{(\alpha-1,\beta-1)}(x). +\end{eqnarray} + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodJacobi} +(1-x)^{\alpha}(1+x)^{\beta}P_n^{(\alpha,\beta)}(x)= +\frac{(-1)^n}{2^nn!}\left(\frac{d}{dx}\right)^n +\left[(1-x)^{n+\alpha}(1+x)^{n+\beta}\right]. +\end{equation} + +\subsection*{Generating functions} +\begin{equation} +\label{GenJacobi1} +\frac{2^{\alpha+\beta}}{R(1+R-t)^{\alpha}(1+R+t)^{\beta}}= +\sum_{n=0}^{\infty}P_n^{(\alpha,\beta)}(x)t^n,\quad R=\sqrt{1-2xt+t^2}. +\end{equation} + +\begin{eqnarray} +\label{GenJacobi2} +& &\hyp{0}{1}{-}{\alpha+1}{\frac{(x-1)t}{2}}\,\hyp{0}{1}{-}{\beta+1}{\frac{(x+1)t}{2}}\nonumber\\ +& &{}=\sum_{n=0}^{\infty}\frac{P_n^{(\alpha,\beta)}(x)}{(\alpha+1)_n(\beta+1)_n}t^n. +\end{eqnarray} + +\begin{eqnarray} +\label{GenJacobi3} +& &(1-t)^{-\alpha-\beta-1}\,\hyp{2}{1}{\frac{1}{2}(\alpha+\beta+1),\frac{1}{2}(\alpha+\beta+2)} +{\alpha+1}{\frac{2(x-1)t}{(1-t)^2}}\nonumber\\ +& &{}=\sum_{n=0}^{\infty}\frac{(\alpha+\beta+1)_n}{(\alpha+1)_n}P_n^{(\alpha,\beta)}(x)t^n. +\end{eqnarray} + +\begin{eqnarray} +\label{GenJacobi4} +& &(1+t)^{-\alpha-\beta-1}\,\hyp{2}{1}{\frac{1}{2}(\alpha+\beta+1),\frac{1}{2}(\alpha+\beta+2)} +{\beta+1}{\frac{2(x+1)t}{(1+t)^2}}\nonumber\\ +& &{}=\sum_{n=0}^{\infty}\frac{(\alpha+\beta+1)_n}{(\beta+1)_n}P_n^{(\alpha,\beta)}(x)t^n. +\end{eqnarray} + +\begin{eqnarray} +\label{GenJacobi5} +& &\hyp{2}{1}{\gamma,\alpha+\beta+1-\gamma}{\alpha+1}{\frac{1-R-t}{2}}\, +\hyp{2}{1}{\gamma,\alpha+\beta+1-\gamma}{\beta+1}{\frac{1-R+t}{2}}\nonumber\\ +& &{}=\sum_{n=0}^{\infty} +\frac{(\gamma)_n(\alpha+\beta+1-\gamma)_n}{(\alpha+1)_n(\beta+1)_n}P_n^{(\alpha,\beta)}(x)t^n, +\quad R=\sqrt{1-2xt+t^2} +\end{eqnarray} +with $\gamma$ arbitrary. + +\subsection*{Limit relations} + +\subsubsection*{Wilson $\rightarrow$ Jacobi} +The Jacobi polynomials can be found from the Wilson polynomials given by (\ref{DefWilson}) by +substituting $a=b=\frac{1}{2}(\alpha+1)$, $c=\frac{1}{2}(\beta+1)+it$, +$d=\frac{1}{2}(\beta+1)-it$ and $x\rightarrow t\sqrt{\frac{1}{2}(1-x)}$ in the +definition (\ref{DefWilson}) of the Wilson polynomials and taking the limit +$t\rightarrow\infty$. In fact we have +$$\lim_{t\rightarrow\infty} +\frac{W_n(\frac{1}{2}(1-x)t^2;\frac{1}{2}(\alpha+1), +\frac{1}{2}(\alpha+1),\frac{1}{2}(\beta+1)+it,\frac{1}{2}(\beta+1)-it)} +{t^{2n}n!}=P_n^{(\alpha,\beta)}(x).$$ + +\subsubsection*{Continuous Hahn $\rightarrow$ Jacobi} +The Jacobi polynomials follow from the continuous Hahn polynomials given by (\ref{DefContHahn}) +by using the substitution $x\rightarrow \frac{1}{2}xt$, $a=\frac{1}{2}(\alpha+1-it)$, +$b=\frac{1}{2}(\beta+1+it)$, $c=\frac{1}{2}(\alpha+1+it)$ and $d=\frac{1}{2}(\beta+1-it)$ +in (\ref{DefContHahn}), division by $t^n$ and the limit $t\rightarrow\infty$: +$$\lim_{t\rightarrow\infty} +\frac{p_n(\frac{1}{2}xt;\frac{1}{2}(\alpha+1-it),\frac{1}{2}(\beta+1+it), +\frac{1}{2}(\alpha+1+it),\frac{1}{2}(\beta+1-it))}{t^n}=P_n^{(\alpha,\beta)}(x).$$ + +\subsubsection*{Hahn $\rightarrow$ Jacobi} +To find the Jacobi polynomials from the Hahn polynomials given by (\ref{DefHahn}) we take +$x\rightarrow Nx$ in (\ref{DefHahn}) and let $N\rightarrow\infty$. In fact we have +$$\lim_{N\rightarrow\infty} +Q_n(Nx;\alpha,\beta,N)=\frac{P_n^{(\alpha,\beta)}(1-2x)}{P_n^{(\alpha,\beta)}(1)}.$$ + +\subsubsection*{Jacobi $\rightarrow$ Laguerre} +The Laguerre polynomials given by (\ref{DefLaguerre}) can be obtained from the Jacobi +polynomials by setting $x\rightarrow 1-2\beta^{-1}x$ and then the limit $\beta\rightarrow\infty$: +\begin{equation} +\lim_{\beta\rightarrow\infty} +P_n^{(\alpha,\beta)}(1-2\beta^{-1}x)=L_n^{(\alpha)}(x). +\end{equation} + +\subsubsection*{Jacobi $\rightarrow$ Bessel} +The Bessel polynomials given by (\ref{DefBessel}) are obtained from the Jacobi polynomials +if we take $\beta=a-\alpha$ and let $\alpha\rightarrow-\infty$: +\begin{equation} +\lim_{\alpha\rightarrow-\infty} +\frac{P_n^{(\alpha,a-\alpha)}(1+\alpha x)}{P_n^{(\alpha,a-\alpha)}(1)}=y_n(x;a). +\end{equation} + +\subsubsection*{Jacobi $\rightarrow$ Hermite} +The Hermite polynomials given by (\ref{DefHermite}) follow from the Jacobi polynomials +by taking $\beta=\alpha$ and letting $\alpha\rightarrow\infty$ in the following way: +\begin{equation} +\lim_{\alpha\rightarrow\infty} +\alpha^{-\frac{1}{2}n}P_n^{(\alpha,\alpha)}(\alpha^{-\frac{1}{2}}x)=\frac{H_n(x)}{2^nn!}. +\end{equation} + +\subsection*{Remarks} +The definition (\ref{DefJacobi}) of the Jacobi polynomials can also be written as: +$$P_n^{(\alpha,\beta)}(x)=\frac{1}{n!}\sum_{k=0}^n\frac{(-n)_k}{k!} +(n+\alpha+\beta+1)_k(\alpha+k+1)_{n-k}\left(\frac{1-x}{2}\right)^k.$$ +In this way the Jacobi polynomials can also be seen as polynomials in the parameters $\alpha$ +and $\beta$. Therefore they can be defined for all $\alpha$ and $\beta$. Then we have the +following connection with the Meixner polynomials given by (\ref{DefMeixner}): +$$\frac{(\beta)_n}{n!}M_n(x;\beta,c)=P_n^{(\beta-1,-n-\beta-x)}((2-c)c^{-1}).$$ + +\noindent +The Jacobi polynomials are related to the pseudo Jacobi polynomials defined by +(\ref{DefPseudoJacobi}) in the following way: +$$P_n(x;\nu,N)=\frac{(-2i)^nn!}{(n-2N-1)_n}P_n^{(-N-1+i\nu,-N-1-i\nu)}(ix).$$ + +\noindent +The Jacobi polynomials are also related to the Gegenbauer (or ultraspherical) polynomials +given by (\ref{DefGegenbauer}) by the quadratic transformations: +$$C_{2n}^{(\lambda)}(x)=\frac{(\lambda)_n}{(\frac{1}{2})_n} +P_n^{(\lambda-\frac{1}{2},-\frac{1}{2})}(2x^2-1)$$ +and +$$C_{2n+1}^{(\lambda)}(x)=\frac{(\lambda)_{n+1}}{(\frac{1}{2})_{n+1}} +xP_n^{(\lambda-\frac{1}{2},\frac{1}{2})}(2x^2-1).$$ +% RS: add begin\label{sec9.8} +% +\paragraph{\large\bf KLSadd: Orthogonality relation}Write the \RHS\ of (9.8.2) as $h_n\,\de_{m,n}$. Then +\begin{equation} +\begin{split} +&\frac{h_n}{h_0}= +\frac{n+\al+\be+1}{2n+\al+\be+1}\, +\frac{(\al+1)_n(\be+1)_n}{(\al+\be+2)_n\,n!}\,,\quad +h_0=\frac{2^{\al+\be+1}\Ga(\al+1)\Ga(\be+1)}{\Ga(\al+\be+2)}\,,\sLP +&\frac{h_n}{h_0\,(P_n^{(\al,\be)}(1))^2}= +\frac{n+\al+\be+1}{2n+\al+\be+1}\, +\frac{(\be+1)_n\,n!}{(\al+1)_n\,(\al+\be+2)_n}\,. +\end{split} +\label{60} +\end{equation} + +In (9.8.3) the numerator factor $\Ga(n+\al+\be+1)$ in the last line should be +$\Ga(\be+1)$. When thus corrected, (9.8.3) can be rewritten as: +\begin{equation} +\begin{split} +&\int_1^\iy P_m^{(\al,\be)}(x)\,P_n^{(\al,\be)}(x)\,(x-1)^\al (x+1)^\be\,dx=h_n\,\de_{m,n}\,,\\ +&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad-1-\be>\al>-1,\quad m,n<-\thalf(\al+\be+1),\\ +&\frac{h_n}{h_0}= +\frac{n+\al+\be+1}{2n+\al+\be+1}\, +\frac{(\al+1)_n(\be+1)_n}{(\al+\be+2)_n\,n!}\,,\quad +h_0=\frac{2^{\al+\be+1}\Ga(\al+1)\Ga(-\al-\be-1)}{\Ga(-\be)}\,. +\end{split} +\label{122} +\end{equation} + +% +\paragraph{\large\bf KLSadd: Symmetry}\begin{equation} +P_n^{(\al,\be)}(-x)=(-1)^n\,P_n^{(\be,\al)}(x). +\label{48} +\end{equation} +Use (9.8.2) and (9.8.5b) or see \mycite{DLMF}{Table 18.6.1}. +% +\paragraph{\large\bf KLSadd: Special values}\begin{equation} +P_n^{(\al,\be)}(1)=\frac{(\al+1)_n}{n!}\,,\quad +P_n^{(\al,\be)}(-1)=\frac{(-1)^n(\be+1)_n}{n!}\,,\quad +\frac{P_n^{(\al,\be)}(-1)}{P_n^{(\al,\be)}(1)}=\frac{(-1)^n(\be+1)_n}{(\al+1)_n}\,. +\label{50} +\end{equation} +Use (9.8.1) and \eqref{48} or see \mycite{DLMF}{Table 18.6.1}. +% +\paragraph{\large\bf KLSadd: Generating functions}Formula (9.8.15) was first obtained by Brafman \myciteKLS{109}. +% +\paragraph{\large\bf KLSadd: Bilateral generating functions}For $0\le r<1$ and $x,y\in[-1,1]$ we have in terms of $F_4$ (see~\eqref{62}): +\begin{align} +&\sum_{n=0}^\iy\frac{(\al+\be+1)_n\,n!}{(\al+1)_n(\be+1)_n}\,r^n\, +P_n^{(\al,\be)}(x)\,P_n^{(\al,\be)}(y) +=\frac1{(1+r)^{\al+\be+1}} +\nonumber\\ +&\qquad\quad\times F_4\Big(\thalf(\al+\be+1),\thalf(\al+\be+2);\al+1,\be+1; +\frac{r(1-x)(1-y)}{(1+r)^2},\frac{r(1+x)(1+y)}{(1+r)^2}\Big), +\label{58}\sLP +&\sum_{n=0}^\iy\frac{2n+\al+\be+1}{n+\al+\be+1} +\frac{(\al+\be+2)_n\,n!}{(\al+1)_n(\be+1)_n}\,r^n\, +P_n^{(\al,\be)}(x)\,P_n^{(\al,\be)}(y) +=\frac{1-r}{(1+r)^{\al+\be+2}}\nonumber\\ +&\qquad\quad\times F_4\Big(\thalf(\al+\be+2),\thalf(\al+\be+3);\al+1,\be+1; +\frac{r(1-x)(1-y)}{(1+r)^2},\frac{r(1+x)(1+y)}{(1+r)^2}\Big). +\label{59} +\end{align} +Formulas \eqref{58} and \eqref{59} were first +given by Bailey \myciteKLS{91}{(2.1), (2.3)}. +See Stanton \myciteKLS{485} for a shorter proof. +(However, in the second line of +\myciteKLS{485}{(1)} $z$ and $Z$ should be interchanged.)$\;$ +As observed in Bailey \myciteKLS{91}{p.10}, \eqref{59} follows +from \eqref{58} +by applying the operator $r^{-\half(\al+\be-1)}\,\frac d{dr}\circ r^{\half(\al+\be+1)}$ +to both sides of \eqref{58}. +In view of \eqref{60}, formula \eqref{59} is the Poisson kernel for Jacobi +polynomials. The \RHS\ of \eqref{59} makes clear that this kernel is positive. +See also the discussion in Askey \myciteKLS{46}{following (2.32)}. +% +\paragraph{\large\bf KLSadd: Quadratic transformations}\begin{align} +\frac{C_{2n}^{(\al+\half)}(x)}{C_{2n}^{(\al+\half)}(1)} +=\frac{P_{2n}^{(\al,\al)}(x)}{P_{2n}^{(\al,\al)}(1)} +&=\frac{P_n^{(\al,-\half)}(2x^2-1)}{P_n^{(\al,-\half)}(1)}\,, +\label{51}\\ +\frac{C_{2n+1}^{(\al+\half)}(x)}{C_{2n+1}^{(\al+\half)}(1)} +=\frac{P_{2n+1}^{(\al,\al)}(x)}{P_{2n+1}^{(\al,\al)}(1)} +&=\frac{x\,P_n^{(\al,\half)}(2x^2-1)}{P_n^{(\al,\half)}(1)}\,. +\label{52} +\end{align} +See p.221, Remarks, last two formulas together with \eqref{50} and \eqref{49}. +Or see \mycite{DLMF}{(18.7.13), (18.7.14)}. +% +\paragraph{\large\bf KLSadd: Differentiation formulas}Each differentiation formula is given in two equivalent forms. +\begin{equation} +\begin{split} +\frac d{dx}\left((1-x)^\al P_n^{(\al,\be)}(x)\right)&= +-(n+\al)\,(1-x)^{\al-1} P_n^{(\al-1,\be+1)}(x),\\ +\left((1-x)\frac d{dx}-\al\right)P_n^{(\al,\be)}(x)&= +-(n+\al)\,P_n^{(\al-1,\be+1)}(x). +\end{split} +\label{68} +\end{equation} +% +\begin{equation} +\begin{split} +\frac d{dx}\left((1+x)^\be P_n^{(\al,\be)}(x)\right)&= +(n+\be)\,(1+x)^{\be-1} P_n^{(\al+1,\be-1)}(x),\\ +\left((1+x)\frac d{dx}+\be\right)P_n^{(\al,\be)}(x)&= +(n+\be)\,P_n^{(\al+1,\be-1)}(x). +\end{split} +\label{69} +\end{equation} +Formulas \eqref{68} and \eqref{69} follow from +\mycite{DLMF}{(15.5.4), (15.5.6)} +together with (9.8.1). They also follow from each other by \eqref{48}. +% +\paragraph{\large\bf KLSadd: Generalized Gegenbauer polynomials}These are defined by +\begin{equation} +S_{2m}^{(\al,\be)}(x):=\const P_m^{(\al,\be)}(2x^2-1),\qquad +S_{2m+1}^{(\al,\be)}(x):=\const x\,P_m^{(\al,\be+1)}(2x^2-1) +\label{70} +\end{equation} +in the notation of \myciteKLS{146}{p.156} +(see also \cite{K27}), while \cite[Section 1.5.2]{K26} +has $C_n^{(\la,\mu)}(x)=\const\allowbreak\times S_n^{(\la-\half,\mu-\half)}(x)$. +For $\al,\be>-1$ we have the orthogonality relation +\begin{equation} +\int_{-1}^1 S_m^{(\al,\be)}(x)\,S_n^{(\al,\be)}(x)\,|x|^{2\be+1}(1-x^2)^\al\,dx +=0\qquad(m\ne n). +\label{71} +\end{equation} +For $\be=\al-1$ generalized Gegenbauer polynomials are limit cases of +continuous $q$-ultraspherical polynomials, see \eqref{176}. + +If we define the {\em Dunkl operator} $T_\mu$ by +\begin{equation} +(T_\mu f)(x):=f'(x)+\mu\,\frac{f(x)-f(-x)}x +\label{72} +\end{equation} +and if we choose the constants in \eqref{70} as +\begin{equation} +S_{2m}^{(\al,\be)}(x)=\frac{(\al+\be+1)_m}{(\be+1)_m}\, P_m^{(\al,\be)}(2x^2-1),\quad +S_{2m+1}^{(\al,\be)}(x)=\frac{(\al+\be+1)_{m+1}}{(\be+1)_{m+1}}\, +x\,P_m^{(\al,\be+1)}(2x^2-1) +\label{73} +\end{equation} +then (see \cite[(1.6)]{K5}) +\begin{equation} +T_{\be+\half}S_n^{(\al,\be)}=2(\al+\be+1)\,S_{n-1}^{(\al+1,\be)}. +\label{74} +\end{equation} +Formula \eqref{74} with \eqref{73} substituted gives rise to two +differentiation formulas involving Jacobi polynomials which are equivalent to +(9.8.7) and \eqref{69}. + +Composition of \eqref{74} with itself gives +\[ +T_{\be+\half}^2S_n^{(\al,\be)}=4(\al+\be+1)(\al+\be+2)\,S_{n-2}^{(\al+2,\be)}, +\] +which is equivalent to the composition of (9.8.7) and \eqref{69}: +\begin{equation} +\left(\frac{d^2}{dx^2}+\frac{2\be+1}x\,\frac d{dx}\right)P_n^{(\al,\be)}(2x^2-1) +=4(n+\al+\be+1)(n+\be)\,P_{n-1}^{(\al+2,\be)}(2x^2-1). +\label{75} +\end{equation} +Formula \eqref{75} was also given in \myciteKLS{322}{(2.4)}. +% +% RS: add end +\subsection*{References} +\cite{Abram}, \cite{Ahmed+82}, \cite{Allaway89}, \cite{NAlSalam66}, \cite{AlSalam64}, +\cite{AlSalamChihara72}, \cite{AndrewsAskey85}, \cite{AndrewsAskeyRoy}, \cite{Askey68}, +\cite{Askey70}, \cite{Askey72}, \cite{Askey73}, \cite{Askey74}, \cite{Askey75}, \cite{Askey78}, +\cite{Askey89I}, \cite{Askey2005}, \cite{AskeyFitch}, \cite{AskeyGasper71I}, \cite{AskeyGasper71II}, +\cite{AskeyGasper76}, \cite{AskeyWainger}, \cite{AskeyWilson85}, \cite{Bailey38}, \cite{Brafman51}, +\cite{Brown}, \cite{Carlitz61II}, \cite{Carlitz67}, \cite{ChenIsmail91}, \cite{Chihara78}, +\cite{Chow+}, \cite{Ciesielski}, \cite{Cooper+}, \cite{DetteStudden92}, \cite{DetteStudden95}, +\cite{DijksmaKoorn}, \cite{DimitrovRafaeli}, \cite{DimitrovRodrigues}, \cite{Doha2002I}, +\cite{Doha2003I}, \cite{Doha2004II}, \cite{Dunkl84}, \cite{ElbertLaforgia87II}, +\cite{ElbertLaforgiaRodono}, \cite{Erdelyi+}, \cite{Faldey}, \cite{FoataLeroux}, +\cite{Gasper69}, \cite{Gasper70I}, \cite{Gasper70II}, \cite{Gasper71I}, \cite{Gasper71II}, +\cite{Gasper72I}, \cite{Gasper72II}, \cite{Gasper73I}, \cite{Gasper73II}, \cite{Gasper74}, +\cite{Gasper77}, \cite{Gatteschi87}, \cite{Gautschi2008}, \cite{Gautschi2009I}, +\cite{Gautschi2009II}, \cite{GautschiLeopardi}, \cite{GawronskiShawyer}, \cite{Godoy+}, +\cite{Grad}, \cite{Hajmirzaahmad94}, \cite{HartmannStephan}, \cite{Horton}, \cite{Ismail74}, +\cite{Ismail77}, \cite{Ismail96}, \cite{Ismail2005II}, \cite{IsmailLi}, +\cite{IsmailMassonRahman}, \cite{Kochneff97I}, \cite{Koekoek99}, \cite{Koekoek2000}, +\cite{Koelink96II}, \cite{Koorn73}, \cite{Koorn74}, \cite{Koorn75}, \cite{Koorn77II}, +\cite{Koorn78}, \cite{Koorn72}, \cite{Koorn85}, \cite{Koorn88}, \cite{KuijlaarsMartinez}, +\cite{Kuijlaars+}, \cite{LabelleYehI}, \cite{LabelleYehII}, \cite{Laine}, \cite{Lesky95II}, +\cite{Lesky96}, \cite{Li96}, \cite{Li97}, \cite{LopezTemme2004}, \cite{Luke}, \cite{Martinez}, +\cite{Mathai}, \cite{Meijer}, \cite{Miller89}, \cite{MoakSaffVarga}, \cite{Nikiforov+}, +\cite{NikiforovUvarov}, \cite{Olver}, \cite{Prasad}, \cite{Rahman76I}, \cite{Rahman76II}, +\cite{Rahman77}, \cite{Rahman81I}, \cite{RahmanShah}, \cite{Rainville}, \cite{Rusev}, +\cite{Shi}, \cite{Srivastava69II}, \cite{Srivastava71}, \cite{Srivastava82}, +\cite{SrivastavaSinghal}, \cite{Stanton80I}, \cite{Stanton90}, \cite{Szego75}, \cite{Temme}, +\cite{Vertesi}, \cite{Wimp87}, \cite{WongZhang94}, \cite{WongZhang2006}, \cite{Zarzo+}, +\cite{Zayed}. + +\section*{Special cases} + +\subsection{Gegenbauer / Ultraspherical}\index{Gegenbauer polynomials} +\index{Ultraspherical polynomials} + +\par + +\subsection*{Hypergeometric representation} +The Gegenbauer (or ultraspherical) polynomials are Jacobi polynomials with +$\alpha=\beta=\lambda-\frac{1}{2}$ and another normalization: +\begin{eqnarray} +\label{DefGegenbauer} +C_n^{(\lambda)}(x)&=&\frac{(2\lambda)_n}{(\lambda+\frac{1}{2})_n} +P_n^{(\lambda-\frac{1}{2},\lambda-\frac{1}{2})}(x)\nonumber\\ +&=&\frac{(2\lambda)_n}{n!}\,\hyp{2}{1}{-n,n+2\lambda} +{\lambda+\frac{1}{2}}{\frac{1-x}{2}},\quad\lambda\neq 0. +\end{eqnarray} + +\subsection*{Orthogonality relation} +\begin{eqnarray} +\label{OrtGegenbauer} +& &\int_{-1}^1(1-x^2)^{\lambda-\frac{1}{2}}C_m^{(\lambda)}(x)C_n^{(\lambda)}(x)\,dx\nonumber\\ +& &{}=\frac{\pi\Gamma(n+2\lambda)2^{1-2\lambda}}{\left\{\Gamma(\lambda)\right\}^2(n+\lambda)n!}\,\delta_{mn}, +\quad\lambda>-\frac{1}{2}\quad\lambda\neq 0. +\end{eqnarray} + +\subsection*{Recurrence relation} +\begin{equation} +\label{RecGegenbauer} +2(n+\lambda)xC_n^{(\lambda)}(x)=(n+1)C_{n+1}^{(\lambda)}(x)+(n+2\lambda-1)C_{n-1}^{(\lambda)}(x). +\end{equation} + +\subsection*{Normalized recurrence relation} +\begin{equation} +\label{NormRecGegenbauer} +xp_n(x)=p_{n+1}(x)+\frac{n(n+2\lambda-1)}{4(n+\lambda-1)(n+\lambda)}p_{n-1}(x), +\end{equation} +where +$$C_n^{(\lambda)}(x)=\frac{2^n(\lambda)_n}{n!}p_n(x).$$ + +\subsection*{Differential equation} +\begin{equation} +\label{dvGegenbauer} +(1-x^2)y''(x)-(2\lambda+1)xy'(x)+n(n+2\lambda)y(x)=0,\quad y(x)=C_n^{(\lambda)}(x). +\end{equation} + +\subsection*{Forward shift operator} +\begin{equation} +\label{shift1Gegenbauer} +\frac{d}{dx}C_n^{(\lambda)}(x)=2\lambda C_{n-1}^{(\lambda+1)}(x). +\end{equation} + +\subsection*{Backward shift operator} +\begin{equation} +\label{shift2GegenbauerI} +(1-x^2)\frac{d}{dx}C_n^{(\lambda)}(x)+(1-2\lambda)xC_n^{(\lambda)}(x)= +-\frac{(n+1)(2\lambda+n-1)}{2(\lambda-1)} C_{n+1}^{(\lambda-1)}(x) +\end{equation} +or equivalently +\begin{eqnarray} +\label{shift2GegenbauerII} +& &\frac{d}{dx}\left[(1-x^2)^{\lambda-\textstyle\frac{1}{2}}C_n^{(\lambda)}(x)\right]\nonumber\\ +& &{}=-\frac{(n+1)(2\lambda+n-1)}{2(\lambda-1)}(1-x^2)^{\lambda-\textstyle\frac{3}{2}}C_{n+1}^{(\lambda-1)}(x). +\end{eqnarray} + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodGegenbauer} +(1-x^2)^{\lambda-\frac{1}{2}}C_n^{(\lambda)}(x)= +\frac{(2\lambda)_n(-1)^n}{(\lambda+\frac{1}{2})_n2^nn!}\left(\frac{d}{dx}\right)^n +\left[(1-x^2)^{\lambda+n-\frac{1}{2}}\right]. +\end{equation} + +\subsection*{Generating functions} +\begin{equation} +\label{GenGegenbauer1} +(1-2xt+t^2)^{-\lambda}=\sum_{n=0}^{\infty}C_n^{(\lambda)}(x)t^n. +\end{equation} + +\begin{equation} +\label{GenGegenbauer2} +R^{-1}\left(\frac{1+R-xt}{2}\right)^{\frac{1}{2}-\lambda}=\sum_{n=0}^{\infty} +\frac{(\lambda+\frac{1}{2})_n}{(2\lambda)_n}C_n^{(\lambda)}(x)t^n, +\quad R=\sqrt{1-2xt+t^2}. +\end{equation} + +\begin{equation} +\label{GenGegenbauer3} +\hyp{0}{1}{-}{\lambda+\frac{1}{2}}{\frac{(x-1)t}{2}}\, +\hyp{0}{1}{-}{\lambda+\frac{1}{2}}{\frac{(x+1)t}{2}} +=\sum_{n=0}^{\infty}\frac{C_n^{(\lambda)}(x)} +{(2\lambda)_n(\lambda+\frac{1}{2})_n}t^n. +\end{equation} + +\begin{equation} +\label{GenGegenbauer4} +\e^{xt}\,\hyp{0}{1}{-}{\lambda+\frac{1}{2}}{\frac{(x^2-1)t^2}{4}}= +\sum_{n=0}^{\infty}\frac{C_n^{(\lambda)}(x)}{(2\lambda)_n}t^n. +\end{equation} + +\begin{eqnarray} +\label{GenGegenbauer5} +& &\hyp{2}{1}{\gamma,2\lambda-\gamma}{\lambda+\frac{1}{2}}{\frac{1-R-t}{2}}\, +\hyp{2}{1}{\gamma,2\lambda-\gamma}{\lambda+\frac{1}{2}}{\frac{1-R+t}{2}}\nonumber\\ +& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n(2\lambda-\gamma)_n} +{(2\lambda)_n(\lambda+\frac{1}{2})_n}C_n^{(\lambda)}(x)t^n, +\quad R=\sqrt{1-2xt+t^2},\quad\textrm{$\gamma$ arbitrary}. +\end{eqnarray} + +\begin{eqnarray} +\label{GenGegenbauer6} +& &(1-xt)^{-\gamma}\,\hyp{2}{1}{\frac{1}{2}\gamma,\frac{1}{2}\gamma+\frac{1}{2}} +{\lambda+\frac{1}{2}}{\frac{(x^2-1)t^2}{(1-xt)^2}}\nonumber\\ +& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n}{(2\lambda)_n}C_n^{(\lambda)}(x)t^n, +\quad\textrm{$\gamma$ arbitrary}. +\end{eqnarray} + +\subsection*{Limit relation} + +\subsubsection*{Gegenbauer / Ultraspherical $\rightarrow$ Hermite} +The Hermite polynomials given by (\ref{DefHermite}) follow from the Gegenbauer (or +ultraspherical) polynomials by taking $\lambda=\alpha+\frac{1}{2}$ and letting +$\alpha\rightarrow\infty$ in the following way: +\begin{equation} +\lim_{\alpha\rightarrow\infty} +\alpha^{-\frac{1}{2}n}C_n^{(\alpha+\frac{1}{2})}(\alpha^{-\frac{1}{2}}x)=\frac{H_n(x)}{n!}. +\end{equation} + +\subsection*{Remarks} +The case $\lambda=0$ needs another normalization. In that case we have the +Chebyshev polynomials of the first kind described in the next subsection. + +\noindent +The Gegenbauer (or ultraspherical) polynomials are related to the Jacobi polynomials +given by (\ref{DefJacobi}) by the quadratic transformations: +$$C_{2n}^{(\lambda)}(x)=\frac{(\lambda)_n}{(\frac{1}{2})_n} +P_n^{(\lambda-\frac{1}{2},-\frac{1}{2})}(2x^2-1)$$ +and +$$C_{2n+1}^{(\lambda)}(x)=\frac{(\lambda)_{n+1}}{(\frac{1}{2})_{n+1}} +xP_n^{(\lambda-\frac{1}{2},\frac{1}{2})}(2x^2-1).$$ +% RS: add begin\label{sec9.8.1} +% +\paragraph{\large\bf KLSadd: Notation}Here the Gegenbauer polynomial is denoted by $C_n^\la$ instead of $C_n^{(\la)}$. +% +\paragraph{\large\bf KLSadd: Orthogonality relation}Write the \RHS\ of (9.8.20) as $h_n\,\de_{m,n}$. Then +\begin{equation} +\frac{h_n}{h_0}= +\frac\la{\la+n}\,\frac{(2\la)_n}{n!}\,,\quad +h_0=\frac{\pi^\half\,\Ga(\la+\thalf)}{\Ga(\la+1)},\quad +\frac{h_n}{h_0\,(C_n^\la(1))^2}= +\frac\la{\la+n}\,\frac{n!}{(2\la)_n}\,. +\label{61} +\end{equation} +% +\paragraph{\large\bf KLSadd: Hypergeometric representation}Beside (9.8.19) we have also +\begin{equation} +C_n^\lambda(x)=\sum_{\ell=0}^{\lfloor n/2\rfloor}\frac{(-1)^{\ell}(\lambda)_{n-\ell}} +{\ell!\;(n-2\ell)!}\,(2x)^{n-2\ell} +=(2x)^{n}\,\frac{(\lambda)_{n}}{n!}\, +\hyp21{-\thalf n,-\thalf n+\thalf}{1-\la-n}{\frac1{x^2}}. +\label{57} +\end{equation} +See \mycite{DLMF}{(18.5.10)}. +% +\paragraph{\large\bf KLSadd: Special value}\begin{equation} +C_n^{\la}(1)=\frac{(2\la)_n}{n!}\,. +\label{49} +\end{equation} +Use (9.8.19) or see \mycite{DLMF}{Table 18.6.1}. +% +\paragraph{\large\bf KLSadd: Expression in terms of Jacobi}% +\begin{equation} +\frac{C_n^\la(x)}{C_n^\la(1)}= +\frac{P_n^{(\la-\half,\la-\half)}(x)}{P_n^{(\la-\half,\la-\half)}(1)}\,,\qquad +C_n^\la(x)=\frac{(2\la)_n}{(\la+\thalf)_n}\,P_n^{(\la-\half,\la-\half)}(x). +\label{65} +\end{equation} +% +\paragraph{\large\bf KLSadd: Re: (9.8.21)}By iteration of recurrence relation (9.8.21): +\begin{multline} +x^2 C_n^\la(x)= +\frac{(n+1)(n+2)}{4(n+\la)(n+\la+1)}\,C_{n+2}^\la(x)+ +\frac{n^2+2n\la+\la-1}{2(n+\la-1)(n+\la+1)}\,C_n^\la(x)\\ ++\frac{(n+2\la-1)(n+2\la-2)}{4(n+\la)(n+\la-1)}\,C_{n-2}^\la(x). +\label{6} +\end{multline} +% +\paragraph{\large\bf KLSadd: Bilateral generating functions}\begin{multline} +\sum_{n=0}^\iy\frac{n!}{(2\la)_n}\,r^n\,C_n^\la(x)\,C_n^\la(y) +=\frac1{(1-2rxy+r^2)^\la}\,\hyp21{\thalf\la,\thalf(\la+1)}{\la+\thalf} +{\frac{4r^2(1-x^2)(1-y^2)}{(1-2rxy+r^2)^2}}\\ +(r\in(-1,1),\;x,y\in[-1,1]). +\label{66} +\end{multline} +For the proof put $\be:=\al$ in \eqref{58}, then use \eqref{63} and \eqref{65}. +The Poisson kernel for Gegenbauer polynomials can be derived in a similar way +from \eqref{59}, or alternatively by applying the operator +$r^{-\la+1}\frac d{dr}\circ r^\la$ to both sides of \eqref{66}: +\begin{multline} +\sum_{n=0}^\iy\frac{\la+n}\la\,\frac{n!}{(2\la)_n}\,r^n\,C_n^\la(x)\,C_n^\la(y) +=\frac{1-r^2}{(1-2rxy+r^2)^{\la+1}}\\ +\times\hyp21{\thalf(\la+1),\thalf(\la+2)}{\la+\thalf} +{\frac{4r^2(1-x^2)(1-y^2)}{(1-2rxy+r^2)^2}}\qquad +(r\in(-1,1),\;x,y\in[-1,1]). +\label{67} +\end{multline} +Formula \eqref{67} was obtained by Gasper \& Rahman \myciteKLS{234}{(4.4)} +as a limit case of their formula for the Poisson kernel for continuous +$q$-ultraspherical polynomials. +% +\paragraph{\large\bf KLSadd: Trigonometric expansions}By \mycite{DLMF}{(18.5.11), (15.8.1)}: +\begin{align} +C_n^{\la}(\cos\tha) +&=\sum_{k=0}^n\frac{(\la)_k(\la)_{n-k}}{k!\,(n-k)!}\,e^{i(n-2k)\tha} +=e^{in\tha}\frac{(\la)_n}{n!}\, +\hyp21{-n,\la}{1-\la-n}{e^{-2i\tha}}\label{103}\\ +&=\frac{(\la)_n}{2^\la n!}\, +e^{-\half i\la\pi}e^{i(n+\la)\tha}\,(\sin\tha)^{-\la}\, +\hyp21{\la,1-\la}{1-\la-n}{\frac{i e^{-i\tha}}{2\sin\tha}}\label{104}\\ +&=\frac{(\la)_n}{n!}\,\sum_{k=0}^\iy\frac{(\la)_k(1-\la)_k}{(1-\la-n)_k k!}\, +\frac{\cos((n-k+\la)\tha+\thalf(k-\la)\pi)}{(2\sin\tha)^{k+\la}}\,.\label{105} +\end{align} +In \eqref{104} and \eqref{105} we require that +$\tfrac16\pi<\tha<\tfrac56\pi$. Then the convergence is absolute for $\la>\thalf$ +and conditional for $0<\la\le\thalf$. + +By \mycite{DLMF}{(14.13.1), (14.3.21), (15.8.1)]}: +\begin{align} +C_n^\la(\cos\tha)&=\frac{2\Ga(\la+\thalf)}{\pi^\half\Ga(\la+1)}\, +\frac{(2\la)_n}{(\la+1)_n}\,(\sin\tha)^{1-2\la}\, +\sum_{k=0}^\iy\frac{(1-\la)_k(n+1)_k}{(n+\la+1)_k k!}\, +\sin\big((2k+n+1)\tha\big) +\label{7}\\ +&=\frac{2\Ga(\la+\thalf)}{\pi^\half\Ga(\la+1)}\, +\frac{(2\la)_n}{(\la+1)_n}\,(\sin\tha)^{1-2\la}\, +\Im\!\!\left(e^{i(n+1)\tha}\,\hyp21{1-\la,n+1}{n+\la+1}{e^{2i\tha}}\right)\nonumber\\ +&=\frac{2^\la\Ga(\la+\thalf)}{\pi^\half\Ga(\la+1)}\, +\frac{(2\la)_n}{(\la+1)_n}\,(\sin\tha)^{-\la}\, +\Re\!\!\left(e^{-\thalf i\la\pi}e^{i(n+\la)\tha}\, +\hyp21{\la,1-\la}{1+\la+n}{\frac{e^{i\tha}}{2i\sin\tha}}\right)\nonumber\\ +&=\frac{2^{2\la}\Ga(\la+\thalf)}{\pi^\half\Ga(\la+1)}\,\frac{(2\la)_n}{(\la+1)_n}\, +\sum_{k=0}^\iy\frac{(\la)_k(1-\la)_k}{(1+\la+n)_k k!}\, +\frac{\cos((n+k+\la)\tha-\thalf(k+\la)\pi)}{(2\sin\tha)^{k+\la}}\,. +\label{106} +\end{align} +We require that $0<\tha<\pi$ in \eqref{7} and $\tfrac16\pi<\tha<\tfrac56\pi$ in +\eqref{106} The convergence is absolute for $\la>\thalf$ and conditional for +$0<\la\le\thalf$. +For $\la\in\Zpos$ the above series terminate after the term with +$k=\la-1$. +Formulas \eqref{7} and \eqref{106} are also given in +\mycite{Sz}{(4.9.22), (4.9.25)}. +% +\paragraph{\large\bf KLSadd: Fourier transform}\begin{equation} +\frac{\Ga(\la+1)}{\Ga(\la+\thalf)\,\Ga(\thalf)}\, +\int_{-1}^1 \frac{C_n^\la(y)}{C_n^\la(1)}\,(1-y^2)^{\la-\half}\, +e^{ixy}\,dy +=i^n\,2^\la\,\Ga(\la+1)\,x^{-\la}\,J_{\la+n}(x). +\label{8} +\end{equation} +See \mycite{DLMF}{(18.17.17) and (18.17.18)}. +% +\paragraph{\large\bf KLSadd: Laplace transforms}\begin{equation} +\frac2{n!\,\Ga(\la)}\, +\int_0^\iy H_n(tx)\,t^{n+2\la-1}\,e^{-t^2}\,dt=C_n^\la(x). +\label{56} +\end{equation} +See Nielsen \cite[p.48, (4) with p.47, (1) and p.28, (10)]{K4} (1918) +or Feldheim \cite[(28)]{K3} (1942). +\begin{equation} +\frac2{\Ga(\la+\thalf)}\,\int_0^1 \frac{C_n^\la(t)}{C_n^\la(1)}\, +(1-t^2)^{\la-\half}\,t^{-1}\,(x/t)^{n+2\la+1}\,e^{-x^2/t^2}\,dt +=2^{-n}\,H_n(x)\,e^{-x^2}\quad(\la>-\thalf). +\label{46} +\end{equation} +Use Askey \& Fitch \cite[(3.29)]{K2} for $\al=\pm\thalf$ together with +\eqref{48}, \eqref{51}, \eqref{52}, \eqref{54} and \eqref{55}. +\paragraph{\large\bf KLSadd: Addition formula}\begin{multline} +R_n^{(\al,\al)}\big(xy+(1-x^2)^\half(1-y^2)^\half t\big) +=\sum_{k=0}^n \frac{(-1)^k(-n)_k\,(n+2\al+1)_k}{2^{2k}((\al+1)_k)^2}\\ +\times(1-x^2)^{k/2} R_{n-k}^{(\al+k,\al+k)}(x)\,(1-y^2)^{k/2} R_{n-k}^{(\al+k,\al+k)}(y)\, +\om_k^{(\al-\half,\al-\half)}\,R_k^{(\al-\half,\al-\half)}(t), +\label{108} +\end{multline} +where +\[ +R_n^{(\al,\be)}(x):=P_n^{(\al,\be)}(x)/P_n^{(\al,\be)}(1),\quad +\om_n^{(\al,\be)}:=\frac{\int_{-1}^1 (1-x)^\al(1+x)^\be\,dx} +{\int_{-1}^1 (R_n^{(\al,\be)}(x))^2\,(1-x)^\al(1+x)^\be\,dx}\,. +\] +% +% RS: add end +\subsection*{References} +\cite{Abram}, \cite{Ahmed+86}, \cite{AndrewsAskeyRoy}, \cite{Area+I}, \cite{Askey67}, \cite{Askey74}, \cite{Askey75}, +\cite{Askey89I}, \cite{AskeyFitch}, \cite{AskeyKoornRahman}, \cite{Berg}, \cite{BilodeauI}, +\cite{BojanovNikolov}, \cite{Brafman51}, \cite{Brafman57I}, \cite{Brown}, +\cite{BustozIsmail82}, \cite{BustozIsmail83}, \cite{BustozIsmail89}, \cite{BustozSavage79}, +\cite{BustozSavage80}, \cite{Carlitz61II}, \cite{Chihara78}, \cite{Common}, \cite{Danese}, +\cite{Dette94}, \cite{DijksmaKoorn}, \cite{DilcherStolarsky}, \cite{Dimitrov96}, +\cite{Dimitrov2003}, \cite{Doha2002II}, \cite{Driver}, \cite{ElbertLaforgia86I}, +\cite{ElbertLaforgia86II}, \cite{ElbertLaforgia90}, \cite{Erdelyi+}, \cite{Exton96}, +\cite{Gasper69}, \cite{Gasper72II}, \cite{Gasper85}, \cite{Grad}, \cite{Ismail74}, +\cite{Ismail2005II}, \cite{Koekoek2000}, \cite{Koorn88}, \cite{Laforgia}, \cite{LewanowiczI}, +\cite{Lorch}, \cite{Mathai}, \cite{Nagel}, \cite{Nikiforov+}, \cite{NikiforovUvarov}, +\cite{RahmanShah}, \cite{Rainville}, \cite{Reimer}, \cite{Sartoretto}, \cite{Srivastava71}, +\cite{Szego75}, \cite{Temme}, \cite{Viswanathan}, \cite{Zayed}. + +\subsection{Chebyshev}\index{Chebyshev polynomials} + +\par + +\subsection*{Hypergeometric representation} +The Chebyshev polynomials of the first kind can be obtained from the Jacobi +polynomials by taking $\alpha=\beta=-\frac{1}{2}$: +\begin{equation} +\label{DefChebyshevI} +T_n(x)=\frac{P_n^{(-\frac{1}{2},-\frac{1}{2})}(x)}{P_n^{(-\frac{1}{2},-\frac{1}{2})}(1)} +=\hyp{2}{1}{-n,n}{\frac{1}{2}}{\frac{1-x}{2}} +\end{equation} +and the Chebyshev polynomials of the second kind can be obtained from the +Jacobi polynomials by taking $\alpha=\beta=\frac{1}{2}$: +\begin{equation} +\label{DefChebyshevII} +U_n(x)=(n+1)\frac{P_n^{(\frac{1}{2},\frac{1}{2})}(x)}{P_n^{(\frac{1}{2},\frac{1}{2})}(1)} +=(n+1)\,\hyp{2}{1}{-n,n+2}{\frac{3}{2}}{\frac{1-x}{2}}. +\end{equation} + +\subsection*{Orthogonality relation} +\begin{equation} +\label{OrtChebyshevI} +\int_{-1}^1(1-x^2)^{-\frac{1}{2}}T_m(x)T_n(x)\,dx= +\left\{\begin{array}{ll} +\displaystyle\frac{\cpi}{2}\,\delta_{mn}, & n\neq 0\\[5mm] +\cpi\,\delta_{mn}, & n=0. +\end{array}\right. +\end{equation} + +\begin{equation} +\label{OrtChebyshevII} +\int_{-1}^1(1-x^2)^{\frac{1}{2}}U_m(x)U_n(x)\,dx=\frac{\cpi}{2}\,\delta_{mn}. +\end{equation} + +\subsection*{Recurrence relations} +\begin{equation} +\label{RecChebyshevI} +2xT_n(x)=T_{n+1}(x)+T_{n-1}(x),\quad T_{0}(x)=1\quad\textrm{and}\quad T_1(x)=x. +\end{equation} + +\begin{equation} +\label{RecChebyshevII} +2xU_n(x)=U_{n+1}(x)+U_{n-1}(x),\quad U_{0}(x)=1\quad\textrm{and}\quad U_1(x)=2x. +\end{equation} + +\subsection*{Normalized recurrence relations} +\begin{equation} +\label{NormRecChebyshevI} +xp_n(x)=p_{n+1}(x)+\frac{1}{4}p_{n-1}(x), +\end{equation} +where +$$T_1(x)=p_1(x)=x\quad\textrm{and}\quad T_n(x)=2^np_n(x),\quad n\neq 1.$$ + +\begin{equation} +\label{NormRecChebyshevII} +xp_n(x)=p_{n+1}(x)+\frac{1}{4}p_{n-1}(x), +\end{equation} +where +$$U_n(x)=2^np_n(x).$$ + +\subsection*{Differential equations} +\begin{equation} +\label{dvChebyshevI} +(1-x^2)y''(x)-xy'(x)+n^2y(x)=0,\quad y(x)=T_n(x). +\end{equation} + +\begin{equation} +\label{dvChebyshevII} +(1-x^2)y''(x)-3xy'(x)+n(n+2)y(x)=0,\quad y(x)=U_n(x). +\end{equation} + +\subsection*{Forward shift operator} +\begin{equation} +\label{shift1Chebyshev} +\frac{d}{dx}T_n(x)=nU_{n-1}(x). +\end{equation} + +\subsection*{Backward shift operator} +\begin{equation} +\label{shift2ChebyshevI} +(1-x^2)\frac{d}{dx}U_n(x)-xU_n(x)=-(n+1)T_{n+1}(x) +\end{equation} +or equivalently +\begin{equation} +\label{shift2ChebyshevII} +\frac{d}{dx}\left[\left(1-x^2\right)^{\frac{1}{2}}U_n(x)\right] +=-(n+1)\left(1-x^2\right)^{-\frac{1}{2}}T_{n+1}(x). +\end{equation} + +\subsection*{Rodrigues-type formulas} +\begin{equation} +\label{RodChebyshevI} +(1-x^2)^{-\frac{1}{2}}T_n(x)=\frac{(-1)^n}{(\frac{1}{2})_n2^n} +\left(\frac{d}{dx}\right)^n\left[(1-x^2)^{n-\frac{1}{2}}\right]. +\end{equation} + +\begin{equation} +\label{RodChebyshevII} +(1-x^2)^{\frac{1}{2}}U_n(x)=\frac{(n+1)(-1)^n}{(\frac{3}{2})_n2^n} +\left(\frac{d}{dx}\right)^n\left[(1-x^2)^{n+\frac{1}{2}}\right]. +\end{equation} + +\subsection*{Generating functions} +\begin{equation} +\label{GenChebyshevI1} +\frac{1-xt}{1-2xt+t^2}=\sum_{n=0}^{\infty}T_n(x)t^n. +\end{equation} + +\begin{equation} +\label{GenChebyshevI2} +R^{-1}\sqrt{\frac{1}{2}(1+R-xt)}=\sum_{n=0}^{\infty} +\frac{\left(\frac{1}{2}\right)_n}{n!}T_n(x)t^n,\quad R=\sqrt{1-2xt+t^2}. +\end{equation} + +\begin{equation} +\label{GenChebyshevI3} +\hyp{0}{1}{-}{\frac{1}{2}}{\frac{(x-1)t}{2}}\, +\hyp{0}{1}{-}{\frac{1}{2}}{\frac{(x+1)t}{2}}= +\sum_{n=0}^{\infty}\frac{T_n(x)}{\left(\frac{1}{2}\right)_nn!}t^n. +\end{equation} + +\begin{equation} +\label{GenChebyshevI4} +\e^{xt}\,\hyp{0}{1}{-}{\frac{1}{2}}{\frac{(x^2-1)t^2}{4}}= +\sum_{n=0}^{\infty}\frac{T_n(x)}{n!}t^n. +\end{equation} + +\begin{eqnarray} +\label{GenChebyshevI5} +& &\hyp{2}{1}{\gamma,-\gamma}{\frac{1}{2}}{\frac{1-R-t}{2}}\, +\hyp{2}{1}{\gamma,-\gamma}{\frac{1}{2}}{\frac{1-R+t}{2}}\nonumber\\ +& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n(-\gamma)_n}{\left(\frac{1}{2}\right)_nn!}T_n(x)t^n, +\quad R=\sqrt{1-2xt+t^2},\quad\textrm{$\gamma$ arbitrary}. +\end{eqnarray} + +\begin{eqnarray} +\label{GenChebyshevI6} +& &(1-xt)^{-\gamma}\,\hyp{2}{1}{\frac{1}{2}\gamma,\frac{1}{2}\gamma+\frac{1}{2}}{\frac{1}{2}} +{\frac{(x^2-1)t^2}{(1-xt)^2}}\nonumber\\ +& &=\sum_{n=0}^{\infty}\frac{(\gamma)_n}{n!}T_n(x)t^n,\quad\textrm{$\gamma$ arbitrary}. +\end{eqnarray} + +\begin{equation} +\label{GenChebyshevII1} +\frac{1}{1-2xt+t^2}=\sum_{n=0}^{\infty}U_n(x)t^n. +\end{equation} + +\begin{equation} +\label{GenChebyshevII2} +\frac{1}{R\sqrt{\frac{1}{2}(1+R-xt)}}=\sum_{n=0}^{\infty} +\frac{\left(\frac{3}{2}\right)_n}{(n+1)!}U_n(x)t^n,\quad R=\sqrt{1-2xt+t^2}. +\end{equation} + +\begin{equation} +\label{GenChebyshevII3} +\hyp{0}{1}{-}{\frac{3}{2}}{\frac{(x-1)t}{2}}\, +\hyp{0}{1}{-}{\frac{3}{2}}{\frac{(x+1)t}{2}}= +\sum_{n=0}^{\infty}\frac{U_n(x)}{\left(\frac{3}{2}\right)_n(n+1)!}t^n. +\end{equation} + +\begin{equation} +\label{GenChebyshevII4} +\e^{xt}\,\hyp{0}{1}{-}{\frac{3}{2}}{\frac{(x^2-1)t^2}{4}}= +\sum_{n=0}^{\infty}\frac{U_n(x)}{(n+1)!}t^n. +\end{equation} + +\begin{eqnarray} +\label{GenChebyshevII5} +& &\hyp{2}{1}{\gamma,2-\gamma}{\frac{3}{2}}{\frac{1-R-t}{2}}\, +\hyp{2}{1}{\gamma,2-\gamma}{\frac{3}{2}}{\frac{1-R+t}{2}}\nonumber\\ +& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n(2-\gamma)_n}{\left(\frac{3}{2}\right)_n(n+1)!}U_n(x)t^n, +\quad R=\sqrt{1-2xt+t^2},\quad\textrm{$\gamma$ arbitrary}. +\end{eqnarray} + +\begin{eqnarray} +\label{GenChebyshevII6} +& &(1-xt)^{-\gamma}\,\hyp{2}{1}{\frac{1}{2}\gamma,\frac{1}{2}\gamma+\frac{1}{2}}{\frac{3}{2}} +{\frac{(x^2-1)t^2}{(1-xt)^2}}\nonumber\\ +& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n}{(n+1)!}U_n(x)t^n,\quad\textrm{$\gamma$ arbitrary}. +\end{eqnarray} + +\subsection*{Remarks} +The Chebyshev polynomials can also be written as: +$$T_n(x)=\cos(n\theta),\quad x=\cos\theta$$ +and +$$U_n(x)=\frac{\sin (n+1)\theta}{\sin\theta},\quad x=\cos\theta.$$ +Further we have +$$U_n(x)=C_n^{(1)}(x)$$ +where $C_n^{(\lambda)}(x)$ denotes the Gegenbauer (or ultraspherical) +polynomial given by (\ref{DefGegenbauer}) in the preceding subsection. +% RS: add begin\label{sec9.8.2} +In addition to the Chebyshev polynomials $T_n$ of the first kind (9.8.35) +and $U_n$ of the second kind (9.8.36), +\begin{align} +T_n(x)&:=\frac{P_n^{(-\half,-\half)}(x)}{P_n^{(-\half,-\half)}(1)} +=\cos(n\tha),\quad x=\cos\tha,\\ +U_n(x)&:=(n+1)\,\frac{P_n^{(\half,\half)}(x)}{P_n^{(\half,\half)}(1)} +=\frac{\sin((n+1)\tha)}{\sin\tha}\,,\quad x=\cos\tha, +\end{align} +we have Chebyshev polynomials $V_n$ {\em of the third kind} +and $W_n$ {\em of the fourth kind}, +\begin{align} +V_n(x)&:=\frac{P_n^{(-\half,\half)}(x)}{P_n^{(-\half,\half)}(1)} +=\frac{\cos((n+\thalf)\tha)}{\cos(\thalf\tha)}\,,\quad x=\cos\tha,\\ +W_n(x)&:=(2n+1)\,\frac{P_n^{(\half,-\half)}(x)}{P_n^{(\half,-\half)}(1)} +=\frac{\sin((n+\thalf)\tha)}{\sin(\thalf\tha)}\,,\quad x=\cos\tha, +\end{align} +see \cite[Section 1.2.3]{K20}. Then there is the symmetry +\begin{equation} +V_n(-x)=(-1)^n W_n(x). +\label{140} +\end{equation} + +The names of Chebyshev polynomials of the third and fourth kind +and the notation $V_n(x)$ are due to Gautschi \cite{K21}. +The notation $W_n(x)$ was first used by Mason \cite{K22}. +Names and notations for Chebyshev polynomials of the third and fourth +kind are interchanged in \mycite{AAR}{Remark 2.5.3} and +\mycite{DLMF}{Table 18.3.1}. +% +% RS: add end +\subsection*{References} +\cite{Abram}, \cite{AskeyFitch}, \cite{AskeyGasperHarris}, \cite{AskeyIsmail76}, +\cite{Bavinck95}, \cite{Chihara78}, \cite{Danese}, \cite{DilcherStolarsky}, +\cite{Erdelyi+}, \cite{Grad}, \cite{HartmannStephan}, \cite{Ismail2005II}, \cite{Koekoek2000}, +\cite{Luke}, \cite{Mathai}, \cite{Nikiforov+}, \cite{NikiforovUvarov}, \cite{Rainville}, +\cite{Rayes+}, \cite{Rivlin}, \cite{SainteViennot}, \cite{Szego75}, \cite{Temme}, +\cite{Wilson70I}, \cite{Zayed}, \cite{Zhang}, \cite{ZhangWang}. + +\subsection{Legendre / Spherical}\index{Legendre polynomials} +\index{Spherical polynomials} + +\par + +\subsection*{Hypergeometric representation} +The Legendre (or spherical) polynomials are Jacobi polynomials with $\alpha=\beta=0$: +\begin{equation} +\label{DefLegendre} +P_n(x)=P_n^{(0,0)}(x)=\hyp{2}{1}{-n,n+1}{1}{\frac{1-x}{2}}. +\end{equation} + +\subsection*{Orthogonality relation} +\begin{equation} +\label{OrtLegendre} +\int_{-1}^1P_m(x)P_n(x)\,dx=\frac{2}{2n+1}\,\delta_{mn}. +\end{equation} + +\subsection*{Recurrence relation} +\begin{equation} +\label{RecLegendre} +(2n+1)xP_n(x)=(n+1)P_{n+1}(x)+nP_{n-1}(x). +\end{equation} + +\subsection*{Normalized recurrence relation} +\begin{equation} +\label{NormRecLegendre} +xp_n(x)=p_{n+1}(x)+\frac{n^2}{(2n-1)(2n+1)}p_{n-1}(x), +\end{equation} +where +$$P_n(x)=\binom{2n}{n}\frac{1}{2^n}p_n(x).$$ + +\subsection*{Differential equation} +\begin{equation} +\label{dvLegendre} +(1-x^2)y''(x)-2xy'(x)+n(n+1)y(x)=0,\quad y(x)=P_n(x). +\end{equation} + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodLegendre} +P_n(x)=\frac{(-1)^n}{2^nn!}\left(\frac{d}{dx}\right)^n\left[(1-x^2)^n\right]. +\end{equation} + +\subsection*{Generating functions} +\begin{equation} +\label{GenLegendre1} +\frac{1}{\sqrt{1-2xt+t^2}}=\sum_{n=0}^{\infty}P_n(x)t^n. +\end{equation} + +\begin{equation} +\label{GenLegendre2} +\hyp{0}{1}{-}{1}{\frac{(x-1)t}{2}}\,\hyp{0}{1}{-}{1}{\frac{(x+1)t}{2}}= +\sum_{n=0}^{\infty}\frac{P_n(x)}{(n!)^2}t^n. +\end{equation} + +\begin{equation} +\label{GenLegendre3} +\e^{xt}\,\hyp{0}{1}{-}{1}{\frac{(x^2-1)t^2}{4}}= +\sum_{n=0}^{\infty}\frac{P_n(x)}{n!}t^n. +\end{equation} + +\begin{eqnarray} +\label{GenLegendre4} +& &\hyp{2}{1}{\gamma,1-\gamma}{1}{\frac{1-R-t}{2}}\, +\hyp{2}{1}{\gamma,1-\gamma}{1}{\frac{1-R+t}{2}}\nonumber\\ +& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n(1-\gamma)_n}{(n!)^2}P_n(x)t^n, +\quad R=\sqrt{1-2xt+t^2},\quad\textrm{$\gamma$ arbitrary}. +\end{eqnarray} + +\begin{eqnarray} +\label{GenLegendre5} +& &(1-xt)^{-\gamma}\,\hyp{2}{1}{\frac{1}{2}\gamma,\frac{1}{2}\gamma+\frac{1}{2}}{1} +{\frac{(x^2-1)t^2}{(1-xt)^2}}\nonumber\\ +& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n}{n!}P_n(x)t^n,\quad\textrm{$\gamma$ arbitrary}. +\end{eqnarray} + +\subsection*{References} +\cite{Abram}, \cite{Alladi}, \cite{AlSalam90}, \cite{Bhonsle}, \cite{Brafman51}, +\cite{Carlitz57II}, \cite{Chihara78}, \cite{Danese}, \cite{Dattoli2001}, +\cite{DilcherStolarsky}, \cite{ElbertLaforgia94}, \cite{Erdelyi+}, \cite{Grad}, +\cite{Mathai}, \cite{Nikiforov+}, \cite{NikiforovUvarov}, \cite{Olver}, \cite{Rainville}, +\cite{Szego75}, \cite{Temme}, \cite{Zayed}. + +\newpage + +\section{Pseudo Jacobi}\index{Pseudo Jacobi polynomials}\index{Jacobi polynomials!Pseudo} + +\par\setcounter{equation}{0} + +\subsection*{Hypergeometric representation} +\begin{eqnarray} +\label{DefPseudoJacobi} +P_n(x;\nu,N)&=&\frac{(-2i)^n(-N+i\nu)_n}{(n-2N-1)_n}\,\hyp{2}{1}{-n,n-2N-1}{-N+i\nu}{\frac{1-ix}{2}}\\ +&=&(x+i)^n\,\hyp{2}{1}{-n,N+1-n-i\nu}{2N+2-2n}{\frac{2}{1-ix}},\quad n=0,1,2,\ldots,N.\nonumber +\end{eqnarray} + +\subsection*{Orthogonality relation} +\begin{eqnarray} +\label{OrtPseudoJacobi} +& &\frac{1}{2\pi}\int_{-\infty}^{\infty}(1+x^2)^{-N-1}\e^{2\nu\arctan x}P_m(x;\nu,N)P_n(x;\nu,N)\,dx\nonumber\\ +& &{}=\frac{\Gamma(2N+1-2n)\Gamma(2N+2-2n)2^{2n-2N-1}n!}{\Gamma(2N+2-n)\left|\Gamma(N+1-n+i\nu)\right|^2}\,\delta_{mn}. +\end{eqnarray} + +\subsection*{Recurrence relation} +\begin{eqnarray} +\label{RecPseudoJacobi} +xP_n(x;\nu,N)&=&P_{n+1}(x;\nu,N)+\frac{(N+1)\nu}{(n-N-1)(n-N)}P_n(x;\nu,N)\nonumber\\ +& &{}\mathindent{}-\frac{n(n-2N-2)}{(2n-2N-3)(n-N-1)^2(2n-2N-1)}\nonumber\\ +& &{}\mathindent\mathindent\times(n-N-1-i\nu)(n-N-1+i\nu)P_{n-1}(x;\nu,N). +\end{eqnarray} + +\subsection*{Normalized recurrence relation} +\begin{eqnarray} +\label{NormRecPseudoJacobi} +xp_n(x)&=&p_{n+1}(x)+\frac{(N+1)\nu}{(n-N-1)(n-N)}p_n(x)\nonumber\\ +& &{}\mathindent{}-\frac{n(n-2N-2)(n-N-1-i\nu)(n-N-1+i\nu)}{(2n-2N-3)(n-N-1)^2(2n-2N-1)}p_{n-1}(x), +\end{eqnarray} +where +$$P_n(x;\nu,N)=p_n(x).$$ + +\subsection*{Differential equation} +\begin{equation} +\label{dvPseudoJacobi} +(1+x^2)y''(x)+2\left(\nu-Nx\right)y'(x)-n(n-2N-1)y(x)=0, +\end{equation} +where +$$y(x)=P_n(x;\nu,N).$$ + +\subsection*{Forward shift operator} +\begin{equation} +\label{shift1PseudoJacobi} +\frac{d}{dx}P_n(x;\nu,N)=nP_{n-1}(x;\nu,N-1). +\end{equation} + +\subsection*{Backward shift operator} +\begin{eqnarray} +\label{shift2PseudoJacobiI} +& &(1+x^2)\frac{d}{dx}P_n(x;\nu,N)+2\left[\nu-(N+1)x\right]P_n(x;\nu,N)\nonumber\\ +& &{}=(n-2N-2)P_{n+1}(x;\nu,N+1) +\end{eqnarray} +or equivalently +\begin{eqnarray} +\label{shift2PseudoJacobiII} +& &\frac{d}{dx}\left[(1+x^2)^{-N-1}\e^{2\nu\arctan x}P_n(x;\nu,N)\right]\nonumber\\ +& &{}=(n-2N-2)(1+x^2)^{-N-2}\e^{2\nu\arctan x}P_{n+1}(x;\nu,N+1). +\end{eqnarray} + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodPseudoJacobi} +P_n(x;\nu,N)=\frac{(1+x^2)^{N+1}\expe^{-2\nu\arctan x}}{(n-2N-1)_n} +\left(\frac{d}{dx}\right)^n\left[(1+x^2)^{n-N-1}\expe^{2\nu\arctan x}\right]. +\end{equation} + +\subsection*{Generating function} +\begin{eqnarray} +\label{GenPseudoJacobi} +& &\left[\hyp{0}{1}{-}{-N+i\nu}{(x+i)t}\,\hyp{0}{1}{-}{-N-i\nu}{(x-i)t}\right]_N\nonumber\\ +& &{}=\sum_{n=0}^N\frac{(n-2N-1)_n}{(-N+i\nu)_n(-N-i\nu)_nn!}P_n(x;\nu,N)t^n. +\end{eqnarray} + +\subsection*{Limit relation} + +\subsubsection*{Continuous Hahn $\rightarrow$ Pseudo Jacobi} +The pseudo Jacobi polynomials follow from the continuous Hahn polynomials given by +(\ref{DefContHahn}) by the substitutions $x\rightarrow xt$, $a=\frac{1}{2}(-N+i\nu-2t)$, +$b=\frac{1}{2}(-N-i\nu+2t)$, $c=\frac{1}{2}(-N-i\nu-2t)$ and $d=\frac{1}{2}(-N+i\nu+2t)$, +division by $t^n$ and the limit $t\rightarrow\infty$: +\begin{eqnarray*} +& &\lim_{t\rightarrow\infty}\frac{p_n(xt;\frac{1}{2}(-N+i\nu-2t),\frac{1}{2}(-N-i\nu+2t), +\frac{1}{2}(-N+i\nu-2t),\frac{1}{2}(-N-i\nu+2t))}{t^n}\\ +& &{}=\frac{(n-2N-1)_n}{n!}P_n(x;\nu,N). +\end{eqnarray*} + +\subsection*{Remarks} +Since we have for $k 0$ & +% $0 < c < 1$ } +\end{equation} + +\subsection*{Recurrence relation} +\begin{eqnarray} +\label{RecMeixner} +(c-1)xM_n(x;\beta,c)&=&c(n+\beta)M_{n+1}(x;\beta,c)\nonumber\\ +& &{}\mathindent{}-\left[n+(n+\beta)c\right]M_n(x;\beta,c)+nM_{n-1}(x;\beta,c). +\end{eqnarray} + +\subsection*{Normalized recurrence relation} +\begin{equation} +\label{NormRecMeixner} +xp_n(x)=p_{n+1}(x)+\frac{n+(n+\beta)c}{1-c}p_n(x)+ +\frac{n(n+\beta-1)c}{(1-c)^2}p_{n-1}(x), +\end{equation} +where +$$M_n(x;\beta,c)=\frac{1}{(\beta)_n}\left(\frac{c-1}{c}\right)^np_n(x).$$ + +\subsection*{Difference equation} +\begin{equation} +\label{dvMeixner} +n(c-1)y(x)=c(x+\beta)y(x+1)-\left[x+(x+\beta)c\right]y(x)+xy(x-1), +\end{equation} +where +$$y(x)=M_n(x;\beta,c).$$ + +\subsection*{Forward shift operator} +\begin{equation} +\label{shift1MeixnerI} +M_n(x+1;\beta,c)-M_n(x;\beta,c)= +\frac{n}{\beta}\left(\frac{c-1}{c}\right)M_{n-1}(x;\beta+1,c) +\end{equation} +or equivalently +\begin{equation} +\label{shift1MeixnerII} +\Delta M_n(x;\beta,c)=\frac{n}{\beta}\left(\frac{c-1}{c}\right)M_{n-1}(x;\beta+1,c). +\end{equation} + +\subsection*{Backward shift operator} +\begin{equation} +\label{shift2MeixnerI} +c(\beta+x-1)M_n(x;\beta,c)-xM_n(x-1;\beta,c)=c(\beta-1)M_{n+1}(x;\beta-1,c) +\end{equation} +or equivalently +\begin{equation} +\label{shift2MeixnerII} +\nabla\left[\frac{(\beta)_xc^x}{x!}M_n(x;\beta,c)\right]= +\frac{(\beta-1)_xc^x}{x!}M_{n+1}(x;\beta-1,c). +\end{equation} + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodMeixner} +\frac{(\beta)_xc^x}{x!}M_n(x;\beta,c)=\nabla^n\left[\frac{(\beta+n)_xc^x}{x!}\right]. +\end{equation} + +\subsection*{Generating functions} +\begin{equation} +\label{GenMeixner1} +\left(1-\frac{t}{c}\right)^x(1-t)^{-x-\beta}= +\sum_{n=0}^{\infty}\frac{(\beta)_n}{n!}M_n(x;\beta,c)t^n. +\end{equation} + +\begin{equation} +\label{GenMeixner2} +\e^t\,\hyp{1}{1}{-x}{\beta}{\left(\frac{1-c}{c}\right)t}= +\sum_{n=0}^{\infty}\frac{M_n(x;\beta,c)}{n!}t^n. +\end{equation} + +\begin{equation} +\label{GenMeixner3} +(1-t)^{-\gamma}\,\hyp{2}{1}{\gamma,-x}{\beta}{\frac{(1-c)t}{c(1-t)}} +=\sum_{n=0}^{\infty}\frac{(\gamma)_n}{n!}M_n(x;\beta,c)t^n, +\quad\textrm{$\gamma$ arbitrary}. +\end{equation} + +\subsection*{Limit relations} + +\subsubsection*{Hahn $\rightarrow$ Meixner} +If we take $\alpha=b-1$, $\beta=N(1-c)c^{-1}$ in the definition (\ref{DefHahn}) +of the Hahn polynomials and let $N\rightarrow\infty$ we find the Meixner polynomials: +$$\lim_{N\rightarrow\infty} +Q_n(x;b-1,N(1-c)c^{-1},N)=M_n(x;b,c).$$ + +\subsubsection*{Dual Hahn $\rightarrow$ Meixner} +To obtain the Meixner polynomials from the dual Hahn polynomials we have to take +$\gamma=\beta-1$ and $\delta=N(1-c)c^{-1}$ in the definition (\ref{DefDualHahn}) of +the dual Hahn polynomials and let $N\rightarrow\infty$: +$$\lim_{N\rightarrow\infty} +R_n(\lambda(x);\beta-1,N(1-c)c^{-1},N)=M_n(x;\beta,c).$$ + +\subsubsection*{Meixner $\rightarrow$ Laguerre} +The Laguerre polynomials given by (\ref{DefLaguerre}) are obtained from the Meixner polynomials +if we take $\beta=\alpha+1$ and $x\rightarrow (1-c)^{-1}x$ and let $c\rightarrow 1$: +\begin{equation} +\lim_{c\rightarrow 1} +M_n((1-c)^{-1}x;\alpha+1,c)=\frac{L_n^{(\alpha)}(x)}{L_n^{(\alpha)}(0)}. +\end{equation} + +\subsubsection*{Meixner $\rightarrow$ Charlier} +The Charlier polynomials given by (\ref{DefCharlier}) are obtained from the Meixner polynomials +if we take $c=(a+\beta)^{-1}a$ and let $\beta\rightarrow\infty$: +\begin{equation} +\lim_{\beta\rightarrow\infty} +M_n(x;\beta,(a+\beta)^{-1}a)=C_n(x;a). +\end{equation} + +\subsection*{Remarks} +The Meixner polynomials are related to the Jacobi polynomials given by (\ref{DefJacobi}) +in the following way: +$$\frac{(\beta)_n}{n!}M_n(x;\beta,c)=P_n^{(\beta-1,-n-\beta-x)}((2-c)c^{-1}).$$ + +\noindent +The Meixner polynomials are also related to the Krawtchouk polynomials given by +(\ref{DefKrawtchouk}) in the following way: +$$K_n(x;p,N)=M_n(x;-N,(p-1)^{-1}p).$$ +% RS: add begin\label{sec9.9} +In this section in \mycite{KLS} the pseudo Jacobi polynomial $P_n(x;\nu,N)$ in (9.9.1) +is considered +for $N\in\ZZ_{\ge0}$ and $n=0,1,\ldots,n$. However, we can more generally take +$-\thalf\be>0\;\;{\rm or}\;\;\al<\be<0). +\] +Then $P_n$ can be expressed as a Meixner polynomial: +\[ +P_n(x)=(-k_2(\al\be)^{-1})_n\,\be^n\, +M_n\left(-\,\frac{x+k_2\al^{-1}}{\al-\be},-k_2(\al\be)^{-1},\be\al^{-1}\right). +\] + +In 1938 Gottlieb \cite[\S2]{K1} introduces polynomials $l_n$ ``of Laguerre type'' +which turn out to be special Meixner polynomials: +$l_n(x)=e^{-n\la} M_n(x;1,e^{-\la})$. +% +\paragraph{\large\bf KLSadd: Uniqueness of orthogonality measure}The coefficient of $p_{n-1}(x)$ in (9.10.4) behaves as $O(n^2)$ as $n\to\iy$. +Hence \eqref{93} holds, by which the orthogonality measure is unique. +% +% RS: add end +\subsection*{References} +\cite{AlSalam90}, \cite{AndrewsAskey85}, \cite{Area+II}, \cite{Askey75}, \cite{Askey89I}, +\cite{AskeyGasper77}, \cite{AskeyWilson85}, \cite{AtakRahmanSuslov}, \cite{Campigotto+}, +\cite{LChiharaStanton}, \cite{Chihara78}, \cite{Dette95}, \cite{Dominici}, \cite{Dunkl76}, +\cite{Dunkl84}, \cite{DunklRamirez}, \cite{Erdelyi+}, \cite{FeinsilverSchott}, +\cite{Gasper73I}, \cite{Gasper74}, \cite{HoareRahman}, \cite{Ismail2005II}, \cite{Karlin58}, +\cite{Koorn82}, \cite{Koorn88}, \cite{LabelleYehI}, \cite{LabelleYehII}, \cite{Lesky62}, +\cite{Lesky89}, \cite{Lesky94I}, \cite{Lesky95II}, \cite{LewanowiczII}, \cite{Nikiforov+}, +\cite{NikiforovUvarov}, \cite{Qiu}, \cite{Rahman78I}, \cite{Rahman79}, \cite{Stanton84}, +\cite{Stanton90}, \cite{Szego75}, \cite{Zarzo+}, \cite{Zeng90}. + + +\section{Laguerre}\index{Laguerre polynomials} + +\par\setcounter{equation}{0} + +\subsection*{Hypergeometric representation} +\begin{equation} +\label{DefLaguerre} +L_n^{(\alpha)}(x)=\frac{(\alpha+1)_n}{n!}\,\hyp{1}{1}{-n}{\alpha+1}{x}. +\end{equation} + +\subsection*{Orthogonality relation} +\begin{equation} +\label{OrtLaguerre} +\int_0^{\infty}\expe^{-x}x^{\alpha}L_m^{(\alpha)}(x)L_n^{(\alpha)}(x)\,dx= +\frac{\Gamma(n+\alpha+1)}{n!}\,\delta_{mn},\quad\alpha > -1. +\end{equation} + +\subsection*{Recurrence relation} +\begin{equation} +\label{RecLaguerre} +(n+1)L_{n+1}^{(\alpha)}(x)-(2n+\alpha+1-x)L_n^{(\alpha)}(x)+(n+\alpha)L_{n-1}^{(\alpha)}(x)=0. +\end{equation} + +\subsection*{Normalized recurrence relation} +\begin{equation} +\label{NormRecLaguerre} +xp_n(x)=p_{n+1}(x)+(2n+\alpha+1)p_n(x)+n(n+\alpha)p_{n-1}(x), +\end{equation} +where +$$L_n^{(\alpha)}(x)=\frac{(-1)^n}{n!}p_n(x).$$ + +\subsection*{Differential equation} +\begin{equation} +\label{dvLaguerre} +xy''(x)+(\alpha+1-x)y'(x)+ny(x)=0,\quad y(x)=L_n^{(\alpha)}(x). +\end{equation} + +\newpage + +\subsection*{Forward shift operator} +\begin{equation} +\label{shift1Laguerre} +\frac{d}{dx}L_n^{(\alpha)}(x)=-L_{n-1}^{(\alpha+1)}(x). +\end{equation} + +\subsection*{Backward shift operator} +\begin{equation} +\label{shift2LaguerreI} +x\frac{d}{dx}L_n^{(\alpha)}(x)+(\alpha-x)L_n^{(\alpha)}(x)=(n+1)L_{n+1}^{(\alpha-1)}(x) +\end{equation} +or equivalently +\begin{equation} +\label{shift2LaguerreII} +\frac{d}{dx}\left[\expe^{-x}x^{\alpha}L_n^{(\alpha)}(x)\right]=(n+1)\expe^{-x}x^{\alpha-1}L^{(\alpha-1)}_{n+1}(x). +\end{equation} + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodLaguerre} +\e^{-x}x^{\alpha}L_n^{(\alpha)}(x)=\frac{1}{n!}\left(\frac{d}{dx}\right)^n\left[\expe^{-x}x^{n+\alpha}\right]. +\end{equation} + +\subsection*{Generating functions} +\begin{equation} +\label{GenLaguerre1} +(1-t)^{-\alpha-1}\exp\left(\frac{xt}{t-1}\right)= +\sum_{n=0}^{\infty}L_n^{(\alpha)}(x)t^n. +\end{equation} + +\begin{equation} +\label{GenLaguerre2} +\e^t\,\hyp{0}{1}{-}{\alpha+1}{-xt} +=\sum_{n=0}^{\infty}\frac{L_n^{(\alpha)}(x)}{(\alpha+1)_n}t^n. +\end{equation} + +\begin{equation} +\label{GenLaguerre3} +(1-t)^{-\gamma}\,\hyp{1}{1}{\gamma}{\alpha+1}{\frac{xt}{t-1}} +=\sum_{n=0}^{\infty}\frac{(\gamma)_n}{(\alpha+1)_n}L_n^{(\alpha)}(x)t^n, +\quad\textrm{$\gamma$ arbitrary}. +\end{equation} + +\subsection*{Limit relations} + +\subsubsection*{Meixner-Pollaczek $\rightarrow$ Laguerre} +The Laguerre polynomials can be obtained from the Meixner-Pollaczek polynomials given by +(\ref{DefMP}) by the substitution $\lambda=\frac{1}{2}(\alpha+1)$, +$x\rightarrow -\frac{1}{2}\phi^{-1}x$ and the limit $\phi\rightarrow 0$: +$$\lim_{\phi\rightarrow 0} +P_n^{(\frac{1}{2}\alpha+\frac{1}{2})}(-\textstyle\frac{1}{2}\phi^{-1}x;\phi)=L_n^{(\alpha)}(x).$$ + +\subsubsection*{Jacobi $\rightarrow$ Laguerre} +The Laguerre polynomials are obtained from the Jacobi polynomials given by (\ref{DefJacobi}) +if we set $x\rightarrow 1-2\beta^{-1}x$ and then take the limit $\beta\rightarrow\infty$: +$$\lim_{\beta\rightarrow\infty} +P_n^{(\alpha,\beta)}(1-2\beta^{-1}x)=L_n^{(\alpha)}(x).$$ + +\subsubsection*{Meixner $\rightarrow$ Laguerre} +If we take $\beta=\alpha+1$ and $x\rightarrow (1-c)^{-1}x$ in the definition +(\ref{DefMeixner}) of the Meixner polynomials and let $c\rightarrow 1$ we obtain +the Laguerre polynomials: +$$\lim_{c\rightarrow 1} +M_n((1-c)^{-1}x;\alpha+1,c)=\frac{L_n^{(\alpha)}(x)}{L_n^{(\alpha)}(0)}.$$ + +\subsubsection*{Laguerre $\rightarrow$ Hermite} +The Hermite polynomials given by (\ref{DefHermite}) can be obtained from the Laguerre +polynomials by taking the limit $\alpha\rightarrow\infty$ in the following way: +\begin{equation} +\lim_{\alpha\rightarrow\infty} +\left(\frac{2}{\alpha}\right)^{\frac{1}{2}n} +L_n^{(\alpha)}((2\alpha)^{\frac{1}{2}}x+\alpha)=\frac{(-1)^n}{n!}H_n(x). +\end{equation} + +\subsection*{Remarks} +The definition (\ref{DefLaguerre}) of the Laguerre polynomials can also be +written as: +$$L_n^{(\alpha)}(x)=\frac{1}{n!}\sum_{k=0}^n\frac{(-n)_k}{k!}(\alpha+k+1)_{n-k}x^k.$$ +In this way the Laguerre polynomials can also be seen as polynomials in the parameter $\alpha$. +Therefore they can be defined for all $\alpha$. + +\noindent +The Laguerre polynomials are related to the Bessel polynomials given by (\ref{DefBessel}) +in the following way: +$$L_n^{(\alpha)}(x)=\frac{(-x)^n}{n!}y_n(2x^{-1};-2n-\alpha-1).$$ + +\noindent +The Laguerre polynomials are related to the Charlier polynomials given by (\ref{DefCharlier}) +in the following way: +$$\frac{(-a)^n}{n!}C_n(x;a)=L_n^{(x-n)}(a).$$ + +\noindent +The Laguerre polynomials and the Hermite polynomials given by (\ref{DefHermite}) are also +connected by the following quadratic transformations: +$$H_{2n}(x)=(-1)^nn!\,2^{2n}L_n^{(-\frac{1}{2})}(x^2)$$ +and +$$H_{2n+1}(x)=(-1)^nn!\,2^{2n+1}xL_n^{(\frac{1}{2})}(x^2).$$ + +\noindent +In combinatorics the Laguerre polynomials with $\alpha=0$ are often called Rook +polynomials. +% RS: add begin\label{sec9.11} +% +\paragraph{\large\bf KLSadd: Special values}By (9.11.1) and the binomial formula: +\begin{equation} +K_n(0;p,N)=1,\qquad +K_n(N;p,N)=(1-p^{-1})^n. +\label{9} +\end{equation} +The self-duality (p.240, Remarks, first formula) +\begin{equation} +K_n(x;p,N)=K_x(n;p,N)\qquad (n,x\in \{0,1,\ldots,N\}) +\label{147} +\end{equation} +combined with \eqref{9} yields: +\begin{equation} +K_N(x;p,N)=(1-p^{-1})^x\qquad(x\in\{0,1,\ldots,N\}). +\label{148} +\end{equation} +% +\paragraph{\large\bf KLSadd: Symmetry}By the orthogonality relation (9.11.2): +\begin{equation} +\frac{K_n(N-x;p,N)}{K_n(N;p,N)}=K_n(x;1-p,N). +\label{10} +\end{equation} +By \eqref{10} and \eqref{147} we have also +\begin{equation} +\frac{K_{N-n}(x;p,N)}{K_N(x;p,N)}=K_n(x;1-p,N) +\qquad(n,x\in\{0,1,\ldots,N\}), +\label{149} +\end{equation} +and, by \eqref{149}, \eqref{10} and \eqref{9}, +\begin{equation} +K_{N-n}(N-x;p,N)=\left(\frac p{p-1}\right)^{n+x-N}K_n(x;p,N) +\qquad(n,x\in\{0,1,\ldots,N\}). +\label{150} +\end{equation} +A particular case of \eqref{10} is: +\begin{equation} +K_n(N-x;\thalf,N)=(-1)^n K_n(x;\thalf,N). +\label{11} +\end{equation} +Hence +\begin{equation} +K_{2m+1}(N;\thalf,2N)=0. +\label{12} +\end{equation} +From (9.11.11): +\begin{equation} +K_{2m}(N;\thalf,2N)=\frac{(\thalf)_m}{(-N+\thalf)_m}\,. +\label{13} +\end{equation} +% +\paragraph{\large\bf KLSadd: Quadratic transformations}\begin{align} +K_{2m}(x+N;\thalf,2N)&=\frac{(\thalf)_m}{(-N+\thalf)_m}\, +R_m(x^2;-\thalf,-\thalf,N), +\label{31}\\ +K_{2m+1}(x+N;\thalf,2N)&=-\,\frac{(\tfrac32)_m}{N\,(-N+\thalf)_m}\, +x\,R_m(x^2-1;\thalf,\thalf,N-1), +\label{33}\\ +K_{2m}(x+N+1;\thalf,2N+1)&=\frac{(\tfrac12)_m}{(-N-\thalf)_m}\, +R_m(x(x+1);-\thalf,\thalf,N), +\label{32}\\ +K_{2m+1}(x+N+1;\thalf,2N+1)&=\frac{(\tfrac32)_m}{(-N-\thalf)_{m+1}}\, +(x+\thalf)\,R_m(x(x+1);\thalf,-\thalf,N), +\label{34} +\end{align} +where $R_m$ is a dual Hahn polynomial (9.6.1). For the proofs use +(9.6.2), (9.11.2), (9.6.4) and (9.11.4). +% +\paragraph{\large\bf KLSadd: Generating functions}\begin{multline} +\sum_{x=0}^N\binom Nx K_m(x;p,N)K_n(x;q,N)z^x\\ +=\left(\frac{p-z+pz}p\right)^m +\left(\frac{q-z+qz}q\right)^n +(1+z)^{N-m-n} +K_m\left(n;-\,\frac{(p-z+pz)(q-z+qz)}z,N\right). +\label{107} +\end{multline} +This follows immediately from Rosengren \cite[(3.5)]{K8}, which goes back +to Meixner \cite{K9}. +% +% RS: add end +\subsection*{References} +\cite{Abdul}, \cite{Abram}, \cite{Ahmed+82}, \cite{Allaway76}, \cite{Allaway91}, +\cite{NAlSalam66}, \cite{AlSalam64}, \cite{AlSalam90}, \cite{AlSalamChihara72}, +\cite{AlSalamChihara76}, \cite{AndrewsAskey85}, \cite{AndrewsAskeyRoy}, \cite{Askey68}, +\cite{Askey75}, \cite{Askey89I}, \cite{Askey2005}, \cite{AskeyGasper76}, \cite{AskeyGasper77}, +\cite{AskeyIsmail76}, \cite{AskeyIsmailKoorn}, \cite{AskeyWilson85}, \cite{Barrett}, +\cite{Bavinck96}, \cite{Brafman51}, \cite{Brafman57II}, \cite{Brenke}, \cite{Brown}, +\cite{BustozSavage79}, \cite{BustozSavage80}, \cite{Carlitz60}, \cite{Carlitz61I}, +\cite{Carlitz61II}, \cite{Carlitz62}, \cite{Carlitz68}, \cite{ChenSrivastava}, +\cite{ChenIsmail91}, \cite{Chihara68II}, \cite{Chihara78}, \cite{Cohen}, \cite{Cooper+}, +\cite{Dattoli2001}, \cite{DetteStudden92}, \cite{DetteStudden95}, \cite{Dimitrov2003}, +\cite{Doha2003II}, \cite{ElbertLaforgia87I}, \cite{Erdelyi}, \cite{Erdelyi+}, \cite{Exton98}, +\cite{Faldey}, \cite{Gasper73II}, \cite{Gasper77}, \cite{Gatteschi2002}, \cite{Gawronski87}, +\cite{Gawronski93}, \cite{Gillis}, \cite{GillisIsmailOffer}, \cite{Godoy+}, \cite{Grad}, +\cite{Hajmirzaahmad95}, \cite{Ismail74}, \cite{Ismail77}, \cite{Ismail2005II}, +\cite{IsmailLetVal88}, \cite{IsmailLi}, \cite{IsmailMassonRahman}, \cite{IsmailMuldoon}, +\cite{IsmailStanton97}, \cite{IsmailTamhankar}, \cite{Jackson}, \cite{Karlin58}, +\cite{Kochneff95}, \cite{Kochneff97II}, \cite{Koekoek2000}, \cite{Koorn77I}, \cite{Koorn78}, +\cite{Koorn85}, \cite{Koorn88}, \cite{Krasikov2003}, \cite{Krasikov2005}, +\cite{KuijlaarsMcLaughlin}, \cite{KwonLittle}, \cite{LabelleYehI}, \cite{LabelleYehII}, +\cite{LabelleYehIII}, \cite{Lee97I}, \cite{Lee97II}, \cite{Lee97III}, \cite{Lesky95II}, +\cite{Lesky96}, \cite{LewanowiczI}, \cite{LopezTemme2004}, \cite{Mathai}, \cite{Meixner}, +\cite{Nikiforov+}, \cite{NikiforovUvarov}, \cite{Olver}, \cite{PerezPinar}, \cite{Pittaluga}, +\cite{Rainville}, \cite{SainteViennot}, \cite{Srivastava66}, \cite{Srivastava69I}, +\cite{Srivastava69II}, \cite{Srivastava70}, \cite{Srivastava71}, \cite{Szego75}, \cite{Temme}, +\cite{Trickovic}, \cite{Trivedi}, \cite{Viennot}. + + +\section{Bessel}\index{Bessel polynomials} + +\par\setcounter{equation}{0} + +\subsection*{Hypergeometric representation} +\begin{eqnarray} +\label{DefBessel} +y_n(x;a)&=&\hyp{2}{0}{-n,n+a+1}{-}{-\frac{x}{2}}\\ +&=&(n+a+1)_n\left(\frac{x}{2}\right)^n\,\hyp{1}{1}{-n}{-2n-a}{\frac{2}{x}}, +\quad n=0,1,2,\ldots,N.\nonumber +\end{eqnarray} + +\subsection*{Orthogonality relation} +\begin{eqnarray} +\label{OrtBessel} +& &\int_0^{\infty}x^a\e^{-\frac{2}{x}}y_m(x;a)y_n(x;a)\,dx\nonumber\\ +& &=-\frac{2^{a+1}}{2n+a+1}\Gamma(-n-a)n!\,\delta_{mn},\quad a<-2N-1. +\end{eqnarray} + +\subsection*{Recurrence relation} +\begin{eqnarray} +\label{RecBessel} +& &2(n+a+1)(2n+a)y_{n+1}(x;a)\nonumber\\ +& &{}=(2n+a+1)\left[2a+(2n+a)(2n+a+2)x\right]y_n(x;a)\nonumber\\ +& &{}\mathindent{}+2n(2n+a+2)y_{n-1}(x;a). +\end{eqnarray} + +\subsection*{Normalized recurrence relation} +\begin{eqnarray} +\label{NormRecBessel} +xp_n(x)&=&p_{n+1}(x)-\frac{2a}{(2n+a)(2n+a+2)}p_n(x)\nonumber\\ +& &{}\mathindent{}-\frac{4n(n+a)}{(2n+a-1)(2n+a)^2(2n+a+1)}p_{n-1}(x), +\end{eqnarray} +where +$$y_n(x;a)=\frac{(n+a+1)_n}{2^n}p_n(x).$$ + +\subsection*{Differential equation} +\begin{equation} +\label{dvBessel} +x^2y''(x)+\left[(a+2)x+2\right]y'(x)-n(n+a+1)y(x)=0,\quad y(x)=y_n(x;a). +\end{equation} + +\subsection*{Forward shift operator} +\begin{equation} +\label{shift1Bessel} +\frac{d}{dx}y_n(x;a)=\frac{n(n+a+1)}{2}y_{n-1}(x;a+2). +\end{equation} + +\subsection*{Backward shift operator} +\begin{equation} +\label{shift2BesselI} +x^2\frac{d}{dx}y_n(x;a)+(ax+2)y_n(x;a)=2y_{n+1}(x;a-2) +\end{equation} +or equivalently +\begin{equation} +\label{shift2BesselII} +\frac{d}{dx}\left[x^a\expe^{-\frac{2}{x}}y_n(x;a)\right]=2x^{a-2}\expe^{-\frac{2}{x}}y_{n+1}(x;a-2). +\end{equation} + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodBessel} +y_n(x;a)=2^{-n}x^{-a}\expe^{\frac{2}{x}}D^n\left(x^{2n+a}\expe^{-\frac{2}{x}}\right). +\end{equation} + +\subsection*{Generating function} +\begin{equation} +\label{GenBessel} +\left(1-2xt\right)^{-\frac{1}{2}}\left(\frac{2}{1+\sqrt{1-2xt}}\right)^a +\exp\left(\frac{2t}{1+\sqrt{1-2xt}}\right)=\sum_{n=0}^{\infty}y_n(x;a)\frac{t^n}{n!}. +\end{equation} + +\subsection*{Limit relation} + +\subsubsection*{Jacobi $\rightarrow$ Bessel} +If we take $\beta=a-\alpha$ in the definition (\ref{DefJacobi}) of the Jacobi polynomials +and let $\alpha\rightarrow -\infty$ we find the Bessel polynomials: +$$\lim_{\alpha\rightarrow -\infty} +\frac{P_n^{(\alpha,a-\alpha)}(1+\alpha x)}{P_n^{(\alpha,a-\alpha)}(1)}=y_n(x;a).$$ + +\subsection*{Remarks} +The following notations are also used for the Bessel polynomials: +$$y_n(x;a,b)=y_n(2b^{-1}x;a)\quad\textrm{and}\quad\theta_n(x;a,b)=x^ny_n(x^{-1};a,b).$$ +However, the Bessel polynomials essentially depend on only one parameter. + +\noindent +The Bessel polynomials are related to the Laguerre polynomials given by (\ref{DefLaguerre}) +in the following way: +$$L_n^{(\alpha)}(x)=\frac{(-x)^n}{n!}y_n(2x^{-1};-2n-\alpha-1).$$ + +\noindent +The special case $a=-2N-2$ of the Bessel polynomials can be obtained from the pseudo Jacobi +polynomials by setting $x\rightarrow\nu x$ in the definition (\ref{DefPseudoJacobi}) of the +pseudo Jacobi polynomials and taking the limit $\nu\rightarrow\infty$ in the following way: +$$\lim\limits_{\nu\rightarrow\infty}\frac{P_n(\nu x;\nu,N)}{\nu^n} +=\frac{2^n}{(n-2N-1)_n}y_n(x;-2N-2).$$ + +\subsection*{References} +\cite{Andrade}, \cite{BergVignat}, \cite{Carlitz57I}, \cite{Dattoli2003}, +\cite{DohaAhmed2004}, \cite{DohaAhmed2006}, \cite{Grosswald}, \cite{Ismail2005II}, +\cite{KrallFrink}, \cite{Lesky98}, \cite{NikiforovUvarov}. + + +\section{Charlier}\index{Charlier polynomials} + +\par\setcounter{equation}{0} + +\subsection*{Hypergeometric representation} +\begin{equation} +\label{DefCharlier} +C_n(x;a)=\hyp{2}{0}{-n,-x}{-}{-\frac{1}{a}}. +\end{equation} + +\subsection*{Orthogonality relation} +\begin{equation} +\label{OrtCharlier} +\sum_{x=0}^{\infty}\frac{a^x}{x!}C_m(x;a)C_n(x;a)= +a^{-n}\expe^an!\,\delta_{mn},\quad a > 0. +\end{equation} + +\subsection*{Recurrence relation} +\begin{equation} +\label{RecCharlier} +-xC_n(x;a)=aC_{n+1}(x;a)-(n+a)C_n(x;a)+nC_{n-1}(x;a). +\end{equation} + +\subsection*{Normalized recurrence relation} +\begin{equation} +\label{NormRecCharlier} +xp_n(x)=p_{n+1}(x)+(n+a)p_n(x)+nap_{n-1}(x), +\end{equation} +where +$$C_n(x;a)=\left(-\frac{1}{a}\right)^np_n(x).$$ + +\subsection*{Difference equation} +\begin{equation} +\label{dvCharlier} +-ny(x)=ay(x+1)-(x+a)y(x)+xy(x-1),\quad y(x)=C_n(x;a). +\end{equation} + +\subsection*{Forward shift operator} +\begin{equation} +\label{shift1CharlierI} +C_n(x+1;a)-C_n(x;a)=-\frac{n}{a}C_{n-1}(x;a) +\end{equation} +or equivalently +\begin{equation} +\label{shift1CharlierII} +\Delta C_n(x;a)=-\frac{n}{a}C_{n-1}(x;a). +\end{equation} + +\subsection*{Backward shift operator} +\begin{equation} +\label{shift2CharlierI} +C_n(x;a)-\frac{x}{a}C_n(x-1;a)=C_{n+1}(x;a) +\end{equation} +or equivalently +\begin{equation} +\label{shift2CharlierII} +\nabla\left[\frac{a^x}{x!}C_n(x;a)\right]=\frac{a^x}{x!}C_{n+1}(x;a). +\end{equation} + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodCharlier} +\frac{a^x}{x!}C_n(x;a)=\nabla^n\left[\frac{a^x}{x!}\right]. +\end{equation} + +\subsection*{Generating function} +\begin{equation} +\label{GenCharlier} +\e^t\left(1-\frac{t}{a}\right)^x=\sum_{n=0}^{\infty}\frac{C_n(x;a)}{n!}t^n. +\end{equation} + +\subsection*{Limit relations} + +\subsubsection*{Meixner $\rightarrow$ Charlier} +If we take $c=(a+\beta)^{-1}a$ in the definition (\ref{DefMeixner}) of the Meixner polynomials +and let $\beta\rightarrow\infty$ we find the Charlier polynomials: +$$\lim_{\beta\rightarrow\infty}M_n(x;\beta,(a+\beta)^{-1}a)=C_n(x;a).$$ + +\subsubsection*{Krawtchouk $\rightarrow$ Charlier} +The Charlier polynomials can be found from the Krawtchouk polynomials given by +(\ref{DefKrawtchouk}) by taking $p=N^{-1}a$ and letting $N\rightarrow\infty$: +$$\lim_{N\rightarrow\infty}K_n(x;N^{-1}a,N)=C_n(x;a).$$ + +\subsubsection*{Charlier $\rightarrow$ Hermite} +The Hermite polynomials given by (\ref{DefHermite}) are obtained from the Charlier polynomials +if we set $x\rightarrow (2a)^{1/2}x+a$ and let $a\rightarrow\infty$. In fact we have +\begin{equation} +\lim_{a\rightarrow\infty} +(2a)^{\frac{1}{2}n}C_n((2a)^{\frac{1}{2}}x+a;a)=(-1)^nH_n(x). +\end{equation} + +\subsection*{Remark} +The Charlier polynomials are related to the Laguerre polynomials given by (\ref{DefLaguerre}) +in the following way: +$$\frac{(-a)^n}{n!}C_n(x;a)=L_n^{(x-n)}(a).$$ +% RS: add begin\label{sec9.12} +\paragraph{\large\bf KLSadd: Notation}Here the Laguerre polynomial is denoted by $L_n^\al$ instead of +$L_n^{(\al)}$. +% +\paragraph{\large\bf KLSadd: Hypergeometric representation}\begin{align} +L_n^\al(x)&= +\frac{(\al+1)_n}{n!}\,\hyp11{-n}{\al+1}x +\label{182}\\ +&=\frac{(-x)^n}{n!} \hyp20{-n,-n-\al}-{-\,\frac1x} +\label{183}\\ +&=\frac{(-x)^n}{n!}\,C_n(n+\al;x), +\label{184} +\end{align} +where $C_n$ in \eqref{184} is a +\hyperref[sec9.14]{Charlier polynomial}. +Formula \eqref{182} is (9.12.1). Then \eqref{183} follows by reversal +of summation. Finally \eqref{184} follows by \eqref{183} and \eqref{179}. +It is also the remark on top of p.244 in \mycite{KLS}, and it is essentially +\myciteKLS{416}{(2.7.10)}. +% +\paragraph{\large\bf KLSadd: Uniqueness of orthogonality measure}The coefficient of $p_{n-1}(x)$ in (9.12.4) behaves as $O(n^2)$ as $n\to\iy$. +Hence \eqref{93} holds, by which the orthogonality measure is unique. +% +\paragraph{\large\bf KLSadd: Special value}\begin{equation} +L_n^{\al}(0)=\frac{(\al+1)_n}{n!}\,. +\label{53} +\end{equation} +Use (9.12.1) or see \mycite{DLMF}{18.6.1)}. +% +\paragraph{\large\bf KLSadd: Quadratic transformations}\begin{align} +H_{2n}(x)&=(-1)^n\,2^{2n}\,n!\,L_n^{-1/2}(x^2), +\label{54}\\ +H_{2n+1}(x)&=(-1)^n\,2^{2n+1}\,n!\,x\,L_n^{1/2}(x^2). +\label{55} +\end{align} +See p.244, Remarks, last two formulas. +Or see \mycite{DLMF}{(18.7.19), (18.7.20)}. +% +\paragraph{\large\bf KLSadd: Fourier transform}\begin{equation} +\frac1{\Ga(\al+1)}\,\int_0^\iy \frac{L_n^\al(y)}{L_n^\al(0)}\, +e^{-y}\,y^\al\,e^{ixy}\,dy= +i^n\,\frac{y^n}{(iy+1)^{n+\al+1}}\,, +\label{14} +\end{equation} +see \mycite{DLMF}{(18.17.34)}. +% +\paragraph{\large\bf KLSadd: Differentiation formulas}Each differentiation formula is given in two equivalent forms. +\begin{equation} +\frac d{dx}\left(x^\al L_n^\al(x)\right)= +(n+\al)\,x^{\al-1} L_n^{\al-1}(x),\qquad +\left(x\frac d{dx}+\al\right)L_n^\al(x)= +(n+\al)\,L_n^{\al-1}(x). +\label{76} +\end{equation} +% +\begin{equation} +\frac d{dx}\left(e^{-x} L_n^\al(x)\right)= +-e^{-x} L_n^{\al+1}(x),\qquad +\left(\frac d{dx}-1\right)L_n^\al(x)= +-L_n^{\al+1}(x). +\label{77} +\end{equation} +% +Formulas \eqref{76} and \eqref{77} follow from +\mycite{DLMF}{(13.3.18), (13.3.20)} +together with (9.12.1). +% +\paragraph{\large\bf KLSadd: Generalized Hermite polynomials}See \myciteKLS{146}{p.156}, \cite[Section 1.5.1]{K26}. +These are defined by +\begin{equation} +H_{2m}^\mu(x):=\const L_m^{\mu-\half}(x^2),\qquad +H_{2m+1}^\mu(x):=\const x\,L_m^{\mu+\half}(x^2). +\label{78} +\end{equation} +Then for $\mu>-\thalf$ we have orthogonality relation +\begin{equation} +\int_{-\iy}^{\iy} H_m^\mu(x)\,H_n^\mu(x)\,|x|^{2\mu}e^{-x^2}\,dx +=0\qquad(m\ne n). +\label{79} +\end{equation} +Let the Dunkl operator $T_\mu$ be defined by \eqref{72}. +If we choose the constants in \eqref{78} as +\begin{equation} +H_{2m}^\mu(x)=\frac{(-1)^m(2m)!}{(\mu+\thalf)_m}\,L_m^{\mu-\half}(x^2),\qquad +H_{2m+1}^\mu(x)=\frac{(-1)^m(2m+1)!}{(\mu+\thalf)_{m+1}}\, + x\,L_m^{\mu+\half}(x^2) + \label{80} +\end{equation} +then (see \cite[(1.6)]{K5}) +\begin{equation} +T_\mu H_n^\mu=2n\,H_{n-1}^\mu. +\label{81} +\end{equation} +Formula \eqref{81} with \eqref{80} substituted gives rise to two +differentiation formulas involving Laguerre polynomials which are equivalent to +(9.12.6) and \eqref{76}. + +Composition of \eqref{81} with itself gives +\[ +T_\mu^2 H_n^\mu=4n(n-1)\,H_{n-2}^\mu, +\] +which is equivalent to the composition of (9.12.6) and \eqref{76}: +\begin{equation} +\left(\frac{d^2}{dx^2}+\frac{2\al+1}x\,\frac d{dx}\right)L_n^\al(x^2) +=-4(n+\al)\,L_{n-1}^\al(x^2). +\label{82} +\end{equation} +% +% RS: add end +\subsection*{References} +\cite{Allaway76}, \cite{NAlSalam66}, \cite{AlSalam90}, \cite{AlSalamChihara76}, +\cite{AlSalamIsmail76}, \cite{AndrewsAskey85}, \cite{Area+II}, \cite{Askey75}, +\cite{AskeyGasper77}, \cite{AskeyWilson85}, \cite{AtakRahmanSuslov}, \cite{Bavinck98}, +\cite{BavinckKoekoek}, \cite{Chihara78}, \cite{Chihara79}, \cite{Dunkl76}, \cite{Erdelyi+}, +\cite{Gasper73I}, \cite{Gasper74}, \cite{Goh}, \cite{HoareRahman}, \cite{IsmailLetVal88}, +\cite{Koekoek2000}, \cite{Koorn88}, \cite{Krasikov2002}, \cite{LabelleYehI}, +\cite{LabelleYehII}, \cite{LabelleYehIII}, \cite{Lesky62}, \cite{Lesky89}, \cite{Lesky94I}, +\cite{Lesky95II}, \cite{LewanowiczII}, \cite{LopezTemme2004}, \cite{Meixner}, \cite{Nikiforov+}, +\cite{NikiforovUvarov}, \cite{Szafraniec}, \cite{Szego75}, \cite{Viennot}, \cite{Zarzo+}, +\cite{Zeng90}. + + +\section{Hermite}\index{Hermite polynomials} + +\par\setcounter{equation}{0} + +\subsection*{Hypergeometric representation} +\begin{equation} +\label{DefHermite} +H_n(x)=(2x)^n\,\hyp{2}{0}{-n/2,-(n-1)/2}{-}{-\frac{1}{x^2}}. +\end{equation} + +\subsection*{Orthogonality relation} +\begin{equation} +\label{OrtHermite} +\frac{1}{\sqrt{\cpi}}\int_{-\infty}^{\infty}\expe^{-x^2}H_m(x)H_n(x)\,dx +=2^nn!\,\delta_{mn}. +\end{equation} + +\subsection*{Recurrence relation} +\begin{equation} +\label{RecHermite} +H_{n+1}(x)-2xH_n(x)+2nH_{n-1}(x)=0. +\end{equation} + +\subsection*{Normalized recurrence relation} +\begin{equation} +\label{NormRecHermite} +xp_n(x)=p_{n+1}(x)+\frac{n}{2}p_{n-1}(x), +\end{equation} +where +$$H_n(x)=2^np_n(x).$$ + +\subsection*{Differential equation} +\begin{equation} +\label{dvHermite} +y''(x)-2xy'(x)+2ny(x)=0,\quad y(x)=H_n(x). +\end{equation} + +\subsection*{Forward shift operator} +\begin{equation} +\label{shift1Hermite} +\frac{d}{dx}H_n(x)=2nH_{n-1}(x). +\end{equation} + +\subsection*{Backward shift operator} +\begin{equation} +\label{shift2HermiteI} +\frac{d}{dx}H_n(x)-2xH_n(x)=-H_{n+1}(x) +\end{equation} +or equivalently +\begin{equation} +\label{shift2HermiteII} +\frac{d}{dx}\left[\expe^{-x^2}H_n(x)\right]=-\expe^{-x^2}H_{n+1}(x). +\end{equation} + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodHermite} +\e^{-x^2}H_n(x)=(-1)^n\left(\frac{d}{dx}\right)^n\left[\expe^{-x^2}\right]. +\end{equation} + +\subsection*{Generating functions} +\begin{equation} +\label{GenHermite1} +\exp\left(2xt-t^2\right)=\sum_{n=0}^{\infty}\frac{H_n(x)}{n!}t^n. +\end{equation} + +\begin{equation} +\label{GenHermite2} +\left\{\begin{array}{l} +\displaystyle\expe^t\cos(2x\sqrt{t})=\sum_{n=0}^{\infty} +\frac{(-1)^n}{(2n)!}H_{2n}(x)t^n\\[5mm] +\displaystyle\frac{\expe^t}{\sqrt{t}}\sin(2x\sqrt{t})=\sum_{n=0}^{\infty} +\frac{(-1)^n}{(2n+1)!}H_{2n+1}(x)t^n. +\end{array}\right. +\end{equation} + +\begin{equation} +\label{GenHermite3} +\left\{\begin{array}{l} +\displaystyle \expe^{-t^2}\cosh(2xt)=\sum_{n=0}^{\infty} +\frac{H_{2n}(x)}{(2n)!}t^{2n}\\[5mm] +\displaystyle\expe^{-t^2}\sinh(2xt)=\sum_{n=0}^{\infty} +\frac{H_{2n+1}(x)}{(2n+1)!}t^{2n+1}. +\end{array}\right. +\end{equation} + +\begin{equation} +\label{GenHermite4} +\left\{\begin{array}{l} +\displaystyle (1+t^2)^{-\gamma}\,\hyp{1}{1}{\gamma}{\frac{1}{2}}{\frac{x^2t^2}{1+t^2}}= +\sum_{n=0}^{\infty}\frac{(\gamma)_n}{(2n)!}H_{2n}(x)t^{2n}\\[5mm] +\displaystyle\frac{xt}{\sqrt{1+t^2}}\, +\hyp{1}{1}{\gamma+\frac{1}{2}}{\frac{3}{2}}{\frac{x^2t^2}{1+t^2}} +=\sum_{n=0}^{\infty}\frac{(\gamma+\frac{1}{2})_n}{(2n+1)!}H_{2n+1}(x)t^{2n+1} +\end{array}\right. +\end{equation} +with $\gamma$ arbitrary. + +\begin{equation} +\label{GenHermite5} +\frac{1+2xt+4t^2}{(1+4t^2)^{\frac{3}{2}}}\exp\left(\frac{4x^2t^2}{1+4t^2}\right) +=\sum_{n=0}^{\infty}\frac{H_n(x)}{\lfloor n/2\rfloor\,!}t^n, +\end{equation} +where $\lfloor n/2\rfloor$ denotes the largest integer smaller than or equal to $n/2$. + +\subsection*{Limit relations} + +\subsubsection*{Meixner-Pollaczek $\rightarrow$ Hermite} +If we take $x\rightarrow (\sin\phi)^{-1}(x\sqrt{\lambda}-\lambda\cos\phi)$ +in the definition (\ref{DefMP}) of the Meixner-Pollaczek polynomials and +then let $\lambda\rightarrow\infty$ we obtain the Hermite polynomials: +$$\lim_{\lambda\rightarrow\infty} +\lambda^{-\frac{1}{2}n}P_n^{(\lambda)} +((\sin\phi)^{-1}(x\sqrt{\lambda}-\lambda\cos\phi);\phi)=\frac{H_n(x)}{n!}.$$ + +\subsubsection*{Jacobi $\rightarrow$ Hermite} +The Hermite polynomials follow from the Jacobi polynomials given by (\ref{DefJacobi}) by +taking $\beta=\alpha$ and letting $\alpha\rightarrow\infty$ in the following way: +$$\lim_{\alpha\rightarrow\infty} +\alpha^{-\frac{1}{2}n}P_n^{(\alpha,\alpha)}(\alpha^{-\frac{1}{2}}x)=\frac{H_n(x)}{2^nn!}.$$ + +\subsubsection*{Gegenbauer / Ultraspherical $\rightarrow$ Hermite} +The Hermite polynomials follow from the Gegenbauer (or ultraspherical) polynomials given by +(\ref{DefGegenbauer}) by taking $\lambda=\alpha+\frac{1}{2}$ and letting +$\alpha\rightarrow\infty$ in the following way: +$$\lim_{\alpha\rightarrow\infty} +\alpha^{-\frac{1}{2}n}C_n^{(\alpha+\frac{1}{2})}(\alpha^{-\frac{1}{2}}x)=\frac{H_n(x)}{n!}.$$ + +\subsubsection*{Krawtchouk $\rightarrow$ Hermite} +The Hermite polynomials follow from the Krawtchouk polynomials given by (\ref{DefKrawtchouk}) +by setting $x\rightarrow pN+x\sqrt{2p(1-p)N}$ and then letting $N\rightarrow\infty$: +$$\lim_{N\rightarrow\infty} +\sqrt{\binom{N}{n}}K_n(pN+x\sqrt{2p(1-p)N};p,N) +=\frac{\displaystyle (-1)^nH_n(x)}{\displaystyle\sqrt{2^nn!\left(\frac{p}{1-p}\right)^n}}.$$ + +\subsubsection*{Laguerre $\rightarrow$ Hermite} +The Hermite polynomials can be obtained from the Laguerre polynomials given by +(\ref{DefLaguerre}) by taking the limit $\alpha\rightarrow\infty$ in the following way: +$$\lim_{\alpha\rightarrow\infty} +\left(\frac{2}{\alpha}\right)^{\frac{1}{2}n} +L_n^{(\alpha)}((2\alpha)^{\frac{1}{2}}x+\alpha)=\frac{(-1)^n}{n!}H_n(x).$$ + +\subsubsection*{Charlier $\rightarrow$ Hermite} +If we set $x\rightarrow (2a)^{1/2}x+a$ in the definition (\ref{DefCharlier}) +of the Charlier polynomials and let $a\rightarrow\infty$ we find the Hermite +polynomials. In fact we have +$$\lim_{a\rightarrow\infty} +(2a)^{\frac{1}{2}n}C_n((2a)^{\frac{1}{2}}x+a;a)=(-1)^nH_n(x).$$ + +\subsection*{Remarks} +The Hermite polynomials can also be written as: +$$\frac{H_n(x)}{n!}=\sum_{k=0}^{\lfloor n/2\rfloor} +\frac{(-1)^k(2x)^{n-2k}}{k!\,(n-2k)!},$$ +where $\lfloor n/2\rfloor$ denotes the largest integer smaller than or equal to $n/2$. + +\noindent +The Hermite polynomials and the Laguerre polynomials given by (\ref{DefLaguerre}) are also +connected by the following quadratic transformations: +$$H_{2n}(x)=(-1)^nn!\,2^{2n}L_n^{(-\frac{1}{2})}(x^2)$$ +and +$$H_{2n+1}(x)=(-1)^nn!\,2^{2n+1}xL_n^{(\frac{1}{2})}(x^2).$$ +% RS: add begin\label{sec9.14} +% +\paragraph{\large\bf KLSadd: Hypergeometric representation}\begin{align} +C_n(x;a)&=\hyp20{-n,-x}-{-\,\frac1a} +\label{179}\\ +&=\frac{(-x)_n}{a^n} \hyp11{-n}{x-n+1}a +\label{180}\\ +&=\frac{n!}{(-a)^n}\,L_n^{x-n}(a), +\label{181} +\end{align} +where $L_n^\al(x)$ is a +\hyperref[sec9.12]{Laguerre polynomial}. +Formula \eqref{179} is (9.14.1). Then \eqref{180} follows by reversal +of the summation. Finally \eqref{181} follows by \eqref{180} and +(9.12.1). It is also the Remark on p.249 of \mycite{KLS}, and it +was earlier given in \myciteKLS{416}{(2.7.10)}. +% +\paragraph{\large\bf KLSadd: Uniqueness of orthogonality measure}The coefficient of $p_{n-1}(x)$ in (9.14.4) behaves as $O(n)$ as $n\to\iy$. +Hence \eqref{93} holds, by which the orthogonality measure is unique. +% +% RS: add end +\subsection*{References} +\label{sec9.15} +% +\paragraph{\large\bf KLSadd: Uniqueness of orthogonality measure}The coefficient of $p_{n-1}(x)$ in (9.15.4) behaves as $O(n)$ as $n\to\iy$. +Hence \eqref{93} holds, by which the orthogonality measure is unique. +% +\paragraph{\large\bf KLSadd: Fourier transforms}\begin{equation} +\frac1{\sqrt{2\pi}}\,\int_{-\iy}^\iy H_n(y)\,e^{-\half y^2}\,e^{ixy}\,dy= +i^n\,H_n(x)\,e^{-\half x^2}, +\label{15} +\end{equation} +see \mycite{AAR}{(6.1.15) and Exercise 6.11}. +\begin{equation} +\frac1{\sqrt\pi}\,\int_{-\iy}^\iy H_n(y)\,e^{-y^2}\,e^{ixy}\,dy= +i^n\,x^n\,e^{-\frac14 x^2}, +\label{16} +\end{equation} +see \mycite{DLMF}{(18.17.35)}. +\begin{equation} +\frac{i^n}{2\sqrt\pi}\,\int_{-\iy}^\iy y^n\,e^{-\frac14 y^2}\,e^{-ixy}\,dy= +H_n(x)\,e^{-x^2}, +\label{17} +\end{equation} +see \mycite{AAR}{(6.1.4)}. +% +\cite{Abram}, \cite{NAlSalam66}, \cite{AlSalam90}, \cite{AlSalamChihara72}, +\cite{AlSalamChihara76}, \cite{AndrewsAskey85}, \cite{AndrewsAskeyRoy}, \cite{Area+I}, +\cite{Askey68}, \cite{Askey75}, \cite{Askey89I}, \cite{AskeyGasper76}, \cite{AskeyWilson85}, +\cite{Azor}, \cite{Berg}, \cite{BilodeauII}, \cite{Brafman51}, \cite{Brafman57II}, +\cite{Brenke}, \cite{CarlitzSrivastava}, \cite{ChenSrivastava}, \cite{Chihara78}, +\cite{Cohen}, \cite{Danese}, \cite{DetteStudden92}, \cite{DetteStudden95}, \cite{Doha2004I}, +\cite{Erdelyi+}, \cite{Faldey}, \cite{Gawronski87}, \cite{Gawronski93}, \cite{Gillis+}, +\cite{Godoy+}, \cite{Grad}, \cite{Ismail74}, \cite{Ismail2005II}, \cite{IsmailStanton97}, +\cite{Koekoek2000}, \cite{Koorn88}, \cite{Krasikov2004}, \cite{Kwon+}, \cite{LabelleYehIII}, +\cite{Lesky95II}, \cite{Lesky96}, \cite{LewanowiczI}, \cite{LopezTemme99}, \cite{Mathai}, +\cite{Meixner}, \cite{Nikiforov+}, \cite{NikiforovUvarov}, \cite{Olver}, \cite{Pittaluga}, +\cite{Rainville}, \cite{SainteViennot}, \cite{Srivastava71}, \cite{SrivastavaMathur}, +\cite{Szego75}, \cite{Temme}, \cite{TemmeLopez2000}, \cite{Trickovic}, \cite{Viennot}, +\cite{Weisner59}, \cite{Wyman}. + +\end{document} diff --git a/KLSadd_insertion/newtempchap14.tex b/KLSadd_insertion/newtempchap14.tex new file mode 100644 index 0000000..880a78e --- /dev/null +++ b/KLSadd_insertion/newtempchap14.tex @@ -0,0 +1,6695 @@ +\documentclass[envcountchap,graybox]{svmono} + +\addtolength{\textwidth}{1mm} + +\usepackage{amsmath,amssymb} + +\usepackage{amsfonts} +%\usepackage{breqn} +\usepackage{DLMFmath} +\usepackage{DRMFfcns} + +\usepackage{mathptmx} +\usepackage{helvet} +\usepackage{courier} + +\usepackage{makeidx} +\usepackage{graphicx} + +\usepackage{multicol} +\usepackage[bottom]{footmisc} + +\usepackage[pdftex]{hyperref} +\usepackage {xparse} +\usepackage{cite} +\makeindex + +\def\bibname{Bibliography} +\def\refname{Bibliography} + +\def\theequation{\thesection.\arabic{equation}} + +\smartqed + +\let\corollary=\undefined +\spnewtheorem{corollary}[theorem]{Corollary}{\bfseries}{\itshape} + +\newcounter{rom} + +\newcommand{\hyp}[5]{\mbox{}_{#1}{F}_{#2} +\left(\genfrac{}{}{0pt}{}{#3}{#4}\,;\,#5\right)} + +\newcommand{\qhyp}[5]{\mbox{}_{#1}{\phi}_{#2} +\left(\genfrac{}{}{0pt}{}{#3}{#4}\,;\,q,\,#5\right)} + +\newcommand{\qhypK}[5]{\,\mbox{}_{#1}\phi_{#2}\!\left( + \genfrac{}{}{0pt}{}{#3}{#4};#5\right)} + +\newcommand{\mathindent}{\hspace{7.5mm}} + +\newcommand{\e}{\textrm{e}} + +\renewcommand{\E}{\textrm{E}} + +\renewcommand{\textfraction}{-1} + +\renewcommand{\Gamma}{\varGamma} + +\renewcommand{\leftlegendglue}{\hfil} + +\settowidth{\tocchpnum}{14\enspace} +\settowidth{\tocsecnum}{14.30\enspace} +\settowidth{\tocsubsecnum}{14.12.1\enspace} + +\makeatletter +\def\cleartoversopage{\clearpage\if@twoside\ifodd\c@page + \hbox{}\newpage\if@twocolumn\hbox{}\newpage\fi + \else\fi\fi} + +\newcommand{\clearemptyversopage}{ + \clearpage{\pagestyle{empty}\cleartoversopage}} +\makeatother + +\oddsidemargin -1.5cm +\topmargin -2.0cm +\textwidth 16.3cm +\newcommand\sa{\smallskipamount} +\newcommand\sLP{\\[\sa]} +\newcommand\sPP{\\[\sa]\indent} +\newcommand\ba{\bigskipamount} +\newcommand\bLP{\\[\ba]} +\newcommand\CC{\mathbb{C}} +\newcommand\RR{\mathbb{R}} +\newcommand\ZZ{\mathbb{Z}} +\newcommand\al\alpha +\newcommand\be\beta +\newcommand\ga\gamma +\newcommand\de\delta +\newcommand\tha\theta +\newcommand\la\lambda +\newcommand\om\omega +\newcommand\Ga{\Gamma} +\newcommand\half{\frac12} +\newcommand\thalf{\tfrac12} +\newcommand\iy\infty +\newcommand\wt{\widetilde} +\newcommand\const{{\rm const.}\,} +\newcommand\Zpos{\ZZ_{>0}} +\newcommand\Znonneg{\ZZ_{\ge0}} +\newcommand{\hyp}[5]{\,\mbox{}_{#1}F_{#2}\!\left( + \genfrac{}{}{0pt}{}{#3}{#4};#5\right)} +\newcommand{\qhyp}[5]{\,\mbox{}_{#1}\phi_{#2}\!\left( + \genfrac{}{}{0pt}{}{#3}{#4};#5\right)} +\newcommand\LHS{left-hand side} +\newcommand\RHS{right-hand side} +\renewcommand\Re{{\rm Re}\,} +\renewcommand\Im{{\rm Im}\,} + +\NewDocumentCommand\mycite{m g}{% + \IfNoValueTF{#2} + {[\hyperlink{#1}{#1}]} + {[\hyperlink{#1}{#1}, #2]}% +} +\newcommand\mybibitem[1]{\bibitem[#1]{#1}\hypertarget{#1}{}} +\textheight 25cm + +\begin{document} + +\author{Roelof Koekoek\\[2.5mm]Peter A. Lesky\\[2.5mm]Ren\'e F. Swarttouw} +\title{Hypergeometric orthogonal polynomials and their\\$q$-analogues} +\subtitle{-- Monograph --} +\maketitle + +\frontmatter + +\large + +\pagenumbering{roman} +\addtocounter{chapter}{13} + +\chapter{Basic hypergeometric orthogonal polynomials} +\label{BasicHyperOrtPol} + + +In this chapter we deal with all families of basic hypergeometric orthogonal polynomials +appearing in the $q$-analogue of the Askey scheme on the pages~\pageref{qscheme1} and +\pageref{qscheme2}. For each family of orthogonal polynomials we state the most important +properties such as a representation as a basic hypergeometric function, orthogonality +relation(s), the three-term recurrence relation, the second-order $q$-difference equation, +the forward shift (or degree lowering) and backward shift (or degree raising) operator, a +Rodrigues-type formula and some generating functions. Throughout this chapter we assume +that $01$ and $b,c,d$ are real or one is real and the other two are complex conjugates,\\ +$\max(|b|,|c|,|d|)<1$ and the pairwise products of $a,b,c$ and $d$ have +absolute value less than $1$, then we have another orthogonality relation +given by: +\begin{eqnarray} +\label{OrtAskeyWilsonII} +& &\frac{1}{2\pi}\int_{-1}^1\frac{w(x)}{\sqrt{1-x^2}}p_m(x;a,b,c,d|q)p_n(x;a,b,c,d|q)\,dx\nonumber\\ +& &{}\mathindent{}+\sum_{\begin{array}{c}\scriptstyle k\\ \scriptstyle 11$ and $b$ and $c$ are real or complex conjugates, +$\max(|b|,|c|)<1$ and the pairwise products of $a,b$ and $c$ have +absolute value less than $1$, then we have another orthogonality relation +given by: +\begin{eqnarray} +\label{OrtContDualqHahnII} +& &\frac{1}{2\pi}\int_{-1}^1\frac{w(x)}{\sqrt{1-x^2}}p_m(x;a,b,c|q)p_n(x;a,b,c|q)\,dx\nonumber\\ +& &{}\mathindent{}+\sum_{\begin{array}{c}\scriptstyle k\\ \scriptstyle 10$) in the definition (\ref{DefBigqJacobi}) +of the big $q$-Jacobi polynomials and take the limit $c\rightarrow -\infty$ we +obtain the $q$-Meixner polynomials given by (\ref{DefqMeixner}): +\begin{equation} +\lim_{c\rightarrow -\infty}P_n(q^{-x};a,-a^{-1}cd^{-1},c;q)=M_n(q^{-x};a,d;q). +\end{equation} + +\subsubsection*{Big $q$-Jacobi $\rightarrow$ Jacobi} +If we set $c=0$, $a=q^{\alpha}$ and $b=q^{\beta}$ in the definition (\ref{DefBigqJacobi}) +of the big $q$-Jacobi polynomials and let $q\rightarrow 1$ we find the Jacobi +polynomials given by (\ref{DefJacobi}): +\begin{equation} +\lim_{q\rightarrow 1}P_n(x;q^{\alpha},q^{\beta},0;q)=\frac{P_n^{(\alpha,\beta)}(2x-1)}{P_n^{(\alpha,\beta)}(1)}. +\end{equation} +If we take $c=-q^{\gamma}$ for arbitrary real $\gamma$ instead of $c=0$ we +find +\begin{equation} +\lim_{q\rightarrow 1}P_n(x;q^{\alpha},q^{\beta},-q^{\gamma};q)=\frac{P_n^{(\alpha,\beta)}(x)}{P_n^{(\alpha,\beta)}(1)}. +\end{equation} + +\subsection*{Remarks} +The big $q$-Jacobi polynomials with $c=0$ and the little +$q$-Jacobi polynomials given by (\ref{DefLittleqJacobi}) are related +in the following way: +$$P_n(x;a,b,0;q)=\frac{(bq;q)_n}{(aq;q)_n}(-1)^na^nq^{n+\binom{n}{2}}p_n(a^{-1}q^{-1}x;b,a|q).$$ + +\noindent +Sometimes the big $q$-Jacobi polynomials are defined in terms of four +parameters instead of three. In fact the polynomials given by the definition +$$P_n(x;a,b,c,d;q)=\qhyp{3}{2}{q^{-n},abq^{n+1},ac^{-1}qx}{aq,-ac^{-1}dq}{q}$$ +are orthogonal on the interval $[-d,c]$ with respect to the weight function +$$\frac{(c^{-1}qx,-d^{-1}qx;q)_{\infty}}{(ac^{-1}qx,-bd^{-1}qx;q)_{\infty}}d_qx.$$ +These polynomials are not really different from those given by +(\ref{DefBigqJacobi}) since we have +$$P_n(x;a,b,c,d;q)=P_n(ac^{-1}qx;a,b,-ac^{-1}d;q)$$ +and +$$P_n(x;a,b,c;q)=P_n(x;a,b,aq,-cq;q).$$ + RS: add begin\label{sec14.4} +The continuous $q$-Hahn polynomials are the special case +of Askey-Wilson polynomials with parameters +$a e^{i\phi},b e^{i\phi},a e^{-i\phi},b e^{-i\phi}$: +\[ +p_n(x;a,b,\phi\,|\, q):= +p_n(x;a e^{i\phi},b e^{i\phi},a e^{-i\phi},b e^{-i\phi}\,|\, q). +\] +In \myciteKLS{72}{(4.29)} and \mycite{GR}{(7.5.43)} +(who write $p_n(x;a,b\,|\,q)$, $x=\cos(\tha+\phi)$) +and in \mycite{KLS}{\S14.4} (who writes $p_n(x;a,b,c,d;q)$, +$x=\cos(\tha+\phi)$) +the parameter +dependence on $\phi$ is incorrectly omitted. + +Since all formulas in \S14.4 are specializations of formulas in \S14.1, +there is no real need to give these specializations explicitly. +In particular, the limit (14.4.15) is in fact a limit from Askey-Wilson to +continuous $q$-Hahn. See also \eqref{177}. +% + +\subsection*{References} +\cite{NAlSalam89}, \cite{AlSalam90}, \cite{AndrewsAskey85}, \cite{AtakKlimyk2004}, +\cite{AtakRahmanSuslov}, \cite{DattaGriffin}, \cite{FloreaniniVinetII}, +\cite{GasperRahman90}, \cite{GrunbaumHaine96}, \cite{Gupta92}, \cite{Hahn}, +\cite{Ismail86I}, \cite{IsmailStanton97}, \cite{IsmailWilson}, \cite{KalninsMiller88}, +\cite{KoelinkE}, \cite{Koorn90II}, \cite{Koorn93}, \cite{Koorn2007}, \cite{Miller89}, +\cite{Nikiforov+}, \cite{NoumiMimachi90II}, \cite{NoumiMimachi90III}, +\cite{NoumiMimachi91}, \cite{Spiridonov97}, \cite{SrivastavaJain90}. + + +\section*{Special case} + +\subsection{Big $q$-Legendre} +\index{Big q-Legendre polynomials@Big $q$-Legendre polynomials} +\index{q-Legendre polynomials@$q$-Legendre polynomials!Big} +\par + +\subsection*{Basic hypergeometric representation} The big $q$-Legendre polynomials are big $q$-Jacobi polynomials +with $a=b=1$: +\begin{equation} +\label{DefBigqLegendre} +P_n(x;c;q)=\qhyp{3}{2}{q^{-n},q^{n+1},x}{q,cq}{q}. +\end{equation} + +\subsection*{Orthogonality relation} +\begin{eqnarray} +\label{OrtBigqLegendre} +& &\int_{cq}^{q}P_m(x;c;q)P_n(x;c;q)\,d_qx\nonumber\\ +& &{}=q(1-c)\frac{(1-q)}{(1-q^{2n+1})} +\frac{(c^{-1}q;q)_n}{(cq;q)_n}(-cq^2)^nq^{\binom{n}{2}}\,\delta_{mn},\quad c<0. +\end{eqnarray} + +\subsection*{Recurrence relation} +\begin{eqnarray} +\label{RecBigqLegendre} +(x-1)P_n(x;c;q)&=&A_nP_{n+1}(x;c;q)-\left(A_n+C_n\right)P_n(x;c;q)\nonumber\\ +& &{}\mathindent{}+C_nP_{n-1}(x;c;q), +\end{eqnarray} +where +$$\left\{\begin{array}{l} +\displaystyle A_n=\frac{(1-q^{n+1})(1-cq^{n+1})}{(1+q^{n+1})(1-q^{2n+1})}\\ +\\ +\displaystyle C_n=-cq^{n+1}\frac{(1-q^n)(1-c^{-1}q^n)}{(1+q^n)(1-q^{2n+1})}. +\end{array}\right.$$ + +\subsection*{Normalized recurrence relation} +\begin{equation} +\label{NormRecBigqLegendre} +xp_n(x)=p_{n+1}(x)+\left[1-(A_n+C_n)\right]p_n(x)+A_{n-1}C_np_{n-1}(x), +\end{equation} +where +$$P_n(x;c;q)=\frac{(q^{n+1};q)_n}{(q,cq;q)_n}p_n(x).$$ + +\subsection*{$q$-Difference equation} +\begin{eqnarray} +\label{dvBigqLegendre} +& &q^{-n}(1-q^n)(1-q^{n+1})x^2y(x)\nonumber\\ +& &{}=B(x)y(qx)-\left[B(x)+D(x)\right]y(x)+D(x)y(q^{-1}x), +\end{eqnarray} +where +$$y(x)=P_n(x;c;q)$$ +and +$$\left\{\begin{array}{l}\displaystyle B(x)=q(x-1)(x-c)\\ +\\ +\displaystyle D(x)=(x-q)(x-cq).\end{array}\right.$$ + +\subsection*{Rodrigues-type formula} +\begin{eqnarray} +\label{RodBigqLegendre} +P_n(x;c;q)&=&\frac{c^nq^{n(n+1)}(1-q)^n}{(q,cq;q)_n} +\left(\mathcal{D}_q\right)^n\left[(q^{-n}x,c^{-1}q^{-n}x;q)_n\right]\\ +&=&\frac{(1-q)^n}{(q,cq;q)_n}\left(\mathcal{D}_q\right)^n +\left[(qx^{-1},cqx^{-1};q)_nx^{2n}\right].\nonumber +\end{eqnarray} + +\subsection*{Generating functions} +\begin{equation} +\label{GenBigqLegendre1} +\qhyp{2}{1}{qx^{-1},0}{q}{xt}\,\qhyp{1}{1}{c^{-1}x}{q}{cqt} +=\sum_{n=0}^{\infty}\frac{(cq;q)_n}{(q,q;q)_n}P_n(x;c;q)t^n. +\end{equation} + +\begin{equation} +\label{GenBigqLegendre2} +\qhyp{2}{1}{cqx^{-1},0}{cq}{xt}\,\qhyp{1}{1}{c^{-1}x}{c^{-1}q}{qt} +=\sum_{n=0}^{\infty}\frac{P_n(x;c;q)}{(c^{-1}q;q)_n}t^n. +\end{equation} + +\subsection*{Limit relations} + +\subsubsection*{Big $q$-Legendre $\rightarrow$ Legendre / Spherical} +If we set $c=0$ in the definition (\ref{DefBigqLegendre}) of the big +$q$-Legendre polynomials and let $q\rightarrow 1$ we simply obtain the Legendre +(or spherical) polynomials given by (\ref{DefLegendre}): +\begin{equation} +\lim_{q\rightarrow 1}P_n(x;0;q)=P_n(2x-1). +\end{equation} +If we take $c=-q^{\gamma}$ for arbitrary real $\gamma$ instead of $c=0$ we +find +\begin{equation} +\lim_{q\rightarrow 1}P_n(x;-q^{\gamma};q)=P_n(x). +\end{equation} + +\subsection*{References} +\cite{Koelink95I}, \cite{Koorn90II}. + + +\section{$q$-Hahn}\index{q-Hahn polynomials@$q$-Hahn polynomials} +\par\setcounter{equation}{0} + +\subsection*{Basic hypergeometric representation} +\begin{equation} +\label{DefqHahn} +Q_n(q^{-x};\alpha,\beta,N|q)=\qhyp{3}{2}{q^{-n},\alpha\beta q^{n+1},q^{-x}} +{\alpha q,q^{-N}}{q},\quad n=0,1,2,\ldots,N. +\end{equation} + +\subsection*{Orthogonality relation} +For $0<\alpha q<1$ and $0<\beta q<1$, or for $\alpha>q^{-N}$ and $\beta>q^{-N}$, we have +\begin{eqnarray} +\label{OrtqHahn} +& &\sum_{x=0}^N\frac{(\alpha q,q^{-N};q)_x}{(q,\beta^{-1}q^{-N};q)_x}(\alpha\beta q)^{-x} +Q_m(q^{-x};\alpha,\beta,N|q)Q_n(q^{-x};\alpha,\beta,N|q)\nonumber\\ +& &{}=\frac{(\alpha\beta q^2;q)_N}{(\beta q;q)_N(\alpha q)^N} +\frac{(q,\alpha\beta q^{N+2},\beta q;q)_n}{(\alpha q,\alpha\beta q,q^{-N};q)_n}\, +\frac{(1-\alpha\beta q)(-\alpha q)^n}{(1-\alpha\beta q^{2n+1})} +q^{\binom{n}{2}-Nn}\,\delta_{mn}. +\end{eqnarray} + +\newpage + +\subsection*{Recurrence relation} +\begin{eqnarray} +\label{RecqHahn} +-\left(1-q^{-x}\right)Q_n(q^{-x})&=&A_nQ_{n+1}(q^{-x})-\left(A_n+C_n\right)Q_n(q^{-x})\nonumber\\ +& &{}\mathindent{}+C_nQ_{n-1}(q^{-x}), +\end{eqnarray} +where +$$Q_n(q^{-x}):=Q_n(q^{-x};\alpha,\beta,N|q)$$ +and +$$\left\{\begin{array}{l} +\displaystyle A_n=\frac{(1-q^{n-N})(1-\alpha q^{n+1})(1-\alpha\beta q^{n+1})}{(1-\alpha\beta q^{2n+1})(1-\alpha\beta q^{2n+2})}\\ +\\ +\displaystyle C_n=-\frac{\alpha q^{n-N}(1-q^n)(1-\alpha\beta q^{n+N+1})(1-\beta q^n)}{(1-\alpha\beta q^{2n})(1-\alpha\beta q^{2n+1})}. +\end{array}\right.$$ + +\subsection*{Normalized recurrence relation} +\begin{equation} +\label{NormRecqHahn} +xp_n(x)=p_{n+1}(x)+\left[1-(A_n+C_n)\right]p_n(x)+A_{n-1}C_np_{n-1}(x), +\end{equation} +where +$$Q_n(q^{-x};\alpha,\beta,N|q)= +\frac{(\alpha\beta q^{n+1};q)_n}{(\alpha q,q^{-N};q)_n}p_n(q^{-x}).$$ + +\subsection*{$q$-Difference equation} +\begin{eqnarray} +\label{dvqHahn} +& &q^{-n}(1-q^n)(1-\alpha\beta q^{n+1})y(x)\nonumber\\ +& &{}=B(x)y(x+1)-\left[B(x)+D(x)\right]y(x)+D(x)y(x-1), +\end{eqnarray} +where +$$y(x)=Q_n(q^{-x};\alpha,\beta,N|q)$$ +and +$$\left\{\begin{array}{l}\displaystyle B(x)=(1-q^{x-N})(1-\alpha q^{x+1})\\ +\\ +\displaystyle D(x)=\alpha q(1-q^x)(\beta-q^{x-N-1}).\end{array}\right.$$ + +\newpage + +\subsection*{Forward shift operator} +\begin{eqnarray} +\label{shift1qHahnI} +& &Q_n(q^{-x-1};\alpha,\beta,N|q)-Q_n(q^{-x};\alpha,\beta,N|q)\nonumber\\ +& &{}=\frac{q^{-n-x}(1-q^n)(1-\alpha\beta q^{n+1})}{(1-\alpha q)(1-q^{-N})} +Q_{n-1}(q^{-x};\alpha q,\beta q,N-1|q) +\end{eqnarray} +or equivalently +\begin{eqnarray} +\label{shift1qHahnII} +\frac{\Delta Q_n(q^{-x};\alpha,\beta,N|q)}{\Delta q^{-x}} +&=&\frac{q^{-n+1}(1-q^n)(1-\alpha\beta q^{n+1})}{(1-q)(1-\alpha q)(1-q^{-N})}\nonumber\\ +& &{}\mathindent{}\times Q_{n-1}(q^{-x};\alpha q,\beta q,N-1|q). +\end{eqnarray} + +\subsection*{Backward shift operator} +\begin{eqnarray} +\label{shift2qHahnI} +& &(1-\alpha q^x)(1-q^{x-N-1})Q_n(q^{-x};\alpha,\beta,N|q)\nonumber\\ +& &{}\mathindent{}-\alpha(1-q^x)(\beta-q^{x-N-1})Q_n(q^{-x+1};\alpha,\beta,N|q)\nonumber\\ +& &{}=q^x(1-\alpha)(1-q^{-N-1})Q_{n+1}(q^{-x};\alpha q^{-1},\beta q^{-1},N+1|q) +\end{eqnarray} +or equivalently +\begin{eqnarray} +\label{shift2qHahnII} +& &\frac{\nabla\left[w(x;\alpha,\beta,N|q)Q_n(q^{-x};\alpha,\beta,N|q)\right]}{\nabla q^{-x}}\nonumber\\ +& &{}=\frac{1}{1-q}w(x;\alpha q^{-1},\beta q^{-1},N+1|q)Q_{n+1}(q^{-x};\alpha q^{-1},\beta q^{-1},N+1|q), +\end{eqnarray} +where +$$w(x;\alpha,\beta,N|q)=\frac{(\alpha q,q^{-N};q)_x}{(q,\beta^{-1}q^{-N};q)_x}(\alpha\beta)^{-x}.$$ + +\subsection*{Rodrigues-type formula} +\begin{eqnarray} +\label{RodqHahn} +& &w(x;\alpha,\beta,N|q)Q_n(q^{-x};\alpha,\beta,N|q)\nonumber\\ +& &{}=(1-q)^n\left(\nabla_q\right)^n\left[w(x;\alpha q^n,\beta q^n,N-n|q)\right], +\end{eqnarray} +where +$$\nabla_q:=\frac{\nabla}{\nabla q^{-x}}.$$ + +\subsection*{Generating functions} For $x=0,1,2,\ldots,N$ we have +\begin{eqnarray} +\label{GenqHahn1} +& &\qhyp{1}{1}{q^{-x}}{\alpha q}{\alpha qt}\,\qhyp{2}{1}{q^{x-N},0}{\beta q}{q^{-x}t}\nonumber\\ +& &{}=\sum_{n=0}^N\frac{(q^{-N};q)_n}{(\beta q,q;q)_n}Q_n(q^{-x};\alpha,\beta,N|q)t^n. +\end{eqnarray} + +\begin{eqnarray} +\label{GenqHahn2} +& &\qhyp{2}{1}{q^{-x},\beta q^{N+1-x}}{0}{-\alpha q^{x-N+1}t}\, +\qhyp{2}{0}{q^{x-N},\alpha q^{x+1}}{-}{-q^{-x}t}\nonumber\\ +& &{}=\sum_{n=0}^N\frac{(\alpha q,q^{-N};q)_n}{(q;q)_n}q^{-\binom{n}{2}}Q_n(q^{-x};\alpha,\beta,N|q)t^n. +\end{eqnarray} + +\subsection*{Limit relations} + +\subsubsection*{$q$-Racah $\rightarrow$ $q$-Hahn} +The $q$-Hahn polynomials follow from the $q$-Racah polynomials by the substitution +$\delta=0$ and $\gamma q=q^{-N}$ in the definition (\ref{DefqRacah}) of the +$q$-Racah polynomials: +$$R_n(\mu(x);\alpha,\beta,q^{-N-1},0|q)=Q_n(q^{-x};\alpha,\beta,N|q).$$ +Another way to obtain the $q$-Hahn polynomials from the $q$-Racah +polynomials is by setting $\gamma=0$ and $\delta=\beta^{-1}q^{-N-1}$ in the definition +(\ref{DefqRacah}): +$$R_n(\mu(x);\alpha,\beta,0,\beta^{-1}q^{-N-1}|q)=Q_n(q^{-x};\alpha,\beta,N|q).$$ +And if we take $\alpha q=q^{-N}$, $\beta\rightarrow\beta\gamma q^{N+1}$ and $\delta=0$ in the +definition (\ref{DefqRacah}) of the $q$-Racah polynomials we find the +$q$-Hahn polynomials given by (\ref{DefqHahn}) in the following way: +$$R_n(\mu(x);q^{-N-1},\beta\gamma q^{N+1},\gamma,0|q)=Q_n(q^{-x};\gamma,\beta,N|q).$$ +Note that $\mu(x)=q^{-x}$ in each case. + +\subsubsection*{$q$-Hahn $\rightarrow$ Little $q$-Jacobi} +If we set $x\rightarrow N-x$ in the definition (\ref{DefqHahn}) of the $q$-Hahn +polynomials and take the limit $N\rightarrow\infty$ we find the little $q$-Jacobi polynomials: +\begin{equation} +\lim _{N\rightarrow\infty} Q_n(q^{x-N};\alpha,\beta,N|q)=p_n(q^x;\alpha,\beta|q), +\end{equation} +where $p_n(q^x;\alpha,\beta|q)$ is given by (\ref{DefLittleqJacobi}). + +\subsubsection*{$q$-Hahn $\rightarrow$ $q$-Meixner} +The $q$-Meixner polynomials given by (\ref{DefqMeixner}) can be obtained from the $q$-Hahn polynomials +by setting $\alpha=b$ and $\beta=-b^{-1}c^{-1}q^{-N-1}$ in the definition (\ref{DefqHahn}) of the +$q$-Hahn polynomials and letting $N\rightarrow\infty$: +\begin{equation} +\lim_{N\rightarrow\infty} +Q_n(q^{-x};b,-b^{-1}c^{-1}q^{-N-1},N|q)=M_n(q^{-x};b,c;q). +\end{equation} + +\subsubsection*{$q$-Hahn $\rightarrow$ Quantum $q$-Krawtchouk} +The quantum $q$-Krawtchouk polynomials given by (\ref{DefQuantumqKrawtchouk}) +simply follow from the $q$-Hahn polynomials by setting $\beta=p$ in the definition (\ref{DefqHahn}) of +the $q$-Hahn polynomials and taking the limit $\alpha\rightarrow\infty$: +\begin{equation} +\lim_{\alpha\rightarrow\infty}Q_n(q^{-x};\alpha,p,N|q)=K_n^{qtm}(q^{-x};p,N;q). +\end{equation} + +\subsubsection*{$q$-Hahn $\rightarrow$ $q$-Krawtchouk} +If we set $\beta=-\alpha^{-1}q^{-1}p$ in the definition (\ref{DefqHahn}) of the $q$-Hahn polynomials +and then let $\alpha\rightarrow 0$ we obtain the $q$-Krawtchouk polynomials given by +(\ref{DefqKrawtchouk}): +\begin{equation} +\lim_{\alpha\rightarrow 0} +Q_n(q^{-x};\alpha,-\alpha^{-1}q^{-1}p,N|q)=K_n(q^{-x};p,N;q). +\end{equation} + +\subsubsection*{$q$-Hahn $\rightarrow$ Affine $q$-Krawtchouk} +The affine $q$-Krawtchouk polynomials given by (\ref{DefAffqKrawtchouk}) +can be obtained from the $q$-Hahn polynomials by the substitution $\alpha=p$ and +$\beta=0$ in (\ref{DefqHahn}): +\begin{equation} +Q_n(q^{-x};p,0,N|q)=K_n^{Aff}(q^{-x};p,N;q). +\end{equation} + +\subsubsection*{$q$-Hahn $\rightarrow$ Hahn} +The Hahn polynomials given by (\ref{DefHahn}) simply follow from the $q$-Hahn +polynomials given by (\ref{DefqHahn}), after setting $\alpha\rightarrow q^{\alpha}$ +and $\beta\rightarrow q^{\beta}$, in the following way: +\begin{equation} +\lim_{q\rightarrow 1}Q_n(q^{-x};q^{\alpha},q^{\beta},N|q)=Q_n(x;\alpha,\beta,N). +\end{equation} + +\subsection*{Remark} +The $q$-Hahn polynomials given by (\ref{DefqHahn}) and the dual $q$-Hahn +polynomials given by (\ref{DefDualqHahn}) are related in the following way: +$$Q_n(q^{-x};\alpha,\beta,N|q)=R_x(\mu(n);\alpha,\beta,N|q),$$ +with +$$\mu(n)=q^{-n}+\alpha\beta q^{n+1}$$ +or +$$R_n(\mu(x);\gamma,\delta,N|q)=Q_x(q^{-n};\gamma,\delta,N|q),$$ +where +$$\mu(x)=q^{-x}+\gamma\delta q^{x+1}.$$ + +\subsection*{References} +\cite{AlSalam90}, \cite{AlvarezRonveaux}, \cite{AndrewsAskey85}, +\cite{AskeyWilson79}, \cite{AskeyWilson85}, \cite{AtakRahmanSuslov}, +\cite{Dunkl78II}, \cite{Fischer95}, \cite{GasperRahman84}, +\cite{GasperRahman90}, \cite{Hahn}, \cite{KalninsMiller88}, +\cite{Koelink96I}, \cite{KoelinkKoorn}, \cite{Koorn89III}, \cite{Koorn90II}, +\cite{Nikiforov+}, \cite{NoumiMimachi90II}, \cite{Rahman82}, +\cite{Stanton80II}, \cite{Stanton84}, \cite{Stanton90}. + + +\section{Dual $q$-Hahn}\index{Dual q-Hahn polynomials@Dual $q$-Hahn polynomials} +\index{q-Hahn polynomials@$q$-Hahn polynomials!Dual} +\par\setcounter{equation}{0} + +\subsection*{Basic hypergeometric representation} +\begin{equation} +\label{DefDualqHahn} +R_n(\mu(x);\gamma,\delta,N|q)= +\qhyp{3}{2}{q^{-n},q^{-x},\gamma\delta q^{x+1}}{\gamma q,q^{-N}}{q},\quad n=0,1,2,\ldots,N, +\end{equation} +where +$$\mu(x):=q^{-x}+\gamma\delta q^{x+1}.$$ + +\newpage + +\subsection*{Orthogonality relation} +For $0<\gamma q<1$ and $0<\delta q<1$, or for $\gamma>q^{-N}$ and $\delta>q^{-N}$, we have +\begin{eqnarray} +\label{OrtDualqHahn} +& &\sum_{x=0}^N\frac{(\gamma q,\gamma\delta q,q^{-N};q)_x}{(q,\gamma\delta q^{N+2},\delta q;q)_x} +\frac{(1-\gamma\delta q^{2x+1})}{(1-\gamma\delta q)(-\gamma q)^x}q^{Nx-\binom{x}{2}}\nonumber\\ +& &{}\mathindent{}\times R_m(\mu(x);\gamma,\delta,N|q)R_n(\mu(x);\gamma,\delta,N|q)\nonumber\\ +& &{}=\frac{(\gamma\delta q^2;q)_N}{(\delta q;q)_N}(\gamma q)^{-N} +\frac{(q,\delta^{-1}q^{-N};q)_n}{(\gamma q,q^{-N};q)_n}(\gamma\delta q)^n\,\delta_{mn}. +\end{eqnarray} + +\subsection*{Recurrence relation} +\begin{eqnarray} +\label{RecDualqHahn} +& &-\left(1-q^{-x}\right)\left(1-\gamma\delta q^{x+1}\right)R_n(\mu(x))\nonumber\\ +& &{}=A_nR_{n+1}(\mu(x))-\left(A_n+C_n\right)R_n(\mu(x))+C_nR_{n-1}(\mu(x)), +\end{eqnarray} +where +$$R_n(\mu(x)):=R_n(\mu(x);\gamma,\delta,N|q)$$ +and +$$\left\{\begin{array}{l} +\displaystyle A_n=\left(1-q^{n-N}\right)\left(1-\gamma q^{n+1}\right)\\ +\\ +\displaystyle C_n=\gamma q\left(1-q^n\right)\left(\delta -q^{n-N-1}\right). +\end{array}\right.$$ + +\subsection*{Normalized recurrence relation} +\begin{eqnarray} +\label{NormRecDualqHahn} +xp_n(x)&=&p_{n+1}(x)+\left[1+\gamma\delta q-(A_n+C_n)\right]p_n(x)\nonumber\\ +& &{}\mathindent{}+\gamma q(1-q^n)(1-\gamma q^n)\nonumber\\ +& &{}\mathindent\mathindent{}\times(1-q^{n-N-1})(\delta-q^{n-N-1})p_{n-1}(x), +\end{eqnarray} +where +$$R_n(\mu(x);\gamma,\delta,N|q)=\frac{1}{(\gamma q,q^{-N};q)_n}p_n(\mu(x)).$$ + +\subsection*{$q$-Difference equation} +\begin{equation} +\label{dvDualqHahn} +q^{-n}(1-q^n)y(x)=B(x)y(x+1)-\left[B(x)+D(x)\right]y(x) ++D(x)y(x-1), +\end{equation} +where +$$y(x)=R_n(\mu(x);\gamma,\delta,N|q)$$ +and +$$\left\{\begin{array}{l}\displaystyle B(x)=\frac{(1-q^{x-N})(1-\gamma q^{x+1})(1-\gamma\delta q^{x+1})} +{(1-\gamma\delta q^{2x+1})(1-\gamma\delta q^{2x+2})}\\ +\\ +\displaystyle D(x)=-\frac{\gamma q^{x-N}(1-q^x)(1-\gamma\delta q^{x+N+1})(1-\delta q^x)} +{(1-\gamma\delta q^{2x})(1-\gamma\delta q^{2x+1})}.\end{array}\right.$$ + +\subsection*{Forward shift operator} +\begin{eqnarray} +\label{shift1DualqHahnI} +& &R_n(\mu(x+1);\gamma,\delta,N|q)-R_n(\mu(x);\gamma,\delta,N|q)\nonumber\\ +& &{}=\frac{q^{-n-x}(1-q^n)(1-\gamma\delta q^{2x+2})}{(1-\gamma q)(1-q^{-N})} +R_{n-1}(\mu(x);\gamma q,\delta,N-1|q) +\end{eqnarray} +or equivalently +\begin{eqnarray} +\label{shift1DualqHahnII} +& &\frac{\Delta R_n(\mu(x);\gamma,\delta,N|q)}{\Delta\mu(x)}\nonumber\\ +& &{}=\frac{q^{-n+1}(1-q^n)}{(1-q)(1-\gamma q)(1-q^{-N})}R_{n-1}(\mu(x);\gamma q,\delta,N-1|q). +\end{eqnarray} + +\subsection*{Backward shift operator} +\begin{eqnarray} +\label{shift2DualqHahnI} +& &(1-\gamma q^x)(1-\gamma\delta q^x)(1-q^{x-N-1})R_n(\mu(x);\gamma,\delta,N|q)\nonumber\\ +& &{}\mathindent{}+\gamma q^{x-N-1}(1-q^x)(1-\gamma\delta q^{x+N+1})(1-\delta q^x)R_n(\mu(x-1);\gamma,\delta,N|q)\nonumber\\ +& &{}=q^x(1-\gamma)(1-q^{-N-1})(1-\gamma\delta q^{2x})R_{n+1}(\mu(x);\gamma q^{-1},\delta,N+1|q) +\end{eqnarray} +or equivalently +\begin{eqnarray} +\label{shift2DualqHahnII} +& &\frac{\nabla\left[w(x;\gamma,\delta,N|q)R_n(\mu(x);\gamma,\delta,N|q)\right]}{\nabla\mu(x)}\nonumber\\ +& &{}=\frac{1}{(1-q)(1-\gamma\delta)}w(x;\gamma q^{-1},\delta,N+1|q)\nonumber\\ +& &{}\mathindent{}\times R_{n+1}(\mu(x);\gamma q^{-1},\delta,N+1|q), +\end{eqnarray} +where +$$w(x;\gamma,\delta,N|q)=\frac{(\gamma q,\gamma\delta q,q^{-N};q)_x} +{(q,\gamma\delta q^{N+2},\delta q;q)_x}(-\gamma^{-1})^x q^{Nx-\binom{x}{2}}.$$ + +\subsection*{Rodrigues-type formula} +\begin{eqnarray} +\label{RodDualqHahn} +& &w(x;\gamma,\delta,N|q)R_n(\mu(x);\gamma,\delta,N|q)\nonumber\\ +& &{}=(1-q)^n(\gamma\delta q;q)_n +\left(\nabla_{\mu}\right)^n\left[w(x;\gamma q^n,\delta,N-n|q)\right], +\end{eqnarray} +where +$$\nabla_{\mu}:=\frac{\nabla}{\nabla\mu(x)}.$$ + +\subsection*{Generating functions} For $x=0,1,2,\ldots,N$ we have +\begin{eqnarray} +\label{GenDualqHahn1} +& &(q^{-N}t;q)_{N-x}\cdot\qhyp{2}{1}{q^{-x},\delta^{-1}q^{-x}}{\gamma q}{\gamma\delta q^{x+1}t}\nonumber\\ +& &{}=\sum_{n=0}^N\frac{(q^{-N};q)_n}{(q;q)_n}R_n(\mu(x);\gamma,\delta,N|q)t^n. +\end{eqnarray} + +\begin{eqnarray} +\label{GenDualqHahn2} +& &(\gamma\delta qt;q)_x\cdot\qhyp{2}{1}{q^{x-N},\gamma q^{x+1}}{\delta^{-1}q^{-N}}{q^{-x}t}\nonumber\\ +& &{}=\sum_{n=0}^N +\frac{(q^{-N},\gamma q;q)_n}{(\delta^{-1}q^{-N},q;q)_n}R_n(\mu(x);\gamma,\delta,N|q)t^n. +\end{eqnarray} + +\subsection*{Limit relations} + +\subsubsection*{$q$-Racah $\rightarrow$ Dual $q$-Hahn} +To obtain the dual $q$-Hahn polynomials from the $q$-Racah polynomials we have to +take $\beta=0$ and $\alpha q=q^{-N}$ in (\ref{DefqRacah}): +$$R_n(\mu(x);q^{-N-1},0,\gamma,\delta|q)=R_n(\mu(x);\gamma,\delta,N|q),$$ +with +$$\mu(x)=q^{-x}+\gamma\delta q^{x+1}.$$ +We may also take $\alpha=0$ and $\beta=\delta^{-1}q^{-N-1}$ in (\ref{DefqRacah}) to obtain +the dual $q$-Hahn polynomials from the $q$-Racah polynomials: +$$R_n(\mu(x);0,\delta^{-1}q^{-N-1},\gamma,\delta|q)=R_n(\mu(x);\gamma,\delta,N|q).$$ +And if we take $\gamma q=q^{-N}$, $\delta\rightarrow\alpha\delta q^{N+1}$ and $\beta=0$ in the +definition (\ref{DefqRacah}) of the $q$-Racah polynomials we find the dual +$q$-Hahn polynomials given by (\ref{DefDualqHahn}) in the following way: +$$R_n(\mu(x);\alpha,0,q^{-N-1},\alpha\delta q^{N+1}|q)=R_n({\tilde \mu}(x);\alpha,\delta,N|q),$$ +with +$${\tilde \mu}(x)=q^{-x}+\alpha\delta q^{x+1}.$$ + +\subsubsection*{Dual $q$-Hahn $\rightarrow$ Affine $q$-Krawtchouk} +The affine $q$-Krawtchouk polynomials given by (\ref{DefAffqKrawtchouk}) +can be obtained from the dual $q$-Hahn polynomials by the substitution $\gamma=p$ +and $\delta=0$ in (\ref{DefDualqHahn}): +\begin{equation} +R_n(\mu(x);p,0,N|q)=K_n^{Aff}(q^{-x};p,N;q). +\end{equation} +Note that $\mu(x)=q^{-x}$ in this case. + +\subsubsection*{Dual $q$-Hahn $\rightarrow$ Dual $q$-Krawtchouk} +The dual $q$-Krawtchouk polynomials given by (\ref{DefDualqKrawtchouk}) can +be obtained from the dual $q$-Hahn polynomials by setting $\delta=c\gamma^{-1}q^{-N-1}$ +in (\ref{DefDualqHahn}) and letting $\gamma\rightarrow 0$: +\begin{equation} +\lim_{\gamma\rightarrow 0} +R_n(\mu(x);\gamma,c\gamma^{-1}q^{-N-1},N|q)=K_n(\lambda(x);c,N|q). +\end{equation} + +\subsubsection*{Dual $q$-Hahn $\rightarrow$ Dual Hahn} +The dual Hahn polynomials given by (\ref{DefDualHahn}) follow from the dual $q$-Hahn +polynomials by simply taking the limit $q\rightarrow 1$ in the definition (\ref{DefDualqHahn}) +of the dual $q$-Hahn polynomials after applying the substitution +$\gamma\rightarrow q^{\gamma}$ and $\delta\rightarrow q^{\delta}$: +\begin{equation} +\lim_{q\rightarrow 1}R_n(\mu(x);q^{\gamma},q^{\delta},N|q)=R_n(\lambda(x);\gamma,\delta,N), +\end{equation} +where +$$\left\{\begin{array}{l} +\displaystyle\mu(x)=q^{-x}+q^{x+\gamma+\delta+1}\\[5mm] +\displaystyle\lambda(x)=x(x+\gamma+\delta+1). +\end{array}\right.$$ + +\subsection*{Remark} +The dual $q$-Hahn polynomials given by (\ref{DefDualqHahn}) and the +$q$-Hahn polynomials given by (\ref{DefqHahn}) are related in the following way: +$$Q_n(q^{-x};\alpha,\beta,N|q)=R_x(\mu(n);\alpha,\beta,N|q),$$ +with +$$\mu(n)=q^{-n}+\alpha\beta q^{n+1}$$ +or +$$R_n(\mu(x);\gamma,\delta,N|q)=Q_x(q^{-n};\gamma,\delta,N|q),$$ +where +$$\mu(x)=q^{-x}+\gamma\delta q^{x+1}.$$ + RS: add begin\label{sec14.5} +% +\paragraph{\large\bf KLSadd: Different notation}See p.442, Remarks: +\begin{equation} +P_n(x;a,b,c,d;q):=P_n(qac^{-1}x;a,b,-ac^{-1}d;q) +=\qhyp32{q^{-n},q^{n+1}ab,qac^{-1}x}{qa,-qac^{-1}d}{q,q}. +\label{123} +\end{equation} +Furthermore, +\begin{equation} +P_n(x;a,b,c,d;q)=P_n(\la x;a,b,\la c,\la d;q), +\label{141} +\end{equation} +\begin{equation} +P_n(x;a,b,c;q)=P_n(-q^{-1}c^{-1}x;a,b,-ac^{-1},1;q) +\label{142} +\end{equation} +% +\paragraph{\large\bf KLSadd: Orthogonality relation}(equivalent to (14.5.2), see also \cite[(2.42), (2.41), (2.36), (2.35)]{K17}). +Let $c,d>0$ and either $a\in (-c/(qd),1/q)$, $b\in(-d/(cq),1/q)$ or +$a/c=-\overline b/d\notin\RR$. Then +\begin{equation} +\int_{-d}^c P_m(x;a,b,c,d;q) P_n(x;a,b,c,d;q)\, +\frac{(qx/c,-qx/d;q)_\iy}{(qax/c,-qbx/d;q)_\iy}\,d_qx=h_n\,\de_{m,n}\,, +\label{124} +\end{equation} +where +\begin{equation} +\frac{h_n}{h_0}=q^{\half n(n-1)}\left(\frac{q^2a^2d}c\right)^n\, +\frac{1-qab}{1-q^{2n+1}ab}\, +\frac{(q,qb,-qbc/d;q)_n}{(qa,qab,-qad/c;q)_n} +\label{125} +\end{equation} +and +\begin{equation} +h_0=(1-q)c\,\frac{(q,-d/c,-qc/d,q^2ab;q)_\iy} +{(qa,qb,-qbc/d,-qad/c;q)_\iy}\,. +\label{126} +\end{equation} +% +\paragraph{\large\bf KLSadd: Other hypergeometric representation and asymptotics}\begin{align} +&P_n(x;a,b,c,d;q) +=\frac{(-qbd^{-1}x;q)_n}{(-q^{-n}a^{-1}cd^{-1};q)_n}\, +\qhyp32{q^{-n},q^{-n}b^{-1},cx^{-1}}{qa,-q^{-n}b^{-1}dx^{-1}}{q,q} +\label{138}\\ +&\qquad=(qac^{-1}x)^n\,\frac{(qb,cx^{-1};q)_n}{(qa,-qac^{-1}d;q)_n}\, +\qhyp32{q^{-n},q^{-n}a^{-1},-qbd^{-1}x}{qb,q^{1-n}c^{-1}x} +{q,-q^{n+1}ac^{-1}d} +\label{132}\\ +&\qquad=(qac^{-1}x)^n\,\frac{(qb,q;q)_n}{(-qac^{-1}d;q)_n}\, +\sum_{k=0}^n\frac{(cx^{-1};q)_{n-k}}{(q,qa;q)_{n-k}}\, +\frac{(-qbd^{-1}x;q)_k}{(qb,q;q)_k}\,(-1)^k q^{\half k(k-1)}(-dx^{-1})^k. +\label{133} +\end{align} +Formula \eqref{138} follows from \eqref{123} by +\mycite{GR}{(III.11)} and next \eqref{132} follows by series inversion +\mycite{GR}{Exercise 1.4(ii)}. +Formulas \eqref{138} and \eqref{133} are also given in +\mycite{Ism}{(18.4.28), (18.4.29)}. +It follows from \eqref{132} or \eqref{133} that +(see \myciteKLS{298}{(1.17)} or \mycite{Ism}{(18.4.31)}) +\begin{equation} +\lim_{n\to\iy}(qac^{-1}x)^{-n} P_n(x;a,b,c,d;q) +=\frac{(cx^{-1},-dx^{-1};q)_\iy}{(-qac^{-1}d,qa;q)_\iy}\,, +\label{134} +\end{equation} +uniformly for $x$ in compact subsets of $\CC\backslash\{0\}$. +(Exclusion of the spectral points $x=cq^m,dq^m$ ($m=0,1,2,\ldots$), +as was done in \myciteKLS{298} and \mycite{Ism}, is not necessary. However, +while \eqref{134} yields 0 at these points, a more refined asymptotics +at these points is given in \myciteKLS{298} and \mycite{Ism}.)$\;$ +For the proof of \eqref{134} use that +\begin{equation} +\lim_{n\to\iy}(qac^{-1}x)^{-n} P_n(x;a,b,c,d;q) +=\frac{(qb,cx^{-1};q)_n}{(qa,-qac^{-1}d;q)_n}\, +\qhyp11{-qbd^{-1}x}{qb}{q,-dx^{-1}}, +\label{135} +\end{equation} +which can be evaluated by \mycite{GR}{(II.5)}. +Formula \eqref{135} follows formally from \eqref{132}, and it follows rigorously, by +dominated convergence, from \eqref{133}. +% +\paragraph{\large\bf KLSadd: Symmetry}(see \cite[\S2.5]{K17}). +\begin{equation} +\frac{P_n(-x;a,b,c,d;q)}{P_n(-d/(qb);a,b,c,d;q)} +=P_n(x;b,a,d,c;q). +\end{equation} +% +\paragraph{\large\bf KLSadd: Special values}\begin{align} +P_n(c/(qa);a,b,c,d;q)&=1,\\ +P_n(-d/(qb);a,b,c,d;q)&=\left(-\,\frac{ad}{bc}\right)^n\, +\frac{(qb,-qbc/d;q)_n}{(qa,-qad/c;q)_n}\,,\\ +P_n(c;a,b,c,d;q)&= +q^{\half n(n+1)}\left(\frac{ad}c\right)^n +\frac{(-qbc/d;q)_n}{(-qad/c;q)_n}\,,\\ +P_n(-d;a,b,c,d;q)&=q^{\half n(n+1)} (-a)^n\,\frac{(qb;q)_n}{(qa;q)_n}\,. +\end{align} +% +\paragraph{\large\bf KLSadd: Quadratic transformations}(see \cite[(2.48), (2.49)]{K17} and \eqref{128}).\\ +These express big $q$-Jacobi polynomials $P_m(x;a,a,1,1;q)$ in terms of little +$q$-Jacobi polynomials (see \S14.12). +\begin{align} +P_{2n}(x;a,a,1,1;q)&=\frac{p_n(x^2;q^{-1},a^2;q^2)}{p_n((qa)^{-2};q^{-1},a^2;q^2)}\,, +\label{130}\\ +P_{2n+1}(x;a,a,1,1;q)&=\frac{qax\,p_n(x^2;q,a^2;q^2)}{p_n((qa)^{-2};q,a^2;q^2)}\,. +\label{131} +\end{align} +Hence, by (14.12.1), \mycite{GR}{Exercise 1.4(ii)} and \eqref{128}, +\begin{align} +P_n(x;a,a,1,1;q)&=\frac{(qa^2;q^2)_n}{(qa^2;q)_n}\,(qax)^n\, +\qhyp21{q^{-n},q^{-n+1}}{q^{-2n+1}a^{-2}}{q^2,(ax)^{-2}} +\label{136}\\ +&=\frac{(q;q)_n}{(qa^2;q)_n}\,(qa)^n\, +\sum_{k=0}^{[\half n]}(-1)^k q^{k(k-1)} +\frac{(qa^2;q^2)_{n-k}}{(q^2;q^2)_k\,(q;q)_{n-2k}}\,x^{n-2k}. +\label{137} +\end{align} +% +\paragraph{\large\bf KLSadd: $q$-Chebyshev polynomials}In \eqref{123}, with $c=d=1$, the cases $a=b=q^{-\half}$ and $a=b=q^\half$ can be considered +as $q$-analogues of the Chebyshev polynomials of the first and second kind, respectively +(\S9.8.2) because of the limit (14.5.17). The quadratic relations \eqref{130}, \eqref{131} +can also be specialized to these cases. The definition of the $q$-Chebyshev polynomials +may vary by normalization and by dilation of argument. They were considered in +\cite{K18}. +By \myciteKLS{24}{p.279} and \eqref{130}, \eqref{131}, the {\em Al-Salam-Ismail polynomials} +$U_n(x;a,b)$ ($q$-dependence suppressed) in the case $a=q$ can be expressed as +$q$-Chebyshev polynomials of the second kind: +\begin{equation*} +U_n(x,q,b)=(q^{-3} b)^{\half n}\,\frac{1-q^{n+1}}{1-q}\, +P_n(b^{-\half}x;q^\half,q^\half,1,1;q). +\end{equation*} +Similarly, by \cite[(5.4), (5.1), (5.3)]{K19} and \eqref{130}, \eqref{131}, Cigler's $q$-Chebyshev +polynomials $T_n(x,s,q)$ and $U_n(x,s,q)$ +can be expressed in terms of the $q$-Chebyshev cases of \eqref{123}: +\begin{align*} +T_n(x,s,q)&=(-s)^{\half n}\,P_n((-qs)^{-\half} x;q^{-\half},q^{-\half},1,1;q),\\ +U_n(x,s,q)&=(-q^{-2}s)^{\half n}\,\frac{1-q^{n+1}}{1-q}\, +P_n((-qs)^{-\half} x;q^{\half},q^{\half},1,1;q). +\end{align*} +% +\paragraph{\large\bf KLSadd: Limit to Discrete $q$-Hermite I}\begin{equation} +\lim_{a\to0} a^{-n}\,P_n(x;a,a,1,1;q)=q^n\,h_n(x;q). +\label{139} +\end{equation} +Here $h_n(x;q)$ is given by (14.28.1). +For the proof of \eqref{139} use \eqref{138}. +% +\paragraph{\large\bf KLSadd: Pseudo big $q$-Jacobi polynomials}Let $a,b,c,d\in\CC$, $z_+>0$, $z_-<0$ such that +$\tfrac{(ax,bx;q)_\iy}{(cx,dx;q)_\iy}>0$ for $x\in z_- q^\ZZ\cup z_+ q^\ZZ$. +Then $(ab)/(qcd)>0$. Assume that $(ab)/(qcd)<1$. +Let $N$ be the largest nonnegative integer such that $q^{2N}>(ab)/(qcd)$. +Then +\begin{multline} +\int_{z_- q^\ZZ\cup z_+ q^\ZZ}P_m(cx;c/b,d/a,c/a;q)\,P_n(cx;c/b,d/a,c/a;q)\, +\frac{(ax,bx;q)_\iy}{(cx,dx;q)_\iy}\,d_qx=h_n\de_{m,n}\\ +(m,n=0,1,\ldots,N), +\label{114} +\end{multline} +where +\begin{equation} +\frac{h_n}{h_0}=(-1)^n\left(\frac{c^2}{ab}\right)^n q^{\half n(n-1)} q^{2n}\, +\frac{(q,qd/a,qd/b;q)_n}{(qcd/(ab),qc/a,qc/b;q)_n}\, +\frac{1-qcd/(ab)}{1-q^{2n+1}cd/(ab)} +\label{115} +\end{equation} +and +\begin{equation} +h_0=\int_{z_- q^\ZZ\cup z_+ q^\ZZ}\frac{(ax,bx;q)_\iy}{(cx,dx;q)_\iy}\,d_qx +=(1-q)z_+\, +\frac{(q,a/c,a/d,b/c,b/d;q)_\iy}{(ab/(qcd);q)_\iy}\, +\frac{\tha(z_-/z_+,cdz_-z_+;q)}{\tha(cz_-,dz_-,cz_+,dz_+;q)}\,. +\label{116} +\end{equation} +See Groenevelt \& Koelink \cite[Prop.~2.2]{K14}. +Formula \eqref{116} was first given by Slater \cite[(5)]{K15} as an evaluation +of a sum of two ${}_2\psi_2$ series. +The same formula is given in Slater \myciteKLS{471}{(7.2.6)} and in +\mycite{GR}{Exercise 5.10}, but in both cases with the same slight error, +see \cite[2nd paragraph after Lemma 2.1]{K14} for correction. +The theta function is given by \eqref{117}. +Note that +\begin{equation} +P_n(cx;c/b,d/a,c/a;q)=P_n(-q^{-1}ax;c/b,d/a,-a/b,1;q). +\label{145} +\end{equation} + +In \cite{K29} the weights of the pseudo big $q$-Jacobi polynomials +occur in certain measures on the space of $N$-point configurations +on the so-called extended Gelfand-Tsetlin graph. +% +\subsubsection*{Limit relations} +\paragraph{\large\bf KLSadd: Pseudo big $q$-Jacobi $\longrightarrow$ Discrete Hermite II}\begin{equation} +\lim_{a\to\iy}i^n q^{\half n(n-1)} P_n(q^{-1}a^{-1}ix;a,a,1,1;q)= +\wt h_n(x;q). +\label{144} +\end{equation} +For the proof use \eqref{137} and \eqref{143}. +Note that $P_n(q^{-1}a^{-1}ix;a,a,1,1;q)$ is obtained from the +\RHS\ of \eqref{144} by replacing $a,b,c,d$ by $-ia^{-1},ia^{-1},i,-i$. +% +\paragraph{\large\bf KLSadd: Pseudo big $q$-Jacobi $\longrightarrow$ Pseudo Jacobi}\begin{equation} +\lim_{q\uparrow1}P_n(iq^{\half(-N-1+i\nu)}x;-q^{-N-1},-q^{-N-1},q^{-N+i\nu-1};q) +=\frac{P_n(x;\nu,N)}{P_n(-i;\nu,N)}\,. +\label{118} +\end{equation} +Here the big $q$-Jacobi polynomial on the \LHS\ equals +$P_n(cx;c/b,d/a,c/a;q)$ with\\ +$a=iq^{\half(N+1-i\nu)}$, $b=-iq^{\half(N+1+i\nu)}$, +$c=iq^{\half(-N-1+i\nu)}$, $d=-iq^{\half(-N-1-i\nu)}$. +% + +\subsection*{References} +\cite{AlvarezSmirnov}, \cite{AndrewsAskey85}, \cite{AskeyWilson79}, +\cite{AskeyWilson85}, \cite{AtakRahmanSuslov}, \cite{GasperRahman90}, +\cite{KoelinkKoorn}, \cite{Nikiforov+}, \cite{Stanton84}. + + +\section{Al-Salam-Chihara}\index{Al-Salam-Chihara polynomials} +\par\setcounter{equation}{0} + +\subsection*{Basic hypergeometric representation} +\begin{eqnarray} +\label{DefAlSalamChihara} +Q_n(x;a,b|q)&=&\frac{(ab;q)_n}{a^n}\, +\qhyp{3}{2}{q^{-n},a\e^{i\theta},a\e^{-i\theta}}{ab,0}{q}\\ +&=&(a\e^{i\theta};q)_n\e^{-in\theta}\,\qhyp{2}{1}{q^{-n},b\e^{-i\theta}} +{a^{-1}q^{-n+1}\e^{-i\theta}}{a^{-1}q\e^{i\theta}}\nonumber\\ +&=&(b\e^{-i\theta};q)_n\e^{in\theta}\,\qhyp{2}{1}{q^{-n},a\e^{i\theta}} +{b^{-1}q^{-n+1}\e^{i\theta}}{b^{-1}q\e^{-i\theta}},\quad x=\cos\theta.\nonumber +\end{eqnarray} + +\subsection*{Orthogonality relation} +If $a$ and $b$ are real or complex conjugates and $\max(|a|,|b|)<1$, then we have the following orthogonality relation +\begin{equation} +\label{OrtAlSalamChiharaI} +\frac{1}{2\pi}\int_{-1}^1\frac{w(x)}{\sqrt{1-x^2}}Q_m(x;a,b|q)Q_n(x;a,b|q)\,dx +=\frac{\,\delta_{mn}}{(q^{n+1},abq^n;q)_{\infty}}, +\end{equation} +where +$$w(x):=w(x;a,b|q)=\left|\frac{(\e^{2i\theta};q)_{\infty}} +{(a\e^{i\theta},b\e^{i\theta};q)_{\infty}}\right|^2 +=\frac{h(x,1)h(x,-1)h(x,q^{\frac{1}{2}})h(x,-q^{\frac{1}{2}})}{h(x,a)h(x,b)},$$ +with +$$h(x,\alpha):=\prod_{k=0}^{\infty}\left(1-2\alpha xq^k+\alpha^2q^{2k}\right) +=\left(\alpha\e^{i\theta},\alpha\e^{-i\theta};q\right)_{\infty},\quad x=\cos\theta.$$ +If $a>1$ and $|ab|<1$, then we have another orthogonality relation given by: +\begin{eqnarray} +\label{OrtAlSalamChiharaII} +& &\frac{1}{2\pi}\int_{-1}^1\frac{w(x)}{\sqrt{1-x^2}}Q_m(x;a,b|q)Q_n(x;a,b|q)\,dx\nonumber\\ +& &{}\mathindent{}+\sum_{\begin{array}{c}\scriptstyle k\\ \scriptstyle 1q^{-N}$;\quad +(iv) $\ga>q^{-N}$, $\de>q^{-N}$;\quad +(v) $01,\;qb0. +\end{eqnarray} + +\subsection*{Recurrence relation} +\begin{eqnarray} +\label{RecqMeixner} +& &q^{2n+1}(1-q^{-x})M_n(q^{-x})\nonumber\\ +& &{}=c(1-bq^{n+1})M_{n+1}(q^{-x})\nonumber\\ +& &{}\mathindent{}-\left[c(1-bq^{n+1})+q(1-q^n)(c+q^n)\right]M_n(q^{-x})\nonumber\\ +& &{}\mathindent\mathindent{}+q(1-q^n)(c+q^n)M_{n-1}(q^{-x}), +\end{eqnarray} +where +$$M_n(q^{-x}):=M_n(q^{-x};b,c;q).$$ + +\subsection*{Normalized recurrence relation} +\begin{eqnarray} +\label{NormRecqMeixner} +xp_n(x)&=&p_{n+1}(x)+ +\left[1+q^{-2n-1}\left\{c(1-bq^{n+1})+q(1-q^n)(c+q^n)\right\}\right]p_n(x)\nonumber\\ +& &{}\mathindent{}+cq^{-4n+1}(1-q^n)(1-bq^n)(c+q^n)p_{n-1}(x), +\end{eqnarray} +where +$$M_n(q^{-x};b,c;q)=\frac{(-1)^nq^{n^2}}{(bq;q)_nc^n}p_n(q^{-x}).$$ + +\subsection*{$q$-Difference equation} +\begin{equation} +\label{dvqMeixner} +-(1-q^n)y(x)=B(x)y(x+1)-\left[B(x)+D(x)\right]y(x)+D(x)y(x-1), +\end{equation} +where +$$y(x)=M_n(q^{-x};b,c;q)$$ +and +$$\left\{\begin{array}{l}\displaystyle B(x)=cq^x(1-bq^{x+1})\\ +\\ +\displaystyle D(x)=(1-q^x)(1+bcq^x).\end{array}\right.$$ + +\subsection*{Forward shift operator} +\begin{eqnarray} +\label{shift1qMeixnerI} +& &M_n(q^{-x-1};b,c;q)-M_n(q^{-x};b,c;q)\nonumber\\ +& &{}=-\frac{q^{-x}(1-q^n)}{c(1-bq)}M_{n-1}(q^{-x};bq,cq^{-1};q) +\end{eqnarray} +or equivalently +\begin{equation} +\label{shift1qMeixnerII} +\frac{\Delta M_n(q^{-x};b,c;q)}{\Delta q^{-x}} +=-\frac{q(1-q^n)}{c(1-q)(1-bq)}M_{n-1}(q^{-x};bq,cq^{-1};q). +\end{equation} + +\subsection*{Backward shift operator} +\begin{eqnarray} +\label{shift2qMeixnerI} +& &cq^x(1-bq^x)M_n(q^{-x};b,c;q)-(1-q^x)(1+bcq^x)M_n(q^{-x+1};b,c;q)\nonumber\\ +& &{}=cq^x(1-b)M_{n+1}(q^{-x};bq^{-1},cq;q) +\end{eqnarray} +or equivalently +\begin{eqnarray} +\label{shift2qMeixnerII} +& &\frac{\nabla\left[w(x;b,c;q)M_n(q^{-x};b,c;q)\right]}{\nabla q^{-x}}\nonumber\\ +& &{}=\frac{1}{1-q}w(x;bq^{-1},cq;q)M_{n+1}(q^{-x};bq^{-1},cq;q), +\end{eqnarray} +where +$$w(x;b,c;q)=\frac{(bq;q)_x}{(q,-bcq;q)_x}c^xq^{\binom{x+1}{2}}.$$ + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodqMeixner} +w(x;b,c;q)M_n(q^{-x};b,c;q)=(1-q)^n\left(\nabla_q\right)^n\left[w(x;bq^n,cq^{-n};q)\right], +\end{equation} +where +$$\nabla_q:=\frac{\nabla}{\nabla q^{-x}}.$$ + +\subsection*{Generating functions} +\begin{equation} +\label{GenqMeixner1} +\frac{1}{(t;q)_{\infty}}\,\qhyp{1}{1}{q^{-x}}{bq}{-c^{-1}qt} +=\sum_{n=0}^{\infty}\frac{M_n(q^{-x};b,c;q)}{(q;q)_n}t^n. +\end{equation} + +\begin{eqnarray} +\label{GenqMeixner2} +& &\frac{1}{(t;q)_{\infty}}\,\qhyp{1}{1}{-b^{-1}c^{-1}q^{-x}}{-c^{-1}q}{bqt}\nonumber\\ +& &{}=\sum_{n=0}^{\infty}\frac{(bq;q)_n}{(-c^{-1}q,q;q)_n}M_n(q^{-x};b,c;q)t^n. +\end{eqnarray} + +\subsection*{Limit relations} + +\subsubsection*{Big $q$-Jacobi $\rightarrow$ $q$-Meixner} +If we set $b=-a^{-1}cd^{-1}$ (with $d>0$) in the definition (\ref{DefBigqJacobi}) +of the big $q$-Jacobi polynomials and take the limit $c\rightarrow -\infty$ we +obtain the $q$-Meixner polynomials given by (\ref{DefqMeixner}): +\begin{equation} +\lim_{c\rightarrow -\infty}P_n(q^{-x};a,-a^{-1}cd^{-1},c;q)=M_n(q^{-x};a,d;q). +\end{equation} + +\subsubsection*{$q$-Hahn $\rightarrow$ $q$-Meixner} +The $q$-Meixner polynomials given by (\ref{DefqMeixner}) can be obtained from the $q$-Hahn polynomials +by setting $\alpha=b$ and $\beta=-b^{-1}c^{-1}q^{-N-1}$ in the definition (\ref{DefqHahn}) of the +$q$-Hahn polynomials and letting $N\rightarrow\infty$: +$$\lim_{N\rightarrow\infty}Q_n(q^{-x};b,-b^{-1}c^{-1}q^{-N-1},N|q)=M_n(q^{-x};b,c;q).$$ + +\subsubsection*{$q$-Meixner $\rightarrow$ $q$-Laguerre} +The $q$-Laguerre polynomials given by (\ref{DefqLaguerre}) can be obtained +from the $q$-Meixner polynomials given by (\ref{DefqMeixner}) by setting +$b=q^{\alpha}$ and $q^{-x}\rightarrow cq^{\alpha}x$ in the definition +(\ref{DefqMeixner}) of the $q$-Meixner polynomials and then taking the limit +$c\rightarrow\infty$: +\begin{equation} +\lim_{c\rightarrow\infty}M_n(cq^{\alpha}x;q^{\alpha},c;q)= +\frac{(q;q)_n}{(q^{\alpha+1};q)_n}L_n^{(\alpha)}(x;q). +\end{equation} + +\subsubsection*{$q$-Meixner $\rightarrow$ $q$-Charlier} +The $q$-Charlier polynomials given by (\ref{DefqCharlier}) can easily be +obtained from the $q$-Meixner given by (\ref{DefqMeixner}) by setting $b=0$ +in the definition (\ref{DefqMeixner}) of the $q$-Meixner polynomials: +\begin{equation} +M_n(x;0,c;q)=C_n(x;c;q). +\end{equation} + +\subsubsection*{$q$-Meixner $\rightarrow$ Al-Salam-Carlitz~II} +The Al-Salam-Carlitz~II polynomials given by (\ref{DefAlSalamCarlitzII}) +can be obtained from the $q$-Meixner polynomials given by (\ref{DefqMeixner}) +by setting $b=-ac^{-1}$ in the definition (\ref{DefqMeixner}) of the +$q$-Meixner polynomials and then taking the limit $c\rightarrow 0$: +\begin{equation} +\lim_{c\rightarrow 0}M_n(x;-ac^{-1},c;q)= +\left(-\frac{1}{a}\right)^nq^{\binom{n}{2}}V_n^{(a)}(x;q). +\end{equation} + +\subsubsection*{$q$-Meixner $\rightarrow$ Meixner} +To find the Meixner polynomials given by (\ref{DefMeixner}) from the $q$-Meixner polynomials given by +(\ref{DefqMeixner}) we set $b=q^{\beta-1}$ and $c\rightarrow (1-c)^{-1}c$ and let $q\rightarrow 1$: +\begin{equation} +\lim_{q\rightarrow 1}M_n(q^{-x};q^{\beta-1},(1-c)^{-1}c;q)=M_n(x;\beta,c). +\end{equation} + +\subsection*{Remarks} +The $q$-Meixner polynomials given by (\ref{DefqMeixner}) and the little +$q$-Jacobi polynomials given by (\ref{DefLittleqJacobi}) are related in the +following way: +$$M_n(q^{-x};b,c;q)=p_n(-c^{-1}q^n;b,b^{-1}q^{-n-x-1}|q).$$ + +\noindent +The $q$-Meixner polynomials and the quantum $q$-Krawtchouk polynomials +given by (\ref{DefQuantumqKrawtchouk}) are related in the following way: +$$K_n^{qtm}(q^{-x};p,N;q)=M_n(q^{-x};q^{-N-1},-p^{-1};q).$$ + +\subsection*{References} +\cite{AlSalam90}, \cite{AlSalamVerma82II}, \cite{AlSalamVerma88}, \cite{AlvarezRonveaux}, +\cite{AtakAtakKlimyk}, \cite{AtakRahmanSuslov}, \cite{Campigotto+}, \cite{GasperRahman90}, +\cite{Hahn}, \cite{Ismail2005I}, \cite{Nikiforov+}, \cite{Smirnov}. + + +\section{Quantum $q$-Krawtchouk} +\index{Quantum q-Krawtchouk polynomials@Quantum $q$-Krawtchouk polynomials} +\index{q-Krawtchouk polynomials@$q$-Krawtchouk polynomials!Quantum} +\par\setcounter{equation}{0} + +\subsection*{Basic hypergeometric representation} +\begin{equation} +\label{DefQuantumqKrawtchouk} +K_n^{qtm}(q^{-x};p,N;q)= +\qhyp{2}{1}{q^{-n},q^{-x}}{q^{-N}}{pq^{n+1}},\quad n=0,1,2,\ldots,N. +\end{equation} + +\subsection*{Orthogonality relation} +\begin{eqnarray} +\label{OrtQuantumqKrawtchouk} +& &\sum_{x=0}^N\frac{(pq;q)_{N-x}}{(q;q)_x(q;q)_{N-x}}(-1)^{N-x}q^{\binom{x}{2}} +K_m^{qtm}(q^{-x};p,N;q)K_n^{qtm}(q^{-x};p,N;q)\nonumber\\ +& &{}=\frac{(-1)^np^N(q;q)_{N-n}(q,pq;q)_n}{(q,q;q)_N} +q^{\binom{N+1}{2}-\binom{n+1}{2}+Nn}\,\delta_{mn},\quad p>q^{-N}. +\end{eqnarray} + +\subsection*{Recurrence relation} +\begin{eqnarray} +\label{RecQuantumqKrawtchouk} +& &-pq^{2n+1}(1-q^{-x})K_n^{qtm}(q^{-x})\nonumber\\ +& &{}=(1-q^{n-N})K_{n+1}^{qtm}(q^{-x})\nonumber\\ +& &{}\mathindent{}-\left[(1-q^{n-N})+q(1-q^n)(1-pq^n)\right]K_n^{qtm}(q^{-x})\nonumber\\ +& &{}\mathindent\mathindent{}+q(1-q^n)(1-pq^n)K_{n-1}^{qtm}(q^{-x}), +\end{eqnarray} +where +$$K_n^{qtm}(q^{-x}):=K_n^{qtm}(q^{-x};p,N;q).$$ + +\subsection*{Normalized recurrence relation} +\begin{eqnarray} +\label{NormRecQuantumqKrawtchouk} +xp_n(x)&=&p_{n+1}(x)+ +\left[1-p^{-1}q^{-2n-1}\left\{(1-q^{n-N})+q(1-q^n)(1-pq^n)\right\}\right]p_n(x)\nonumber\\ +& &{}\mathindent{}+p^{-2}q^{-4n+1}(1-q^n)(1-pq^n)(1-q^{n-N-1})p_{n-1}(x), +\end{eqnarray} +where +$$K_n^{qtm}(q^{-x};p,N;q)=\frac{p^nq^{n^2}}{(q^{-N};q)_n}p_n(q^{-x}).$$ + +\subsection*{$q$-Difference equation} +\begin{equation} +\label{dvQuantumqKrawtchouk} +-p(1-q^n)y(x)=B(x)y(x+1)-\left[B(x)+D(x)\right]y(x)+D(x)y(x-1), +\end{equation} +where +$$y(x)=K_n^{qtm}(q^{-x};p,N;q)$$ +and +$$\left\{\begin{array}{l}\displaystyle B(x)=-q^x(1-q^{x-N})\\ +\\ +\displaystyle D(x)=(1-q^x)(p-q^{x-N-1}).\end{array}\right.$$ + +\subsection*{Forward shift operator} +\begin{eqnarray} +\label{shift1QuantumqKrawtchoukI} +& &K_n^{qtm}(q^{-x-1};p,N;q)-K_n^{qtm}(q^{-x};p,N;q)\nonumber\\ +& &{}=\frac{pq^{-x}(1-q^n)}{1-q^{-N}}K_{n-1}^{qtm}(q^{-x};pq,N-1;q) +\end{eqnarray} +or equivalently +\begin{equation} +\label{shift1QuantumqKrawtchoukII} +\frac{\Delta K_n^{qtm}(q^{-x};p,N;q)}{\Delta q^{-x}}= +\frac{pq(1-q^n)}{(1-q)(1-q^{-N})}K_{n-1}^{qtm}(q^{-x};pq,N-1;q). +\end{equation} + +\subsection*{Backward shift operator} +\begin{eqnarray} +\label{shift2QuantumqKrawtchoukI} +& &(1-q^{x-N-1})K_n^{qtm}(q^{-x};p,N;q)\nonumber\\ +& &{}\mathindent{}+q^{-x}(1-q^x)(p-q^{x-N-1})K_n^{qtm}(q^{-x+1};p,N;q)\nonumber\\ +& &{}=(1-q^{-N-1})K_{n+1}^{qtm}(q^{-x};pq^{-1},N+1;q) +\end{eqnarray} +or equivalently +\begin{eqnarray} +\label{shift2QuantumqKrawtchoukII} +& &\frac{\nabla\left[w(x;p,N;q)K_n^{qtm}(q^{-x};p,N;q)\right]}{\nabla q^{-x}}\nonumber\\ +& &{}=\frac{1}{1-q}w(x;pq^{-1},N+1;q)K_{n+1}^{qtm}(q^{-x};pq^{-1},N+1;q), +\end{eqnarray} +where +$$w(x;p,N;q)=\frac{(q^{-N};q)_x}{(q,p^{-1}q^{-N};q)_x}(-p)^{-x}q^{\binom{x+1}{2}}.$$ + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodQuantumqKrawtchouk} +w(x;p,N;q)K_n^{qtm}(q^{-x};p,N;q)=(1-q)^n\left(\nabla_q\right)^n\left[w(x;pq^n,N-n;q)\right], +\end{equation} +where +$$\nabla_q:=\frac{\nabla}{\nabla q^{-x}}.$$ + +\subsection*{Generating functions} For $x=0,1,2,\ldots,N$ we have +\begin{eqnarray} +\label{GenQuantumqKrawtchouk1} +& &(q^{x-N}t;q)_{N-x}\cdot\qhyp{2}{1}{q^{-x},pq^{N+1-x}}{0}{q^{x-N}t}\nonumber\\ +& &{}=\sum_{n=0}^N\frac{(q^{-N};q)_n}{(q;q)_n}K_n^{qtm}(q^{-x};p,N;q)t^n. +\end{eqnarray} + +\begin{eqnarray} +\label{GenQuantumqKrawtchouk2} +& &(q^{-x}t;q)_x\cdot\qhyp{2}{1}{q^{x-N},0}{pq}{q^{-x}t}\nonumber\\ +& &{}=\sum_{n=0}^N\frac{(q^{-N};q)_n}{(pq,q;q)_n}K_n^{qtm}(q^{-x};p,N;q)t^n. +\end{eqnarray} + +\subsection*{Limit relations} + +\subsubsection*{$q$-Hahn $\rightarrow$ Quantum $q$-Krawtchouk} +The quantum $q$-Krawtchouk polynomials given by (\ref{DefQuantumqKrawtchouk}) +simply follow from the $q$-Hahn polynomials by setting $\beta=p$ in the definition (\ref{DefqHahn}) of +the $q$-Hahn polynomials and taking the limit $\alpha\rightarrow\infty$: +$$\lim_{\alpha\rightarrow\infty}Q_n(q^{-x};\alpha,p,N|q)=K_n^{qtm}(q^{-x};p,N;q).$$ + +\subsubsection*{Quantum $q$-Krawtchouk $\rightarrow$ Al-Salam-Carlitz~II} +If we set $p=a^{-1}q^{-N-1}$ in the definition (\ref{DefQuantumqKrawtchouk}) +of the quantum $q$-Krawtchouk polynomials and let $N\rightarrow\infty$ we +obtain the Al-Salam-Carlitz~II polynomials given by +(\ref{DefAlSalamCarlitzII}). In fact we have +\begin{equation} +\lim_{N\rightarrow\infty}K_n^{qtm}(x;a^{-1}q^{-N-1},N;q)= +\left(-\frac{1}{a}\right)^nq^{\binom{n}{2}}V_n^{(a)}(x;q). +\end{equation} + +\subsubsection*{Quantum $q$-Krawtchouk $\rightarrow$ Krawtchouk} +The Krawtchouk polynomials given by (\ref{DefKrawtchouk}) easily follow from +the quantum $q$-Krawtchouk polynomials given by (\ref{DefQuantumqKrawtchouk}) +in the following way: +\begin{equation} +\lim_{q\rightarrow 1}K_n^{qtm}(q^{-x};p,N;q)=K_n(x;p^{-1},N). +\end{equation} + +\subsection*{Remarks} +The quantum $q$-Krawtchouk polynomials given by +(\ref{DefQuantumqKrawtchouk}) and the $q$-Meixner polynomials given by +(\ref{DefqMeixner}) are related in the following way: +$$K_n^{qtm}(q^{-x};p,N;q)=M_n(q^{-x};q^{-N-1},-p^{-1};q).$$ + +\noindent +The quantum $q$-Krawtchouk polynomials are related to the affine +$q$-Krawtchouk polynomials given by (\ref{DefAffqKrawtchouk}) +by the transformation $q\leftrightarrow q^{-1}$ in the following way: +$$K_n^{qtm}(q^x;p,N;q^{-1})=(p^{-1}q;q)_n\left(-\frac{p}{q}\right)^nq^{-\binom{n}{2}} +K_n^{Aff}(q^{x-N};p^{-1},N;q).$$ + RS: add begin\label{sec14.12} +% +\paragraph{\large\bf KLSadd: Notation}Here the little $q$-Jacobi polynomial is denoted by +$p_n(x;a,b;q)$ instead of +$p_n(x;a,b\,|\, q)$. +% +\paragraph{\large\bf KLSadd: Special values}(see \cite[\S2.4]{K17}). +\begin{align} +p_n(0;a,b;q)&=1,\label{127}\\ +p_n(q^{-1}b^{-1};a,b;q)&=(-qb)^{-n}\,q^{-\half n(n-1)}\,\frac{(qb;q)_n}{(qa;q)_n}\,,\label{128}\\ +p_n(1;a,b;q)&=(-a)^n\,q^{\half n(n+1)}\,\frac{(qb;q)_n}{(qa;q)_n}\,.\label{129} +\end{align} +% + +\subsection*{References} +\cite{GasperRahman90}, \cite{Koorn89III}, \cite{Koorn90II}, \cite{Smirnov}. + + +\section{$q$-Krawtchouk}\index{q-Krawtchouk polynomials@$q$-Krawtchouk polynomials} +\par\setcounter{equation}{0} + +\subsection*{Basic hypergeometric representation} For $n=0,1,2,\ldots,N$ we have +\begin{eqnarray} +\label{DefqKrawtchouk} +K_n(q^{-x};p,N;q)&=&\qhyp{3}{2}{q^{-n},q^{-x},-pq^n}{q^{-N},0}{q}\\ +&=&\frac{(q^{x-N};q)_n}{(q^{-N};q)_nq^{nx}}\, +\qhyp{2}{1}{q^{-n},q^{-x}}{q^{N-x-n+1}}{-pq^{n+N+1}}.\nonumber +\end{eqnarray} + +\subsection*{Orthogonality relation} +\begin{eqnarray} +\label{OrtqKrawtchouk} +& &\sum_{x=0}^N\frac{(q^{-N};q)_x}{(q;q)_x}(-p)^{-x}K_m(q^{-x};p,N;q)K_n(q^{-x};p,N;q)\nonumber\\ +& &{}=\frac{(q,-pq^{N+1};q)_n}{(-p,q^{-N};q)_n}\frac{(1+p)}{(1+pq^{2n})}\nonumber\\ +& &{}\mathindent{}\times (-pq;q)_Np^{-N}q^{-\binom{N+1}{2}} +\left(-pq^{-N}\right)^nq^{n^2}\,\delta_{mn},\quad p>0. +\end{eqnarray} + +\subsection*{Recurrence relation} +\begin{eqnarray} +\label{RecqKrawtchouk} +-\left(1-q^{-x}\right)K_n(q^{-x})&=&A_nK_{n+1}(q^{-x})-\left(A_n+C_n\right)K_n(q^{-x})\nonumber\\ +& &{}\mathindent{}+C_nK_{n-1}(q^{-x}), +\end{eqnarray} +where +$$K_n(q^{-x}):=K_n(q^{-x};p,N;q)$$ +and +$$\left\{\begin{array}{l} +\displaystyle A_n=\frac{(1-q^{n-N})(1+pq^n)}{(1+pq^{2n})(1+pq^{2n+1})}\\ +\\ +\displaystyle C_n=-pq^{2n-N-1}\frac{(1+pq^{n+N})(1-q^n)}{(1+pq^{2n-1})(1+pq^{2n})}. +\end{array}\right.$$ + +\subsection*{Normalized recurrence relation} +\begin{equation} +\label{NormRecqKrawtchouk} +xp_n(x)=p_{n+1}(x)+\left[1-(A_n+C_n)\right]p_n(x)+A_{n-1}C_np_{n-1}(x), +\end{equation} +where +$$K_n(q^{-x};p,N;q)=\frac{(-pq^n;q)_n}{(q^{-N};q)_n}p_n(q^{-x}).$$ + +\subsection*{$q$-Difference equation} +\begin{eqnarray} +\label{dvqKrawtchouk} +& &q^{-n}(1-q^n)(1+pq^n)y(x)\nonumber\\ +& &{}=(1-q^{x-N})y(x+1)-\left[(1-q^{x-N})-p(1-q^x)\right]y(x)\nonumber\\ +& &{}\mathindent{}-p(1-q^x)y(x-1), +\end{eqnarray} +where +$$y(x)=K_n(q^{-x};p,N;q).$$ + +\subsection*{Forward shift operator} +\begin{eqnarray} +\label{shift1qKrawtchoukI} +& &K_n(q^{-x-1};p,N;q)-K_n(q^{-x};p,N;q)\nonumber\\ +& &{}=\frac{q^{-n-x}(1-q^n)(1+pq^n)}{1-q^{-N}}K_{n-1}(q^{-x};pq^2,N-1;q) +\end{eqnarray} +or equivalently +\begin{equation} +\label{shift1qKrawtchoukII} +\frac{\Delta K_n(q^{-x};p,N;q)}{\Delta q^{-x}}= +\frac{q^{-n+1}(1-q^n)(1+pq^n)}{(1-q)(1-q^{-N})}K_{n-1}(q^{-x};pq^2,N-1;q). +\end{equation} + +\subsection*{Backward shift operator} +\begin{eqnarray} +\label{shift2qKrawtchoukI} +& &(1-q^{x-N-1})K_n(q^{-x};p,N;q)+pq^{-1}(1-q^x)K_n(q^{-x+1};p,N;q)\nonumber\\ +& &{}=q^x(1-q^{-N-1})K_{n+1}(q^{-x};pq^{-2},N+1;q) +\end{eqnarray} +or equivalently +\begin{eqnarray} +\label{shift2qKrawtchoukII} +& &\frac{\nabla\left[w(x;p,N;q)K_n(q^{-x};p,N;q)\right]}{\nabla q^{-x}}\nonumber\\ +& &{}=\frac{1}{1-q}w(x;pq^{-2},N+1;q)K_{n+1}(q^{-x};pq^{-2},N+1;q), +\end{eqnarray} +where +$$w(x;p,N;q)=\frac{(q^{-N};q)_x}{(q;q)_x}\left(-\frac{q}{p}\right)^x.$$ + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodqKrawtchouk} +w(x;p,N;q)K_n(q^{-x};p,N;q)=(1-q)^n\left(\nabla_q\right)^n\left[w(x;pq^{2n},N-n;q)\right], +\end{equation} +where +$$\nabla_q:=\frac{\nabla}{\nabla q^{-x}}.$$ + +\subsection*{Generating function} For $x=0,1,2,\ldots,N$ we have +\begin{eqnarray} +\label{GenqKrawtchouk} +& &\qhyp{1}{1}{q^{-x}}{0}{pqt}\,\qhyp{2}{0}{q^{x-N},0}{-}{-q^{-x}t}\nonumber\\ +& &{}=\sum_{n=0}^N\frac{(q^{-N};q)_n}{(q;q)_n}q^{-\binom{n}{2}}K_n(q^{-x};p,N;q)t^n. +\end{eqnarray} + +\subsection*{Limit relations} + +\subsubsection*{$q$-Racah $\rightarrow$ $q$-Krawtchouk} +The $q$-Krawtchouk polynomials given by (\ref{DefqKrawtchouk}) can be obtained from +the $q$-Racah polynomials by setting $\alpha q=q^{-N}$, $\beta=-pq^N$ and +$\gamma=\delta=0$ in the definition (\ref{DefqRacah}) of the $q$-Racah polynomials: +$$R_n(q^{-x};q^{-N-1},-pq^N,0,0|q)=K_n(q^{-x};p,N;q).$$ +Note that $\mu(x)=q^{-x}$ in this case. + +\subsubsection*{$q$-Hahn $\rightarrow$ $q$-Krawtchouk} +If we set $\beta=-\alpha^{-1}q^{-1}p$ in the definition (\ref{DefqHahn}) of the $q$-Hahn polynomials +and then let $\alpha\rightarrow 0$ we obtain the $q$-Krawtchouk polynomials given by +(\ref{DefqKrawtchouk}): +$$\lim_{\alpha\rightarrow 0} +Q_n(q^{-x};\alpha,-\alpha^{-1}q^{-1}p,N|q)=K_n(q^{-x};p,N;q).$$ + +\subsubsection*{$q$-Krawtchouk $\rightarrow$ $q$-Bessel} +If we set $x\rightarrow N-x$ in the definition (\ref{DefqKrawtchouk}) of the +$q$-Krawtchouk polynomials and then take the limit $N\rightarrow\infty$ we +obtain the $q$-Bessel polynomials given by (\ref{DefqBessel}): +\begin{equation} +\lim_{N\rightarrow\infty}K_n(q^{x-N};p,N;q)=y_n(q^x;p;q). +\end{equation} + +\subsubsection*{$q$-Krawtchouk $\rightarrow$ $q$-Charlier} +By setting $p=a^{-1}q^{-N}$ in the definition (\ref{DefqKrawtchouk}) of the +$q$-Krawtchouk polynomials and then taking the limit $N\rightarrow\infty$ we +obtain the $q$-Charlier polynomials given by (\ref{DefqCharlier}): +\begin{equation} +\lim_{N\rightarrow\infty}K_n(q^{-x};a^{-1}q^{-N},N;q)=C_n(q^{-x};a;q). +\end{equation} + +\subsubsection*{$q$-Krawtchouk $\rightarrow$ Krawtchouk} +If we take the limit $q\rightarrow 1$ in the definition (\ref{DefqKrawtchouk}) of the $q$-Krawtchouk +polynomials we simply find the Krawtchouk polynomials given by (\ref{DefKrawtchouk}) in the +following way: +\begin{equation} +\lim_{q\rightarrow 1}K_n(q^{-x};p,N;q)=K_n(x;(p+1)^{-1},N). +\end{equation} + +\subsection*{Remark} +The $q$-Krawtchouk polynomials given by (\ref{DefqKrawtchouk}) and the +dual $q$-Krawtchouk polynomials given by (\ref{DefDualqKrawtchouk}) are +related in the following way: +$$K_n(q^{-x};p,N;q)=K_x(\lambda(n);-pq^N,N|q)$$ +with +$$\lambda(n)=q^{-n}-pq^n$$ +or +$$K_n(\lambda(x);c,N|q)=K_x(q^{-n};-cq^{-N},N;q)$$ +with +$$\lambda(x)=q^{-x}+cq^{x-N}.$$ + +\subsection*{References} +\cite{AlvarezRonveaux}, \cite{AskeyWilson79}, \cite{AtakRahmanSuslov}, +\cite{Campigotto+}, \cite{GasperRahman90}, \cite{Nikiforov+}, +\cite{NoumiMimachi91}, \cite{Stanton80III}, \cite{Stanton84}. + + +\newpage + +\section{Affine $q$-Krawtchouk} +\index{Affine q-Krawtchouk polynomials@Affine $q$-Krawtchouk polynomials} +\index{q-Krawtchouk polynomials@$q$-Krawtchouk polynomials!Affine} +\par\setcounter{equation}{0} + +\subsection*{Basic hypergeometric representation} +\begin{eqnarray} +\label{DefAffqKrawtchouk} +K_n^{Aff}(q^{-x};p,N;q)&=&\qhyp{3}{2}{q^{-n},0,q^{-x}}{pq,q^{-N}}{q}\\ +&=&\frac{(-pq)^nq^{\binom{n}{2}}}{(pq;q)_n}\, +\qhyp{2}{1}{q^{-n},q^{x-N}}{q^{-N}}{\frac{q^{-x}}{p}},\quad n=0,1,2,\ldots,N.\nonumber +\end{eqnarray} + +\subsection*{Orthogonality relation} +\begin{eqnarray} +\label{OrtAffqKrawtchouk} +& &\sum_{x=0}^N\frac{(pq;q)_x(q;q)_N}{(q;q)_x(q;q)_{N-x}}(pq)^{-x}K_m^{Aff}(q^{-x};p,N;q)K_n^{Aff}(q^{-x};p,N;q)\nonumber\\ +& &{}=(pq)^{n-N}\frac{(q;q)_n(q;q)_{N-n}}{(pq;q)_n(q;q)_N}\,\delta_{mn},\quad 01$ +if $p>q^{-1}$. +% +\paragraph{\large\bf KLSadd: History}The origin of the name of the quantum $q$-Krawtchouk polynomials +is by their interpretation +as matrix elements of irreducible corepresentations of (the quantized +function algebra of) the quantum group $SU_q(2)$ considered +with respect to its quantum subgroup $U(1)$. The orthogonality +relation and dual orthogonality relation of these polynomials +are an expression of the unitarity of these corepresentations. +See for instance \myciteKLS{343}{Section 6}. +% + +\subsection*{References} +\cite{AtakRahmanSuslov}, \cite{LChiharaStanton}, \cite{Delsarte}, +\cite{DelsarteGoethals}, \cite{Dunkl78II}, \cite{FlorisKoelink}, +\cite{GasperRahman90}, \cite{Stanton84}. + + +\section{Dual $q$-Krawtchouk} +\index{Dual q-Krawtchouk polynomials@Dual $q$-Krawtchouk polynomials} +\index{q-Krawtchouk polynomials@$q$-Krawtchouk polynomials!Dual} +\par\setcounter{equation}{0} + +\subsection*{Basic hypergeometric representation} +\begin{eqnarray} +\label{DefDualqKrawtchouk} +K_n(\lambda(x);c,N|q)&=&\qhyp{3}{2}{q^{-n},q^{-x},cq^{x-N}}{q^{-N},0}{q}\\ +&=&\frac{(q^{x-N};q)_n}{(q^{-N};q)_nq^{nx}}\, +\qhyp{2}{1}{q^{-n},q^{-x}}{q^{N-x-n+1}}{cq^{x+1}},\quad n=0,1,2,\ldots,N,\nonumber +\end{eqnarray} +where +$$\lambda(x):=q^{-x}+cq^{x-N}.$$ + +\subsection*{Orthogonality relation} +\begin{eqnarray} +\label{OrtDualqKrawtchouk} +& &\sum_{x=0}^N\frac{(cq^{-N},q^{-N};q)_x}{(q,cq;q)_x} +\frac{(1-cq^{2x-N})}{(1-cq^{-N})}c^{-x}q^{x(2N-x)}K_m(\lambda(x))K_n(\lambda(x))\nonumber\\ +& &{}=(c^{-1};q)_N\frac{(q;q)_n}{(q^{-N};q)_n}(cq^{-N})^n\,\delta_{mn},\quad c<0, +\end{eqnarray} +where +$$K_n(\lambda(x)):=K_n(\lambda(x);c,N|q).$$ + +\newpage + +\subsection*{Recurrence relation} +\begin{eqnarray} +\label{RecDualqKrawtchouk} +& &-(1-q^{-x})(1-cq^{x-N})K_n(\lambda(x))\nonumber\\ +& &{}=(1-q^{n-N})K_{n+1}(\lambda(x))\nonumber\\ +& &{}\mathindent{}-\left[(1-q^{n-N})+cq^{-N}(1-q^n)\right]K_n(\lambda(x))\nonumber\\ +& &{}\mathindent\mathindent{}+cq^{-N}(1-q^n)K_{n-1}(\lambda(x)), +\end{eqnarray} +where +$$K_n(\lambda(x)):=K_n(\lambda(x);c,N|q).$$ + +\subsection*{Normalized recurrence relation} +\begin{eqnarray} +\label{NormRecDualqKrawtchouk} +xp_n(x)&=&p_{n+1}(x)+(1+c)q^{n-N}p_n(x)\nonumber\\ +& &{}\mathindent{}+cq^{-N}(1-q^n)(1-q^{n-N-1})p_{n-1}(x), +\end{eqnarray} +where +$$K_n(\lambda(x);c,N|q)=\frac{1}{(q^{-N};q)_n}p_n(\lambda(x)).$$ + +\subsection*{$q$-Difference equation} +\begin{equation} +\label{dvDualqKrawtchouk} +q^{-n}(1-q^n)y(x)=B(x)y(x+1)-\left[B(x)+D(x)\right]y(x)+D(x)y(x-1), +\end{equation} +where +$$y(x)=K_n(\lambda(x);c,N|q)$$ +and +$$\left\{\begin{array}{l}\displaystyle B(x)=\frac{(1-q^{x-N})(1-cq^{x-N})}{(1-cq^{2x-N})(1-cq^{2x-N+1})}\\ +\\ +\displaystyle D(x)=cq^{2x-2N-1}\frac{(1-q^x)(1-cq^x)}{(1-cq^{2x-N-1})(1-cq^{2x-N})}.\end{array}\right.$$ + +\newpage + +\subsection*{Forward shift operator} +\begin{eqnarray} +\label{shift1DualqKrawtchoukI} +& &K_n(\lambda(x+1);c,N|q)-K_n(\lambda(x);c,N|q)\nonumber\\ +& &{}=\frac{q^{-n-x}(1-q^n)(1-cq^{2x-N+1})}{1-q^{-N}} +K_{n-1}(\lambda(x);c,N-1|q) +\end{eqnarray} +or equivalently +\begin{equation} +\label{shift1DualqKrawtchoukII} +\frac{\Delta K_n(\lambda(x);c,N|q)}{\Delta\lambda(x)}= +\frac{q^{-n+1}(1-q^n)}{(1-q)(1-q^{-N})}K_{n-1}(\lambda(x);c,N-1|q). +\end{equation} + +\subsection*{Backward shift operator} +\begin{eqnarray} +\label{shift2DualqKrawtchoukI} +& &(1-q^{x-N-1})(1-cq^{x-N-1})K_n(\lambda(x);c,N|q)\nonumber\\ +& &{}\mathindent{}-cq^{2(x-N-1)}(1-q^x)(1-cq^x)K_n(\lambda(x-1);c,N|q)\nonumber\\ +& &{}=q^x(1-q^{-N-1})(1-cq^{2x-N-1})K_{n+1}(\lambda(x);c,N+1|q) +\end{eqnarray} +or equivalently +\begin{eqnarray} +\label{shift2DualqKrawtchoukII} +& &\frac{\nabla\left[w(x;c,N|q)K_n(\lambda(x);c,N|q)\right]}{\nabla\lambda(x)}\nonumber\\ +& &{}=\frac{1}{(1-q)(1-cq^{-N-1})}w(x;c,N+1|q)K_{n+1}(\lambda(x);c,N+1|q), +\end{eqnarray} +where +$$w(x;c,N|q)=\frac{(q^{-N},cq^{-N};q)_x}{(q,cq;q)_x}c^{-x}q^{2Nx-x(x-1)}.$$ + +\subsection*{Rodrigues-type formula} +\begin{eqnarray} +\label{RodDualqKrawtchouk} +& &w(x;c,N|q)K_n(\lambda(x);c,N|q)\nonumber\\ +& &{}=(1-q)^n(cq^{-N};q)_n\left(\nabla_{\lambda}\right)^n\left[w(x;c,N-n|q)\right], +\end{eqnarray} +where +$$\nabla_{\lambda}:=\frac{\nabla}{\nabla\lambda(x)}.$$ + +\subsection*{Generating function} For $x=0,1,2,\ldots,N$ we have +\begin{equation} +\label{GenDualqKrawtchouk} +(cq^{-N}t;q)_x\cdot (q^{-N}t;q)_{N-x}=\sum_{n=0}^N +\frac{(q^{-N};q)_n}{(q;q)_n}K_n(\lambda(x);c,N|q)t^n. +\end{equation} + +\subsection*{Limit relations} + +\subsubsection*{$q$-Racah $\rightarrow$ Dual $q$-Krawtchouk} +The dual $q$-Krawtchouk polynomials given by (\ref{DefDualqKrawtchouk}) easily +follow from the $q$-Racah polynomials given by (\ref{DefqRacah}) by using the +substitutions $\alpha=\beta=0$, $\gamma q=q^{-N}$ and $\delta=c$: +$$R_n(\mu(x);0,0,q^{-N-1},c|q)=K_n(\lambda(x);c,N|q).$$ +Note that +$$\mu(x)=\lambda(x)=q^{-x}+cq^{x-N}.$$ + +\subsubsection*{Dual $q$-Hahn $\rightarrow$ Dual $q$-Krawtchouk} +The dual $q$-Krawtchouk polynomials given by (\ref{DefDualqKrawtchouk}) can +be obtained from the dual $q$-Hahn polynomials by setting $\delta=c\gamma^{-1}q^{-N-1}$ +in (\ref{DefDualqHahn}) and letting $\gamma\rightarrow 0$: +$$\lim_{\gamma\rightarrow 0} +R_n(\mu(x);\gamma,c\gamma^{-1}q^{-N-1},N|q)=K_n(\lambda(x);c,N|q).$$ + +\subsubsection*{Dual $q$-Krawtchouk $\rightarrow$ Al-Salam-Carlitz~I} +If we set $c=a^{-1}$ in the definition (\ref{DefDualqKrawtchouk}) +of the dual $q$-Krawtchouk polynomials and take the limit +$N\rightarrow\infty$ we simply obtain the Al-Salam-Carlitz~I polynomials +given by (\ref{DefAlSalamCarlitzI}): +\begin{equation} +\lim_{N\rightarrow\infty}K_n(\lambda(x);a^{-1},N|q)= +\left(-\frac{1}{a}\right)^nq^{-\binom{n}{2}}U_n^{(a)}(q^x;q). +\end{equation} +Note that $\lambda(x)=q^{-x}+a^{-1}q^{x-N}$. + +\subsubsection*{Dual $q$-Krawtchouk $\rightarrow$ Krawtchouk} +If we set $c=1-p^{-1}$ in the definition (\ref{DefDualqKrawtchouk}) of the +dual $q$-Krawtchouk polynomials and take the limit $q\rightarrow 1$ we simply find +the Krawtchouk polynomials given by (\ref{DefKrawtchouk}): +\begin{equation} +\lim_{q\rightarrow 1}K_n(\lambda(x);1-p^{-1},N|q)=K_n(x;p,N). +\end{equation} + +\subsection*{Remark} +The dual $q$-Krawtchouk polynomials given by (\ref{DefDualqKrawtchouk}) +and the $q$-Krawtchouk polynomials given by (\ref{DefqKrawtchouk}) are +related in the following way: +$$K_n(q^{-x};p,N;q)=K_x(\lambda(n);-pq^N,N|q)$$ +with +$$\lambda(n)=q^{-n}-pq^n$$ +or +$$K_n(\lambda(x);c,N|q)=K_x(q^{-n};-cq^{-N},N;q)$$ +with +$$\lambda(x)=q^{-x}+cq^{x-N}.$$ + RS: add begin\label{sec14.16} +% +\paragraph{\large\bf KLSadd: $q$-Hypergeometric representation}For $n=0,1,\ldots,N$ +(see (14.16.1)): +\begin{align} +K_n^{\rm Aff}(y;p,N;q) +&=\frac1{(p^{-1}q^{-1};q^{-1})_n}\,\qhyp21{q^{-n},q^{-N}y^{-1}}{q^{-N}}{q,p^{-1}y} +\label{156}\\ +&=\qhyp32{q^{-n},y,0}{q^{-N},pq}{q,q}. +\label{157} +\end{align} +% +\paragraph{\large\bf KLSadd: Self-duality}By \eqref{157}: +\begin{equation} +K_n^{\rm Aff}(q^{-x};p,N;q)=K_x^{\rm Aff}(q^{-n};p,N;q)\qquad +(n,x\in\{0,1,\ldots,N\}). +\label{168} +\end{equation} +% +\paragraph{\large\bf KLSadd: Special values}By \eqref{156} and \mycite{GR}{(II.4)}: +\begin{equation} +K_n^{\rm Aff}(1;p,N;q)=1,\qquad +K_n^{\rm Aff}(q^{-N};p,N;q)=\frac1{((pq)^{-1};q^{-1})_n}\,. +\label{165} +\end{equation} +By \eqref{165} and \eqref{168} we have also +\begin{equation} +K_N^{\rm Aff}(q^{-x};p,N;q)=\frac1{((pq)^{-1};q^{-1})_x}\,. +\label{170} +\end{equation} +% +\paragraph{\large\bf KLSadd: Limit for $q\to1$ to Krawtchouk}\begin{align} +\lim_{q\to1} K_n^{\rm Aff}(1+(1-q)x;p,N;q)&=K_n(x;1-p,N), +\label{162}\\ +\lim_{q\to1} K_n^{\rm Aff}(q^{-x};p,N;q)&=K_n(x;1-p,N). +\label{166} +\end{align} +% +\paragraph{\large\bf KLSadd: A relation between quantum and affine $q$-Krawtchouk}By \eqref{152}, \eqref{156}, \eqref{165} and \eqref{168} +we have for $x\in\{0,1,\ldots,N\}$: +\begin{align} +K_{N-n}^{\rm qtm}(q^{-x};p^{-1}q^{-N-1},N;q) +&=\frac{K_x^{\rm Aff}(q^{-n};p,N;q)}{K_x^{\rm Aff}(q^{-N};p,N;q)} +\label{171}\\ +&=\frac{K_n^{\rm Aff}(q^{-x};p,N;q)}{K_N^{\rm Aff}(q^{-x};p,N;q)}\,. +\label{172} +\end{align} +Formula \eqref{171} is given in \cite[formula after (12)]{K24} +and \cite[(59)]{K25}. +In view of \eqref{164} and \eqref{166} +formula \eqref{172} has \eqref{149} as a limit case for +$q\to 1$. +% +\paragraph{\large\bf KLSadd: Affine $q^{-1}By \eqref{156}, \eqref{165}, +\eqref{154} and \eqref{152} (see also p.505, first formula): +\begin{align} +\frac{K_n^{\rm Aff}(y;p,N;q^{-1})}{K_n^{\rm Aff}(q^N;p,N;q^{-1})} +&=\qhyp21{q^{-n},q^{-N}y}{q^{-N}}{q,p^{-1}q^{n+1}} +\label{158}\\ +&=K_n^{\rm qtm}(q^{-N}y;p^{-1},N;q). +\label{159} +\end{align} +Formula \eqref{159} is equivalent to \eqref{160}. +Just as for \eqref{160}, it tends after suitable substitutions to +\eqref{10} as $q\to1$. + +The orthogonality relation (14.16.2) holds with positive weights for $q>1$ +if $01$, then we have another orthogonality relation given by: +\begin{eqnarray} +\label{OrtContBigqHermite2} +& &\frac{1}{2\pi}\int_{-1}^1\frac{w(x)}{\sqrt{1-x^2}}H_m(x;a|q)H_n(x;a|q)\,dx\nonumber\\ +& &{}\mathindent{}+\sum_{\begin{array}{c}\scriptstyle k\\ \scriptstyle 1-1 +\end{eqnarray} +and +\begin{eqnarray} +\label{OrtqLaguerre2} +& &\sum_{k=-\infty}^{\infty}\frac{q^{k\alpha+k}}{(-cq^k;q)_{\infty}}L_m^{(\alpha)}(cq^k;q)L_n^{(\alpha)}(cq^k;q)\nonumber\\ +& &{}=\frac{(q,-cq^{\alpha+1},-c^{-1}q^{-\alpha};q)_{\infty}} +{(q^{\alpha+1},-c,-c^{-1}q;q)_{\infty}}\frac{(q^{\alpha+1};q)_n}{(q;q)_nq^n}\,\delta_{mn}, +\quad\alpha>-1,\quad c>0. +\end{eqnarray} +For $c=1$ the latter orthogonality relation can also be written as +\begin{eqnarray} +\label{OrtqLaguerre3} +& &\int_0^{\infty}\frac{x^{\alpha}}{(-x;q)_{\infty}}L_m^{(\alpha)}(x;q)L_n^{(\alpha)}(x;q)\,d_qx\nonumber\\ +& &{}=\frac{1-q}{2}\,\frac{(q,-q^{\alpha+1},-q^{-\alpha};q)_{\infty}} +{(q^{\alpha+1},-q,-q;q)_{\infty}}\frac{(q^{\alpha+1};q)_n}{(q;q)_nq^n}\,\delta_{mn},\quad\alpha>-1. +\end{eqnarray} + +\newpage + +\subsection*{Recurrence relation} +\begin{eqnarray} +\label{RecqLaguerre} +-q^{2n+\alpha+1}xL_n^{(\alpha)}(x;q)&=&(1-q^{n+1})L_{n+1}^{(\alpha)}(x;q)\nonumber\\ +& &{}\mathindent{}-\left[(1-q^{n+1})+q(1-q^{n+\alpha})\right]L_n^{(\alpha)}(x;q)\nonumber\\ +& &{}\mathindent\mathindent{}+q(1-q^{n+\alpha})L_{n-1}^{(\alpha)}(x;q). +\end{eqnarray} + +\subsection*{Normalized recurrence relation} +\begin{eqnarray} +\label{NormRecqLaguerre} +xp_n(x)&=&p_{n+1}(x)+q^{-2n-\alpha-1}\left[(1-q^{n+1})+q(1-q^{n+\alpha})\right]p_n(x)\nonumber\\ +& &{}\mathindent{}+q^{-4n-2\alpha+1}(1-q^n)(1-q^{n+\alpha})p_{n-1}(x), +\end{eqnarray} +where +$$L_n^{(\alpha)}(x;q)=\frac{(-1)^nq^{n(n+\alpha)}}{(q;q)_n}p_n(x).$$ + +\subsection*{$q$-Difference equation} +\begin{equation} +\label{dvqLaguerre} +-q^{\alpha}(1-q^n)xy(x)=q^{\alpha}(1+x)y(qx)-\left[1+q^{\alpha}(1+x)\right]y(x)+y(q^{-1}x), +\end{equation} +where +$$y(x)=L_n^{(\alpha)}(x;q).$$ + +\subsection*{Forward shift operator} +\begin{equation} +\label{shift1qLaguerreI} +L_n^{(\alpha)}(x;q)-L_n^{(\alpha)}(qx;q)=-q^{\alpha+1}xL_{n-1}^{(\alpha+1)}(qx;q) +\end{equation} +or equivalently +\begin{equation} +\label{shift1qLaguerreII} +\mathcal{D}_qL_n^{(\alpha)}(x;q)=-\frac{q^{\alpha+1}}{1-q}L_{n-1}^{(\alpha+1)}(qx;q). +\end{equation} + +\subsection*{Backward shift operator} +\begin{equation} +\label{shift2qLaguerreI} +L_n^{(\alpha)}(x;q)-q^{\alpha}(1+x)L_n^{(\alpha)}(qx;q)=(1-q^{n+1})L_{n+1}^{(\alpha-1)}(x;q) +\end{equation} +or equivalently +\begin{equation} +\label{shift2qLaguerreII} +\mathcal{D}_q\left[w(x;\alpha;q)L_n^{(\alpha)}(x;q)\right]= +\frac{1-q^{n+1}}{1-q}w(x;\alpha-1;q)L_{n+1}^{(\alpha-1)}(x;q), +\end{equation} +where +$$w(x;\alpha;q)=\frac{x^{\alpha}}{(-x;q)_{\infty}}.$$ + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodqLaguerre} +w(x;\alpha;q)L_n^{(\alpha)}(x;q)= +\frac{(1-q)^n}{(q;q)_n}\left(\mathcal{D}_q\right)^n\left[w(x;\alpha+n;q)\right]. +\end{equation} + +\subsection*{Generating functions} +\begin{equation} +\label{GenqLaguerre1} +\frac{1}{(t;q)_{\infty}}\,\qhyp{1}{1}{-x}{0}{q^{\alpha+1}t} +=\sum_{n=0}^{\infty}L_n^{(\alpha)}(x;q)t^n. +\end{equation} + +\begin{equation} +\label{GenqLaguerre2} +\frac{1}{(t;q)_{\infty}}\,\qhyp{0}{1}{-}{q^{\alpha+1}}{-q^{\alpha+1}xt} +=\sum_{n=0}^{\infty}\frac{L_n^{(\alpha)}(x;q)}{(q^{\alpha+1};q)_n}t^n. +\end{equation} + +\begin{equation} +\label{GenqLaguerre3} +(t;q)_{\infty}\cdot\qhyp{0}{2}{-}{q^{\alpha+1},t}{-q^{\alpha+1}xt} +=\sum_{n=0}^{\infty}\frac{(-1)^nq^{\binom{n}{2}}}{(q^{\alpha+1};q)_n}L_n^{(\alpha)}(x;q)t^n. +\end{equation} + +\begin{eqnarray} +\label{GenqLaguerre4} +& &\frac{(\gamma t;q)_{\infty}}{(t;q)_{\infty}}\,\qhyp{1}{2}{\gamma}{q^{\alpha+1},\gamma t}{-q^{\alpha+1}xt}\nonumber\\ +& &{}=\sum_{n=0}^{\infty}\frac{(\gamma;q)_n}{(q^{\alpha+1};q)_n}L_n^{(\alpha)}(x;q)t^n, +\quad\textrm{$\gamma$ arbitrary}. +\end{eqnarray} + +\subsection*{Limit relations} + +\subsubsection*{Little $q$-Jacobi $\rightarrow$ $q$-Laguerre} +If we substitute $a=q^{\alpha}$ and $x\rightarrow -b^{-1}q^{-1}x$ in the definition +(\ref{DefLittleqJacobi}) of the little $q$-Jacobi polynomials and then take the limit +$b\rightarrow -\infty$ we find the $q$-Laguerre polynomials given by (\ref{DefqLaguerre}): +$$\lim_{b\rightarrow -\infty}p_n(-b^{-1}q^{-1}x;q^{\alpha},b|q)= +\frac{(q;q)_n}{(q^{\alpha+1};q)_n}L_n^{(\alpha)}(x;q).$$ + +\subsubsection*{$q$-Meixner $\rightarrow$ $q$-Laguerre} +The $q$-Laguerre polynomials given by (\ref{DefqLaguerre}) can be obtained +from the $q$-Meixner polynomials given by (\ref{DefqMeixner}) by setting +$b=q^{\alpha}$ and $q^{-x}\rightarrow cq^{\alpha}x$ in the definition +(\ref{DefqMeixner}) of the $q$-Meixner polynomials and then taking the limit +$c\rightarrow\infty$: +$$\lim_{c\rightarrow\infty}M_n(cq^{\alpha}x;q^{\alpha},c;q)= +\frac{(q;q)_n}{(q^{\alpha+1};q)_n}L_n^{(\alpha)}(x;q).$$ + +\subsubsection*{$q$-Laguerre $\rightarrow$ Stieltjes-Wigert} +If we set $x\rightarrow xq^{-\alpha}$ in the definition +(\ref{DefqLaguerre}) of the $q$-Laguerre polynomials and take the limit +$\alpha\rightarrow\infty$ we simply obtain the Stieltjes-Wigert polynomials given +by (\ref{DefStieltjesWigert}): +\begin{equation} +\lim_{\alpha\rightarrow\infty}L_n^{(\alpha)}\left(xq^{-\alpha};q\right)=S_n(x;q). +\end{equation} + +\subsubsection*{$q$-Laguerre $\rightarrow$ Laguerre / Charlier} +If we set $x\rightarrow (1-q)x$ in the definition (\ref{DefqLaguerre}) +of the $q$-Laguerre polynomials and take the limit $q\rightarrow 1$ we obtain +the Laguerre polynomials given by (\ref{DefLaguerre}): +\begin{equation} +\lim_{q\rightarrow 1}L_n^{(\alpha)}((1-q)x;q)=L_n^{(\alpha)}(x). +\end{equation} + +If we set $x\rightarrow -q^{-x}$ and $q^{\alpha}=a^{-1}(q-1)^{-1}$ (or +$\alpha=-(\ln q)^{-1}\ln (q-1)a$) in the definition (\ref{DefqLaguerre}) of the +$q$-Laguerre polynomials, multiply by $(q;q)_n$, and take the limit +$q\rightarrow 1$ we obtain the Charlier polynomials given by (\ref{DefCharlier}): +\begin{equation} +\lim_{q\rightarrow 1}\,(q;q)_nL_n^{(\alpha)}(-q^{-x};q)=C_n(x;a), +\end{equation} +where +$$q^{\alpha}=\frac{1}{a(q-1)}\quad\textrm{or}\quad\alpha=-\frac{\ln (q-1)a}{\ln q}.$$ + +\subsection*{Remarks} +The $q$-Laguerre polynomials are sometimes called the +generalized Stieltjes-Wigert polynomials. + +If we replace $q$ by $q^{-1}$ we obtain the little $q$-Laguerre +(or Wall) polynomials given by (\ref{DefLittleqLaguerre}) in the following +way: +$$L_n^{(\alpha)}(x;q^{-1})=\frac{(q^{\alpha+1};q)_n}{(q;q)_nq^{n\alpha}}p_n(-x;q^{\alpha}|q).$$ + +\noindent +The $q$-Laguerre polynomials given by (\ref{DefqLaguerre}) and the $q$-Bessel polynomials +given by (\ref{DefqBessel}) are related in the following way: +$$\frac{y_n(q^x;a;q)}{(q;q)_n}=L_n^{(x-n)}(aq^n;q).$$ + +\noindent +The $q$-Laguerre polynomials given by (\ref{DefqLaguerre}) and the +$q$-Charlier polynomials given by (\ref{DefqCharlier}) are related in the +following way: +$$\frac{C_n(-x;-q^{-\alpha};q)}{(q;q)_n}=L_n^{(\alpha)}(x;q).$$ + +\noindent +Since the Stieltjes and Hamburger moment problems corresponding to the +$q$-Laguerre polynomials are indeterminate there exist many different weight +functions. + RS: add begin\label{sec14.20} +% +\paragraph{\large\bf KLSadd: Notation}Here the little $q$-Laguerre polynomial is denoted by +$p_n(x;a;q)$ instead of +$p_n(x;a\,|\, q)$. +% +\paragraph{\large\bf KLSadd: Re: (14.20.11)}The \RHS\ of this generating function converges for $|xt|<1$. +We can rewrite the \LHS\ by use of the transformation +\begin{equation*} +\qhyp21{0,0}c{q,z}=\frac1{(z;q)_\iy}\,\qhyp01-c{q,cz}. +\end{equation*} +Then we obtain: +\begin{equation} +(t;q)_\iy\,\qhyp21{0,0}{aq}{q,xt} +=\sum_{n=0}^\iy\frac{(-1)^n\,q^{\half n(n-1)}}{(q;q)_n}\, +p_n(x;a;q)\,t^n\qquad(|xt|<1). +\label{35} +\end{equation} +% +\subsubsection*{Expansion of $x^n$} +Divide both sides of \eqref{35} by $(t;q)_\iy$. Then coefficients of the +same power of $t$ on both sides must be equal. We obtain: +\begin{equation} +x^n=(a;q)_n\,\sum_{k=0}^n \frac{(q^{-n};q)_k}{(q;q)_k}\,q^{nk}\,p_k(x;a;q). +\label{36} +\end{equation} +% +\subsubsection*{Quadratic transformations} +Little $q$-Laguerre polynomials $p_n(x;a;q)$ with $a=q^{\pm\half}$ are +related to discrete $q$-Hermite I polynomials $h_n(x;q)$: +\begin{align} +p_n(x^2;q^{-1};q^2)&= +\frac{(-1)^n q^{-n(n-1)}}{(q;q^2)_n}\,h_{2n}(x;q), +\label{28}\\ +xp_n(x^2;q;q^2)&= +\frac{(-1)^n q^{-n(n-1)}}{(q^3;q^2)_n}\,h_{2n+1}(x;q). +\label{29} +\end{align} +% + +\subsection*{References} +\cite{NAlSalam89}, \cite{AlSalam90}, \cite{Askey86}, \cite{Askey89I}, \cite{AskeyWilson85}, +\cite{AtakAtakI}, \cite{ChenIsmailMuttalib}, \cite{Chihara68II}, \cite{Chihara78}, +\cite{Chihara79}, \cite{Chris}, \cite{Exton77}, \cite{GasperRahman90}, \cite{GrunbaumHaine96}, +\cite{Ismail2005I}, \cite{IsmailRahman98}, \cite{Jain95}, \cite{Moak}. + + +\section{$q$-Bessel} +\index{q-Bessel polynomials@$q$-Bessel polynomials} +\par\setcounter{equation}{0} + +\subsection*{Basic hypergeometric representation} +\begin{eqnarray} +\label{DefqBessel} +y_n(x;a;q)&=&\qhyp{2}{1}{q^{-n},-aq^n}{0}{qx}\\ +&=&(q^{-n+1}x;q)_n\cdot\qhyp{1}{1}{q^{-n}}{q^{-n+1}x}{-aq^{n+1}x}\nonumber\\ +&=&\left(-aq^nx\right)^n\cdot\qhyp{2}{1}{q^{-n},x^{-1}}{0}{-\frac{q^{-n+1}}{a}}.\nonumber +\end{eqnarray} + +\newpage + +\subsection*{Orthogonality relation} +\begin{eqnarray} +\label{OrtqBessel} +& &\sum_{k=0}^{\infty}\frac{a^k}{(q;q)_k}q^{\binom{k+1}{2}}y_m(q^k;a;q)y_n(q^k;a;q)\nonumber\\ +& &{}=(q;q)_n(-aq^n;q)_{\infty}\frac{a^nq^{\binom{n+1}{2}}}{(1+aq^{2n})}\,\delta_{mn},\quad a>0. +\end{eqnarray} + +\subsection*{Recurrence relation} +\begin{equation} +\label{RecqBessel} +-xy_n(x;a;q)=A_ny_{n+1}(x;a;q)-(A_n+C_n)y_n(x;a;q)+C_ny_{n-1}(x;a;q), +\end{equation} +where +$$\left\{\begin{array}{l}\displaystyle A_n=q^n\frac{(1+aq^n)}{(1+aq^{2n})(1+aq^{2n+1})}\\ +\\ +\displaystyle C_n=aq^{2n-1}\frac{(1-q^n)}{(1+aq^{2n-1})(1+aq^{2n})}.\end{array}\right.$$ + +\subsection*{Normalized recurrence relation} +\begin{equation} +\label{NormRecqBessel} +xp_n(x)=p_{n+1}(x)+(A_n+C_n)p_n(x)+A_{n-1}C_np_{n-1}(x), +\end{equation} +where +$$y_n(x;a;q)=(-1)^nq^{-\binom{n}{2}}(-aq^n;q)_np_n(x).$$ + +\subsection*{$q$-Difference equation} +\begin{eqnarray} +\label{dvqBessel} +& &-q^{-n}(1-q^n)(1+aq^n)xy(x)\nonumber\\ +& &{}=axy(qx)-(ax+1-x)y(x)+(1-x)y(q^{-1}x), +\end{eqnarray} +where +$$y(x)=y_n(x;a;q).$$ + +\newpage + +\subsection*{Forward shift operator} +\begin{equation} +\label{shift1qBesselI} +y_n(x;a;q)-y_n(qx;a;q)=-q^{-n+1}(1-q^n)(1+aq^n)xy_{n-1}(x;aq^2;q) +\end{equation} +or equivalently +\begin{equation} +\label{shift1qBesselII} +\mathcal{D}_qy_n(x;a;q)=-\frac{q^{-n+1}(1-q^n)(1+aq^n)}{1-q}y_{n-1}(x;aq^2;q). +\end{equation} + +\subsection*{Backward shift operator} +\begin{equation} +\label{shift2qBesselI} +aq^{x-1}y_n(q^x;a;q)-(1-q^x)y_n(q^{x-1}x;a;q)=-y_{n+1}(q^x;aq^{-2};q) +\end{equation} +or equivalently +\begin{equation} +\label{shift2qBesselII} +\frac{\nabla\left[w(x;a;q)y_n(q^x;a;q)\right]}{\nabla q^x}= +\frac{q^2}{a(1-q)}w(x;aq^{-2};q)y_{n+1}(q^x;aq^{-2};q), +\end{equation} +where +$$w(x;a;q)=\frac{a^xq^{\binom{x}{2}}}{(q;q)_x}.$$ + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodqBessel} +w(x;a;q)y_n(q^x;a;q)=a^n(1-q)^nq^{n(n-1)}\left(\nabla_q\right)^n\left[w(x;aq^{2n};q)\right], +\end{equation} +where +$$\nabla_q:=\frac{\nabla}{\nabla q^x}.$$ + +\subsection*{Generating functions} +\begin{eqnarray} +\label{GenqBessel1} +& &\qhyp{0}{1}{-}{0}{-aq^{x+1}t}\,\qhyp{2}{0}{q^{-x},0}{-}{q^xt}\nonumber\\ +& &{}=\sum_{n=0}^{\infty}\frac{y_n(q^x;a;q)}{(q;q)_n}t^n,\quad x=0,1,2,\ldots. +\end{eqnarray} + +\begin{equation} +\label{GenqBessel2} +\frac{(t;q)_{\infty}}{(xt;q)_{\infty}}\,\qhyp{1}{3}{xt}{0,0,t}{-aqxt}= +\sum_{n=0}^{\infty}\frac{(-1)^nq^{\binom{n}{2}}}{(q;q)_n}y_n(x;a;q)t^n. +\end{equation} + +\subsection*{Limit relations} + +\subsubsection*{Little $q$-Jacobi $\rightarrow$ $q$-Bessel} +If we set $b\rightarrow -a^{-1}q^{-1}b$ in the definition (\ref{DefLittleqJacobi}) of +the little $q$-Jacobi polynomials and then take the limit $a\rightarrow 0$ we obtain +the $q$-Bessel polynomials given by (\ref{DefqBessel}): +$$\lim_{a\rightarrow 0}p_n(x;a,-a^{-1}q^{-1}b|q)=y_n(x;b;q).$$ + +\subsubsection*{$q$-Krawtchouk $\rightarrow$ $q$-Bessel} +If we set $x\rightarrow N-x$ in the definition (\ref{DefqKrawtchouk}) of the +$q$-Krawtchouk polynomials and then take the limit $N\rightarrow\infty$ we +obtain the $q$-Bessel polynomials given by (\ref{DefqBessel}): +$$\lim_{N\rightarrow\infty}K_n(q^{x-N};p,N;q)=y_n(q^x;p;q).$$ + +\subsubsection*{$q$-Bessel $\rightarrow$ Stieltjes-Wigert} +The Stieltjes-Wigert polynomials given by (\ref{DefStieltjesWigert}) can be obtained +from the $q$-Bessel polynomials by setting $x\rightarrow a^{-1}x$ in the definition +(\ref{DefqBessel}) of the $q$-Bessel polynomials and then taking the limit +$a\rightarrow\infty$. In fact we have +\begin{equation} +\lim_{a\rightarrow\infty}y_n(a^{-1}x;a;q)=(q;q)_nS_n(x;q). +\end{equation} + +\subsubsection*{$q$-Bessel $\rightarrow$ Bessel} +If we set $x\rightarrow -\frac{1}{2}(1-q)^{-1}x$ and $a\rightarrow -q^{a+1}$ in the definition +(\ref{DefqBessel}) of the $q$-Bessel polynomials and take the limit $q\rightarrow 1$ +we find the Bessel polynomials given by (\ref{DefBessel}): +\begin{equation} +\lim_{q\rightarrow 1}y_n(-\textstyle\frac{1}{2}(1-q)^{-1}x;-q^{a+1};q)=y_n(x;a). +\end{equation} + +\subsubsection*{$q$-Bessel $\rightarrow$ Charlier} +If we set $x\rightarrow q^x$ and $a\rightarrow a(1-q)$ in the definition (\ref{DefqBessel}) +of the $q$-Bessel polynomials and take the limit $q\rightarrow 1$ we find the Charlier +polynomials given by (\ref{DefCharlier}): +\begin{equation} +\lim_{q\rightarrow 1}\frac{y_n(q^x;a(1-q);q)}{(q-1)^n}=a^nC_n(x;a). +\end{equation} + +\subsection*{Remark} +In \cite{Koekoek94} and \cite{Koekoek98} these $q$-Bessel polynomials were called +\emph{alternative $q$-Charlier polynomials}. + +\noindent +The $q$-Bessel polynomials given by (\ref{DefqBessel}) and the $q$-Laguerre +polynomials given by (\ref{DefqLaguerre}) are related in the following way: +$$\frac{y_n(q^x;a;q)}{(q;q)_n}=L_n^{(x-n)}(aq^n;q).$$ + +\subsection*{Reference} +\cite{DattaGriffin}. + + +\section{$q$-Charlier}\index{q-Charlier polynomials@$q$-Charlier polynomials} +\par\setcounter{equation}{0} + +\subsection*{Basic hypergeometric representation} +\begin{eqnarray} +\label{DefqCharlier} +C_n(q^{-x};a;q)&=&\qhyp{2}{1}{q^{-n},q^{-x}}{0}{-\frac{q^{n+1}}{a}}\\ +&=&(-a^{-1}q;q)_n\cdot\qhyp{1}{1}{q^{-n}}{-a^{-1}q}{-\frac{q^{n+1-x}}{a}}.\nonumber +\end{eqnarray} + +\subsection*{Orthogonality relation} +\begin{eqnarray} +\label{OrtqCharlier} +& &\sum_{x=0}^{\infty}\frac{a^x}{(q;q)_x}q^{\binom{x}{2}}C_m(q^{-x};a;q)C_n(q^{-x};a;q)\nonumber\\ +& &{}=q^{-n}(-a;q)_{\infty}(-a^{-1}q,q;q)_n\,\delta_{mn},\quad a>0. +\end{eqnarray} + +\newpage + +\subsection*{Recurrence relation} +\begin{eqnarray} +\label{RecqCharlier} +& &q^{2n+1}(1-q^{-x})C_n(q^{-x})\nonumber\\ +& &{}=aC_{n+1}(q^{-x})-\left[a+q(1-q^n)(a+q^n)\right]C_n(q^{-x})\nonumber\\ +& &{}\mathindent{}+q(1-q^n)(a+q^n)C_{n-1}(q^{-x}), +\end{eqnarray} +where +$$C_n(q^{-x}):=C_n(q^{-x};a;q).$$ + +\subsection*{Normalized recurrence relation} +\begin{eqnarray} +\label{NormRecqCharlier} +xp_n(x)&=&p_{n+1}(x)+\left[1+q^{-2n-1}\left\{a+q(1-q^n)(a+q^n)\right\}\right]p_n(x)\nonumber\\ +& &{}\mathindent{}+aq^{-4n+1}(1-q^n)(a+q^n)p_{n-1}(x), +\end{eqnarray} +where +$$C_n(q^{-x};a;q)=\frac{(-1)^nq^{n^2}}{a^n}p_n(q^{-x}).$$ + +\subsection*{$q$-Difference equation} +\begin{equation} +\label{dvqCharlier} +q^ny(x)=aq^xy(x+1)-q^x(a-1)y(x)+(1-q^x)y(x-1), +\end{equation} +where +$$y(x)=C_n(q^{-x};a;q).$$ + +\subsection*{Forward shift operator} +\begin{equation} +\label{shift1qCharlierI} +C_n(q^{-x-1};a;q)-C_n(q^{-x};a;q)=-a^{-1}q^{-x}(1-q^n)C_{n-1}(q^{-x};aq^{-1};q) +\end{equation} +or equivalently +\begin{equation} +\label{shift1qCharlierII} +\frac{\Delta C_n(q^{-x};a;q)}{\Delta q^{-x}}=-\frac{q(1-q^n)}{a(1-q)}C_{n-1}(q^{-x};aq^{-1};q). +\end{equation} + +\newpage + +\subsection*{Backward shift operator} +\begin{equation} +\label{shift2qCharlierI} +C_n(q^{-x};a;q)-a^{-1}q^{-x}(1-q^x)C_n(q^{-x+1};a;q)=C_{n+1}(q^{-x};aq;q) +\end{equation} +or equivalently +\begin{equation} +\label{shift2qCharlierII} +\frac{\nabla\left[w(x;a;q)C_n(q^{-x};a;q)\right]}{\nabla q^{-x}} +=\frac{1}{1-q}w(x;aq;q)C_{n+1}(q^{-x};aq;q), +\end{equation} +where +$$w(x;a;q)=\frac{a^xq^{\binom{x+1}{2}}}{(q;q)_x}.$$ + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodqCharlier} +w(x;a;q)C_n(q^{-x};a;q)=(1-q)^n\left(\nabla_q\right)^n\left[w(x;aq^{-n};q)\right], +\end{equation} +where +$$\nabla_q:=\frac{\nabla}{\nabla q^{-x}}.$$ + +\subsection*{Generating functions} +\begin{equation} +\label{GenqCharlier1} +\frac{1}{(t;q)_{\infty}}\,\qhyp{1}{1}{q^{-x}}{0}{-a^{-1}qt} +=\sum_{n=0}^{\infty}\frac{C_n(q^{-x};a;q)}{(q;q)_n}t^n. +\end{equation} + +\begin{equation} +\label{GenqCharlier2} +\frac{1}{(t;q)_{\infty}}\,\qhyp{0}{1}{-}{-a^{-1}q}{-a^{-1}q^{-x+1}t} +=\sum_{n=0}^{\infty}\frac{C_n(q^{-x};a;q)}{(-a^{-1}q,q;q)_n}t^n. +\end{equation} + +\subsection*{Limit relations} + +\subsubsection*{$q$-Meixner $\rightarrow$ $q$-Charlier} +The $q$-Charlier polynomials given by (\ref{DefqCharlier}) can easily be +obtained from the $q$-Meixner given by (\ref{DefqMeixner}) by setting $b=0$ +in the definition (\ref{DefqMeixner}) of the $q$-Meixner polynomials: +\begin{equation} +M_n(x;0,c;q)=C_n(x;c;q). +\end{equation} + +\subsubsection*{$q$-Krawtchouk $\rightarrow$ $q$-Charlier} +By setting $p=a^{-1}q^{-N}$ in the definition (\ref{DefqKrawtchouk}) of the +$q$-Krawtchouk polynomials and then taking the limit $N\rightarrow\infty$ we +obtain the $q$-Charlier polynomials given by (\ref{DefqCharlier}): +$$\lim_{N\rightarrow\infty}K_n(q^{-x};a^{-1}q^{-N},N;q)=C_n(q^{-x};a;q).$$ + +\subsubsection*{$q$-Charlier $\rightarrow$ Stieltjes-Wigert} +If we set $q^{-x}\rightarrow ax$ in the definition (\ref{DefqCharlier}) of the +$q$-Charlier polynomials and take the limit $a\rightarrow\infty$ we obtain +the Stieltjes-Wigert polynomials given by (\ref{DefStieltjesWigert}) in the +following way: +\begin{equation} +\lim_{a\rightarrow\infty}C_n(ax;a;q)=(q;q)_nS_n(x;q). +\end{equation} + +\subsubsection*{$q$-Charlier $\rightarrow$ Charlier} +If we set $a\rightarrow a(1-q)$ in the definition (\ref{DefqCharlier}) +of the $q$-Charlier polynomials and take the limit $q\rightarrow 1$ we obtain the +Charlier polynomials given by (\ref{DefCharlier}): +\begin{equation} +\lim_{q\rightarrow 1}C_n(q^{-x};a(1-q);q)=C_n(x;a). +\end{equation} + +\subsection*{Remark} +The $q$-Charlier polynomials given by (\ref{DefqCharlier}) and the +$q$-Laguerre polynomials given by (\ref{DefqLaguerre}) are related in the +following way: +$$\frac{C_n(-x;-q^{-\alpha};q)}{(q;q)_n}=L_n^{(\alpha)}(x;q).$$ + +\subsection*{References} +\cite{AlvarezRonveaux}, \cite{AtakRahmanSuslov}, \cite{GasperRahman90}, +\cite{Hahn}, \cite{Koelink96III}, \cite{Nikiforov+}, \cite{Zeng95}. + + +\section{Al-Salam-Carlitz~I}\index{Al-Salam-Carlitz~I polynomials} +\par\setcounter{equation}{0} + +\subsection*{Basic hypergeometric representation} +\begin{equation} +\label{DefAlSalamCarlitzI} +U_n^{(a)}(x;q)=(-a)^nq^{\binom{n}{2}}\,\qhyp{2}{1}{q^{-n},x^{-1}}{0}{\frac{qx}{a}}. +\end{equation} + +\subsection*{Orthogonality relation} +\begin{eqnarray} +\label{OrtAlSalamCarlitzI} +& &\int_a^1(qx,a^{-1}qx;q)_{\infty}U_m^{(a)}(x;q)U_n^{(a)}(x;q)\,d_qx\nonumber\\ +& &{}=(-a)^n(1-q)(q;q)_n(q,a,a^{-1}q;q)_{\infty}q^{\binom{n}{2}}\,\delta_{mn},\quad a<0. +\end{eqnarray} + +\subsection*{Recurrence relation} +\begin{equation} +\label{RecAlSalamCarlitzI} +xU_n^{(a)}(x;q)=U_{n+1}^{(a)}(x;q)+(a+1)q^nU_n^{(a)}(x;q) +-aq^{n-1}(1-q^n)U_{n-1}^{(a)}(x;q). +\end{equation} + +\subsection*{Normalized recurrence relation} +\begin{equation} +\label{NormRecAlSalamCarlitzI} +xp_n(x)=p_{n+1}(x)+(a+1)q^np_n(x)-aq^{n-1}(1-q^n)p_{n-1}(x), +\end{equation} +where +$$U_n^{(a)}(x;q)=p_n(x).$$ + +\subsection*{$q$-Difference equation} +\begin{eqnarray} +\label{dvAlSalamCarlitzI} +(1-q^n)x^2y(x)&=&aq^{n-1}y(qx)-\left[aq^{n-1}+q^n(1-x)(a-x)\right]y(x)\nonumber\\ +& &{}\mathindent{}+q^n(1-x)(a-x)y(q^{-1}x), +\end{eqnarray} +where +$$y(x)=U_n^{(a)}(x;q).$$ + +\subsection*{Forward shift operator} +\begin{equation} +\label{shift1AlSalamCarlitzI-I} +U_n^{(a)}(x;q)-U_n^{(a)}(qx;q)=(1-q^n)xU_{n-1}^{(a)}(x;q) +\end{equation} +or equivalently +\begin{equation} +\label{shift1AlSalamCarlitzI-II} +\mathcal{D}_qU_n^{(a)}(x;q)=\frac{1-q^n}{1-q}U_{n-1}^{(a)}(x;q). +\end{equation} + +\subsection*{Backward shift operator} +\begin{equation} +\label{shift2AlSalamCarlitzI-I} +aU_n^{(a)}(x;q)-(1-x)(a-x)U_n^{(a)}(q^{-1}x;q)=-q^{-n}xU_{n+1}^{(a)}(x;q) +\end{equation} +or equivalently +\begin{equation} +\label{shift2AlSalamCarlitzI-II} +\mathcal{D}_{q^{-1}}\left[w(x;a;q)U_n^{(a)}(x;q)\right]= +\frac{q^{-n+1}}{a(1-q)}w(x;a;q)U_{n+1}^{(a)}(x;q), +\end{equation} +where +$$w(x;a;q)=(qx,a^{-1}qx;q)_{\infty}.$$ + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodAlSalamCarlitzI} +w(x;a;q)U_n^{(a)}(x;q)=a^nq^{\frac{1}{2}n(n-3)}(1-q)^n +\left(\mathcal{D}_{q^{-1}}\right)^n\left[w(x;a;q)\right]. +\end{equation} + +\subsection*{Generating function} +\begin{equation} +\label{GenAlSalamCarlitzI} +\frac{(t,at;q)_{\infty}}{(xt;q)_{\infty}}= +\sum_{n=0}^{\infty}\frac{U_n^{(a)}(x;q)}{(q;q)_n}t^n. +\end{equation} + +\subsection*{Limit relations} + +\subsubsection*{Big $q$-Laguerre $\rightarrow$ Al-Salam-Carlitz~I} +If we set $x\rightarrow aqx$ and $b\rightarrow ab$ in the definition +(\ref{DefBigqLaguerre}) of the big $q$-Laguerre polynomials and take the +limit $a\rightarrow 0$ we obtain the Al-Salam-Carlitz~I polynomials given by +(\ref{DefAlSalamCarlitzI}): +$$\lim_{a\rightarrow 0}\frac{P_n(aqx;a,ab;q)}{a^n}=q^nU_n^{(b)}(x;q).$$ + +\subsubsection*{Dual $q$-Krawtchouk $\rightarrow$ Al-Salam-Carlitz~I} +If we set $c=a^{-1}$ in the definition (\ref{DefDualqKrawtchouk}) +of the dual $q$-Krawtchouk polynomials and take the limit +$N\rightarrow\infty$ we simply obtain the Al-Salam-Carlitz~I polynomials +given by (\ref{DefAlSalamCarlitzI}): +$$\lim_{N\rightarrow\infty}K_n(\lambda(x);a^{-1},N|q)= +\left(-\frac{1}{a}\right)^nq^{-\binom{n}{2}}U_n^{(a)}(q^x;q).$$ +Note that $\lambda(x)=q^{-x}+a^{-1}q^{x-N}$. + +\subsubsection*{Al-Salam-Carlitz~I $\rightarrow$ Discrete $q$-Hermite~I} +The discrete $q$-Hermite~I polynomials given by +(\ref{DefDiscreteqHermiteI}) can easily be obtained from the +Al-Salam-Carlitz~I polynomials given by (\ref{DefAlSalamCarlitzI}) by the +substitution $a=-1$: +\begin{equation} +U_n^{(-1)}(x;q)=h_n(x;q). +\end{equation} + +\subsubsection*{Al-Salam-Carlitz~I $\rightarrow$ Charlier / Hermite} +If we set $a\rightarrow a(q-1)$ and $x\rightarrow q^x$ in the definition +(\ref{DefAlSalamCarlitzI}) of the Al-Salam-Carlitz~I polynomials and take the +limit $q\rightarrow 1$ after dividing by $a^n(1-q)^n$ we obtain the Charlier +polynomials given by (\ref{DefCharlier}): +\begin{equation} +\lim_{q\rightarrow 1}\frac{U_n^{(a(q-1))}(q^x;q)}{(1-q)^n}=a^nC_n(x;a). +\end{equation} + +If we set $x\rightarrow x\sqrt{1-q^2}$ and $a\rightarrow a\sqrt{1-q^2}-1$ +in the definition (\ref{DefAlSalamCarlitzI}) of the Al-Salam-Carlitz~I +polynomials, divide by $(1-q^2)^{\frac{n}{2}}$, and let $q$ tend to $1$ we +obtain the Hermite polynomials given by (\ref{DefHermite}) with shifted +argument. In fact we have +\begin{equation} +\lim_{q\rightarrow 1}\frac{U_n^{(a\sqrt{1-q^2}-1)}(x\sqrt{1-q^2};q)} +{(1-q^2)^{\frac{n}{2}}}=\frac{H_n(x-a)}{2^n}. +\end{equation} + +\subsection*{Remark} +The Al-Salam-Carlitz~I polynomials are related to the Al-Salam-Carlitz~II +polynomials given by (\ref{DefAlSalamCarlitzII}) in the following way: +$$U_n^{(a)}(x;q^{-1})=V_n^{(a)}(x;q).$$ + +\subsection*{References} +\cite{AlSalam90}, \cite{AlSalamCarlitz}, \cite{AlSalamChihara76}, \cite{AskeySuslovII}, +\cite{AtakRahmanSuslov}, \cite{Chihara68II}, \cite{Chihara78}, \cite{DattaGriffin}, +\cite{Dehesa}, \cite{DohaAhmed2005}, \cite{GasperRahman90}, \cite{Ismail85I}, +\cite{IsmailMuldoon}, \cite{Kim}, \cite{Zeng95}. + + +\section{Al-Salam-Carlitz~II}\index{Al-Salam-Carlitz~II polynomials} +\par\setcounter{equation}{0} + +\subsection*{Basic hypergeometric representation} +\begin{equation} +\label{DefAlSalamCarlitzII} +V_n^{(a)}(x;q)= +(-a)^nq^{-\binom{n}{2}}\,\qhyp{2}{0}{q^{-n},x}{-}{\frac{q^n}{a}}. +\end{equation} + +\subsection*{Orthogonality relation} +\begin{eqnarray} +\label{OrtAlSalamCarlitzII} +& &\sum_{k=0}^{\infty}\frac{q^{k^2}a^k}{(q;q)_k(aq;q)_k} +V_m^{(a)}(q^{-k};q)V_n^{(a)}(q^{-k};q)\nonumber\\ +& &{}=\frac{(q;q)_na^n}{(aq;q)_{\infty}q^{n^2}}\,\delta_{mn},\quad 00,\quad\textrm{with}\quad +\gamma^2=-\frac{1}{2\ln q}.$$ + RS: add begin\label{sec14.21} +% +\paragraph{\large\bf KLSadd: Notation}Here the $q$-Laguerre polynomial is denoted by $L_n^\al(x;q)$ instead of +$L_n^{(\al)}(x;q)$. +% +\subsubsection*{Orthogonality relation} +(14.21.2) can be rewritten with simplified \RHS: +\begin{equation} +\int_0^\iy L_m^{\al}(x;q)\,L_n^{\al}(x;q)\,\frac{x^\al}{(-x;q)_\iy}\,dx=h_n\,\de_{m,n} +\qquad(\al>-1) +\label{119} +\end{equation} +with +\begin{equation} +\frac{h_n}{h_0}=\frac{(q^{\al+1};q)_n}{(q;q)_n q^n},\qquad +h_0=-\,\frac{(q^{-\al};q)_\iy}{(q;q)_\iy}\,\frac\pi{\sin(\pi\al)}\,. +\label{120} +\end{equation} +The expression for $h_0$ (which is Askey's $q$-gamma evaluation +\cite[(4.2)]{K16}) +should be interpreted by continuity in $\al$ for +$\al\in\Znonneg$. +Explicitly we can write +\begin{equation} +h_n=q^{-\half\al(\al+1)}\,(q;q)_\al\,\log(q^{-1})\qquad(\al\in\Znonneg). +\label{121} +\end{equation} +% +\subsubsection*{Expansion of $x^n$} +\begin{equation} +x^n=q^{-\half n(n+2\al+1)}\,(q^{\al+1};q)_n\, +\sum_{k=0}^n\frac{(q^{-n};q)_k}{(q^{\al+1};q)_k}\,q^k\,L_k^\al(x;q). +\label{37} +\end{equation} +This follows from \eqref{36} by the equality given in the Remark at the end +of \S14.20. Alternatively, it can be derived in the same way as \eqref{36} +from the generating function (14.21.14). +% +\subsubsection*{Quadratic transformations} +$q$-Laguerre polynomials $L_n^\al(x;q)$ with $\al=\pm\half$ are +related to discrete $q$-Hermite II polynomials $\wt h_n(x;q)$: +\begin{align} +L_n^{-1/2}(x^2;q^2)&= +\frac{(-1)^n q^{2n^2-n}}{(q^2;q^2)_n}\,\wt h_{2n}(x;q), +\label{38}\\ +xL_n^{1/2}(x^2;q^2)&= +\frac{(-1)^n q^{2n^2+n}}{(q^2;q^2)_n}\,\wt h_{2n+1}(x;q). +\label{39} +\end{align} +These follows from \eqref{28} and \eqref{29}, respectively, by applying +the equalities given in the Remarks at the end of \S14.20 and \S14.28. +% + +\subsection*{References} +\cite{Askey86}, \cite{Askey89I}, \cite{AtakAtakIII}, \cite{Chihara70}, \cite{Chihara78}, +\cite{Dehesa}, \cite{Ismail2005I}, \cite{Nikiforov+}, \cite{Stieltjes}, \cite{Szego75}, +\cite{ValentAssche}, \cite{Wigert}. + + +\section{Discrete $q$-Hermite~I} +\index{Discrete q-Hermite~I polynomials@Discrete $q$-Hermite~I polynomials} +\index{q-Hermite~I polynomials@$q$-Hermite~I polynomials!Discrete} +\par\setcounter{equation}{0} + +\subsection*{Basic hypergeometric representation} +The discrete $q$-Hermite~I polynomials are Al-Salam-Carlitz~I polynomials +with $a=-1$: +\begin{eqnarray} +\label{DefDiscreteqHermiteI} +h_n(x;q)=U_n^{(-1)}(x;q)&=&q^{\binom{n}{2}}\,\qhyp{2}{1}{q^{-n},x^{-1}}{0}{-qx}\\ +&=&x^n\qhypK{2}{0}{q^{-n},q^{-n+1}}{0}{q^2,\frac{q^{2n-1}}{x^2}}\nonumber +\end{eqnarray} + +\subsection*{Orthogonality relation} +\begin{eqnarray} +\label{OrtDiscreteqHermiteI} +& &\int_{-1}^1(qx,-qx;q)_{\infty}h_m(x;q)h_n(x;q)\,d_qx\nonumber\\ +& &{}=(1-q)(q;q)_n(q,-1,-q;q)_{\infty}q^{\binom{n}{2}}\,\delta_{mn}. +\end{eqnarray} + +\subsection*{Recurrence relation} +\begin{equation} +\label{RecDiscreteqHermiteI} +xh_n(x;q)=h_{n+1}(x;q)+q^{n-1}(1-q^n)h_{n-1}(x;q). +\end{equation} + +\subsection*{Normalized recurrence relation} +\begin{equation} +\label{NormRecDiscreteqHermiteI} +xp_n(x)=p_{n+1}(x)+q^{n-1}(1-q^n)p_{n-1}(x), +\end{equation} +where +$$h_n(x;q)=p_n(x).$$ + +\subsection*{$q$-Difference equation} +\begin{equation} +\label{dvDiscreteqHermiteI} +-q^{-n+1}x^2y(x)=y(qx)-(1+q)y(x)+q(1-x^2)y(q^{-1}x), +\end{equation} +where +$$y(x)=h_n(x;q).$$ + +\subsection*{Forward shift operator} +\begin{equation} +\label{shift1DiscreteqHermiteI-I} +h_n(x;q)-h_n(qx;q)=(1-q^n)xh_{n-1}(x;q) +\end{equation} +or equivalently +\begin{equation} +\label{shift1DiscreteqHermiteI-II} +\mathcal{D}_qh_n(x;q)=\frac{1-q^n}{1-q}h_{n-1}(x;q). +\end{equation} + +\subsection*{Backward shift operator} +\begin{equation} +\label{shift2DiscreteqHermiteI-I} +h_n(x;q)-(1-x^2)h_n(q^{-1}x;q)=q^{-n}xh_{n+1}(x;q) +\end{equation} +or equivalently +\begin{equation} +\label{shift2DiscreteqHermiteI-II} +\mathcal{D}_{q^{-1}}\left[w(x;q)h_n(x;q)\right] +=-\frac{q^{-n+1}}{1-q}w(x;q)h_{n+1}(x;q), +\end{equation} +where +$$w(x;q)=(qx,-qx;q)_{\infty}.$$ + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodDiscreteqHermiteI} +w(x;q)h_n(x;q)=(q-1)^nq^{\frac{1}{2}n(n-3)} +\left(\mathcal{D}_{q^{-1}}\right)^n\left[w(x;q)\right]. +\end{equation} + +\subsection*{Generating function} +\begin{equation} +\label{GenDiscreteqHermiteI} +\frac{(t^2;q^2)_{\infty}}{(xt;q)_{\infty}}= +\sum_{n=0}^{\infty}\frac{h_n(x;q)}{(q;q)_n}t^n. +\end{equation} + +\subsection*{Limit relations} + +\subsubsection*{Al-Salam-Carlitz~I $\rightarrow$ Discrete $q$-Hermite~I} +The discrete $q$-Hermite~I polynomials given by +(\ref{DefDiscreteqHermiteI}) can easily be obtained from the +Al-Salam-Carlitz~I polynomials given by (\ref{DefAlSalamCarlitzI}) by the +substitution $a=-1$: +$$U_n^{(-1)}(x;q)=h_n(x;q).$$ + +\subsubsection*{Discrete $q$-Hermite~I $\rightarrow$ Hermite} +The Hermite polynomials given by (\ref{DefHermite}) can be found +from the discrete $q$-Hermite~I polynomials given by +(\ref{DefDiscreteqHermiteI}) in the following way: +\begin{equation} +\lim_{q\rightarrow 1}\frac{h_n(x\sqrt{1-q^2};q)} +{(1-q^2)^{\frac{n}{2}}}=\frac{H_n(x)}{2^n}. +\end{equation} + +\subsection*{Remark} The discrete $q$-Hermite~I polynomials are related to the +discrete $q$-Hermite~II polynomials given by (\ref{DefDiscreteqHermiteII}) +in the following way: +$$h_n(ix;q^{-1})=i^n{\tilde h}_n(x;q).$$ + +\subsection*{References} +\cite{AlSalam90}, \cite{AlSalamCarlitz}, \cite{AtakRahmanSuslov}, +\cite{BergIsmail}, \cite{BustozIsmail82}, \cite{GasperRahman90}, \cite{Hahn}, +\cite{Koorn97}. + + +\newpage + +\section{Discrete $q$-Hermite~II} +\index{Discrete q-Hermite~II polynomials@Discrete $q$-Hermite~II polynomials} +\index{q-Hermite~II polynomials@$q$-Hermite~II polynomials!Discrete} +\par\setcounter{equation}{0} + +\subsection*{Basic hypergeometric representation} +The discrete $q$-Hermite~II polynomials are Al-Salam-Carlitz~II polynomials +with $a=-1$: +\begin{eqnarray} +\label{DefDiscreteqHermiteII} +{\tilde h}_n(x;q)=i^{-n}V_n^{(-1)}(ix;q) +&=&i^{-n}q^{-\binom{n}{2}}\,\qhyp{2}{0}{q^{-n},ix}{-}{-q^n}\\ +&=&x^n +\qhypK{2}{1}{q^{-n},q^{-n+1}}{0}{q^2,-\frac{q^2}{x^2}}\nonumber +\end{eqnarray} + +\subsection*{Orthogonality relation} +\begin{eqnarray} +\label{OrtDiscreteqHermiteIIa} +& &\sum_{k=-\infty}^{\infty}\left[{\tilde h}_m(cq^k;q) +{\tilde h}_n(cq^k;q)+{\tilde h}_m(-cq^k;q){\tilde h}_n(-cq^k;q)\right] +w(cq^k;q)q^k\nonumber\\ +& &{}=2\frac{(q^2,-c^2q,-c^{-2}q;q^2)_{\infty}}{(q,-c^2,-c^{-2}q^2;q^2)_{\infty}} +\frac{(q;q)_n}{q^{n^2}}\,\delta_{mn},\quad c>0, +\end{eqnarray} +where +$$w(x;q)=\frac{1}{(ix,-ix;q)_{\infty}}=\frac{1}{(-x^2;q^2)_{\infty}}.$$ +For $c=1$ this orthogonality relation can also be written as +\begin{equation} +\label{OrtDiscreteqHermiteIIb} +\int_{-\infty}^{\infty}\frac{{\tilde h}_m(x;q){\tilde h}_n(x;q)}{(-x^2;q^2)_{\infty}}\,d_qx +=\frac{\left(q^2,-q,-q;q^2\right)_{\infty}}{\left(q^3,-q^2,-q^2;q^2\right)_{\infty}} +\frac{(q;q)_n}{q^{n^2}}\,\delta_{mn}. +\end{equation} + +\subsection*{Recurrence relation} +\begin{equation} +\label{RecDiscreteqHermiteII} +x{\tilde h}_n(x;q)={\tilde h}_{n+1}(x;q)+q^{-2n+1}(1-q^n){\tilde h}_{n-1}(x;q). +\end{equation} + +\subsection*{Normalized recurrence relation} +\begin{equation} +\label{NormRecDiscreteqHermiteII} +xp_n(x)=p_{n+1}(x)+q^{-2n+1}(1-q^n)p_{n-1}(x), +\end{equation} +where +$${\tilde h}_n(x;q)=p_n(x).$$ + +\subsection*{$q$-Difference equation} +\begin{eqnarray} +\label{dvDiscreteqHermiteII} +& &-(1-q^n)x^2{\tilde h}_n(x;q)\nonumber\\ +& &{}=(1+x^2){\tilde h}_n(qx;q)-(1+x^2+q){\tilde h}_n(x;q)+q{\tilde h}_n(q^{-1}x;q). +\end{eqnarray} + +\subsection*{Forward shift operator} +\begin{equation} +\label{shift1DiscreteqHermiteII-I} +\tilde{h}_n(x;q)-\tilde{h}_n(qx;q)=q^{-n+1}(1-q^n)x\tilde{h}_{n-1}(qx;q) +\end{equation} +or equivalently +\begin{equation} +\label{shift1DiscreteqHermiteII-II} +\mathcal{D}_q\tilde{h}_n(x;q)=\frac{q^{-n+1}(1-q^n)}{1-q}\tilde{h}_{n-1}(qx;q). +\end{equation} + +\subsection*{Backward shift operator} +\begin{equation} +\label{shift2DiscreteqHermiteII-I} +\tilde{h}_n(x;q)-(1+x^2)\tilde{h}_n(qx;q)=-q^nx\tilde{h}_{n+1}(x;q) +\end{equation} +or equivalently +\begin{equation} +\label{shift2DiscreteqHermiteII-II} +\mathcal{D}_q\left[w(x;q)\tilde{h}_n(x;q)\right] +=-\frac{q^n}{1-q}w(x;q)\tilde{h}_{n+1}(x;q). +\end{equation} + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodDiscreteqHermiteII} +w(x;q)\tilde{h}_n(x;q)=(q-1)^nq^{-\binom{n}{2}} +\left(\mathcal{D}_q\right)^n\left[w(x;q)\right]. +\end{equation} + +\subsection*{Generating functions} +\begin{equation} +\label{GenDiscreteqHermiteII1} +\frac{(-xt;q)_{\infty}}{(-t^2;q^2)_{\infty}}=\sum_{n=0}^{\infty} +\frac{q^{\binom{n}{2}}}{(q;q)_n}{\tilde h}_n(x;q)t^n. +\end{equation} + +\begin{equation} +\label{GenDiscreteqHermiteII2} +(-it;q)_{\infty}\cdot\qhyp{1}{1}{ix}{-it}{it}=\sum_{n=0}^{\infty} +\frac{(-1)^nq^{n(n-1)}}{(q;q)_n}{\tilde h}_n(x;q)t^n. +\end{equation} + +\subsection*{Limit relations} + +\subsubsection*{Al-Salam-Carlitz~II $\rightarrow$ Discrete $q$-Hermite~II} +The discrete $q$-Hermite~II polynomials given by +(\ref{DefDiscreteqHermiteII}) follow from the Al-Salam-Carlitz~II +polynomials given by (\ref{DefAlSalamCarlitzII}) by the substitution $a=-1$ +in the following way: +$$i^{-n}V_n^{(-1)}(ix;q)={\tilde h}_n(x;q).$$ + +\subsubsection*{Discrete $q$-Hermite~II $\rightarrow$ Hermite} +The Hermite polynomials given by (\ref{DefHermite}) can be found +from the discrete $q$-Hermite~II polynomials given by +(\ref{DefDiscreteqHermiteII}) in the following way: +\begin{equation} +\lim_{q\rightarrow 1}\frac{{\tilde h}_n(x\sqrt{1-q^2};q)} +{(1-q^2)^{\frac{n}{2}}}=\frac{H_n(x)}{2^n}. +\end{equation} + +\subsection*{Remark} The discrete $q$-Hermite~II polynomials are related to the +discrete $q$-Hermite~I polynomials given by (\ref{DefDiscreteqHermiteI}) +in the following way: +$${\tilde h}_n(x;q^{-1})=i^{-n}h_n(ix;q).$$ + +\subsection*{References} +\cite{BergIsmail}, \cite{Koorn97}. + +\end{document} diff --git a/KLSadd_insertion/svmono.cls b/KLSadd_insertion/svmono.cls new file mode 100644 index 0000000..2341bee --- /dev/null +++ b/KLSadd_insertion/svmono.cls @@ -0,0 +1,1690 @@ +% SVMONO DOCUMENT CLASS -- version 3.9 (23-Jul-03) +% Springer Verlag global LaTeX2e support for monographs +%% +%% +%% \CharacterTable +%% {Upper-case \A\B\C\D\E\F\G\H\I\J\K\L\M\N\O\P\Q\R\S\T\U\V\W\X\Y\Z +%% Lower-case \a\b\c\d\e\f\g\h\i\j\k\l\m\n\o\p\q\r\s\t\u\v\w\x\y\z +%% Digits \0\1\2\3\4\5\6\7\8\9 +%% Exclamation \! 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Solidus \/ +%% Colon \: Semicolon \; Less than \< +%% Equals \= Greater than \> Question mark \? +%% Commercial at \@ Left bracket \[ Backslash \\ +%% Right bracket \] Circumflex \^ Underscore \_ +%% Grave accent \` Left brace \{ Vertical bar \| +%% Right brace \} Tilde \~} +%% +\NeedsTeXFormat{LaTeX2e}[1995/12/01] +\ProvidesClass{svmono}[2003/07/23 3.9 +^^JSpringer Verlag global LaTeX document class for monographs] + +% Options +% citations +\DeclareOption{natbib}{\ExecuteOptions{oribibl}% +\AtEndOfClass{% Loading package 'NATBIB' +\RequirePackage{natbib} +% Changing some parameters of NATBIB +\setlength{\bibhang}{\parindent} +%\setlength{\bibsep}{0mm} +\let\bibfont=\small +\def\@biblabel#1{#1.} +\newcommand{\etal}{\textit{et al}.} +%\bibpunct[,]{(}{)}{;}{a}{}{,}}} +}} +% Springer environment +\let\if@spthms\iftrue +\DeclareOption{nospthms}{\let\if@spthms\iffalse} +% +\let\envankh\@empty % no anchor for "theorems" +% +\let\if@envcntreset\iffalse % environment counter is not reset +\let\if@envcntresetsect=\iffalse % reset each section? +\DeclareOption{envcountresetchap}{\let\if@envcntreset\iftrue} +\DeclareOption{envcountresetsect}{\let\if@envcntreset\iftrue +\let\if@envcntresetsect=\iftrue} +% +\let\if@envcntsame\iffalse % NOT all environments work like "Theorem", + % each using its own counter +\DeclareOption{envcountsame}{\let\if@envcntsame\iftrue} +% +\let\if@envcntshowhiercnt=\iffalse % do not show hierarchy counter at all +% +% enhance theorem counter +\DeclareOption{envcountchap}{\def\envankh{chapter}% show \thechapter along with theorem number +\let\if@envcntshowhiercnt=\iftrue +\ExecuteOptions{envcountreset}} +% +\DeclareOption{envcountsect}{\def\envankh{section}% show \thesection along with theorem number +\let\if@envcntshowhiercnt=\iftrue +\ExecuteOptions{envcountreset}} +% +% languages +\let\switcht@@therlang\relax +\let\svlanginfo\relax +\def\ds@deutsch{\def\switcht@@therlang{\switcht@deutsch}% +\gdef\svlanginfo{\typeout{Man spricht deutsch.}\global\let\svlanginfo\relax}} +\def\ds@francais{\def\switcht@@therlang{\switcht@francais}% +\gdef\svlanginfo{\typeout{On parle francais.}\global\let\svlanginfo\relax}} +% +\AtBeginDocument{\@ifpackageloaded{babel}{% +\@ifundefined{extrasenglish}{}{\addto\extrasenglish{\switcht@albion}}% +\@ifundefined{extrasUKenglish}{}{\addto\extrasUKenglish{\switcht@albion}}% +\@ifundefined{extrasfrenchb}{}{\addto\extrasfrenchb{\switcht@francais}}% +\@ifundefined{extrasgerman}{}{\addto\extrasgerman{\switcht@deutsch}}% +}{\switcht@@therlang}% +} +% numbering style of floats, equations +\newif\if@numart \@numartfalse +\DeclareOption{numart}{\@numarttrue} +\def\set@numbering{\if@numart\else\num@book\fi} +\AtEndOfClass{\set@numbering} +% style for vectors +\DeclareOption{vecphys}{\def\vec@style{phys}} +\DeclareOption{vecarrow}{\def\vec@style{arrow}} +% running heads +\let\if@runhead\iftrue +\DeclareOption{norunningheads}{\let\if@runhead\iffalse} +% referee option +\let\if@referee\iffalse +\def\makereferee{\def\baselinestretch{2}\selectfont +\newbox\refereebox +\setbox\refereebox=\vbox to\z@{\vskip0.5cm% + \hbox to\textwidth{\normalsize\tt\hrulefill\lower0.5ex + \hbox{\kern5\p@ referee's copy\kern5\p@}\hrulefill}\vss}% +\def\@oddfoot{\copy\refereebox}\let\@evenfoot=\@oddfoot} +\DeclareOption{referee}{\let\if@referee\iftrue +\AtBeginDocument{\makereferee\small\normalsize}} +% modification of thebibliography +\let\if@openbib\iffalse +\DeclareOption{openbib}{\let\if@openbib\iftrue} +% LaTeX standard, sectionwise references +\DeclareOption{oribibl}{\let\oribibl=Y} +\DeclareOption{sectrefs}{\let\secbibl=Y} +% +% footinfo option (provides an informatory line on every page) +\def\SpringerMacroPackageNameA{svmono.cls} +% \thetime, \thedate and \timstamp are macros to include +% time, date (or both) of the TeX run in the document +\def\maketimestamp{\count255=\time +\divide\count255 by 60\relax +\edef\thetime{\the\count255:}% +\multiply\count255 by-60\relax +\advance\count255 by\time +\edef\thetime{\thetime\ifnum\count255<10 0\fi\the\count255} +\edef\thedate{\number\day-\ifcase\month\or Jan\or Feb\or Mar\or + Apr\or May\or Jun\or Jul\or Aug\or Sep\or Oct\or + Nov\or Dec\fi-\number\year} +\def\timstamp{\hbox to\hsize{\tt\hfil\thedate\hfil\thetime\hfil}}} +\maketimestamp +% +% \footinfo generates a info footline on every page containing +% pagenumber, jobname, macroname, and timestamp +\DeclareOption{footinfo}{\AtBeginDocument{\maketimestamp + \def\@oddfoot{\footnotesize\tt Page: \thepage\hfil job: \jobname\hfil + macro: \SpringerMacroPackageNameA\hfil + date/time: \thedate/\thetime}% + \let\@evenfoot=\@oddfoot}} +% +% start new chapter on any page +\newif\if@openright \@openrighttrue +\DeclareOption{openany}{\@openrightfalse} +% +% no size changing allowed +\DeclareOption{11pt}{\OptionNotUsed} +\DeclareOption{12pt}{\OptionNotUsed} +% options for the article class +\def\@rticle@options{10pt,twoside} +% fleqn +\DeclareOption{fleqn}{\def\@rticle@options{10pt,twoside,fleqn}% +\AtEndOfClass{\let\leftlegendglue\relax}% +\AtBeginDocument{\mathindent\parindent}} +% hanging sectioning titles +\let\if@sechang\iffalse +\DeclareOption{sechang}{\let\if@sechang\iftrue} +\def\ClassInfoNoLine#1#2{% + \ClassInfo{#1}{#2\@gobble}% +} +\let\SVMonoOpt\@empty +\DeclareOption*{\InputIfFileExists{sv\CurrentOption.clo}{% +\global\let\SVMonoOpt\CurrentOption}{% +\ClassWarning{Springer-SVMono}{Specified option or subpackage +"\CurrentOption" \MessageBreak not found +passing it to article class \MessageBreak +-}\PassOptionsToClass{\CurrentOption}{article}% +}} +\ProcessOptions\relax +\ifx\SVMonoOpt\@empty\relax +\ClassInfoNoLine{Springer-SVMono}{extra/valid Springer sub-package +\MessageBreak not found in option list - using "global" style}{} +\fi +\LoadClass[\@rticle@options]{article} +\raggedbottom + +% various sizes and settings for monographs + +%\setlength{\textwidth}{28pc} % 11.8cm +\setlength{\textwidth}{43pc} % 11.8cm +%\setlength{\textheight}{12pt}\multiply\textheight by 45\relax +%\setlength{\textheight}{540\p@} +\setlength{\textheight}{720\p@} +%\setlength{\topmargin}{0cm} +\setlength{\topmargin}{-2cm} +%\setlength\oddsidemargin {63\p@} +%\setlength\evensidemargin {63\p@} +\setlength\oddsidemargin {-28\p@} +\setlength\evensidemargin {-28\p@} +\setlength\marginparwidth{90\p@} +\setlength\headsep {12\p@} + +\setlength{\parindent}{15\p@} +\setlength{\parskip}{\z@ \@plus \p@} +\setlength{\hfuzz}{2\p@} +\setlength{\arraycolsep}{1.5\p@} + +\frenchspacing + +\tolerance=500 + +\predisplaypenalty=0 +\clubpenalty=10000 +\widowpenalty=10000 + +\setlength\footnotesep{7.7\p@} + +\newdimen\betweenumberspace % dimension for space between +\betweenumberspace=5\p@ % number and text of titles +\newdimen\headlineindent % dimension for space of +\headlineindent=2.5cc % number and gap of running heads + +% fonts, sizes, and the like +\renewcommand\small{% + \@setfontsize\small\@ixpt{11}% + \abovedisplayskip 8.5\p@ \@plus3\p@ \@minus4\p@ + \abovedisplayshortskip \z@ \@plus2\p@ + \belowdisplayshortskip 4\p@ \@plus2\p@ \@minus2\p@ + \def\@listi{\leftmargin\leftmargini + \parsep \z@ \@plus\p@ \@minus\p@ + \topsep 6\p@ \@plus2\p@ \@minus4\p@ + \itemsep\z@}% + \belowdisplayskip \abovedisplayskip +} +% +\let\footnotesize=\small +% +\newenvironment{petit}{\par\addvspace{6\p@}\small}{\par\addvspace{6\p@}} +% + +% modification of automatic positioning of floating objects +\setlength\@fptop{\z@ } +\setlength\@fpsep{12\p@ } +\setlength\@fpbot{\z@ \@plus 1fil } +\def\textfraction{.01} +\def\floatpagefraction{.8} +\setlength{\intextsep}{20\p@ \@plus 2\p@ \@minus 2\p@} +\setcounter{topnumber}{4} +\def\topfraction{.9} +\setcounter{bottomnumber}{2} +\def\bottomfraction{.7} +\setcounter{totalnumber}{6} +% +% size and style of headings +\newcommand{\partsize}{\Large} +\newcommand{\partstyle}{\bfseries\boldmath} +\newcommand{\chapsize}{\Large} +\newcommand{\chapstyle}{\bfseries\boldmath} +\newcommand{\secsize}{\large} +\newcommand{\secstyle}{\bfseries\boldmath} +\newcommand{\subsecsize}{\normalsize} +\newcommand{\subsecstyle}{\bfseries\boldmath} + +\def\newendpage {\par + \ifdim \pagetotal>\topskip + \else\thispagestyle{empty} + \fi + \vfil\penalty -\@M} +\def\clearpage{% + \ifvmode + \ifnum \@dbltopnum =\m@ne + \ifdim \pagetotal <\topskip + \hbox{} + \fi + \fi + \fi + \newendpage + \write\m@ne{} + \vbox{} + \penalty -\@Mi +} +\def\cleardoublepage{\clearpage\if@twoside \ifodd\c@page\else + \hbox{}\newendpage\if@twocolumn\hbox{}\newpage\fi\fi\fi} + +\newcommand{\clearemptydoublepage}{% + \newpage{\pagestyle{empty}\cleardoublepage}} +\newcommand{\startnewpage}{\if@openright\clearemptydoublepage\else\clearpage\fi} + +% redefinition of \part +\renewcommand\part{\clearemptydoublepage + \thispagestyle{empty} + \if@twocolumn + \onecolumn + \@tempswatrue + \else + \@tempswafalse + \fi + \@ifundefined{thispagecropped}{}{\thispagecropped} + \secdef\@part\@spart} + +\def\@part[#1]#2{\ifnum \c@secnumdepth >-2\relax + \refstepcounter{part} + \addcontentsline{toc}{part}{\partname\ + \thepart\thechapterend\hskip\betweenumberspace + #1}\else + \addcontentsline{toc}{part}{#1}\fi + \markboth{}{} + {\raggedleft + \ifnum \c@secnumdepth >-2\relax + \normalfont\partstyle\partsize\vrule height 34pt width 0pt depth 0pt% + \partname\ \thepart\llap{\smash{\lower 5pt\hbox to\textwidth{\hrulefill}}} + \par + \vskip 128.3\p@ \fi + #2\par}\@endpart} +% +% \@endpart finishes the part page +% +\def\@endpart{\vfil\newpage + \if@twoside + \hbox{} + \thispagestyle{empty} + \newpage + \fi + \if@tempswa + \twocolumn + \fi} +% +\def\@spart#1{{\raggedleft + \normalfont\partsize\partstyle + #1\par}\@endpart} +% +% (re)define sectioning +\setcounter{secnumdepth}{2} + +\def\seccounterend{} +\def\seccountergap{\hskip\betweenumberspace} +\def\@seccntformat#1{\csname the#1\endcsname\seccounterend\seccountergap\ignorespaces} +% +\let\firstmark=\botmark +% +\@ifundefined{thechapterend}{\def\thechapterend{}}{} +% +\if@sechang + \def\sec@hangfrom#1{\setbox\@tempboxa\hbox{#1}% + \hangindent\wd\@tempboxa\noindent\box\@tempboxa} +\else + \def\sec@hangfrom#1{\setbox\@tempboxa\hbox{#1}% + \hangindent\z@\noindent\box\@tempboxa} +\fi + +\def\chap@hangfrom#1{\noindent\vrule height 34pt width 0pt depth 0pt +\rlap{\smash{\lower 5pt\hbox to\textwidth{\hrulefill}}}\hbox{#1} +\vskip10pt} +\def\schap@hangfrom{\chap@hangfrom{}} + +\newcounter{chapter} +% +\@addtoreset{section}{chapter} +\@addtoreset{footnote}{chapter} + +\newif\if@mainmatter \@mainmattertrue +\newcommand\frontmatter{\startnewpage + \@mainmatterfalse\pagenumbering{Roman} + \setcounter{page}{5}} +% +\newcommand\mainmatter{\clearemptydoublepage + \@mainmattertrue\pagenumbering{arabic}} +% +\newcommand\backmatter{\clearemptydoublepage\@mainmatterfalse} + +\def\@chapapp{\chaptername} + +\newdimen\chapstarthookwidth +\newcommand\chapstarthook[2][0.66\textwidth]{% +\setlength{\chapstarthookwidth}{#1}% +\gdef\chapst@rthook{#2}} + +\newcommand{\processchapstarthook}{\@ifundefined{chapst@rthook}{}{% + \setbox0=\hbox{\vbox{ + \begin{flushright} + \begin{minipage}{\chapstarthookwidth} + \vrule\@width\z@\@height21\p@\@depth\z@ + \normalfont\small\itshape\chapst@rthook + \end{minipage} + \end{flushright}}}% + \@tempdima=\pagetotal + \advance\@tempdima by\ht0 + \ifdim\@tempdima<106\p@ + \multiply\@tempdima by-1 + \advance\@tempdima by106\p@ + \vskip\@tempdima + \fi + \box0\par + \global\let\chapst@rthook=\undefined}} + +\newcommand\chapter{\startnewpage + \@ifundefined{thispagecropped}{}{\thispagecropped} + \thispagestyle{empty}% + \global\@topnum\z@ + \@afterindentfalse + \secdef\@chapter\@schapter} + +\def\@chapter[#1]#2{\ifnum \c@secnumdepth >\m@ne + \refstepcounter{chapter}% + \if@mainmatter + \typeout{\@chapapp\space\thechapter.}% + \addcontentsline{toc}{chapter}{\protect + \numberline{\thechapter\thechapterend}#1}% + \else + \addcontentsline{toc}{chapter}{#1}% + \fi + \else + \addcontentsline{toc}{chapter}{#1}% + \fi + \chaptermark{#1}% + \addtocontents{lof}{\protect\addvspace{10\p@}}% + \addtocontents{lot}{\protect\addvspace{10\p@}}% + \if@twocolumn + \@topnewpage[\@makechapterhead{#2}]% + \else + \@makechapterhead{#2}% + \@afterheading + \fi} + +\def\@schapter#1{\if@twocolumn + \@topnewpage[\@makeschapterhead{#1}]% + \else + \@makeschapterhead{#1}% + \@afterheading + \fi} + +%%changes position and layout of numbered chapter headings +\def\@makechapterhead#1{{\parindent\z@\raggedright\normalfont + \hyphenpenalty \@M + \interlinepenalty\@M + \chapsize\chapstyle + \chap@hangfrom{\thechapter\thechapterend\hskip\betweenumberspace}%!!! + \ignorespaces#1\par\nobreak + \processchapstarthook + \ifdim\pagetotal>157\p@ + \vskip 11\p@ + \else + \@tempdima=168\p@\advance\@tempdima by-\pagetotal + \vskip\@tempdima + \fi}} + +%%changes position and layout of unnumbered chapter headings +\def\@makeschapterhead#1{{\parindent \z@ \raggedright\normalfont + \hyphenpenalty \@M + \interlinepenalty\@M + \chapsize\chapstyle + \schap@hangfrom + \ignorespaces#1\par\nobreak + \processchapstarthook + \ifdim\pagetotal>157\p@ + \vskip 11\p@ + \else + \@tempdima=168\p@\advance\@tempdima by-\pagetotal + \vskip\@tempdima + \fi}} + +% predefined unnumbered headings +\newcommand{\preface}[1][\prefacename]{\chapter*{#1}\markboth{#1}{#1}} + +% measures and setting of sections +\renewcommand\section{\@startsection{section}{1}{\z@}% + {-24\p@ \@plus -4\p@ \@minus -4\p@}% + {12\p@ \@plus 4\p@ \@minus 4\p@}% + {\normalfont\secsize\secstyle + \rightskip=\z@ \@plus 8em\pretolerance=10000 }} +\renewcommand\subsection{\@startsection{subsection}{2}{\z@}% + {-17\p@ \@plus -4\p@ \@minus -4\p@}% + {10\p@ \@plus 4\p@ \@minus 4\p@}% + {\normalfont\subsecsize\subsecstyle + \rightskip=\z@ \@plus 8em\pretolerance=10000 }} +\renewcommand\subsubsection{\@startsection{subsubsection}{3}{\z@}% + {-17\p@ \@plus -4\p@ \@minus -4\p@}% + {10\p@ \@plus 4\p@ \@minus 4\p@}% + {\normalfont\normalsize\subsecstyle + \rightskip=\z@ \@plus 8em\pretolerance=10000 }} +\renewcommand\paragraph{\@startsection{paragraph}{4}{\z@}% + {-10\p@ \@plus -4\p@ \@minus -4\p@}% + {10\p@ \@plus 4\p@ \@minus 4\p@}% + {\normalfont\normalsize\itshape + \rightskip=\z@ \@plus 8em\pretolerance=10000 }} +\def\subparagraph{\@startsection{subparagraph}{5}{\z@}% + {-5.388\p@ \@plus-4\p@ \@minus-4\p@}{-5\p@}{\normalfont\normalsize\itshape}} + +% Appendix +\renewcommand\appendix{\par + \stepcounter{chapter} + \setcounter{chapter}{0} + \stepcounter{section} + \setcounter{section}{0} + \setcounter{equation}{0} + \setcounter{figure}{0} + \setcounter{table}{0} + \setcounter{footnote}{0} + \def\@chapapp{\appendixname}% + \renewcommand\thechapter{\@Alph\c@chapter}} + +% definition of sections +% \hyphenpenalty and \raggedright added, so that there is no +% hyphenation and the text is set ragged-right in sectioning + +\def\runinsep{} +\def\aftertext{\unskip\runinsep} +% +\def\thesection{\thechapter.\arabic{section}} +\def\thesubsection{\thesection.\arabic{subsection}} +\def\thesubsubsection{\thesubsection.\arabic{subsubsection}} +\def\theparagraph{\thesubsubsection.\arabic{paragraph}} +\def\thesubparagraph{\theparagraph.\arabic{subparagraph}} +\def\chaptermark#1{} +% +\def\@ssect#1#2#3#4#5{% + \@tempskipa #3\relax + \ifdim \@tempskipa>\z@ + \begingroup + #4{% + \@hangfrom{\hskip #1}% + \raggedright + \hyphenpenalty \@M + \interlinepenalty \@M #5\@@par}% + \endgroup + \else + \def\@svsechd{#4{\hskip #1\relax #5}}% + \fi + \@xsect{#3}} +% +\def\@sect#1#2#3#4#5#6[#7]#8{% + \ifnum #2>\c@secnumdepth + \let\@svsec\@empty + \else + \refstepcounter{#1}% + \protected@edef\@svsec{\@seccntformat{#1}\relax}% + \fi + \@tempskipa #5\relax + \ifdim \@tempskipa>\z@ + \begingroup #6\relax + \sec@hangfrom{\hskip #3\relax\@svsec}% + {\raggedright + \hyphenpenalty \@M + \interlinepenalty \@M #8\@@par}% + \endgroup + \csname #1mark\endcsname{#7\seccounterend}% + \addcontentsline{toc}{#1}{\ifnum #2>\c@secnumdepth + \else + \protect\numberline{\csname the#1\endcsname\seccounterend}% + \fi + #7}% + \else + \def\@svsechd{% + #6\hskip #3\relax + \@svsec #8\aftertext\ignorespaces + \csname #1mark\endcsname{#7}% + \addcontentsline{toc}{#1}{% + \ifnum #2>\c@secnumdepth \else + \protect\numberline{\csname the#1\endcsname\seccounterend}% + \fi + #7}}% + \fi + \@xsect{#5}} + +% figures and tables are processed in small print +\def \@floatboxreset {% + \reset@font + \small + \@setnobreak + \@setminipage +} +\def\fps@figure{htbp} +\def\fps@table{htbp} + +% Frame for paste-in figures or tables +\def\mpicplace#1#2{% #1 =width #2 =height +\vbox{\hbox to #1{\vrule\@width \fboxrule \@height #2\hfill}}} + +% labels of enumerate +\renewcommand\labelenumii{\theenumii)} +\renewcommand\theenumii{\@alph\c@enumii} + +% labels of itemize +\renewcommand\labelitemi{\textbullet} +\renewcommand\labelitemii{\textendash} +\let\labelitemiii=\labelitemiv + +% labels of description +\renewcommand*\descriptionlabel[1]{\hspace\labelsep #1\hfil} + +% fixed indentation for standard itemize-environment +\newdimen\svitemindent \setlength{\svitemindent}{\parindent} + + +% make indentations changeable + +\def\setitemindent#1{\settowidth{\labelwidth}{#1}% + \let\setit@m=Y% + \leftmargini\labelwidth + \advance\leftmargini\labelsep + \def\@listi{\leftmargin\leftmargini + \labelwidth\leftmargini\advance\labelwidth by -\labelsep + \parsep=\parskip + \topsep=\medskipamount + \itemsep=\parskip \advance\itemsep by -\parsep}} +\def\setitemitemindent#1{\settowidth{\labelwidth}{#1}% + \let\setit@m=Y% + \leftmarginii\labelwidth + \advance\leftmarginii\labelsep +\def\@listii{\leftmargin\leftmarginii + \labelwidth\leftmarginii\advance\labelwidth by -\labelsep + \parsep=\parskip + \topsep=\z@ + \itemsep=\parskip \advance\itemsep by -\parsep}} +% +% adjusted environment "description" +% if an optional parameter (at the first two levels of lists) +% is present, its width is considered to be the widest mark +% throughout the current list. +\def\description{\@ifnextchar[{\@describe}{\list{}{\labelwidth\z@ + \itemindent-\leftmargin \let\makelabel\descriptionlabel}}} +% +\def\describelabel#1{#1\hfil} +\def\@describe[#1]{\relax\ifnum\@listdepth=0 +\setitemindent{#1}\else\ifnum\@listdepth=1 +\setitemitemindent{#1}\fi\fi +\list{--}{\let\makelabel\describelabel}} +% +\def\itemize{% + \ifnum \@itemdepth >\thr@@\@toodeep\else + \advance\@itemdepth\@ne + \ifx\setit@m\undefined + \ifnum \@itemdepth=1 \leftmargini=\svitemindent + \labelwidth\leftmargini\advance\labelwidth-\labelsep + \leftmarginii=\leftmargini \leftmarginiii=\leftmargini + \fi + \fi + \edef\@itemitem{labelitem\romannumeral\the\@itemdepth}% + \expandafter\list + \csname\@itemitem\endcsname + {\def\makelabel##1{\rlap{##1}\hss}}% + \fi} +% +\newdimen\verbatimindent \verbatimindent\parindent +\def\verbatim{\advance\@totalleftmargin by\verbatimindent +\@verbatim \frenchspacing\@vobeyspaces \@xverbatim} + +% +% special signs and characters +\newcommand{\D}{\mathrm{d}} +\newcommand{\E}{\mathrm{e}} +\let\eul=\E +\newcommand{\I}{{\rm i}} +\let\imag=\I +% +% the definition of uppercase Greek characters +% Springer likes them as italics to depict variables +\DeclareMathSymbol{\Gamma}{\mathalpha}{letters}{"00} +\DeclareMathSymbol{\Delta}{\mathalpha}{letters}{"01} +\DeclareMathSymbol{\Theta}{\mathalpha}{letters}{"02} +\DeclareMathSymbol{\Lambda}{\mathalpha}{letters}{"03} +\DeclareMathSymbol{\Xi}{\mathalpha}{letters}{"04} +\DeclareMathSymbol{\Pi}{\mathalpha}{letters}{"05} +\DeclareMathSymbol{\Sigma}{\mathalpha}{letters}{"06} +\DeclareMathSymbol{\Upsilon}{\mathalpha}{letters}{"07} +\DeclareMathSymbol{\Phi}{\mathalpha}{letters}{"08} +\DeclareMathSymbol{\Psi}{\mathalpha}{letters}{"09} +\DeclareMathSymbol{\Omega}{\mathalpha}{letters}{"0A} +% the upright forms are defined here as \var +\DeclareMathSymbol{\varGamma}{\mathalpha}{operators}{"00} +\DeclareMathSymbol{\varDelta}{\mathalpha}{operators}{"01} +\DeclareMathSymbol{\varTheta}{\mathalpha}{operators}{"02} +\DeclareMathSymbol{\varLambda}{\mathalpha}{operators}{"03} +\DeclareMathSymbol{\varXi}{\mathalpha}{operators}{"04} +\DeclareMathSymbol{\varPi}{\mathalpha}{operators}{"05} +\DeclareMathSymbol{\varSigma}{\mathalpha}{operators}{"06} +\DeclareMathSymbol{\varUpsilon}{\mathalpha}{operators}{"07} +\DeclareMathSymbol{\varPhi}{\mathalpha}{operators}{"08} +\DeclareMathSymbol{\varPsi}{\mathalpha}{operators}{"09} +\DeclareMathSymbol{\varOmega}{\mathalpha}{operators}{"0A} +% Upright Lower Case Greek letters without using a new MathAlphabet +\newcommand{\greeksym}[1]{\usefont{U}{psy}{m}{n}#1} +\newcommand{\greeksymbold}[1]{{\usefont{U}{psy}{b}{n}#1}} +\newcommand{\allmodesymb}[2]{\relax\ifmmode{\mathchoice +{\mbox{\fontsize{\tf@size}{\tf@size}#1{#2}}} +{\mbox{\fontsize{\tf@size}{\tf@size}#1{#2}}} +{\mbox{\fontsize{\sf@size}{\sf@size}#1{#2}}} +{\mbox{\fontsize{\ssf@size}{\ssf@size}#1{#2}}}} +\else +\mbox{#1{#2}}\fi} +% Definition of lower case Greek letters +\newcommand{\ualpha}{\allmodesymb{\greeksym}{a}} +\newcommand{\ubeta}{\allmodesymb{\greeksym}{b}} +\newcommand{\uchi}{\allmodesymb{\greeksym}{c}} +\newcommand{\udelta}{\allmodesymb{\greeksym}{d}} +\newcommand{\ugamma}{\allmodesymb{\greeksym}{g}} +\newcommand{\umu}{\allmodesymb{\greeksym}{m}} +\newcommand{\unu}{\allmodesymb{\greeksym}{n}} +\newcommand{\upi}{\allmodesymb{\greeksym}{p}} +\newcommand{\utau}{\allmodesymb{\greeksym}{t}} +% redefines the \vec accent to a bold character - if desired +\def\fig@type{arrow}% temporarily abused +\ifx\vec@style\fig@type\else +\@ifundefined{vec@style}{% + \def\vec#1{\ensuremath{\mathchoice + {\mbox{\boldmath$\displaystyle\mathbf{#1}$}} + {\mbox{\boldmath$\textstyle\mathbf{#1}$}} + {\mbox{\boldmath$\scriptstyle\mathbf{#1}$}} + {\mbox{\boldmath$\scriptscriptstyle\mathbf{#1}$}}}}% +} +{\def\vec#1{\ensuremath{\mathchoice + {\mbox{\boldmath$\displaystyle#1$}} + {\mbox{\boldmath$\textstyle#1$}} + {\mbox{\boldmath$\scriptstyle#1$}} + {\mbox{\boldmath$\scriptscriptstyle#1$}}}}% +} +\fi +% tensor +\def\tens#1{\relax\ifmmode\mathsf{#1}\else\textsf{#1}\fi} + +% end of proof symbol +\newcommand\qedsymbol{\hbox{\rlap{$\sqcap$}$\sqcup$}} +\newcommand\qed{\relax\ifmmode\else\unskip\quad\fi\qedsymbol} +\newcommand\smartqed{\renewcommand\qed{\relax\ifmmode\qedsymbol\else + {\unskip\nobreak\hfil\penalty50\hskip1em\null\nobreak\hfil\qedsymbol + \parfillskip=\z@\finalhyphendemerits=0\endgraf}\fi}} +% +\def\num@book{% +\renewcommand\thesection{\thechapter.\@arabic\c@section}% +\renewcommand\thesubsection{\thesection.\@arabic\c@subsection}% +\renewcommand\theequation{\thechapter.\@arabic\c@equation}% +\renewcommand\thefigure{\thechapter.\@arabic\c@figure}% +\renewcommand\thetable{\thechapter.\@arabic\c@table}% +\@addtoreset{section}{chapter}% +\@addtoreset{figure}{chapter}% +\@addtoreset{table}{chapter}% +\@addtoreset{equation}{chapter}} +% +% Ragged bottom for the actual page +\def\thisbottomragged{\def\@textbottom{\vskip\z@ \@plus.0001fil +\global\let\@textbottom\relax}} + +% This is texte.tex +% it defines various texts and their translations +% called up with documentstyle options +\def\switcht@albion{% +\def\abstractname{Summary.}% +\def\ackname{Acknowledgement.}% +\def\andname{and}% +\def\bibname{References}% +\def\lastandname{, and}% +\def\appendixname{Appendix}% +\def\chaptername{Chapter}% +\def\claimname{Claim}% +\def\conjecturename{Conjecture}% +\def\contentsname{Contents}% +\def\corollaryname{Corollary}% +\def\definitionname{Definition}% +\def\examplename{Example}% +\def\exercisename{Exercise}% +\def\figurename{Fig.}% +\def\keywordname{{\bf Key words:}}% +\def\indexname{Index}% +\def\lemmaname{Lemma}% +\def\contriblistname{List of Contributors}% +\def\listfigurename{List of Figures}% +\def\listtablename{List of Tables}% +\def\mailname{{\it Correspondence to\/}:}% +\def\noteaddname{Note added in proof}% +\def\notename{Note}% +\def\partname{Part}% +\def\prefacename{Preface}% +\def\problemname{Problem}% +\def\proofname{Proof}% +\def\propertyname{Property}% +\def\propositionname{Proposition}% +\def\questionname{Question}% +\def\refname{References}% +\def\remarkname{Remark}% +\def\seename{see}% +\def\solutionname{Solution}% +\def\subclassname{{\it Subject Classifications\/}:}% +\def\tablename{Table}% +\def\theoremname{Theorem}} +\switcht@albion +% Names of theorem like environments are already defined +% but must be translated if another language is chosen +% +% French section +\def\switcht@francais{\svlanginfo + \def\abstractname{R\'esum\'e.}% + \def\ackname{Remerciements.}% + \def\andname{et}% + \def\lastandname{ et}% + \def\appendixname{Appendice}% + \def\bibname{Bibliographie}% + \def\chaptername{Chapitre}% + \def\claimname{Pr\'etention}% + \def\conjecturename{Hypoth\`ese}% + \def\contentsname{Table des mati\`eres}% + \def\corollaryname{Corollaire}% + \def\definitionname{D\'efinition}% + \def\examplename{Exemple}% + \def\exercisename{Exercice}% + \def\figurename{Fig.}% + \def\keywordname{{\bf Mots-cl\'e:}}% + \def\indexname{Index}% + \def\lemmaname{Lemme}% + \def\contriblistname{Liste des contributeurs}% + \def\listfigurename{Liste des figures}% + \def\listtablename{Liste des tables}% + \def\mailname{{\it Correspondence to\/}:}% + \def\noteaddname{Note ajout\'ee \`a l'\'epreuve}% + \def\notename{Remarque}% + \def\partname{Partie}% + \def\prefacename{Avant-propos}% ou Pr\'eface + \def\problemname{Probl\`eme}% + \def\proofname{Preuve}% + \def\propertyname{Caract\'eristique}% +%\def\propositionname{Proposition}% + \def\questionname{Question}% + \def\refname{Lit\'erature}% + \def\remarkname{Remarque}% + \def\seename{voir}% + \def\solutionname{Solution}% + \def\subclassname{{\it Subject Classifications\/}:}% + \def\tablename{Tableau}% + \def\theoremname{Th\'eor\`eme}% +} +% +% German section +\def\switcht@deutsch{\svlanginfo + \def\abstractname{Zusammenfassung.}% + \def\ackname{Danksagung.}% + \def\andname{und}% + \def\lastandname{ und}% + \def\appendixname{Anhang}% + \def\bibname{Literaturverzeichnis}% + \def\chaptername{Kapitel}% + \def\claimname{Behauptung}% + \def\conjecturename{Hypothese}% + \def\contentsname{Inhaltsverzeichnis}% + \def\corollaryname{Korollar}% +%\def\definitionname{Definition}% + \def\examplename{Beispiel}% + \def\exercisename{\"Ubung}% + \def\figurename{Abb.}% + \def\keywordname{{\bf Schl\"usselw\"orter:}}% + \def\indexname{Sachverzeichnis}% +%\def\lemmaname{Lemma}% + \def\contriblistname{Mitarbeiter}% + \def\listfigurename{Abbildungsverzeichnis}% + \def\listtablename{Tabellenverzeichnis}% + \def\mailname{{\it Correspondence to\/}:}% + \def\noteaddname{Nachtrag}% + \def\notename{Anmerkung}% + \def\partname{Teil}% + \def\prefacename{Vorwort}% +%\def\problemname{Problem}% + \def\proofname{Beweis}% + \def\propertyname{Eigenschaft}% +%\def\propositionname{Proposition}% + \def\questionname{Frage}% + \def\refname{Literaturverzeichnis}% + \def\remarkname{Anmerkung}% + \def\seename{siehe}% + \def\solutionname{L\"osung}% + \def\subclassname{{\it Subject Classifications\/}:}% + \def\tablename{Tabelle}% +%\def\theoremname{Theorem}% +} + +\def\getsto{\mathrel{\mathchoice {\vcenter{\offinterlineskip +\halign{\hfil +$\displaystyle##$\hfil\cr\gets\cr\to\cr}}} +{\vcenter{\offinterlineskip\halign{\hfil$\textstyle##$\hfil\cr\gets +\cr\to\cr}}} +{\vcenter{\offinterlineskip\halign{\hfil$\scriptstyle##$\hfil\cr\gets +\cr\to\cr}}} +{\vcenter{\offinterlineskip\halign{\hfil$\scriptscriptstyle##$\hfil\cr +\gets\cr\to\cr}}}}} +\def\lid{\mathrel{\mathchoice {\vcenter{\offinterlineskip\halign{\hfil +$\displaystyle##$\hfil\cr<\cr\noalign{\vskip1.2\p@}=\cr}}} +{\vcenter{\offinterlineskip\halign{\hfil$\textstyle##$\hfil\cr<\cr +\noalign{\vskip1.2\p@}=\cr}}} +{\vcenter{\offinterlineskip\halign{\hfil$\scriptstyle##$\hfil\cr<\cr +\noalign{\vskip\p@}=\cr}}} +{\vcenter{\offinterlineskip\halign{\hfil$\scriptscriptstyle##$\hfil\cr +<\cr +\noalign{\vskip0.9\p@}=\cr}}}}} +\def\gid{\mathrel{\mathchoice {\vcenter{\offinterlineskip\halign{\hfil +$\displaystyle##$\hfil\cr>\cr\noalign{\vskip1.2\p@}=\cr}}} +{\vcenter{\offinterlineskip\halign{\hfil$\textstyle##$\hfil\cr>\cr +\noalign{\vskip1.2\p@}=\cr}}} +{\vcenter{\offinterlineskip\halign{\hfil$\scriptstyle##$\hfil\cr>\cr +\noalign{\vskip\p@}=\cr}}} +{\vcenter{\offinterlineskip\halign{\hfil$\scriptscriptstyle##$\hfil\cr +>\cr +\noalign{\vskip0.9\p@}=\cr}}}}} +\def\grole{\mathrel{\mathchoice {\vcenter{\offinterlineskip +\halign{\hfil +$\displaystyle##$\hfil\cr>\cr\noalign{\vskip-\p@}<\cr}}} +{\vcenter{\offinterlineskip\halign{\hfil$\textstyle##$\hfil\cr +>\cr\noalign{\vskip-\p@}<\cr}}} +{\vcenter{\offinterlineskip\halign{\hfil$\scriptstyle##$\hfil\cr +>\cr\noalign{\vskip-0.8\p@}<\cr}}} +{\vcenter{\offinterlineskip\halign{\hfil$\scriptscriptstyle##$\hfil\cr +>\cr\noalign{\vskip-0.3\p@}<\cr}}}}} +\def\bbbr{{\rm I\!R}} %reelle Zahlen +\def\bbbm{{\rm I\!M}} +\def\bbbn{{\rm I\!N}} %natuerliche Zahlen +\def\bbbf{{\rm I\!F}} +\def\bbbh{{\rm I\!H}} +\def\bbbk{{\rm I\!K}} +\def\bbbp{{\rm I\!P}} +\def\bbbone{{\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l} +{\rm 1\mskip-4.5mu l} {\rm 1\mskip-5mu l}}} +\def\bbbc{{\mathchoice {\setbox0=\hbox{$\displaystyle\rm C$}\hbox{\hbox +to\z@{\kern0.4\wd0\vrule\@height0.9\ht0\hss}\box0}} +{\setbox0=\hbox{$\textstyle\rm C$}\hbox{\hbox +to\z@{\kern0.4\wd0\vrule\@height0.9\ht0\hss}\box0}} +{\setbox0=\hbox{$\scriptstyle\rm C$}\hbox{\hbox +to\z@{\kern0.4\wd0\vrule\@height0.9\ht0\hss}\box0}} +{\setbox0=\hbox{$\scriptscriptstyle\rm C$}\hbox{\hbox +to\z@{\kern0.4\wd0\vrule\@height0.9\ht0\hss}\box0}}}} +\def\bbbq{{\mathchoice {\setbox0=\hbox{$\displaystyle\rm +Q$}\hbox{\raise +0.15\ht0\hbox to\z@{\kern0.4\wd0\vrule\@height0.8\ht0\hss}\box0}} +{\setbox0=\hbox{$\textstyle\rm Q$}\hbox{\raise +0.15\ht0\hbox to\z@{\kern0.4\wd0\vrule\@height0.8\ht0\hss}\box0}} +{\setbox0=\hbox{$\scriptstyle\rm Q$}\hbox{\raise +0.15\ht0\hbox to\z@{\kern0.4\wd0\vrule\@height0.7\ht0\hss}\box0}} +{\setbox0=\hbox{$\scriptscriptstyle\rm Q$}\hbox{\raise +0.15\ht0\hbox to\z@{\kern0.4\wd0\vrule\@height0.7\ht0\hss}\box0}}}} +\def\bbbt{{\mathchoice {\setbox0=\hbox{$\displaystyle\rm +T$}\hbox{\hbox to\z@{\kern0.3\wd0\vrule\@height0.9\ht0\hss}\box0}} +{\setbox0=\hbox{$\textstyle\rm T$}\hbox{\hbox +to\z@{\kern0.3\wd0\vrule\@height0.9\ht0\hss}\box0}} +{\setbox0=\hbox{$\scriptstyle\rm T$}\hbox{\hbox +to\z@{\kern0.3\wd0\vrule\@height0.9\ht0\hss}\box0}} +{\setbox0=\hbox{$\scriptscriptstyle\rm T$}\hbox{\hbox +to\z@{\kern0.3\wd0\vrule\@height0.9\ht0\hss}\box0}}}} +\def\bbbs{{\mathchoice +{\setbox0=\hbox{$\displaystyle \rm S$}\hbox{\raise0.5\ht0\hbox +to\z@{\kern0.35\wd0\vrule\@height0.45\ht0\hss}\hbox +to\z@{\kern0.55\wd0\vrule\@height0.5\ht0\hss}\box0}} +{\setbox0=\hbox{$\textstyle \rm S$}\hbox{\raise0.5\ht0\hbox +to\z@{\kern0.35\wd0\vrule\@height0.45\ht0\hss}\hbox +to\z@{\kern0.55\wd0\vrule\@height0.5\ht0\hss}\box0}} +{\setbox0=\hbox{$\scriptstyle \rm S$}\hbox{\raise0.5\ht0\hbox +to\z@{\kern0.35\wd0\vrule\@height0.45\ht0\hss}\raise0.05\ht0\hbox +to\z@{\kern0.5\wd0\vrule\@height0.45\ht0\hss}\box0}} +{\setbox0=\hbox{$\scriptscriptstyle\rm S$}\hbox{\raise0.5\ht0\hbox +to\z@{\kern0.4\wd0\vrule\@height0.45\ht0\hss}\raise0.05\ht0\hbox +to\z@{\kern0.55\wd0\vrule\@height0.45\ht0\hss}\box0}}}} +\def\bbbz{{\mathchoice {\hbox{$\textstyle\sf Z\kern-0.4em Z$}} +{\hbox{$\textstyle\sf Z\kern-0.4em Z$}} +{\hbox{$\scriptstyle\sf Z\kern-0.3em Z$}} +{\hbox{$\scriptscriptstyle\sf Z\kern-0.2em Z$}}}} + +\let\ts\, + +\setlength \labelsep {5\p@} +\setlength\leftmargini {17\p@} +\setlength\leftmargin {\leftmargini} +\setlength\leftmarginii {\leftmargini} +\setlength\leftmarginiii {\leftmargini} +\setlength\leftmarginiv {\leftmargini} +\setlength\labelwidth {\leftmargini} +\addtolength\labelwidth{-\labelsep} + +\def\@listI{\leftmargin\leftmargini + \parsep=\parskip + \topsep=\medskipamount + \itemsep=\parskip \advance\itemsep by -\parsep} +\let\@listi\@listI +\@listi + +\def\@listii{\leftmargin\leftmarginii + \labelwidth\leftmarginii + \advance\labelwidth by -\labelsep + \parsep=\parskip + \topsep=\z@ + \itemsep=\parskip + \advance\itemsep by -\parsep} + +\def\@listiii{\leftmargin\leftmarginiii + \labelwidth\leftmarginiii\advance\labelwidth by -\labelsep + \parsep=\parskip + \topsep=\z@ + \itemsep=\parskip + \advance\itemsep by -\parsep + \partopsep=\topsep} + +\setlength\arraycolsep{1.5\p@} +\setlength\tabcolsep{1.5\p@} + +\def\tableofcontents{\@restonecolfalse\if@twocolumn\@restonecoltrue\onecolumn + \fi\chapter*{\contentsname \@mkboth{{\contentsname}}{{\contentsname}}} + \@starttoc{toc}\if@restonecol\twocolumn\fi} + +\setcounter{tocdepth}{2} + +\def\l@part#1#2{\addpenalty{\@secpenalty}% + \addvspace{2em \@plus\p@}% + \begingroup + \parindent \z@ + \rightskip \z@ \@plus 5em + \hrule\vskip5\p@ + \bfseries\boldmath + \leavevmode + #1\par + \vskip5\p@ + \hrule + \vskip\p@ + \nobreak + \endgroup} + +\def\@dotsep{2} + +\def\addnumcontentsmark#1#2#3{% +\addtocontents{#1}{\protect\contentsline{#2}{\protect\numberline + {\thechapter}#3}{\thepage}}} +\def\addcontentsmark#1#2#3{% +\addtocontents{#1}{\protect\contentsline{#2}{#3}{\thepage}}} +\def\addcontentsmarkwop#1#2#3{% +\addtocontents{#1}{\protect\contentsline{#2}{#3}{0}}} + +\def\@adcmk[#1]{\ifcase #1 \or +\def\@gtempa{\addnumcontentsmark}% + \or \def\@gtempa{\addcontentsmark}% + \or \def\@gtempa{\addcontentsmarkwop}% + \fi\@gtempa{toc}{chapter}} +\def\addtocmark{\@ifnextchar[{\@adcmk}{\@adcmk[3]}} + +\def\l@chapter#1#2{\par\addpenalty{-\@highpenalty} + \addvspace{1.0em \@plus \p@} + \@tempdima \tocchpnum \begingroup + \parindent \z@ \rightskip \@tocrmarg + \advance\rightskip by \z@ \@plus 2cm + \parfillskip -\rightskip \pretolerance=10000 + \leavevmode \advance\leftskip\@tempdima \hskip -\leftskip + {\bfseries\boldmath#1}\ifx0#2\hfil\null + \else + \nobreak + \leaders\hbox{$\m@th \mkern \@dotsep mu.\mkern + \@dotsep mu$}\hfill + \nobreak\hbox to\@pnumwidth{\hss #2}% + \fi\par + \penalty\@highpenalty \endgroup} + +\newdimen\tocchpnum +\newdimen\tocsecnum +\newdimen\tocsectotal +\newdimen\tocsubsecnum +\newdimen\tocsubsectotal +\newdimen\tocsubsubsecnum +\newdimen\tocsubsubsectotal +\newdimen\tocparanum +\newdimen\tocparatotal +\newdimen\tocsubparanum +\tocchpnum=20\p@ % chapter {\bf 88.} \@plus 5.3\p@ +\tocsecnum=22.5\p@ % section 88.8. plus 4.722\p@ +\tocsubsecnum=30.5\p@ % subsection 88.8.8 plus 4.944\p@ +\tocsubsubsecnum=38\p@ % subsubsection 88.8.8.8 plus 4.666\p@ +\tocparanum=45\p@ % paragraph 88.8.8.8.8 plus 3.888\p@ +\tocsubparanum=53\p@ % subparagraph 88.8.8.8.8.8 plus 4.11\p@ +\def\calctocindent{% +\tocsectotal=\tocchpnum +\advance\tocsectotal by\tocsecnum +\tocsubsectotal=\tocsectotal +\advance\tocsubsectotal by\tocsubsecnum +\tocsubsubsectotal=\tocsubsectotal +\advance\tocsubsubsectotal by\tocsubsubsecnum +\tocparatotal=\tocsubsubsectotal +\advance\tocparatotal by\tocparanum} +\calctocindent + +\def\@dottedtocline#1#2#3#4#5{% + \ifnum #1>\c@tocdepth \else + \vskip \z@ \@plus.2\p@ + {\leftskip #2\relax \rightskip \@tocrmarg \advance\rightskip by \z@ \@plus 2cm + \parfillskip -\rightskip \pretolerance=10000 + \parindent #2\relax\@afterindenttrue + \interlinepenalty\@M + \leavevmode + \@tempdima #3\relax + \advance\leftskip \@tempdima \null\nobreak\hskip -\leftskip + {#4}\nobreak + \leaders\hbox{$\m@th + \mkern \@dotsep mu\hbox{.}\mkern \@dotsep + mu$}\hfill + \nobreak + \hb@xt@\@pnumwidth{\hfil\normalfont \normalcolor #5}% + \par}% + \fi} +% +\def\l@section{\@dottedtocline{1}{\tocchpnum}{\tocsecnum}} +\def\l@subsection{\@dottedtocline{2}{\tocsectotal}{\tocsubsecnum}} +\def\l@subsubsection{\@dottedtocline{3}{\tocsubsectotal}{\tocsubsubsecnum}} +\def\l@paragraph{\@dottedtocline{4}{\tocsubsubsectotal}{\tocparanum}} +\def\l@subparagraph{\@dottedtocline{5}{\tocparatotal}{\tocsubparanum}} + +\renewcommand\listoffigures{% + \chapter*{\listfigurename + \@mkboth{\listfigurename}{\listfigurename}}% + \@starttoc{lof}% + } + +\renewcommand\listoftables{% + \chapter*{\listtablename + \@mkboth{\listtablename}{\listtablename}}% + \@starttoc{lot}% + } + +\renewcommand\footnoterule{% + \kern-3\p@ + \hrule\@width 50\p@ + \kern2.6\p@} + +\newdimen\foot@parindent +\foot@parindent 10.83\p@ + +\long\def\@makefntext#1{\@setpar{\@@par\@tempdima \hsize + \advance\@tempdima-\foot@parindent\parshape\@ne\foot@parindent + \@tempdima}\par + \parindent \foot@parindent\noindent \hbox to \z@{% + \hss\hss$^{\@thefnmark}$ }#1} + +\if@spthms +% Definition of the "\spnewtheorem" command. +% +% Usage: +% +% \spnewtheorem{env_nam}{caption}[within]{cap_font}{body_font} +% or \spnewtheorem{env_nam}[numbered_like]{caption}{cap_font}{body_font} +% or \spnewtheorem*{env_nam}{caption}{cap_font}{body_font} +% +% New is "cap_font" and "body_font". It stands for +% fontdefinition of the caption and the text itself. +% +% "\spnewtheorem*" gives a theorem without number. +% +% A defined spnewthoerem environment is used as described +% by Lamport. +% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\def\@thmcountersep{.} +\def\@thmcounterend{.} +\newcommand\nocaption{\noexpand\@gobble} +\newdimen\spthmsep \spthmsep=3pt + +\def\spnewtheorem{\@ifstar{\@sthm}{\@Sthm}} + +% definition of \spnewtheorem with number + +\def\@spnthm#1#2{% + \@ifnextchar[{\@spxnthm{#1}{#2}}{\@spynthm{#1}{#2}}} +\def\@Sthm#1{\@ifnextchar[{\@spothm{#1}}{\@spnthm{#1}}} + +\def\@spxnthm#1#2[#3]#4#5{\expandafter\@ifdefinable\csname #1\endcsname + {\@definecounter{#1}\@addtoreset{#1}{#3}% + \expandafter\xdef\csname the#1\endcsname{\expandafter\noexpand + \csname the#3\endcsname \noexpand\@thmcountersep \@thmcounter{#1}}% + \expandafter\xdef\csname #1name\endcsname{#2}% + \global\@namedef{#1}{\@spthm{#1}{\csname #1name\endcsname}{#4}{#5}}% + \global\@namedef{end#1}{\@endtheorem}}} + +\def\@spynthm#1#2#3#4{\expandafter\@ifdefinable\csname #1\endcsname + {\@definecounter{#1}% + \expandafter\xdef\csname the#1\endcsname{\@thmcounter{#1}}% + \expandafter\xdef\csname #1name\endcsname{#2}% + \global\@namedef{#1}{\@spthm{#1}{\csname #1name\endcsname}{#3}{#4}}% + \global\@namedef{end#1}{\@endtheorem}}} + +\def\@spothm#1[#2]#3#4#5{% + \@ifundefined{c@#2}{\@latexerr{No theorem environment `#2' defined}\@eha}% + {\expandafter\@ifdefinable\csname #1\endcsname + {\global\@namedef{the#1}{\@nameuse{the#2}}% + \expandafter\xdef\csname #1name\endcsname{#3}% + \global\@namedef{#1}{\@spthm{#2}{\csname #1name\endcsname}{#4}{#5}}% + \global\@namedef{end#1}{\@endtheorem}}}} + +\def\@spthm#1#2#3#4{\topsep 7\p@ \@plus2\p@ \@minus4\p@ +\labelsep=\spthmsep\refstepcounter{#1}% +\@ifnextchar[{\@spythm{#1}{#2}{#3}{#4}}{\@spxthm{#1}{#2}{#3}{#4}}} + +\def\@spxthm#1#2#3#4{\@spbegintheorem{#2}{\csname the#1\endcsname}{#3}{#4}% + \ignorespaces} + +\def\@spythm#1#2#3#4[#5]{\@spopargbegintheorem{#2}{\csname + the#1\endcsname}{#5}{#3}{#4}\ignorespaces} + +\def\normalthmheadings{\def\@spbegintheorem##1##2##3##4{\trivlist + \item[\hskip\labelsep{##3##1\ ##2\@thmcounterend}]##4} +\def\@spopargbegintheorem##1##2##3##4##5{\trivlist + \item[\hskip\labelsep{##4##1\ ##2}]{##4(##3)\@thmcounterend\ }##5}} +\normalthmheadings + +\def\reversethmheadings{\def\@spbegintheorem##1##2##3##4{\trivlist + \item[\hskip\labelsep{##3##2\ ##1\@thmcounterend}]##4} +\def\@spopargbegintheorem##1##2##3##4##5{\trivlist + \item[\hskip\labelsep{##4##2\ ##1}]{##4(##3)\@thmcounterend\ }##5}} + +% definition of \spnewtheorem* without number + +\def\@sthm#1#2{\@Ynthm{#1}{#2}} + +\def\@Ynthm#1#2#3#4{\expandafter\@ifdefinable\csname #1\endcsname + {\global\@namedef{#1}{\@Thm{\csname #1name\endcsname}{#3}{#4}}% + \expandafter\xdef\csname #1name\endcsname{#2}% + \global\@namedef{end#1}{\@endtheorem}}} + +\def\@Thm#1#2#3{\topsep 7\p@ \@plus2\p@ \@minus4\p@ +\@ifnextchar[{\@Ythm{#1}{#2}{#3}}{\@Xthm{#1}{#2}{#3}}} + +\def\@Xthm#1#2#3{\@Begintheorem{#1}{#2}{#3}\ignorespaces} + +\def\@Ythm#1#2#3[#4]{\@Opargbegintheorem{#1} + {#4}{#2}{#3}\ignorespaces} + +\def\@Begintheorem#1#2#3{#3\trivlist + \item[\hskip\labelsep{#2#1\@thmcounterend}]} + +\def\@Opargbegintheorem#1#2#3#4{#4\trivlist + \item[\hskip\labelsep{#3#1}]{#3(#2)\@thmcounterend\ }} + +% initialize theorem environment + +\if@envcntshowhiercnt % show hierarchy counter + \def\@thmcountersep{.} + \spnewtheorem{theorem}{Theorem}[\envankh]{\bfseries}{\itshape} + \@addtoreset{theorem}{chapter} +\else % theorem counter only + \spnewtheorem{theorem}{Theorem}{\bfseries}{\itshape} + \if@envcntreset + \@addtoreset{theorem}{chapter} + \if@envcntresetsect + \@addtoreset{theorem}{section} + \fi + \fi +\fi + +%definition of divers theorem environments +\spnewtheorem*{claim}{Claim}{\itshape}{\rmfamily} +\spnewtheorem*{proof}{Proof}{\itshape}{\rmfamily} +% +\if@envcntsame % all environments like "Theorem" - using its counter + \def\spn@wtheorem#1#2#3#4{\@spothm{#1}[theorem]{#2}{#3}{#4}} +\else % all environments with their own counter + \if@envcntshowhiercnt % show hierarchy counter + \def\spn@wtheorem#1#2#3#4{\@spxnthm{#1}{#2}[\envankh]{#3}{#4}} + \else % environment counter only + \if@envcntreset % environment counter is reset each section + \if@envcntresetsect + \def\spn@wtheorem#1#2#3#4{\@spynthm{#1}{#2}{#3}{#4} + \@addtoreset{#1}{chapter}\@addtoreset{#1}{section}} + \else + \def\spn@wtheorem#1#2#3#4{\@spynthm{#1}{#2}{#3}{#4} + \@addtoreset{#1}{chapter}} + \fi + \else + \let\spn@wtheorem=\@spynthm + \fi + \fi +\fi +% +\let\spdefaulttheorem=\spn@wtheorem +% +\spn@wtheorem{case}{Case}{\itshape}{\rmfamily} +\spn@wtheorem{conjecture}{Conjecture}{\itshape}{\rmfamily} +\spn@wtheorem{corollary}{Corollary}{\bfseries}{\itshape} +\spn@wtheorem{definition}{Definition}{\bfseries}{\itshape} +\spn@wtheorem{example}{Example}{\itshape}{\rmfamily} +\spn@wtheorem{exercise}{Exercise}{\bfseries}{\rmfamily} +\spn@wtheorem{lemma}{Lemma}{\bfseries}{\itshape} +\spn@wtheorem{note}{Note}{\itshape}{\rmfamily} +\spn@wtheorem{problem}{Problem}{\bfseries}{\rmfamily} +\spn@wtheorem{property}{Property}{\itshape}{\rmfamily} +\spn@wtheorem{proposition}{Proposition}{\bfseries}{\itshape} +\spn@wtheorem{question}{Question}{\itshape}{\rmfamily} +\spn@wtheorem{solution}{Solution}{\bfseries}{\rmfamily} +\spn@wtheorem{remark}{Remark}{\itshape}{\rmfamily} +% +\newenvironment{theopargself} + {\def\@spopargbegintheorem##1##2##3##4##5{\trivlist + \item[\hskip\labelsep{##4##1\ ##2}]{##4##3\@thmcounterend\ }##5} + \def\@Opargbegintheorem##1##2##3##4{##4\trivlist + \item[\hskip\labelsep{##3##1}]{##3##2\@thmcounterend\ }}}{} +\newenvironment{theopargself*} + {\def\@spopargbegintheorem##1##2##3##4##5{\trivlist + \item[\hskip\labelsep{##4##1\ ##2}]{\hspace*{-\labelsep}##4##3\@thmcounterend}##5} + \def\@Opargbegintheorem##1##2##3##4{##4\trivlist + \item[\hskip\labelsep{##3##1}]{\hspace*{-\labelsep}##3##2\@thmcounterend}}}{} +\fi + +\def\@takefromreset#1#2{% + \def\@tempa{#1}% + \let\@tempd\@elt + \def\@elt##1{% + \def\@tempb{##1}% + \ifx\@tempa\@tempb\else + \@addtoreset{##1}{#2}% + \fi}% + \expandafter\expandafter\let\expandafter\@tempc\csname cl@#2\endcsname + \expandafter\def\csname cl@#2\endcsname{}% + \@tempc + \let\@elt\@tempd} + +% redefininition of the captions for "figure" and "table" environments +% +\@ifundefined{floatlegendstyle}{\def\floatlegendstyle{\bfseries}}{} +\def\floatcounterend{.\ } +\def\capstrut{\vrule\@width\z@\@height\topskip} +\@ifundefined{captionstyle}{\def\captionstyle{\normalfont\small}}{} +\@ifundefined{instindent}{\newdimen\instindent}{} + +\long\def\@caption#1[#2]#3{\par\addcontentsline{\csname + ext@#1\endcsname}{#1}{\protect\numberline{\csname + the#1\endcsname}{\ignorespaces #2}}\begingroup + \@parboxrestore\if@minipage\@setminipage\fi + \@makecaption{\csname fnum@#1\endcsname}{\ignorespaces #3}\par + \endgroup} + +\def\twocaptionwidth#1#2{\def\first@capwidth{#1}\def\second@capwidth{#2}} +% Default: .46\textwidth +\twocaptionwidth{.46\textwidth}{.46\textwidth} + +\def\leftcaption{\refstepcounter\@captype\@dblarg% + {\@leftcaption\@captype}} + +\def\rightcaption{\refstepcounter\@captype\@dblarg% + {\@rightcaption\@captype}} + +\long\def\@leftcaption#1[#2]#3{\addcontentsline{\csname + ext@#1\endcsname}{#1}{\protect\numberline{\csname + the#1\endcsname}{\ignorespaces #2}}\begingroup + \@parboxrestore + \vskip\figcapgap + \@maketwocaptions{\csname fnum@#1\endcsname}{\ignorespaces #3}% + {\first@capwidth}\ignorespaces\hspace{.073\textwidth}\hfill% + \endgroup} + +\long\def\@rightcaption#1[#2]#3{\addcontentsline{\csname + ext@#1\endcsname}{#1}{\protect\numberline{\csname + the#1\endcsname}{\ignorespaces #2}}\begingroup + \@parboxrestore + \@maketwocaptions{\csname fnum@#1\endcsname}{\ignorespaces #3}% + {\second@capwidth}\par + \endgroup} + +\long\def\@maketwocaptions#1#2#3{% + \parbox[t]{#3}{{\floatlegendstyle #1\floatcounterend}#2}} + +\def\fig@pos{l} +\newcommand{\leftfigure}[2][\fig@pos]{\makebox[.4635\textwidth][#1]{#2}} +\let\rightfigure\leftfigure + +\newdimen\figgap\figgap=0.5cm % hgap between figure and sidecaption +% +\long\def\@makesidecaption#1#2{% + \setbox0=\vbox{\hsize=\@tempdima + \captionstyle{\floatlegendstyle + #1\floatcounterend}#2}% + \ifdim\instindent<\z@ + \ifdim\ht0>-\instindent + \advance\instindent by\ht0 + \typeout{^^JClass-Warning: Legend of \string\sidecaption\space for + \@captype\space\csname the\@captype\endcsname + ^^Jis \the\instindent\space taller than the corresponding float - + ^^Jyou'd better switch the environment. }% + \instindent\z@ + \fi + \else + \ifdim\ht0<\instindent + \advance\instindent by-\ht0 + \advance\instindent by-\dp0\relax + \advance\instindent by\topskip + \advance\instindent by-11\p@ + \else + \advance\instindent by-\ht0 + \instindent=-\instindent + \typeout{^^JClass-Warning: Legend of \string\sidecaption\space for + \@captype\space\csname the\@captype\endcsname + ^^Jis \the\instindent\space taller than the corresponding float - + ^^Jyou'd better switch the environment. }% + \instindent\z@ + \fi + \fi + \parbox[b]{\@tempdima}{\captionstyle{\floatlegendstyle + #1\floatcounterend}#2% + \ifdim\instindent>\z@ \\ + \vrule\@width\z@\@height\instindent + \@depth\z@ + \fi}} +\def\sidecaption{\@ifnextchar[\sidec@ption{\sidec@ption[b]}} +\def\sidec@ption[#1]#2\caption{% +\setbox\@tempboxa=\hbox{\ignorespaces#2\unskip}% +\if@twocolumn + \ifdim\hsize<\textwidth\else + \ifdim\wd\@tempboxa<\columnwidth + \typeout{Double column float fits into single column - + ^^Jyou'd better switch the environment. }% + \fi + \fi +\fi + \instindent=\ht\@tempboxa + \advance\instindent by\dp\@tempboxa +\if t#1 +\else + \instindent=-\instindent +\fi +\@tempdima=\hsize +\advance\@tempdima by-\figgap +\advance\@tempdima by-\wd\@tempboxa +\ifdim\@tempdima<3cm + \ClassWarning{SVMono}{\string\sidecaption: No sufficient room for the legend; + ^^Jusing normal \string\caption}% + \unhbox\@tempboxa + \let\@capcommand=\@caption +\else + \ifdim\@tempdima<4.5cm + \ClassWarning{SVMono}{\string\sidecaption: Room for the legend very narrow; + ^^Jusing \string\raggedright}% + \toks@\expandafter{\captionstyle\sloppy + \rightskip=\z@\@plus6mm\relax}% + \def\captionstyle{\the\toks@}% + \fi + \let\@capcommand=\@sidecaption + \leavevmode + \unhbox\@tempboxa + \hfill +\fi +\refstepcounter\@captype +\@dblarg{\@capcommand\@captype}} +\long\def\@sidecaption#1[#2]#3{\addcontentsline{\csname + ext@#1\endcsname}{#1}{\protect\numberline{\csname + the#1\endcsname}{\ignorespaces #2}}\begingroup + \@parboxrestore + \@makesidecaption{\csname fnum@#1\endcsname}{\ignorespaces #3}\par + \endgroup} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\def\fig@type{figure} + +\def\leftlegendglue{\hfil} +\newdimen\figcapgap\figcapgap=5\p@ % vgap between figure and caption +\newdimen\tabcapgap\tabcapgap=5.5\p@ % vgap between caption and table + +\long\def\@makecaption#1#2{% + \captionstyle + \ifx\@captype\fig@type + \vskip\figcapgap + \fi + \setbox\@tempboxa\hbox{{\floatlegendstyle #1\floatcounterend}% + \capstrut #2}% + \ifdim \wd\@tempboxa >\hsize + {\floatlegendstyle #1\floatcounterend}\capstrut #2\par + \else + \hbox to\hsize{\leftlegendglue\unhbox\@tempboxa\hfil}% + \fi + \ifx\@captype\fig@type\else + \vskip\tabcapgap + \fi} + +\newcounter{merk} + +\def\endfigure{\resetsubfig\end@float} + +\@namedef{endfigure*}{\resetsubfig\end@dblfloat} + +\def\resetsubfig{\global\let\last@subfig=\undefined} + +\def\r@setsubfig{\xdef\last@subfig{\number\value{figure}}% +\setcounter{figure}{\value{merk}}% +\setcounter{merk}{0}} + +\def\subfigures{\refstepcounter{figure}% + \@tempcnta=\value{merk}% + \setcounter{merk}{\value{figure}}% + \setcounter{figure}{\the\@tempcnta}% + \def\thefigure{\if@numart\else\thechapter.\fi + \@arabic\c@merk\alph{figure}}% + \let\resetsubfig=\r@setsubfig} + +\def\samenumber{\addtocounter{\@captype}{-1}% +\@ifundefined{last@subfig}{}{\setcounter{merk}{\last@subfig}}} + +% redefinition of the "bibliography" environment +% +\def\biblstarthook#1{\gdef\biblst@rthook{#1}} +% +\AtBeginDocument{% +\ifx\secbibl\undefined + \def\bibsection{\chapter*{\refname}\markboth{\refname}{\refname}% + \addcontentsline{toc}{chapter}{\refname}% + \csname biblst@rthook\endcsname} +\else + \def\bibsection{\section*{\refname}\markright{\refname}% + \addcontentsline{toc}{section}{\refname}% + \csname biblst@rthook\endcsname} +\fi} +\ifx\oribibl\undefined % Springer way of life + \renewenvironment{thebibliography}[1]{\bibsection + \global\let\biblst@rthook=\undefined + \def\@biblabel##1{##1.} + \small + \list{\@biblabel{\@arabic\c@enumiv}}% + {\settowidth\labelwidth{\@biblabel{#1}}% + \leftmargin\labelwidth + \advance\leftmargin\labelsep + \if@openbib + \advance\leftmargin\bibindent + \itemindent -\bibindent + \listparindent \itemindent + \parsep \z@ + \fi + \usecounter{enumiv}% + \let\p@enumiv\@empty + \renewcommand\theenumiv{\@arabic\c@enumiv}}% + \if@openbib + \renewcommand\newblock{\par}% + \else + \renewcommand\newblock{\hskip .11em \@plus.33em \@minus.07em}% + \fi + \sloppy\clubpenalty4000\widowpenalty4000% + \sfcode`\.=\@m} + {\def\@noitemerr + {\@latex@warning{Empty `thebibliography' environment}}% + \endlist} + \def\@lbibitem[#1]#2{\item[{[#1]}\hfill]\if@filesw + {\let\protect\noexpand\immediate + \write\@auxout{\string\bibcite{#2}{#1}}}\fi\ignorespaces} +\else % original bibliography is required + \let\bibname=\refname + \renewenvironment{thebibliography}[1] + {\chapter*{\bibname + \@mkboth{\bibname}{\bibname}}% + \list{\@biblabel{\@arabic\c@enumiv}}% + {\settowidth\labelwidth{\@biblabel{#1}}% + \leftmargin\labelwidth + \advance\leftmargin\labelsep + \@openbib@code + \usecounter{enumiv}% + \let\p@enumiv\@empty + \renewcommand\theenumiv{\@arabic\c@enumiv}}% + \sloppy + \clubpenalty4000 + \@clubpenalty \clubpenalty + \widowpenalty4000% + \sfcode`\.\@m} + {\def\@noitemerr + {\@latex@warning{Empty `thebibliography' environment}}% + \endlist} +\fi + +\let\if@threecolind\iffalse +\def\threecolindex{\let\if@threecolind\iftrue} +\def\indexstarthook#1{\gdef\indexst@rthook{#1}} +\renewenvironment{theindex} + {\if@twocolumn + \@restonecolfalse + \else + \@restonecoltrue + \fi + \columnseprule \z@ + \columnsep 1cc + \@nobreaktrue + \if@threecolind + \begin{multicols}{3}[\chapter*{\indexname}% + \else + \begin{multicols}{2}[\chapter*{\indexname}% + \fi + {\csname indexst@rthook\endcsname}]% + \global\let\indexst@rthook=\undefined + \markboth{\indexname}{\indexname}% + \addcontentsline{toc}{chapter}{\indexname}% + \flushbottom + \parindent\z@ + \rightskip\z@ \@plus 40\p@ + \parskip\z@ \@plus .3\p@\relax + \flushbottom + \let\item\@idxitem + \def\,{\relax\ifmmode\mskip\thinmuskip + \else\hskip0.2em\ignorespaces\fi}% + \normalfont\small} + {\end{multicols} + \global\let\if@threecolind\iffalse + \if@restonecol\onecolumn\else\clearpage\fi} + +\def\idxquad{\hskip 10\p@}% space that divides entry from number + +\def\@idxitem{\par\setbox0=\hbox{--\,--\,--\enspace}% + \hangindent\wd0\relax} + +\def\subitem{\par\noindent\setbox0=\hbox{--\enspace}% second order + \kern\wd0\setbox0=\hbox{--\,--\,--\enspace}% + \hangindent\wd0\relax}% indexentry + +\def\subsubitem{\par\noindent\setbox0=\hbox{--\,--\enspace}% third order + \kern\wd0\setbox0=\hbox{--\,--\,--\enspace}% + \hangindent\wd0\relax}% indexentry + +\def\indexspace{\par \vskip 10\p@ \@plus5\p@ \@minus3\p@\relax} + +\def\subtitle#1{\gdef\@subtitle{#1}} +\def\@subtitle{} + +\def\maketitle{\par + \begingroup + \def\thefootnote{\fnsymbol{footnote}}% + \def\@makefnmark{\hbox + to\z@{$\m@th^{\@thefnmark}$\hss}}% + \if@twocolumn + \twocolumn[\@maketitle]% + \else \newpage + \global\@topnum\z@ % Prevents figures from going at top of page. + \@maketitle \fi\thispagestyle{empty}\@thanks + \par\penalty -\@M + \endgroup + \setcounter{footnote}{0}% + \let\maketitle\relax + \let\@maketitle\relax + \gdef\@thanks{}\gdef\@author{}\gdef\@title{}\let\thanks\relax} + +\def\@maketitle{\newpage + \null + \vskip 2em % Vertical space above title. +\begingroup + \def\and{\unskip, } + \parindent=\z@ + \pretolerance=10000 + \rightskip=\z@ \@plus 3cm + {\LARGE % each author set in \LARGE + \lineskip .5em + \@author + \par}% + \vskip 2cm % Vertical space after author. + {\Huge \@title \par}% % Title set in \Huge size. + \vskip 1cm % Vertical space after title. + \if!\@subtitle!\else + {\LARGE\ignorespaces\@subtitle \par} + \vskip 1cm % Vertical space after subtitle. + \fi + \if!\@date!\else + {\large \@date}% % Date set in \large size. + \par + \vskip 1.5em % Vertical space after date. + \fi + \vfill + {\Large Springer\par} + \vskip 5\p@ + \large + Berlin\enspace Heidelberg\enspace New\kern0.1em York\\ + Hong\thinspace Kong\enspace London\\ + Milan\enspace Paris\enspace Tokyo\par +\endgroup} + +% Useful environments +\newenvironment{acknowledgement}{\par\addvspace{17\p@}\small\rm +\trivlist\item[\hskip\labelsep{\it\ackname}]} +{\endtrivlist\addvspace{6\p@}} +% +\newenvironment{noteadd}{\par\addvspace{17\p@}\small\rm +\trivlist\item[\hskip\labelsep{\it\noteaddname}]} +{\endtrivlist\addvspace{6\p@}} +% +\renewenvironment{abstract}{% + \advance\topsep by0.35cm\relax\small + \labelwidth=\z@ + \listparindent=\z@ + \itemindent\listparindent + \trivlist\item[\hskip\labelsep\bfseries\abstractname]% + \if!\abstractname!\hskip-\labelsep\fi + } + {\endtrivlist} + +% define the running headings of a twoside text +\def\runheadsize{\small} +\def\runheadstyle{\rmfamily\upshape} +\def\customizhead{\hspace{\headlineindent}} + +\def\ps@headings{\let\@mkboth\markboth + \let\@oddfoot\@empty\let\@evenfoot\@empty + \def\@evenhead{\runheadsize\runheadstyle\rlap{\thepage}\customizhead + \leftmark\hfil} + \def\@oddhead{\hfil\runheadsize\runheadstyle\rightmark\customizhead + \llap{\thepage}} + \def\chaptermark##1{\markboth{{\ifnum\c@secnumdepth>\m@ne + \thechapter\thechapterend\hskip\betweenumberspace\fi ##1}}{{\ifnum %!!! + \c@secnumdepth>\m@ne\thechapter\thechapterend\hskip\betweenumberspace\fi ##1}}}%!!! + \def\sectionmark##1{\markright{{\ifnum\c@secnumdepth>\z@ + \thesection\seccounterend\hskip\betweenumberspace\fi ##1}}}} + +\def\ps@myheadings{\let\@mkboth\@gobbletwo + \let\@oddfoot\@empty\let\@evenfoot\@empty + \def\@evenhead{\runheadsize\runheadstyle\rlap{\thepage}\customizhead + \leftmark\hfil} + \def\@oddhead{\hfil\runheadsize\runheadstyle\rightmark\customizhead + \llap{\thepage}} + \let\chaptermark\@gobble + \let\sectionmark\@gobble + \let\subsectionmark\@gobble} + + +\ps@headings + +\endinput +%end of file svmono.cls diff --git a/KLSadd_insertion/tempchap09.tex b/KLSadd_insertion/tempchap09.tex new file mode 100644 index 0000000..71f90cb --- /dev/null +++ b/KLSadd_insertion/tempchap09.tex @@ -0,0 +1,3926 @@ +\documentclass[envcountchap,graybox]{svmono} + +\addtolength{\textwidth}{1mm} + +\usepackage{amsmath,amssymb} + +\usepackage{amsfonts} +%\usepackage{breqn} +\usepackage{DLMFmath} +\usepackage{DRMFfcns} + +\usepackage{mathptmx} +\usepackage{helvet} +\usepackage{courier} + +\usepackage{makeidx} +\usepackage{graphicx} + +\usepackage{multicol} +\usepackage[bottom]{footmisc} + +\makeindex + +\def\bibname{Bibliography} +\def\refname{Bibliography} + +\def\theequation{\thesection.\arabic{equation}} + +\smartqed + +\let\corollary=\undefined +\spnewtheorem{corollary}[theorem]{Corollary}{\bfseries}{\itshape} + +\newcounter{rom} + +\newcommand{\hyp}[5]{\mbox{}_{#1}{F}_{#2} +\left(\genfrac{}{}{0pt}{}{#3}{#4}\,;\,#5\right)} + +\newcommand{\qhyp}[5]{\mbox{}_{#1}{\phi}_{#2} +\left(\genfrac{}{}{0pt}{}{#3}{#4}\,;\,q,\,#5\right)} + +\newcommand{\mathindent}{\hspace{7.5mm}} + +\newcommand{\e}{\textrm{e}} + +\renewcommand{\E}{\textrm{E}} + +\renewcommand{\textfraction}{-1} + +\renewcommand{\Gamma}{\varGamma} + +\renewcommand{\leftlegendglue}{\hfil} + +\settowidth{\tocchpnum}{14\enspace} +\settowidth{\tocsecnum}{14.30\enspace} +\settowidth{\tocsubsecnum}{14.12.1\enspace} + +\makeatletter +\def\cleartoversopage{\clearpage\if@twoside\ifodd\c@page + \hbox{}\newpage\if@twocolumn\hbox{}\newpage\fi + \else\fi\fi} + +\newcommand{\clearemptyversopage}{ + \clearpage{\pagestyle{empty}\cleartoversopage}} +\makeatother + +\oddsidemargin -1.5cm +\topmargin -2.0cm +\textwidth 16.3cm +\textheight 25cm + +\begin{document} + +\author{Roelof Koekoek\\[2.5mm]Peter A. Lesky\\[2.5mm]Ren\'e F. Swarttouw} +\title{Hypergeometric orthogonal polynomials and their\\$q$-analogues} +\subtitle{-- Monograph --} +\maketitle + +\frontmatter + +\large + +\addtocounter{chapter}{8} +\pagenumbering{roman} +\chapter{Hypergeometric orthogonal polynomials} +\label{HyperOrtPol} + + +In this chapter we deal with all families of hypergeometric orthogonal polynomials +appearing in the Askey scheme on page~\pageref{scheme}. For each family of orthogonal +polynomials we state the most important properties such as a representation as a +hypergeometric function, orthogonality relation(s), the three-term recurrence relation, +the second-order differential or difference equation, the forward shift (or degree lowering) +and backward shift (or degree raising) operator, a Rodrigues-type formula and some generating +functions. In each case we use the notation which seems to be most common in the literature. +Moreover, in each case we mention the connection between various families by stating the +appropriate limit relations. See also \cite{Terwilliger2006} for an algebraic approach of +this Askey scheme and \cite{TemmeLopez2001} for a view from asymptotic analysis. +For notations the reader is referred to chapter~\ref{Definitions}. + +\section{Wilson}\index{Wilson polynomials} + +\par + +\subsection*{Hypergeometric representation} +\begin{eqnarray} +\label{DefWilson} +& &\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\nonumber\\ +& &{}=\hyp{4}{3}{-n,n+a+b+c+d-1,a+ix,a-ix}{a+b,a+c,a+d}{1}. +\end{eqnarray} + +\newpage + +\subsection*{Orthogonality relation} +If $Re(a,b,c,d)>0$ and non-real parameters occur in conjugate pairs, then +\begin{eqnarray} +\label{OrthIWilson} +& &\frac{1}{2\pi}\int_0^{\infty} +\left|\frac{\Gamma(a+ix)\Gamma(b+ix)\Gamma(c+ix)\Gamma(d+ix)}{\Gamma(2ix)}\right|^2\nonumber\\ +& &{}\mathindent\times W_m(x^2;a,b,c,d)W_n(x^2;a,b,c,d)\,dx\nonumber\\ +& &{}=\frac{\Gamma(n+a+b)\cdots\Gamma(n+c+d)}{\Gamma(2n+a+b+c+d)}(n+a+b+c+d-1)_nn!\,\delta_{mn}, +\end{eqnarray} +where +\begin{eqnarray*} +& &\Gamma(n+a+b)\cdots\Gamma(n+c+d)\\ +& &{}=\Gamma(n+a+b)\Gamma(n+a+c)\Gamma(n+a+d)\Gamma(n+b+c)\Gamma(n+b+d)\Gamma(n+c+d). +\end{eqnarray*} +If $a<0$ and $a+b$, $a+c$, $a+d$ are positive or a pair of complex conjugates +occur with positive real parts, then +\begin{eqnarray} +\label{OrtIIWilson} +& &\frac{1}{2\pi}\int_0^{\infty}\left|\frac{\Gamma(a+ix)\Gamma(b+ix)\Gamma(c+ix)\Gamma(d+ix)}{\Gamma(2ix)}\right|^2\nonumber\\ +& &{}\mathindent\mathindent\times W_m(x^2;a,b,c,d)W_n(x^2;a,b,c,d)\,dx\nonumber\\ +& &{}\mathindent{}+\frac{\Gamma(a+b)\Gamma(a+c)\Gamma(a+d)\Gamma(b-a)\Gamma(c-a)\Gamma(d-a)}{\Gamma(-2a)}\nonumber\\ +& &{}\mathindent\mathindent{}\times\sum_{\begin{array}{c} +{\scriptstyle k=0,1,2\ldots}\\{\scriptstyle a+k<0}\end{array}} +\frac{(2a)_k(a+1)_k(a+b)_k(a+c)_k(a+d)_k}{(a)_k(a-b+1)_k(a-c+1)_k(a-d+1)_kk!}\nonumber\\ +& &{}\mathindent\mathindent\mathindent\mathindent{}\times W_m(-(a+k)^2;a,b,c,d)W_n(-(a+k)^2;a,b,c,d)\nonumber\\ +& &{}=\frac{\Gamma(n+a+b)\cdots\Gamma(n+c+d)}{\Gamma(2n+a+b+c+d)}(n+a+b+c+d-1)_nn!\,\delta_{mn}. +\end{eqnarray} + +\subsection*{Recurrence relation} +\begin{equation} +\label{RecWilson} +-\left(a^2+x^2\right){\tilde{W}}_n(x^2) +=A_n{\tilde{W}}_{n+1}(x^2)-\left(A_n+C_n\right){\tilde{W}}_n(x^2)+C_n{\tilde{W}}_{n-1}(x^2), +\end{equation} +where +$${\tilde{W}}_n(x^2):={\tilde{W}}_n(x^2;a,b,c,d)=\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}$$ +and +$$\left\{\begin{array}{l} +\displaystyle A_n=\frac{(n+a+b+c+d-1)(n+a+b)(n+a+c)(n+a+d)}{(2n+a+b+c+d-1)(2n+a+b+c+d)}\\[5mm] +\displaystyle C_n=\frac{n(n+b+c-1)(n+b+d-1)(n+c+d-1)}{(2n+a+b+c+d-2)(2n+a+b+c+d-1)}. +\end{array}\right.$$ + +\subsection*{Normalized recurrence relation} +\begin{equation} +\label{NormRecWilson} +xp_n(x)=p_{n+1}(x)+(A_n+C_n-a^2)p_n(x)+A_{n-1}C_np_{n-1}(x), +\end{equation} +where +$$W_n(x^2;a,b,c,d)=(-1)^n(n+a+b+c+d-1)_np_n(x^2).$$ + +\subsection*{Difference equation} +\begin{eqnarray} +\label{dvWilson} +& &n(n+a+b+c+d-1)y(x)\nonumber\\ +& &{}=B(x)y(x+i)-\left[B(x)+D(x)\right]y(x)+D(x)y(x-i), +\end{eqnarray} +where +$$y(x)=W_n(x^2;a,b,c,d)$$ +and +$$\left\{\begin{array}{l} +\displaystyle B(x)=\frac{(a-ix)(b-ix)(c-ix)(d-ix)}{2ix(2ix-1)}\\[5mm] +\displaystyle D(x)=\frac{(a+ix)(b+ix)(c+ix)(d+ix)}{2ix(2ix+1)}. +\end{array}\right.$$ + +\subsection*{Forward shift operator} +\begin{eqnarray} +\label{shift1WilsonI} +& &W_n((x+\textstyle\frac{1}{2}i)^2;a,b,c,d) +-W_n((x-\textstyle\frac{1}{2}i)^2;a,b,c,d)\nonumber\\ +& &{}=-2inx(n+a+b+c+d-1)W_{n-1}(x^2;a+\textstyle\frac{1}{2},b+\textstyle\frac{1}{2}, +c+\textstyle\frac{1}{2},d+\textstyle\frac{1}{2}) +\end{eqnarray} +or equivalently +\begin{eqnarray} +\label{shift1WilsonII} +& &\frac{\delta W_n(x^2;a,b,c,d)}{\delta x^2}\nonumber\\ +& &{}=-n(n+a+b+c+d-1)W_{n-1}(x^2;a+\textstyle\frac{1}{2},b+\textstyle\frac{1}{2}, +c+\textstyle\frac{1}{2},d+\textstyle\frac{1}{2}). +\end{eqnarray} + +\newpage + +\subsection*{Backward shift operator} +\begin{eqnarray} +\label{shift2WilsonI} +& &(a-\textstyle\frac{1}{2}-ix)(b-\textstyle\frac{1}{2}-ix) +(c-\textstyle\frac{1}{2}-ix)(d-\textstyle\frac{1}{2}-ix) +W_n((x+\textstyle\frac{1}{2}i)^2;a,b,c,d)\nonumber\\ +& &{}\mathindent{}-(a-\textstyle\frac{1}{2}+ix)(b-\textstyle\frac{1}{2}+ix) +(c-\textstyle\frac{1}{2}+ix)(d-\textstyle\frac{1}{2}+ix) +W_n((x-\textstyle\frac{1}{2}i)^2;a,b,c,d)\nonumber\\ +& &{}=-2ix W_{n+1}(x^2;a-\textstyle\frac{1}{2},b-\textstyle\frac{1}{2}, +c-\textstyle\frac{1}{2},d-\textstyle\frac{1}{2}) +\end{eqnarray} +or equivalently +\begin{eqnarray} +\label{shift2WilsonII} +& &\frac{\delta\left[\omega(x;a,b,c,d)W_n(x^2;a,b,c,d)\right]}{\delta x^2}\nonumber\\ +& &{}=\omega(x;a-\textstyle\frac{1}{2},b-\textstyle\frac{1}{2}, +c-\textstyle\frac{1}{2},d-\textstyle\frac{1}{2}) +W_{n+1}(x^2;a-\textstyle\frac{1}{2},b-\textstyle\frac{1}{2},c-\textstyle\frac{1}{2},d-\textstyle\frac{1}{2}), +\end{eqnarray} +where +$$\omega(x;a,b,c,d):=\frac{1}{2ix}\left|\frac{\Gamma(a+ix)\Gamma(b+ix)\Gamma(c+ix)\Gamma(d+ix)}{\Gamma(2ix)}\right|^2.$$ + +\subsection*{Rodrigues-type formula} +\begin{eqnarray} +\label{RodWilson} +& &\omega(x;a,b,c,d)W_n(x^2;a,b,c,d)\nonumber\\ +& &{}=\left(\frac{\delta}{\delta x^2}\right)^n +\left[\omega(x;a+\textstyle\frac{1}{2}n,b+\textstyle\frac{1}{2}n, +c+\textstyle\frac{1}{2}n,d+\textstyle\frac{1}{2}n)\right]. +\end{eqnarray} + +\subsection*{Generating functions} +\begin{equation} +\label{GenWilson1} +\hyp{2}{1}{a+ix,b+ix}{a+b}{t}\,\hyp{2}{1}{c-ix,d-ix}{c+d}{t}= +\sum_{n=0}^{\infty}\frac{W_n(x^2;a,b,c,d)t^n}{(a+b)_n(c+d)_nn!}. +\end{equation} + +\begin{equation} +\label{GenWilson2} +\hyp{2}{1}{a+ix,c+ix}{a+c}{t}\,\hyp{2}{1}{b-ix,d-ix}{b+d}{t}= +\sum_{n=0}^{\infty}\frac{W_n(x^2;a,b,c,d)t^n}{(a+c)_n(b+d)_nn!}. +\end{equation} + +\begin{equation} +\label{GenWilson3} +\hyp{2}{1}{a+ix,d+ix}{a+d}{t}\,\hyp{2}{1}{b-ix,c-ix}{b+c}{t}= +\sum_{n=0}^{\infty}\frac{W_n(x^2;a,b,c,d)t^n}{(a+d)_n(b+c)_nn!}. +\end{equation} + +\begin{eqnarray} +\label{GenWilson4} +& &(1-t)^{1-a-b-c-d}\nonumber\\ +& &{}\mathindent\times\hyp{4}{3}{\frac{1}{2}(a+b+c+d-1),\frac{1}{2}(a+b+c+d),a+ix,a-ix} +{a+b,a+c,a+d}{-\frac{4t}{(1-t)^2}}\nonumber\\ +& &{}=\sum_{n=0}^{\infty}\frac{(a+b+c+d-1)_n}{(a+b)_n(a+c)_n(a+d)_nn!}W_n(x^2;a,b,c,d)t^n. +\end{eqnarray} + +\subsection*{Limit relations} + +\subsubsection*{Wilson $\rightarrow$ Continuous dual Hahn} +The continuous dual Hahn polynomials given by (\ref{DefContDualHahn}) can be found from the +Wilson polynomials by dividing by $(a+d)_n$ and letting $d\rightarrow\infty$: +\begin{equation} +\lim_{d\rightarrow\infty}\frac{W_n(x^2;a,b,c,d)}{(a+d)_n}=S_n(x^2;a,b,c). +\end{equation} + +\subsubsection*{Wilson $\rightarrow$ Continuous Hahn} +The continuous Hahn polynomials given by (\ref{DefContHahn}) are obtained from the Wilson +polynomials by the substitutions $a\rightarrow a-it$, $b\rightarrow b-it$, $c\rightarrow c+it$, +$d\rightarrow d+it$ and $x\rightarrow x+t$ and the limit $t\rightarrow\infty$ in the following +way: +\begin{equation} +\lim_{t\rightarrow\infty} +\frac{W_n((x+t)^2;a-it,b-it,c+it,d+it)}{(-2t)^nn!}=p_n(x;a,b,c,d). +\end{equation} + +\subsubsection*{Wilson $\rightarrow$ Jacobi} +The Jacobi polynomials given by (\ref{DefJacobi}) can be found from the Wilson +polynomials by substituting $a=b=\frac{1}{2}(\alpha+1)$, $c=\frac{1}{2}(\beta+1)+it$, +$d=\frac{1}{2}(\beta+1)-it$ and $x\rightarrow t\sqrt{\frac{1}{2}(1-x)}$ in the +definition (\ref{DefWilson}) of the Wilson polynomials and taking the limit +$t\rightarrow\infty$. In fact we have +\begin{eqnarray} +& &\lim_{t\rightarrow\infty}\frac{W_n(\frac{1}{2}(1-x)t^2;\frac{1}{2}(\alpha+1), +\frac{1}{2}(\alpha+1),\frac{1}{2}(\beta+1)+it,\frac{1}{2}(\beta+1)-it)}{t^{2n}n!}\nonumber\\ +& &{}=P_n^{(\alpha,\beta)}(x). +\end{eqnarray} + +\subsection*{Remarks} +Note that for $k0$, $c=\bar{a}$ and $d=\bar{b}$, then +\begin{eqnarray} +\label{OrtContHahn} +& &\frac{1}{2\pi}\int_{-\infty}^{\infty}\Gamma(a+ix)\Gamma(b+ix)\Gamma(c-ix)\Gamma(d-ix) +p_m(x;a,b,c,d)p_n(x;a,b,c,d)\,dx\nonumber\\ +& &{}=\frac{\Gamma(n+a+c)\Gamma(n+a+d)\Gamma(n+b+c)\Gamma(n+b+d)} +{(2n+a+b+c+d-1)\Gamma(n+a+b+c+d-1)n!}\,\delta_{mn}. +\end{eqnarray} + +\subsection*{Recurrence relation} +\begin{equation} +\label{RecContHahn} +(a+ix)\tilde{p}_n(x)=A_n\tilde{p}_{n+1}(x)-\left(A_n+C_n\right)\tilde{p}_n(x)+C_n\tilde{p}_{n-1}(x), +\end{equation} +where +$$\tilde{p}_n(x):=\tilde{p}_n(x;a,b,c,d)=\frac{n!}{i^n(a+c)_n(a+d)_n}p_n(x;a,b,c,d)$$ +and +$$\left\{\begin{array}{l} +\displaystyle A_n=-\frac{(n+a+b+c+d-1)(n+a+c)(n+a+d)}{(2n+a+b+c+d-1)(2n+a+b+c+d)}\\[5mm] +\displaystyle C_n=\frac{n(n+b+c-1)(n+b+d-1)}{(2n+a+b+c+d-2)(2n+a+b+c+d-1)}. +\end{array}\right.$$ + +\subsection*{Normalized recurrence relation} +\begin{equation} +\label{NormRecContHahn} +xp_n(x)=p_{n+1}(x)+i(A_n+C_n+a)p_n(x)-A_{n-1}C_np_{n-1}(x), +\end{equation} +where +$$p_n(x;a,b,c,d)=\frac{(n+a+b+c+d-1)_n}{n!}p_n(x).$$ + +\subsection*{Difference equation} +\begin{eqnarray} +\label{dvContHahn} +& &n(n+a+b+c+d-1)y(x)\nonumber\\ +& &{}=B(x)y(x+i)-\left[B(x)+D(x)\right]y(x)+D(x)y(x-i), +\end{eqnarray} +where +$$y(x)=p_n(x;a,b,c,d)$$ +and +$$\left\{\begin{array}{l} +\displaystyle B(x)=(c-ix)(d-ix)\\[5mm] +\displaystyle D(x)=(a+ix)(b+ix). +\end{array}\right.$$ + +\subsection*{Forward shift operator} +\begin{eqnarray} +\label{shift1ContHahnI} +& &p_n(x+\textstyle\frac{1}{2}i;a,b,c,d)-p_n(x-\textstyle\frac{1}{2}i;a,b,c,d)\nonumber\\ +& &{}=i(n+a+b+c+d-1)p_{n-1}(x;a+\textstyle\frac{1}{2}, +b+\textstyle\frac{1}{2},c+\textstyle\frac{1}{2},d+\textstyle\frac{1}{2}) +\end{eqnarray} +or equivalently +\begin{equation} +\label{shift1ContHahnII} +\frac{\delta p_n(x;a,b,c,d)}{\delta x}=(n+a+b+c+d-1) +p_{n-1}(x;a+\textstyle\frac{1}{2},b+\textstyle\frac{1}{2}, +c+\textstyle\frac{1}{2},d+\textstyle\frac{1}{2}). +\end{equation} + +\subsection*{Backward shift operator} +\begin{eqnarray} +\label{shift2ContHahnI} +& &(c-\textstyle\frac{1}{2}-ix)(d-\textstyle\frac{1}{2}-ix) +p_n(x+\textstyle\frac{1}{2}i;a,b,c,d)\nonumber\\ +& &{}\mathindent{}-(a-\textstyle\frac{1}{2}+ix)(b-\textstyle\frac{1}{2}+ix) +p_n(x-\textstyle\frac{1}{2}i;a,b,c,d)\nonumber\\ +& &{}=\frac{n+1}{i}p_{n+1}(x;a-\textstyle\frac{1}{2}, +b-\textstyle\frac{1}{2},c-\textstyle\frac{1}{2},d-\textstyle\frac{1}{2}) +\end{eqnarray} +or equivalently +\begin{eqnarray} +\label{shift2ContHahnII} +& &\frac{\delta\left[\omega(x;a,b,c,d)p_n(x;a,b,c,d)\right]}{\delta x}\nonumber\\ +& &{}=-(n+1)\omega(x;a-\textstyle\frac{1}{2},b-\textstyle\frac{1}{2}, +c-\textstyle\frac{1}{2},d-\textstyle\frac{1}{2})\nonumber\\ +& &{}\mathindent\times p_{n+1}(x;a-\textstyle\frac{1}{2},b-\textstyle\frac{1}{2},c-\textstyle\frac{1}{2},d-\textstyle\frac{1}{2}), +\end{eqnarray} +where +$$\omega(x;a,b,c,d)=\Gamma(a+ix)\Gamma(b+ix)\Gamma(c-ix)\Gamma(d-ix).$$ + +\subsection*{Rodrigues-type formula} +\begin{eqnarray} +\label{RodContHahn} +& &\omega(x;a,b,c,d)p_n(x;a,b,c,d)\nonumber\\ +& &{}=\frac{(-1)^n}{n!}\left(\frac{\delta}{\delta x}\right)^n +\left[\omega(x;a+\textstyle\frac{1}{2}n,b+\textstyle\frac{1}{2}n, +c+\textstyle\frac{1}{2}n,d+\textstyle\frac{1}{2}n)\right]. +\end{eqnarray} + +\subsection*{Generating functions} +\begin{equation} +\label{GenContHahn1} +\hyp{1}{1}{a+ix}{a+c}{-it}\,\hyp{1}{1}{d-ix}{b+d}{it}= +\sum_{n=0}^{\infty}\frac{p_n(x;a,b,c,d)}{(a+c)_n(b+d)_n}t^n. +\end{equation} + +\begin{equation} +\label{GenContHahn2} +\hyp{1}{1}{a+ix}{a+d}{-it}\,\hyp{1}{1}{c-ix}{b+c}{it}= +\sum_{n=0}^{\infty}\frac{p_n(x;a,b,c,d)}{(a+d)_n(b+c)_n}t^n. +\end{equation} + +\begin{eqnarray} +\label{GenContHahn3} +& &(1-t)^{1-a-b-c-d}\,\hyp{3}{2}{\frac{1}{2}(a+b+c+d-1),\frac{1}{2}(a+b+c+d),a+ix} +{a+c,a+d}{-\frac{4t}{(1-t)^2}}\nonumber\\ +& &{}=\sum_{n=0}^{\infty} +\frac{(a+b+c+d-1)_n}{(a+c)_n(a+d)_ni^n}p_n(x;a,b,c,d)t^n. +\end{eqnarray} + +\subsection*{Limit relations} + +\subsubsection*{Wilson $\rightarrow$ Continuous Hahn} +The continuous Hahn polynomials are obtained from the Wilson polynomials given by +(\ref{DefWilson}) by the substitution $a\rightarrow a-it$, $b\rightarrow b-it$, +$c\rightarrow c+it$, $d\rightarrow d+it$ and $x\rightarrow x+t$ and the limit +$t\rightarrow\infty$ in the following way: +$$\lim_{t\rightarrow\infty} +\frac{W_n((x+t)^2;a-it,b-it,c+it,d+it)}{(-2t)^nn!}=p_n(x;a,b,c,d).$$ + +\subsubsection*{Continuous Hahn $\rightarrow$ Meixner-Pollaczek} +The Meixner-Pollaczek polynomials given by (\ref{DefMP}) can be obtained from the +continuous Hahn polynomials by setting $x\rightarrow x+t$, $a=\lambda-it$, $c=\lambda+it$ +and $b=d=t\tan\phi$ and taking the limit $t\rightarrow\infty$: +\begin{equation} +\lim_{t\rightarrow\infty}\frac{p_n(x+t;\lambda-it,t\tan\phi,\lambda+it,t\tan\phi)}{t^n} +=\frac{P_n^{(\lambda)}(x;\phi)}{(\cos\phi)^n}. +\end{equation} + +\subsubsection*{Continuous Hahn $\rightarrow$ Jacobi} +The Jacobi polynomials given by (\ref{DefJacobi}) follow from the continuous Hahn polynomials +by the substitution $x\rightarrow \frac{1}{2}xt$, $a=\frac{1}{2}(\alpha+1-it)$, +$b=\frac{1}{2}(\beta+1+it)$, $c=\frac{1}{2}(\alpha+1+it)$ and $d=\frac{1}{2}(\beta+1-it)$, +division by $t^n$ and the limit $t\rightarrow\infty$: +\begin{eqnarray} +& &\lim_{t\rightarrow\infty} +\frac{p_n(\frac{1}{2}xt;\frac{1}{2}(\alpha+1-it),\frac{1}{2}(\beta+1+it), +\frac{1}{2}(\alpha+1+it),\frac{1}{2}(\beta+1-it))}{t^n}\nonumber\\ +& &{}=P_n^{(\alpha,\beta)}(x). +\end{eqnarray} + +\subsubsection*{Continuous Hahn $\rightarrow$ Pseudo Jacobi} +The pseudo Jacobi polynomials given by (\ref{DefPseudoJacobi}) follow from the continuous +Hahn polynomials by the substitution $x\rightarrow xt$, $a=\frac{1}{2}(-N+i\nu-2t)$, +$b=\frac{1}{2}(-N-i\nu+2t)$, $c=\frac{1}{2}(-N-i\nu-2t)$ and $d=\frac{1}{2}(-N+i\nu+2t)$, +division by $t^n$ and the limit $t\rightarrow\infty$: +\begin{eqnarray} +& &\lim_{t\rightarrow\infty}\frac{p_n(xt;\frac{1}{2}(-N+i\nu-2t),\frac{1}{2}(-N-i\nu+2t), +\frac{1}{2}(-N+i\nu-2t),\frac{1}{2}(-N-i\nu+2t))}{t^n}\nonumber\\ +& &{}=\frac{(n-2N-1)_n}{n!}P_n(x;\nu,N). +\end{eqnarray} + +\subsection*{Remark} +Since we have for $k0$ and $(c,d)=(\overline a,\overline b)$ or $(\overline b,\overline a)$.\\ +Thus, under these assumptions, the continuous Hahn polynomial +$p_n(x;a,b,c,d)$ +is symmetric in $a,b$ and in $c,d$. +This follows from the orthogonality relation (9.4.2) +together with the value of its coefficient of $x^n$ given in (9.4.4b).\\ +As a consequence, it is sufficient to give generating function (9.4.11). Then the generating +function (9.4.12) will follow by symmetry in the parameters. +% +\paragraph{Uniqueness of orthogonality measure} +The coefficient of $p_{n-1}(x)$ in (9.4.4) behaves as $O(n^2)$ as $n\to\iy$. +Hence \eqref{93} holds, by which the orthogonality measure is unique. +% +\paragraph{Special cases} +In the following special case there is a reduction to +Meixner-Pollaczek: +\begin{equation} +p_n(x;a,a+\thalf,a,a+\thalf)= +\frac{(2a)_n (2a+\thalf)_n}{(4a)_n}\,P_n^{(2a)}(2x;\thalf\pi). +\end{equation} +See \myciteKLS{342}{(2.6)} (note that in \myciteKLS{342}{(2.3)} the +Meixner-Pollaczek polynonmials are defined different from (9.7.1), +without a constant factor in front). + +For $0-1$ and $\beta>-1$, or for $\alpha<-N$ and $\beta<-N$, we have +\begin{eqnarray} +\label{OrtHahn} +& &\sum_{x=0}^N\binom{\alpha +x}{x}\binom{\beta+N-x}{N-x}Q_m(x;\alpha,\beta,N)Q_n(x;\alpha,\beta,N)\nonumber\\ +& &{}=\frac{(-1)^n(n+\alpha+\beta+1)_{N+1}(\beta+1)_nn!}{(2n+\alpha+\beta+1)(\alpha+1)_n(-N)_nN!}\,\delta_{mn}. +\end{eqnarray} + +\subsection*{Recurrence relation} +\begin{equation} +\label{RecHahn} +-xQ_n(x)=A_nQ_{n+1}(x)-\left(A_n+C_n\right)Q_n(x)+C_nQ_{n-1}(x), +\end{equation} +where +$$Q_n(x):=Q_n(x;\alpha,\beta,N)$$ +and +$$\left\{\begin{array}{l} +\displaystyle A_n=\frac{(n+\alpha+\beta+1)(n+\alpha+1)(N-n)}{(2n+\alpha+\beta+1)(2n+\alpha+\beta+2)}\\[5mm] +\displaystyle C_n=\frac{n(n+\alpha+\beta+N+1)(n+\beta)}{(2n+\alpha+\beta)(2n+\alpha+\beta+1)}. +\end{array}\right.$$ + +\subsection*{Normalized recurrence relation} +\begin{equation} +\label{NormRecHahn} +xp_n(x)=p_{n+1}(x)+\left(A_n+C_n\right)p_n(x)+A_{n-1}C_np_{n-1}(x), +\end{equation} +where +$$Q_n(x;\alpha,\beta,N)=\frac{(n+\alpha+\beta+1)_n}{(\alpha+1)_n(-N)_n}p_n(x).$$ + +\subsection*{Difference equation} +\begin{equation} +\label{dvHahn} +n(n+\alpha+\beta+1)y(x)=B(x)y(x+1)-\left[B(x)+D(x)\right]y(x)+D(x)y(x-1), +\end{equation} +where +$$y(x)=Q_n(x;\alpha,\beta,N)$$ +and +$$\left\{\begin{array}{l} +\displaystyle B(x)=(x+\alpha+1)(x-N)\\[5mm] +\displaystyle D(x)=x(x-\beta-N-1). +\end{array}\right.$$ + +\subsection*{Forward shift operator} +\begin{eqnarray} +\label{shift1HahnI} +& &Q_n(x+1;\alpha,\beta,N)-Q_n(x;\alpha,\beta,N)\nonumber\\ +& &{}=-\frac{n(n+\alpha+\beta+1)}{(\alpha+1)N}Q_{n-1}(x;\alpha+1,\beta+1,N-1) +\end{eqnarray} +or equivalently +\begin{equation} +\label{shift1HahnII} +\Delta Q_n(x;\alpha,\beta,N)=-\frac{n(n+\alpha+\beta+1)}{(\alpha+1)N}Q_{n-1}(x;\alpha+1,\beta+1,N-1). +\end{equation} + +\subsection*{Backward shift operator} +\begin{eqnarray} +\label{shift2HahnI} +& &(x+\alpha)(N+1-x)Q_n(x;\alpha,\beta,N)-x(\beta+N+1-x)Q_n(x-1;\alpha,\beta,N)\nonumber\\ +& &{}=\alpha(N+1)Q_{n+1}(x;\alpha-1,\beta-1,N+1) +\end{eqnarray} +or equivalently +\begin{eqnarray} +\label{shift2HahnII} +& &\nabla\left[\omega(x;\alpha,\beta,N)Q_n(x;\alpha,\beta,N)\right]\nonumber\\ +& &{}=\frac{N+1}{\beta}\omega(x;\alpha-1,\beta-1,N+1)Q_{n+1}(x;\alpha-1,\beta-1,N+1), +\end{eqnarray} +where +$$\omega(x;\alpha,\beta,N)=\binom{\alpha+x}{x}\binom{\beta+N-x}{N-x}.$$ + +\subsection*{Rodrigues-type formula} +\begin{eqnarray} +\label{RodHahn} +& &\omega(x;\alpha,\beta,N)Q_n(x;\alpha,\beta,N)\nonumber\\ +& &{}=\frac{(-1)^n(\beta+1)_n}{(-N)_n}\nabla^n\left[\omega(x;\alpha+n,\beta+n,N-n)\right]. +\end{eqnarray} + +\subsection*{Generating functions} +For $x=0,1,2,\ldots,N$ we have +\begin{equation} +\label{GenHahn1} +\hyp{1}{1}{-x}{\alpha+1}{-t}\,\hyp{1}{1}{x-N}{\beta+1}{t} +=\sum_{n=0}^N\frac{(-N)_n}{(\beta+1)_nn!}Q_n(x;\alpha,\beta,N)t^n. +\end{equation} + +\begin{eqnarray} +\label{GenHahn2} +& &\hyp{2}{0}{-x,-x+\beta+N+1}{-}{-t}\,\hyp{2}{0}{x-N,x+\alpha+1}{-}{t}\nonumber\\ +& &{}=\sum_{n=0}^N\frac{(-N)_n(\alpha+1)_n}{n!}Q_n(x;\alpha,\beta,N)t^n. +\end{eqnarray} + +\begin{eqnarray} +\label{GenHahn3} +& &\left[(1-t)^{-\alpha-\beta-1}\,\hyp{3}{2}{\frac{1}{2}(\alpha+\beta+1),\frac{1}{2}(\alpha+\beta+2),-x} +{\alpha+1,-N}{-\frac{4t}{(1-t)^2}}\right]_N\nonumber\\ +& &{}=\sum_{n=0}^N\frac{(\alpha+\beta+1)_n}{n!}Q_n(x;\alpha,\beta,N)t^n. +\end{eqnarray} + +\subsection*{Limit relations} + +\subsubsection*{Racah $\rightarrow$ Hahn} +If we take $\gamma+1=-N$ and let $\delta\rightarrow\infty$ in the definition (\ref{DefRacah}) +of the Racah polynomials, we obtain the Hahn polynomials. Hence +$$\lim_{\delta\rightarrow\infty} +R_n(\lambda(x);\alpha,\beta,-N-1,\delta)=Q_n(x;\alpha,\beta,N).$$ +And if we take $\delta=-\beta-N-1$ and let $\gamma\rightarrow\infty$ in the definition (\ref{DefRacah}) +of the Racah polynomials, we also obtain the Hahn polynomials: +$$\lim_{\gamma\rightarrow\infty} +R_n(\lambda(x);\alpha,\beta,\gamma,-\beta-N-1)=Q_n(x;\alpha,\beta,N).$$ +Another way to do this is to take $\alpha+1=-N$ and $\beta\rightarrow\beta+\gamma+N+1$ in +the definition (\ref{DefRacah}) of the Racah polynomials and then take the limit +$\delta\rightarrow\infty$. In that case we obtain the Hahn polynomials in the following way: +$$\lim_{\delta\rightarrow\infty} +R_n(\lambda(x);-N-1,\beta+\gamma+N+1,\gamma,\delta)=Q_n(x;\gamma,\beta,N).$$ + +\subsubsection*{Hahn $\rightarrow$ Jacobi} +To find the Jacobi polynomials given by (\ref{DefJacobi}) from the Hahn polynomials we take +$x\rightarrow Nx$ and let $N\rightarrow\infty$. In fact we have +\begin{equation} +\lim_{N\rightarrow\infty} +Q_n(Nx;\alpha,\beta,N)=\frac{P_n^{(\alpha,\beta)}(1-2x)}{P_n^{(\alpha,\beta)}(1)}. +\end{equation} + +\subsubsection*{Hahn $\rightarrow$ Meixner} +The Meixner polynomials given by (\ref{DefMeixner}) can be obtained from the Hahn polynomials +by taking $\alpha=b-1$, $\beta=N(1-c)c^{-1}$ and letting $N\rightarrow\infty$: +\begin{equation} +\lim_{N\rightarrow\infty} +Q_n(x;b-1,N(1-c)c^{-1},N)=M_n(x;b,c). +\end{equation} + +\subsubsection*{Hahn $\rightarrow$ Krawtchouk} +The Krawtchouk polynomials given by (\ref{DefKrawtchouk}) are obtained from the Hahn polynomials +if we take $\alpha=pt$ and $\beta=(1-p)t$ and let $t\rightarrow\infty$: +\begin{equation} +\lim_{t\rightarrow\infty}Q_n(x;pt,(1-p)t,N)=K_n(x;p,N). +\end{equation} + +\subsection*{Remark} +If we interchange the role of $x$ and $n$ in (\ref{DefHahn}) we obtain the +dual Hahn polynomials given by (\ref{DefDualHahn}). + +\noindent +Since +$$Q_n(x;\alpha,\beta,N)=R_x(\lambda(n);\alpha,\beta,N)$$ +we obtain the dual orthogonality relation for the Hahn polynomials from the +orthogonality relation (\ref{OrtDualHahn}) of the dual Hahn polynomials: +\begin{eqnarray*} +& &\sum_{n=0}^N\frac{(2n+\alpha+\beta+1)(\alpha+1)_n(-N)_nN!}{(-1)^n(n+\alpha+\beta+1)_{N+1}(\beta+1)_nn!} +Q_n(x;\alpha,\beta,N)Q_n(y;\alpha,\beta,N)\\ +& &{}=\frac{\delta_{xy}}{\dbinom{\alpha+x}{x}\dbinom{\beta+N-x}{N-x}},\quad x,y \in \{0,1,2,\ldots,N\}. +\end{eqnarray*} + +\subsection*{References} +\label{sec9.5} +% +\paragraph{Special values} +\begin{equation} +Q_n(0;\al,\be,N)=1,\quad +Q_n(N;\al,\be,N)=\frac{(-1)^n(\be+1)_n}{(\al+1)_n}\,. +\label{95} +\end{equation} +Use (9.5.1) and compare with (9.8.1) and \eqref{50}. + +From (9.5.3) and \eqref{1} it follows that +\begin{equation} +Q_{2n}(N;\al,\al,2N)=\frac{(\thalf)_n(N+\al+1)_n}{(-N+\thalf)_n(\al+1)_n}\,. +\label{30} +\end{equation} +From (9.5.1) and \mycite{DLMF}{(15.4.24)} it follows that +\begin{equation} +Q_N(x;\al,\be,N)=\frac{(-N-\be)_x}{(\al+1)_x}\qquad(x=0,1,\ldots,N). +\label{44} +\end{equation} +% +\paragraph{Symmetries} +By the orthogonality relation (9.5.2): +\begin{equation} +\frac{Q_n(N-x;\al,\be,N)}{Q_n(N;\al,\be,N)}=Q_n(x;\be,\al,N), +\label{96} +\end{equation} +It follows from \eqref{97} and \eqref{45} that +\begin{equation} +\frac{Q_{N-n}(x;\al,\be,N)}{Q_N(x;\al,\be,N)} +=Q_n(x;-N-\be-1,-N-\al-1,N) +\qquad(x=0,1,\ldots,N). +\label{100} +\end{equation} +% +\paragraph{Duality} +The Remark on p.208 gives the duality between Hahn and dual Hahn polynomials: +% +\begin{equation} +Q_n(x;\al,\be,N)=R_x(n(n+\al+\be+1);\al,\be,N)\quad(n,x\in\{0,1,\ldots N\}). +\label{45} +\end{equation} +% +\cite{AlSalam90}, \cite{AndrewsAskey85}, \cite{Area+II}, \cite{Askey75}, +\cite{Askey89I}, \cite{Askey2005}, \cite{AskeyGasper77}, \cite{AskeyWilson85}, +\cite{AtakRahmanSuslov}, \cite{AtakSuslov88}, \cite{Chihara78}, +\cite{Ciesielski}, \cite{Cooper+}, \cite{Dette95}, \cite{Dunkl76}, +\cite{Dunkl78I}, \cite{Gasper73I}, \cite{Gasper74}, \cite{HoareRahman}, +\cite{Ismail77}, \cite{Ismail2005II}, \cite{Karlin61}, \cite{Koorn81}, \cite{Koorn88}, +\cite{LabelleYehI}, \cite{LabelleYehII}, \cite{Laine}, \cite{Lesky62}, +\cite{Lesky88}, \cite{Lesky89}, \cite{Lesky94I}, \cite{Lesky95II}, +\cite{LewanowiczII}, \cite{Neuman}, \cite{Nikiforov+}, \cite{NikiforovUvarov}, +\cite{Rahman76III}, \cite{Rahman78I}, \cite{Rahman78II}, \cite{Rahman81III}, +\cite{Sablonniere}, \cite{Stanton84}, \cite{Stanton90}, \cite{Wilson80}, \cite{Wilson70II}, +\cite{Zarzo+}. + + +\section{Dual Hahn}\index{Dual Hahn polynomials}\index{Hahn polynomials!Dual} + +\par\setcounter{equation}{0} + +\subsection*{Hypergeometric representation} +\begin{equation} +\label{DefDualHahn} +R_n(\lambda(x);\gamma,\delta,N)= +\hyp{3}{2}{-n,-x,x+\gamma+\delta+1}{\gamma+1,-N}{1},\quad n=0,1,2,\ldots,N, +\end{equation} +where +$$\lambda(x)=x(x+\gamma+\delta+1).$$ + +\subsection*{Orthogonality relation} +For $\gamma>-1$ and $\delta>-1$, or for $\gamma<-N$ and $\delta<-N$, we have +\begin{eqnarray} +\label{OrtDualHahn} +& &\sum_{x=0}^N\frac{(2x+\gamma+\delta+1)(\gamma+1)_x(-N)_xN!}{(-1)^x(x+\gamma+\delta+1)_{N+1}(\delta+1)_xx!} +R_m(\lambda(x);\gamma,\delta,N)R_n(\lambda(x);\gamma,\delta,N)\nonumber\\ +& &{}=\frac{\,\delta_{mn}}{\dbinom{\gamma+n}{n}\dbinom{\delta+N-n}{N-n}}. +\end{eqnarray} + +\subsection*{Recurrence relation} +\begin{equation} +\label{RecDualHahn} +\lambda(x)R_n(\lambda(x)) +=A_nR_{n+1}(\lambda(x))-\left(A_n+C_n\right)R_n(\lambda(x))+C_nR_{n-1}(\lambda(x)), +\end{equation} +where +$$R_n(\lambda(x)):=R_n(\lambda(x);\gamma,\delta,N)$$ +and +$$\left\{\begin{array}{l} +\displaystyle A_n=(n+\gamma+1)(n-N)\\[5mm] +\displaystyle C_n=n(n-\delta-N-1). +\end{array}\right.$$ + +\subsection*{Normalized recurrence relation} +\begin{equation} +\label{NormRecDualHahn} +xp_n(x)=p_{n+1}(x)-(A_n+C_n)p_n(x)+A_{n-1}C_np_{n-1}(x), +\end{equation} +where +$$R_n(\lambda(x);\gamma,\delta,N)=\frac{1}{(\gamma+1)_n(-N)_n}p_n(\lambda(x)).$$ + +\subsection*{Difference equation} +\begin{equation} +\label{dvDualHahn} +-ny(x)=B(x)y(x+1)-\left[B(x)+D(x)\right]y(x)+D(x)y(x-1), +\end{equation} +where +$$y(x)=R_n(\lambda(x);\gamma,\delta,N)$$ +and +$$\left\{\begin{array}{l} +\displaystyle B(x)=\frac{(x+\gamma+1)(x+\gamma+\delta+1)(N-x)}{(2x+\gamma+\delta+1)(2x+\gamma+\delta+2)}\\[5mm] +\displaystyle D(x)=\frac{x(x+\gamma+\delta+N+1)(x+\delta)}{(2x+\gamma+\delta)(2x+\gamma+\delta+1)}. +\end{array}\right.$$ + +\subsection*{Forward shift operator} +\begin{eqnarray} +\label{shift1DualHahnI} +& &R_n(\lambda(x+1);\gamma,\delta,N)-R_n(\lambda(x);\gamma,\delta,N)\nonumber\\ +& &{}=-\frac{n(2x+\gamma+\delta+2)}{(\gamma+1)N}R_{n-1}(\lambda(x);\gamma+1,\delta,N-1) +\end{eqnarray} +or equivalently +\begin{equation} +\label{shift1DualHahnII} +\frac{\Delta R_n(\lambda(x);\gamma,\delta,N)}{\Delta\lambda(x)}= +-\frac{n}{(\gamma+1)N}R_{n-1}(\lambda(x);\gamma+1,\delta,N-1). +\end{equation} + +\subsection*{Backward shift operator} +\begin{eqnarray} +\label{shift2DualHahnI} +& &(x+\gamma)(x+\gamma+\delta)(N+1-x)R_n(\lambda(x);\gamma,\delta,N)\nonumber\\ +& &{}\mathindent{}-x(x+\gamma+\delta+N+1)(x+\delta)R_n(\lambda(x-1);\gamma,\delta,N)\nonumber\\ +& &{}=\gamma(N+1)(2x+\gamma+\delta)R_{n+1}(\lambda(x);\gamma-1,\delta,N+1) +\end{eqnarray} +or equivalently +\begin{eqnarray} +\label{shift2DualHahnII} +& &\frac{\nabla\left[\omega(x;\gamma,\delta,N)R_n(\lambda(x);\gamma,\delta,N)\right]}{\nabla\lambda(x)}\nonumber\\ +& &{}=\frac{1}{\gamma+\delta}\omega(x;\gamma-1,\delta,N+1)R_{n+1}(\lambda(x);\gamma-1,\delta,N+1), +\end{eqnarray} +where +$$\omega(x;\gamma,\delta,N)=\frac{(-1)^x(\gamma+1)_x(\gamma+\delta+1)_x(-N)_x} +{(\gamma+\delta+N+2)_x(\delta+1)_xx!}.$$ + +\newpage + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodDualHahn} +\omega(x;\gamma,\delta,N)R_n(\lambda(x);\gamma,\delta,N) +=(\gamma+\delta+1)_n\left(\nabla_{\lambda}\right)^n\left[\omega(x;\gamma+n,\delta,N-n)\right], +\end{equation} +where +$$\nabla_{\lambda}:=\frac{\nabla}{\nabla\lambda(x)}.$$ + +\subsection*{Generating functions} +For $x=0,1,2,\ldots,N$ we have +\begin{equation} +\label{GenDualHahn1} +(1-t)^{N-x}\,\hyp{2}{1}{-x,-x-\delta}{\gamma+1}{t} +=\sum_{n=0}^N\frac{(-N)_n}{n!}R_n(\lambda(x);\gamma,\delta,N)t^n. +\end{equation} + +\begin{eqnarray} +\label{GenDualHahn2} +& &(1-t)^x\,\hyp{2}{1}{x-N,x+\gamma+1}{-\delta-N}{t}\nonumber\\ +& &=\sum_{n=0}^N\frac{(\gamma+1)_n(-N)_n}{(-\delta-N)_nn!}R_n(\lambda(x);\gamma,\delta,N)t^n. +\end{eqnarray} + +\begin{equation} +\label{GenDualHahn3} +\left[\expe^t\,\hyp{2}{2}{-x,x+\gamma+\delta+1}{\gamma+1,-N}{-t}\right]_N=\sum_{n=0}^N +\frac{R_n(\lambda(x);\gamma,\delta,N)}{n!}t^n. +\end{equation} + +\begin{eqnarray} +\label{GenDualHahn4} +& &\left[(1-t)^{-\epsilon}\,\hyp{3}{2}{\epsilon,-x,x+\gamma+\delta+1}{\gamma+1,-N}{\frac{t}{t-1}}\right]_N\nonumber\\ +& &{}=\sum_{n=0}^N\frac{(\epsilon)_n}{n!}R_n(\lambda(x);\gamma,\delta,N)t^n, +\quad\textrm{$\epsilon$ arbitrary}. +\end{eqnarray} + +\subsection*{Limit relations} + +\subsubsection*{Racah $\rightarrow$ Dual Hahn} +If we take $\alpha+1=-N$ and let $\beta\rightarrow\infty$ in the definition (\ref{DefRacah}) +of the Racah polynomials, then we obtain the dual Hahn polynomials: +$$\lim_{\beta\rightarrow\infty} +R_n(\lambda(x);-N-1,\beta,\gamma,\delta)=R_n(\lambda(x);\gamma,\delta,N).$$ +And if we take $\beta=-\delta-N-1$ and let $\alpha\rightarrow\infty$ in the definition (\ref{DefRacah}) +of the Racah polynomials, then we also obtain the dual Hahn polynomials: +$$\lim_{\alpha\rightarrow\infty} +R_n(\lambda(x);\alpha,-\delta-N-1,\gamma,\delta)=R_n(\lambda(x);\gamma,\delta,N).$$ +Finally, if we take $\gamma+1=-N$ and $\delta\rightarrow\alpha+\delta+N+1$ in the definition +(\ref{DefRacah}) of the Racah polynomials and take the limit +$\beta\rightarrow\infty$ we find the dual Hahn polynomials in the following way: +$$\lim_{\beta\rightarrow\infty} +R_n(\lambda(x);\alpha,\beta,-N-1,\alpha+\delta+N+1)=R_n(\lambda(x);\alpha,\delta,N).$$ + +\subsubsection*{Dual Hahn $\rightarrow$ Meixner} +The Meixner polynomials given by (\ref{DefMeixner}) are obtained from the dual Hahn polynomials +if we take $\gamma=\beta-1$ and $\delta=N(1-c)c^{-1}$ and let $N\rightarrow\infty$: +\begin{equation} +\lim_{N\rightarrow\infty} +R_n(\lambda(x);\beta-1,N(1-c)c^{-1},N)=M_n(x;\beta,c). +\end{equation} + +\subsubsection*{Dual Hahn $\rightarrow$ Krawtchouk} +The Krawtchouk polynomials given by (\ref{DefKrawtchouk}) can be obtained from the dual +Hahn polynomials by setting $\gamma=pt$, $\delta=(1-p)t$ and letting $t\rightarrow\infty$: +\begin{equation} +\lim_{t\rightarrow\infty}R_n(\lambda(x);pt,(1-p)t,N)=K_n(x;p,N). +\end{equation} + +\subsection*{Remark} +If we interchange the role of $x$ and $n$ in the definition (\ref{DefDualHahn}) +of the dual Hahn polynomials we obtain the Hahn polynomials given by (\ref{DefHahn}). + +\noindent +Since +$$R_n(\lambda(x);\gamma,\delta,N)=Q_x(n;\gamma,\delta,N)$$ +we obtain the dual orthogonality relation for the dual Hahn polynomials +from the orthogonality relation (\ref{OrtHahn}) for the Hahn polynomials: +\begin{eqnarray*} +& &\sum_{n=0}^N\binom{\gamma+n}{n}\binom{\delta+N-n}{N-n} R_n(\lambda(x);\gamma,\delta,N)R_n(\lambda(y);\gamma,\delta,N)\\ +& &{}=\frac{(-1)^x(x+\gamma+\delta+1)_{N+1}(\delta+1)_xx!} +{(2x+\gamma+\delta+1)(\gamma+1)_x(-N)_xN!}\delta_{xy},\quad x,y \in \{0,1,2,\ldots,N\}. +\end{eqnarray*} + +\subsection*{References} +\label{sec9.6} +% +\paragraph{Special values} +By \eqref{44} and \eqref{45} we have +\begin{equation} +R_n(N(N+\ga+\de+1);\ga,\de,N)=\frac{(-N-\de)_n}{(\ga+1)_n}\,. +\label{47} +\end{equation} +It follows from \eqref{95} and \eqref{45} that +\begin{equation} +R_N(x(x+\ga+\de+1);\ga,\de,N) +=\frac{(-1)^x(\de+1)_x}{(\ga+1)_x}\qquad(x=0,1,\ldots,N). +\label{101} +\end{equation} +% +\paragraph{Symmetries} +Write the weight in (9.6.2) as +\begin{equation} +w_x(\al,\be,N):=N!\,\frac{2x+\ga+\de+1}{(x+\ga+\de+1)_{N+1}}\, +\frac{(\ga+1)_x}{(\de+1)_x}\,\binom Nx. +\label{98} +\end{equation} +Then +\begin{equation} +(\de+1)_N\,w_{N-x}(\ga,\de,N)= +(-\ga-N)_N\,w_x(-\de-N-1,-\ga-N-1,N). +\label{99} +\end{equation} +Hence, by (9.6.2), +\begin{equation} +\frac{R_n((N-x)(N-x+\ga+\de+1);\ga,\de,N)}{R_n(N(N+\ga+\de+1);\ga,\de,N)} +=R_n(x(x-2N-\ga-\de-1);-N-\de-1,-N-\ga-1,N). +\label{97} +\end{equation} +Alternatively, \eqref{97} follows from (9.6.1) and +\mycite{DLMF}{(16.4.11)}. + +It follows from \eqref{96} and \eqref{45} that +\begin{equation} +\frac{R_{N-n}(x(x+\ga+\de+1);\ga,\de,N)} +{R_N(x(x+\ga+\de+1);\ga,\de,N)} +=R_n(x(x+\ga+\de+1);\de,\ga,N)\qquad(x=0,1,\ldots,N). +\label{102} +\end{equation} +% +\paragraph{Re: (9.6.11).} +The generating function (9.6.11) can be written in a more conceptual way as +\begin{equation} +(1-t)^x\,\hyp21{x-N,x+\ga+1}{-\de-N}t=\frac{N!}{(\de+1)_N}\, +\sum_{n=0}^N \om_n\,R_n(\la(x);\ga,\de,N)\,t^n, +\label{2} +\end{equation} +where +\begin{equation} +\om_n:=\binom{\ga+n}n \binom{\de+N-n}{N-n}, +\label{3} +\end{equation} +i.e., the denominator on the \RHS\ of (9.6.2). +By the duality between Hahn polynomials and dual Hahn polynomials (see \eqref{45}) the above generating function can be rewritten in +terms of Hahn polynomials: +\begin{equation} +(1-t)^n\,\hyp21{n-N,n+\al+1}{-\be-N}t=\frac{N!}{(\be+1)_N}\, +\sum_{x=0}^N w_x\,Q_n(x;\al,\be,N)\,t^x, +\label{4} +\end{equation} +where +\begin{equation} +w_x:=\binom{\al+x}x \binom{\be+N-x}{N-x}, +\label{5} +\end{equation} +i.e., the weight occurring in the orthogonality relation (9.5.2) +for Hahn polynomials. +\paragraph{Re: (9.6.15).} +There should be a closing bracket before the equality sign. +% +\cite{Askey2005}, \cite{AskeyWilson85}, \cite{AtakRahmanSuslov}, \cite{AtakSuslov88}, +\cite{Ismail2005II}, \cite{Karlin61}, \cite{Koorn81}, \cite{Koorn88}, \cite{Lesky93}, +\cite{Lesky94I}, \cite{Lesky95I}, \cite{Lesky95II}, \cite{Nikiforov+}, \cite{NikiforovUvarov}, +\cite{Rahman81II}, \cite{Stanton84}, \cite{Wilson80}. + + +\section{Meixner-Pollaczek}\index{Meixner-Pollaczek polynomials} + +\par\setcounter{equation}{0} + +\subsection*{Hypergeometric representation} +\begin{equation} +\label{DefMP} +P_n^{(\lambda)}(x;\phi)=\frac{(2\lambda)_n}{n!}\expe^{in\phi}\, +\hyp{2}{1}{-n,\lambda+ix}{2\lambda}{1-\expe^{-2i\phi}}. +\end{equation} + +\subsection*{Orthogonality relation} +\begin{equation} +\label{OrtMP} +\frac{1}{2\pi}\int_{-\infty}^{\infty} +\e^{(2\phi-\pi)x}\left|\Gamma(\lambda+ix)\right|^2 +P_m^{(\lambda)}(x;\phi)P_n^{(\lambda)}(x;\phi)\,dx +{}=\frac{\Gamma(n+2\lambda)}{(2\sin\phi)^{2\lambda}n!}\,\delta_{mn}, +% \constraint{ +% $\lambda > 0$ & +% $0 < \phi < \pi$ } +\end{equation} + +\subsection*{Recurrence relation} +\begin{eqnarray} +\label{RecMP} +& &(n+1)P_{n+1}^{(\lambda)}(x;\phi)-2\left[x\sin\phi+(n+\lambda)\cos\phi\right] +P_n^{(\lambda)}(x;\phi)\nonumber\\ +& &{}\mathindent{}+(n+2\lambda-1)P_{n-1}^{(\lambda)}(x;\phi)=0. +\end{eqnarray} + +\subsection*{Normalized recurrence relation} +\begin{equation} +\label{NormRecMP} +xp_n(x)=p_{n+1}(x)-\left(\frac{n+\lambda}{\tan\phi}\right)p_n(x)+ +\frac{n(n+2\lambda-1)}{4\sin^2\phi}p_{n-1}(x), +\end{equation} +where +$$P_n^{(\lambda)}(x;\phi)=\frac{(2\sin\phi)^n}{n!}p_n(x).$$ + +\subsection*{Difference equation} +\begin{eqnarray} +\label{dvMP} +& &\e^{i\phi}(\lambda-ix)y(x+i)+2i\left[x\cos\phi-(n+\lambda)\sin\phi\right]y(x)\nonumber\\ +& &{}\mathindent{}-\e^{-i\phi}(\lambda+ix)y(x-i)=0,\quad y(x)=P_n^{(\lambda)}(x;\phi). +\end{eqnarray} + +\subsection*{Forward shift operator} +\begin{equation} +\label{shift1MPI} +P_n^{(\lambda)}(x+\textstyle\frac{1}{2}i;\phi)- +P_n^{(\lambda)}(x-\textstyle\frac{1}{2}i;\phi)=(\expe^{i\phi}-\expe^{-i\phi}) +P_{n-1}^{(\lambda+\frac{1}{2})}(x;\phi) +\end{equation} +or equivalently +\begin{equation} +\label{shift1MPII} +\frac{\delta P_n^{(\lambda)}(x;\phi)}{\delta x} +=2\sin\phi\,P_{n-1}^{(\lambda+\frac{1}{2})}(x;\phi). +\end{equation} + +\subsection*{Backward shift operator} +\begin{eqnarray} +\label{shift2MPI} +& &\e^{i\phi}(\lambda-\textstyle\frac{1}{2}-ix) +P_n^{(\lambda)}(x+\textstyle\frac{1}{2}i;\phi)+ +\e^{-i\phi}(\lambda-\textstyle\frac{1}{2}+ix) +P_n^{(\lambda)}(x-\textstyle\frac{1}{2}i;\phi)\nonumber\\ +& &{}=(n+1)P_{n+1}^{(\lambda-\frac{1}{2})}(x;\phi) +\end{eqnarray} +or equivalently +\begin{equation} +\label{shift2MPII} +\frac{\delta\left[\omega(x;\lambda,\phi)P_n^{(\lambda)}(x;\phi)\right]}{\delta x}= +-(n+1)\omega(x;\lambda-\textstyle\frac{1}{2},\phi) +P_{n+1}^{(\lambda-\frac{1}{2})}(x;\phi), +\end{equation} +where +$$\omega(x;\lambda,\phi)=\Gamma(\lambda+ix)\Gamma(\lambda-ix)\e^{(2\phi-\pi)x}.$$ + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodMP} +\omega(x;\lambda,\phi)P_n^{(\lambda)}(x;\phi)=\frac{(-1)^n}{n!} +\left(\frac{\delta}{\delta x}\right)^n\left[\omega(x;\lambda+\textstyle\frac{1}{2}n,\phi)\right]. +\end{equation} + +\newpage + +\subsection*{Generating functions} +\begin{equation} +\label{GenMP1} +(1-\expe^{i\phi}t)^{-\lambda+ix}(1-\expe^{-i\phi}t)^{-\lambda-ix}= +\sum_{n=0}^{\infty}P_n^{(\lambda)}(x;\phi)t^n. +\end{equation} + +\begin{equation} +\label{GenMP2} +\e^t\,\hyp{1}{1}{\lambda+ix}{2\lambda}{(\expe^{-2i\phi}-1)t}= +\sum_{n=0}^{\infty}\frac{P_n^{(\lambda)}(x;\phi)}{(2\lambda)_n\expe^{in\phi}}t^n. +\end{equation} + +\begin{eqnarray} +\label{GenMP3} +& &(1-t)^{-\gamma}\,\hyp{2}{1}{\gamma,\lambda+ix}{2\lambda}{\frac{(1-\e^{-2i\phi})t}{t-1}}\nonumber\\ +& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n}{(2\lambda)_n}\frac{P_n^{(\lambda)}(x;\phi)}{\e^{in\phi}}t^n, +\quad\textrm{$\gamma$ arbitrary}. +\end{eqnarray} + +\subsection*{Limit relations} + +\subsubsection*{Continuous dual Hahn $\rightarrow$ Meixner-Pollaczek} +The Meixner-Pollaczek polynomials can be obtained from the continuous dual Hahn polynomials +given by (\ref{DefContDualHahn}) by the substitutions $x\rightarrow x-t$, $a=\lambda+it$, +$b=\lambda-it$ and $c=t\cot\phi$ and the limit $t\rightarrow\infty$: +$$\lim_{t\rightarrow\infty}\frac{S_n((x-t)^2;\lambda+it,\lambda-it,t\cot\phi)}{t^nn!} +=\frac{P_n^{(\lambda)}(x;\phi)}{(\sin\phi)^n}.$$ + +\subsubsection*{Continuous Hahn $\rightarrow$ Meixner-Pollaczek} +By setting $x\rightarrow x+t$, $a=\lambda-it$, $c=\lambda+it$ and $b=d=t\tan\phi$ in the +definition (\ref{DefContHahn}) of the continuous Hahn polynomials and taking the limit +$t\rightarrow\infty$ we obtain the Meixner-Pollaczek polynomials: +$$\lim_{t\rightarrow\infty}\frac{p_n(x+t;\lambda-it,t\tan\phi,\lambda+it,t\tan\phi)}{t^nn!} +=\frac{P_n^{(\lambda)}(x;\phi)}{(\cos\phi)^n}.$$ + +\newpage + +\subsubsection*{Meixner-Pollaczek $\rightarrow$ Laguerre} +The Laguerre polynomials given by (\ref{DefLaguerre}) can be obtained from the +Meixner-Pollaczek polynomials by the substitution $\lambda=\frac{1}{2}(\alpha+1)$, +$x\rightarrow -\frac{1}{2}\phi^{-1}x$ and the limit $\phi\rightarrow 0$: +\begin{equation} +\lim_{\phi\rightarrow 0} +P_n^{(\frac{1}{2}\alpha+\frac{1}{2})}(-\textstyle\frac{1}{2}\phi^{-1}x;\phi)=L_n^{(\alpha)}(x). +\end{equation} + +\subsubsection*{Meixner-Pollaczek $\rightarrow$ Hermite} +The Hermite polynomials given by (\ref{DefHermite}) are obtained from the Meixner-Pollaczek +polynomials if we substitute $x\rightarrow (\sin\phi)^{-1}(x\sqrt{\lambda}-\lambda\cos\phi)$ +and then let $\lambda\rightarrow\infty$: +\begin{equation} +\lim_{\lambda\rightarrow\infty} +\lambda^{-\frac{1}{2}n}P_n^{(\lambda)} +((\sin\phi)^{-1}(x\sqrt{\lambda}-\lambda\cos\phi);\phi)=\frac{H_n(x)}{n!}. +\end{equation} + +\subsection*{Remark} +Since we have for $k-1$ and $\beta>-1$ we have +\begin{eqnarray} +\label{OrtJacobi1} +& &\int_{-1}^1(1-x)^{\alpha}(1+x)^{\beta}P_m^{(\alpha,\beta)}(x)P_n^{(\alpha,\beta)}(x)\,dx\nonumber\\ +& &{}=\frac{2^{\alpha+\beta+1}}{2n+\alpha+\beta+1}\frac{\Gamma(n+\alpha+1)\Gamma(n+\beta+1)}{\Gamma(n+\alpha+\beta+1)n!}\,\delta_{mn}. +\end{eqnarray} +For $\alpha+\beta<-2N-1$, $\beta>-1$ and $m,n\in\{0,1,2,\ldots,N\}$ we also have +\begin{eqnarray} +\label{OrtJacobi2} +& &\int_1^{\infty}(x+1)^{\alpha}(x-1)^{\beta}P_m^{(\alpha,\beta)}(-x)P_n^{(\alpha,\beta)}(-x)\,dx\nonumber\\ +& &{}=-\frac{2^{\alpha+\beta+1}}{2n+\alpha+\beta+1}\frac{\Gamma(-n-\alpha-\beta)\Gamma(n+\alpha+\beta+1)}{\Gamma(-n-\alpha)n!}\,\delta_{mn}. +\end{eqnarray} + +\subsection*{Recurrence relation} +\begin{eqnarray} +\label{RecJacobi} +xP_n^{(\alpha,\beta)}(x)&=&\frac{2(n+1)(n+\alpha+\beta+1)}{(2n+\alpha+\beta+1)(2n+\alpha+\beta+2)}P_{n+1}^{(\alpha,\beta)}(x)\nonumber\\ +& &{}\mathindent{}+\frac{\beta^2-\alpha^2}{(2n+\alpha+\beta)(2n+\alpha+\beta+2)}P_n^{(\alpha,\beta)}(x)\nonumber\\ +& &{}\mathindent\mathindent{}+\frac{2(n+\alpha)(n+\beta)}{(2n+\alpha+\beta)(2n+\alpha+\beta+1)}P_{n-1}^{(\alpha,\beta)}(x). +\end{eqnarray} + +\subsection*{Normalized recurrence relation} +\begin{eqnarray} +\label{NormRecJacobi} +xp_n(x)&=&p_{n+1}(x)+\frac{\beta^2-\alpha^2}{(2n+\alpha+\beta)(2n+\alpha+\beta+2)}p_n(x)\nonumber\\ +& &{}\mathindent{}+\frac{4n(n+\alpha)(n+\beta)(n+\alpha+\beta)} +{(2n+\alpha+\beta-1)(2n+\alpha+\beta)^2(2n+\alpha+\beta+1)}p_{n-1}(x) +\end{eqnarray} +where +$$P_n^{(\alpha,\beta)}(x)=\frac{(n+\alpha+\beta+1)_n}{2^nn!}p_n(x).$$ + +\subsection*{Differential equation} +\begin{eqnarray} +\label{dvJacobi} +& &(1-x^2)y''(x)+\left[\beta-\alpha-(\alpha+\beta+2)x\right]y'(x)\nonumber\\ +& &{}\mathindent{}+n(n+\alpha+\beta+1)y(x)=0,\quad y(x)=P_n^{(\alpha,\beta)}(x). +\end{eqnarray} + +\subsection*{Forward shift operator} +\begin{equation} +\label{shift1Jacobi} +\frac{d}{dx}P_n^{(\alpha,\beta)}(x)=\frac{n+\alpha+\beta+1}{2}P_{n-1}^{(\alpha+1,\beta+1)}(x). +\end{equation} + +\subsection*{Backward shift operator} +\begin{eqnarray} +\label{shift2JacobiI} +& &(1-x^2)\frac{d}{dx}P_n^{(\alpha,\beta)}(x)+ +\left[(\beta-\alpha)-(\alpha+\beta)x\right]P_n^{(\alpha,\beta)}(x)\nonumber\\ +& &{}=-2(n+1)P_{n+1}^{(\alpha-1,\beta-1)}(x) +\end{eqnarray} +or equivalently +\begin{eqnarray} +\label{shift2JacobiII} +& &\frac{d}{dx}\left[(1-x)^\alpha(1+x)^\beta P_n^{(\alpha,\beta)}(x)\right]\nonumber\\ +& &{}=-2(n+1)(1-x)^{\alpha-1}(1+x)^{\beta-1}P_{n+1}^{(\alpha-1,\beta-1)}(x). +\end{eqnarray} + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodJacobi} +(1-x)^{\alpha}(1+x)^{\beta}P_n^{(\alpha,\beta)}(x)= +\frac{(-1)^n}{2^nn!}\left(\frac{d}{dx}\right)^n +\left[(1-x)^{n+\alpha}(1+x)^{n+\beta}\right]. +\end{equation} + +\subsection*{Generating functions} +\begin{equation} +\label{GenJacobi1} +\frac{2^{\alpha+\beta}}{R(1+R-t)^{\alpha}(1+R+t)^{\beta}}= +\sum_{n=0}^{\infty}P_n^{(\alpha,\beta)}(x)t^n,\quad R=\sqrt{1-2xt+t^2}. +\end{equation} + +\begin{eqnarray} +\label{GenJacobi2} +& &\hyp{0}{1}{-}{\alpha+1}{\frac{(x-1)t}{2}}\,\hyp{0}{1}{-}{\beta+1}{\frac{(x+1)t}{2}}\nonumber\\ +& &{}=\sum_{n=0}^{\infty}\frac{P_n^{(\alpha,\beta)}(x)}{(\alpha+1)_n(\beta+1)_n}t^n. +\end{eqnarray} + +\begin{eqnarray} +\label{GenJacobi3} +& &(1-t)^{-\alpha-\beta-1}\,\hyp{2}{1}{\frac{1}{2}(\alpha+\beta+1),\frac{1}{2}(\alpha+\beta+2)} +{\alpha+1}{\frac{2(x-1)t}{(1-t)^2}}\nonumber\\ +& &{}=\sum_{n=0}^{\infty}\frac{(\alpha+\beta+1)_n}{(\alpha+1)_n}P_n^{(\alpha,\beta)}(x)t^n. +\end{eqnarray} + +\begin{eqnarray} +\label{GenJacobi4} +& &(1+t)^{-\alpha-\beta-1}\,\hyp{2}{1}{\frac{1}{2}(\alpha+\beta+1),\frac{1}{2}(\alpha+\beta+2)} +{\beta+1}{\frac{2(x+1)t}{(1+t)^2}}\nonumber\\ +& &{}=\sum_{n=0}^{\infty}\frac{(\alpha+\beta+1)_n}{(\beta+1)_n}P_n^{(\alpha,\beta)}(x)t^n. +\end{eqnarray} + +\begin{eqnarray} +\label{GenJacobi5} +& &\hyp{2}{1}{\gamma,\alpha+\beta+1-\gamma}{\alpha+1}{\frac{1-R-t}{2}}\, +\hyp{2}{1}{\gamma,\alpha+\beta+1-\gamma}{\beta+1}{\frac{1-R+t}{2}}\nonumber\\ +& &{}=\sum_{n=0}^{\infty} +\frac{(\gamma)_n(\alpha+\beta+1-\gamma)_n}{(\alpha+1)_n(\beta+1)_n}P_n^{(\alpha,\beta)}(x)t^n, +\quad R=\sqrt{1-2xt+t^2} +\end{eqnarray} +with $\gamma$ arbitrary. + +\subsection*{Limit relations} + +\subsubsection*{Wilson $\rightarrow$ Jacobi} +The Jacobi polynomials can be found from the Wilson polynomials given by (\ref{DefWilson}) by +substituting $a=b=\frac{1}{2}(\alpha+1)$, $c=\frac{1}{2}(\beta+1)+it$, +$d=\frac{1}{2}(\beta+1)-it$ and $x\rightarrow t\sqrt{\frac{1}{2}(1-x)}$ in the +definition (\ref{DefWilson}) of the Wilson polynomials and taking the limit +$t\rightarrow\infty$. In fact we have +$$\lim_{t\rightarrow\infty} +\frac{W_n(\frac{1}{2}(1-x)t^2;\frac{1}{2}(\alpha+1), +\frac{1}{2}(\alpha+1),\frac{1}{2}(\beta+1)+it,\frac{1}{2}(\beta+1)-it)} +{t^{2n}n!}=P_n^{(\alpha,\beta)}(x).$$ + +\subsubsection*{Continuous Hahn $\rightarrow$ Jacobi} +The Jacobi polynomials follow from the continuous Hahn polynomials given by (\ref{DefContHahn}) +by using the substitution $x\rightarrow \frac{1}{2}xt$, $a=\frac{1}{2}(\alpha+1-it)$, +$b=\frac{1}{2}(\beta+1+it)$, $c=\frac{1}{2}(\alpha+1+it)$ and $d=\frac{1}{2}(\beta+1-it)$ +in (\ref{DefContHahn}), division by $t^n$ and the limit $t\rightarrow\infty$: +$$\lim_{t\rightarrow\infty} +\frac{p_n(\frac{1}{2}xt;\frac{1}{2}(\alpha+1-it),\frac{1}{2}(\beta+1+it), +\frac{1}{2}(\alpha+1+it),\frac{1}{2}(\beta+1-it))}{t^n}=P_n^{(\alpha,\beta)}(x).$$ + +\subsubsection*{Hahn $\rightarrow$ Jacobi} +To find the Jacobi polynomials from the Hahn polynomials given by (\ref{DefHahn}) we take +$x\rightarrow Nx$ in (\ref{DefHahn}) and let $N\rightarrow\infty$. In fact we have +$$\lim_{N\rightarrow\infty} +Q_n(Nx;\alpha,\beta,N)=\frac{P_n^{(\alpha,\beta)}(1-2x)}{P_n^{(\alpha,\beta)}(1)}.$$ + +\subsubsection*{Jacobi $\rightarrow$ Laguerre} +The Laguerre polynomials given by (\ref{DefLaguerre}) can be obtained from the Jacobi +polynomials by setting $x\rightarrow 1-2\beta^{-1}x$ and then the limit $\beta\rightarrow\infty$: +\begin{equation} +\lim_{\beta\rightarrow\infty} +P_n^{(\alpha,\beta)}(1-2\beta^{-1}x)=L_n^{(\alpha)}(x). +\end{equation} + +\subsubsection*{Jacobi $\rightarrow$ Bessel} +The Bessel polynomials given by (\ref{DefBessel}) are obtained from the Jacobi polynomials +if we take $\beta=a-\alpha$ and let $\alpha\rightarrow-\infty$: +\begin{equation} +\lim_{\alpha\rightarrow-\infty} +\frac{P_n^{(\alpha,a-\alpha)}(1+\alpha x)}{P_n^{(\alpha,a-\alpha)}(1)}=y_n(x;a). +\end{equation} + +\subsubsection*{Jacobi $\rightarrow$ Hermite} +The Hermite polynomials given by (\ref{DefHermite}) follow from the Jacobi polynomials +by taking $\beta=\alpha$ and letting $\alpha\rightarrow\infty$ in the following way: +\begin{equation} +\lim_{\alpha\rightarrow\infty} +\alpha^{-\frac{1}{2}n}P_n^{(\alpha,\alpha)}(\alpha^{-\frac{1}{2}}x)=\frac{H_n(x)}{2^nn!}. +\end{equation} + +\subsection*{Remarks} +The definition (\ref{DefJacobi}) of the Jacobi polynomials can also be written as: +$$P_n^{(\alpha,\beta)}(x)=\frac{1}{n!}\sum_{k=0}^n\frac{(-n)_k}{k!} +(n+\alpha+\beta+1)_k(\alpha+k+1)_{n-k}\left(\frac{1-x}{2}\right)^k.$$ +In this way the Jacobi polynomials can also be seen as polynomials in the parameters $\alpha$ +and $\beta$. Therefore they can be defined for all $\alpha$ and $\beta$. Then we have the +following connection with the Meixner polynomials given by (\ref{DefMeixner}): +$$\frac{(\beta)_n}{n!}M_n(x;\beta,c)=P_n^{(\beta-1,-n-\beta-x)}((2-c)c^{-1}).$$ + +\noindent +The Jacobi polynomials are related to the pseudo Jacobi polynomials defined by +(\ref{DefPseudoJacobi}) in the following way: +$$P_n(x;\nu,N)=\frac{(-2i)^nn!}{(n-2N-1)_n}P_n^{(-N-1+i\nu,-N-1-i\nu)}(ix).$$ + +\noindent +The Jacobi polynomials are also related to the Gegenbauer (or ultraspherical) polynomials +given by (\ref{DefGegenbauer}) by the quadratic transformations: +$$C_{2n}^{(\lambda)}(x)=\frac{(\lambda)_n}{(\frac{1}{2})_n} +P_n^{(\lambda-\frac{1}{2},-\frac{1}{2})}(2x^2-1)$$ +and +$$C_{2n+1}^{(\lambda)}(x)=\frac{(\lambda)_{n+1}}{(\frac{1}{2})_{n+1}} +xP_n^{(\lambda-\frac{1}{2},\frac{1}{2})}(2x^2-1).$$ + +\subsection*{References} +\label{sec9.8} +% +\paragraph{Orthogonality relation} +Write the \RHS\ of (9.8.2) as $h_n\,\de_{m,n}$. Then +\begin{equation} +\begin{split} +&\frac{h_n}{h_0}= +\frac{n+\al+\be+1}{2n+\al+\be+1}\, +\frac{(\al+1)_n(\be+1)_n}{(\al+\be+2)_n\,n!}\,,\quad +h_0=\frac{2^{\al+\be+1}\Ga(\al+1)\Ga(\be+1)}{\Ga(\al+\be+2)}\,,\sLP +&\frac{h_n}{h_0\,(P_n^{(\al,\be)}(1))^2}= +\frac{n+\al+\be+1}{2n+\al+\be+1}\, +\frac{(\be+1)_n\,n!}{(\al+1)_n\,(\al+\be+2)_n}\,. +\end{split} +\label{60} +\end{equation} + +In (9.8.3) the numerator factor $\Ga(n+\al+\be+1)$ in the last line should be +$\Ga(\be+1)$. When thus corrected, (9.8.3) can be rewritten as: +\begin{equation} +\begin{split} +&\int_1^\iy P_m^{(\al,\be)}(x)\,P_n^{(\al,\be)}(x)\,(x-1)^\al (x+1)^\be\,dx=h_n\,\de_{m,n}\,,\\ +&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad-1-\be>\al>-1,\quad m,n<-\thalf(\al+\be+1),\\ +&\frac{h_n}{h_0}= +\frac{n+\al+\be+1}{2n+\al+\be+1}\, +\frac{(\al+1)_n(\be+1)_n}{(\al+\be+2)_n\,n!}\,,\quad +h_0=\frac{2^{\al+\be+1}\Ga(\al+1)\Ga(-\al-\be-1)}{\Ga(-\be)}\,. +\end{split} +\label{122} +\end{equation} + +% +\paragraph{Symmetry} +\begin{equation} +P_n^{(\al,\be)}(-x)=(-1)^n\,P_n^{(\be,\al)}(x). +\label{48} +\end{equation} +Use (9.8.2) and (9.8.5b) or see \mycite{DLMF}{Table 18.6.1}. +% +\paragraph{Special values} +\begin{equation} +P_n^{(\al,\be)}(1)=\frac{(\al+1)_n}{n!}\,,\quad +P_n^{(\al,\be)}(-1)=\frac{(-1)^n(\be+1)_n}{n!}\,,\quad +\frac{P_n^{(\al,\be)}(-1)}{P_n^{(\al,\be)}(1)}=\frac{(-1)^n(\be+1)_n}{(\al+1)_n}\,. +\label{50} +\end{equation} +Use (9.8.1) and \eqref{48} or see \mycite{DLMF}{Table 18.6.1}. +% +\paragraph{Generating functions} +Formula (9.8.15) was first obtained by Brafman \myciteKLS{109}. +% +\paragraph{Bilateral generating functions} +For $0\le r<1$ and $x,y\in[-1,1]$ we have in terms of $F_4$ (see~\eqref{62}): +\begin{align} +&\sum_{n=0}^\iy\frac{(\al+\be+1)_n\,n!}{(\al+1)_n(\be+1)_n}\,r^n\, +P_n^{(\al,\be)}(x)\,P_n^{(\al,\be)}(y) +=\frac1{(1+r)^{\al+\be+1}} +\nonumber\\ +&\qquad\quad\times F_4\Big(\thalf(\al+\be+1),\thalf(\al+\be+2);\al+1,\be+1; +\frac{r(1-x)(1-y)}{(1+r)^2},\frac{r(1+x)(1+y)}{(1+r)^2}\Big), +\label{58}\sLP +&\sum_{n=0}^\iy\frac{2n+\al+\be+1}{n+\al+\be+1} +\frac{(\al+\be+2)_n\,n!}{(\al+1)_n(\be+1)_n}\,r^n\, +P_n^{(\al,\be)}(x)\,P_n^{(\al,\be)}(y) +=\frac{1-r}{(1+r)^{\al+\be+2}}\nonumber\\ +&\qquad\quad\times F_4\Big(\thalf(\al+\be+2),\thalf(\al+\be+3);\al+1,\be+1; +\frac{r(1-x)(1-y)}{(1+r)^2},\frac{r(1+x)(1+y)}{(1+r)^2}\Big). +\label{59} +\end{align} +Formulas \eqref{58} and \eqref{59} were first +given by Bailey \myciteKLS{91}{(2.1), (2.3)}. +See Stanton \myciteKLS{485} for a shorter proof. +(However, in the second line of +\myciteKLS{485}{(1)} $z$ and $Z$ should be interchanged.)$\;$ +As observed in Bailey \myciteKLS{91}{p.10}, \eqref{59} follows +from \eqref{58} +by applying the operator $r^{-\half(\al+\be-1)}\,\frac d{dr}\circ r^{\half(\al+\be+1)}$ +to both sides of \eqref{58}. +In view of \eqref{60}, formula \eqref{59} is the Poisson kernel for Jacobi +polynomials. The \RHS\ of \eqref{59} makes clear that this kernel is positive. +See also the discussion in Askey \myciteKLS{46}{following (2.32)}. +% +\paragraph{Quadratic transformations} +\begin{align} +\frac{C_{2n}^{(\al+\half)}(x)}{C_{2n}^{(\al+\half)}(1)} +=\frac{P_{2n}^{(\al,\al)}(x)}{P_{2n}^{(\al,\al)}(1)} +&=\frac{P_n^{(\al,-\half)}(2x^2-1)}{P_n^{(\al,-\half)}(1)}\,, +\label{51}\\ +\frac{C_{2n+1}^{(\al+\half)}(x)}{C_{2n+1}^{(\al+\half)}(1)} +=\frac{P_{2n+1}^{(\al,\al)}(x)}{P_{2n+1}^{(\al,\al)}(1)} +&=\frac{x\,P_n^{(\al,\half)}(2x^2-1)}{P_n^{(\al,\half)}(1)}\,. +\label{52} +\end{align} +See p.221, Remarks, last two formulas together with \eqref{50} and \eqref{49}. +Or see \mycite{DLMF}{(18.7.13), (18.7.14)}. +% +\paragraph{Differentiation formulas} +Each differentiation formula is given in two equivalent forms. +\begin{equation} +\begin{split} +\frac d{dx}\left((1-x)^\al P_n^{(\al,\be)}(x)\right)&= +-(n+\al)\,(1-x)^{\al-1} P_n^{(\al-1,\be+1)}(x),\\ +\left((1-x)\frac d{dx}-\al\right)P_n^{(\al,\be)}(x)&= +-(n+\al)\,P_n^{(\al-1,\be+1)}(x). +\end{split} +\label{68} +\end{equation} +% +\begin{equation} +\begin{split} +\frac d{dx}\left((1+x)^\be P_n^{(\al,\be)}(x)\right)&= +(n+\be)\,(1+x)^{\be-1} P_n^{(\al+1,\be-1)}(x),\\ +\left((1+x)\frac d{dx}+\be\right)P_n^{(\al,\be)}(x)&= +(n+\be)\,P_n^{(\al+1,\be-1)}(x). +\end{split} +\label{69} +\end{equation} +Formulas \eqref{68} and \eqref{69} follow from +\mycite{DLMF}{(15.5.4), (15.5.6)} +together with (9.8.1). They also follow from each other by \eqref{48}. +% +\paragraph{Generalized Gegenbauer polynomials} +These are defined by +\begin{equation} +S_{2m}^{(\al,\be)}(x):=\const P_m^{(\al,\be)}(2x^2-1),\qquad +S_{2m+1}^{(\al,\be)}(x):=\const x\,P_m^{(\al,\be+1)}(2x^2-1) +\label{70} +\end{equation} +in the notation of \myciteKLS{146}{p.156} +(see also \cite{K27}), while \cite[Section 1.5.2]{K26} +has $C_n^{(\la,\mu)}(x)=\const\allowbreak\times S_n^{(\la-\half,\mu-\half)}(x)$. +For $\al,\be>-1$ we have the orthogonality relation +\begin{equation} +\int_{-1}^1 S_m^{(\al,\be)}(x)\,S_n^{(\al,\be)}(x)\,|x|^{2\be+1}(1-x^2)^\al\,dx +=0\qquad(m\ne n). +\label{71} +\end{equation} +For $\be=\al-1$ generalized Gegenbauer polynomials are limit cases of +continuous $q$-ultraspherical polynomials, see \eqref{176}. + +If we define the {\em Dunkl operator} $T_\mu$ by +\begin{equation} +(T_\mu f)(x):=f'(x)+\mu\,\frac{f(x)-f(-x)}x +\label{72} +\end{equation} +and if we choose the constants in \eqref{70} as +\begin{equation} +S_{2m}^{(\al,\be)}(x)=\frac{(\al+\be+1)_m}{(\be+1)_m}\, P_m^{(\al,\be)}(2x^2-1),\quad +S_{2m+1}^{(\al,\be)}(x)=\frac{(\al+\be+1)_{m+1}}{(\be+1)_{m+1}}\, +x\,P_m^{(\al,\be+1)}(2x^2-1) +\label{73} +\end{equation} +then (see \cite[(1.6)]{K5}) +\begin{equation} +T_{\be+\half}S_n^{(\al,\be)}=2(\al+\be+1)\,S_{n-1}^{(\al+1,\be)}. +\label{74} +\end{equation} +Formula \eqref{74} with \eqref{73} substituted gives rise to two +differentiation formulas involving Jacobi polynomials which are equivalent to +(9.8.7) and \eqref{69}. + +Composition of \eqref{74} with itself gives +\[ +T_{\be+\half}^2S_n^{(\al,\be)}=4(\al+\be+1)(\al+\be+2)\,S_{n-2}^{(\al+2,\be)}, +\] +which is equivalent to the composition of (9.8.7) and \eqref{69}: +\begin{equation} +\left(\frac{d^2}{dx^2}+\frac{2\be+1}x\,\frac d{dx}\right)P_n^{(\al,\be)}(2x^2-1) +=4(n+\al+\be+1)(n+\be)\,P_{n-1}^{(\al+2,\be)}(2x^2-1). +\label{75} +\end{equation} +Formula \eqref{75} was also given in \myciteKLS{322}{(2.4)}. +% +\cite{Abram}, \cite{Ahmed+82}, \cite{Allaway89}, \cite{NAlSalam66}, \cite{AlSalam64}, +\cite{AlSalamChihara72}, \cite{AndrewsAskey85}, \cite{AndrewsAskeyRoy}, \cite{Askey68}, +\cite{Askey70}, \cite{Askey72}, \cite{Askey73}, \cite{Askey74}, \cite{Askey75}, \cite{Askey78}, +\cite{Askey89I}, \cite{Askey2005}, \cite{AskeyFitch}, \cite{AskeyGasper71I}, \cite{AskeyGasper71II}, +\cite{AskeyGasper76}, \cite{AskeyWainger}, \cite{AskeyWilson85}, \cite{Bailey38}, \cite{Brafman51}, +\cite{Brown}, \cite{Carlitz61II}, \cite{Carlitz67}, \cite{ChenIsmail91}, \cite{Chihara78}, +\cite{Chow+}, \cite{Ciesielski}, \cite{Cooper+}, \cite{DetteStudden92}, \cite{DetteStudden95}, +\cite{DijksmaKoorn}, \cite{DimitrovRafaeli}, \cite{DimitrovRodrigues}, \cite{Doha2002I}, +\cite{Doha2003I}, \cite{Doha2004II}, \cite{Dunkl84}, \cite{ElbertLaforgia87II}, +\cite{ElbertLaforgiaRodono}, \cite{Erdelyi+}, \cite{Faldey}, \cite{FoataLeroux}, +\cite{Gasper69}, \cite{Gasper70I}, \cite{Gasper70II}, \cite{Gasper71I}, \cite{Gasper71II}, +\cite{Gasper72I}, \cite{Gasper72II}, \cite{Gasper73I}, \cite{Gasper73II}, \cite{Gasper74}, +\cite{Gasper77}, \cite{Gatteschi87}, \cite{Gautschi2008}, \cite{Gautschi2009I}, +\cite{Gautschi2009II}, \cite{GautschiLeopardi}, \cite{GawronskiShawyer}, \cite{Godoy+}, +\cite{Grad}, \cite{Hajmirzaahmad94}, \cite{HartmannStephan}, \cite{Horton}, \cite{Ismail74}, +\cite{Ismail77}, \cite{Ismail96}, \cite{Ismail2005II}, \cite{IsmailLi}, +\cite{IsmailMassonRahman}, \cite{Kochneff97I}, \cite{Koekoek99}, \cite{Koekoek2000}, +\cite{Koelink96II}, \cite{Koorn73}, \cite{Koorn74}, \cite{Koorn75}, \cite{Koorn77II}, +\cite{Koorn78}, \cite{Koorn72}, \cite{Koorn85}, \cite{Koorn88}, \cite{KuijlaarsMartinez}, +\cite{Kuijlaars+}, \cite{LabelleYehI}, \cite{LabelleYehII}, \cite{Laine}, \cite{Lesky95II}, +\cite{Lesky96}, \cite{Li96}, \cite{Li97}, \cite{LopezTemme2004}, \cite{Luke}, \cite{Martinez}, +\cite{Mathai}, \cite{Meijer}, \cite{Miller89}, \cite{MoakSaffVarga}, \cite{Nikiforov+}, +\cite{NikiforovUvarov}, \cite{Olver}, \cite{Prasad}, \cite{Rahman76I}, \cite{Rahman76II}, +\cite{Rahman77}, \cite{Rahman81I}, \cite{RahmanShah}, \cite{Rainville}, \cite{Rusev}, +\cite{Shi}, \cite{Srivastava69II}, \cite{Srivastava71}, \cite{Srivastava82}, +\cite{SrivastavaSinghal}, \cite{Stanton80I}, \cite{Stanton90}, \cite{Szego75}, \cite{Temme}, +\cite{Vertesi}, \cite{Wimp87}, \cite{WongZhang94}, \cite{WongZhang2006}, \cite{Zarzo+}, +\cite{Zayed}. + +\section*{Special cases} + +\subsection{Gegenbauer / Ultraspherical}\index{Gegenbauer polynomials} +\index{Ultraspherical polynomials} + +\par + +\subsection*{Hypergeometric representation} +The Gegenbauer (or ultraspherical) polynomials are Jacobi polynomials with +$\alpha=\beta=\lambda-\frac{1}{2}$ and another normalization: +\begin{eqnarray} +\label{DefGegenbauer} +C_n^{(\lambda)}(x)&=&\frac{(2\lambda)_n}{(\lambda+\frac{1}{2})_n} +P_n^{(\lambda-\frac{1}{2},\lambda-\frac{1}{2})}(x)\nonumber\\ +&=&\frac{(2\lambda)_n}{n!}\,\hyp{2}{1}{-n,n+2\lambda} +{\lambda+\frac{1}{2}}{\frac{1-x}{2}},\quad\lambda\neq 0. +\end{eqnarray} + +\subsection*{Orthogonality relation} +\begin{eqnarray} +\label{OrtGegenbauer} +& &\int_{-1}^1(1-x^2)^{\lambda-\frac{1}{2}}C_m^{(\lambda)}(x)C_n^{(\lambda)}(x)\,dx\nonumber\\ +& &{}=\frac{\pi\Gamma(n+2\lambda)2^{1-2\lambda}}{\left\{\Gamma(\lambda)\right\}^2(n+\lambda)n!}\,\delta_{mn}, +\quad\lambda>-\frac{1}{2}\quad\lambda\neq 0. +\end{eqnarray} + +\subsection*{Recurrence relation} +\begin{equation} +\label{RecGegenbauer} +2(n+\lambda)xC_n^{(\lambda)}(x)=(n+1)C_{n+1}^{(\lambda)}(x)+(n+2\lambda-1)C_{n-1}^{(\lambda)}(x). +\end{equation} + +\subsection*{Normalized recurrence relation} +\begin{equation} +\label{NormRecGegenbauer} +xp_n(x)=p_{n+1}(x)+\frac{n(n+2\lambda-1)}{4(n+\lambda-1)(n+\lambda)}p_{n-1}(x), +\end{equation} +where +$$C_n^{(\lambda)}(x)=\frac{2^n(\lambda)_n}{n!}p_n(x).$$ + +\subsection*{Differential equation} +\begin{equation} +\label{dvGegenbauer} +(1-x^2)y''(x)-(2\lambda+1)xy'(x)+n(n+2\lambda)y(x)=0,\quad y(x)=C_n^{(\lambda)}(x). +\end{equation} + +\subsection*{Forward shift operator} +\begin{equation} +\label{shift1Gegenbauer} +\frac{d}{dx}C_n^{(\lambda)}(x)=2\lambda C_{n-1}^{(\lambda+1)}(x). +\end{equation} + +\subsection*{Backward shift operator} +\begin{equation} +\label{shift2GegenbauerI} +(1-x^2)\frac{d}{dx}C_n^{(\lambda)}(x)+(1-2\lambda)xC_n^{(\lambda)}(x)= +-\frac{(n+1)(2\lambda+n-1)}{2(\lambda-1)} C_{n+1}^{(\lambda-1)}(x) +\end{equation} +or equivalently +\begin{eqnarray} +\label{shift2GegenbauerII} +& &\frac{d}{dx}\left[(1-x^2)^{\lambda-\textstyle\frac{1}{2}}C_n^{(\lambda)}(x)\right]\nonumber\\ +& &{}=-\frac{(n+1)(2\lambda+n-1)}{2(\lambda-1)}(1-x^2)^{\lambda-\textstyle\frac{3}{2}}C_{n+1}^{(\lambda-1)}(x). +\end{eqnarray} + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodGegenbauer} +(1-x^2)^{\lambda-\frac{1}{2}}C_n^{(\lambda)}(x)= +\frac{(2\lambda)_n(-1)^n}{(\lambda+\frac{1}{2})_n2^nn!}\left(\frac{d}{dx}\right)^n +\left[(1-x^2)^{\lambda+n-\frac{1}{2}}\right]. +\end{equation} + +\subsection*{Generating functions} +\begin{equation} +\label{GenGegenbauer1} +(1-2xt+t^2)^{-\lambda}=\sum_{n=0}^{\infty}C_n^{(\lambda)}(x)t^n. +\end{equation} + +\begin{equation} +\label{GenGegenbauer2} +R^{-1}\left(\frac{1+R-xt}{2}\right)^{\frac{1}{2}-\lambda}=\sum_{n=0}^{\infty} +\frac{(\lambda+\frac{1}{2})_n}{(2\lambda)_n}C_n^{(\lambda)}(x)t^n, +\quad R=\sqrt{1-2xt+t^2}. +\end{equation} + +\begin{equation} +\label{GenGegenbauer3} +\hyp{0}{1}{-}{\lambda+\frac{1}{2}}{\frac{(x-1)t}{2}}\, +\hyp{0}{1}{-}{\lambda+\frac{1}{2}}{\frac{(x+1)t}{2}} +=\sum_{n=0}^{\infty}\frac{C_n^{(\lambda)}(x)} +{(2\lambda)_n(\lambda+\frac{1}{2})_n}t^n. +\end{equation} + +\begin{equation} +\label{GenGegenbauer4} +\e^{xt}\,\hyp{0}{1}{-}{\lambda+\frac{1}{2}}{\frac{(x^2-1)t^2}{4}}= +\sum_{n=0}^{\infty}\frac{C_n^{(\lambda)}(x)}{(2\lambda)_n}t^n. +\end{equation} + +\begin{eqnarray} +\label{GenGegenbauer5} +& &\hyp{2}{1}{\gamma,2\lambda-\gamma}{\lambda+\frac{1}{2}}{\frac{1-R-t}{2}}\, +\hyp{2}{1}{\gamma,2\lambda-\gamma}{\lambda+\frac{1}{2}}{\frac{1-R+t}{2}}\nonumber\\ +& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n(2\lambda-\gamma)_n} +{(2\lambda)_n(\lambda+\frac{1}{2})_n}C_n^{(\lambda)}(x)t^n, +\quad R=\sqrt{1-2xt+t^2},\quad\textrm{$\gamma$ arbitrary}. +\end{eqnarray} + +\begin{eqnarray} +\label{GenGegenbauer6} +& &(1-xt)^{-\gamma}\,\hyp{2}{1}{\frac{1}{2}\gamma,\frac{1}{2}\gamma+\frac{1}{2}} +{\lambda+\frac{1}{2}}{\frac{(x^2-1)t^2}{(1-xt)^2}}\nonumber\\ +& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n}{(2\lambda)_n}C_n^{(\lambda)}(x)t^n, +\quad\textrm{$\gamma$ arbitrary}. +\end{eqnarray} + +\subsection*{Limit relation} + +\subsubsection*{Gegenbauer / Ultraspherical $\rightarrow$ Hermite} +The Hermite polynomials given by (\ref{DefHermite}) follow from the Gegenbauer (or +ultraspherical) polynomials by taking $\lambda=\alpha+\frac{1}{2}$ and letting +$\alpha\rightarrow\infty$ in the following way: +\begin{equation} +\lim_{\alpha\rightarrow\infty} +\alpha^{-\frac{1}{2}n}C_n^{(\alpha+\frac{1}{2})}(\alpha^{-\frac{1}{2}}x)=\frac{H_n(x)}{n!}. +\end{equation} + +\subsection*{Remarks} +The case $\lambda=0$ needs another normalization. In that case we have the +Chebyshev polynomials of the first kind described in the next subsection. + +\noindent +The Gegenbauer (or ultraspherical) polynomials are related to the Jacobi polynomials +given by (\ref{DefJacobi}) by the quadratic transformations: +$$C_{2n}^{(\lambda)}(x)=\frac{(\lambda)_n}{(\frac{1}{2})_n} +P_n^{(\lambda-\frac{1}{2},-\frac{1}{2})}(2x^2-1)$$ +and +$$C_{2n+1}^{(\lambda)}(x)=\frac{(\lambda)_{n+1}}{(\frac{1}{2})_{n+1}} +xP_n^{(\lambda-\frac{1}{2},\frac{1}{2})}(2x^2-1).$$ + +\subsection*{References} +\label{sec9.8.1} +% +\paragraph{Notation} +Here the Gegenbauer polynomial is denoted by $C_n^\la$ instead of $C_n^{(\la)}$. +% +\paragraph{Orthogonality relation} +Write the \RHS\ of (9.8.20) as $h_n\,\de_{m,n}$. Then +\begin{equation} +\frac{h_n}{h_0}= +\frac\la{\la+n}\,\frac{(2\la)_n}{n!}\,,\quad +h_0=\frac{\pi^\half\,\Ga(\la+\thalf)}{\Ga(\la+1)},\quad +\frac{h_n}{h_0\,(C_n^\la(1))^2}= +\frac\la{\la+n}\,\frac{n!}{(2\la)_n}\,. +\label{61} +\end{equation} +% +\paragraph{Hypergeometric representation} +Beside (9.8.19) we have also +\begin{equation} +C_n^\lambda(x)=\sum_{\ell=0}^{\lfloor n/2\rfloor}\frac{(-1)^{\ell}(\lambda)_{n-\ell}} +{\ell!\;(n-2\ell)!}\,(2x)^{n-2\ell} +=(2x)^{n}\,\frac{(\lambda)_{n}}{n!}\, +\hyp21{-\thalf n,-\thalf n+\thalf}{1-\la-n}{\frac1{x^2}}. +\label{57} +\end{equation} +See \mycite{DLMF}{(18.5.10)}. +% +\paragraph{Special value} +\begin{equation} +C_n^{\la}(1)=\frac{(2\la)_n}{n!}\,. +\label{49} +\end{equation} +Use (9.8.19) or see \mycite{DLMF}{Table 18.6.1}. +% +\paragraph{Expression in terms of Jacobi} +% +\begin{equation} +\frac{C_n^\la(x)}{C_n^\la(1)}= +\frac{P_n^{(\la-\half,\la-\half)}(x)}{P_n^{(\la-\half,\la-\half)}(1)}\,,\qquad +C_n^\la(x)=\frac{(2\la)_n}{(\la+\thalf)_n}\,P_n^{(\la-\half,\la-\half)}(x). +\label{65} +\end{equation} +% +\paragraph{Re: (9.8.21)} +By iteration of recurrence relation (9.8.21): +\begin{multline} +x^2 C_n^\la(x)= +\frac{(n+1)(n+2)}{4(n+\la)(n+\la+1)}\,C_{n+2}^\la(x)+ +\frac{n^2+2n\la+\la-1}{2(n+\la-1)(n+\la+1)}\,C_n^\la(x)\\ ++\frac{(n+2\la-1)(n+2\la-2)}{4(n+\la)(n+\la-1)}\,C_{n-2}^\la(x). +\label{6} +\end{multline} +% +\paragraph{Bilateral generating functions} +\begin{multline} +\sum_{n=0}^\iy\frac{n!}{(2\la)_n}\,r^n\,C_n^\la(x)\,C_n^\la(y) +=\frac1{(1-2rxy+r^2)^\la}\,\hyp21{\thalf\la,\thalf(\la+1)}{\la+\thalf} +{\frac{4r^2(1-x^2)(1-y^2)}{(1-2rxy+r^2)^2}}\\ +(r\in(-1,1),\;x,y\in[-1,1]). +\label{66} +\end{multline} +For the proof put $\be:=\al$ in \eqref{58}, then use \eqref{63} and \eqref{65}. +The Poisson kernel for Gegenbauer polynomials can be derived in a similar way +from \eqref{59}, or alternatively by applying the operator +$r^{-\la+1}\frac d{dr}\circ r^\la$ to both sides of \eqref{66}: +\begin{multline} +\sum_{n=0}^\iy\frac{\la+n}\la\,\frac{n!}{(2\la)_n}\,r^n\,C_n^\la(x)\,C_n^\la(y) +=\frac{1-r^2}{(1-2rxy+r^2)^{\la+1}}\\ +\times\hyp21{\thalf(\la+1),\thalf(\la+2)}{\la+\thalf} +{\frac{4r^2(1-x^2)(1-y^2)}{(1-2rxy+r^2)^2}}\qquad +(r\in(-1,1),\;x,y\in[-1,1]). +\label{67} +\end{multline} +Formula \eqref{67} was obtained by Gasper \& Rahman \myciteKLS{234}{(4.4)} +as a limit case of their formula for the Poisson kernel for continuous +$q$-ultraspherical polynomials. +% +\paragraph{Trigonometric expansions} +By \mycite{DLMF}{(18.5.11), (15.8.1)}: +\begin{align} +C_n^{\la}(\cos\tha) +&=\sum_{k=0}^n\frac{(\la)_k(\la)_{n-k}}{k!\,(n-k)!}\,e^{i(n-2k)\tha} +=e^{in\tha}\frac{(\la)_n}{n!}\, +\hyp21{-n,\la}{1-\la-n}{e^{-2i\tha}}\label{103}\\ +&=\frac{(\la)_n}{2^\la n!}\, +e^{-\half i\la\pi}e^{i(n+\la)\tha}\,(\sin\tha)^{-\la}\, +\hyp21{\la,1-\la}{1-\la-n}{\frac{i e^{-i\tha}}{2\sin\tha}}\label{104}\\ +&=\frac{(\la)_n}{n!}\,\sum_{k=0}^\iy\frac{(\la)_k(1-\la)_k}{(1-\la-n)_k k!}\, +\frac{\cos((n-k+\la)\tha+\thalf(k-\la)\pi)}{(2\sin\tha)^{k+\la}}\,.\label{105} +\end{align} +In \eqref{104} and \eqref{105} we require that +$\tfrac16\pi<\tha<\tfrac56\pi$. Then the convergence is absolute for $\la>\thalf$ +and conditional for $0<\la\le\thalf$. + +By \mycite{DLMF}{(14.13.1), (14.3.21), (15.8.1)]}: +\begin{align} +C_n^\la(\cos\tha)&=\frac{2\Ga(\la+\thalf)}{\pi^\half\Ga(\la+1)}\, +\frac{(2\la)_n}{(\la+1)_n}\,(\sin\tha)^{1-2\la}\, +\sum_{k=0}^\iy\frac{(1-\la)_k(n+1)_k}{(n+\la+1)_k k!}\, +\sin\big((2k+n+1)\tha\big) +\label{7}\\ +&=\frac{2\Ga(\la+\thalf)}{\pi^\half\Ga(\la+1)}\, +\frac{(2\la)_n}{(\la+1)_n}\,(\sin\tha)^{1-2\la}\, +\Im\!\!\left(e^{i(n+1)\tha}\,\hyp21{1-\la,n+1}{n+\la+1}{e^{2i\tha}}\right)\nonumber\\ +&=\frac{2^\la\Ga(\la+\thalf)}{\pi^\half\Ga(\la+1)}\, +\frac{(2\la)_n}{(\la+1)_n}\,(\sin\tha)^{-\la}\, +\Re\!\!\left(e^{-\thalf i\la\pi}e^{i(n+\la)\tha}\, +\hyp21{\la,1-\la}{1+\la+n}{\frac{e^{i\tha}}{2i\sin\tha}}\right)\nonumber\\ +&=\frac{2^{2\la}\Ga(\la+\thalf)}{\pi^\half\Ga(\la+1)}\,\frac{(2\la)_n}{(\la+1)_n}\, +\sum_{k=0}^\iy\frac{(\la)_k(1-\la)_k}{(1+\la+n)_k k!}\, +\frac{\cos((n+k+\la)\tha-\thalf(k+\la)\pi)}{(2\sin\tha)^{k+\la}}\,. +\label{106} +\end{align} +We require that $0<\tha<\pi$ in \eqref{7} and $\tfrac16\pi<\tha<\tfrac56\pi$ in +\eqref{106} The convergence is absolute for $\la>\thalf$ and conditional for +$0<\la\le\thalf$. +For $\la\in\Zpos$ the above series terminate after the term with +$k=\la-1$. +Formulas \eqref{7} and \eqref{106} are also given in +\mycite{Sz}{(4.9.22), (4.9.25)}. +% +\paragraph{Fourier transform} +\begin{equation} +\frac{\Ga(\la+1)}{\Ga(\la+\thalf)\,\Ga(\thalf)}\, +\int_{-1}^1 \frac{C_n^\la(y)}{C_n^\la(1)}\,(1-y^2)^{\la-\half}\, +e^{ixy}\,dy +=i^n\,2^\la\,\Ga(\la+1)\,x^{-\la}\,J_{\la+n}(x). +\label{8} +\end{equation} +See \mycite{DLMF}{(18.17.17) and (18.17.18)}. +% +\paragraph{Laplace transforms} +\begin{equation} +\frac2{n!\,\Ga(\la)}\, +\int_0^\iy H_n(tx)\,t^{n+2\la-1}\,e^{-t^2}\,dt=C_n^\la(x). +\label{56} +\end{equation} +See Nielsen \cite[p.48, (4) with p.47, (1) and p.28, (10)]{K4} (1918) +or Feldheim \cite[(28)]{K3} (1942). +\begin{equation} +\frac2{\Ga(\la+\thalf)}\,\int_0^1 \frac{C_n^\la(t)}{C_n^\la(1)}\, +(1-t^2)^{\la-\half}\,t^{-1}\,(x/t)^{n+2\la+1}\,e^{-x^2/t^2}\,dt +=2^{-n}\,H_n(x)\,e^{-x^2}\quad(\la>-\thalf). +\label{46} +\end{equation} +Use Askey \& Fitch \cite[(3.29)]{K2} for $\al=\pm\thalf$ together with +\eqref{48}, \eqref{51}, \eqref{52}, \eqref{54} and \eqref{55}. +\paragraph{Addition formula} (see \mycite{AAR}{(9.8.5$'$)]}) +\begin{multline} +R_n^{(\al,\al)}\big(xy+(1-x^2)^\half(1-y^2)^\half t\big) +=\sum_{k=0}^n \frac{(-1)^k(-n)_k\,(n+2\al+1)_k}{2^{2k}((\al+1)_k)^2}\\ +\times(1-x^2)^{k/2} R_{n-k}^{(\al+k,\al+k)}(x)\,(1-y^2)^{k/2} R_{n-k}^{(\al+k,\al+k)}(y)\, +\om_k^{(\al-\half,\al-\half)}\,R_k^{(\al-\half,\al-\half)}(t), +\label{108} +\end{multline} +where +\[ +R_n^{(\al,\be)}(x):=P_n^{(\al,\be)}(x)/P_n^{(\al,\be)}(1),\quad +\om_n^{(\al,\be)}:=\frac{\int_{-1}^1 (1-x)^\al(1+x)^\be\,dx} +{\int_{-1}^1 (R_n^{(\al,\be)}(x))^2\,(1-x)^\al(1+x)^\be\,dx}\,. +\] +% +\cite{Abram}, \cite{Ahmed+86}, \cite{AndrewsAskeyRoy}, \cite{Area+I}, \cite{Askey67}, \cite{Askey74}, \cite{Askey75}, +\cite{Askey89I}, \cite{AskeyFitch}, \cite{AskeyKoornRahman}, \cite{Berg}, \cite{BilodeauI}, +\cite{BojanovNikolov}, \cite{Brafman51}, \cite{Brafman57I}, \cite{Brown}, +\cite{BustozIsmail82}, \cite{BustozIsmail83}, \cite{BustozIsmail89}, \cite{BustozSavage79}, +\cite{BustozSavage80}, \cite{Carlitz61II}, \cite{Chihara78}, \cite{Common}, \cite{Danese}, +\cite{Dette94}, \cite{DijksmaKoorn}, \cite{DilcherStolarsky}, \cite{Dimitrov96}, +\cite{Dimitrov2003}, \cite{Doha2002II}, \cite{Driver}, \cite{ElbertLaforgia86I}, +\cite{ElbertLaforgia86II}, \cite{ElbertLaforgia90}, \cite{Erdelyi+}, \cite{Exton96}, +\cite{Gasper69}, \cite{Gasper72II}, \cite{Gasper85}, \cite{Grad}, \cite{Ismail74}, +\cite{Ismail2005II}, \cite{Koekoek2000}, \cite{Koorn88}, \cite{Laforgia}, \cite{LewanowiczI}, +\cite{Lorch}, \cite{Mathai}, \cite{Nagel}, \cite{Nikiforov+}, \cite{NikiforovUvarov}, +\cite{RahmanShah}, \cite{Rainville}, \cite{Reimer}, \cite{Sartoretto}, \cite{Srivastava71}, +\cite{Szego75}, \cite{Temme}, \cite{Viswanathan}, \cite{Zayed}. + +\subsection{Chebyshev}\index{Chebyshev polynomials} + +\par + +\subsection*{Hypergeometric representation} +The Chebyshev polynomials of the first kind can be obtained from the Jacobi +polynomials by taking $\alpha=\beta=-\frac{1}{2}$: +\begin{equation} +\label{DefChebyshevI} +T_n(x)=\frac{P_n^{(-\frac{1}{2},-\frac{1}{2})}(x)}{P_n^{(-\frac{1}{2},-\frac{1}{2})}(1)} +=\hyp{2}{1}{-n,n}{\frac{1}{2}}{\frac{1-x}{2}} +\end{equation} +and the Chebyshev polynomials of the second kind can be obtained from the +Jacobi polynomials by taking $\alpha=\beta=\frac{1}{2}$: +\begin{equation} +\label{DefChebyshevII} +U_n(x)=(n+1)\frac{P_n^{(\frac{1}{2},\frac{1}{2})}(x)}{P_n^{(\frac{1}{2},\frac{1}{2})}(1)} +=(n+1)\,\hyp{2}{1}{-n,n+2}{\frac{3}{2}}{\frac{1-x}{2}}. +\end{equation} + +\subsection*{Orthogonality relation} +\begin{equation} +\label{OrtChebyshevI} +\int_{-1}^1(1-x^2)^{-\frac{1}{2}}T_m(x)T_n(x)\,dx= +\left\{\begin{array}{ll} +\displaystyle\frac{\cpi}{2}\,\delta_{mn}, & n\neq 0\\[5mm] +\cpi\,\delta_{mn}, & n=0. +\end{array}\right. +\end{equation} + +\begin{equation} +\label{OrtChebyshevII} +\int_{-1}^1(1-x^2)^{\frac{1}{2}}U_m(x)U_n(x)\,dx=\frac{\cpi}{2}\,\delta_{mn}. +\end{equation} + +\subsection*{Recurrence relations} +\begin{equation} +\label{RecChebyshevI} +2xT_n(x)=T_{n+1}(x)+T_{n-1}(x),\quad T_{0}(x)=1\quad\textrm{and}\quad T_1(x)=x. +\end{equation} + +\begin{equation} +\label{RecChebyshevII} +2xU_n(x)=U_{n+1}(x)+U_{n-1}(x),\quad U_{0}(x)=1\quad\textrm{and}\quad U_1(x)=2x. +\end{equation} + +\subsection*{Normalized recurrence relations} +\begin{equation} +\label{NormRecChebyshevI} +xp_n(x)=p_{n+1}(x)+\frac{1}{4}p_{n-1}(x), +\end{equation} +where +$$T_1(x)=p_1(x)=x\quad\textrm{and}\quad T_n(x)=2^np_n(x),\quad n\neq 1.$$ + +\begin{equation} +\label{NormRecChebyshevII} +xp_n(x)=p_{n+1}(x)+\frac{1}{4}p_{n-1}(x), +\end{equation} +where +$$U_n(x)=2^np_n(x).$$ + +\subsection*{Differential equations} +\begin{equation} +\label{dvChebyshevI} +(1-x^2)y''(x)-xy'(x)+n^2y(x)=0,\quad y(x)=T_n(x). +\end{equation} + +\begin{equation} +\label{dvChebyshevII} +(1-x^2)y''(x)-3xy'(x)+n(n+2)y(x)=0,\quad y(x)=U_n(x). +\end{equation} + +\subsection*{Forward shift operator} +\begin{equation} +\label{shift1Chebyshev} +\frac{d}{dx}T_n(x)=nU_{n-1}(x). +\end{equation} + +\subsection*{Backward shift operator} +\begin{equation} +\label{shift2ChebyshevI} +(1-x^2)\frac{d}{dx}U_n(x)-xU_n(x)=-(n+1)T_{n+1}(x) +\end{equation} +or equivalently +\begin{equation} +\label{shift2ChebyshevII} +\frac{d}{dx}\left[\left(1-x^2\right)^{\frac{1}{2}}U_n(x)\right] +=-(n+1)\left(1-x^2\right)^{-\frac{1}{2}}T_{n+1}(x). +\end{equation} + +\subsection*{Rodrigues-type formulas} +\begin{equation} +\label{RodChebyshevI} +(1-x^2)^{-\frac{1}{2}}T_n(x)=\frac{(-1)^n}{(\frac{1}{2})_n2^n} +\left(\frac{d}{dx}\right)^n\left[(1-x^2)^{n-\frac{1}{2}}\right]. +\end{equation} + +\begin{equation} +\label{RodChebyshevII} +(1-x^2)^{\frac{1}{2}}U_n(x)=\frac{(n+1)(-1)^n}{(\frac{3}{2})_n2^n} +\left(\frac{d}{dx}\right)^n\left[(1-x^2)^{n+\frac{1}{2}}\right]. +\end{equation} + +\subsection*{Generating functions} +\begin{equation} +\label{GenChebyshevI1} +\frac{1-xt}{1-2xt+t^2}=\sum_{n=0}^{\infty}T_n(x)t^n. +\end{equation} + +\begin{equation} +\label{GenChebyshevI2} +R^{-1}\sqrt{\frac{1}{2}(1+R-xt)}=\sum_{n=0}^{\infty} +\frac{\left(\frac{1}{2}\right)_n}{n!}T_n(x)t^n,\quad R=\sqrt{1-2xt+t^2}. +\end{equation} + +\begin{equation} +\label{GenChebyshevI3} +\hyp{0}{1}{-}{\frac{1}{2}}{\frac{(x-1)t}{2}}\, +\hyp{0}{1}{-}{\frac{1}{2}}{\frac{(x+1)t}{2}}= +\sum_{n=0}^{\infty}\frac{T_n(x)}{\left(\frac{1}{2}\right)_nn!}t^n. +\end{equation} + +\begin{equation} +\label{GenChebyshevI4} +\e^{xt}\,\hyp{0}{1}{-}{\frac{1}{2}}{\frac{(x^2-1)t^2}{4}}= +\sum_{n=0}^{\infty}\frac{T_n(x)}{n!}t^n. +\end{equation} + +\begin{eqnarray} +\label{GenChebyshevI5} +& &\hyp{2}{1}{\gamma,-\gamma}{\frac{1}{2}}{\frac{1-R-t}{2}}\, +\hyp{2}{1}{\gamma,-\gamma}{\frac{1}{2}}{\frac{1-R+t}{2}}\nonumber\\ +& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n(-\gamma)_n}{\left(\frac{1}{2}\right)_nn!}T_n(x)t^n, +\quad R=\sqrt{1-2xt+t^2},\quad\textrm{$\gamma$ arbitrary}. +\end{eqnarray} + +\begin{eqnarray} +\label{GenChebyshevI6} +& &(1-xt)^{-\gamma}\,\hyp{2}{1}{\frac{1}{2}\gamma,\frac{1}{2}\gamma+\frac{1}{2}}{\frac{1}{2}} +{\frac{(x^2-1)t^2}{(1-xt)^2}}\nonumber\\ +& &=\sum_{n=0}^{\infty}\frac{(\gamma)_n}{n!}T_n(x)t^n,\quad\textrm{$\gamma$ arbitrary}. +\end{eqnarray} + +\begin{equation} +\label{GenChebyshevII1} +\frac{1}{1-2xt+t^2}=\sum_{n=0}^{\infty}U_n(x)t^n. +\end{equation} + +\begin{equation} +\label{GenChebyshevII2} +\frac{1}{R\sqrt{\frac{1}{2}(1+R-xt)}}=\sum_{n=0}^{\infty} +\frac{\left(\frac{3}{2}\right)_n}{(n+1)!}U_n(x)t^n,\quad R=\sqrt{1-2xt+t^2}. +\end{equation} + +\begin{equation} +\label{GenChebyshevII3} +\hyp{0}{1}{-}{\frac{3}{2}}{\frac{(x-1)t}{2}}\, +\hyp{0}{1}{-}{\frac{3}{2}}{\frac{(x+1)t}{2}}= +\sum_{n=0}^{\infty}\frac{U_n(x)}{\left(\frac{3}{2}\right)_n(n+1)!}t^n. +\end{equation} + +\begin{equation} +\label{GenChebyshevII4} +\e^{xt}\,\hyp{0}{1}{-}{\frac{3}{2}}{\frac{(x^2-1)t^2}{4}}= +\sum_{n=0}^{\infty}\frac{U_n(x)}{(n+1)!}t^n. +\end{equation} + +\begin{eqnarray} +\label{GenChebyshevII5} +& &\hyp{2}{1}{\gamma,2-\gamma}{\frac{3}{2}}{\frac{1-R-t}{2}}\, +\hyp{2}{1}{\gamma,2-\gamma}{\frac{3}{2}}{\frac{1-R+t}{2}}\nonumber\\ +& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n(2-\gamma)_n}{\left(\frac{3}{2}\right)_n(n+1)!}U_n(x)t^n, +\quad R=\sqrt{1-2xt+t^2},\quad\textrm{$\gamma$ arbitrary}. +\end{eqnarray} + +\begin{eqnarray} +\label{GenChebyshevII6} +& &(1-xt)^{-\gamma}\,\hyp{2}{1}{\frac{1}{2}\gamma,\frac{1}{2}\gamma+\frac{1}{2}}{\frac{3}{2}} +{\frac{(x^2-1)t^2}{(1-xt)^2}}\nonumber\\ +& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n}{(n+1)!}U_n(x)t^n,\quad\textrm{$\gamma$ arbitrary}. +\end{eqnarray} + +\subsection*{Remarks} +The Chebyshev polynomials can also be written as: +$$T_n(x)=\cos(n\theta),\quad x=\cos\theta$$ +and +$$U_n(x)=\frac{\sin (n+1)\theta}{\sin\theta},\quad x=\cos\theta.$$ +Further we have +$$U_n(x)=C_n^{(1)}(x)$$ +where $C_n^{(\lambda)}(x)$ denotes the Gegenbauer (or ultraspherical) +polynomial given by (\ref{DefGegenbauer}) in the preceding subsection. + +\subsection*{References} +\label{sec9.8.2} +In addition to the Chebyshev polynomials $T_n$ of the first kind (9.8.35) +and $U_n$ of the second kind (9.8.36), +\begin{align} +T_n(x)&:=\frac{P_n^{(-\half,-\half)}(x)}{P_n^{(-\half,-\half)}(1)} +=\cos(n\tha),\quad x=\cos\tha,\\ +U_n(x)&:=(n+1)\,\frac{P_n^{(\half,\half)}(x)}{P_n^{(\half,\half)}(1)} +=\frac{\sin((n+1)\tha)}{\sin\tha}\,,\quad x=\cos\tha, +\end{align} +we have Chebyshev polynomials $V_n$ {\em of the third kind} +and $W_n$ {\em of the fourth kind}, +\begin{align} +V_n(x)&:=\frac{P_n^{(-\half,\half)}(x)}{P_n^{(-\half,\half)}(1)} +=\frac{\cos((n+\thalf)\tha)}{\cos(\thalf\tha)}\,,\quad x=\cos\tha,\\ +W_n(x)&:=(2n+1)\,\frac{P_n^{(\half,-\half)}(x)}{P_n^{(\half,-\half)}(1)} +=\frac{\sin((n+\thalf)\tha)}{\sin(\thalf\tha)}\,,\quad x=\cos\tha, +\end{align} +see \cite[Section 1.2.3]{K20}. Then there is the symmetry +\begin{equation} +V_n(-x)=(-1)^n W_n(x). +\label{140} +\end{equation} + +The names of Chebyshev polynomials of the third and fourth kind +and the notation $V_n(x)$ are due to Gautschi \cite{K21}. +The notation $W_n(x)$ was first used by Mason \cite{K22}. +Names and notations for Chebyshev polynomials of the third and fourth +kind are interchanged in \mycite{AAR}{Remark 2.5.3} and +\mycite{DLMF}{Table 18.3.1}. +% +\cite{Abram}, \cite{AskeyFitch}, \cite{AskeyGasperHarris}, \cite{AskeyIsmail76}, +\cite{Bavinck95}, \cite{Chihara78}, \cite{Danese}, \cite{DilcherStolarsky}, +\cite{Erdelyi+}, \cite{Grad}, \cite{HartmannStephan}, \cite{Ismail2005II}, \cite{Koekoek2000}, +\cite{Luke}, \cite{Mathai}, \cite{Nikiforov+}, \cite{NikiforovUvarov}, \cite{Rainville}, +\cite{Rayes+}, \cite{Rivlin}, \cite{SainteViennot}, \cite{Szego75}, \cite{Temme}, +\cite{Wilson70I}, \cite{Zayed}, \cite{Zhang}, \cite{ZhangWang}. + +\subsection{Legendre / Spherical}\index{Legendre polynomials} +\index{Spherical polynomials} + +\par + +\subsection*{Hypergeometric representation} +The Legendre (or spherical) polynomials are Jacobi polynomials with $\alpha=\beta=0$: +\begin{equation} +\label{DefLegendre} +P_n(x)=P_n^{(0,0)}(x)=\hyp{2}{1}{-n,n+1}{1}{\frac{1-x}{2}}. +\end{equation} + +\subsection*{Orthogonality relation} +\begin{equation} +\label{OrtLegendre} +\int_{-1}^1P_m(x)P_n(x)\,dx=\frac{2}{2n+1}\,\delta_{mn}. +\end{equation} + +\subsection*{Recurrence relation} +\begin{equation} +\label{RecLegendre} +(2n+1)xP_n(x)=(n+1)P_{n+1}(x)+nP_{n-1}(x). +\end{equation} + +\subsection*{Normalized recurrence relation} +\begin{equation} +\label{NormRecLegendre} +xp_n(x)=p_{n+1}(x)+\frac{n^2}{(2n-1)(2n+1)}p_{n-1}(x), +\end{equation} +where +$$P_n(x)=\binom{2n}{n}\frac{1}{2^n}p_n(x).$$ + +\subsection*{Differential equation} +\begin{equation} +\label{dvLegendre} +(1-x^2)y''(x)-2xy'(x)+n(n+1)y(x)=0,\quad y(x)=P_n(x). +\end{equation} + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodLegendre} +P_n(x)=\frac{(-1)^n}{2^nn!}\left(\frac{d}{dx}\right)^n\left[(1-x^2)^n\right]. +\end{equation} + +\subsection*{Generating functions} +\begin{equation} +\label{GenLegendre1} +\frac{1}{\sqrt{1-2xt+t^2}}=\sum_{n=0}^{\infty}P_n(x)t^n. +\end{equation} + +\begin{equation} +\label{GenLegendre2} +\hyp{0}{1}{-}{1}{\frac{(x-1)t}{2}}\,\hyp{0}{1}{-}{1}{\frac{(x+1)t}{2}}= +\sum_{n=0}^{\infty}\frac{P_n(x)}{(n!)^2}t^n. +\end{equation} + +\begin{equation} +\label{GenLegendre3} +\e^{xt}\,\hyp{0}{1}{-}{1}{\frac{(x^2-1)t^2}{4}}= +\sum_{n=0}^{\infty}\frac{P_n(x)}{n!}t^n. +\end{equation} + +\begin{eqnarray} +\label{GenLegendre4} +& &\hyp{2}{1}{\gamma,1-\gamma}{1}{\frac{1-R-t}{2}}\, +\hyp{2}{1}{\gamma,1-\gamma}{1}{\frac{1-R+t}{2}}\nonumber\\ +& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n(1-\gamma)_n}{(n!)^2}P_n(x)t^n, +\quad R=\sqrt{1-2xt+t^2},\quad\textrm{$\gamma$ arbitrary}. +\end{eqnarray} + +\begin{eqnarray} +\label{GenLegendre5} +& &(1-xt)^{-\gamma}\,\hyp{2}{1}{\frac{1}{2}\gamma,\frac{1}{2}\gamma+\frac{1}{2}}{1} +{\frac{(x^2-1)t^2}{(1-xt)^2}}\nonumber\\ +& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n}{n!}P_n(x)t^n,\quad\textrm{$\gamma$ arbitrary}. +\end{eqnarray} + +\subsection*{References} +\cite{Abram}, \cite{Alladi}, \cite{AlSalam90}, \cite{Bhonsle}, \cite{Brafman51}, +\cite{Carlitz57II}, \cite{Chihara78}, \cite{Danese}, \cite{Dattoli2001}, +\cite{DilcherStolarsky}, \cite{ElbertLaforgia94}, \cite{Erdelyi+}, \cite{Grad}, +\cite{Mathai}, \cite{Nikiforov+}, \cite{NikiforovUvarov}, \cite{Olver}, \cite{Rainville}, +\cite{Szego75}, \cite{Temme}, \cite{Zayed}. + +\newpage + +\section{Pseudo Jacobi}\index{Pseudo Jacobi polynomials}\index{Jacobi polynomials!Pseudo} + +\par\setcounter{equation}{0} + +\subsection*{Hypergeometric representation} +\begin{eqnarray} +\label{DefPseudoJacobi} +P_n(x;\nu,N)&=&\frac{(-2i)^n(-N+i\nu)_n}{(n-2N-1)_n}\,\hyp{2}{1}{-n,n-2N-1}{-N+i\nu}{\frac{1-ix}{2}}\\ +&=&(x+i)^n\,\hyp{2}{1}{-n,N+1-n-i\nu}{2N+2-2n}{\frac{2}{1-ix}},\quad n=0,1,2,\ldots,N.\nonumber +\end{eqnarray} + +\subsection*{Orthogonality relation} +\begin{eqnarray} +\label{OrtPseudoJacobi} +& &\frac{1}{2\pi}\int_{-\infty}^{\infty}(1+x^2)^{-N-1}\e^{2\nu\arctan x}P_m(x;\nu,N)P_n(x;\nu,N)\,dx\nonumber\\ +& &{}=\frac{\Gamma(2N+1-2n)\Gamma(2N+2-2n)2^{2n-2N-1}n!}{\Gamma(2N+2-n)\left|\Gamma(N+1-n+i\nu)\right|^2}\,\delta_{mn}. +\end{eqnarray} + +\subsection*{Recurrence relation} +\begin{eqnarray} +\label{RecPseudoJacobi} +xP_n(x;\nu,N)&=&P_{n+1}(x;\nu,N)+\frac{(N+1)\nu}{(n-N-1)(n-N)}P_n(x;\nu,N)\nonumber\\ +& &{}\mathindent{}-\frac{n(n-2N-2)}{(2n-2N-3)(n-N-1)^2(2n-2N-1)}\nonumber\\ +& &{}\mathindent\mathindent\times(n-N-1-i\nu)(n-N-1+i\nu)P_{n-1}(x;\nu,N). +\end{eqnarray} + +\subsection*{Normalized recurrence relation} +\begin{eqnarray} +\label{NormRecPseudoJacobi} +xp_n(x)&=&p_{n+1}(x)+\frac{(N+1)\nu}{(n-N-1)(n-N)}p_n(x)\nonumber\\ +& &{}\mathindent{}-\frac{n(n-2N-2)(n-N-1-i\nu)(n-N-1+i\nu)}{(2n-2N-3)(n-N-1)^2(2n-2N-1)}p_{n-1}(x), +\end{eqnarray} +where +$$P_n(x;\nu,N)=p_n(x).$$ + +\subsection*{Differential equation} +\begin{equation} +\label{dvPseudoJacobi} +(1+x^2)y''(x)+2\left(\nu-Nx\right)y'(x)-n(n-2N-1)y(x)=0, +\end{equation} +where +$$y(x)=P_n(x;\nu,N).$$ + +\subsection*{Forward shift operator} +\begin{equation} +\label{shift1PseudoJacobi} +\frac{d}{dx}P_n(x;\nu,N)=nP_{n-1}(x;\nu,N-1). +\end{equation} + +\subsection*{Backward shift operator} +\begin{eqnarray} +\label{shift2PseudoJacobiI} +& &(1+x^2)\frac{d}{dx}P_n(x;\nu,N)+2\left[\nu-(N+1)x\right]P_n(x;\nu,N)\nonumber\\ +& &{}=(n-2N-2)P_{n+1}(x;\nu,N+1) +\end{eqnarray} +or equivalently +\begin{eqnarray} +\label{shift2PseudoJacobiII} +& &\frac{d}{dx}\left[(1+x^2)^{-N-1}\e^{2\nu\arctan x}P_n(x;\nu,N)\right]\nonumber\\ +& &{}=(n-2N-2)(1+x^2)^{-N-2}\e^{2\nu\arctan x}P_{n+1}(x;\nu,N+1). +\end{eqnarray} + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodPseudoJacobi} +P_n(x;\nu,N)=\frac{(1+x^2)^{N+1}\expe^{-2\nu\arctan x}}{(n-2N-1)_n} +\left(\frac{d}{dx}\right)^n\left[(1+x^2)^{n-N-1}\expe^{2\nu\arctan x}\right]. +\end{equation} + +\subsection*{Generating function} +\begin{eqnarray} +\label{GenPseudoJacobi} +& &\left[\hyp{0}{1}{-}{-N+i\nu}{(x+i)t}\,\hyp{0}{1}{-}{-N-i\nu}{(x-i)t}\right]_N\nonumber\\ +& &{}=\sum_{n=0}^N\frac{(n-2N-1)_n}{(-N+i\nu)_n(-N-i\nu)_nn!}P_n(x;\nu,N)t^n. +\end{eqnarray} + +\subsection*{Limit relation} + +\subsubsection*{Continuous Hahn $\rightarrow$ Pseudo Jacobi} +The pseudo Jacobi polynomials follow from the continuous Hahn polynomials given by +(\ref{DefContHahn}) by the substitutions $x\rightarrow xt$, $a=\frac{1}{2}(-N+i\nu-2t)$, +$b=\frac{1}{2}(-N-i\nu+2t)$, $c=\frac{1}{2}(-N-i\nu-2t)$ and $d=\frac{1}{2}(-N+i\nu+2t)$, +division by $t^n$ and the limit $t\rightarrow\infty$: +\begin{eqnarray*} +& &\lim_{t\rightarrow\infty}\frac{p_n(xt;\frac{1}{2}(-N+i\nu-2t),\frac{1}{2}(-N-i\nu+2t), +\frac{1}{2}(-N+i\nu-2t),\frac{1}{2}(-N-i\nu+2t))}{t^n}\\ +& &{}=\frac{(n-2N-1)_n}{n!}P_n(x;\nu,N). +\end{eqnarray*} + +\subsection*{Remarks} +Since we have for $k 0$ & +% $0 < c < 1$ } +\end{equation} + +\subsection*{Recurrence relation} +\begin{eqnarray} +\label{RecMeixner} +(c-1)xM_n(x;\beta,c)&=&c(n+\beta)M_{n+1}(x;\beta,c)\nonumber\\ +& &{}\mathindent{}-\left[n+(n+\beta)c\right]M_n(x;\beta,c)+nM_{n-1}(x;\beta,c). +\end{eqnarray} + +\subsection*{Normalized recurrence relation} +\begin{equation} +\label{NormRecMeixner} +xp_n(x)=p_{n+1}(x)+\frac{n+(n+\beta)c}{1-c}p_n(x)+ +\frac{n(n+\beta-1)c}{(1-c)^2}p_{n-1}(x), +\end{equation} +where +$$M_n(x;\beta,c)=\frac{1}{(\beta)_n}\left(\frac{c-1}{c}\right)^np_n(x).$$ + +\subsection*{Difference equation} +\begin{equation} +\label{dvMeixner} +n(c-1)y(x)=c(x+\beta)y(x+1)-\left[x+(x+\beta)c\right]y(x)+xy(x-1), +\end{equation} +where +$$y(x)=M_n(x;\beta,c).$$ + +\subsection*{Forward shift operator} +\begin{equation} +\label{shift1MeixnerI} +M_n(x+1;\beta,c)-M_n(x;\beta,c)= +\frac{n}{\beta}\left(\frac{c-1}{c}\right)M_{n-1}(x;\beta+1,c) +\end{equation} +or equivalently +\begin{equation} +\label{shift1MeixnerII} +\Delta M_n(x;\beta,c)=\frac{n}{\beta}\left(\frac{c-1}{c}\right)M_{n-1}(x;\beta+1,c). +\end{equation} + +\subsection*{Backward shift operator} +\begin{equation} +\label{shift2MeixnerI} +c(\beta+x-1)M_n(x;\beta,c)-xM_n(x-1;\beta,c)=c(\beta-1)M_{n+1}(x;\beta-1,c) +\end{equation} +or equivalently +\begin{equation} +\label{shift2MeixnerII} +\nabla\left[\frac{(\beta)_xc^x}{x!}M_n(x;\beta,c)\right]= +\frac{(\beta-1)_xc^x}{x!}M_{n+1}(x;\beta-1,c). +\end{equation} + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodMeixner} +\frac{(\beta)_xc^x}{x!}M_n(x;\beta,c)=\nabla^n\left[\frac{(\beta+n)_xc^x}{x!}\right]. +\end{equation} + +\subsection*{Generating functions} +\begin{equation} +\label{GenMeixner1} +\left(1-\frac{t}{c}\right)^x(1-t)^{-x-\beta}= +\sum_{n=0}^{\infty}\frac{(\beta)_n}{n!}M_n(x;\beta,c)t^n. +\end{equation} + +\begin{equation} +\label{GenMeixner2} +\e^t\,\hyp{1}{1}{-x}{\beta}{\left(\frac{1-c}{c}\right)t}= +\sum_{n=0}^{\infty}\frac{M_n(x;\beta,c)}{n!}t^n. +\end{equation} + +\begin{equation} +\label{GenMeixner3} +(1-t)^{-\gamma}\,\hyp{2}{1}{\gamma,-x}{\beta}{\frac{(1-c)t}{c(1-t)}} +=\sum_{n=0}^{\infty}\frac{(\gamma)_n}{n!}M_n(x;\beta,c)t^n, +\quad\textrm{$\gamma$ arbitrary}. +\end{equation} + +\subsection*{Limit relations} + +\subsubsection*{Hahn $\rightarrow$ Meixner} +If we take $\alpha=b-1$, $\beta=N(1-c)c^{-1}$ in the definition (\ref{DefHahn}) +of the Hahn polynomials and let $N\rightarrow\infty$ we find the Meixner polynomials: +$$\lim_{N\rightarrow\infty} +Q_n(x;b-1,N(1-c)c^{-1},N)=M_n(x;b,c).$$ + +\subsubsection*{Dual Hahn $\rightarrow$ Meixner} +To obtain the Meixner polynomials from the dual Hahn polynomials we have to take +$\gamma=\beta-1$ and $\delta=N(1-c)c^{-1}$ in the definition (\ref{DefDualHahn}) of +the dual Hahn polynomials and let $N\rightarrow\infty$: +$$\lim_{N\rightarrow\infty} +R_n(\lambda(x);\beta-1,N(1-c)c^{-1},N)=M_n(x;\beta,c).$$ + +\subsubsection*{Meixner $\rightarrow$ Laguerre} +The Laguerre polynomials given by (\ref{DefLaguerre}) are obtained from the Meixner polynomials +if we take $\beta=\alpha+1$ and $x\rightarrow (1-c)^{-1}x$ and let $c\rightarrow 1$: +\begin{equation} +\lim_{c\rightarrow 1} +M_n((1-c)^{-1}x;\alpha+1,c)=\frac{L_n^{(\alpha)}(x)}{L_n^{(\alpha)}(0)}. +\end{equation} + +\subsubsection*{Meixner $\rightarrow$ Charlier} +The Charlier polynomials given by (\ref{DefCharlier}) are obtained from the Meixner polynomials +if we take $c=(a+\beta)^{-1}a$ and let $\beta\rightarrow\infty$: +\begin{equation} +\lim_{\beta\rightarrow\infty} +M_n(x;\beta,(a+\beta)^{-1}a)=C_n(x;a). +\end{equation} + +\subsection*{Remarks} +The Meixner polynomials are related to the Jacobi polynomials given by (\ref{DefJacobi}) +in the following way: +$$\frac{(\beta)_n}{n!}M_n(x;\beta,c)=P_n^{(\beta-1,-n-\beta-x)}((2-c)c^{-1}).$$ + +\noindent +The Meixner polynomials are also related to the Krawtchouk polynomials given by +(\ref{DefKrawtchouk}) in the following way: +$$K_n(x;p,N)=M_n(x;-N,(p-1)^{-1}p).$$ + +\subsection*{References} +\label{sec9.9} +In this section in \mycite{KLS} the pseudo Jacobi polynomial $P_n(x;\nu,N)$ in (9.9.1) +is considered +for $N\in\ZZ_{\ge0}$ and $n=0,1,\ldots,n$. However, we can more generally take +$-\thalf\be>0\;\;{\rm or}\;\;\al<\be<0). +\] +Then $P_n$ can be expressed as a Meixner polynomial: +\[ +P_n(x)=(-k_2(\al\be)^{-1})_n\,\be^n\, +M_n\left(-\,\frac{x+k_2\al^{-1}}{\al-\be},-k_2(\al\be)^{-1},\be\al^{-1}\right). +\] + +In 1938 Gottlieb \cite[\S2]{K1} introduces polynomials $l_n$ ``of Laguerre type'' +which turn out to be special Meixner polynomials: +$l_n(x)=e^{-n\la} M_n(x;1,e^{-\la})$. +% +\paragraph{Uniqueness of orthogonality measure} +The coefficient of $p_{n-1}(x)$ in (9.10.4) behaves as $O(n^2)$ as $n\to\iy$. +Hence \eqref{93} holds, by which the orthogonality measure is unique. +% +\cite{AlSalam90}, \cite{AndrewsAskey85}, \cite{Area+II}, \cite{Askey75}, \cite{Askey89I}, +\cite{AskeyGasper77}, \cite{AskeyWilson85}, \cite{AtakRahmanSuslov}, \cite{Campigotto+}, +\cite{LChiharaStanton}, \cite{Chihara78}, \cite{Dette95}, \cite{Dominici}, \cite{Dunkl76}, +\cite{Dunkl84}, \cite{DunklRamirez}, \cite{Erdelyi+}, \cite{FeinsilverSchott}, +\cite{Gasper73I}, \cite{Gasper74}, \cite{HoareRahman}, \cite{Ismail2005II}, \cite{Karlin58}, +\cite{Koorn82}, \cite{Koorn88}, \cite{LabelleYehI}, \cite{LabelleYehII}, \cite{Lesky62}, +\cite{Lesky89}, \cite{Lesky94I}, \cite{Lesky95II}, \cite{LewanowiczII}, \cite{Nikiforov+}, +\cite{NikiforovUvarov}, \cite{Qiu}, \cite{Rahman78I}, \cite{Rahman79}, \cite{Stanton84}, +\cite{Stanton90}, \cite{Szego75}, \cite{Zarzo+}, \cite{Zeng90}. + + +\section{Laguerre}\index{Laguerre polynomials} + +\par\setcounter{equation}{0} + +\subsection*{Hypergeometric representation} +\begin{equation} +\label{DefLaguerre} +L_n^{(\alpha)}(x)=\frac{(\alpha+1)_n}{n!}\,\hyp{1}{1}{-n}{\alpha+1}{x}. +\end{equation} + +\subsection*{Orthogonality relation} +\begin{equation} +\label{OrtLaguerre} +\int_0^{\infty}\expe^{-x}x^{\alpha}L_m^{(\alpha)}(x)L_n^{(\alpha)}(x)\,dx= +\frac{\Gamma(n+\alpha+1)}{n!}\,\delta_{mn},\quad\alpha > -1. +\end{equation} + +\subsection*{Recurrence relation} +\begin{equation} +\label{RecLaguerre} +(n+1)L_{n+1}^{(\alpha)}(x)-(2n+\alpha+1-x)L_n^{(\alpha)}(x)+(n+\alpha)L_{n-1}^{(\alpha)}(x)=0. +\end{equation} + +\subsection*{Normalized recurrence relation} +\begin{equation} +\label{NormRecLaguerre} +xp_n(x)=p_{n+1}(x)+(2n+\alpha+1)p_n(x)+n(n+\alpha)p_{n-1}(x), +\end{equation} +where +$$L_n^{(\alpha)}(x)=\frac{(-1)^n}{n!}p_n(x).$$ + +\subsection*{Differential equation} +\begin{equation} +\label{dvLaguerre} +xy''(x)+(\alpha+1-x)y'(x)+ny(x)=0,\quad y(x)=L_n^{(\alpha)}(x). +\end{equation} + +\newpage + +\subsection*{Forward shift operator} +\begin{equation} +\label{shift1Laguerre} +\frac{d}{dx}L_n^{(\alpha)}(x)=-L_{n-1}^{(\alpha+1)}(x). +\end{equation} + +\subsection*{Backward shift operator} +\begin{equation} +\label{shift2LaguerreI} +x\frac{d}{dx}L_n^{(\alpha)}(x)+(\alpha-x)L_n^{(\alpha)}(x)=(n+1)L_{n+1}^{(\alpha-1)}(x) +\end{equation} +or equivalently +\begin{equation} +\label{shift2LaguerreII} +\frac{d}{dx}\left[\expe^{-x}x^{\alpha}L_n^{(\alpha)}(x)\right]=(n+1)\expe^{-x}x^{\alpha-1}L^{(\alpha-1)}_{n+1}(x). +\end{equation} + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodLaguerre} +\e^{-x}x^{\alpha}L_n^{(\alpha)}(x)=\frac{1}{n!}\left(\frac{d}{dx}\right)^n\left[\expe^{-x}x^{n+\alpha}\right]. +\end{equation} + +\subsection*{Generating functions} +\begin{equation} +\label{GenLaguerre1} +(1-t)^{-\alpha-1}\exp\left(\frac{xt}{t-1}\right)= +\sum_{n=0}^{\infty}L_n^{(\alpha)}(x)t^n. +\end{equation} + +\begin{equation} +\label{GenLaguerre2} +\e^t\,\hyp{0}{1}{-}{\alpha+1}{-xt} +=\sum_{n=0}^{\infty}\frac{L_n^{(\alpha)}(x)}{(\alpha+1)_n}t^n. +\end{equation} + +\begin{equation} +\label{GenLaguerre3} +(1-t)^{-\gamma}\,\hyp{1}{1}{\gamma}{\alpha+1}{\frac{xt}{t-1}} +=\sum_{n=0}^{\infty}\frac{(\gamma)_n}{(\alpha+1)_n}L_n^{(\alpha)}(x)t^n, +\quad\textrm{$\gamma$ arbitrary}. +\end{equation} + +\subsection*{Limit relations} + +\subsubsection*{Meixner-Pollaczek $\rightarrow$ Laguerre} +The Laguerre polynomials can be obtained from the Meixner-Pollaczek polynomials given by +(\ref{DefMP}) by the substitution $\lambda=\frac{1}{2}(\alpha+1)$, +$x\rightarrow -\frac{1}{2}\phi^{-1}x$ and the limit $\phi\rightarrow 0$: +$$\lim_{\phi\rightarrow 0} +P_n^{(\frac{1}{2}\alpha+\frac{1}{2})}(-\textstyle\frac{1}{2}\phi^{-1}x;\phi)=L_n^{(\alpha)}(x).$$ + +\subsubsection*{Jacobi $\rightarrow$ Laguerre} +The Laguerre polynomials are obtained from the Jacobi polynomials given by (\ref{DefJacobi}) +if we set $x\rightarrow 1-2\beta^{-1}x$ and then take the limit $\beta\rightarrow\infty$: +$$\lim_{\beta\rightarrow\infty} +P_n^{(\alpha,\beta)}(1-2\beta^{-1}x)=L_n^{(\alpha)}(x).$$ + +\subsubsection*{Meixner $\rightarrow$ Laguerre} +If we take $\beta=\alpha+1$ and $x\rightarrow (1-c)^{-1}x$ in the definition +(\ref{DefMeixner}) of the Meixner polynomials and let $c\rightarrow 1$ we obtain +the Laguerre polynomials: +$$\lim_{c\rightarrow 1} +M_n((1-c)^{-1}x;\alpha+1,c)=\frac{L_n^{(\alpha)}(x)}{L_n^{(\alpha)}(0)}.$$ + +\subsubsection*{Laguerre $\rightarrow$ Hermite} +The Hermite polynomials given by (\ref{DefHermite}) can be obtained from the Laguerre +polynomials by taking the limit $\alpha\rightarrow\infty$ in the following way: +\begin{equation} +\lim_{\alpha\rightarrow\infty} +\left(\frac{2}{\alpha}\right)^{\frac{1}{2}n} +L_n^{(\alpha)}((2\alpha)^{\frac{1}{2}}x+\alpha)=\frac{(-1)^n}{n!}H_n(x). +\end{equation} + +\subsection*{Remarks} +The definition (\ref{DefLaguerre}) of the Laguerre polynomials can also be +written as: +$$L_n^{(\alpha)}(x)=\frac{1}{n!}\sum_{k=0}^n\frac{(-n)_k}{k!}(\alpha+k+1)_{n-k}x^k.$$ +In this way the Laguerre polynomials can also be seen as polynomials in the parameter $\alpha$. +Therefore they can be defined for all $\alpha$. + +\noindent +The Laguerre polynomials are related to the Bessel polynomials given by (\ref{DefBessel}) +in the following way: +$$L_n^{(\alpha)}(x)=\frac{(-x)^n}{n!}y_n(2x^{-1};-2n-\alpha-1).$$ + +\noindent +The Laguerre polynomials are related to the Charlier polynomials given by (\ref{DefCharlier}) +in the following way: +$$\frac{(-a)^n}{n!}C_n(x;a)=L_n^{(x-n)}(a).$$ + +\noindent +The Laguerre polynomials and the Hermite polynomials given by (\ref{DefHermite}) are also +connected by the following quadratic transformations: +$$H_{2n}(x)=(-1)^nn!\,2^{2n}L_n^{(-\frac{1}{2})}(x^2)$$ +and +$$H_{2n+1}(x)=(-1)^nn!\,2^{2n+1}xL_n^{(\frac{1}{2})}(x^2).$$ + +\noindent +In combinatorics the Laguerre polynomials with $\alpha=0$ are often called Rook +polynomials. + +\subsection*{References} +\label{sec9.11} +% +\paragraph{Special values} +By (9.11.1) and the binomial formula: +\begin{equation} +K_n(0;p,N)=1,\qquad +K_n(N;p,N)=(1-p^{-1})^n. +\label{9} +\end{equation} +The self-duality (p.240, Remarks, first formula) +\begin{equation} +K_n(x;p,N)=K_x(n;p,N)\qquad (n,x\in \{0,1,\ldots,N\}) +\label{147} +\end{equation} +combined with \eqref{9} yields: +\begin{equation} +K_N(x;p,N)=(1-p^{-1})^x\qquad(x\in\{0,1,\ldots,N\}). +\label{148} +\end{equation} +% +\paragraph{Symmetry} +By the orthogonality relation (9.11.2): +\begin{equation} +\frac{K_n(N-x;p,N)}{K_n(N;p,N)}=K_n(x;1-p,N). +\label{10} +\end{equation} +By \eqref{10} and \eqref{147} we have also +\begin{equation} +\frac{K_{N-n}(x;p,N)}{K_N(x;p,N)}=K_n(x;1-p,N) +\qquad(n,x\in\{0,1,\ldots,N\}), +\label{149} +\end{equation} +and, by \eqref{149}, \eqref{10} and \eqref{9}, +\begin{equation} +K_{N-n}(N-x;p,N)=\left(\frac p{p-1}\right)^{n+x-N}K_n(x;p,N) +\qquad(n,x\in\{0,1,\ldots,N\}). +\label{150} +\end{equation} +A particular case of \eqref{10} is: +\begin{equation} +K_n(N-x;\thalf,N)=(-1)^n K_n(x;\thalf,N). +\label{11} +\end{equation} +Hence +\begin{equation} +K_{2m+1}(N;\thalf,2N)=0. +\label{12} +\end{equation} +From (9.11.11): +\begin{equation} +K_{2m}(N;\thalf,2N)=\frac{(\thalf)_m}{(-N+\thalf)_m}\,. +\label{13} +\end{equation} +% +\paragraph{Quadratic transformations} +\begin{align} +K_{2m}(x+N;\thalf,2N)&=\frac{(\thalf)_m}{(-N+\thalf)_m}\, +R_m(x^2;-\thalf,-\thalf,N), +\label{31}\\ +K_{2m+1}(x+N;\thalf,2N)&=-\,\frac{(\tfrac32)_m}{N\,(-N+\thalf)_m}\, +x\,R_m(x^2-1;\thalf,\thalf,N-1), +\label{33}\\ +K_{2m}(x+N+1;\thalf,2N+1)&=\frac{(\tfrac12)_m}{(-N-\thalf)_m}\, +R_m(x(x+1);-\thalf,\thalf,N), +\label{32}\\ +K_{2m+1}(x+N+1;\thalf,2N+1)&=\frac{(\tfrac32)_m}{(-N-\thalf)_{m+1}}\, +(x+\thalf)\,R_m(x(x+1);\thalf,-\thalf,N), +\label{34} +\end{align} +where $R_m$ is a dual Hahn polynomial (9.6.1). For the proofs use +(9.6.2), (9.11.2), (9.6.4) and (9.11.4). +% +\paragraph{Generating functions} +\begin{multline} +\sum_{x=0}^N\binom Nx K_m(x;p,N)K_n(x;q,N)z^x\\ +=\left(\frac{p-z+pz}p\right)^m +\left(\frac{q-z+qz}q\right)^n +(1+z)^{N-m-n} +K_m\left(n;-\,\frac{(p-z+pz)(q-z+qz)}z,N\right). +\label{107} +\end{multline} +This follows immediately from Rosengren \cite[(3.5)]{K8}, which goes back +to Meixner \cite{K9}. +% +\cite{Abdul}, \cite{Abram}, \cite{Ahmed+82}, \cite{Allaway76}, \cite{Allaway91}, +\cite{NAlSalam66}, \cite{AlSalam64}, \cite{AlSalam90}, \cite{AlSalamChihara72}, +\cite{AlSalamChihara76}, \cite{AndrewsAskey85}, \cite{AndrewsAskeyRoy}, \cite{Askey68}, +\cite{Askey75}, \cite{Askey89I}, \cite{Askey2005}, \cite{AskeyGasper76}, \cite{AskeyGasper77}, +\cite{AskeyIsmail76}, \cite{AskeyIsmailKoorn}, \cite{AskeyWilson85}, \cite{Barrett}, +\cite{Bavinck96}, \cite{Brafman51}, \cite{Brafman57II}, \cite{Brenke}, \cite{Brown}, +\cite{BustozSavage79}, \cite{BustozSavage80}, \cite{Carlitz60}, \cite{Carlitz61I}, +\cite{Carlitz61II}, \cite{Carlitz62}, \cite{Carlitz68}, \cite{ChenSrivastava}, +\cite{ChenIsmail91}, \cite{Chihara68II}, \cite{Chihara78}, \cite{Cohen}, \cite{Cooper+}, +\cite{Dattoli2001}, \cite{DetteStudden92}, \cite{DetteStudden95}, \cite{Dimitrov2003}, +\cite{Doha2003II}, \cite{ElbertLaforgia87I}, \cite{Erdelyi}, \cite{Erdelyi+}, \cite{Exton98}, +\cite{Faldey}, \cite{Gasper73II}, \cite{Gasper77}, \cite{Gatteschi2002}, \cite{Gawronski87}, +\cite{Gawronski93}, \cite{Gillis}, \cite{GillisIsmailOffer}, \cite{Godoy+}, \cite{Grad}, +\cite{Hajmirzaahmad95}, \cite{Ismail74}, \cite{Ismail77}, \cite{Ismail2005II}, +\cite{IsmailLetVal88}, \cite{IsmailLi}, \cite{IsmailMassonRahman}, \cite{IsmailMuldoon}, +\cite{IsmailStanton97}, \cite{IsmailTamhankar}, \cite{Jackson}, \cite{Karlin58}, +\cite{Kochneff95}, \cite{Kochneff97II}, \cite{Koekoek2000}, \cite{Koorn77I}, \cite{Koorn78}, +\cite{Koorn85}, \cite{Koorn88}, \cite{Krasikov2003}, \cite{Krasikov2005}, +\cite{KuijlaarsMcLaughlin}, \cite{KwonLittle}, \cite{LabelleYehI}, \cite{LabelleYehII}, +\cite{LabelleYehIII}, \cite{Lee97I}, \cite{Lee97II}, \cite{Lee97III}, \cite{Lesky95II}, +\cite{Lesky96}, \cite{LewanowiczI}, \cite{LopezTemme2004}, \cite{Mathai}, \cite{Meixner}, +\cite{Nikiforov+}, \cite{NikiforovUvarov}, \cite{Olver}, \cite{PerezPinar}, \cite{Pittaluga}, +\cite{Rainville}, \cite{SainteViennot}, \cite{Srivastava66}, \cite{Srivastava69I}, +\cite{Srivastava69II}, \cite{Srivastava70}, \cite{Srivastava71}, \cite{Szego75}, \cite{Temme}, +\cite{Trickovic}, \cite{Trivedi}, \cite{Viennot}. + + +\section{Bessel}\index{Bessel polynomials} + +\par\setcounter{equation}{0} + +\subsection*{Hypergeometric representation} +\begin{eqnarray} +\label{DefBessel} +y_n(x;a)&=&\hyp{2}{0}{-n,n+a+1}{-}{-\frac{x}{2}}\\ +&=&(n+a+1)_n\left(\frac{x}{2}\right)^n\,\hyp{1}{1}{-n}{-2n-a}{\frac{2}{x}}, +\quad n=0,1,2,\ldots,N.\nonumber +\end{eqnarray} + +\subsection*{Orthogonality relation} +\begin{eqnarray} +\label{OrtBessel} +& &\int_0^{\infty}x^a\e^{-\frac{2}{x}}y_m(x;a)y_n(x;a)\,dx\nonumber\\ +& &=-\frac{2^{a+1}}{2n+a+1}\Gamma(-n-a)n!\,\delta_{mn},\quad a<-2N-1. +\end{eqnarray} + +\subsection*{Recurrence relation} +\begin{eqnarray} +\label{RecBessel} +& &2(n+a+1)(2n+a)y_{n+1}(x;a)\nonumber\\ +& &{}=(2n+a+1)\left[2a+(2n+a)(2n+a+2)x\right]y_n(x;a)\nonumber\\ +& &{}\mathindent{}+2n(2n+a+2)y_{n-1}(x;a). +\end{eqnarray} + +\subsection*{Normalized recurrence relation} +\begin{eqnarray} +\label{NormRecBessel} +xp_n(x)&=&p_{n+1}(x)-\frac{2a}{(2n+a)(2n+a+2)}p_n(x)\nonumber\\ +& &{}\mathindent{}-\frac{4n(n+a)}{(2n+a-1)(2n+a)^2(2n+a+1)}p_{n-1}(x), +\end{eqnarray} +where +$$y_n(x;a)=\frac{(n+a+1)_n}{2^n}p_n(x).$$ + +\subsection*{Differential equation} +\begin{equation} +\label{dvBessel} +x^2y''(x)+\left[(a+2)x+2\right]y'(x)-n(n+a+1)y(x)=0,\quad y(x)=y_n(x;a). +\end{equation} + +\subsection*{Forward shift operator} +\begin{equation} +\label{shift1Bessel} +\frac{d}{dx}y_n(x;a)=\frac{n(n+a+1)}{2}y_{n-1}(x;a+2). +\end{equation} + +\subsection*{Backward shift operator} +\begin{equation} +\label{shift2BesselI} +x^2\frac{d}{dx}y_n(x;a)+(ax+2)y_n(x;a)=2y_{n+1}(x;a-2) +\end{equation} +or equivalently +\begin{equation} +\label{shift2BesselII} +\frac{d}{dx}\left[x^a\expe^{-\frac{2}{x}}y_n(x;a)\right]=2x^{a-2}\expe^{-\frac{2}{x}}y_{n+1}(x;a-2). +\end{equation} + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodBessel} +y_n(x;a)=2^{-n}x^{-a}\expe^{\frac{2}{x}}D^n\left(x^{2n+a}\expe^{-\frac{2}{x}}\right). +\end{equation} + +\subsection*{Generating function} +\begin{equation} +\label{GenBessel} +\left(1-2xt\right)^{-\frac{1}{2}}\left(\frac{2}{1+\sqrt{1-2xt}}\right)^a +\exp\left(\frac{2t}{1+\sqrt{1-2xt}}\right)=\sum_{n=0}^{\infty}y_n(x;a)\frac{t^n}{n!}. +\end{equation} + +\subsection*{Limit relation} + +\subsubsection*{Jacobi $\rightarrow$ Bessel} +If we take $\beta=a-\alpha$ in the definition (\ref{DefJacobi}) of the Jacobi polynomials +and let $\alpha\rightarrow -\infty$ we find the Bessel polynomials: +$$\lim_{\alpha\rightarrow -\infty} +\frac{P_n^{(\alpha,a-\alpha)}(1+\alpha x)}{P_n^{(\alpha,a-\alpha)}(1)}=y_n(x;a).$$ + +\subsection*{Remarks} +The following notations are also used for the Bessel polynomials: +$$y_n(x;a,b)=y_n(2b^{-1}x;a)\quad\textrm{and}\quad\theta_n(x;a,b)=x^ny_n(x^{-1};a,b).$$ +However, the Bessel polynomials essentially depend on only one parameter. + +\noindent +The Bessel polynomials are related to the Laguerre polynomials given by (\ref{DefLaguerre}) +in the following way: +$$L_n^{(\alpha)}(x)=\frac{(-x)^n}{n!}y_n(2x^{-1};-2n-\alpha-1).$$ + +\noindent +The special case $a=-2N-2$ of the Bessel polynomials can be obtained from the pseudo Jacobi +polynomials by setting $x\rightarrow\nu x$ in the definition (\ref{DefPseudoJacobi}) of the +pseudo Jacobi polynomials and taking the limit $\nu\rightarrow\infty$ in the following way: +$$\lim\limits_{\nu\rightarrow\infty}\frac{P_n(\nu x;\nu,N)}{\nu^n} +=\frac{2^n}{(n-2N-1)_n}y_n(x;-2N-2).$$ + +\subsection*{References} +\cite{Andrade}, \cite{BergVignat}, \cite{Carlitz57I}, \cite{Dattoli2003}, +\cite{DohaAhmed2004}, \cite{DohaAhmed2006}, \cite{Grosswald}, \cite{Ismail2005II}, +\cite{KrallFrink}, \cite{Lesky98}, \cite{NikiforovUvarov}. + + +\section{Charlier}\index{Charlier polynomials} + +\par\setcounter{equation}{0} + +\subsection*{Hypergeometric representation} +\begin{equation} +\label{DefCharlier} +C_n(x;a)=\hyp{2}{0}{-n,-x}{-}{-\frac{1}{a}}. +\end{equation} + +\subsection*{Orthogonality relation} +\begin{equation} +\label{OrtCharlier} +\sum_{x=0}^{\infty}\frac{a^x}{x!}C_m(x;a)C_n(x;a)= +a^{-n}\expe^an!\,\delta_{mn},\quad a > 0. +\end{equation} + +\subsection*{Recurrence relation} +\begin{equation} +\label{RecCharlier} +-xC_n(x;a)=aC_{n+1}(x;a)-(n+a)C_n(x;a)+nC_{n-1}(x;a). +\end{equation} + +\subsection*{Normalized recurrence relation} +\begin{equation} +\label{NormRecCharlier} +xp_n(x)=p_{n+1}(x)+(n+a)p_n(x)+nap_{n-1}(x), +\end{equation} +where +$$C_n(x;a)=\left(-\frac{1}{a}\right)^np_n(x).$$ + +\subsection*{Difference equation} +\begin{equation} +\label{dvCharlier} +-ny(x)=ay(x+1)-(x+a)y(x)+xy(x-1),\quad y(x)=C_n(x;a). +\end{equation} + +\subsection*{Forward shift operator} +\begin{equation} +\label{shift1CharlierI} +C_n(x+1;a)-C_n(x;a)=-\frac{n}{a}C_{n-1}(x;a) +\end{equation} +or equivalently +\begin{equation} +\label{shift1CharlierII} +\Delta C_n(x;a)=-\frac{n}{a}C_{n-1}(x;a). +\end{equation} + +\subsection*{Backward shift operator} +\begin{equation} +\label{shift2CharlierI} +C_n(x;a)-\frac{x}{a}C_n(x-1;a)=C_{n+1}(x;a) +\end{equation} +or equivalently +\begin{equation} +\label{shift2CharlierII} +\nabla\left[\frac{a^x}{x!}C_n(x;a)\right]=\frac{a^x}{x!}C_{n+1}(x;a). +\end{equation} + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodCharlier} +\frac{a^x}{x!}C_n(x;a)=\nabla^n\left[\frac{a^x}{x!}\right]. +\end{equation} + +\subsection*{Generating function} +\begin{equation} +\label{GenCharlier} +\e^t\left(1-\frac{t}{a}\right)^x=\sum_{n=0}^{\infty}\frac{C_n(x;a)}{n!}t^n. +\end{equation} + +\subsection*{Limit relations} + +\subsubsection*{Meixner $\rightarrow$ Charlier} +If we take $c=(a+\beta)^{-1}a$ in the definition (\ref{DefMeixner}) of the Meixner polynomials +and let $\beta\rightarrow\infty$ we find the Charlier polynomials: +$$\lim_{\beta\rightarrow\infty}M_n(x;\beta,(a+\beta)^{-1}a)=C_n(x;a).$$ + +\subsubsection*{Krawtchouk $\rightarrow$ Charlier} +The Charlier polynomials can be found from the Krawtchouk polynomials given by +(\ref{DefKrawtchouk}) by taking $p=N^{-1}a$ and letting $N\rightarrow\infty$: +$$\lim_{N\rightarrow\infty}K_n(x;N^{-1}a,N)=C_n(x;a).$$ + +\subsubsection*{Charlier $\rightarrow$ Hermite} +The Hermite polynomials given by (\ref{DefHermite}) are obtained from the Charlier polynomials +if we set $x\rightarrow (2a)^{1/2}x+a$ and let $a\rightarrow\infty$. In fact we have +\begin{equation} +\lim_{a\rightarrow\infty} +(2a)^{\frac{1}{2}n}C_n((2a)^{\frac{1}{2}}x+a;a)=(-1)^nH_n(x). +\end{equation} + +\subsection*{Remark} +The Charlier polynomials are related to the Laguerre polynomials given by (\ref{DefLaguerre}) +in the following way: +$$\frac{(-a)^n}{n!}C_n(x;a)=L_n^{(x-n)}(a).$$ + +\subsection*{References} +\label{sec9.12} +\paragraph{Notation} +Here the Laguerre polynomial is denoted by $L_n^\al$ instead of +$L_n^{(\al)}$. +% +\paragraph{Hypergeometric representation} +\begin{align} +L_n^\al(x)&= +\frac{(\al+1)_n}{n!}\,\hyp11{-n}{\al+1}x +\label{182}\\ +&=\frac{(-x)^n}{n!} \hyp20{-n,-n-\al}-{-\,\frac1x} +\label{183}\\ +&=\frac{(-x)^n}{n!}\,C_n(n+\al;x), +\label{184} +\end{align} +where $C_n$ in \eqref{184} is a +\hyperref[sec9.14]{Charlier polynomial}. +Formula \eqref{182} is (9.12.1). Then \eqref{183} follows by reversal +of summation. Finally \eqref{184} follows by \eqref{183} and \eqref{179}. +It is also the remark on top of p.244 in \mycite{KLS}, and it is essentially +\myciteKLS{416}{(2.7.10)}. +% +\paragraph{Uniqueness of orthogonality measure} +The coefficient of $p_{n-1}(x)$ in (9.12.4) behaves as $O(n^2)$ as $n\to\iy$. +Hence \eqref{93} holds, by which the orthogonality measure is unique. +% +\paragraph{Special value} +\begin{equation} +L_n^{\al}(0)=\frac{(\al+1)_n}{n!}\,. +\label{53} +\end{equation} +Use (9.12.1) or see \mycite{DLMF}{18.6.1)}. +% +\paragraph{Quadratic transformations} +\begin{align} +H_{2n}(x)&=(-1)^n\,2^{2n}\,n!\,L_n^{-1/2}(x^2), +\label{54}\\ +H_{2n+1}(x)&=(-1)^n\,2^{2n+1}\,n!\,x\,L_n^{1/2}(x^2). +\label{55} +\end{align} +See p.244, Remarks, last two formulas. +Or see \mycite{DLMF}{(18.7.19), (18.7.20)}. +% +\paragraph{Fourier transform} +\begin{equation} +\frac1{\Ga(\al+1)}\,\int_0^\iy \frac{L_n^\al(y)}{L_n^\al(0)}\, +e^{-y}\,y^\al\,e^{ixy}\,dy= +i^n\,\frac{y^n}{(iy+1)^{n+\al+1}}\,, +\label{14} +\end{equation} +see \mycite{DLMF}{(18.17.34)}. +% +\paragraph{Differentiation formulas} +Each differentiation formula is given in two equivalent forms. +\begin{equation} +\frac d{dx}\left(x^\al L_n^\al(x)\right)= +(n+\al)\,x^{\al-1} L_n^{\al-1}(x),\qquad +\left(x\frac d{dx}+\al\right)L_n^\al(x)= +(n+\al)\,L_n^{\al-1}(x). +\label{76} +\end{equation} +% +\begin{equation} +\frac d{dx}\left(e^{-x} L_n^\al(x)\right)= +-e^{-x} L_n^{\al+1}(x),\qquad +\left(\frac d{dx}-1\right)L_n^\al(x)= +-L_n^{\al+1}(x). +\label{77} +\end{equation} +% +Formulas \eqref{76} and \eqref{77} follow from +\mycite{DLMF}{(13.3.18), (13.3.20)} +together with (9.12.1). +% +\paragraph{Generalized Hermite polynomials} +See \myciteKLS{146}{p.156}, \cite[Section 1.5.1]{K26}. +These are defined by +\begin{equation} +H_{2m}^\mu(x):=\const L_m^{\mu-\half}(x^2),\qquad +H_{2m+1}^\mu(x):=\const x\,L_m^{\mu+\half}(x^2). +\label{78} +\end{equation} +Then for $\mu>-\thalf$ we have orthogonality relation +\begin{equation} +\int_{-\iy}^{\iy} H_m^\mu(x)\,H_n^\mu(x)\,|x|^{2\mu}e^{-x^2}\,dx +=0\qquad(m\ne n). +\label{79} +\end{equation} +Let the Dunkl operator $T_\mu$ be defined by \eqref{72}. +If we choose the constants in \eqref{78} as +\begin{equation} +H_{2m}^\mu(x)=\frac{(-1)^m(2m)!}{(\mu+\thalf)_m}\,L_m^{\mu-\half}(x^2),\qquad +H_{2m+1}^\mu(x)=\frac{(-1)^m(2m+1)!}{(\mu+\thalf)_{m+1}}\, + x\,L_m^{\mu+\half}(x^2) + \label{80} +\end{equation} +then (see \cite[(1.6)]{K5}) +\begin{equation} +T_\mu H_n^\mu=2n\,H_{n-1}^\mu. +\label{81} +\end{equation} +Formula \eqref{81} with \eqref{80} substituted gives rise to two +differentiation formulas involving Laguerre polynomials which are equivalent to +(9.12.6) and \eqref{76}. + +Composition of \eqref{81} with itself gives +\[ +T_\mu^2 H_n^\mu=4n(n-1)\,H_{n-2}^\mu, +\] +which is equivalent to the composition of (9.12.6) and \eqref{76}: +\begin{equation} +\left(\frac{d^2}{dx^2}+\frac{2\al+1}x\,\frac d{dx}\right)L_n^\al(x^2) +=-4(n+\al)\,L_{n-1}^\al(x^2). +\label{82} +\end{equation} +% +\cite{Allaway76}, \cite{NAlSalam66}, \cite{AlSalam90}, \cite{AlSalamChihara76}, +\cite{AlSalamIsmail76}, \cite{AndrewsAskey85}, \cite{Area+II}, \cite{Askey75}, +\cite{AskeyGasper77}, \cite{AskeyWilson85}, \cite{AtakRahmanSuslov}, \cite{Bavinck98}, +\cite{BavinckKoekoek}, \cite{Chihara78}, \cite{Chihara79}, \cite{Dunkl76}, \cite{Erdelyi+}, +\cite{Gasper73I}, \cite{Gasper74}, \cite{Goh}, \cite{HoareRahman}, \cite{IsmailLetVal88}, +\cite{Koekoek2000}, \cite{Koorn88}, \cite{Krasikov2002}, \cite{LabelleYehI}, +\cite{LabelleYehII}, \cite{LabelleYehIII}, \cite{Lesky62}, \cite{Lesky89}, \cite{Lesky94I}, +\cite{Lesky95II}, \cite{LewanowiczII}, \cite{LopezTemme2004}, \cite{Meixner}, \cite{Nikiforov+}, +\cite{NikiforovUvarov}, \cite{Szafraniec}, \cite{Szego75}, \cite{Viennot}, \cite{Zarzo+}, +\cite{Zeng90}. + + +\section{Hermite}\index{Hermite polynomials} + +\par\setcounter{equation}{0} + +\subsection*{Hypergeometric representation} +\begin{equation} +\label{DefHermite} +H_n(x)=(2x)^n\,\hyp{2}{0}{-n/2,-(n-1)/2}{-}{-\frac{1}{x^2}}. +\end{equation} + +\subsection*{Orthogonality relation} +\begin{equation} +\label{OrtHermite} +\frac{1}{\sqrt{\cpi}}\int_{-\infty}^{\infty}\expe^{-x^2}H_m(x)H_n(x)\,dx +=2^nn!\,\delta_{mn}. +\end{equation} + +\subsection*{Recurrence relation} +\begin{equation} +\label{RecHermite} +H_{n+1}(x)-2xH_n(x)+2nH_{n-1}(x)=0. +\end{equation} + +\subsection*{Normalized recurrence relation} +\begin{equation} +\label{NormRecHermite} +xp_n(x)=p_{n+1}(x)+\frac{n}{2}p_{n-1}(x), +\end{equation} +where +$$H_n(x)=2^np_n(x).$$ + +\subsection*{Differential equation} +\begin{equation} +\label{dvHermite} +y''(x)-2xy'(x)+2ny(x)=0,\quad y(x)=H_n(x). +\end{equation} + +\subsection*{Forward shift operator} +\begin{equation} +\label{shift1Hermite} +\frac{d}{dx}H_n(x)=2nH_{n-1}(x). +\end{equation} + +\subsection*{Backward shift operator} +\begin{equation} +\label{shift2HermiteI} +\frac{d}{dx}H_n(x)-2xH_n(x)=-H_{n+1}(x) +\end{equation} +or equivalently +\begin{equation} +\label{shift2HermiteII} +\frac{d}{dx}\left[\expe^{-x^2}H_n(x)\right]=-\expe^{-x^2}H_{n+1}(x). +\end{equation} + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodHermite} +\e^{-x^2}H_n(x)=(-1)^n\left(\frac{d}{dx}\right)^n\left[\expe^{-x^2}\right]. +\end{equation} + +\subsection*{Generating functions} +\begin{equation} +\label{GenHermite1} +\exp\left(2xt-t^2\right)=\sum_{n=0}^{\infty}\frac{H_n(x)}{n!}t^n. +\end{equation} + +\begin{equation} +\label{GenHermite2} +\left\{\begin{array}{l} +\displaystyle\expe^t\cos(2x\sqrt{t})=\sum_{n=0}^{\infty} +\frac{(-1)^n}{(2n)!}H_{2n}(x)t^n\\[5mm] +\displaystyle\frac{\expe^t}{\sqrt{t}}\sin(2x\sqrt{t})=\sum_{n=0}^{\infty} +\frac{(-1)^n}{(2n+1)!}H_{2n+1}(x)t^n. +\end{array}\right. +\end{equation} + +\begin{equation} +\label{GenHermite3} +\left\{\begin{array}{l} +\displaystyle \expe^{-t^2}\cosh(2xt)=\sum_{n=0}^{\infty} +\frac{H_{2n}(x)}{(2n)!}t^{2n}\\[5mm] +\displaystyle\expe^{-t^2}\sinh(2xt)=\sum_{n=0}^{\infty} +\frac{H_{2n+1}(x)}{(2n+1)!}t^{2n+1}. +\end{array}\right. +\end{equation} + +\begin{equation} +\label{GenHermite4} +\left\{\begin{array}{l} +\displaystyle (1+t^2)^{-\gamma}\,\hyp{1}{1}{\gamma}{\frac{1}{2}}{\frac{x^2t^2}{1+t^2}}= +\sum_{n=0}^{\infty}\frac{(\gamma)_n}{(2n)!}H_{2n}(x)t^{2n}\\[5mm] +\displaystyle\frac{xt}{\sqrt{1+t^2}}\, +\hyp{1}{1}{\gamma+\frac{1}{2}}{\frac{3}{2}}{\frac{x^2t^2}{1+t^2}} +=\sum_{n=0}^{\infty}\frac{(\gamma+\frac{1}{2})_n}{(2n+1)!}H_{2n+1}(x)t^{2n+1} +\end{array}\right. +\end{equation} +with $\gamma$ arbitrary. + +\begin{equation} +\label{GenHermite5} +\frac{1+2xt+4t^2}{(1+4t^2)^{\frac{3}{2}}}\exp\left(\frac{4x^2t^2}{1+4t^2}\right) +=\sum_{n=0}^{\infty}\frac{H_n(x)}{\lfloor n/2\rfloor\,!}t^n, +\end{equation} +where $\lfloor n/2\rfloor$ denotes the largest integer smaller than or equal to $n/2$. + +\subsection*{Limit relations} + +\subsubsection*{Meixner-Pollaczek $\rightarrow$ Hermite} +If we take $x\rightarrow (\sin\phi)^{-1}(x\sqrt{\lambda}-\lambda\cos\phi)$ +in the definition (\ref{DefMP}) of the Meixner-Pollaczek polynomials and +then let $\lambda\rightarrow\infty$ we obtain the Hermite polynomials: +$$\lim_{\lambda\rightarrow\infty} +\lambda^{-\frac{1}{2}n}P_n^{(\lambda)} +((\sin\phi)^{-1}(x\sqrt{\lambda}-\lambda\cos\phi);\phi)=\frac{H_n(x)}{n!}.$$ + +\subsubsection*{Jacobi $\rightarrow$ Hermite} +The Hermite polynomials follow from the Jacobi polynomials given by (\ref{DefJacobi}) by +taking $\beta=\alpha$ and letting $\alpha\rightarrow\infty$ in the following way: +$$\lim_{\alpha\rightarrow\infty} +\alpha^{-\frac{1}{2}n}P_n^{(\alpha,\alpha)}(\alpha^{-\frac{1}{2}}x)=\frac{H_n(x)}{2^nn!}.$$ + +\subsubsection*{Gegenbauer / Ultraspherical $\rightarrow$ Hermite} +The Hermite polynomials follow from the Gegenbauer (or ultraspherical) polynomials given by +(\ref{DefGegenbauer}) by taking $\lambda=\alpha+\frac{1}{2}$ and letting +$\alpha\rightarrow\infty$ in the following way: +$$\lim_{\alpha\rightarrow\infty} +\alpha^{-\frac{1}{2}n}C_n^{(\alpha+\frac{1}{2})}(\alpha^{-\frac{1}{2}}x)=\frac{H_n(x)}{n!}.$$ + +\subsubsection*{Krawtchouk $\rightarrow$ Hermite} +The Hermite polynomials follow from the Krawtchouk polynomials given by (\ref{DefKrawtchouk}) +by setting $x\rightarrow pN+x\sqrt{2p(1-p)N}$ and then letting $N\rightarrow\infty$: +$$\lim_{N\rightarrow\infty} +\sqrt{\binom{N}{n}}K_n(pN+x\sqrt{2p(1-p)N};p,N) +=\frac{\displaystyle (-1)^nH_n(x)}{\displaystyle\sqrt{2^nn!\left(\frac{p}{1-p}\right)^n}}.$$ + +\subsubsection*{Laguerre $\rightarrow$ Hermite} +The Hermite polynomials can be obtained from the Laguerre polynomials given by +(\ref{DefLaguerre}) by taking the limit $\alpha\rightarrow\infty$ in the following way: +$$\lim_{\alpha\rightarrow\infty} +\left(\frac{2}{\alpha}\right)^{\frac{1}{2}n} +L_n^{(\alpha)}((2\alpha)^{\frac{1}{2}}x+\alpha)=\frac{(-1)^n}{n!}H_n(x).$$ + +\subsubsection*{Charlier $\rightarrow$ Hermite} +If we set $x\rightarrow (2a)^{1/2}x+a$ in the definition (\ref{DefCharlier}) +of the Charlier polynomials and let $a\rightarrow\infty$ we find the Hermite +polynomials. In fact we have +$$\lim_{a\rightarrow\infty} +(2a)^{\frac{1}{2}n}C_n((2a)^{\frac{1}{2}}x+a;a)=(-1)^nH_n(x).$$ + +\subsection*{Remarks} +The Hermite polynomials can also be written as: +$$\frac{H_n(x)}{n!}=\sum_{k=0}^{\lfloor n/2\rfloor} +\frac{(-1)^k(2x)^{n-2k}}{k!\,(n-2k)!},$$ +where $\lfloor n/2\rfloor$ denotes the largest integer smaller than or equal to $n/2$. + +\noindent +The Hermite polynomials and the Laguerre polynomials given by (\ref{DefLaguerre}) are also +connected by the following quadratic transformations: +$$H_{2n}(x)=(-1)^nn!\,2^{2n}L_n^{(-\frac{1}{2})}(x^2)$$ +and +$$H_{2n+1}(x)=(-1)^nn!\,2^{2n+1}xL_n^{(\frac{1}{2})}(x^2).$$ + +\subsection*{References} +\label{sec9.14} +% +\paragraph{Hypergeometric representation} +\begin{align} +C_n(x;a)&=\hyp20{-n,-x}-{-\,\frac1a} +\label{179}\\ +&=\frac{(-x)_n}{a^n} \hyp11{-n}{x-n+1}a +\label{180}\\ +&=\frac{n!}{(-a)^n}\,L_n^{x-n}(a), +\label{181} +\end{align} +where $L_n^\al(x)$ is a +\hyperref[sec9.12]{Laguerre polynomial}. +Formula \eqref{179} is (9.14.1). Then \eqref{180} follows by reversal +of the summation. Finally \eqref{181} follows by \eqref{180} and +(9.12.1). It is also the Remark on p.249 of \mycite{KLS}, and it +was earlier given in \myciteKLS{416}{(2.7.10)}. +% +\paragraph{Uniqueness of orthogonality measure} +The coefficient of $p_{n-1}(x)$ in (9.14.4) behaves as $O(n)$ as $n\to\iy$. +Hence \eqref{93} holds, by which the orthogonality measure is unique. +% +\cite{Abram}, \cite{NAlSalam66}, \cite{AlSalam90}, \cite{AlSalamChihara72}, +\cite{AlSalamChihara76}, \cite{AndrewsAskey85}, \cite{AndrewsAskeyRoy}, \cite{Area+I}, +\cite{Askey68}, \cite{Askey75}, \cite{Askey89I}, \cite{AskeyGasper76}, \cite{AskeyWilson85}, +\cite{Azor}, \cite{Berg}, \cite{BilodeauII}, \cite{Brafman51}, \cite{Brafman57II}, +\cite{Brenke}, \cite{CarlitzSrivastava}, \cite{ChenSrivastava}, \cite{Chihara78}, +\cite{Cohen}, \cite{Danese}, \cite{DetteStudden92}, \cite{DetteStudden95}, \cite{Doha2004I}, +\cite{Erdelyi+}, \cite{Faldey}, \cite{Gawronski87}, \cite{Gawronski93}, \cite{Gillis+}, +\cite{Godoy+}, \cite{Grad}, \cite{Ismail74}, \cite{Ismail2005II}, \cite{IsmailStanton97}, +\cite{Koekoek2000}, \cite{Koorn88}, \cite{Krasikov2004}, \cite{Kwon+}, \cite{LabelleYehIII}, +\cite{Lesky95II}, \cite{Lesky96}, \cite{LewanowiczI}, \cite{LopezTemme99}, \cite{Mathai}, +\cite{Meixner}, \cite{Nikiforov+}, \cite{NikiforovUvarov}, \cite{Olver}, \cite{Pittaluga}, +\cite{Rainville}, \cite{SainteViennot}, \cite{Srivastava71}, \cite{SrivastavaMathur}, +\cite{Szego75}, \cite{Temme}, \cite{TemmeLopez2000}, \cite{Trickovic}, \cite{Viennot}, +\cite{Weisner59}, \cite{Wyman}. + +\end{document} diff --git a/KLSadd_insertion/tempchap14.tex b/KLSadd_insertion/tempchap14.tex new file mode 100644 index 0000000..0b5f396 --- /dev/null +++ b/KLSadd_insertion/tempchap14.tex @@ -0,0 +1,5734 @@ +\documentclass[envcountchap,graybox]{svmono} + +\addtolength{\textwidth}{1mm} + +\usepackage{amsmath,amssymb} + +\usepackage{amsfonts} +%\usepackage{breqn} +\usepackage{DLMFmath} +\usepackage{DRMFfcns} + +\usepackage{mathptmx} +\usepackage{helvet} +\usepackage{courier} + +\usepackage{makeidx} +\usepackage{graphicx} + +\usepackage{multicol} +\usepackage[bottom]{footmisc} + +\makeindex + +\def\bibname{Bibliography} +\def\refname{Bibliography} + +\def\theequation{\thesection.\arabic{equation}} + +\smartqed + +\let\corollary=\undefined +\spnewtheorem{corollary}[theorem]{Corollary}{\bfseries}{\itshape} + +\newcounter{rom} + +\newcommand{\hyp}[5]{\mbox{}_{#1}{F}_{#2} +\left(\genfrac{}{}{0pt}{}{#3}{#4}\,;\,#5\right)} + +\newcommand{\qhyp}[5]{\mbox{}_{#1}{\phi}_{#2} +\left(\genfrac{}{}{0pt}{}{#3}{#4}\,;\,q,\,#5\right)} + +\newcommand{\qhypK}[5]{\,\mbox{}_{#1}\phi_{#2}\!\left( + \genfrac{}{}{0pt}{}{#3}{#4};#5\right)} + +\newcommand{\mathindent}{\hspace{7.5mm}} + +\newcommand{\e}{\textrm{e}} + +\renewcommand{\E}{\textrm{E}} + +\renewcommand{\textfraction}{-1} + +\renewcommand{\Gamma}{\varGamma} + +\renewcommand{\leftlegendglue}{\hfil} + +\settowidth{\tocchpnum}{14\enspace} +\settowidth{\tocsecnum}{14.30\enspace} +\settowidth{\tocsubsecnum}{14.12.1\enspace} + +\makeatletter +\def\cleartoversopage{\clearpage\if@twoside\ifodd\c@page + \hbox{}\newpage\if@twocolumn\hbox{}\newpage\fi + \else\fi\fi} + +\newcommand{\clearemptyversopage}{ + \clearpage{\pagestyle{empty}\cleartoversopage}} +\makeatother + +\oddsidemargin -1.5cm +\topmargin -2.0cm +\textwidth 16.3cm +\textheight 25cm + +\begin{document} + +\author{Roelof Koekoek\\[2.5mm]Peter A. Lesky\\[2.5mm]Ren\'e F. Swarttouw} +\title{Hypergeometric orthogonal polynomials and their\\$q$-analogues} +\subtitle{-- Monograph --} +\maketitle + +\frontmatter + +\large + +\pagenumbering{roman} +\addtocounter{chapter}{13} + +\chapter{Basic hypergeometric orthogonal polynomials} +\label{BasicHyperOrtPol} + + +In this chapter we deal with all families of basic hypergeometric orthogonal polynomials +appearing in the $q$-analogue of the Askey scheme on the pages~\pageref{qscheme1} and +\pageref{qscheme2}. For each family of orthogonal polynomials we state the most important +properties such as a representation as a basic hypergeometric function, orthogonality +relation(s), the three-term recurrence relation, the second-order $q$-difference equation, +the forward shift (or degree lowering) and backward shift (or degree raising) operator, a +Rodrigues-type formula and some generating functions. Throughout this chapter we assume +that $01$ and $b,c,d$ are real or one is real and the other two are complex conjugates,\\ +$\max(|b|,|c|,|d|)<1$ and the pairwise products of $a,b,c$ and $d$ have +absolute value less than $1$, then we have another orthogonality relation +given by: +\begin{eqnarray} +\label{OrtAskeyWilsonII} +& &\frac{1}{2\pi}\int_{-1}^1\frac{w(x)}{\sqrt{1-x^2}}p_m(x;a,b,c,d|q)p_n(x;a,b,c,d|q)\,dx\nonumber\\ +& &{}\mathindent{}+\sum_{\begin{array}{c}\scriptstyle k\\ \scriptstyle 11$ and $b$ and $c$ are real or complex conjugates, +$\max(|b|,|c|)<1$ and the pairwise products of $a,b$ and $c$ have +absolute value less than $1$, then we have another orthogonality relation +given by: +\begin{eqnarray} +\label{OrtContDualqHahnII} +& &\frac{1}{2\pi}\int_{-1}^1\frac{w(x)}{\sqrt{1-x^2}}p_m(x;a,b,c|q)p_n(x;a,b,c|q)\,dx\nonumber\\ +& &{}\mathindent{}+\sum_{\begin{array}{c}\scriptstyle k\\ \scriptstyle 10$) in the definition (\ref{DefBigqJacobi}) +of the big $q$-Jacobi polynomials and take the limit $c\rightarrow -\infty$ we +obtain the $q$-Meixner polynomials given by (\ref{DefqMeixner}): +\begin{equation} +\lim_{c\rightarrow -\infty}P_n(q^{-x};a,-a^{-1}cd^{-1},c;q)=M_n(q^{-x};a,d;q). +\end{equation} + +\subsubsection*{Big $q$-Jacobi $\rightarrow$ Jacobi} +If we set $c=0$, $a=q^{\alpha}$ and $b=q^{\beta}$ in the definition (\ref{DefBigqJacobi}) +of the big $q$-Jacobi polynomials and let $q\rightarrow 1$ we find the Jacobi +polynomials given by (\ref{DefJacobi}): +\begin{equation} +\lim_{q\rightarrow 1}P_n(x;q^{\alpha},q^{\beta},0;q)=\frac{P_n^{(\alpha,\beta)}(2x-1)}{P_n^{(\alpha,\beta)}(1)}. +\end{equation} +If we take $c=-q^{\gamma}$ for arbitrary real $\gamma$ instead of $c=0$ we +find +\begin{equation} +\lim_{q\rightarrow 1}P_n(x;q^{\alpha},q^{\beta},-q^{\gamma};q)=\frac{P_n^{(\alpha,\beta)}(x)}{P_n^{(\alpha,\beta)}(1)}. +\end{equation} + +\subsection*{Remarks} +The big $q$-Jacobi polynomials with $c=0$ and the little +$q$-Jacobi polynomials given by (\ref{DefLittleqJacobi}) are related +in the following way: +$$P_n(x;a,b,0;q)=\frac{(bq;q)_n}{(aq;q)_n}(-1)^na^nq^{n+\binom{n}{2}}p_n(a^{-1}q^{-1}x;b,a|q).$$ + +\noindent +Sometimes the big $q$-Jacobi polynomials are defined in terms of four +parameters instead of three. In fact the polynomials given by the definition +$$P_n(x;a,b,c,d;q)=\qhyp{3}{2}{q^{-n},abq^{n+1},ac^{-1}qx}{aq,-ac^{-1}dq}{q}$$ +are orthogonal on the interval $[-d,c]$ with respect to the weight function +$$\frac{(c^{-1}qx,-d^{-1}qx;q)_{\infty}}{(ac^{-1}qx,-bd^{-1}qx;q)_{\infty}}d_qx.$$ +These polynomials are not really different from those given by +(\ref{DefBigqJacobi}) since we have +$$P_n(x;a,b,c,d;q)=P_n(ac^{-1}qx;a,b,-ac^{-1}d;q)$$ +and +$$P_n(x;a,b,c;q)=P_n(x;a,b,aq,-cq;q).$$ + +\subsection*{References} +\cite{NAlSalam89}, \cite{AlSalam90}, \cite{AndrewsAskey85}, \cite{AtakKlimyk2004}, +\cite{AtakRahmanSuslov}, \cite{DattaGriffin}, \cite{FloreaniniVinetII}, +\cite{GasperRahman90}, \cite{GrunbaumHaine96}, \cite{Gupta92}, \cite{Hahn}, +\cite{Ismail86I}, \cite{IsmailStanton97}, \cite{IsmailWilson}, \cite{KalninsMiller88}, +\cite{KoelinkE}, \cite{Koorn90II}, \cite{Koorn93}, \cite{Koorn2007}, \cite{Miller89}, +\cite{Nikiforov+}, \cite{NoumiMimachi90II}, \cite{NoumiMimachi90III}, +\cite{NoumiMimachi91}, \cite{Spiridonov97}, \cite{SrivastavaJain90}. + + +\section*{Special case} + +\subsection{Big $q$-Legendre} +\index{Big q-Legendre polynomials@Big $q$-Legendre polynomials} +\index{q-Legendre polynomials@$q$-Legendre polynomials!Big} +\par + +\subsection*{Basic hypergeometric representation} The big $q$-Legendre polynomials are big $q$-Jacobi polynomials +with $a=b=1$: +\begin{equation} +\label{DefBigqLegendre} +P_n(x;c;q)=\qhyp{3}{2}{q^{-n},q^{n+1},x}{q,cq}{q}. +\end{equation} + +\subsection*{Orthogonality relation} +\begin{eqnarray} +\label{OrtBigqLegendre} +& &\int_{cq}^{q}P_m(x;c;q)P_n(x;c;q)\,d_qx\nonumber\\ +& &{}=q(1-c)\frac{(1-q)}{(1-q^{2n+1})} +\frac{(c^{-1}q;q)_n}{(cq;q)_n}(-cq^2)^nq^{\binom{n}{2}}\,\delta_{mn},\quad c<0. +\end{eqnarray} + +\subsection*{Recurrence relation} +\begin{eqnarray} +\label{RecBigqLegendre} +(x-1)P_n(x;c;q)&=&A_nP_{n+1}(x;c;q)-\left(A_n+C_n\right)P_n(x;c;q)\nonumber\\ +& &{}\mathindent{}+C_nP_{n-1}(x;c;q), +\end{eqnarray} +where +$$\left\{\begin{array}{l} +\displaystyle A_n=\frac{(1-q^{n+1})(1-cq^{n+1})}{(1+q^{n+1})(1-q^{2n+1})}\\ +\\ +\displaystyle C_n=-cq^{n+1}\frac{(1-q^n)(1-c^{-1}q^n)}{(1+q^n)(1-q^{2n+1})}. +\end{array}\right.$$ + +\subsection*{Normalized recurrence relation} +\begin{equation} +\label{NormRecBigqLegendre} +xp_n(x)=p_{n+1}(x)+\left[1-(A_n+C_n)\right]p_n(x)+A_{n-1}C_np_{n-1}(x), +\end{equation} +where +$$P_n(x;c;q)=\frac{(q^{n+1};q)_n}{(q,cq;q)_n}p_n(x).$$ + +\subsection*{$q$-Difference equation} +\begin{eqnarray} +\label{dvBigqLegendre} +& &q^{-n}(1-q^n)(1-q^{n+1})x^2y(x)\nonumber\\ +& &{}=B(x)y(qx)-\left[B(x)+D(x)\right]y(x)+D(x)y(q^{-1}x), +\end{eqnarray} +where +$$y(x)=P_n(x;c;q)$$ +and +$$\left\{\begin{array}{l}\displaystyle B(x)=q(x-1)(x-c)\\ +\\ +\displaystyle D(x)=(x-q)(x-cq).\end{array}\right.$$ + +\subsection*{Rodrigues-type formula} +\begin{eqnarray} +\label{RodBigqLegendre} +P_n(x;c;q)&=&\frac{c^nq^{n(n+1)}(1-q)^n}{(q,cq;q)_n} +\left(\mathcal{D}_q\right)^n\left[(q^{-n}x,c^{-1}q^{-n}x;q)_n\right]\\ +&=&\frac{(1-q)^n}{(q,cq;q)_n}\left(\mathcal{D}_q\right)^n +\left[(qx^{-1},cqx^{-1};q)_nx^{2n}\right].\nonumber +\end{eqnarray} + +\subsection*{Generating functions} +\begin{equation} +\label{GenBigqLegendre1} +\qhyp{2}{1}{qx^{-1},0}{q}{xt}\,\qhyp{1}{1}{c^{-1}x}{q}{cqt} +=\sum_{n=0}^{\infty}\frac{(cq;q)_n}{(q,q;q)_n}P_n(x;c;q)t^n. +\end{equation} + +\begin{equation} +\label{GenBigqLegendre2} +\qhyp{2}{1}{cqx^{-1},0}{cq}{xt}\,\qhyp{1}{1}{c^{-1}x}{c^{-1}q}{qt} +=\sum_{n=0}^{\infty}\frac{P_n(x;c;q)}{(c^{-1}q;q)_n}t^n. +\end{equation} + +\subsection*{Limit relations} + +\subsubsection*{Big $q$-Legendre $\rightarrow$ Legendre / Spherical} +If we set $c=0$ in the definition (\ref{DefBigqLegendre}) of the big +$q$-Legendre polynomials and let $q\rightarrow 1$ we simply obtain the Legendre +(or spherical) polynomials given by (\ref{DefLegendre}): +\begin{equation} +\lim_{q\rightarrow 1}P_n(x;0;q)=P_n(2x-1). +\end{equation} +If we take $c=-q^{\gamma}$ for arbitrary real $\gamma$ instead of $c=0$ we +find +\begin{equation} +\lim_{q\rightarrow 1}P_n(x;-q^{\gamma};q)=P_n(x). +\end{equation} + +\subsection*{References} +\cite{Koelink95I}, \cite{Koorn90II}. + + +\section{$q$-Hahn}\index{q-Hahn polynomials@$q$-Hahn polynomials} +\par\setcounter{equation}{0} + +\subsection*{Basic hypergeometric representation} +\begin{equation} +\label{DefqHahn} +Q_n(q^{-x};\alpha,\beta,N|q)=\qhyp{3}{2}{q^{-n},\alpha\beta q^{n+1},q^{-x}} +{\alpha q,q^{-N}}{q},\quad n=0,1,2,\ldots,N. +\end{equation} + +\subsection*{Orthogonality relation} +For $0<\alpha q<1$ and $0<\beta q<1$, or for $\alpha>q^{-N}$ and $\beta>q^{-N}$, we have +\begin{eqnarray} +\label{OrtqHahn} +& &\sum_{x=0}^N\frac{(\alpha q,q^{-N};q)_x}{(q,\beta^{-1}q^{-N};q)_x}(\alpha\beta q)^{-x} +Q_m(q^{-x};\alpha,\beta,N|q)Q_n(q^{-x};\alpha,\beta,N|q)\nonumber\\ +& &{}=\frac{(\alpha\beta q^2;q)_N}{(\beta q;q)_N(\alpha q)^N} +\frac{(q,\alpha\beta q^{N+2},\beta q;q)_n}{(\alpha q,\alpha\beta q,q^{-N};q)_n}\, +\frac{(1-\alpha\beta q)(-\alpha q)^n}{(1-\alpha\beta q^{2n+1})} +q^{\binom{n}{2}-Nn}\,\delta_{mn}. +\end{eqnarray} + +\newpage + +\subsection*{Recurrence relation} +\begin{eqnarray} +\label{RecqHahn} +-\left(1-q^{-x}\right)Q_n(q^{-x})&=&A_nQ_{n+1}(q^{-x})-\left(A_n+C_n\right)Q_n(q^{-x})\nonumber\\ +& &{}\mathindent{}+C_nQ_{n-1}(q^{-x}), +\end{eqnarray} +where +$$Q_n(q^{-x}):=Q_n(q^{-x};\alpha,\beta,N|q)$$ +and +$$\left\{\begin{array}{l} +\displaystyle A_n=\frac{(1-q^{n-N})(1-\alpha q^{n+1})(1-\alpha\beta q^{n+1})}{(1-\alpha\beta q^{2n+1})(1-\alpha\beta q^{2n+2})}\\ +\\ +\displaystyle C_n=-\frac{\alpha q^{n-N}(1-q^n)(1-\alpha\beta q^{n+N+1})(1-\beta q^n)}{(1-\alpha\beta q^{2n})(1-\alpha\beta q^{2n+1})}. +\end{array}\right.$$ + +\subsection*{Normalized recurrence relation} +\begin{equation} +\label{NormRecqHahn} +xp_n(x)=p_{n+1}(x)+\left[1-(A_n+C_n)\right]p_n(x)+A_{n-1}C_np_{n-1}(x), +\end{equation} +where +$$Q_n(q^{-x};\alpha,\beta,N|q)= +\frac{(\alpha\beta q^{n+1};q)_n}{(\alpha q,q^{-N};q)_n}p_n(q^{-x}).$$ + +\subsection*{$q$-Difference equation} +\begin{eqnarray} +\label{dvqHahn} +& &q^{-n}(1-q^n)(1-\alpha\beta q^{n+1})y(x)\nonumber\\ +& &{}=B(x)y(x+1)-\left[B(x)+D(x)\right]y(x)+D(x)y(x-1), +\end{eqnarray} +where +$$y(x)=Q_n(q^{-x};\alpha,\beta,N|q)$$ +and +$$\left\{\begin{array}{l}\displaystyle B(x)=(1-q^{x-N})(1-\alpha q^{x+1})\\ +\\ +\displaystyle D(x)=\alpha q(1-q^x)(\beta-q^{x-N-1}).\end{array}\right.$$ + +\newpage + +\subsection*{Forward shift operator} +\begin{eqnarray} +\label{shift1qHahnI} +& &Q_n(q^{-x-1};\alpha,\beta,N|q)-Q_n(q^{-x};\alpha,\beta,N|q)\nonumber\\ +& &{}=\frac{q^{-n-x}(1-q^n)(1-\alpha\beta q^{n+1})}{(1-\alpha q)(1-q^{-N})} +Q_{n-1}(q^{-x};\alpha q,\beta q,N-1|q) +\end{eqnarray} +or equivalently +\begin{eqnarray} +\label{shift1qHahnII} +\frac{\Delta Q_n(q^{-x};\alpha,\beta,N|q)}{\Delta q^{-x}} +&=&\frac{q^{-n+1}(1-q^n)(1-\alpha\beta q^{n+1})}{(1-q)(1-\alpha q)(1-q^{-N})}\nonumber\\ +& &{}\mathindent{}\times Q_{n-1}(q^{-x};\alpha q,\beta q,N-1|q). +\end{eqnarray} + +\subsection*{Backward shift operator} +\begin{eqnarray} +\label{shift2qHahnI} +& &(1-\alpha q^x)(1-q^{x-N-1})Q_n(q^{-x};\alpha,\beta,N|q)\nonumber\\ +& &{}\mathindent{}-\alpha(1-q^x)(\beta-q^{x-N-1})Q_n(q^{-x+1};\alpha,\beta,N|q)\nonumber\\ +& &{}=q^x(1-\alpha)(1-q^{-N-1})Q_{n+1}(q^{-x};\alpha q^{-1},\beta q^{-1},N+1|q) +\end{eqnarray} +or equivalently +\begin{eqnarray} +\label{shift2qHahnII} +& &\frac{\nabla\left[w(x;\alpha,\beta,N|q)Q_n(q^{-x};\alpha,\beta,N|q)\right]}{\nabla q^{-x}}\nonumber\\ +& &{}=\frac{1}{1-q}w(x;\alpha q^{-1},\beta q^{-1},N+1|q)Q_{n+1}(q^{-x};\alpha q^{-1},\beta q^{-1},N+1|q), +\end{eqnarray} +where +$$w(x;\alpha,\beta,N|q)=\frac{(\alpha q,q^{-N};q)_x}{(q,\beta^{-1}q^{-N};q)_x}(\alpha\beta)^{-x}.$$ + +\subsection*{Rodrigues-type formula} +\begin{eqnarray} +\label{RodqHahn} +& &w(x;\alpha,\beta,N|q)Q_n(q^{-x};\alpha,\beta,N|q)\nonumber\\ +& &{}=(1-q)^n\left(\nabla_q\right)^n\left[w(x;\alpha q^n,\beta q^n,N-n|q)\right], +\end{eqnarray} +where +$$\nabla_q:=\frac{\nabla}{\nabla q^{-x}}.$$ + +\subsection*{Generating functions} For $x=0,1,2,\ldots,N$ we have +\begin{eqnarray} +\label{GenqHahn1} +& &\qhyp{1}{1}{q^{-x}}{\alpha q}{\alpha qt}\,\qhyp{2}{1}{q^{x-N},0}{\beta q}{q^{-x}t}\nonumber\\ +& &{}=\sum_{n=0}^N\frac{(q^{-N};q)_n}{(\beta q,q;q)_n}Q_n(q^{-x};\alpha,\beta,N|q)t^n. +\end{eqnarray} + +\begin{eqnarray} +\label{GenqHahn2} +& &\qhyp{2}{1}{q^{-x},\beta q^{N+1-x}}{0}{-\alpha q^{x-N+1}t}\, +\qhyp{2}{0}{q^{x-N},\alpha q^{x+1}}{-}{-q^{-x}t}\nonumber\\ +& &{}=\sum_{n=0}^N\frac{(\alpha q,q^{-N};q)_n}{(q;q)_n}q^{-\binom{n}{2}}Q_n(q^{-x};\alpha,\beta,N|q)t^n. +\end{eqnarray} + +\subsection*{Limit relations} + +\subsubsection*{$q$-Racah $\rightarrow$ $q$-Hahn} +The $q$-Hahn polynomials follow from the $q$-Racah polynomials by the substitution +$\delta=0$ and $\gamma q=q^{-N}$ in the definition (\ref{DefqRacah}) of the +$q$-Racah polynomials: +$$R_n(\mu(x);\alpha,\beta,q^{-N-1},0|q)=Q_n(q^{-x};\alpha,\beta,N|q).$$ +Another way to obtain the $q$-Hahn polynomials from the $q$-Racah +polynomials is by setting $\gamma=0$ and $\delta=\beta^{-1}q^{-N-1}$ in the definition +(\ref{DefqRacah}): +$$R_n(\mu(x);\alpha,\beta,0,\beta^{-1}q^{-N-1}|q)=Q_n(q^{-x};\alpha,\beta,N|q).$$ +And if we take $\alpha q=q^{-N}$, $\beta\rightarrow\beta\gamma q^{N+1}$ and $\delta=0$ in the +definition (\ref{DefqRacah}) of the $q$-Racah polynomials we find the +$q$-Hahn polynomials given by (\ref{DefqHahn}) in the following way: +$$R_n(\mu(x);q^{-N-1},\beta\gamma q^{N+1},\gamma,0|q)=Q_n(q^{-x};\gamma,\beta,N|q).$$ +Note that $\mu(x)=q^{-x}$ in each case. + +\subsubsection*{$q$-Hahn $\rightarrow$ Little $q$-Jacobi} +If we set $x\rightarrow N-x$ in the definition (\ref{DefqHahn}) of the $q$-Hahn +polynomials and take the limit $N\rightarrow\infty$ we find the little $q$-Jacobi polynomials: +\begin{equation} +\lim _{N\rightarrow\infty} Q_n(q^{x-N};\alpha,\beta,N|q)=p_n(q^x;\alpha,\beta|q), +\end{equation} +where $p_n(q^x;\alpha,\beta|q)$ is given by (\ref{DefLittleqJacobi}). + +\subsubsection*{$q$-Hahn $\rightarrow$ $q$-Meixner} +The $q$-Meixner polynomials given by (\ref{DefqMeixner}) can be obtained from the $q$-Hahn polynomials +by setting $\alpha=b$ and $\beta=-b^{-1}c^{-1}q^{-N-1}$ in the definition (\ref{DefqHahn}) of the +$q$-Hahn polynomials and letting $N\rightarrow\infty$: +\begin{equation} +\lim_{N\rightarrow\infty} +Q_n(q^{-x};b,-b^{-1}c^{-1}q^{-N-1},N|q)=M_n(q^{-x};b,c;q). +\end{equation} + +\subsubsection*{$q$-Hahn $\rightarrow$ Quantum $q$-Krawtchouk} +The quantum $q$-Krawtchouk polynomials given by (\ref{DefQuantumqKrawtchouk}) +simply follow from the $q$-Hahn polynomials by setting $\beta=p$ in the definition (\ref{DefqHahn}) of +the $q$-Hahn polynomials and taking the limit $\alpha\rightarrow\infty$: +\begin{equation} +\lim_{\alpha\rightarrow\infty}Q_n(q^{-x};\alpha,p,N|q)=K_n^{qtm}(q^{-x};p,N;q). +\end{equation} + +\subsubsection*{$q$-Hahn $\rightarrow$ $q$-Krawtchouk} +If we set $\beta=-\alpha^{-1}q^{-1}p$ in the definition (\ref{DefqHahn}) of the $q$-Hahn polynomials +and then let $\alpha\rightarrow 0$ we obtain the $q$-Krawtchouk polynomials given by +(\ref{DefqKrawtchouk}): +\begin{equation} +\lim_{\alpha\rightarrow 0} +Q_n(q^{-x};\alpha,-\alpha^{-1}q^{-1}p,N|q)=K_n(q^{-x};p,N;q). +\end{equation} + +\subsubsection*{$q$-Hahn $\rightarrow$ Affine $q$-Krawtchouk} +The affine $q$-Krawtchouk polynomials given by (\ref{DefAffqKrawtchouk}) +can be obtained from the $q$-Hahn polynomials by the substitution $\alpha=p$ and +$\beta=0$ in (\ref{DefqHahn}): +\begin{equation} +Q_n(q^{-x};p,0,N|q)=K_n^{Aff}(q^{-x};p,N;q). +\end{equation} + +\subsubsection*{$q$-Hahn $\rightarrow$ Hahn} +The Hahn polynomials given by (\ref{DefHahn}) simply follow from the $q$-Hahn +polynomials given by (\ref{DefqHahn}), after setting $\alpha\rightarrow q^{\alpha}$ +and $\beta\rightarrow q^{\beta}$, in the following way: +\begin{equation} +\lim_{q\rightarrow 1}Q_n(q^{-x};q^{\alpha},q^{\beta},N|q)=Q_n(x;\alpha,\beta,N). +\end{equation} + +\subsection*{Remark} +The $q$-Hahn polynomials given by (\ref{DefqHahn}) and the dual $q$-Hahn +polynomials given by (\ref{DefDualqHahn}) are related in the following way: +$$Q_n(q^{-x};\alpha,\beta,N|q)=R_x(\mu(n);\alpha,\beta,N|q),$$ +with +$$\mu(n)=q^{-n}+\alpha\beta q^{n+1}$$ +or +$$R_n(\mu(x);\gamma,\delta,N|q)=Q_x(q^{-n};\gamma,\delta,N|q),$$ +where +$$\mu(x)=q^{-x}+\gamma\delta q^{x+1}.$$ + +\subsection*{References} +\cite{AlSalam90}, \cite{AlvarezRonveaux}, \cite{AndrewsAskey85}, +\cite{AskeyWilson79}, \cite{AskeyWilson85}, \cite{AtakRahmanSuslov}, +\cite{Dunkl78II}, \cite{Fischer95}, \cite{GasperRahman84}, +\cite{GasperRahman90}, \cite{Hahn}, \cite{KalninsMiller88}, +\cite{Koelink96I}, \cite{KoelinkKoorn}, \cite{Koorn89III}, \cite{Koorn90II}, +\cite{Nikiforov+}, \cite{NoumiMimachi90II}, \cite{Rahman82}, +\cite{Stanton80II}, \cite{Stanton84}, \cite{Stanton90}. + + +\section{Dual $q$-Hahn}\index{Dual q-Hahn polynomials@Dual $q$-Hahn polynomials} +\index{q-Hahn polynomials@$q$-Hahn polynomials!Dual} +\par\setcounter{equation}{0} + +\subsection*{Basic hypergeometric representation} +\begin{equation} +\label{DefDualqHahn} +R_n(\mu(x);\gamma,\delta,N|q)= +\qhyp{3}{2}{q^{-n},q^{-x},\gamma\delta q^{x+1}}{\gamma q,q^{-N}}{q},\quad n=0,1,2,\ldots,N, +\end{equation} +where +$$\mu(x):=q^{-x}+\gamma\delta q^{x+1}.$$ + +\newpage + +\subsection*{Orthogonality relation} +For $0<\gamma q<1$ and $0<\delta q<1$, or for $\gamma>q^{-N}$ and $\delta>q^{-N}$, we have +\begin{eqnarray} +\label{OrtDualqHahn} +& &\sum_{x=0}^N\frac{(\gamma q,\gamma\delta q,q^{-N};q)_x}{(q,\gamma\delta q^{N+2},\delta q;q)_x} +\frac{(1-\gamma\delta q^{2x+1})}{(1-\gamma\delta q)(-\gamma q)^x}q^{Nx-\binom{x}{2}}\nonumber\\ +& &{}\mathindent{}\times R_m(\mu(x);\gamma,\delta,N|q)R_n(\mu(x);\gamma,\delta,N|q)\nonumber\\ +& &{}=\frac{(\gamma\delta q^2;q)_N}{(\delta q;q)_N}(\gamma q)^{-N} +\frac{(q,\delta^{-1}q^{-N};q)_n}{(\gamma q,q^{-N};q)_n}(\gamma\delta q)^n\,\delta_{mn}. +\end{eqnarray} + +\subsection*{Recurrence relation} +\begin{eqnarray} +\label{RecDualqHahn} +& &-\left(1-q^{-x}\right)\left(1-\gamma\delta q^{x+1}\right)R_n(\mu(x))\nonumber\\ +& &{}=A_nR_{n+1}(\mu(x))-\left(A_n+C_n\right)R_n(\mu(x))+C_nR_{n-1}(\mu(x)), +\end{eqnarray} +where +$$R_n(\mu(x)):=R_n(\mu(x);\gamma,\delta,N|q)$$ +and +$$\left\{\begin{array}{l} +\displaystyle A_n=\left(1-q^{n-N}\right)\left(1-\gamma q^{n+1}\right)\\ +\\ +\displaystyle C_n=\gamma q\left(1-q^n\right)\left(\delta -q^{n-N-1}\right). +\end{array}\right.$$ + +\subsection*{Normalized recurrence relation} +\begin{eqnarray} +\label{NormRecDualqHahn} +xp_n(x)&=&p_{n+1}(x)+\left[1+\gamma\delta q-(A_n+C_n)\right]p_n(x)\nonumber\\ +& &{}\mathindent{}+\gamma q(1-q^n)(1-\gamma q^n)\nonumber\\ +& &{}\mathindent\mathindent{}\times(1-q^{n-N-1})(\delta-q^{n-N-1})p_{n-1}(x), +\end{eqnarray} +where +$$R_n(\mu(x);\gamma,\delta,N|q)=\frac{1}{(\gamma q,q^{-N};q)_n}p_n(\mu(x)).$$ + +\subsection*{$q$-Difference equation} +\begin{equation} +\label{dvDualqHahn} +q^{-n}(1-q^n)y(x)=B(x)y(x+1)-\left[B(x)+D(x)\right]y(x) ++D(x)y(x-1), +\end{equation} +where +$$y(x)=R_n(\mu(x);\gamma,\delta,N|q)$$ +and +$$\left\{\begin{array}{l}\displaystyle B(x)=\frac{(1-q^{x-N})(1-\gamma q^{x+1})(1-\gamma\delta q^{x+1})} +{(1-\gamma\delta q^{2x+1})(1-\gamma\delta q^{2x+2})}\\ +\\ +\displaystyle D(x)=-\frac{\gamma q^{x-N}(1-q^x)(1-\gamma\delta q^{x+N+1})(1-\delta q^x)} +{(1-\gamma\delta q^{2x})(1-\gamma\delta q^{2x+1})}.\end{array}\right.$$ + +\subsection*{Forward shift operator} +\begin{eqnarray} +\label{shift1DualqHahnI} +& &R_n(\mu(x+1);\gamma,\delta,N|q)-R_n(\mu(x);\gamma,\delta,N|q)\nonumber\\ +& &{}=\frac{q^{-n-x}(1-q^n)(1-\gamma\delta q^{2x+2})}{(1-\gamma q)(1-q^{-N})} +R_{n-1}(\mu(x);\gamma q,\delta,N-1|q) +\end{eqnarray} +or equivalently +\begin{eqnarray} +\label{shift1DualqHahnII} +& &\frac{\Delta R_n(\mu(x);\gamma,\delta,N|q)}{\Delta\mu(x)}\nonumber\\ +& &{}=\frac{q^{-n+1}(1-q^n)}{(1-q)(1-\gamma q)(1-q^{-N})}R_{n-1}(\mu(x);\gamma q,\delta,N-1|q). +\end{eqnarray} + +\subsection*{Backward shift operator} +\begin{eqnarray} +\label{shift2DualqHahnI} +& &(1-\gamma q^x)(1-\gamma\delta q^x)(1-q^{x-N-1})R_n(\mu(x);\gamma,\delta,N|q)\nonumber\\ +& &{}\mathindent{}+\gamma q^{x-N-1}(1-q^x)(1-\gamma\delta q^{x+N+1})(1-\delta q^x)R_n(\mu(x-1);\gamma,\delta,N|q)\nonumber\\ +& &{}=q^x(1-\gamma)(1-q^{-N-1})(1-\gamma\delta q^{2x})R_{n+1}(\mu(x);\gamma q^{-1},\delta,N+1|q) +\end{eqnarray} +or equivalently +\begin{eqnarray} +\label{shift2DualqHahnII} +& &\frac{\nabla\left[w(x;\gamma,\delta,N|q)R_n(\mu(x);\gamma,\delta,N|q)\right]}{\nabla\mu(x)}\nonumber\\ +& &{}=\frac{1}{(1-q)(1-\gamma\delta)}w(x;\gamma q^{-1},\delta,N+1|q)\nonumber\\ +& &{}\mathindent{}\times R_{n+1}(\mu(x);\gamma q^{-1},\delta,N+1|q), +\end{eqnarray} +where +$$w(x;\gamma,\delta,N|q)=\frac{(\gamma q,\gamma\delta q,q^{-N};q)_x} +{(q,\gamma\delta q^{N+2},\delta q;q)_x}(-\gamma^{-1})^x q^{Nx-\binom{x}{2}}.$$ + +\subsection*{Rodrigues-type formula} +\begin{eqnarray} +\label{RodDualqHahn} +& &w(x;\gamma,\delta,N|q)R_n(\mu(x);\gamma,\delta,N|q)\nonumber\\ +& &{}=(1-q)^n(\gamma\delta q;q)_n +\left(\nabla_{\mu}\right)^n\left[w(x;\gamma q^n,\delta,N-n|q)\right], +\end{eqnarray} +where +$$\nabla_{\mu}:=\frac{\nabla}{\nabla\mu(x)}.$$ + +\subsection*{Generating functions} For $x=0,1,2,\ldots,N$ we have +\begin{eqnarray} +\label{GenDualqHahn1} +& &(q^{-N}t;q)_{N-x}\cdot\qhyp{2}{1}{q^{-x},\delta^{-1}q^{-x}}{\gamma q}{\gamma\delta q^{x+1}t}\nonumber\\ +& &{}=\sum_{n=0}^N\frac{(q^{-N};q)_n}{(q;q)_n}R_n(\mu(x);\gamma,\delta,N|q)t^n. +\end{eqnarray} + +\begin{eqnarray} +\label{GenDualqHahn2} +& &(\gamma\delta qt;q)_x\cdot\qhyp{2}{1}{q^{x-N},\gamma q^{x+1}}{\delta^{-1}q^{-N}}{q^{-x}t}\nonumber\\ +& &{}=\sum_{n=0}^N +\frac{(q^{-N},\gamma q;q)_n}{(\delta^{-1}q^{-N},q;q)_n}R_n(\mu(x);\gamma,\delta,N|q)t^n. +\end{eqnarray} + +\subsection*{Limit relations} + +\subsubsection*{$q$-Racah $\rightarrow$ Dual $q$-Hahn} +To obtain the dual $q$-Hahn polynomials from the $q$-Racah polynomials we have to +take $\beta=0$ and $\alpha q=q^{-N}$ in (\ref{DefqRacah}): +$$R_n(\mu(x);q^{-N-1},0,\gamma,\delta|q)=R_n(\mu(x);\gamma,\delta,N|q),$$ +with +$$\mu(x)=q^{-x}+\gamma\delta q^{x+1}.$$ +We may also take $\alpha=0$ and $\beta=\delta^{-1}q^{-N-1}$ in (\ref{DefqRacah}) to obtain +the dual $q$-Hahn polynomials from the $q$-Racah polynomials: +$$R_n(\mu(x);0,\delta^{-1}q^{-N-1},\gamma,\delta|q)=R_n(\mu(x);\gamma,\delta,N|q).$$ +And if we take $\gamma q=q^{-N}$, $\delta\rightarrow\alpha\delta q^{N+1}$ and $\beta=0$ in the +definition (\ref{DefqRacah}) of the $q$-Racah polynomials we find the dual +$q$-Hahn polynomials given by (\ref{DefDualqHahn}) in the following way: +$$R_n(\mu(x);\alpha,0,q^{-N-1},\alpha\delta q^{N+1}|q)=R_n({\tilde \mu}(x);\alpha,\delta,N|q),$$ +with +$${\tilde \mu}(x)=q^{-x}+\alpha\delta q^{x+1}.$$ + +\subsubsection*{Dual $q$-Hahn $\rightarrow$ Affine $q$-Krawtchouk} +The affine $q$-Krawtchouk polynomials given by (\ref{DefAffqKrawtchouk}) +can be obtained from the dual $q$-Hahn polynomials by the substitution $\gamma=p$ +and $\delta=0$ in (\ref{DefDualqHahn}): +\begin{equation} +R_n(\mu(x);p,0,N|q)=K_n^{Aff}(q^{-x};p,N;q). +\end{equation} +Note that $\mu(x)=q^{-x}$ in this case. + +\subsubsection*{Dual $q$-Hahn $\rightarrow$ Dual $q$-Krawtchouk} +The dual $q$-Krawtchouk polynomials given by (\ref{DefDualqKrawtchouk}) can +be obtained from the dual $q$-Hahn polynomials by setting $\delta=c\gamma^{-1}q^{-N-1}$ +in (\ref{DefDualqHahn}) and letting $\gamma\rightarrow 0$: +\begin{equation} +\lim_{\gamma\rightarrow 0} +R_n(\mu(x);\gamma,c\gamma^{-1}q^{-N-1},N|q)=K_n(\lambda(x);c,N|q). +\end{equation} + +\subsubsection*{Dual $q$-Hahn $\rightarrow$ Dual Hahn} +The dual Hahn polynomials given by (\ref{DefDualHahn}) follow from the dual $q$-Hahn +polynomials by simply taking the limit $q\rightarrow 1$ in the definition (\ref{DefDualqHahn}) +of the dual $q$-Hahn polynomials after applying the substitution +$\gamma\rightarrow q^{\gamma}$ and $\delta\rightarrow q^{\delta}$: +\begin{equation} +\lim_{q\rightarrow 1}R_n(\mu(x);q^{\gamma},q^{\delta},N|q)=R_n(\lambda(x);\gamma,\delta,N), +\end{equation} +where +$$\left\{\begin{array}{l} +\displaystyle\mu(x)=q^{-x}+q^{x+\gamma+\delta+1}\\[5mm] +\displaystyle\lambda(x)=x(x+\gamma+\delta+1). +\end{array}\right.$$ + +\subsection*{Remark} +The dual $q$-Hahn polynomials given by (\ref{DefDualqHahn}) and the +$q$-Hahn polynomials given by (\ref{DefqHahn}) are related in the following way: +$$Q_n(q^{-x};\alpha,\beta,N|q)=R_x(\mu(n);\alpha,\beta,N|q),$$ +with +$$\mu(n)=q^{-n}+\alpha\beta q^{n+1}$$ +or +$$R_n(\mu(x);\gamma,\delta,N|q)=Q_x(q^{-n};\gamma,\delta,N|q),$$ +where +$$\mu(x)=q^{-x}+\gamma\delta q^{x+1}.$$ + +\subsection*{References} +\cite{AlvarezSmirnov}, \cite{AndrewsAskey85}, \cite{AskeyWilson79}, +\cite{AskeyWilson85}, \cite{AtakRahmanSuslov}, \cite{GasperRahman90}, +\cite{KoelinkKoorn}, \cite{Nikiforov+}, \cite{Stanton84}. + + +\section{Al-Salam-Chihara}\index{Al-Salam-Chihara polynomials} +\par\setcounter{equation}{0} + +\subsection*{Basic hypergeometric representation} +\begin{eqnarray} +\label{DefAlSalamChihara} +Q_n(x;a,b|q)&=&\frac{(ab;q)_n}{a^n}\, +\qhyp{3}{2}{q^{-n},a\e^{i\theta},a\e^{-i\theta}}{ab,0}{q}\\ +&=&(a\e^{i\theta};q)_n\e^{-in\theta}\,\qhyp{2}{1}{q^{-n},b\e^{-i\theta}} +{a^{-1}q^{-n+1}\e^{-i\theta}}{a^{-1}q\e^{i\theta}}\nonumber\\ +&=&(b\e^{-i\theta};q)_n\e^{in\theta}\,\qhyp{2}{1}{q^{-n},a\e^{i\theta}} +{b^{-1}q^{-n+1}\e^{i\theta}}{b^{-1}q\e^{-i\theta}},\quad x=\cos\theta.\nonumber +\end{eqnarray} + +\subsection*{Orthogonality relation} +If $a$ and $b$ are real or complex conjugates and $\max(|a|,|b|)<1$, then we have the following orthogonality relation +\begin{equation} +\label{OrtAlSalamChiharaI} +\frac{1}{2\pi}\int_{-1}^1\frac{w(x)}{\sqrt{1-x^2}}Q_m(x;a,b|q)Q_n(x;a,b|q)\,dx +=\frac{\,\delta_{mn}}{(q^{n+1},abq^n;q)_{\infty}}, +\end{equation} +where +$$w(x):=w(x;a,b|q)=\left|\frac{(\e^{2i\theta};q)_{\infty}} +{(a\e^{i\theta},b\e^{i\theta};q)_{\infty}}\right|^2 +=\frac{h(x,1)h(x,-1)h(x,q^{\frac{1}{2}})h(x,-q^{\frac{1}{2}})}{h(x,a)h(x,b)},$$ +with +$$h(x,\alpha):=\prod_{k=0}^{\infty}\left(1-2\alpha xq^k+\alpha^2q^{2k}\right) +=\left(\alpha\e^{i\theta},\alpha\e^{-i\theta};q\right)_{\infty},\quad x=\cos\theta.$$ +If $a>1$ and $|ab|<1$, then we have another orthogonality relation given by: +\begin{eqnarray} +\label{OrtAlSalamChiharaII} +& &\frac{1}{2\pi}\int_{-1}^1\frac{w(x)}{\sqrt{1-x^2}}Q_m(x;a,b|q)Q_n(x;a,b|q)\,dx\nonumber\\ +& &{}\mathindent{}+\sum_{\begin{array}{c}\scriptstyle k\\ \scriptstyle 10. +\end{eqnarray} + +\subsection*{Recurrence relation} +\begin{eqnarray} +\label{RecqMeixner} +& &q^{2n+1}(1-q^{-x})M_n(q^{-x})\nonumber\\ +& &{}=c(1-bq^{n+1})M_{n+1}(q^{-x})\nonumber\\ +& &{}\mathindent{}-\left[c(1-bq^{n+1})+q(1-q^n)(c+q^n)\right]M_n(q^{-x})\nonumber\\ +& &{}\mathindent\mathindent{}+q(1-q^n)(c+q^n)M_{n-1}(q^{-x}), +\end{eqnarray} +where +$$M_n(q^{-x}):=M_n(q^{-x};b,c;q).$$ + +\subsection*{Normalized recurrence relation} +\begin{eqnarray} +\label{NormRecqMeixner} +xp_n(x)&=&p_{n+1}(x)+ +\left[1+q^{-2n-1}\left\{c(1-bq^{n+1})+q(1-q^n)(c+q^n)\right\}\right]p_n(x)\nonumber\\ +& &{}\mathindent{}+cq^{-4n+1}(1-q^n)(1-bq^n)(c+q^n)p_{n-1}(x), +\end{eqnarray} +where +$$M_n(q^{-x};b,c;q)=\frac{(-1)^nq^{n^2}}{(bq;q)_nc^n}p_n(q^{-x}).$$ + +\subsection*{$q$-Difference equation} +\begin{equation} +\label{dvqMeixner} +-(1-q^n)y(x)=B(x)y(x+1)-\left[B(x)+D(x)\right]y(x)+D(x)y(x-1), +\end{equation} +where +$$y(x)=M_n(q^{-x};b,c;q)$$ +and +$$\left\{\begin{array}{l}\displaystyle B(x)=cq^x(1-bq^{x+1})\\ +\\ +\displaystyle D(x)=(1-q^x)(1+bcq^x).\end{array}\right.$$ + +\subsection*{Forward shift operator} +\begin{eqnarray} +\label{shift1qMeixnerI} +& &M_n(q^{-x-1};b,c;q)-M_n(q^{-x};b,c;q)\nonumber\\ +& &{}=-\frac{q^{-x}(1-q^n)}{c(1-bq)}M_{n-1}(q^{-x};bq,cq^{-1};q) +\end{eqnarray} +or equivalently +\begin{equation} +\label{shift1qMeixnerII} +\frac{\Delta M_n(q^{-x};b,c;q)}{\Delta q^{-x}} +=-\frac{q(1-q^n)}{c(1-q)(1-bq)}M_{n-1}(q^{-x};bq,cq^{-1};q). +\end{equation} + +\subsection*{Backward shift operator} +\begin{eqnarray} +\label{shift2qMeixnerI} +& &cq^x(1-bq^x)M_n(q^{-x};b,c;q)-(1-q^x)(1+bcq^x)M_n(q^{-x+1};b,c;q)\nonumber\\ +& &{}=cq^x(1-b)M_{n+1}(q^{-x};bq^{-1},cq;q) +\end{eqnarray} +or equivalently +\begin{eqnarray} +\label{shift2qMeixnerII} +& &\frac{\nabla\left[w(x;b,c;q)M_n(q^{-x};b,c;q)\right]}{\nabla q^{-x}}\nonumber\\ +& &{}=\frac{1}{1-q}w(x;bq^{-1},cq;q)M_{n+1}(q^{-x};bq^{-1},cq;q), +\end{eqnarray} +where +$$w(x;b,c;q)=\frac{(bq;q)_x}{(q,-bcq;q)_x}c^xq^{\binom{x+1}{2}}.$$ + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodqMeixner} +w(x;b,c;q)M_n(q^{-x};b,c;q)=(1-q)^n\left(\nabla_q\right)^n\left[w(x;bq^n,cq^{-n};q)\right], +\end{equation} +where +$$\nabla_q:=\frac{\nabla}{\nabla q^{-x}}.$$ + +\subsection*{Generating functions} +\begin{equation} +\label{GenqMeixner1} +\frac{1}{(t;q)_{\infty}}\,\qhyp{1}{1}{q^{-x}}{bq}{-c^{-1}qt} +=\sum_{n=0}^{\infty}\frac{M_n(q^{-x};b,c;q)}{(q;q)_n}t^n. +\end{equation} + +\begin{eqnarray} +\label{GenqMeixner2} +& &\frac{1}{(t;q)_{\infty}}\,\qhyp{1}{1}{-b^{-1}c^{-1}q^{-x}}{-c^{-1}q}{bqt}\nonumber\\ +& &{}=\sum_{n=0}^{\infty}\frac{(bq;q)_n}{(-c^{-1}q,q;q)_n}M_n(q^{-x};b,c;q)t^n. +\end{eqnarray} + +\subsection*{Limit relations} + +\subsubsection*{Big $q$-Jacobi $\rightarrow$ $q$-Meixner} +If we set $b=-a^{-1}cd^{-1}$ (with $d>0$) in the definition (\ref{DefBigqJacobi}) +of the big $q$-Jacobi polynomials and take the limit $c\rightarrow -\infty$ we +obtain the $q$-Meixner polynomials given by (\ref{DefqMeixner}): +\begin{equation} +\lim_{c\rightarrow -\infty}P_n(q^{-x};a,-a^{-1}cd^{-1},c;q)=M_n(q^{-x};a,d;q). +\end{equation} + +\subsubsection*{$q$-Hahn $\rightarrow$ $q$-Meixner} +The $q$-Meixner polynomials given by (\ref{DefqMeixner}) can be obtained from the $q$-Hahn polynomials +by setting $\alpha=b$ and $\beta=-b^{-1}c^{-1}q^{-N-1}$ in the definition (\ref{DefqHahn}) of the +$q$-Hahn polynomials and letting $N\rightarrow\infty$: +$$\lim_{N\rightarrow\infty}Q_n(q^{-x};b,-b^{-1}c^{-1}q^{-N-1},N|q)=M_n(q^{-x};b,c;q).$$ + +\subsubsection*{$q$-Meixner $\rightarrow$ $q$-Laguerre} +The $q$-Laguerre polynomials given by (\ref{DefqLaguerre}) can be obtained +from the $q$-Meixner polynomials given by (\ref{DefqMeixner}) by setting +$b=q^{\alpha}$ and $q^{-x}\rightarrow cq^{\alpha}x$ in the definition +(\ref{DefqMeixner}) of the $q$-Meixner polynomials and then taking the limit +$c\rightarrow\infty$: +\begin{equation} +\lim_{c\rightarrow\infty}M_n(cq^{\alpha}x;q^{\alpha},c;q)= +\frac{(q;q)_n}{(q^{\alpha+1};q)_n}L_n^{(\alpha)}(x;q). +\end{equation} + +\subsubsection*{$q$-Meixner $\rightarrow$ $q$-Charlier} +The $q$-Charlier polynomials given by (\ref{DefqCharlier}) can easily be +obtained from the $q$-Meixner given by (\ref{DefqMeixner}) by setting $b=0$ +in the definition (\ref{DefqMeixner}) of the $q$-Meixner polynomials: +\begin{equation} +M_n(x;0,c;q)=C_n(x;c;q). +\end{equation} + +\subsubsection*{$q$-Meixner $\rightarrow$ Al-Salam-Carlitz~II} +The Al-Salam-Carlitz~II polynomials given by (\ref{DefAlSalamCarlitzII}) +can be obtained from the $q$-Meixner polynomials given by (\ref{DefqMeixner}) +by setting $b=-ac^{-1}$ in the definition (\ref{DefqMeixner}) of the +$q$-Meixner polynomials and then taking the limit $c\rightarrow 0$: +\begin{equation} +\lim_{c\rightarrow 0}M_n(x;-ac^{-1},c;q)= +\left(-\frac{1}{a}\right)^nq^{\binom{n}{2}}V_n^{(a)}(x;q). +\end{equation} + +\subsubsection*{$q$-Meixner $\rightarrow$ Meixner} +To find the Meixner polynomials given by (\ref{DefMeixner}) from the $q$-Meixner polynomials given by +(\ref{DefqMeixner}) we set $b=q^{\beta-1}$ and $c\rightarrow (1-c)^{-1}c$ and let $q\rightarrow 1$: +\begin{equation} +\lim_{q\rightarrow 1}M_n(q^{-x};q^{\beta-1},(1-c)^{-1}c;q)=M_n(x;\beta,c). +\end{equation} + +\subsection*{Remarks} +The $q$-Meixner polynomials given by (\ref{DefqMeixner}) and the little +$q$-Jacobi polynomials given by (\ref{DefLittleqJacobi}) are related in the +following way: +$$M_n(q^{-x};b,c;q)=p_n(-c^{-1}q^n;b,b^{-1}q^{-n-x-1}|q).$$ + +\noindent +The $q$-Meixner polynomials and the quantum $q$-Krawtchouk polynomials +given by (\ref{DefQuantumqKrawtchouk}) are related in the following way: +$$K_n^{qtm}(q^{-x};p,N;q)=M_n(q^{-x};q^{-N-1},-p^{-1};q).$$ + +\subsection*{References} +\cite{AlSalam90}, \cite{AlSalamVerma82II}, \cite{AlSalamVerma88}, \cite{AlvarezRonveaux}, +\cite{AtakAtakKlimyk}, \cite{AtakRahmanSuslov}, \cite{Campigotto+}, \cite{GasperRahman90}, +\cite{Hahn}, \cite{Ismail2005I}, \cite{Nikiforov+}, \cite{Smirnov}. + + +\section{Quantum $q$-Krawtchouk} +\index{Quantum q-Krawtchouk polynomials@Quantum $q$-Krawtchouk polynomials} +\index{q-Krawtchouk polynomials@$q$-Krawtchouk polynomials!Quantum} +\par\setcounter{equation}{0} + +\subsection*{Basic hypergeometric representation} +\begin{equation} +\label{DefQuantumqKrawtchouk} +K_n^{qtm}(q^{-x};p,N;q)= +\qhyp{2}{1}{q^{-n},q^{-x}}{q^{-N}}{pq^{n+1}},\quad n=0,1,2,\ldots,N. +\end{equation} + +\subsection*{Orthogonality relation} +\begin{eqnarray} +\label{OrtQuantumqKrawtchouk} +& &\sum_{x=0}^N\frac{(pq;q)_{N-x}}{(q;q)_x(q;q)_{N-x}}(-1)^{N-x}q^{\binom{x}{2}} +K_m^{qtm}(q^{-x};p,N;q)K_n^{qtm}(q^{-x};p,N;q)\nonumber\\ +& &{}=\frac{(-1)^np^N(q;q)_{N-n}(q,pq;q)_n}{(q,q;q)_N} +q^{\binom{N+1}{2}-\binom{n+1}{2}+Nn}\,\delta_{mn},\quad p>q^{-N}. +\end{eqnarray} + +\subsection*{Recurrence relation} +\begin{eqnarray} +\label{RecQuantumqKrawtchouk} +& &-pq^{2n+1}(1-q^{-x})K_n^{qtm}(q^{-x})\nonumber\\ +& &{}=(1-q^{n-N})K_{n+1}^{qtm}(q^{-x})\nonumber\\ +& &{}\mathindent{}-\left[(1-q^{n-N})+q(1-q^n)(1-pq^n)\right]K_n^{qtm}(q^{-x})\nonumber\\ +& &{}\mathindent\mathindent{}+q(1-q^n)(1-pq^n)K_{n-1}^{qtm}(q^{-x}), +\end{eqnarray} +where +$$K_n^{qtm}(q^{-x}):=K_n^{qtm}(q^{-x};p,N;q).$$ + +\subsection*{Normalized recurrence relation} +\begin{eqnarray} +\label{NormRecQuantumqKrawtchouk} +xp_n(x)&=&p_{n+1}(x)+ +\left[1-p^{-1}q^{-2n-1}\left\{(1-q^{n-N})+q(1-q^n)(1-pq^n)\right\}\right]p_n(x)\nonumber\\ +& &{}\mathindent{}+p^{-2}q^{-4n+1}(1-q^n)(1-pq^n)(1-q^{n-N-1})p_{n-1}(x), +\end{eqnarray} +where +$$K_n^{qtm}(q^{-x};p,N;q)=\frac{p^nq^{n^2}}{(q^{-N};q)_n}p_n(q^{-x}).$$ + +\subsection*{$q$-Difference equation} +\begin{equation} +\label{dvQuantumqKrawtchouk} +-p(1-q^n)y(x)=B(x)y(x+1)-\left[B(x)+D(x)\right]y(x)+D(x)y(x-1), +\end{equation} +where +$$y(x)=K_n^{qtm}(q^{-x};p,N;q)$$ +and +$$\left\{\begin{array}{l}\displaystyle B(x)=-q^x(1-q^{x-N})\\ +\\ +\displaystyle D(x)=(1-q^x)(p-q^{x-N-1}).\end{array}\right.$$ + +\subsection*{Forward shift operator} +\begin{eqnarray} +\label{shift1QuantumqKrawtchoukI} +& &K_n^{qtm}(q^{-x-1};p,N;q)-K_n^{qtm}(q^{-x};p,N;q)\nonumber\\ +& &{}=\frac{pq^{-x}(1-q^n)}{1-q^{-N}}K_{n-1}^{qtm}(q^{-x};pq,N-1;q) +\end{eqnarray} +or equivalently +\begin{equation} +\label{shift1QuantumqKrawtchoukII} +\frac{\Delta K_n^{qtm}(q^{-x};p,N;q)}{\Delta q^{-x}}= +\frac{pq(1-q^n)}{(1-q)(1-q^{-N})}K_{n-1}^{qtm}(q^{-x};pq,N-1;q). +\end{equation} + +\subsection*{Backward shift operator} +\begin{eqnarray} +\label{shift2QuantumqKrawtchoukI} +& &(1-q^{x-N-1})K_n^{qtm}(q^{-x};p,N;q)\nonumber\\ +& &{}\mathindent{}+q^{-x}(1-q^x)(p-q^{x-N-1})K_n^{qtm}(q^{-x+1};p,N;q)\nonumber\\ +& &{}=(1-q^{-N-1})K_{n+1}^{qtm}(q^{-x};pq^{-1},N+1;q) +\end{eqnarray} +or equivalently +\begin{eqnarray} +\label{shift2QuantumqKrawtchoukII} +& &\frac{\nabla\left[w(x;p,N;q)K_n^{qtm}(q^{-x};p,N;q)\right]}{\nabla q^{-x}}\nonumber\\ +& &{}=\frac{1}{1-q}w(x;pq^{-1},N+1;q)K_{n+1}^{qtm}(q^{-x};pq^{-1},N+1;q), +\end{eqnarray} +where +$$w(x;p,N;q)=\frac{(q^{-N};q)_x}{(q,p^{-1}q^{-N};q)_x}(-p)^{-x}q^{\binom{x+1}{2}}.$$ + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodQuantumqKrawtchouk} +w(x;p,N;q)K_n^{qtm}(q^{-x};p,N;q)=(1-q)^n\left(\nabla_q\right)^n\left[w(x;pq^n,N-n;q)\right], +\end{equation} +where +$$\nabla_q:=\frac{\nabla}{\nabla q^{-x}}.$$ + +\subsection*{Generating functions} For $x=0,1,2,\ldots,N$ we have +\begin{eqnarray} +\label{GenQuantumqKrawtchouk1} +& &(q^{x-N}t;q)_{N-x}\cdot\qhyp{2}{1}{q^{-x},pq^{N+1-x}}{0}{q^{x-N}t}\nonumber\\ +& &{}=\sum_{n=0}^N\frac{(q^{-N};q)_n}{(q;q)_n}K_n^{qtm}(q^{-x};p,N;q)t^n. +\end{eqnarray} + +\begin{eqnarray} +\label{GenQuantumqKrawtchouk2} +& &(q^{-x}t;q)_x\cdot\qhyp{2}{1}{q^{x-N},0}{pq}{q^{-x}t}\nonumber\\ +& &{}=\sum_{n=0}^N\frac{(q^{-N};q)_n}{(pq,q;q)_n}K_n^{qtm}(q^{-x};p,N;q)t^n. +\end{eqnarray} + +\subsection*{Limit relations} + +\subsubsection*{$q$-Hahn $\rightarrow$ Quantum $q$-Krawtchouk} +The quantum $q$-Krawtchouk polynomials given by (\ref{DefQuantumqKrawtchouk}) +simply follow from the $q$-Hahn polynomials by setting $\beta=p$ in the definition (\ref{DefqHahn}) of +the $q$-Hahn polynomials and taking the limit $\alpha\rightarrow\infty$: +$$\lim_{\alpha\rightarrow\infty}Q_n(q^{-x};\alpha,p,N|q)=K_n^{qtm}(q^{-x};p,N;q).$$ + +\subsubsection*{Quantum $q$-Krawtchouk $\rightarrow$ Al-Salam-Carlitz~II} +If we set $p=a^{-1}q^{-N-1}$ in the definition (\ref{DefQuantumqKrawtchouk}) +of the quantum $q$-Krawtchouk polynomials and let $N\rightarrow\infty$ we +obtain the Al-Salam-Carlitz~II polynomials given by +(\ref{DefAlSalamCarlitzII}). In fact we have +\begin{equation} +\lim_{N\rightarrow\infty}K_n^{qtm}(x;a^{-1}q^{-N-1},N;q)= +\left(-\frac{1}{a}\right)^nq^{\binom{n}{2}}V_n^{(a)}(x;q). +\end{equation} + +\subsubsection*{Quantum $q$-Krawtchouk $\rightarrow$ Krawtchouk} +The Krawtchouk polynomials given by (\ref{DefKrawtchouk}) easily follow from +the quantum $q$-Krawtchouk polynomials given by (\ref{DefQuantumqKrawtchouk}) +in the following way: +\begin{equation} +\lim_{q\rightarrow 1}K_n^{qtm}(q^{-x};p,N;q)=K_n(x;p^{-1},N). +\end{equation} + +\subsection*{Remarks} +The quantum $q$-Krawtchouk polynomials given by +(\ref{DefQuantumqKrawtchouk}) and the $q$-Meixner polynomials given by +(\ref{DefqMeixner}) are related in the following way: +$$K_n^{qtm}(q^{-x};p,N;q)=M_n(q^{-x};q^{-N-1},-p^{-1};q).$$ + +\noindent +The quantum $q$-Krawtchouk polynomials are related to the affine +$q$-Krawtchouk polynomials given by (\ref{DefAffqKrawtchouk}) +by the transformation $q\leftrightarrow q^{-1}$ in the following way: +$$K_n^{qtm}(q^x;p,N;q^{-1})=(p^{-1}q;q)_n\left(-\frac{p}{q}\right)^nq^{-\binom{n}{2}} +K_n^{Aff}(q^{x-N};p^{-1},N;q).$$ + +\subsection*{References} +\cite{GasperRahman90}, \cite{Koorn89III}, \cite{Koorn90II}, \cite{Smirnov}. + + +\section{$q$-Krawtchouk}\index{q-Krawtchouk polynomials@$q$-Krawtchouk polynomials} +\par\setcounter{equation}{0} + +\subsection*{Basic hypergeometric representation} For $n=0,1,2,\ldots,N$ we have +\begin{eqnarray} +\label{DefqKrawtchouk} +K_n(q^{-x};p,N;q)&=&\qhyp{3}{2}{q^{-n},q^{-x},-pq^n}{q^{-N},0}{q}\\ +&=&\frac{(q^{x-N};q)_n}{(q^{-N};q)_nq^{nx}}\, +\qhyp{2}{1}{q^{-n},q^{-x}}{q^{N-x-n+1}}{-pq^{n+N+1}}.\nonumber +\end{eqnarray} + +\subsection*{Orthogonality relation} +\begin{eqnarray} +\label{OrtqKrawtchouk} +& &\sum_{x=0}^N\frac{(q^{-N};q)_x}{(q;q)_x}(-p)^{-x}K_m(q^{-x};p,N;q)K_n(q^{-x};p,N;q)\nonumber\\ +& &{}=\frac{(q,-pq^{N+1};q)_n}{(-p,q^{-N};q)_n}\frac{(1+p)}{(1+pq^{2n})}\nonumber\\ +& &{}\mathindent{}\times (-pq;q)_Np^{-N}q^{-\binom{N+1}{2}} +\left(-pq^{-N}\right)^nq^{n^2}\,\delta_{mn},\quad p>0. +\end{eqnarray} + +\subsection*{Recurrence relation} +\begin{eqnarray} +\label{RecqKrawtchouk} +-\left(1-q^{-x}\right)K_n(q^{-x})&=&A_nK_{n+1}(q^{-x})-\left(A_n+C_n\right)K_n(q^{-x})\nonumber\\ +& &{}\mathindent{}+C_nK_{n-1}(q^{-x}), +\end{eqnarray} +where +$$K_n(q^{-x}):=K_n(q^{-x};p,N;q)$$ +and +$$\left\{\begin{array}{l} +\displaystyle A_n=\frac{(1-q^{n-N})(1+pq^n)}{(1+pq^{2n})(1+pq^{2n+1})}\\ +\\ +\displaystyle C_n=-pq^{2n-N-1}\frac{(1+pq^{n+N})(1-q^n)}{(1+pq^{2n-1})(1+pq^{2n})}. +\end{array}\right.$$ + +\subsection*{Normalized recurrence relation} +\begin{equation} +\label{NormRecqKrawtchouk} +xp_n(x)=p_{n+1}(x)+\left[1-(A_n+C_n)\right]p_n(x)+A_{n-1}C_np_{n-1}(x), +\end{equation} +where +$$K_n(q^{-x};p,N;q)=\frac{(-pq^n;q)_n}{(q^{-N};q)_n}p_n(q^{-x}).$$ + +\subsection*{$q$-Difference equation} +\begin{eqnarray} +\label{dvqKrawtchouk} +& &q^{-n}(1-q^n)(1+pq^n)y(x)\nonumber\\ +& &{}=(1-q^{x-N})y(x+1)-\left[(1-q^{x-N})-p(1-q^x)\right]y(x)\nonumber\\ +& &{}\mathindent{}-p(1-q^x)y(x-1), +\end{eqnarray} +where +$$y(x)=K_n(q^{-x};p,N;q).$$ + +\subsection*{Forward shift operator} +\begin{eqnarray} +\label{shift1qKrawtchoukI} +& &K_n(q^{-x-1};p,N;q)-K_n(q^{-x};p,N;q)\nonumber\\ +& &{}=\frac{q^{-n-x}(1-q^n)(1+pq^n)}{1-q^{-N}}K_{n-1}(q^{-x};pq^2,N-1;q) +\end{eqnarray} +or equivalently +\begin{equation} +\label{shift1qKrawtchoukII} +\frac{\Delta K_n(q^{-x};p,N;q)}{\Delta q^{-x}}= +\frac{q^{-n+1}(1-q^n)(1+pq^n)}{(1-q)(1-q^{-N})}K_{n-1}(q^{-x};pq^2,N-1;q). +\end{equation} + +\subsection*{Backward shift operator} +\begin{eqnarray} +\label{shift2qKrawtchoukI} +& &(1-q^{x-N-1})K_n(q^{-x};p,N;q)+pq^{-1}(1-q^x)K_n(q^{-x+1};p,N;q)\nonumber\\ +& &{}=q^x(1-q^{-N-1})K_{n+1}(q^{-x};pq^{-2},N+1;q) +\end{eqnarray} +or equivalently +\begin{eqnarray} +\label{shift2qKrawtchoukII} +& &\frac{\nabla\left[w(x;p,N;q)K_n(q^{-x};p,N;q)\right]}{\nabla q^{-x}}\nonumber\\ +& &{}=\frac{1}{1-q}w(x;pq^{-2},N+1;q)K_{n+1}(q^{-x};pq^{-2},N+1;q), +\end{eqnarray} +where +$$w(x;p,N;q)=\frac{(q^{-N};q)_x}{(q;q)_x}\left(-\frac{q}{p}\right)^x.$$ + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodqKrawtchouk} +w(x;p,N;q)K_n(q^{-x};p,N;q)=(1-q)^n\left(\nabla_q\right)^n\left[w(x;pq^{2n},N-n;q)\right], +\end{equation} +where +$$\nabla_q:=\frac{\nabla}{\nabla q^{-x}}.$$ + +\subsection*{Generating function} For $x=0,1,2,\ldots,N$ we have +\begin{eqnarray} +\label{GenqKrawtchouk} +& &\qhyp{1}{1}{q^{-x}}{0}{pqt}\,\qhyp{2}{0}{q^{x-N},0}{-}{-q^{-x}t}\nonumber\\ +& &{}=\sum_{n=0}^N\frac{(q^{-N};q)_n}{(q;q)_n}q^{-\binom{n}{2}}K_n(q^{-x};p,N;q)t^n. +\end{eqnarray} + +\subsection*{Limit relations} + +\subsubsection*{$q$-Racah $\rightarrow$ $q$-Krawtchouk} +The $q$-Krawtchouk polynomials given by (\ref{DefqKrawtchouk}) can be obtained from +the $q$-Racah polynomials by setting $\alpha q=q^{-N}$, $\beta=-pq^N$ and +$\gamma=\delta=0$ in the definition (\ref{DefqRacah}) of the $q$-Racah polynomials: +$$R_n(q^{-x};q^{-N-1},-pq^N,0,0|q)=K_n(q^{-x};p,N;q).$$ +Note that $\mu(x)=q^{-x}$ in this case. + +\subsubsection*{$q$-Hahn $\rightarrow$ $q$-Krawtchouk} +If we set $\beta=-\alpha^{-1}q^{-1}p$ in the definition (\ref{DefqHahn}) of the $q$-Hahn polynomials +and then let $\alpha\rightarrow 0$ we obtain the $q$-Krawtchouk polynomials given by +(\ref{DefqKrawtchouk}): +$$\lim_{\alpha\rightarrow 0} +Q_n(q^{-x};\alpha,-\alpha^{-1}q^{-1}p,N|q)=K_n(q^{-x};p,N;q).$$ + +\subsubsection*{$q$-Krawtchouk $\rightarrow$ $q$-Bessel} +If we set $x\rightarrow N-x$ in the definition (\ref{DefqKrawtchouk}) of the +$q$-Krawtchouk polynomials and then take the limit $N\rightarrow\infty$ we +obtain the $q$-Bessel polynomials given by (\ref{DefqBessel}): +\begin{equation} +\lim_{N\rightarrow\infty}K_n(q^{x-N};p,N;q)=y_n(q^x;p;q). +\end{equation} + +\subsubsection*{$q$-Krawtchouk $\rightarrow$ $q$-Charlier} +By setting $p=a^{-1}q^{-N}$ in the definition (\ref{DefqKrawtchouk}) of the +$q$-Krawtchouk polynomials and then taking the limit $N\rightarrow\infty$ we +obtain the $q$-Charlier polynomials given by (\ref{DefqCharlier}): +\begin{equation} +\lim_{N\rightarrow\infty}K_n(q^{-x};a^{-1}q^{-N},N;q)=C_n(q^{-x};a;q). +\end{equation} + +\subsubsection*{$q$-Krawtchouk $\rightarrow$ Krawtchouk} +If we take the limit $q\rightarrow 1$ in the definition (\ref{DefqKrawtchouk}) of the $q$-Krawtchouk +polynomials we simply find the Krawtchouk polynomials given by (\ref{DefKrawtchouk}) in the +following way: +\begin{equation} +\lim_{q\rightarrow 1}K_n(q^{-x};p,N;q)=K_n(x;(p+1)^{-1},N). +\end{equation} + +\subsection*{Remark} +The $q$-Krawtchouk polynomials given by (\ref{DefqKrawtchouk}) and the +dual $q$-Krawtchouk polynomials given by (\ref{DefDualqKrawtchouk}) are +related in the following way: +$$K_n(q^{-x};p,N;q)=K_x(\lambda(n);-pq^N,N|q)$$ +with +$$\lambda(n)=q^{-n}-pq^n$$ +or +$$K_n(\lambda(x);c,N|q)=K_x(q^{-n};-cq^{-N},N;q)$$ +with +$$\lambda(x)=q^{-x}+cq^{x-N}.$$ + +\subsection*{References} +\cite{AlvarezRonveaux}, \cite{AskeyWilson79}, \cite{AtakRahmanSuslov}, +\cite{Campigotto+}, \cite{GasperRahman90}, \cite{Nikiforov+}, +\cite{NoumiMimachi91}, \cite{Stanton80III}, \cite{Stanton84}. + + +\newpage + +\section{Affine $q$-Krawtchouk} +\index{Affine q-Krawtchouk polynomials@Affine $q$-Krawtchouk polynomials} +\index{q-Krawtchouk polynomials@$q$-Krawtchouk polynomials!Affine} +\par\setcounter{equation}{0} + +\subsection*{Basic hypergeometric representation} +\begin{eqnarray} +\label{DefAffqKrawtchouk} +K_n^{Aff}(q^{-x};p,N;q)&=&\qhyp{3}{2}{q^{-n},0,q^{-x}}{pq,q^{-N}}{q}\\ +&=&\frac{(-pq)^nq^{\binom{n}{2}}}{(pq;q)_n}\, +\qhyp{2}{1}{q^{-n},q^{x-N}}{q^{-N}}{\frac{q^{-x}}{p}},\quad n=0,1,2,\ldots,N.\nonumber +\end{eqnarray} + +\subsection*{Orthogonality relation} +\begin{eqnarray} +\label{OrtAffqKrawtchouk} +& &\sum_{x=0}^N\frac{(pq;q)_x(q;q)_N}{(q;q)_x(q;q)_{N-x}}(pq)^{-x}K_m^{Aff}(q^{-x};p,N;q)K_n^{Aff}(q^{-x};p,N;q)\nonumber\\ +& &{}=(pq)^{n-N}\frac{(q;q)_n(q;q)_{N-n}}{(pq;q)_n(q;q)_N}\,\delta_{mn},\quad 01$, then we have another orthogonality relation given by: +\begin{eqnarray} +\label{OrtContBigqHermite2} +& &\frac{1}{2\pi}\int_{-1}^1\frac{w(x)}{\sqrt{1-x^2}}H_m(x;a|q)H_n(x;a|q)\,dx\nonumber\\ +& &{}\mathindent{}+\sum_{\begin{array}{c}\scriptstyle k\\ \scriptstyle 1-1 +\end{eqnarray} +and +\begin{eqnarray} +\label{OrtqLaguerre2} +& &\sum_{k=-\infty}^{\infty}\frac{q^{k\alpha+k}}{(-cq^k;q)_{\infty}}L_m^{(\alpha)}(cq^k;q)L_n^{(\alpha)}(cq^k;q)\nonumber\\ +& &{}=\frac{(q,-cq^{\alpha+1},-c^{-1}q^{-\alpha};q)_{\infty}} +{(q^{\alpha+1},-c,-c^{-1}q;q)_{\infty}}\frac{(q^{\alpha+1};q)_n}{(q;q)_nq^n}\,\delta_{mn}, +\quad\alpha>-1,\quad c>0. +\end{eqnarray} +For $c=1$ the latter orthogonality relation can also be written as +\begin{eqnarray} +\label{OrtqLaguerre3} +& &\int_0^{\infty}\frac{x^{\alpha}}{(-x;q)_{\infty}}L_m^{(\alpha)}(x;q)L_n^{(\alpha)}(x;q)\,d_qx\nonumber\\ +& &{}=\frac{1-q}{2}\,\frac{(q,-q^{\alpha+1},-q^{-\alpha};q)_{\infty}} +{(q^{\alpha+1},-q,-q;q)_{\infty}}\frac{(q^{\alpha+1};q)_n}{(q;q)_nq^n}\,\delta_{mn},\quad\alpha>-1. +\end{eqnarray} + +\newpage + +\subsection*{Recurrence relation} +\begin{eqnarray} +\label{RecqLaguerre} +-q^{2n+\alpha+1}xL_n^{(\alpha)}(x;q)&=&(1-q^{n+1})L_{n+1}^{(\alpha)}(x;q)\nonumber\\ +& &{}\mathindent{}-\left[(1-q^{n+1})+q(1-q^{n+\alpha})\right]L_n^{(\alpha)}(x;q)\nonumber\\ +& &{}\mathindent\mathindent{}+q(1-q^{n+\alpha})L_{n-1}^{(\alpha)}(x;q). +\end{eqnarray} + +\subsection*{Normalized recurrence relation} +\begin{eqnarray} +\label{NormRecqLaguerre} +xp_n(x)&=&p_{n+1}(x)+q^{-2n-\alpha-1}\left[(1-q^{n+1})+q(1-q^{n+\alpha})\right]p_n(x)\nonumber\\ +& &{}\mathindent{}+q^{-4n-2\alpha+1}(1-q^n)(1-q^{n+\alpha})p_{n-1}(x), +\end{eqnarray} +where +$$L_n^{(\alpha)}(x;q)=\frac{(-1)^nq^{n(n+\alpha)}}{(q;q)_n}p_n(x).$$ + +\subsection*{$q$-Difference equation} +\begin{equation} +\label{dvqLaguerre} +-q^{\alpha}(1-q^n)xy(x)=q^{\alpha}(1+x)y(qx)-\left[1+q^{\alpha}(1+x)\right]y(x)+y(q^{-1}x), +\end{equation} +where +$$y(x)=L_n^{(\alpha)}(x;q).$$ + +\subsection*{Forward shift operator} +\begin{equation} +\label{shift1qLaguerreI} +L_n^{(\alpha)}(x;q)-L_n^{(\alpha)}(qx;q)=-q^{\alpha+1}xL_{n-1}^{(\alpha+1)}(qx;q) +\end{equation} +or equivalently +\begin{equation} +\label{shift1qLaguerreII} +\mathcal{D}_qL_n^{(\alpha)}(x;q)=-\frac{q^{\alpha+1}}{1-q}L_{n-1}^{(\alpha+1)}(qx;q). +\end{equation} + +\subsection*{Backward shift operator} +\begin{equation} +\label{shift2qLaguerreI} +L_n^{(\alpha)}(x;q)-q^{\alpha}(1+x)L_n^{(\alpha)}(qx;q)=(1-q^{n+1})L_{n+1}^{(\alpha-1)}(x;q) +\end{equation} +or equivalently +\begin{equation} +\label{shift2qLaguerreII} +\mathcal{D}_q\left[w(x;\alpha;q)L_n^{(\alpha)}(x;q)\right]= +\frac{1-q^{n+1}}{1-q}w(x;\alpha-1;q)L_{n+1}^{(\alpha-1)}(x;q), +\end{equation} +where +$$w(x;\alpha;q)=\frac{x^{\alpha}}{(-x;q)_{\infty}}.$$ + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodqLaguerre} +w(x;\alpha;q)L_n^{(\alpha)}(x;q)= +\frac{(1-q)^n}{(q;q)_n}\left(\mathcal{D}_q\right)^n\left[w(x;\alpha+n;q)\right]. +\end{equation} + +\subsection*{Generating functions} +\begin{equation} +\label{GenqLaguerre1} +\frac{1}{(t;q)_{\infty}}\,\qhyp{1}{1}{-x}{0}{q^{\alpha+1}t} +=\sum_{n=0}^{\infty}L_n^{(\alpha)}(x;q)t^n. +\end{equation} + +\begin{equation} +\label{GenqLaguerre2} +\frac{1}{(t;q)_{\infty}}\,\qhyp{0}{1}{-}{q^{\alpha+1}}{-q^{\alpha+1}xt} +=\sum_{n=0}^{\infty}\frac{L_n^{(\alpha)}(x;q)}{(q^{\alpha+1};q)_n}t^n. +\end{equation} + +\begin{equation} +\label{GenqLaguerre3} +(t;q)_{\infty}\cdot\qhyp{0}{2}{-}{q^{\alpha+1},t}{-q^{\alpha+1}xt} +=\sum_{n=0}^{\infty}\frac{(-1)^nq^{\binom{n}{2}}}{(q^{\alpha+1};q)_n}L_n^{(\alpha)}(x;q)t^n. +\end{equation} + +\begin{eqnarray} +\label{GenqLaguerre4} +& &\frac{(\gamma t;q)_{\infty}}{(t;q)_{\infty}}\,\qhyp{1}{2}{\gamma}{q^{\alpha+1},\gamma t}{-q^{\alpha+1}xt}\nonumber\\ +& &{}=\sum_{n=0}^{\infty}\frac{(\gamma;q)_n}{(q^{\alpha+1};q)_n}L_n^{(\alpha)}(x;q)t^n, +\quad\textrm{$\gamma$ arbitrary}. +\end{eqnarray} + +\subsection*{Limit relations} + +\subsubsection*{Little $q$-Jacobi $\rightarrow$ $q$-Laguerre} +If we substitute $a=q^{\alpha}$ and $x\rightarrow -b^{-1}q^{-1}x$ in the definition +(\ref{DefLittleqJacobi}) of the little $q$-Jacobi polynomials and then take the limit +$b\rightarrow -\infty$ we find the $q$-Laguerre polynomials given by (\ref{DefqLaguerre}): +$$\lim_{b\rightarrow -\infty}p_n(-b^{-1}q^{-1}x;q^{\alpha},b|q)= +\frac{(q;q)_n}{(q^{\alpha+1};q)_n}L_n^{(\alpha)}(x;q).$$ + +\subsubsection*{$q$-Meixner $\rightarrow$ $q$-Laguerre} +The $q$-Laguerre polynomials given by (\ref{DefqLaguerre}) can be obtained +from the $q$-Meixner polynomials given by (\ref{DefqMeixner}) by setting +$b=q^{\alpha}$ and $q^{-x}\rightarrow cq^{\alpha}x$ in the definition +(\ref{DefqMeixner}) of the $q$-Meixner polynomials and then taking the limit +$c\rightarrow\infty$: +$$\lim_{c\rightarrow\infty}M_n(cq^{\alpha}x;q^{\alpha},c;q)= +\frac{(q;q)_n}{(q^{\alpha+1};q)_n}L_n^{(\alpha)}(x;q).$$ + +\subsubsection*{$q$-Laguerre $\rightarrow$ Stieltjes-Wigert} +If we set $x\rightarrow xq^{-\alpha}$ in the definition +(\ref{DefqLaguerre}) of the $q$-Laguerre polynomials and take the limit +$\alpha\rightarrow\infty$ we simply obtain the Stieltjes-Wigert polynomials given +by (\ref{DefStieltjesWigert}): +\begin{equation} +\lim_{\alpha\rightarrow\infty}L_n^{(\alpha)}\left(xq^{-\alpha};q\right)=S_n(x;q). +\end{equation} + +\subsubsection*{$q$-Laguerre $\rightarrow$ Laguerre / Charlier} +If we set $x\rightarrow (1-q)x$ in the definition (\ref{DefqLaguerre}) +of the $q$-Laguerre polynomials and take the limit $q\rightarrow 1$ we obtain +the Laguerre polynomials given by (\ref{DefLaguerre}): +\begin{equation} +\lim_{q\rightarrow 1}L_n^{(\alpha)}((1-q)x;q)=L_n^{(\alpha)}(x). +\end{equation} + +If we set $x\rightarrow -q^{-x}$ and $q^{\alpha}=a^{-1}(q-1)^{-1}$ (or +$\alpha=-(\ln q)^{-1}\ln (q-1)a$) in the definition (\ref{DefqLaguerre}) of the +$q$-Laguerre polynomials, multiply by $(q;q)_n$, and take the limit +$q\rightarrow 1$ we obtain the Charlier polynomials given by (\ref{DefCharlier}): +\begin{equation} +\lim_{q\rightarrow 1}\,(q;q)_nL_n^{(\alpha)}(-q^{-x};q)=C_n(x;a), +\end{equation} +where +$$q^{\alpha}=\frac{1}{a(q-1)}\quad\textrm{or}\quad\alpha=-\frac{\ln (q-1)a}{\ln q}.$$ + +\subsection*{Remarks} +The $q$-Laguerre polynomials are sometimes called the +generalized Stieltjes-Wigert polynomials. + +If we replace $q$ by $q^{-1}$ we obtain the little $q$-Laguerre +(or Wall) polynomials given by (\ref{DefLittleqLaguerre}) in the following +way: +$$L_n^{(\alpha)}(x;q^{-1})=\frac{(q^{\alpha+1};q)_n}{(q;q)_nq^{n\alpha}}p_n(-x;q^{\alpha}|q).$$ + +\noindent +The $q$-Laguerre polynomials given by (\ref{DefqLaguerre}) and the $q$-Bessel polynomials +given by (\ref{DefqBessel}) are related in the following way: +$$\frac{y_n(q^x;a;q)}{(q;q)_n}=L_n^{(x-n)}(aq^n;q).$$ + +\noindent +The $q$-Laguerre polynomials given by (\ref{DefqLaguerre}) and the +$q$-Charlier polynomials given by (\ref{DefqCharlier}) are related in the +following way: +$$\frac{C_n(-x;-q^{-\alpha};q)}{(q;q)_n}=L_n^{(\alpha)}(x;q).$$ + +\noindent +Since the Stieltjes and Hamburger moment problems corresponding to the +$q$-Laguerre polynomials are indeterminate there exist many different weight +functions. + +\subsection*{References} +\cite{NAlSalam89}, \cite{AlSalam90}, \cite{Askey86}, \cite{Askey89I}, \cite{AskeyWilson85}, +\cite{AtakAtakI}, \cite{ChenIsmailMuttalib}, \cite{Chihara68II}, \cite{Chihara78}, +\cite{Chihara79}, \cite{Chris}, \cite{Exton77}, \cite{GasperRahman90}, \cite{GrunbaumHaine96}, +\cite{Ismail2005I}, \cite{IsmailRahman98}, \cite{Jain95}, \cite{Moak}. + + +\section{$q$-Bessel} +\index{q-Bessel polynomials@$q$-Bessel polynomials} +\par\setcounter{equation}{0} + +\subsection*{Basic hypergeometric representation} +\begin{eqnarray} +\label{DefqBessel} +y_n(x;a;q)&=&\qhyp{2}{1}{q^{-n},-aq^n}{0}{qx}\\ +&=&(q^{-n+1}x;q)_n\cdot\qhyp{1}{1}{q^{-n}}{q^{-n+1}x}{-aq^{n+1}x}\nonumber\\ +&=&\left(-aq^nx\right)^n\cdot\qhyp{2}{1}{q^{-n},x^{-1}}{0}{-\frac{q^{-n+1}}{a}}.\nonumber +\end{eqnarray} + +\newpage + +\subsection*{Orthogonality relation} +\begin{eqnarray} +\label{OrtqBessel} +& &\sum_{k=0}^{\infty}\frac{a^k}{(q;q)_k}q^{\binom{k+1}{2}}y_m(q^k;a;q)y_n(q^k;a;q)\nonumber\\ +& &{}=(q;q)_n(-aq^n;q)_{\infty}\frac{a^nq^{\binom{n+1}{2}}}{(1+aq^{2n})}\,\delta_{mn},\quad a>0. +\end{eqnarray} + +\subsection*{Recurrence relation} +\begin{equation} +\label{RecqBessel} +-xy_n(x;a;q)=A_ny_{n+1}(x;a;q)-(A_n+C_n)y_n(x;a;q)+C_ny_{n-1}(x;a;q), +\end{equation} +where +$$\left\{\begin{array}{l}\displaystyle A_n=q^n\frac{(1+aq^n)}{(1+aq^{2n})(1+aq^{2n+1})}\\ +\\ +\displaystyle C_n=aq^{2n-1}\frac{(1-q^n)}{(1+aq^{2n-1})(1+aq^{2n})}.\end{array}\right.$$ + +\subsection*{Normalized recurrence relation} +\begin{equation} +\label{NormRecqBessel} +xp_n(x)=p_{n+1}(x)+(A_n+C_n)p_n(x)+A_{n-1}C_np_{n-1}(x), +\end{equation} +where +$$y_n(x;a;q)=(-1)^nq^{-\binom{n}{2}}(-aq^n;q)_np_n(x).$$ + +\subsection*{$q$-Difference equation} +\begin{eqnarray} +\label{dvqBessel} +& &-q^{-n}(1-q^n)(1+aq^n)xy(x)\nonumber\\ +& &{}=axy(qx)-(ax+1-x)y(x)+(1-x)y(q^{-1}x), +\end{eqnarray} +where +$$y(x)=y_n(x;a;q).$$ + +\newpage + +\subsection*{Forward shift operator} +\begin{equation} +\label{shift1qBesselI} +y_n(x;a;q)-y_n(qx;a;q)=-q^{-n+1}(1-q^n)(1+aq^n)xy_{n-1}(x;aq^2;q) +\end{equation} +or equivalently +\begin{equation} +\label{shift1qBesselII} +\mathcal{D}_qy_n(x;a;q)=-\frac{q^{-n+1}(1-q^n)(1+aq^n)}{1-q}y_{n-1}(x;aq^2;q). +\end{equation} + +\subsection*{Backward shift operator} +\begin{equation} +\label{shift2qBesselI} +aq^{x-1}y_n(q^x;a;q)-(1-q^x)y_n(q^{x-1}x;a;q)=-y_{n+1}(q^x;aq^{-2};q) +\end{equation} +or equivalently +\begin{equation} +\label{shift2qBesselII} +\frac{\nabla\left[w(x;a;q)y_n(q^x;a;q)\right]}{\nabla q^x}= +\frac{q^2}{a(1-q)}w(x;aq^{-2};q)y_{n+1}(q^x;aq^{-2};q), +\end{equation} +where +$$w(x;a;q)=\frac{a^xq^{\binom{x}{2}}}{(q;q)_x}.$$ + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodqBessel} +w(x;a;q)y_n(q^x;a;q)=a^n(1-q)^nq^{n(n-1)}\left(\nabla_q\right)^n\left[w(x;aq^{2n};q)\right], +\end{equation} +where +$$\nabla_q:=\frac{\nabla}{\nabla q^x}.$$ + +\subsection*{Generating functions} +\begin{eqnarray} +\label{GenqBessel1} +& &\qhyp{0}{1}{-}{0}{-aq^{x+1}t}\,\qhyp{2}{0}{q^{-x},0}{-}{q^xt}\nonumber\\ +& &{}=\sum_{n=0}^{\infty}\frac{y_n(q^x;a;q)}{(q;q)_n}t^n,\quad x=0,1,2,\ldots. +\end{eqnarray} + +\begin{equation} +\label{GenqBessel2} +\frac{(t;q)_{\infty}}{(xt;q)_{\infty}}\,\qhyp{1}{3}{xt}{0,0,t}{-aqxt}= +\sum_{n=0}^{\infty}\frac{(-1)^nq^{\binom{n}{2}}}{(q;q)_n}y_n(x;a;q)t^n. +\end{equation} + +\subsection*{Limit relations} + +\subsubsection*{Little $q$-Jacobi $\rightarrow$ $q$-Bessel} +If we set $b\rightarrow -a^{-1}q^{-1}b$ in the definition (\ref{DefLittleqJacobi}) of +the little $q$-Jacobi polynomials and then take the limit $a\rightarrow 0$ we obtain +the $q$-Bessel polynomials given by (\ref{DefqBessel}): +$$\lim_{a\rightarrow 0}p_n(x;a,-a^{-1}q^{-1}b|q)=y_n(x;b;q).$$ + +\subsubsection*{$q$-Krawtchouk $\rightarrow$ $q$-Bessel} +If we set $x\rightarrow N-x$ in the definition (\ref{DefqKrawtchouk}) of the +$q$-Krawtchouk polynomials and then take the limit $N\rightarrow\infty$ we +obtain the $q$-Bessel polynomials given by (\ref{DefqBessel}): +$$\lim_{N\rightarrow\infty}K_n(q^{x-N};p,N;q)=y_n(q^x;p;q).$$ + +\subsubsection*{$q$-Bessel $\rightarrow$ Stieltjes-Wigert} +The Stieltjes-Wigert polynomials given by (\ref{DefStieltjesWigert}) can be obtained +from the $q$-Bessel polynomials by setting $x\rightarrow a^{-1}x$ in the definition +(\ref{DefqBessel}) of the $q$-Bessel polynomials and then taking the limit +$a\rightarrow\infty$. In fact we have +\begin{equation} +\lim_{a\rightarrow\infty}y_n(a^{-1}x;a;q)=(q;q)_nS_n(x;q). +\end{equation} + +\subsubsection*{$q$-Bessel $\rightarrow$ Bessel} +If we set $x\rightarrow -\frac{1}{2}(1-q)^{-1}x$ and $a\rightarrow -q^{a+1}$ in the definition +(\ref{DefqBessel}) of the $q$-Bessel polynomials and take the limit $q\rightarrow 1$ +we find the Bessel polynomials given by (\ref{DefBessel}): +\begin{equation} +\lim_{q\rightarrow 1}y_n(-\textstyle\frac{1}{2}(1-q)^{-1}x;-q^{a+1};q)=y_n(x;a). +\end{equation} + +\subsubsection*{$q$-Bessel $\rightarrow$ Charlier} +If we set $x\rightarrow q^x$ and $a\rightarrow a(1-q)$ in the definition (\ref{DefqBessel}) +of the $q$-Bessel polynomials and take the limit $q\rightarrow 1$ we find the Charlier +polynomials given by (\ref{DefCharlier}): +\begin{equation} +\lim_{q\rightarrow 1}\frac{y_n(q^x;a(1-q);q)}{(q-1)^n}=a^nC_n(x;a). +\end{equation} + +\subsection*{Remark} +In \cite{Koekoek94} and \cite{Koekoek98} these $q$-Bessel polynomials were called +\emph{alternative $q$-Charlier polynomials}. + +\noindent +The $q$-Bessel polynomials given by (\ref{DefqBessel}) and the $q$-Laguerre +polynomials given by (\ref{DefqLaguerre}) are related in the following way: +$$\frac{y_n(q^x;a;q)}{(q;q)_n}=L_n^{(x-n)}(aq^n;q).$$ + +\subsection*{Reference} +\cite{DattaGriffin}. + + +\section{$q$-Charlier}\index{q-Charlier polynomials@$q$-Charlier polynomials} +\par\setcounter{equation}{0} + +\subsection*{Basic hypergeometric representation} +\begin{eqnarray} +\label{DefqCharlier} +C_n(q^{-x};a;q)&=&\qhyp{2}{1}{q^{-n},q^{-x}}{0}{-\frac{q^{n+1}}{a}}\\ +&=&(-a^{-1}q;q)_n\cdot\qhyp{1}{1}{q^{-n}}{-a^{-1}q}{-\frac{q^{n+1-x}}{a}}.\nonumber +\end{eqnarray} + +\subsection*{Orthogonality relation} +\begin{eqnarray} +\label{OrtqCharlier} +& &\sum_{x=0}^{\infty}\frac{a^x}{(q;q)_x}q^{\binom{x}{2}}C_m(q^{-x};a;q)C_n(q^{-x};a;q)\nonumber\\ +& &{}=q^{-n}(-a;q)_{\infty}(-a^{-1}q,q;q)_n\,\delta_{mn},\quad a>0. +\end{eqnarray} + +\newpage + +\subsection*{Recurrence relation} +\begin{eqnarray} +\label{RecqCharlier} +& &q^{2n+1}(1-q^{-x})C_n(q^{-x})\nonumber\\ +& &{}=aC_{n+1}(q^{-x})-\left[a+q(1-q^n)(a+q^n)\right]C_n(q^{-x})\nonumber\\ +& &{}\mathindent{}+q(1-q^n)(a+q^n)C_{n-1}(q^{-x}), +\end{eqnarray} +where +$$C_n(q^{-x}):=C_n(q^{-x};a;q).$$ + +\subsection*{Normalized recurrence relation} +\begin{eqnarray} +\label{NormRecqCharlier} +xp_n(x)&=&p_{n+1}(x)+\left[1+q^{-2n-1}\left\{a+q(1-q^n)(a+q^n)\right\}\right]p_n(x)\nonumber\\ +& &{}\mathindent{}+aq^{-4n+1}(1-q^n)(a+q^n)p_{n-1}(x), +\end{eqnarray} +where +$$C_n(q^{-x};a;q)=\frac{(-1)^nq^{n^2}}{a^n}p_n(q^{-x}).$$ + +\subsection*{$q$-Difference equation} +\begin{equation} +\label{dvqCharlier} +q^ny(x)=aq^xy(x+1)-q^x(a-1)y(x)+(1-q^x)y(x-1), +\end{equation} +where +$$y(x)=C_n(q^{-x};a;q).$$ + +\subsection*{Forward shift operator} +\begin{equation} +\label{shift1qCharlierI} +C_n(q^{-x-1};a;q)-C_n(q^{-x};a;q)=-a^{-1}q^{-x}(1-q^n)C_{n-1}(q^{-x};aq^{-1};q) +\end{equation} +or equivalently +\begin{equation} +\label{shift1qCharlierII} +\frac{\Delta C_n(q^{-x};a;q)}{\Delta q^{-x}}=-\frac{q(1-q^n)}{a(1-q)}C_{n-1}(q^{-x};aq^{-1};q). +\end{equation} + +\newpage + +\subsection*{Backward shift operator} +\begin{equation} +\label{shift2qCharlierI} +C_n(q^{-x};a;q)-a^{-1}q^{-x}(1-q^x)C_n(q^{-x+1};a;q)=C_{n+1}(q^{-x};aq;q) +\end{equation} +or equivalently +\begin{equation} +\label{shift2qCharlierII} +\frac{\nabla\left[w(x;a;q)C_n(q^{-x};a;q)\right]}{\nabla q^{-x}} +=\frac{1}{1-q}w(x;aq;q)C_{n+1}(q^{-x};aq;q), +\end{equation} +where +$$w(x;a;q)=\frac{a^xq^{\binom{x+1}{2}}}{(q;q)_x}.$$ + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodqCharlier} +w(x;a;q)C_n(q^{-x};a;q)=(1-q)^n\left(\nabla_q\right)^n\left[w(x;aq^{-n};q)\right], +\end{equation} +where +$$\nabla_q:=\frac{\nabla}{\nabla q^{-x}}.$$ + +\subsection*{Generating functions} +\begin{equation} +\label{GenqCharlier1} +\frac{1}{(t;q)_{\infty}}\,\qhyp{1}{1}{q^{-x}}{0}{-a^{-1}qt} +=\sum_{n=0}^{\infty}\frac{C_n(q^{-x};a;q)}{(q;q)_n}t^n. +\end{equation} + +\begin{equation} +\label{GenqCharlier2} +\frac{1}{(t;q)_{\infty}}\,\qhyp{0}{1}{-}{-a^{-1}q}{-a^{-1}q^{-x+1}t} +=\sum_{n=0}^{\infty}\frac{C_n(q^{-x};a;q)}{(-a^{-1}q,q;q)_n}t^n. +\end{equation} + +\subsection*{Limit relations} + +\subsubsection*{$q$-Meixner $\rightarrow$ $q$-Charlier} +The $q$-Charlier polynomials given by (\ref{DefqCharlier}) can easily be +obtained from the $q$-Meixner given by (\ref{DefqMeixner}) by setting $b=0$ +in the definition (\ref{DefqMeixner}) of the $q$-Meixner polynomials: +\begin{equation} +M_n(x;0,c;q)=C_n(x;c;q). +\end{equation} + +\subsubsection*{$q$-Krawtchouk $\rightarrow$ $q$-Charlier} +By setting $p=a^{-1}q^{-N}$ in the definition (\ref{DefqKrawtchouk}) of the +$q$-Krawtchouk polynomials and then taking the limit $N\rightarrow\infty$ we +obtain the $q$-Charlier polynomials given by (\ref{DefqCharlier}): +$$\lim_{N\rightarrow\infty}K_n(q^{-x};a^{-1}q^{-N},N;q)=C_n(q^{-x};a;q).$$ + +\subsubsection*{$q$-Charlier $\rightarrow$ Stieltjes-Wigert} +If we set $q^{-x}\rightarrow ax$ in the definition (\ref{DefqCharlier}) of the +$q$-Charlier polynomials and take the limit $a\rightarrow\infty$ we obtain +the Stieltjes-Wigert polynomials given by (\ref{DefStieltjesWigert}) in the +following way: +\begin{equation} +\lim_{a\rightarrow\infty}C_n(ax;a;q)=(q;q)_nS_n(x;q). +\end{equation} + +\subsubsection*{$q$-Charlier $\rightarrow$ Charlier} +If we set $a\rightarrow a(1-q)$ in the definition (\ref{DefqCharlier}) +of the $q$-Charlier polynomials and take the limit $q\rightarrow 1$ we obtain the +Charlier polynomials given by (\ref{DefCharlier}): +\begin{equation} +\lim_{q\rightarrow 1}C_n(q^{-x};a(1-q);q)=C_n(x;a). +\end{equation} + +\subsection*{Remark} +The $q$-Charlier polynomials given by (\ref{DefqCharlier}) and the +$q$-Laguerre polynomials given by (\ref{DefqLaguerre}) are related in the +following way: +$$\frac{C_n(-x;-q^{-\alpha};q)}{(q;q)_n}=L_n^{(\alpha)}(x;q).$$ + +\subsection*{References} +\cite{AlvarezRonveaux}, \cite{AtakRahmanSuslov}, \cite{GasperRahman90}, +\cite{Hahn}, \cite{Koelink96III}, \cite{Nikiforov+}, \cite{Zeng95}. + + +\section{Al-Salam-Carlitz~I}\index{Al-Salam-Carlitz~I polynomials} +\par\setcounter{equation}{0} + +\subsection*{Basic hypergeometric representation} +\begin{equation} +\label{DefAlSalamCarlitzI} +U_n^{(a)}(x;q)=(-a)^nq^{\binom{n}{2}}\,\qhyp{2}{1}{q^{-n},x^{-1}}{0}{\frac{qx}{a}}. +\end{equation} + +\subsection*{Orthogonality relation} +\begin{eqnarray} +\label{OrtAlSalamCarlitzI} +& &\int_a^1(qx,a^{-1}qx;q)_{\infty}U_m^{(a)}(x;q)U_n^{(a)}(x;q)\,d_qx\nonumber\\ +& &{}=(-a)^n(1-q)(q;q)_n(q,a,a^{-1}q;q)_{\infty}q^{\binom{n}{2}}\,\delta_{mn},\quad a<0. +\end{eqnarray} + +\subsection*{Recurrence relation} +\begin{equation} +\label{RecAlSalamCarlitzI} +xU_n^{(a)}(x;q)=U_{n+1}^{(a)}(x;q)+(a+1)q^nU_n^{(a)}(x;q) +-aq^{n-1}(1-q^n)U_{n-1}^{(a)}(x;q). +\end{equation} + +\subsection*{Normalized recurrence relation} +\begin{equation} +\label{NormRecAlSalamCarlitzI} +xp_n(x)=p_{n+1}(x)+(a+1)q^np_n(x)-aq^{n-1}(1-q^n)p_{n-1}(x), +\end{equation} +where +$$U_n^{(a)}(x;q)=p_n(x).$$ + +\subsection*{$q$-Difference equation} +\begin{eqnarray} +\label{dvAlSalamCarlitzI} +(1-q^n)x^2y(x)&=&aq^{n-1}y(qx)-\left[aq^{n-1}+q^n(1-x)(a-x)\right]y(x)\nonumber\\ +& &{}\mathindent{}+q^n(1-x)(a-x)y(q^{-1}x), +\end{eqnarray} +where +$$y(x)=U_n^{(a)}(x;q).$$ + +\subsection*{Forward shift operator} +\begin{equation} +\label{shift1AlSalamCarlitzI-I} +U_n^{(a)}(x;q)-U_n^{(a)}(qx;q)=(1-q^n)xU_{n-1}^{(a)}(x;q) +\end{equation} +or equivalently +\begin{equation} +\label{shift1AlSalamCarlitzI-II} +\mathcal{D}_qU_n^{(a)}(x;q)=\frac{1-q^n}{1-q}U_{n-1}^{(a)}(x;q). +\end{equation} + +\subsection*{Backward shift operator} +\begin{equation} +\label{shift2AlSalamCarlitzI-I} +aU_n^{(a)}(x;q)-(1-x)(a-x)U_n^{(a)}(q^{-1}x;q)=-q^{-n}xU_{n+1}^{(a)}(x;q) +\end{equation} +or equivalently +\begin{equation} +\label{shift2AlSalamCarlitzI-II} +\mathcal{D}_{q^{-1}}\left[w(x;a;q)U_n^{(a)}(x;q)\right]= +\frac{q^{-n+1}}{a(1-q)}w(x;a;q)U_{n+1}^{(a)}(x;q), +\end{equation} +where +$$w(x;a;q)=(qx,a^{-1}qx;q)_{\infty}.$$ + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodAlSalamCarlitzI} +w(x;a;q)U_n^{(a)}(x;q)=a^nq^{\frac{1}{2}n(n-3)}(1-q)^n +\left(\mathcal{D}_{q^{-1}}\right)^n\left[w(x;a;q)\right]. +\end{equation} + +\subsection*{Generating function} +\begin{equation} +\label{GenAlSalamCarlitzI} +\frac{(t,at;q)_{\infty}}{(xt;q)_{\infty}}= +\sum_{n=0}^{\infty}\frac{U_n^{(a)}(x;q)}{(q;q)_n}t^n. +\end{equation} + +\subsection*{Limit relations} + +\subsubsection*{Big $q$-Laguerre $\rightarrow$ Al-Salam-Carlitz~I} +If we set $x\rightarrow aqx$ and $b\rightarrow ab$ in the definition +(\ref{DefBigqLaguerre}) of the big $q$-Laguerre polynomials and take the +limit $a\rightarrow 0$ we obtain the Al-Salam-Carlitz~I polynomials given by +(\ref{DefAlSalamCarlitzI}): +$$\lim_{a\rightarrow 0}\frac{P_n(aqx;a,ab;q)}{a^n}=q^nU_n^{(b)}(x;q).$$ + +\subsubsection*{Dual $q$-Krawtchouk $\rightarrow$ Al-Salam-Carlitz~I} +If we set $c=a^{-1}$ in the definition (\ref{DefDualqKrawtchouk}) +of the dual $q$-Krawtchouk polynomials and take the limit +$N\rightarrow\infty$ we simply obtain the Al-Salam-Carlitz~I polynomials +given by (\ref{DefAlSalamCarlitzI}): +$$\lim_{N\rightarrow\infty}K_n(\lambda(x);a^{-1},N|q)= +\left(-\frac{1}{a}\right)^nq^{-\binom{n}{2}}U_n^{(a)}(q^x;q).$$ +Note that $\lambda(x)=q^{-x}+a^{-1}q^{x-N}$. + +\subsubsection*{Al-Salam-Carlitz~I $\rightarrow$ Discrete $q$-Hermite~I} +The discrete $q$-Hermite~I polynomials given by +(\ref{DefDiscreteqHermiteI}) can easily be obtained from the +Al-Salam-Carlitz~I polynomials given by (\ref{DefAlSalamCarlitzI}) by the +substitution $a=-1$: +\begin{equation} +U_n^{(-1)}(x;q)=h_n(x;q). +\end{equation} + +\subsubsection*{Al-Salam-Carlitz~I $\rightarrow$ Charlier / Hermite} +If we set $a\rightarrow a(q-1)$ and $x\rightarrow q^x$ in the definition +(\ref{DefAlSalamCarlitzI}) of the Al-Salam-Carlitz~I polynomials and take the +limit $q\rightarrow 1$ after dividing by $a^n(1-q)^n$ we obtain the Charlier +polynomials given by (\ref{DefCharlier}): +\begin{equation} +\lim_{q\rightarrow 1}\frac{U_n^{(a(q-1))}(q^x;q)}{(1-q)^n}=a^nC_n(x;a). +\end{equation} + +If we set $x\rightarrow x\sqrt{1-q^2}$ and $a\rightarrow a\sqrt{1-q^2}-1$ +in the definition (\ref{DefAlSalamCarlitzI}) of the Al-Salam-Carlitz~I +polynomials, divide by $(1-q^2)^{\frac{n}{2}}$, and let $q$ tend to $1$ we +obtain the Hermite polynomials given by (\ref{DefHermite}) with shifted +argument. In fact we have +\begin{equation} +\lim_{q\rightarrow 1}\frac{U_n^{(a\sqrt{1-q^2}-1)}(x\sqrt{1-q^2};q)} +{(1-q^2)^{\frac{n}{2}}}=\frac{H_n(x-a)}{2^n}. +\end{equation} + +\subsection*{Remark} +The Al-Salam-Carlitz~I polynomials are related to the Al-Salam-Carlitz~II +polynomials given by (\ref{DefAlSalamCarlitzII}) in the following way: +$$U_n^{(a)}(x;q^{-1})=V_n^{(a)}(x;q).$$ + +\subsection*{References} +\cite{AlSalam90}, \cite{AlSalamCarlitz}, \cite{AlSalamChihara76}, \cite{AskeySuslovII}, +\cite{AtakRahmanSuslov}, \cite{Chihara68II}, \cite{Chihara78}, \cite{DattaGriffin}, +\cite{Dehesa}, \cite{DohaAhmed2005}, \cite{GasperRahman90}, \cite{Ismail85I}, +\cite{IsmailMuldoon}, \cite{Kim}, \cite{Zeng95}. + + +\section{Al-Salam-Carlitz~II}\index{Al-Salam-Carlitz~II polynomials} +\par\setcounter{equation}{0} + +\subsection*{Basic hypergeometric representation} +\begin{equation} +\label{DefAlSalamCarlitzII} +V_n^{(a)}(x;q)= +(-a)^nq^{-\binom{n}{2}}\,\qhyp{2}{0}{q^{-n},x}{-}{\frac{q^n}{a}}. +\end{equation} + +\subsection*{Orthogonality relation} +\begin{eqnarray} +\label{OrtAlSalamCarlitzII} +& &\sum_{k=0}^{\infty}\frac{q^{k^2}a^k}{(q;q)_k(aq;q)_k} +V_m^{(a)}(q^{-k};q)V_n^{(a)}(q^{-k};q)\nonumber\\ +& &{}=\frac{(q;q)_na^n}{(aq;q)_{\infty}q^{n^2}}\,\delta_{mn},\quad 00,\quad\textrm{with}\quad +\gamma^2=-\frac{1}{2\ln q}.$$ + +\subsection*{References} +\cite{Askey86}, \cite{Askey89I}, \cite{AtakAtakIII}, \cite{Chihara70}, \cite{Chihara78}, +\cite{Dehesa}, \cite{Ismail2005I}, \cite{Nikiforov+}, \cite{Stieltjes}, \cite{Szego75}, +\cite{ValentAssche}, \cite{Wigert}. + + +\section{Discrete $q$-Hermite~I} +\index{Discrete q-Hermite~I polynomials@Discrete $q$-Hermite~I polynomials} +\index{q-Hermite~I polynomials@$q$-Hermite~I polynomials!Discrete} +\par\setcounter{equation}{0} + +\subsection*{Basic hypergeometric representation} +The discrete $q$-Hermite~I polynomials are Al-Salam-Carlitz~I polynomials +with $a=-1$: +\begin{eqnarray} +\label{DefDiscreteqHermiteI} +h_n(x;q)=U_n^{(-1)}(x;q)&=&q^{\binom{n}{2}}\,\qhyp{2}{1}{q^{-n},x^{-1}}{0}{-qx}\\ +&=&x^n\qhypK{2}{0}{q^{-n},q^{-n+1}}{0}{q^2,\frac{q^{2n-1}}{x^2}}\nonumber +\end{eqnarray} + +\subsection*{Orthogonality relation} +\begin{eqnarray} +\label{OrtDiscreteqHermiteI} +& &\int_{-1}^1(qx,-qx;q)_{\infty}h_m(x;q)h_n(x;q)\,d_qx\nonumber\\ +& &{}=(1-q)(q;q)_n(q,-1,-q;q)_{\infty}q^{\binom{n}{2}}\,\delta_{mn}. +\end{eqnarray} + +\subsection*{Recurrence relation} +\begin{equation} +\label{RecDiscreteqHermiteI} +xh_n(x;q)=h_{n+1}(x;q)+q^{n-1}(1-q^n)h_{n-1}(x;q). +\end{equation} + +\subsection*{Normalized recurrence relation} +\begin{equation} +\label{NormRecDiscreteqHermiteI} +xp_n(x)=p_{n+1}(x)+q^{n-1}(1-q^n)p_{n-1}(x), +\end{equation} +where +$$h_n(x;q)=p_n(x).$$ + +\subsection*{$q$-Difference equation} +\begin{equation} +\label{dvDiscreteqHermiteI} +-q^{-n+1}x^2y(x)=y(qx)-(1+q)y(x)+q(1-x^2)y(q^{-1}x), +\end{equation} +where +$$y(x)=h_n(x;q).$$ + +\subsection*{Forward shift operator} +\begin{equation} +\label{shift1DiscreteqHermiteI-I} +h_n(x;q)-h_n(qx;q)=(1-q^n)xh_{n-1}(x;q) +\end{equation} +or equivalently +\begin{equation} +\label{shift1DiscreteqHermiteI-II} +\mathcal{D}_qh_n(x;q)=\frac{1-q^n}{1-q}h_{n-1}(x;q). +\end{equation} + +\subsection*{Backward shift operator} +\begin{equation} +\label{shift2DiscreteqHermiteI-I} +h_n(x;q)-(1-x^2)h_n(q^{-1}x;q)=q^{-n}xh_{n+1}(x;q) +\end{equation} +or equivalently +\begin{equation} +\label{shift2DiscreteqHermiteI-II} +\mathcal{D}_{q^{-1}}\left[w(x;q)h_n(x;q)\right] +=-\frac{q^{-n+1}}{1-q}w(x;q)h_{n+1}(x;q), +\end{equation} +where +$$w(x;q)=(qx,-qx;q)_{\infty}.$$ + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodDiscreteqHermiteI} +w(x;q)h_n(x;q)=(q-1)^nq^{\frac{1}{2}n(n-3)} +\left(\mathcal{D}_{q^{-1}}\right)^n\left[w(x;q)\right]. +\end{equation} + +\subsection*{Generating function} +\begin{equation} +\label{GenDiscreteqHermiteI} +\frac{(t^2;q^2)_{\infty}}{(xt;q)_{\infty}}= +\sum_{n=0}^{\infty}\frac{h_n(x;q)}{(q;q)_n}t^n. +\end{equation} + +\subsection*{Limit relations} + +\subsubsection*{Al-Salam-Carlitz~I $\rightarrow$ Discrete $q$-Hermite~I} +The discrete $q$-Hermite~I polynomials given by +(\ref{DefDiscreteqHermiteI}) can easily be obtained from the +Al-Salam-Carlitz~I polynomials given by (\ref{DefAlSalamCarlitzI}) by the +substitution $a=-1$: +$$U_n^{(-1)}(x;q)=h_n(x;q).$$ + +\subsubsection*{Discrete $q$-Hermite~I $\rightarrow$ Hermite} +The Hermite polynomials given by (\ref{DefHermite}) can be found +from the discrete $q$-Hermite~I polynomials given by +(\ref{DefDiscreteqHermiteI}) in the following way: +\begin{equation} +\lim_{q\rightarrow 1}\frac{h_n(x\sqrt{1-q^2};q)} +{(1-q^2)^{\frac{n}{2}}}=\frac{H_n(x)}{2^n}. +\end{equation} + +\subsection*{Remark} The discrete $q$-Hermite~I polynomials are related to the +discrete $q$-Hermite~II polynomials given by (\ref{DefDiscreteqHermiteII}) +in the following way: +$$h_n(ix;q^{-1})=i^n{\tilde h}_n(x;q).$$ + +\subsection*{References} +\cite{AlSalam90}, \cite{AlSalamCarlitz}, \cite{AtakRahmanSuslov}, +\cite{BergIsmail}, \cite{BustozIsmail82}, \cite{GasperRahman90}, \cite{Hahn}, +\cite{Koorn97}. + + +\newpage + +\section{Discrete $q$-Hermite~II} +\index{Discrete q-Hermite~II polynomials@Discrete $q$-Hermite~II polynomials} +\index{q-Hermite~II polynomials@$q$-Hermite~II polynomials!Discrete} +\par\setcounter{equation}{0} + +\subsection*{Basic hypergeometric representation} +The discrete $q$-Hermite~II polynomials are Al-Salam-Carlitz~II polynomials +with $a=-1$: +\begin{eqnarray} +\label{DefDiscreteqHermiteII} +{\tilde h}_n(x;q)=i^{-n}V_n^{(-1)}(ix;q) +&=&i^{-n}q^{-\binom{n}{2}}\,\qhyp{2}{0}{q^{-n},ix}{-}{-q^n}\\ +&=&x^n +\qhypK{2}{1}{q^{-n},q^{-n+1}}{0}{q^2,-\frac{q^2}{x^2}}\nonumber +\end{eqnarray} + +\subsection*{Orthogonality relation} +\begin{eqnarray} +\label{OrtDiscreteqHermiteIIa} +& &\sum_{k=-\infty}^{\infty}\left[{\tilde h}_m(cq^k;q) +{\tilde h}_n(cq^k;q)+{\tilde h}_m(-cq^k;q){\tilde h}_n(-cq^k;q)\right] +w(cq^k;q)q^k\nonumber\\ +& &{}=2\frac{(q^2,-c^2q,-c^{-2}q;q^2)_{\infty}}{(q,-c^2,-c^{-2}q^2;q^2)_{\infty}} +\frac{(q;q)_n}{q^{n^2}}\,\delta_{mn},\quad c>0, +\end{eqnarray} +where +$$w(x;q)=\frac{1}{(ix,-ix;q)_{\infty}}=\frac{1}{(-x^2;q^2)_{\infty}}.$$ +For $c=1$ this orthogonality relation can also be written as +\begin{equation} +\label{OrtDiscreteqHermiteIIb} +\int_{-\infty}^{\infty}\frac{{\tilde h}_m(x;q){\tilde h}_n(x;q)}{(-x^2;q^2)_{\infty}}\,d_qx +=\frac{\left(q^2,-q,-q;q^2\right)_{\infty}}{\left(q^3,-q^2,-q^2;q^2\right)_{\infty}} +\frac{(q;q)_n}{q^{n^2}}\,\delta_{mn}. +\end{equation} + +\subsection*{Recurrence relation} +\begin{equation} +\label{RecDiscreteqHermiteII} +x{\tilde h}_n(x;q)={\tilde h}_{n+1}(x;q)+q^{-2n+1}(1-q^n){\tilde h}_{n-1}(x;q). +\end{equation} + +\subsection*{Normalized recurrence relation} +\begin{equation} +\label{NormRecDiscreteqHermiteII} +xp_n(x)=p_{n+1}(x)+q^{-2n+1}(1-q^n)p_{n-1}(x), +\end{equation} +where +$${\tilde h}_n(x;q)=p_n(x).$$ + +\subsection*{$q$-Difference equation} +\begin{eqnarray} +\label{dvDiscreteqHermiteII} +& &-(1-q^n)x^2{\tilde h}_n(x;q)\nonumber\\ +& &{}=(1+x^2){\tilde h}_n(qx;q)-(1+x^2+q){\tilde h}_n(x;q)+q{\tilde h}_n(q^{-1}x;q). +\end{eqnarray} + +\subsection*{Forward shift operator} +\begin{equation} +\label{shift1DiscreteqHermiteII-I} +\tilde{h}_n(x;q)-\tilde{h}_n(qx;q)=q^{-n+1}(1-q^n)x\tilde{h}_{n-1}(qx;q) +\end{equation} +or equivalently +\begin{equation} +\label{shift1DiscreteqHermiteII-II} +\mathcal{D}_q\tilde{h}_n(x;q)=\frac{q^{-n+1}(1-q^n)}{1-q}\tilde{h}_{n-1}(qx;q). +\end{equation} + +\subsection*{Backward shift operator} +\begin{equation} +\label{shift2DiscreteqHermiteII-I} +\tilde{h}_n(x;q)-(1+x^2)\tilde{h}_n(qx;q)=-q^nx\tilde{h}_{n+1}(x;q) +\end{equation} +or equivalently +\begin{equation} +\label{shift2DiscreteqHermiteII-II} +\mathcal{D}_q\left[w(x;q)\tilde{h}_n(x;q)\right] +=-\frac{q^n}{1-q}w(x;q)\tilde{h}_{n+1}(x;q). +\end{equation} + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodDiscreteqHermiteII} +w(x;q)\tilde{h}_n(x;q)=(q-1)^nq^{-\binom{n}{2}} +\left(\mathcal{D}_q\right)^n\left[w(x;q)\right]. +\end{equation} + +\subsection*{Generating functions} +\begin{equation} +\label{GenDiscreteqHermiteII1} +\frac{(-xt;q)_{\infty}}{(-t^2;q^2)_{\infty}}=\sum_{n=0}^{\infty} +\frac{q^{\binom{n}{2}}}{(q;q)_n}{\tilde h}_n(x;q)t^n. +\end{equation} + +\begin{equation} +\label{GenDiscreteqHermiteII2} +(-it;q)_{\infty}\cdot\qhyp{1}{1}{ix}{-it}{it}=\sum_{n=0}^{\infty} +\frac{(-1)^nq^{n(n-1)}}{(q;q)_n}{\tilde h}_n(x;q)t^n. +\end{equation} + +\subsection*{Limit relations} + +\subsubsection*{Al-Salam-Carlitz~II $\rightarrow$ Discrete $q$-Hermite~II} +The discrete $q$-Hermite~II polynomials given by +(\ref{DefDiscreteqHermiteII}) follow from the Al-Salam-Carlitz~II +polynomials given by (\ref{DefAlSalamCarlitzII}) by the substitution $a=-1$ +in the following way: +$$i^{-n}V_n^{(-1)}(ix;q)={\tilde h}_n(x;q).$$ + +\subsubsection*{Discrete $q$-Hermite~II $\rightarrow$ Hermite} +The Hermite polynomials given by (\ref{DefHermite}) can be found +from the discrete $q$-Hermite~II polynomials given by +(\ref{DefDiscreteqHermiteII}) in the following way: +\begin{equation} +\lim_{q\rightarrow 1}\frac{{\tilde h}_n(x\sqrt{1-q^2};q)} +{(1-q^2)^{\frac{n}{2}}}=\frac{H_n(x)}{2^n}. +\end{equation} + +\subsection*{Remark} The discrete $q$-Hermite~II polynomials are related to the +discrete $q$-Hermite~I polynomials given by (\ref{DefDiscreteqHermiteI}) +in the following way: +$${\tilde h}_n(x;q^{-1})=i^{-n}h_n(ix;q).$$ + +\subsection*{References} +\cite{BergIsmail}, \cite{Koorn97}. + +\end{document} diff --git a/KLSadd_insertion/updateChapters.py b/KLSadd_insertion/updateChapters.py new file mode 100644 index 0000000..f7f40fe --- /dev/null +++ b/KLSadd_insertion/updateChapters.py @@ -0,0 +1,117 @@ +""" +Rahul Shah +self-taught python don't kill me I know its bad +2/5/16 +Re-write of linetest.py so people other than me can read it + +This project was/is being written to streamline the update process of the book used +for the online repository, updated via the KLSadd addendum file which only affects chapters +9 and 14 + +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +NOTE! AS OF 2/8/16 THIS FILE IS NOT YET UPDATED TO THE FULL CAPACITY OF THE PREVIOUS FILE. IF FURTHER DEVELOPMENT IS NEEDED, REFER TO THE +linetest.py FILE IF THIS FILE DOES NOT ADDRESS THE NEW/EXTRA GOALS. This means that XCITE PARSE etc HAS NOT BEEN ADDED + +Current update status: incomplete + +Goals:change the book chapter files to include paragraphs from the addendum, and after that insert some edits so they are intelligently integrated into the chapter +""" + +#start out by reading KLSadd.tex to get all of the paragraphs that must be added to the chapter files +#also keep track of which chapter each one is in + +#variables: chapNums for the chapter number each section belongs to paras will hold the sections that will be copied over +#chapNums is needed to know which file to open (9 or 14) +#mathPeople is just the name for the sections that are used, like Wilson, Racah, etc. I know some are not people, its just a var name +#yes I like lists +chapNums = [] +paras = [] +mathPeople = [] + +#method to find the indices of the reference paragraphs +def findReferences(str): + references = [] + index = -1 + canAdd = False + for word in str: + index+=1 + #check sections and subsections + if("section{" in word or "subsection*{" in word): + w = word[word.find("{")+1: word.find("}")] + if(w not in mathPeople): + canAdd = True + if("\\subsection*{References}" in word and canAdd == True): + references.append(index) + canAdd = False + return references + +#method to change file string(actually a list right now), returns string to be written to file +def fixChapter(str, references, p): + #str is the file string, references is the specific references for the file, and p is the paras variable(not sure if needed) + #TODO: OPTIMIZE + count = 0 #count is used to represent the values in count + for i in references: + #add + str[i-3] += p[count] + count+=1 + #I actually don't remember why I had to do this but I think you need it + str[i-1] += p[count] + return str + +#open the KLSadd file to do things with +with open("KLSadd.tex", "r") as add: + #store the file as a string + addendum = add.readlines() + index = 0 + indexes = [] + for word in addendum: + index+=1 + if("." in word and "\\subsection*{" in word): + name = word[word.find(" "): word.find("}")] + mathPeople.append(name) + if("9." in word): + chapNums.append(9) + if("14." in word): + chapNums.append(9) + #get the index + indexes.append(index-1) + #now indexes holds all of the places there is a section + #using these indexes, get all of the words in between and add that to the paras[] + for i in range(len(indexes)-1): + str = ''.join(addendum[indexes[i]: indexes[i+1]-1]) + paras.append(str) + + #paras now holds the paragraphs that need to go into the chapter files, but they need to go in the appropriate + #section(like Wilson, Racah, Hahn, etc.) so we use the mathPeople variable + #we can use the section names to place the relevant paragraphs in the right place + + #as of 2/8/16 the paragraphs will go before the References paragraph of the relevant section + #parse both files 9 and 14 as strings + + #chapter 9 + with open("tempchap9.tex", "r") as ch9: + entire9 = ch9.readlines() #reads in as a list of strings + ch9.close() + + #chapter 14 + with open("tempchap14.tex", "r") as ch14: + entire14 = ch14.readlines() + ch14.close() + #call the findReferences method to find the index of the References paragraph in the two file strings + #two variables for the references lists one for chapter 9 one for chapter 14 + references9 = findReferences(entire9) + references14 = findReferences(entire14) + + #call the fixChapter method to get a list with the addendum paragraphs added in + str9 = ''.join(fixChapter(entire9, references9, paras)) + str14 = ''.join(fixChapter(entire14, references14, paras)) + + #write to file + #new output files for safety + with open("updated9.tex","w") as temp9: + temp9.write(str9) + temp9.close() + + with open("updated14.tex", "w") as temp14: + temp14.write(str9) + diff --git a/KLSadd_insertion/updated14.tex b/KLSadd_insertion/updated14.tex new file mode 100644 index 0000000..e1b34b5 --- /dev/null +++ b/KLSadd_insertion/updated14.tex @@ -0,0 +1,4162 @@ +\documentclass[envcountchap,graybox]{svmono} + +\addtolength{\textwidth}{1mm} + +\usepackage{amsmath,amssymb} + +\usepackage{amsfonts} +%\usepackage{breqn} +\usepackage{DLMFmath} +\usepackage{DRMFfcns} + +\usepackage{mathptmx} +\usepackage{helvet} +\usepackage{courier} + +\usepackage{makeidx} +\usepackage{graphicx} + +\usepackage{multicol} +\usepackage[bottom]{footmisc} + +\makeindex + +\def\bibname{Bibliography} +\def\refname{Bibliography} + +\def\theequation{\thesection.\arabic{equation}} + +\smartqed + +\let\corollary=\undefined +\spnewtheorem{corollary}[theorem]{Corollary}{\bfseries}{\itshape} + +\newcounter{rom} + +\newcommand{\hyp}[5]{\mbox{}_{#1}{F}_{#2} +\left(\genfrac{}{}{0pt}{}{#3}{#4}\,;\,#5\right)} + +\newcommand{\qhyp}[5]{\mbox{}_{#1}{\phi}_{#2} +\left(\genfrac{}{}{0pt}{}{#3}{#4}\,;\,q,\,#5\right)} + +\newcommand{\mathindent}{\hspace{7.5mm}} + +\newcommand{\e}{\textrm{e}} + +\renewcommand{\E}{\textrm{E}} + +\renewcommand{\textfraction}{-1} + +\renewcommand{\Gamma}{\varGamma} + +\renewcommand{\leftlegendglue}{\hfil} + +\settowidth{\tocchpnum}{14\enspace} +\settowidth{\tocsecnum}{14.30\enspace} +\settowidth{\tocsubsecnum}{14.12.1\enspace} + +\makeatletter +\def\cleartoversopage{\clearpage\if@twoside\ifodd\c@page + \hbox{}\newpage\if@twocolumn\hbox{}\newpage\fi + \else\fi\fi} + +\newcommand{\clearemptyversopage}{ + \clearpage{\pagestyle{empty}\cleartoversopage}} +\makeatother + +\oddsidemargin -1.5cm +\topmargin -2.0cm +\textwidth 16.3cm +\textheight 25cm + +\begin{document} + +\author{Roelof Koekoek\\[2.5mm]Peter A. Lesky\\[2.5mm]Ren\'e F. Swarttouw} +\title{Hypergeometric orthogonal polynomials and their\\$q$-analogues} +\subtitle{-- Monograph --} +\maketitle + +\frontmatter + +\large + +\addtocounter{chapter}{8} +\pagenumbering{roman} +\chapter{Hypergeometric orthogonal polynomials} +\label{HyperOrtPol} + + +In this chapter we deal with all families of hypergeometric orthogonal polynomials +appearing in the Askey scheme on page~\pageref{scheme}. For each family of orthogonal +polynomials we state the most important properties such as a representation as a +hypergeometric function, orthogonality relation(s), the three-term recurrence relation, +the second-order differential or difference equation, the forward shift (or degree lowering) +and backward shift (or degree raising) operator, a Rodrigues-type formula and some generating +functions. In each case we use the notation which seems to be most common in the literature. +Moreover, in each case we mention the connection between various families by stating the +appropriate limit relations. See also \cite{Terwilliger2006} for an algebraic approach of +this Askey scheme and \cite{TemmeLopez2001} for a view from asymptotic analysis. +For notations the reader is referred to chapter~\ref{Definitions}. + +\section{Wilson}\index{Wilson polynomials} + +\par + +\subsection*{Hypergeometric representation} +\begin{eqnarray} +\label{DefWilson} +& &\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\nonumber\\ +& &{}=\hyp{4}{3}{-n,n+a+b+c+d-1,a+ix,a-ix}{a+b,a+c,a+d}{1}. +\end{eqnarray} + +\newpage + +\subsection*{Orthogonality relation} +If $Re(a,b,c,d)>0$ and non-real parameters occur in conjugate pairs, then +\begin{eqnarray} +\label{OrthIWilson} +& &\frac{1}{2\pi}\int_0^{\infty} +\left|\frac{\Gamma(a+ix)\Gamma(b+ix)\Gamma(c+ix)\Gamma(d+ix)}{\Gamma(2ix)}\right|^2\nonumber\\ +& &{}\mathindent\times W_m(x^2;a,b,c,d)W_n(x^2;a,b,c,d)\,dx\nonumber\\ +& &{}=\frac{\Gamma(n+a+b)\cdots\Gamma(n+c+d)}{\Gamma(2n+a+b+c+d)}(n+a+b+c+d-1)_nn!\,\delta_{mn}, +\end{eqnarray} +where +\begin{eqnarray*} +& &\Gamma(n+a+b)\cdots\Gamma(n+c+d)\\ +& &{}=\Gamma(n+a+b)\Gamma(n+a+c)\Gamma(n+a+d)\Gamma(n+b+c)\Gamma(n+b+d)\Gamma(n+c+d). +\end{eqnarray*} +If $a<0$ and $a+b$, $a+c$, $a+d$ are positive or a pair of complex conjugates +occur with positive real parts, then +\begin{eqnarray} +\label{OrtIIWilson} +& &\frac{1}{2\pi}\int_0^{\infty}\left|\frac{\Gamma(a+ix)\Gamma(b+ix)\Gamma(c+ix)\Gamma(d+ix)}{\Gamma(2ix)}\right|^2\nonumber\\ +& &{}\mathindent\mathindent\times W_m(x^2;a,b,c,d)W_n(x^2;a,b,c,d)\,dx\nonumber\\ +& &{}\mathindent{}+\frac{\Gamma(a+b)\Gamma(a+c)\Gamma(a+d)\Gamma(b-a)\Gamma(c-a)\Gamma(d-a)}{\Gamma(-2a)}\nonumber\\ +& &{}\mathindent\mathindent{}\times\sum_{\begin{array}{c} +{\scriptstyle k=0,1,2\ldots}\\{\scriptstyle a+k<0}\end{array}} +\frac{(2a)_k(a+1)_k(a+b)_k(a+c)_k(a+d)_k}{(a)_k(a-b+1)_k(a-c+1)_k(a-d+1)_kk!}\nonumber\\ +& &{}\mathindent\mathindent\mathindent\mathindent{}\times W_m(-(a+k)^2;a,b,c,d)W_n(-(a+k)^2;a,b,c,d)\nonumber\\ +& &{}=\frac{\Gamma(n+a+b)\cdots\Gamma(n+c+d)}{\Gamma(2n+a+b+c+d)}(n+a+b+c+d-1)_nn!\,\delta_{mn}. +\end{eqnarray} + +\subsection*{Recurrence relation} +\begin{equation} +\label{RecWilson} +-\left(a^2+x^2\right){\tilde{W}}_n(x^2) +=A_n{\tilde{W}}_{n+1}(x^2)-\left(A_n+C_n\right){\tilde{W}}_n(x^2)+C_n{\tilde{W}}_{n-1}(x^2), +\end{equation} +where +$${\tilde{W}}_n(x^2):={\tilde{W}}_n(x^2;a,b,c,d)=\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}$$ +and +$$\left\{\begin{array}{l} +\displaystyle A_n=\frac{(n+a+b+c+d-1)(n+a+b)(n+a+c)(n+a+d)}{(2n+a+b+c+d-1)(2n+a+b+c+d)}\\[5mm] +\displaystyle C_n=\frac{n(n+b+c-1)(n+b+d-1)(n+c+d-1)}{(2n+a+b+c+d-2)(2n+a+b+c+d-1)}. +\end{array}\right.$$ + +\subsection*{Normalized recurrence relation} +\begin{equation} +\label{NormRecWilson} +xp_n(x)=p_{n+1}(x)+(A_n+C_n-a^2)p_n(x)+A_{n-1}C_np_{n-1}(x), +\end{equation} +where +$$W_n(x^2;a,b,c,d)=(-1)^n(n+a+b+c+d-1)_np_n(x^2).$$ + +\subsection*{Difference equation} +\begin{eqnarray} +\label{dvWilson} +& &n(n+a+b+c+d-1)y(x)\nonumber\\ +& &{}=B(x)y(x+i)-\left[B(x)+D(x)\right]y(x)+D(x)y(x-i), +\end{eqnarray} +where +$$y(x)=W_n(x^2;a,b,c,d)$$ +and +$$\left\{\begin{array}{l} +\displaystyle B(x)=\frac{(a-ix)(b-ix)(c-ix)(d-ix)}{2ix(2ix-1)}\\[5mm] +\displaystyle D(x)=\frac{(a+ix)(b+ix)(c+ix)(d+ix)}{2ix(2ix+1)}. +\end{array}\right.$$ + +\subsection*{Forward shift operator} +\begin{eqnarray} +\label{shift1WilsonI} +& &W_n((x+\textstyle\frac{1}{2}i)^2;a,b,c,d) +-W_n((x-\textstyle\frac{1}{2}i)^2;a,b,c,d)\nonumber\\ +& &{}=-2inx(n+a+b+c+d-1)W_{n-1}(x^2;a+\textstyle\frac{1}{2},b+\textstyle\frac{1}{2}, +c+\textstyle\frac{1}{2},d+\textstyle\frac{1}{2}) +\end{eqnarray} +or equivalently +\begin{eqnarray} +\label{shift1WilsonII} +& &\frac{\delta W_n(x^2;a,b,c,d)}{\delta x^2}\nonumber\\ +& &{}=-n(n+a+b+c+d-1)W_{n-1}(x^2;a+\textstyle\frac{1}{2},b+\textstyle\frac{1}{2}, +c+\textstyle\frac{1}{2},d+\textstyle\frac{1}{2}). +\end{eqnarray} + +\newpage + +\subsection*{Backward shift operator} +\begin{eqnarray} +\label{shift2WilsonI} +& &(a-\textstyle\frac{1}{2}-ix)(b-\textstyle\frac{1}{2}-ix) +(c-\textstyle\frac{1}{2}-ix)(d-\textstyle\frac{1}{2}-ix) +W_n((x+\textstyle\frac{1}{2}i)^2;a,b,c,d)\nonumber\\ +& &{}\mathindent{}-(a-\textstyle\frac{1}{2}+ix)(b-\textstyle\frac{1}{2}+ix) +(c-\textstyle\frac{1}{2}+ix)(d-\textstyle\frac{1}{2}+ix) +W_n((x-\textstyle\frac{1}{2}i)^2;a,b,c,d)\nonumber\\ +& &{}=-2ix W_{n+1}(x^2;a-\textstyle\frac{1}{2},b-\textstyle\frac{1}{2}, +c-\textstyle\frac{1}{2},d-\textstyle\frac{1}{2}) +\end{eqnarray} +or equivalently +\begin{eqnarray} +\label{shift2WilsonII} +& &\frac{\delta\left[\omega(x;a,b,c,d)W_n(x^2;a,b,c,d)\right]}{\delta x^2}\nonumber\\ +& &{}=\omega(x;a-\textstyle\frac{1}{2},b-\textstyle\frac{1}{2}, +c-\textstyle\frac{1}{2},d-\textstyle\frac{1}{2}) +W_{n+1}(x^2;a-\textstyle\frac{1}{2},b-\textstyle\frac{1}{2},c-\textstyle\frac{1}{2},d-\textstyle\frac{1}{2}), +\end{eqnarray} +where +$$\omega(x;a,b,c,d):=\frac{1}{2ix}\left|\frac{\Gamma(a+ix)\Gamma(b+ix)\Gamma(c+ix)\Gamma(d+ix)}{\Gamma(2ix)}\right|^2.$$ + +\subsection*{Rodrigues-type formula} +\begin{eqnarray} +\label{RodWilson} +& &\omega(x;a,b,c,d)W_n(x^2;a,b,c,d)\nonumber\\ +& &{}=\left(\frac{\delta}{\delta x^2}\right)^n +\left[\omega(x;a+\textstyle\frac{1}{2}n,b+\textstyle\frac{1}{2}n, +c+\textstyle\frac{1}{2}n,d+\textstyle\frac{1}{2}n)\right]. +\end{eqnarray} + +\subsection*{Generating functions} +\begin{equation} +\label{GenWilson1} +\hyp{2}{1}{a+ix,b+ix}{a+b}{t}\,\hyp{2}{1}{c-ix,d-ix}{c+d}{t}= +\sum_{n=0}^{\infty}\frac{W_n(x^2;a,b,c,d)t^n}{(a+b)_n(c+d)_nn!}. +\end{equation} + +\begin{equation} +\label{GenWilson2} +\hyp{2}{1}{a+ix,c+ix}{a+c}{t}\,\hyp{2}{1}{b-ix,d-ix}{b+d}{t}= +\sum_{n=0}^{\infty}\frac{W_n(x^2;a,b,c,d)t^n}{(a+c)_n(b+d)_nn!}. +\end{equation} + +\begin{equation} +\label{GenWilson3} +\hyp{2}{1}{a+ix,d+ix}{a+d}{t}\,\hyp{2}{1}{b-ix,c-ix}{b+c}{t}= +\sum_{n=0}^{\infty}\frac{W_n(x^2;a,b,c,d)t^n}{(a+d)_n(b+c)_nn!}. +\end{equation} + +\begin{eqnarray} +\label{GenWilson4} +& &(1-t)^{1-a-b-c-d}\nonumber\\ +& &{}\mathindent\times\hyp{4}{3}{\frac{1}{2}(a+b+c+d-1),\frac{1}{2}(a+b+c+d),a+ix,a-ix} +{a+b,a+c,a+d}{-\frac{4t}{(1-t)^2}}\nonumber\\ +& &{}=\sum_{n=0}^{\infty}\frac{(a+b+c+d-1)_n}{(a+b)_n(a+c)_n(a+d)_nn!}W_n(x^2;a,b,c,d)t^n. +\end{eqnarray} + +\subsection*{Limit relations} + +\subsubsection*{Wilson $\rightarrow$ Continuous dual Hahn} +The continuous dual Hahn polynomials given by (\ref{DefContDualHahn}) can be found from the +Wilson polynomials by dividing by $(a+d)_n$ and letting $d\rightarrow\infty$: +\begin{equation} +\lim_{d\rightarrow\infty}\frac{W_n(x^2;a,b,c,d)}{(a+d)_n}=S_n(x^2;a,b,c). +\end{equation} + +\subsubsection*{Wilson $\rightarrow$ Continuous Hahn} +The continuous Hahn polynomials given by (\ref{DefContHahn}) are obtained from the Wilson +polynomials by the substitutions $a\rightarrow a-it$, $b\rightarrow b-it$, $c\rightarrow c+it$, +$d\rightarrow d+it$ and $x\rightarrow x+t$ and the limit $t\rightarrow\infty$ in the following +way: +\begin{equation} +\lim_{t\rightarrow\infty} +\frac{W_n((x+t)^2;a-it,b-it,c+it,d+it)}{(-2t)^nn!}=p_n(x;a,b,c,d). +\end{equation} + +\subsubsection*{Wilson $\rightarrow$ Jacobi} +The Jacobi polynomials given by (\ref{DefJacobi}) can be found from the Wilson +polynomials by substituting $a=b=\frac{1}{2}(\alpha+1)$, $c=\frac{1}{2}(\beta+1)+it$, +$d=\frac{1}{2}(\beta+1)-it$ and $x\rightarrow t\sqrt{\frac{1}{2}(1-x)}$ in the +definition (\ref{DefWilson}) of the Wilson polynomials and taking the limit +$t\rightarrow\infty$. In fact we have +\begin{eqnarray} +& &\lim_{t\rightarrow\infty}\frac{W_n(\frac{1}{2}(1-x)t^2;\frac{1}{2}(\alpha+1), +\frac{1}{2}(\alpha+1),\frac{1}{2}(\beta+1)+it,\frac{1}{2}(\beta+1)-it)}{t^{2n}n!}\nonumber\\ +& &{}=P_n^{(\alpha,\beta)}(x). +\end{eqnarray} + +\subsection*{Remarks} +Note that for $k0$, $c=\bar{a}$ and $d=\bar{b}$, then +\begin{eqnarray} +\label{OrtContHahn} +& &\frac{1}{2\pi}\int_{-\infty}^{\infty}\Gamma(a+ix)\Gamma(b+ix)\Gamma(c-ix)\Gamma(d-ix) +p_m(x;a,b,c,d)p_n(x;a,b,c,d)\,dx\nonumber\\ +& &{}=\frac{\Gamma(n+a+c)\Gamma(n+a+d)\Gamma(n+b+c)\Gamma(n+b+d)} +{(2n+a+b+c+d-1)\Gamma(n+a+b+c+d-1)n!}\,\delta_{mn}. +\end{eqnarray} + +\subsection*{Recurrence relation} +\begin{equation} +\label{RecContHahn} +(a+ix)\tilde{p}_n(x)=A_n\tilde{p}_{n+1}(x)-\left(A_n+C_n\right)\tilde{p}_n(x)+C_n\tilde{p}_{n-1}(x), +\end{equation} +where +$$\tilde{p}_n(x):=\tilde{p}_n(x;a,b,c,d)=\frac{n!}{i^n(a+c)_n(a+d)_n}p_n(x;a,b,c,d)$$ +and +$$\left\{\begin{array}{l} +\displaystyle A_n=-\frac{(n+a+b+c+d-1)(n+a+c)(n+a+d)}{(2n+a+b+c+d-1)(2n+a+b+c+d)}\\[5mm] +\displaystyle C_n=\frac{n(n+b+c-1)(n+b+d-1)}{(2n+a+b+c+d-2)(2n+a+b+c+d-1)}. +\end{array}\right.$$ + +\subsection*{Normalized recurrence relation} +\begin{equation} +\label{NormRecContHahn} +xp_n(x)=p_{n+1}(x)+i(A_n+C_n+a)p_n(x)-A_{n-1}C_np_{n-1}(x), +\end{equation} +where +$$p_n(x;a,b,c,d)=\frac{(n+a+b+c+d-1)_n}{n!}p_n(x).$$ + +\subsection*{Difference equation} +\begin{eqnarray} +\label{dvContHahn} +& &n(n+a+b+c+d-1)y(x)\nonumber\\ +& &{}=B(x)y(x+i)-\left[B(x)+D(x)\right]y(x)+D(x)y(x-i), +\end{eqnarray} +where +$$y(x)=p_n(x;a,b,c,d)$$ +and +$$\left\{\begin{array}{l} +\displaystyle B(x)=(c-ix)(d-ix)\\[5mm] +\displaystyle D(x)=(a+ix)(b+ix). +\end{array}\right.$$ + +\subsection*{Forward shift operator} +\begin{eqnarray} +\label{shift1ContHahnI} +& &p_n(x+\textstyle\frac{1}{2}i;a,b,c,d)-p_n(x-\textstyle\frac{1}{2}i;a,b,c,d)\nonumber\\ +& &{}=i(n+a+b+c+d-1)p_{n-1}(x;a+\textstyle\frac{1}{2}, +b+\textstyle\frac{1}{2},c+\textstyle\frac{1}{2},d+\textstyle\frac{1}{2}) +\end{eqnarray} +or equivalently +\begin{equation} +\label{shift1ContHahnII} +\frac{\delta p_n(x;a,b,c,d)}{\delta x}=(n+a+b+c+d-1) +p_{n-1}(x;a+\textstyle\frac{1}{2},b+\textstyle\frac{1}{2}, +c+\textstyle\frac{1}{2},d+\textstyle\frac{1}{2}). +\end{equation} + +\subsection*{Backward shift operator} +\begin{eqnarray} +\label{shift2ContHahnI} +& &(c-\textstyle\frac{1}{2}-ix)(d-\textstyle\frac{1}{2}-ix) +p_n(x+\textstyle\frac{1}{2}i;a,b,c,d)\nonumber\\ +& &{}\mathindent{}-(a-\textstyle\frac{1}{2}+ix)(b-\textstyle\frac{1}{2}+ix) +p_n(x-\textstyle\frac{1}{2}i;a,b,c,d)\nonumber\\ +& &{}=\frac{n+1}{i}p_{n+1}(x;a-\textstyle\frac{1}{2}, +b-\textstyle\frac{1}{2},c-\textstyle\frac{1}{2},d-\textstyle\frac{1}{2}) +\end{eqnarray} +or equivalently +\begin{eqnarray} +\label{shift2ContHahnII} +& &\frac{\delta\left[\omega(x;a,b,c,d)p_n(x;a,b,c,d)\right]}{\delta x}\nonumber\\ +& &{}=-(n+1)\omega(x;a-\textstyle\frac{1}{2},b-\textstyle\frac{1}{2}, +c-\textstyle\frac{1}{2},d-\textstyle\frac{1}{2})\nonumber\\ +& &{}\mathindent\times p_{n+1}(x;a-\textstyle\frac{1}{2},b-\textstyle\frac{1}{2},c-\textstyle\frac{1}{2},d-\textstyle\frac{1}{2}), +\end{eqnarray} +where +$$\omega(x;a,b,c,d)=\Gamma(a+ix)\Gamma(b+ix)\Gamma(c-ix)\Gamma(d-ix).$$ + +\subsection*{Rodrigues-type formula} +\begin{eqnarray} +\label{RodContHahn} +& &\omega(x;a,b,c,d)p_n(x;a,b,c,d)\nonumber\\ +& &{}=\frac{(-1)^n}{n!}\left(\frac{\delta}{\delta x}\right)^n +\left[\omega(x;a+\textstyle\frac{1}{2}n,b+\textstyle\frac{1}{2}n, +c+\textstyle\frac{1}{2}n,d+\textstyle\frac{1}{2}n)\right]. +\end{eqnarray} + +\subsection*{Generating functions} +\begin{equation} +\label{GenContHahn1} +\hyp{1}{1}{a+ix}{a+c}{-it}\,\hyp{1}{1}{d-ix}{b+d}{it}= +\sum_{n=0}^{\infty}\frac{p_n(x;a,b,c,d)}{(a+c)_n(b+d)_n}t^n. +\end{equation} + +\begin{equation} +\label{GenContHahn2} +\hyp{1}{1}{a+ix}{a+d}{-it}\,\hyp{1}{1}{c-ix}{b+c}{it}= +\sum_{n=0}^{\infty}\frac{p_n(x;a,b,c,d)}{(a+d)_n(b+c)_n}t^n. +\end{equation} + +\begin{eqnarray} +\label{GenContHahn3} +& &(1-t)^{1-a-b-c-d}\,\hyp{3}{2}{\frac{1}{2}(a+b+c+d-1),\frac{1}{2}(a+b+c+d),a+ix} +{a+c,a+d}{-\frac{4t}{(1-t)^2}}\nonumber\\ +& &{}=\sum_{n=0}^{\infty} +\frac{(a+b+c+d-1)_n}{(a+c)_n(a+d)_ni^n}p_n(x;a,b,c,d)t^n. +\end{eqnarray} + +\subsection*{Limit relations} + +\subsubsection*{Wilson $\rightarrow$ Continuous Hahn} +The continuous Hahn polynomials are obtained from the Wilson polynomials given by +(\ref{DefWilson}) by the substitution $a\rightarrow a-it$, $b\rightarrow b-it$, +$c\rightarrow c+it$, $d\rightarrow d+it$ and $x\rightarrow x+t$ and the limit +$t\rightarrow\infty$ in the following way: +$$\lim_{t\rightarrow\infty} +\frac{W_n((x+t)^2;a-it,b-it,c+it,d+it)}{(-2t)^nn!}=p_n(x;a,b,c,d).$$ + +\subsubsection*{Continuous Hahn $\rightarrow$ Meixner-Pollaczek} +The Meixner-Pollaczek polynomials given by (\ref{DefMP}) can be obtained from the +continuous Hahn polynomials by setting $x\rightarrow x+t$, $a=\lambda-it$, $c=\lambda+it$ +and $b=d=t\tan\phi$ and taking the limit $t\rightarrow\infty$: +\begin{equation} +\lim_{t\rightarrow\infty}\frac{p_n(x+t;\lambda-it,t\tan\phi,\lambda+it,t\tan\phi)}{t^n} +=\frac{P_n^{(\lambda)}(x;\phi)}{(\cos\phi)^n}. +\end{equation} + +\subsubsection*{Continuous Hahn $\rightarrow$ Jacobi} +The Jacobi polynomials given by (\ref{DefJacobi}) follow from the continuous Hahn polynomials +by the substitution $x\rightarrow \frac{1}{2}xt$, $a=\frac{1}{2}(\alpha+1-it)$, +$b=\frac{1}{2}(\beta+1+it)$, $c=\frac{1}{2}(\alpha+1+it)$ and $d=\frac{1}{2}(\beta+1-it)$, +division by $t^n$ and the limit $t\rightarrow\infty$: +\begin{eqnarray} +& &\lim_{t\rightarrow\infty} +\frac{p_n(\frac{1}{2}xt;\frac{1}{2}(\alpha+1-it),\frac{1}{2}(\beta+1+it), +\frac{1}{2}(\alpha+1+it),\frac{1}{2}(\beta+1-it))}{t^n}\nonumber\\ +& &{}=P_n^{(\alpha,\beta)}(x). +\end{eqnarray} + +\subsubsection*{Continuous Hahn $\rightarrow$ Pseudo Jacobi} +The pseudo Jacobi polynomials given by (\ref{DefPseudoJacobi}) follow from the continuous +Hahn polynomials by the substitution $x\rightarrow xt$, $a=\frac{1}{2}(-N+i\nu-2t)$, +$b=\frac{1}{2}(-N-i\nu+2t)$, $c=\frac{1}{2}(-N-i\nu-2t)$ and $d=\frac{1}{2}(-N+i\nu+2t)$, +division by $t^n$ and the limit $t\rightarrow\infty$: +\begin{eqnarray} +& &\lim_{t\rightarrow\infty}\frac{p_n(xt;\frac{1}{2}(-N+i\nu-2t),\frac{1}{2}(-N-i\nu+2t), +\frac{1}{2}(-N+i\nu-2t),\frac{1}{2}(-N-i\nu+2t))}{t^n}\nonumber\\ +& &{}=\frac{(n-2N-1)_n}{n!}P_n(x;\nu,N). +\end{eqnarray} + +\subsection*{Remark} +Since we have for $k0$ and $(c,d)=(\overline a,\overline b)$ or $(\overline b,\overline a)$.\\ +Thus, under these assumptions, the continuous Hahn polynomial +$p_n(x;a,b,c,d)$ +is symmetric in $a,b$ and in $c,d$. +This follows from the orthogonality relation (9.4.2) +together with the value of its coefficient of $x^n$ given in (9.4.4b).\\ +As a consequence, it is sufficient to give generating function (9.4.11). Then the generating +function (9.4.12) will follow by symmetry in the parameters. +% +\paragraph{Uniqueness of orthogonality measure} +The coefficient of $p_{n-1}(x)$ in (9.4.4) behaves as $O(n^2)$ as $n\to\iy$. +Hence \eqref{93} holds, by which the orthogonality measure is unique. +% +\paragraph{Special cases} +In the following special case there is a reduction to +Meixner-Pollaczek: +\begin{equation} +p_n(x;a,a+\thalf,a,a+\thalf)= +\frac{(2a)_n (2a+\thalf)_n}{(4a)_n}\,P_n^{(2a)}(2x;\thalf\pi). +\end{equation} +See \myciteKLS{342}{(2.6)} (note that in \myciteKLS{342}{(2.3)} the +Meixner-Pollaczek polynonmials are defined different from (9.7.1), +without a constant factor in front). + +For $0-1$ and $\beta>-1$, or for $\alpha<-N$ and $\beta<-N$, we have +\begin{eqnarray} +\label{OrtHahn} +& &\sum_{x=0}^N\binom{\alpha +x}{x}\binom{\beta+N-x}{N-x}Q_m(x;\alpha,\beta,N)Q_n(x;\alpha,\beta,N)\nonumber\\ +& &{}=\frac{(-1)^n(n+\alpha+\beta+1)_{N+1}(\beta+1)_nn!}{(2n+\alpha+\beta+1)(\alpha+1)_n(-N)_nN!}\,\delta_{mn}. +\end{eqnarray} + +\subsection*{Recurrence relation} +\begin{equation} +\label{RecHahn} +-xQ_n(x)=A_nQ_{n+1}(x)-\left(A_n+C_n\right)Q_n(x)+C_nQ_{n-1}(x), +\end{equation} +where +$$Q_n(x):=Q_n(x;\alpha,\beta,N)$$ +and +$$\left\{\begin{array}{l} +\displaystyle A_n=\frac{(n+\alpha+\beta+1)(n+\alpha+1)(N-n)}{(2n+\alpha+\beta+1)(2n+\alpha+\beta+2)}\\[5mm] +\displaystyle C_n=\frac{n(n+\alpha+\beta+N+1)(n+\beta)}{(2n+\alpha+\beta)(2n+\alpha+\beta+1)}. +\end{array}\right.$$ + +\subsection*{Normalized recurrence relation} +\begin{equation} +\label{NormRecHahn} +xp_n(x)=p_{n+1}(x)+\left(A_n+C_n\right)p_n(x)+A_{n-1}C_np_{n-1}(x), +\end{equation} +where +$$Q_n(x;\alpha,\beta,N)=\frac{(n+\alpha+\beta+1)_n}{(\alpha+1)_n(-N)_n}p_n(x).$$ + +\subsection*{Difference equation} +\begin{equation} +\label{dvHahn} +n(n+\alpha+\beta+1)y(x)=B(x)y(x+1)-\left[B(x)+D(x)\right]y(x)+D(x)y(x-1), +\end{equation} +where +$$y(x)=Q_n(x;\alpha,\beta,N)$$ +and +$$\left\{\begin{array}{l} +\displaystyle B(x)=(x+\alpha+1)(x-N)\\[5mm] +\displaystyle D(x)=x(x-\beta-N-1). +\end{array}\right.$$ + +\subsection*{Forward shift operator} +\begin{eqnarray} +\label{shift1HahnI} +& &Q_n(x+1;\alpha,\beta,N)-Q_n(x;\alpha,\beta,N)\nonumber\\ +& &{}=-\frac{n(n+\alpha+\beta+1)}{(\alpha+1)N}Q_{n-1}(x;\alpha+1,\beta+1,N-1) +\end{eqnarray} +or equivalently +\begin{equation} +\label{shift1HahnII} +\Delta Q_n(x;\alpha,\beta,N)=-\frac{n(n+\alpha+\beta+1)}{(\alpha+1)N}Q_{n-1}(x;\alpha+1,\beta+1,N-1). +\end{equation} + +\subsection*{Backward shift operator} +\begin{eqnarray} +\label{shift2HahnI} +& &(x+\alpha)(N+1-x)Q_n(x;\alpha,\beta,N)-x(\beta+N+1-x)Q_n(x-1;\alpha,\beta,N)\nonumber\\ +& &{}=\alpha(N+1)Q_{n+1}(x;\alpha-1,\beta-1,N+1) +\end{eqnarray} +or equivalently +\begin{eqnarray} +\label{shift2HahnII} +& &\nabla\left[\omega(x;\alpha,\beta,N)Q_n(x;\alpha,\beta,N)\right]\nonumber\\ +& &{}=\frac{N+1}{\beta}\omega(x;\alpha-1,\beta-1,N+1)Q_{n+1}(x;\alpha-1,\beta-1,N+1), +\end{eqnarray} +where +$$\omega(x;\alpha,\beta,N)=\binom{\alpha+x}{x}\binom{\beta+N-x}{N-x}.$$ + +\subsection*{Rodrigues-type formula} +\begin{eqnarray} +\label{RodHahn} +& &\omega(x;\alpha,\beta,N)Q_n(x;\alpha,\beta,N)\nonumber\\ +& &{}=\frac{(-1)^n(\beta+1)_n}{(-N)_n}\nabla^n\left[\omega(x;\alpha+n,\beta+n,N-n)\right]. +\end{eqnarray} + +\subsection*{Generating functions} +For $x=0,1,2,\ldots,N$ we have +\begin{equation} +\label{GenHahn1} +\hyp{1}{1}{-x}{\alpha+1}{-t}\,\hyp{1}{1}{x-N}{\beta+1}{t} +=\sum_{n=0}^N\frac{(-N)_n}{(\beta+1)_nn!}Q_n(x;\alpha,\beta,N)t^n. +\end{equation} + +\begin{eqnarray} +\label{GenHahn2} +& &\hyp{2}{0}{-x,-x+\beta+N+1}{-}{-t}\,\hyp{2}{0}{x-N,x+\alpha+1}{-}{t}\nonumber\\ +& &{}=\sum_{n=0}^N\frac{(-N)_n(\alpha+1)_n}{n!}Q_n(x;\alpha,\beta,N)t^n. +\end{eqnarray} + +\begin{eqnarray} +\label{GenHahn3} +& &\left[(1-t)^{-\alpha-\beta-1}\,\hyp{3}{2}{\frac{1}{2}(\alpha+\beta+1),\frac{1}{2}(\alpha+\beta+2),-x} +{\alpha+1,-N}{-\frac{4t}{(1-t)^2}}\right]_N\nonumber\\ +& &{}=\sum_{n=0}^N\frac{(\alpha+\beta+1)_n}{n!}Q_n(x;\alpha,\beta,N)t^n. +\end{eqnarray} + +\subsection*{Limit relations} + +\subsubsection*{Racah $\rightarrow$ Hahn} +If we take $\gamma+1=-N$ and let $\delta\rightarrow\infty$ in the definition (\ref{DefRacah}) +of the Racah polynomials, we obtain the Hahn polynomials. Hence +$$\lim_{\delta\rightarrow\infty} +R_n(\lambda(x);\alpha,\beta,-N-1,\delta)=Q_n(x;\alpha,\beta,N).$$ +And if we take $\delta=-\beta-N-1$ and let $\gamma\rightarrow\infty$ in the definition (\ref{DefRacah}) +of the Racah polynomials, we also obtain the Hahn polynomials: +$$\lim_{\gamma\rightarrow\infty} +R_n(\lambda(x);\alpha,\beta,\gamma,-\beta-N-1)=Q_n(x;\alpha,\beta,N).$$ +Another way to do this is to take $\alpha+1=-N$ and $\beta\rightarrow\beta+\gamma+N+1$ in +the definition (\ref{DefRacah}) of the Racah polynomials and then take the limit +$\delta\rightarrow\infty$. In that case we obtain the Hahn polynomials in the following way: +$$\lim_{\delta\rightarrow\infty} +R_n(\lambda(x);-N-1,\beta+\gamma+N+1,\gamma,\delta)=Q_n(x;\gamma,\beta,N).$$ + +\subsubsection*{Hahn $\rightarrow$ Jacobi} +To find the Jacobi polynomials given by (\ref{DefJacobi}) from the Hahn polynomials we take +$x\rightarrow Nx$ and let $N\rightarrow\infty$. In fact we have +\begin{equation} +\lim_{N\rightarrow\infty} +Q_n(Nx;\alpha,\beta,N)=\frac{P_n^{(\alpha,\beta)}(1-2x)}{P_n^{(\alpha,\beta)}(1)}. +\end{equation} + +\subsubsection*{Hahn $\rightarrow$ Meixner} +The Meixner polynomials given by (\ref{DefMeixner}) can be obtained from the Hahn polynomials +by taking $\alpha=b-1$, $\beta=N(1-c)c^{-1}$ and letting $N\rightarrow\infty$: +\begin{equation} +\lim_{N\rightarrow\infty} +Q_n(x;b-1,N(1-c)c^{-1},N)=M_n(x;b,c). +\end{equation} + +\subsubsection*{Hahn $\rightarrow$ Krawtchouk} +The Krawtchouk polynomials given by (\ref{DefKrawtchouk}) are obtained from the Hahn polynomials +if we take $\alpha=pt$ and $\beta=(1-p)t$ and let $t\rightarrow\infty$: +\begin{equation} +\lim_{t\rightarrow\infty}Q_n(x;pt,(1-p)t,N)=K_n(x;p,N). +\end{equation} + +\subsection*{Remark} +If we interchange the role of $x$ and $n$ in (\ref{DefHahn}) we obtain the +dual Hahn polynomials given by (\ref{DefDualHahn}). + +\noindent +Since +$$Q_n(x;\alpha,\beta,N)=R_x(\lambda(n);\alpha,\beta,N)$$ +we obtain the dual orthogonality relation for the Hahn polynomials from the +orthogonality relation (\ref{OrtDualHahn}) of the dual Hahn polynomials: +\begin{eqnarray*} +& &\sum_{n=0}^N\frac{(2n+\alpha+\beta+1)(\alpha+1)_n(-N)_nN!}{(-1)^n(n+\alpha+\beta+1)_{N+1}(\beta+1)_nn!} +Q_n(x;\alpha,\beta,N)Q_n(y;\alpha,\beta,N)\\ +& &{}=\frac{\delta_{xy}}{\dbinom{\alpha+x}{x}\dbinom{\beta+N-x}{N-x}},\quad x,y \in \{0,1,2,\ldots,N\}. +\subsection*{9.5 Hahn} +\label{sec9.5} +% +\paragraph{Special values} +\begin{equation} +Q_n(0;\al,\be,N)=1,\quad +Q_n(N;\al,\be,N)=\frac{(-1)^n(\be+1)_n}{(\al+1)_n}\,. +\label{95} +\end{equation} +Use (9.5.1) and compare with (9.8.1) and \eqref{50}. + +From (9.5.3) and \eqref{1} it follows that +\begin{equation} +Q_{2n}(N;\al,\al,2N)=\frac{(\thalf)_n(N+\al+1)_n}{(-N+\thalf)_n(\al+1)_n}\,. +\label{30} +\end{equation} +From (9.5.1) and \mycite{DLMF}{(15.4.24)} it follows that +\begin{equation} +Q_N(x;\al,\be,N)=\frac{(-N-\be)_x}{(\al+1)_x}\qquad(x=0,1,\ldots,N). +\label{44} +\end{equation} +% +\paragraph{Symmetries} +By the orthogonality relation (9.5.2): +\begin{equation} +\frac{Q_n(N-x;\al,\be,N)}{Q_n(N;\al,\be,N)}=Q_n(x;\be,\al,N), +\label{96} +\end{equation} +It follows from \eqref{97} and \eqref{45} that +\begin{equation} +\frac{Q_{N-n}(x;\al,\be,N)}{Q_N(x;\al,\be,N)} +=Q_n(x;-N-\be-1,-N-\al-1,N) +\qquad(x=0,1,\ldots,N). +\label{100} +\end{equation} +% +\paragraph{Duality} +The Remark on p.208 gives the duality between Hahn and dual Hahn polynomials: +% +\begin{equation} +Q_n(x;\al,\be,N)=R_x(n(n+\al+\be+1);\al,\be,N)\quad(n,x\in\{0,1,\ldots N\}). +\label{45} +\end{equation} +\end{eqnarray*} + +\subsection*{References} +\cite{AlSalam90}, \cite{AndrewsAskey85}, \cite{Area+II}, \cite{Askey75}, +\cite{Askey89I}, \cite{Askey2005}, \cite{AskeyGasper77}, \cite{AskeyWilson85}, +\cite{AtakRahmanSuslov}, \cite{AtakSuslov88}, \cite{Chihara78}, +\cite{Ciesielski}, \cite{Cooper+}, \cite{Dette95}, \cite{Dunkl76}, +\cite{Dunkl78I}, \cite{Gasper73I}, \cite{Gasper74}, \cite{HoareRahman}, +\cite{Ismail77}, \cite{Ismail2005II}, \cite{Karlin61}, \cite{Koorn81}, \cite{Koorn88}, +\cite{LabelleYehI}, \cite{LabelleYehII}, \cite{Laine}, \cite{Lesky62}, +\cite{Lesky88}, \cite{Lesky89}, \cite{Lesky94I}, \cite{Lesky95II}, +\cite{LewanowiczII}, \cite{Neuman}, \cite{Nikiforov+}, \cite{NikiforovUvarov}, +\cite{Rahman76III}, \cite{Rahman78I}, \cite{Rahman78II}, \cite{Rahman81III}, +\cite{Sablonniere}, \cite{Stanton84}, \cite{Stanton90}, \cite{Wilson80}, \cite{Wilson70II}, +\cite{Zarzo+}. + + +\section{Dual Hahn}\index{Dual Hahn polynomials}\index{Hahn polynomials!Dual} + +\par\setcounter{equation}{0} + +\subsection*{Hypergeometric representation} +\begin{equation} +\label{DefDualHahn} +R_n(\lambda(x);\gamma,\delta,N)= +\hyp{3}{2}{-n,-x,x+\gamma+\delta+1}{\gamma+1,-N}{1},\quad n=0,1,2,\ldots,N, +\end{equation} +where +$$\lambda(x)=x(x+\gamma+\delta+1).$$ + +\subsection*{Orthogonality relation} +For $\gamma>-1$ and $\delta>-1$, or for $\gamma<-N$ and $\delta<-N$, we have +\begin{eqnarray} +\label{OrtDualHahn} +& &\sum_{x=0}^N\frac{(2x+\gamma+\delta+1)(\gamma+1)_x(-N)_xN!}{(-1)^x(x+\gamma+\delta+1)_{N+1}(\delta+1)_xx!} +R_m(\lambda(x);\gamma,\delta,N)R_n(\lambda(x);\gamma,\delta,N)\nonumber\\ +& &{}=\frac{\,\delta_{mn}}{\dbinom{\gamma+n}{n}\dbinom{\delta+N-n}{N-n}}. +\end{eqnarray} + +\subsection*{Recurrence relation} +\begin{equation} +\label{RecDualHahn} +\lambda(x)R_n(\lambda(x)) +=A_nR_{n+1}(\lambda(x))-\left(A_n+C_n\right)R_n(\lambda(x))+C_nR_{n-1}(\lambda(x)), +\end{equation} +where +$$R_n(\lambda(x)):=R_n(\lambda(x);\gamma,\delta,N)$$ +and +$$\left\{\begin{array}{l} +\displaystyle A_n=(n+\gamma+1)(n-N)\\[5mm] +\displaystyle C_n=n(n-\delta-N-1). +\end{array}\right.$$ + +\subsection*{Normalized recurrence relation} +\begin{equation} +\label{NormRecDualHahn} +xp_n(x)=p_{n+1}(x)-(A_n+C_n)p_n(x)+A_{n-1}C_np_{n-1}(x), +\end{equation} +where +$$R_n(\lambda(x);\gamma,\delta,N)=\frac{1}{(\gamma+1)_n(-N)_n}p_n(\lambda(x)).$$ + +\subsection*{Difference equation} +\begin{equation} +\label{dvDualHahn} +-ny(x)=B(x)y(x+1)-\left[B(x)+D(x)\right]y(x)+D(x)y(x-1), +\end{equation} +where +$$y(x)=R_n(\lambda(x);\gamma,\delta,N)$$ +and +$$\left\{\begin{array}{l} +\displaystyle B(x)=\frac{(x+\gamma+1)(x+\gamma+\delta+1)(N-x)}{(2x+\gamma+\delta+1)(2x+\gamma+\delta+2)}\\[5mm] +\displaystyle D(x)=\frac{x(x+\gamma+\delta+N+1)(x+\delta)}{(2x+\gamma+\delta)(2x+\gamma+\delta+1)}. +\end{array}\right.$$ + +\subsection*{Forward shift operator} +\begin{eqnarray} +\label{shift1DualHahnI} +& &R_n(\lambda(x+1);\gamma,\delta,N)-R_n(\lambda(x);\gamma,\delta,N)\nonumber\\ +& &{}=-\frac{n(2x+\gamma+\delta+2)}{(\gamma+1)N}R_{n-1}(\lambda(x);\gamma+1,\delta,N-1) +\end{eqnarray} +or equivalently +\begin{equation} +\label{shift1DualHahnII} +\frac{\Delta R_n(\lambda(x);\gamma,\delta,N)}{\Delta\lambda(x)}= +-\frac{n}{(\gamma+1)N}R_{n-1}(\lambda(x);\gamma+1,\delta,N-1). +\end{equation} + +\subsection*{Backward shift operator} +\begin{eqnarray} +\label{shift2DualHahnI} +& &(x+\gamma)(x+\gamma+\delta)(N+1-x)R_n(\lambda(x);\gamma,\delta,N)\nonumber\\ +& &{}\mathindent{}-x(x+\gamma+\delta+N+1)(x+\delta)R_n(\lambda(x-1);\gamma,\delta,N)\nonumber\\ +& &{}=\gamma(N+1)(2x+\gamma+\delta)R_{n+1}(\lambda(x);\gamma-1,\delta,N+1) +\end{eqnarray} +or equivalently +\begin{eqnarray} +\label{shift2DualHahnII} +& &\frac{\nabla\left[\omega(x;\gamma,\delta,N)R_n(\lambda(x);\gamma,\delta,N)\right]}{\nabla\lambda(x)}\nonumber\\ +& &{}=\frac{1}{\gamma+\delta}\omega(x;\gamma-1,\delta,N+1)R_{n+1}(\lambda(x);\gamma-1,\delta,N+1), +\end{eqnarray} +where +$$\omega(x;\gamma,\delta,N)=\frac{(-1)^x(\gamma+1)_x(\gamma+\delta+1)_x(-N)_x} +{(\gamma+\delta+N+2)_x(\delta+1)_xx!}.$$ + +\newpage + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodDualHahn} +\omega(x;\gamma,\delta,N)R_n(\lambda(x);\gamma,\delta,N) +=(\gamma+\delta+1)_n\left(\nabla_{\lambda}\right)^n\left[\omega(x;\gamma+n,\delta,N-n)\right], +\end{equation} +where +$$\nabla_{\lambda}:=\frac{\nabla}{\nabla\lambda(x)}.$$ + +\subsection*{Generating functions} +For $x=0,1,2,\ldots,N$ we have +\begin{equation} +\label{GenDualHahn1} +(1-t)^{N-x}\,\hyp{2}{1}{-x,-x-\delta}{\gamma+1}{t} +=\sum_{n=0}^N\frac{(-N)_n}{n!}R_n(\lambda(x);\gamma,\delta,N)t^n. +\end{equation} + +\begin{eqnarray} +\label{GenDualHahn2} +& &(1-t)^x\,\hyp{2}{1}{x-N,x+\gamma+1}{-\delta-N}{t}\nonumber\\ +& &=\sum_{n=0}^N\frac{(\gamma+1)_n(-N)_n}{(-\delta-N)_nn!}R_n(\lambda(x);\gamma,\delta,N)t^n. +\end{eqnarray} + +\begin{equation} +\label{GenDualHahn3} +\left[\expe^t\,\hyp{2}{2}{-x,x+\gamma+\delta+1}{\gamma+1,-N}{-t}\right]_N=\sum_{n=0}^N +\frac{R_n(\lambda(x);\gamma,\delta,N)}{n!}t^n. +\end{equation} + +\begin{eqnarray} +\label{GenDualHahn4} +& &\left[(1-t)^{-\epsilon}\,\hyp{3}{2}{\epsilon,-x,x+\gamma+\delta+1}{\gamma+1,-N}{\frac{t}{t-1}}\right]_N\nonumber\\ +& &{}=\sum_{n=0}^N\frac{(\epsilon)_n}{n!}R_n(\lambda(x);\gamma,\delta,N)t^n, +\quad\textrm{$\epsilon$ arbitrary}. +\end{eqnarray} + +\subsection*{Limit relations} + +\subsubsection*{Racah $\rightarrow$ Dual Hahn} +If we take $\alpha+1=-N$ and let $\beta\rightarrow\infty$ in the definition (\ref{DefRacah}) +of the Racah polynomials, then we obtain the dual Hahn polynomials: +$$\lim_{\beta\rightarrow\infty} +R_n(\lambda(x);-N-1,\beta,\gamma,\delta)=R_n(\lambda(x);\gamma,\delta,N).$$ +And if we take $\beta=-\delta-N-1$ and let $\alpha\rightarrow\infty$ in the definition (\ref{DefRacah}) +of the Racah polynomials, then we also obtain the dual Hahn polynomials: +$$\lim_{\alpha\rightarrow\infty} +R_n(\lambda(x);\alpha,-\delta-N-1,\gamma,\delta)=R_n(\lambda(x);\gamma,\delta,N).$$ +Finally, if we take $\gamma+1=-N$ and $\delta\rightarrow\alpha+\delta+N+1$ in the definition +(\ref{DefRacah}) of the Racah polynomials and take the limit +$\beta\rightarrow\infty$ we find the dual Hahn polynomials in the following way: +$$\lim_{\beta\rightarrow\infty} +R_n(\lambda(x);\alpha,\beta,-N-1,\alpha+\delta+N+1)=R_n(\lambda(x);\alpha,\delta,N).$$ + +\subsubsection*{Dual Hahn $\rightarrow$ Meixner} +The Meixner polynomials given by (\ref{DefMeixner}) are obtained from the dual Hahn polynomials +if we take $\gamma=\beta-1$ and $\delta=N(1-c)c^{-1}$ and let $N\rightarrow\infty$: +\begin{equation} +\lim_{N\rightarrow\infty} +R_n(\lambda(x);\beta-1,N(1-c)c^{-1},N)=M_n(x;\beta,c). +\end{equation} + +\subsubsection*{Dual Hahn $\rightarrow$ Krawtchouk} +The Krawtchouk polynomials given by (\ref{DefKrawtchouk}) can be obtained from the dual +Hahn polynomials by setting $\gamma=pt$, $\delta=(1-p)t$ and letting $t\rightarrow\infty$: +\begin{equation} +\lim_{t\rightarrow\infty}R_n(\lambda(x);pt,(1-p)t,N)=K_n(x;p,N). +\end{equation} + +\subsection*{Remark} +If we interchange the role of $x$ and $n$ in the definition (\ref{DefDualHahn}) +of the dual Hahn polynomials we obtain the Hahn polynomials given by (\ref{DefHahn}). + +\noindent +Since +$$R_n(\lambda(x);\gamma,\delta,N)=Q_x(n;\gamma,\delta,N)$$ +we obtain the dual orthogonality relation for the dual Hahn polynomials +from the orthogonality relation (\ref{OrtHahn}) for the Hahn polynomials: +\begin{eqnarray*} +& &\sum_{n=0}^N\binom{\gamma+n}{n}\binom{\delta+N-n}{N-n} R_n(\lambda(x);\gamma,\delta,N)R_n(\lambda(y);\gamma,\delta,N)\\ +& &{}=\frac{(-1)^x(x+\gamma+\delta+1)_{N+1}(\delta+1)_xx!} +{(2x+\gamma+\delta+1)(\gamma+1)_x(-N)_xN!}\delta_{xy},\quad x,y \in \{0,1,2,\ldots,N\}. +\subsection*{9.6 Dual Hahn} +\label{sec9.6} +% +\paragraph{Special values} +By \eqref{44} and \eqref{45} we have +\begin{equation} +R_n(N(N+\ga+\de+1);\ga,\de,N)=\frac{(-N-\de)_n}{(\ga+1)_n}\,. +\label{47} +\end{equation} +It follows from \eqref{95} and \eqref{45} that +\begin{equation} +R_N(x(x+\ga+\de+1);\ga,\de,N) +=\frac{(-1)^x(\de+1)_x}{(\ga+1)_x}\qquad(x=0,1,\ldots,N). +\label{101} +\end{equation} +% +\paragraph{Symmetries} +Write the weight in (9.6.2) as +\begin{equation} +w_x(\al,\be,N):=N!\,\frac{2x+\ga+\de+1}{(x+\ga+\de+1)_{N+1}}\, +\frac{(\ga+1)_x}{(\de+1)_x}\,\binom Nx. +\label{98} +\end{equation} +Then +\begin{equation} +(\de+1)_N\,w_{N-x}(\ga,\de,N)= +(-\ga-N)_N\,w_x(-\de-N-1,-\ga-N-1,N). +\label{99} +\end{equation} +Hence, by (9.6.2), +\begin{equation} +\frac{R_n((N-x)(N-x+\ga+\de+1);\ga,\de,N)}{R_n(N(N+\ga+\de+1);\ga,\de,N)} +=R_n(x(x-2N-\ga-\de-1);-N-\de-1,-N-\ga-1,N). +\label{97} +\end{equation} +Alternatively, \eqref{97} follows from (9.6.1) and +\mycite{DLMF}{(16.4.11)}. + +It follows from \eqref{96} and \eqref{45} that +\begin{equation} +\frac{R_{N-n}(x(x+\ga+\de+1);\ga,\de,N)} +{R_N(x(x+\ga+\de+1);\ga,\de,N)} +=R_n(x(x+\ga+\de+1);\de,\ga,N)\qquad(x=0,1,\ldots,N). +\label{102} +\end{equation} +% +\paragraph{Re: (9.6.11).} +The generating function (9.6.11) can be written in a more conceptual way as +\begin{equation} +(1-t)^x\,\hyp21{x-N,x+\ga+1}{-\de-N}t=\frac{N!}{(\de+1)_N}\, +\sum_{n=0}^N \om_n\,R_n(\la(x);\ga,\de,N)\,t^n, +\label{2} +\end{equation} +where +\begin{equation} +\om_n:=\binom{\ga+n}n \binom{\de+N-n}{N-n}, +\label{3} +\end{equation} +i.e., the denominator on the \RHS\ of (9.6.2). +By the duality between Hahn polynomials and dual Hahn polynomials (see \eqref{45}) the above generating function can be rewritten in +terms of Hahn polynomials: +\begin{equation} +(1-t)^n\,\hyp21{n-N,n+\al+1}{-\be-N}t=\frac{N!}{(\be+1)_N}\, +\sum_{x=0}^N w_x\,Q_n(x;\al,\be,N)\,t^x, +\label{4} +\end{equation} +where +\begin{equation} +w_x:=\binom{\al+x}x \binom{\be+N-x}{N-x}, +\label{5} +\end{equation} +i.e., the weight occurring in the orthogonality relation (9.5.2) +for Hahn polynomials. +\paragraph{Re: (9.6.15).} +There should be a closing bracket before the equality sign. +\end{eqnarray*} + +\subsection*{References} +\cite{Askey2005}, \cite{AskeyWilson85}, \cite{AtakRahmanSuslov}, \cite{AtakSuslov88}, +\cite{Ismail2005II}, \cite{Karlin61}, \cite{Koorn81}, \cite{Koorn88}, \cite{Lesky93}, +\cite{Lesky94I}, \cite{Lesky95I}, \cite{Lesky95II}, \cite{Nikiforov+}, \cite{NikiforovUvarov}, +\cite{Rahman81II}, \cite{Stanton84}, \cite{Wilson80}. + + +\section{Meixner-Pollaczek}\index{Meixner-Pollaczek polynomials} + +\par\setcounter{equation}{0} + +\subsection*{Hypergeometric representation} +\begin{equation} +\label{DefMP} +P_n^{(\lambda)}(x;\phi)=\frac{(2\lambda)_n}{n!}\expe^{in\phi}\, +\hyp{2}{1}{-n,\lambda+ix}{2\lambda}{1-\expe^{-2i\phi}}. +\end{equation} + +\subsection*{Orthogonality relation} +\begin{equation} +\label{OrtMP} +\frac{1}{2\pi}\int_{-\infty}^{\infty} +\e^{(2\phi-\pi)x}\left|\Gamma(\lambda+ix)\right|^2 +P_m^{(\lambda)}(x;\phi)P_n^{(\lambda)}(x;\phi)\,dx +{}=\frac{\Gamma(n+2\lambda)}{(2\sin\phi)^{2\lambda}n!}\,\delta_{mn}, +% \constraint{ +% $\lambda > 0$ & +% $0 < \phi < \pi$ } +\end{equation} + +\subsection*{Recurrence relation} +\begin{eqnarray} +\label{RecMP} +& &(n+1)P_{n+1}^{(\lambda)}(x;\phi)-2\left[x\sin\phi+(n+\lambda)\cos\phi\right] +P_n^{(\lambda)}(x;\phi)\nonumber\\ +& &{}\mathindent{}+(n+2\lambda-1)P_{n-1}^{(\lambda)}(x;\phi)=0. +\end{eqnarray} + +\subsection*{Normalized recurrence relation} +\begin{equation} +\label{NormRecMP} +xp_n(x)=p_{n+1}(x)-\left(\frac{n+\lambda}{\tan\phi}\right)p_n(x)+ +\frac{n(n+2\lambda-1)}{4\sin^2\phi}p_{n-1}(x), +\end{equation} +where +$$P_n^{(\lambda)}(x;\phi)=\frac{(2\sin\phi)^n}{n!}p_n(x).$$ + +\subsection*{Difference equation} +\begin{eqnarray} +\label{dvMP} +& &\e^{i\phi}(\lambda-ix)y(x+i)+2i\left[x\cos\phi-(n+\lambda)\sin\phi\right]y(x)\nonumber\\ +& &{}\mathindent{}-\e^{-i\phi}(\lambda+ix)y(x-i)=0,\quad y(x)=P_n^{(\lambda)}(x;\phi). +\end{eqnarray} + +\subsection*{Forward shift operator} +\begin{equation} +\label{shift1MPI} +P_n^{(\lambda)}(x+\textstyle\frac{1}{2}i;\phi)- +P_n^{(\lambda)}(x-\textstyle\frac{1}{2}i;\phi)=(\expe^{i\phi}-\expe^{-i\phi}) +P_{n-1}^{(\lambda+\frac{1}{2})}(x;\phi) +\end{equation} +or equivalently +\begin{equation} +\label{shift1MPII} +\frac{\delta P_n^{(\lambda)}(x;\phi)}{\delta x} +=2\sin\phi\,P_{n-1}^{(\lambda+\frac{1}{2})}(x;\phi). +\end{equation} + +\subsection*{Backward shift operator} +\begin{eqnarray} +\label{shift2MPI} +& &\e^{i\phi}(\lambda-\textstyle\frac{1}{2}-ix) +P_n^{(\lambda)}(x+\textstyle\frac{1}{2}i;\phi)+ +\e^{-i\phi}(\lambda-\textstyle\frac{1}{2}+ix) +P_n^{(\lambda)}(x-\textstyle\frac{1}{2}i;\phi)\nonumber\\ +& &{}=(n+1)P_{n+1}^{(\lambda-\frac{1}{2})}(x;\phi) +\end{eqnarray} +or equivalently +\begin{equation} +\label{shift2MPII} +\frac{\delta\left[\omega(x;\lambda,\phi)P_n^{(\lambda)}(x;\phi)\right]}{\delta x}= +-(n+1)\omega(x;\lambda-\textstyle\frac{1}{2},\phi) +P_{n+1}^{(\lambda-\frac{1}{2})}(x;\phi), +\end{equation} +where +$$\omega(x;\lambda,\phi)=\Gamma(\lambda+ix)\Gamma(\lambda-ix)\e^{(2\phi-\pi)x}.$$ + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodMP} +\omega(x;\lambda,\phi)P_n^{(\lambda)}(x;\phi)=\frac{(-1)^n}{n!} +\left(\frac{\delta}{\delta x}\right)^n\left[\omega(x;\lambda+\textstyle\frac{1}{2}n,\phi)\right]. +\end{equation} + +\newpage + +\subsection*{Generating functions} +\begin{equation} +\label{GenMP1} +(1-\expe^{i\phi}t)^{-\lambda+ix}(1-\expe^{-i\phi}t)^{-\lambda-ix}= +\sum_{n=0}^{\infty}P_n^{(\lambda)}(x;\phi)t^n. +\end{equation} + +\begin{equation} +\label{GenMP2} +\e^t\,\hyp{1}{1}{\lambda+ix}{2\lambda}{(\expe^{-2i\phi}-1)t}= +\sum_{n=0}^{\infty}\frac{P_n^{(\lambda)}(x;\phi)}{(2\lambda)_n\expe^{in\phi}}t^n. +\end{equation} + +\begin{eqnarray} +\label{GenMP3} +& &(1-t)^{-\gamma}\,\hyp{2}{1}{\gamma,\lambda+ix}{2\lambda}{\frac{(1-\e^{-2i\phi})t}{t-1}}\nonumber\\ +& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n}{(2\lambda)_n}\frac{P_n^{(\lambda)}(x;\phi)}{\e^{in\phi}}t^n, +\quad\textrm{$\gamma$ arbitrary}. +\end{eqnarray} + +\subsection*{Limit relations} + +\subsubsection*{Continuous dual Hahn $\rightarrow$ Meixner-Pollaczek} +The Meixner-Pollaczek polynomials can be obtained from the continuous dual Hahn polynomials +given by (\ref{DefContDualHahn}) by the substitutions $x\rightarrow x-t$, $a=\lambda+it$, +$b=\lambda-it$ and $c=t\cot\phi$ and the limit $t\rightarrow\infty$: +$$\lim_{t\rightarrow\infty}\frac{S_n((x-t)^2;\lambda+it,\lambda-it,t\cot\phi)}{t^nn!} +=\frac{P_n^{(\lambda)}(x;\phi)}{(\sin\phi)^n}.$$ + +\subsubsection*{Continuous Hahn $\rightarrow$ Meixner-Pollaczek} +By setting $x\rightarrow x+t$, $a=\lambda-it$, $c=\lambda+it$ and $b=d=t\tan\phi$ in the +definition (\ref{DefContHahn}) of the continuous Hahn polynomials and taking the limit +$t\rightarrow\infty$ we obtain the Meixner-Pollaczek polynomials: +$$\lim_{t\rightarrow\infty}\frac{p_n(x+t;\lambda-it,t\tan\phi,\lambda+it,t\tan\phi)}{t^nn!} +=\frac{P_n^{(\lambda)}(x;\phi)}{(\cos\phi)^n}.$$ + +\newpage + +\subsubsection*{Meixner-Pollaczek $\rightarrow$ Laguerre} +The Laguerre polynomials given by (\ref{DefLaguerre}) can be obtained from the +Meixner-Pollaczek polynomials by the substitution $\lambda=\frac{1}{2}(\alpha+1)$, +$x\rightarrow -\frac{1}{2}\phi^{-1}x$ and the limit $\phi\rightarrow 0$: +\begin{equation} +\lim_{\phi\rightarrow 0} +P_n^{(\frac{1}{2}\alpha+\frac{1}{2})}(-\textstyle\frac{1}{2}\phi^{-1}x;\phi)=L_n^{(\alpha)}(x). +\end{equation} + +\subsubsection*{Meixner-Pollaczek $\rightarrow$ Hermite} +The Hermite polynomials given by (\ref{DefHermite}) are obtained from the Meixner-Pollaczek +polynomials if we substitute $x\rightarrow (\sin\phi)^{-1}(x\sqrt{\lambda}-\lambda\cos\phi)$ +and then let $\lambda\rightarrow\infty$: +\begin{equation} +\lim_{\lambda\rightarrow\infty} +\lambda^{-\frac{1}{2}n}P_n^{(\lambda)} +((\sin\phi)^{-1}(x\sqrt{\lambda}-\lambda\cos\phi);\phi)=\frac{H_n(x)}{n!}. +\end{equation} + +\subsection*{Remark} +Since we have for $k-1$ and $\beta>-1$ we have +\begin{eqnarray} +\label{OrtJacobi1} +& &\int_{-1}^1(1-x)^{\alpha}(1+x)^{\beta}P_m^{(\alpha,\beta)}(x)P_n^{(\alpha,\beta)}(x)\,dx\nonumber\\ +& &{}=\frac{2^{\alpha+\beta+1}}{2n+\alpha+\beta+1}\frac{\Gamma(n+\alpha+1)\Gamma(n+\beta+1)}{\Gamma(n+\alpha+\beta+1)n!}\,\delta_{mn}. +\end{eqnarray} +For $\alpha+\beta<-2N-1$, $\beta>-1$ and $m,n\in\{0,1,2,\ldots,N\}$ we also have +\begin{eqnarray} +\label{OrtJacobi2} +& &\int_1^{\infty}(x+1)^{\alpha}(x-1)^{\beta}P_m^{(\alpha,\beta)}(-x)P_n^{(\alpha,\beta)}(-x)\,dx\nonumber\\ +& &{}=-\frac{2^{\alpha+\beta+1}}{2n+\alpha+\beta+1}\frac{\Gamma(-n-\alpha-\beta)\Gamma(n+\alpha+\beta+1)}{\Gamma(-n-\alpha)n!}\,\delta_{mn}. +\end{eqnarray} + +\subsection*{Recurrence relation} +\begin{eqnarray} +\label{RecJacobi} +xP_n^{(\alpha,\beta)}(x)&=&\frac{2(n+1)(n+\alpha+\beta+1)}{(2n+\alpha+\beta+1)(2n+\alpha+\beta+2)}P_{n+1}^{(\alpha,\beta)}(x)\nonumber\\ +& &{}\mathindent{}+\frac{\beta^2-\alpha^2}{(2n+\alpha+\beta)(2n+\alpha+\beta+2)}P_n^{(\alpha,\beta)}(x)\nonumber\\ +& &{}\mathindent\mathindent{}+\frac{2(n+\alpha)(n+\beta)}{(2n+\alpha+\beta)(2n+\alpha+\beta+1)}P_{n-1}^{(\alpha,\beta)}(x). +\end{eqnarray} + +\subsection*{Normalized recurrence relation} +\begin{eqnarray} +\label{NormRecJacobi} +xp_n(x)&=&p_{n+1}(x)+\frac{\beta^2-\alpha^2}{(2n+\alpha+\beta)(2n+\alpha+\beta+2)}p_n(x)\nonumber\\ +& &{}\mathindent{}+\frac{4n(n+\alpha)(n+\beta)(n+\alpha+\beta)} +{(2n+\alpha+\beta-1)(2n+\alpha+\beta)^2(2n+\alpha+\beta+1)}p_{n-1}(x) +\end{eqnarray} +where +$$P_n^{(\alpha,\beta)}(x)=\frac{(n+\alpha+\beta+1)_n}{2^nn!}p_n(x).$$ + +\subsection*{Differential equation} +\begin{eqnarray} +\label{dvJacobi} +& &(1-x^2)y''(x)+\left[\beta-\alpha-(\alpha+\beta+2)x\right]y'(x)\nonumber\\ +& &{}\mathindent{}+n(n+\alpha+\beta+1)y(x)=0,\quad y(x)=P_n^{(\alpha,\beta)}(x). +\end{eqnarray} + +\subsection*{Forward shift operator} +\begin{equation} +\label{shift1Jacobi} +\frac{d}{dx}P_n^{(\alpha,\beta)}(x)=\frac{n+\alpha+\beta+1}{2}P_{n-1}^{(\alpha+1,\beta+1)}(x). +\end{equation} + +\subsection*{Backward shift operator} +\begin{eqnarray} +\label{shift2JacobiI} +& &(1-x^2)\frac{d}{dx}P_n^{(\alpha,\beta)}(x)+ +\left[(\beta-\alpha)-(\alpha+\beta)x\right]P_n^{(\alpha,\beta)}(x)\nonumber\\ +& &{}=-2(n+1)P_{n+1}^{(\alpha-1,\beta-1)}(x) +\end{eqnarray} +or equivalently +\begin{eqnarray} +\label{shift2JacobiII} +& &\frac{d}{dx}\left[(1-x)^\alpha(1+x)^\beta P_n^{(\alpha,\beta)}(x)\right]\nonumber\\ +& &{}=-2(n+1)(1-x)^{\alpha-1}(1+x)^{\beta-1}P_{n+1}^{(\alpha-1,\beta-1)}(x). +\end{eqnarray} + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodJacobi} +(1-x)^{\alpha}(1+x)^{\beta}P_n^{(\alpha,\beta)}(x)= +\frac{(-1)^n}{2^nn!}\left(\frac{d}{dx}\right)^n +\left[(1-x)^{n+\alpha}(1+x)^{n+\beta}\right]. +\end{equation} + +\subsection*{Generating functions} +\begin{equation} +\label{GenJacobi1} +\frac{2^{\alpha+\beta}}{R(1+R-t)^{\alpha}(1+R+t)^{\beta}}= +\sum_{n=0}^{\infty}P_n^{(\alpha,\beta)}(x)t^n,\quad R=\sqrt{1-2xt+t^2}. +\end{equation} + +\begin{eqnarray} +\label{GenJacobi2} +& &\hyp{0}{1}{-}{\alpha+1}{\frac{(x-1)t}{2}}\,\hyp{0}{1}{-}{\beta+1}{\frac{(x+1)t}{2}}\nonumber\\ +& &{}=\sum_{n=0}^{\infty}\frac{P_n^{(\alpha,\beta)}(x)}{(\alpha+1)_n(\beta+1)_n}t^n. +\end{eqnarray} + +\begin{eqnarray} +\label{GenJacobi3} +& &(1-t)^{-\alpha-\beta-1}\,\hyp{2}{1}{\frac{1}{2}(\alpha+\beta+1),\frac{1}{2}(\alpha+\beta+2)} +{\alpha+1}{\frac{2(x-1)t}{(1-t)^2}}\nonumber\\ +& &{}=\sum_{n=0}^{\infty}\frac{(\alpha+\beta+1)_n}{(\alpha+1)_n}P_n^{(\alpha,\beta)}(x)t^n. +\end{eqnarray} + +\begin{eqnarray} +\label{GenJacobi4} +& &(1+t)^{-\alpha-\beta-1}\,\hyp{2}{1}{\frac{1}{2}(\alpha+\beta+1),\frac{1}{2}(\alpha+\beta+2)} +{\beta+1}{\frac{2(x+1)t}{(1+t)^2}}\nonumber\\ +& &{}=\sum_{n=0}^{\infty}\frac{(\alpha+\beta+1)_n}{(\beta+1)_n}P_n^{(\alpha,\beta)}(x)t^n. +\end{eqnarray} + +\begin{eqnarray} +\label{GenJacobi5} +& &\hyp{2}{1}{\gamma,\alpha+\beta+1-\gamma}{\alpha+1}{\frac{1-R-t}{2}}\, +\hyp{2}{1}{\gamma,\alpha+\beta+1-\gamma}{\beta+1}{\frac{1-R+t}{2}}\nonumber\\ +& &{}=\sum_{n=0}^{\infty} +\frac{(\gamma)_n(\alpha+\beta+1-\gamma)_n}{(\alpha+1)_n(\beta+1)_n}P_n^{(\alpha,\beta)}(x)t^n, +\quad R=\sqrt{1-2xt+t^2} +\end{eqnarray} +with $\gamma$ arbitrary. + +\subsection*{Limit relations} + +\subsubsection*{Wilson $\rightarrow$ Jacobi} +The Jacobi polynomials can be found from the Wilson polynomials given by (\ref{DefWilson}) by +substituting $a=b=\frac{1}{2}(\alpha+1)$, $c=\frac{1}{2}(\beta+1)+it$, +$d=\frac{1}{2}(\beta+1)-it$ and $x\rightarrow t\sqrt{\frac{1}{2}(1-x)}$ in the +definition (\ref{DefWilson}) of the Wilson polynomials and taking the limit +$t\rightarrow\infty$. In fact we have +$$\lim_{t\rightarrow\infty} +\frac{W_n(\frac{1}{2}(1-x)t^2;\frac{1}{2}(\alpha+1), +\frac{1}{2}(\alpha+1),\frac{1}{2}(\beta+1)+it,\frac{1}{2}(\beta+1)-it)} +{t^{2n}n!}=P_n^{(\alpha,\beta)}(x).$$ + +\subsubsection*{Continuous Hahn $\rightarrow$ Jacobi} +The Jacobi polynomials follow from the continuous Hahn polynomials given by (\ref{DefContHahn}) +by using the substitution $x\rightarrow \frac{1}{2}xt$, $a=\frac{1}{2}(\alpha+1-it)$, +$b=\frac{1}{2}(\beta+1+it)$, $c=\frac{1}{2}(\alpha+1+it)$ and $d=\frac{1}{2}(\beta+1-it)$ +in (\ref{DefContHahn}), division by $t^n$ and the limit $t\rightarrow\infty$: +$$\lim_{t\rightarrow\infty} +\frac{p_n(\frac{1}{2}xt;\frac{1}{2}(\alpha+1-it),\frac{1}{2}(\beta+1+it), +\frac{1}{2}(\alpha+1+it),\frac{1}{2}(\beta+1-it))}{t^n}=P_n^{(\alpha,\beta)}(x).$$ + +\subsubsection*{Hahn $\rightarrow$ Jacobi} +To find the Jacobi polynomials from the Hahn polynomials given by (\ref{DefHahn}) we take +$x\rightarrow Nx$ in (\ref{DefHahn}) and let $N\rightarrow\infty$. In fact we have +$$\lim_{N\rightarrow\infty} +Q_n(Nx;\alpha,\beta,N)=\frac{P_n^{(\alpha,\beta)}(1-2x)}{P_n^{(\alpha,\beta)}(1)}.$$ + +\subsubsection*{Jacobi $\rightarrow$ Laguerre} +The Laguerre polynomials given by (\ref{DefLaguerre}) can be obtained from the Jacobi +polynomials by setting $x\rightarrow 1-2\beta^{-1}x$ and then the limit $\beta\rightarrow\infty$: +\begin{equation} +\lim_{\beta\rightarrow\infty} +P_n^{(\alpha,\beta)}(1-2\beta^{-1}x)=L_n^{(\alpha)}(x). +\end{equation} + +\subsubsection*{Jacobi $\rightarrow$ Bessel} +The Bessel polynomials given by (\ref{DefBessel}) are obtained from the Jacobi polynomials +if we take $\beta=a-\alpha$ and let $\alpha\rightarrow-\infty$: +\begin{equation} +\lim_{\alpha\rightarrow-\infty} +\frac{P_n^{(\alpha,a-\alpha)}(1+\alpha x)}{P_n^{(\alpha,a-\alpha)}(1)}=y_n(x;a). +\end{equation} + +\subsubsection*{Jacobi $\rightarrow$ Hermite} +The Hermite polynomials given by (\ref{DefHermite}) follow from the Jacobi polynomials +by taking $\beta=\alpha$ and letting $\alpha\rightarrow\infty$ in the following way: +\begin{equation} +\lim_{\alpha\rightarrow\infty} +\alpha^{-\frac{1}{2}n}P_n^{(\alpha,\alpha)}(\alpha^{-\frac{1}{2}}x)=\frac{H_n(x)}{2^nn!}. +\end{equation} + +\subsection*{Remarks} +The definition (\ref{DefJacobi}) of the Jacobi polynomials can also be written as: +$$P_n^{(\alpha,\beta)}(x)=\frac{1}{n!}\sum_{k=0}^n\frac{(-n)_k}{k!} +(n+\alpha+\beta+1)_k(\alpha+k+1)_{n-k}\left(\frac{1-x}{2}\right)^k.$$ +In this way the Jacobi polynomials can also be seen as polynomials in the parameters $\alpha$ +and $\beta$. Therefore they can be defined for all $\alpha$ and $\beta$. Then we have the +following connection with the Meixner polynomials given by (\ref{DefMeixner}): +$$\frac{(\beta)_n}{n!}M_n(x;\beta,c)=P_n^{(\beta-1,-n-\beta-x)}((2-c)c^{-1}).$$ + +\noindent +The Jacobi polynomials are related to the pseudo Jacobi polynomials defined by +(\ref{DefPseudoJacobi}) in the following way: +$$P_n(x;\nu,N)=\frac{(-2i)^nn!}{(n-2N-1)_n}P_n^{(-N-1+i\nu,-N-1-i\nu)}(ix).$$ + +\noindent +The Jacobi polynomials are also related to the Gegenbauer (or ultraspherical) polynomials +given by (\ref{DefGegenbauer}) by the quadratic transformations: +$$C_{2n}^{(\lambda)}(x)=\frac{(\lambda)_n}{(\frac{1}{2})_n} +P_n^{(\lambda-\frac{1}{2},-\frac{1}{2})}(2x^2-1)$$ +and +$$C_{2n+1}^{(\lambda)}(x)=\frac{(\lambda)_{n+1}}{(\frac{1}{2})_{n+1}} +\subsection*{9.8 Jacobi} +\label{sec9.8} +% +\paragraph{Orthogonality relation} +Write the \RHS\ of (9.8.2) as $h_n\,\de_{m,n}$. Then +\begin{equation} +\begin{split} +&\frac{h_n}{h_0}= +\frac{n+\al+\be+1}{2n+\al+\be+1}\, +\frac{(\al+1)_n(\be+1)_n}{(\al+\be+2)_n\,n!}\,,\quad +h_0=\frac{2^{\al+\be+1}\Ga(\al+1)\Ga(\be+1)}{\Ga(\al+\be+2)}\,,\sLP +&\frac{h_n}{h_0\,(P_n^{(\al,\be)}(1))^2}= +\frac{n+\al+\be+1}{2n+\al+\be+1}\, +\frac{(\be+1)_n\,n!}{(\al+1)_n\,(\al+\be+2)_n}\,. +\end{split} +\label{60} +\end{equation} + +In (9.8.3) the numerator factor $\Ga(n+\al+\be+1)$ in the last line should be +$\Ga(\be+1)$. When thus corrected, (9.8.3) can be rewritten as: +\begin{equation} +\begin{split} +&\int_1^\iy P_m^{(\al,\be)}(x)\,P_n^{(\al,\be)}(x)\,(x-1)^\al (x+1)^\be\,dx=h_n\,\de_{m,n}\,,\\ +&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad-1-\be>\al>-1,\quad m,n<-\thalf(\al+\be+1),\\ +&\frac{h_n}{h_0}= +\frac{n+\al+\be+1}{2n+\al+\be+1}\, +\frac{(\al+1)_n(\be+1)_n}{(\al+\be+2)_n\,n!}\,,\quad +h_0=\frac{2^{\al+\be+1}\Ga(\al+1)\Ga(-\al-\be-1)}{\Ga(-\be)}\,. +\end{split} +\label{122} +\end{equation} + +% +\paragraph{Symmetry} +\begin{equation} +P_n^{(\al,\be)}(-x)=(-1)^n\,P_n^{(\be,\al)}(x). +\label{48} +\end{equation} +Use (9.8.2) and (9.8.5b) or see \mycite{DLMF}{Table 18.6.1}. +% +\paragraph{Special values} +\begin{equation} +P_n^{(\al,\be)}(1)=\frac{(\al+1)_n}{n!}\,,\quad +P_n^{(\al,\be)}(-1)=\frac{(-1)^n(\be+1)_n}{n!}\,,\quad +\frac{P_n^{(\al,\be)}(-1)}{P_n^{(\al,\be)}(1)}=\frac{(-1)^n(\be+1)_n}{(\al+1)_n}\,. +\label{50} +\end{equation} +Use (9.8.1) and \eqref{48} or see \mycite{DLMF}{Table 18.6.1}. +% +\paragraph{Generating functions} +Formula (9.8.15) was first obtained by Brafman \myciteKLS{109}. +% +\paragraph{Bilateral generating functions} +For $0\le r<1$ and $x,y\in[-1,1]$ we have in terms of $F_4$ (see~\eqref{62}): +\begin{align} +&\sum_{n=0}^\iy\frac{(\al+\be+1)_n\,n!}{(\al+1)_n(\be+1)_n}\,r^n\, +P_n^{(\al,\be)}(x)\,P_n^{(\al,\be)}(y) +=\frac1{(1+r)^{\al+\be+1}} +\nonumber\\ +&\qquad\quad\times F_4\Big(\thalf(\al+\be+1),\thalf(\al+\be+2);\al+1,\be+1; +\frac{r(1-x)(1-y)}{(1+r)^2},\frac{r(1+x)(1+y)}{(1+r)^2}\Big), +\label{58}\sLP +&\sum_{n=0}^\iy\frac{2n+\al+\be+1}{n+\al+\be+1} +\frac{(\al+\be+2)_n\,n!}{(\al+1)_n(\be+1)_n}\,r^n\, +P_n^{(\al,\be)}(x)\,P_n^{(\al,\be)}(y) +=\frac{1-r}{(1+r)^{\al+\be+2}}\nonumber\\ +&\qquad\quad\times F_4\Big(\thalf(\al+\be+2),\thalf(\al+\be+3);\al+1,\be+1; +\frac{r(1-x)(1-y)}{(1+r)^2},\frac{r(1+x)(1+y)}{(1+r)^2}\Big). +\label{59} +\end{align} +Formulas \eqref{58} and \eqref{59} were first +given by Bailey \myciteKLS{91}{(2.1), (2.3)}. +See Stanton \myciteKLS{485} for a shorter proof. +(However, in the second line of +\myciteKLS{485}{(1)} $z$ and $Z$ should be interchanged.)$\;$ +As observed in Bailey \myciteKLS{91}{p.10}, \eqref{59} follows +from \eqref{58} +by applying the operator $r^{-\half(\al+\be-1)}\,\frac d{dr}\circ r^{\half(\al+\be+1)}$ +to both sides of \eqref{58}. +In view of \eqref{60}, formula \eqref{59} is the Poisson kernel for Jacobi +polynomials. The \RHS\ of \eqref{59} makes clear that this kernel is positive. +See also the discussion in Askey \myciteKLS{46}{following (2.32)}. +% +\paragraph{Quadratic transformations} +\begin{align} +\frac{C_{2n}^{(\al+\half)}(x)}{C_{2n}^{(\al+\half)}(1)} +=\frac{P_{2n}^{(\al,\al)}(x)}{P_{2n}^{(\al,\al)}(1)} +&=\frac{P_n^{(\al,-\half)}(2x^2-1)}{P_n^{(\al,-\half)}(1)}\,, +\label{51}\\ +\frac{C_{2n+1}^{(\al+\half)}(x)}{C_{2n+1}^{(\al+\half)}(1)} +=\frac{P_{2n+1}^{(\al,\al)}(x)}{P_{2n+1}^{(\al,\al)}(1)} +&=\frac{x\,P_n^{(\al,\half)}(2x^2-1)}{P_n^{(\al,\half)}(1)}\,. +\label{52} +\end{align} +See p.221, Remarks, last two formulas together with \eqref{50} and \eqref{49}. +Or see \mycite{DLMF}{(18.7.13), (18.7.14)}. +% +\paragraph{Differentiation formulas} +Each differentiation formula is given in two equivalent forms. +\begin{equation} +\begin{split} +\frac d{dx}\left((1-x)^\al P_n^{(\al,\be)}(x)\right)&= +-(n+\al)\,(1-x)^{\al-1} P_n^{(\al-1,\be+1)}(x),\\ +\left((1-x)\frac d{dx}-\al\right)P_n^{(\al,\be)}(x)&= +-(n+\al)\,P_n^{(\al-1,\be+1)}(x). +\end{split} +\label{68} +\end{equation} +% +\begin{equation} +\begin{split} +\frac d{dx}\left((1+x)^\be P_n^{(\al,\be)}(x)\right)&= +(n+\be)\,(1+x)^{\be-1} P_n^{(\al+1,\be-1)}(x),\\ +\left((1+x)\frac d{dx}+\be\right)P_n^{(\al,\be)}(x)&= +(n+\be)\,P_n^{(\al+1,\be-1)}(x). +\end{split} +\label{69} +\end{equation} +Formulas \eqref{68} and \eqref{69} follow from +\mycite{DLMF}{(15.5.4), (15.5.6)} +together with (9.8.1). They also follow from each other by \eqref{48}. +% +\paragraph{Generalized Gegenbauer polynomials} +These are defined by +\begin{equation} +S_{2m}^{(\al,\be)}(x):=\const P_m^{(\al,\be)}(2x^2-1),\qquad +S_{2m+1}^{(\al,\be)}(x):=\const x\,P_m^{(\al,\be+1)}(2x^2-1) +\label{70} +\end{equation} +in the notation of \myciteKLS{146}{p.156} +(see also \cite{K27}), while \cite[Section 1.5.2]{K26} +has $C_n^{(\la,\mu)}(x)=\const\allowbreak\times S_n^{(\la-\half,\mu-\half)}(x)$. +For $\al,\be>-1$ we have the orthogonality relation +\begin{equation} +\int_{-1}^1 S_m^{(\al,\be)}(x)\,S_n^{(\al,\be)}(x)\,|x|^{2\be+1}(1-x^2)^\al\,dx +=0\qquad(m\ne n). +\label{71} +\end{equation} +For $\be=\al-1$ generalized Gegenbauer polynomials are limit cases of +continuous $q$-ultraspherical polynomials, see \eqref{176}. + +If we define the {\em Dunkl operator} $T_\mu$ by +\begin{equation} +(T_\mu f)(x):=f'(x)+\mu\,\frac{f(x)-f(-x)}x +\label{72} +\end{equation} +and if we choose the constants in \eqref{70} as +\begin{equation} +S_{2m}^{(\al,\be)}(x)=\frac{(\al+\be+1)_m}{(\be+1)_m}\, P_m^{(\al,\be)}(2x^2-1),\quad +S_{2m+1}^{(\al,\be)}(x)=\frac{(\al+\be+1)_{m+1}}{(\be+1)_{m+1}}\, +x\,P_m^{(\al,\be+1)}(2x^2-1) +\label{73} +\end{equation} +then (see \cite[(1.6)]{K5}) +\begin{equation} +T_{\be+\half}S_n^{(\al,\be)}=2(\al+\be+1)\,S_{n-1}^{(\al+1,\be)}. +\label{74} +\end{equation} +Formula \eqref{74} with \eqref{73} substituted gives rise to two +differentiation formulas involving Jacobi polynomials which are equivalent to +(9.8.7) and \eqref{69}. + +Composition of \eqref{74} with itself gives +\[ +T_{\be+\half}^2S_n^{(\al,\be)}=4(\al+\be+1)(\al+\be+2)\,S_{n-2}^{(\al+2,\be)}, +\] +which is equivalent to the composition of (9.8.7) and \eqref{69}: +\begin{equation} +\left(\frac{d^2}{dx^2}+\frac{2\be+1}x\,\frac d{dx}\right)P_n^{(\al,\be)}(2x^2-1) +=4(n+\al+\be+1)(n+\be)\,P_{n-1}^{(\al+2,\be)}(2x^2-1). +\label{75} +\end{equation} +Formula \eqref{75} was also given in \myciteKLS{322}{(2.4)}. +xP_n^{(\lambda-\frac{1}{2},\frac{1}{2})}(2x^2-1).$$ + +\subsection*{References} +\cite{Abram}, \cite{Ahmed+82}, \cite{Allaway89}, \cite{NAlSalam66}, \cite{AlSalam64}, +\cite{AlSalamChihara72}, \cite{AndrewsAskey85}, \cite{AndrewsAskeyRoy}, \cite{Askey68}, +\cite{Askey70}, \cite{Askey72}, \cite{Askey73}, \cite{Askey74}, \cite{Askey75}, \cite{Askey78}, +\cite{Askey89I}, \cite{Askey2005}, \cite{AskeyFitch}, \cite{AskeyGasper71I}, \cite{AskeyGasper71II}, +\cite{AskeyGasper76}, \cite{AskeyWainger}, \cite{AskeyWilson85}, \cite{Bailey38}, \cite{Brafman51}, +\cite{Brown}, \cite{Carlitz61II}, \cite{Carlitz67}, \cite{ChenIsmail91}, \cite{Chihara78}, +\cite{Chow+}, \cite{Ciesielski}, \cite{Cooper+}, \cite{DetteStudden92}, \cite{DetteStudden95}, +\cite{DijksmaKoorn}, \cite{DimitrovRafaeli}, \cite{DimitrovRodrigues}, \cite{Doha2002I}, +\cite{Doha2003I}, \cite{Doha2004II}, \cite{Dunkl84}, \cite{ElbertLaforgia87II}, +\cite{ElbertLaforgiaRodono}, \cite{Erdelyi+}, \cite{Faldey}, \cite{FoataLeroux}, +\cite{Gasper69}, \cite{Gasper70I}, \cite{Gasper70II}, \cite{Gasper71I}, \cite{Gasper71II}, +\cite{Gasper72I}, \cite{Gasper72II}, \cite{Gasper73I}, \cite{Gasper73II}, \cite{Gasper74}, +\cite{Gasper77}, \cite{Gatteschi87}, \cite{Gautschi2008}, \cite{Gautschi2009I}, +\cite{Gautschi2009II}, \cite{GautschiLeopardi}, \cite{GawronskiShawyer}, \cite{Godoy+}, +\cite{Grad}, \cite{Hajmirzaahmad94}, \cite{HartmannStephan}, \cite{Horton}, \cite{Ismail74}, +\cite{Ismail77}, \cite{Ismail96}, \cite{Ismail2005II}, \cite{IsmailLi}, +\cite{IsmailMassonRahman}, \cite{Kochneff97I}, \cite{Koekoek99}, \cite{Koekoek2000}, +\cite{Koelink96II}, \cite{Koorn73}, \cite{Koorn74}, \cite{Koorn75}, \cite{Koorn77II}, +\cite{Koorn78}, \cite{Koorn72}, \cite{Koorn85}, \cite{Koorn88}, \cite{KuijlaarsMartinez}, +\cite{Kuijlaars+}, \cite{LabelleYehI}, \cite{LabelleYehII}, \cite{Laine}, \cite{Lesky95II}, +\cite{Lesky96}, \cite{Li96}, \cite{Li97}, \cite{LopezTemme2004}, \cite{Luke}, \cite{Martinez}, +\cite{Mathai}, \cite{Meijer}, \cite{Miller89}, \cite{MoakSaffVarga}, \cite{Nikiforov+}, +\cite{NikiforovUvarov}, \cite{Olver}, \cite{Prasad}, \cite{Rahman76I}, \cite{Rahman76II}, +\cite{Rahman77}, \cite{Rahman81I}, \cite{RahmanShah}, \cite{Rainville}, \cite{Rusev}, +\cite{Shi}, \cite{Srivastava69II}, \cite{Srivastava71}, \cite{Srivastava82}, +\cite{SrivastavaSinghal}, \cite{Stanton80I}, \cite{Stanton90}, \cite{Szego75}, \cite{Temme}, +\cite{Vertesi}, \cite{Wimp87}, \cite{WongZhang94}, \cite{WongZhang2006}, \cite{Zarzo+}, +\cite{Zayed}. + +\section*{Special cases} + +\subsection{Gegenbauer / Ultraspherical}\index{Gegenbauer polynomials} +\index{Ultraspherical polynomials} + +\par + +\subsection*{Hypergeometric representation} +The Gegenbauer (or ultraspherical) polynomials are Jacobi polynomials with +$\alpha=\beta=\lambda-\frac{1}{2}$ and another normalization: +\begin{eqnarray} +\label{DefGegenbauer} +C_n^{(\lambda)}(x)&=&\frac{(2\lambda)_n}{(\lambda+\frac{1}{2})_n} +P_n^{(\lambda-\frac{1}{2},\lambda-\frac{1}{2})}(x)\nonumber\\ +&=&\frac{(2\lambda)_n}{n!}\,\hyp{2}{1}{-n,n+2\lambda} +{\lambda+\frac{1}{2}}{\frac{1-x}{2}},\quad\lambda\neq 0. +\end{eqnarray} + +\subsection*{Orthogonality relation} +\begin{eqnarray} +\label{OrtGegenbauer} +& &\int_{-1}^1(1-x^2)^{\lambda-\frac{1}{2}}C_m^{(\lambda)}(x)C_n^{(\lambda)}(x)\,dx\nonumber\\ +& &{}=\frac{\pi\Gamma(n+2\lambda)2^{1-2\lambda}}{\left\{\Gamma(\lambda)\right\}^2(n+\lambda)n!}\,\delta_{mn}, +\quad\lambda>-\frac{1}{2}\quad\lambda\neq 0. +\end{eqnarray} + +\subsection*{Recurrence relation} +\begin{equation} +\label{RecGegenbauer} +2(n+\lambda)xC_n^{(\lambda)}(x)=(n+1)C_{n+1}^{(\lambda)}(x)+(n+2\lambda-1)C_{n-1}^{(\lambda)}(x). +\end{equation} + +\subsection*{Normalized recurrence relation} +\begin{equation} +\label{NormRecGegenbauer} +xp_n(x)=p_{n+1}(x)+\frac{n(n+2\lambda-1)}{4(n+\lambda-1)(n+\lambda)}p_{n-1}(x), +\end{equation} +where +$$C_n^{(\lambda)}(x)=\frac{2^n(\lambda)_n}{n!}p_n(x).$$ + +\subsection*{Differential equation} +\begin{equation} +\label{dvGegenbauer} +(1-x^2)y''(x)-(2\lambda+1)xy'(x)+n(n+2\lambda)y(x)=0,\quad y(x)=C_n^{(\lambda)}(x). +\end{equation} + +\subsection*{Forward shift operator} +\begin{equation} +\label{shift1Gegenbauer} +\frac{d}{dx}C_n^{(\lambda)}(x)=2\lambda C_{n-1}^{(\lambda+1)}(x). +\end{equation} + +\subsection*{Backward shift operator} +\begin{equation} +\label{shift2GegenbauerI} +(1-x^2)\frac{d}{dx}C_n^{(\lambda)}(x)+(1-2\lambda)xC_n^{(\lambda)}(x)= +-\frac{(n+1)(2\lambda+n-1)}{2(\lambda-1)} C_{n+1}^{(\lambda-1)}(x) +\end{equation} +or equivalently +\begin{eqnarray} +\label{shift2GegenbauerII} +& &\frac{d}{dx}\left[(1-x^2)^{\lambda-\textstyle\frac{1}{2}}C_n^{(\lambda)}(x)\right]\nonumber\\ +& &{}=-\frac{(n+1)(2\lambda+n-1)}{2(\lambda-1)}(1-x^2)^{\lambda-\textstyle\frac{3}{2}}C_{n+1}^{(\lambda-1)}(x). +\end{eqnarray} + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodGegenbauer} +(1-x^2)^{\lambda-\frac{1}{2}}C_n^{(\lambda)}(x)= +\frac{(2\lambda)_n(-1)^n}{(\lambda+\frac{1}{2})_n2^nn!}\left(\frac{d}{dx}\right)^n +\left[(1-x^2)^{\lambda+n-\frac{1}{2}}\right]. +\end{equation} + +\subsection*{Generating functions} +\begin{equation} +\label{GenGegenbauer1} +(1-2xt+t^2)^{-\lambda}=\sum_{n=0}^{\infty}C_n^{(\lambda)}(x)t^n. +\end{equation} + +\begin{equation} +\label{GenGegenbauer2} +R^{-1}\left(\frac{1+R-xt}{2}\right)^{\frac{1}{2}-\lambda}=\sum_{n=0}^{\infty} +\frac{(\lambda+\frac{1}{2})_n}{(2\lambda)_n}C_n^{(\lambda)}(x)t^n, +\quad R=\sqrt{1-2xt+t^2}. +\end{equation} + +\begin{equation} +\label{GenGegenbauer3} +\hyp{0}{1}{-}{\lambda+\frac{1}{2}}{\frac{(x-1)t}{2}}\, +\hyp{0}{1}{-}{\lambda+\frac{1}{2}}{\frac{(x+1)t}{2}} +=\sum_{n=0}^{\infty}\frac{C_n^{(\lambda)}(x)} +{(2\lambda)_n(\lambda+\frac{1}{2})_n}t^n. +\end{equation} + +\begin{equation} +\label{GenGegenbauer4} +\e^{xt}\,\hyp{0}{1}{-}{\lambda+\frac{1}{2}}{\frac{(x^2-1)t^2}{4}}= +\sum_{n=0}^{\infty}\frac{C_n^{(\lambda)}(x)}{(2\lambda)_n}t^n. +\end{equation} + +\begin{eqnarray} +\label{GenGegenbauer5} +& &\hyp{2}{1}{\gamma,2\lambda-\gamma}{\lambda+\frac{1}{2}}{\frac{1-R-t}{2}}\, +\hyp{2}{1}{\gamma,2\lambda-\gamma}{\lambda+\frac{1}{2}}{\frac{1-R+t}{2}}\nonumber\\ +& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n(2\lambda-\gamma)_n} +{(2\lambda)_n(\lambda+\frac{1}{2})_n}C_n^{(\lambda)}(x)t^n, +\quad R=\sqrt{1-2xt+t^2},\quad\textrm{$\gamma$ arbitrary}. +\end{eqnarray} + +\begin{eqnarray} +\label{GenGegenbauer6} +& &(1-xt)^{-\gamma}\,\hyp{2}{1}{\frac{1}{2}\gamma,\frac{1}{2}\gamma+\frac{1}{2}} +{\lambda+\frac{1}{2}}{\frac{(x^2-1)t^2}{(1-xt)^2}}\nonumber\\ +& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n}{(2\lambda)_n}C_n^{(\lambda)}(x)t^n, +\quad\textrm{$\gamma$ arbitrary}. +\end{eqnarray} + +\subsection*{Limit relation} + +\subsubsection*{Gegenbauer / Ultraspherical $\rightarrow$ Hermite} +The Hermite polynomials given by (\ref{DefHermite}) follow from the Gegenbauer (or +ultraspherical) polynomials by taking $\lambda=\alpha+\frac{1}{2}$ and letting +$\alpha\rightarrow\infty$ in the following way: +\begin{equation} +\lim_{\alpha\rightarrow\infty} +\alpha^{-\frac{1}{2}n}C_n^{(\alpha+\frac{1}{2})}(\alpha^{-\frac{1}{2}}x)=\frac{H_n(x)}{n!}. +\end{equation} + +\subsection*{Remarks} +The case $\lambda=0$ needs another normalization. In that case we have the +Chebyshev polynomials of the first kind described in the next subsection. + +\noindent +The Gegenbauer (or ultraspherical) polynomials are related to the Jacobi polynomials +given by (\ref{DefJacobi}) by the quadratic transformations: +$$C_{2n}^{(\lambda)}(x)=\frac{(\lambda)_n}{(\frac{1}{2})_n} +P_n^{(\lambda-\frac{1}{2},-\frac{1}{2})}(2x^2-1)$$ +and +$$C_{2n+1}^{(\lambda)}(x)=\frac{(\lambda)_{n+1}}{(\frac{1}{2})_{n+1}} +\subsection*{9.8.1 Gegenbauer / Ultraspherical} +\label{sec9.8.1} +% +\paragraph{Notation} +Here the Gegenbauer polynomial is denoted by $C_n^\la$ instead of $C_n^{(\la)}$. +% +\paragraph{Orthogonality relation} +Write the \RHS\ of (9.8.20) as $h_n\,\de_{m,n}$. Then +\begin{equation} +\frac{h_n}{h_0}= +\frac\la{\la+n}\,\frac{(2\la)_n}{n!}\,,\quad +h_0=\frac{\pi^\half\,\Ga(\la+\thalf)}{\Ga(\la+1)},\quad +\frac{h_n}{h_0\,(C_n^\la(1))^2}= +\frac\la{\la+n}\,\frac{n!}{(2\la)_n}\,. +\label{61} +\end{equation} +% +\paragraph{Hypergeometric representation} +Beside (9.8.19) we have also +\begin{equation} +C_n^\lambda(x)=\sum_{\ell=0}^{\lfloor n/2\rfloor}\frac{(-1)^{\ell}(\lambda)_{n-\ell}} +{\ell!\;(n-2\ell)!}\,(2x)^{n-2\ell} +=(2x)^{n}\,\frac{(\lambda)_{n}}{n!}\, +\hyp21{-\thalf n,-\thalf n+\thalf}{1-\la-n}{\frac1{x^2}}. +\label{57} +\end{equation} +See \mycite{DLMF}{(18.5.10)}. +% +\paragraph{Special value} +\begin{equation} +C_n^{\la}(1)=\frac{(2\la)_n}{n!}\,. +\label{49} +\end{equation} +Use (9.8.19) or see \mycite{DLMF}{Table 18.6.1}. +% +\paragraph{Expression in terms of Jacobi} +% +\begin{equation} +\frac{C_n^\la(x)}{C_n^\la(1)}= +\frac{P_n^{(\la-\half,\la-\half)}(x)}{P_n^{(\la-\half,\la-\half)}(1)}\,,\qquad +C_n^\la(x)=\frac{(2\la)_n}{(\la+\thalf)_n}\,P_n^{(\la-\half,\la-\half)}(x). +\label{65} +\end{equation} +% +\paragraph{Re: (9.8.21)} +By iteration of recurrence relation (9.8.21): +\begin{multline} +x^2 C_n^\la(x)= +\frac{(n+1)(n+2)}{4(n+\la)(n+\la+1)}\,C_{n+2}^\la(x)+ +\frac{n^2+2n\la+\la-1}{2(n+\la-1)(n+\la+1)}\,C_n^\la(x)\\ ++\frac{(n+2\la-1)(n+2\la-2)}{4(n+\la)(n+\la-1)}\,C_{n-2}^\la(x). +\label{6} +\end{multline} +% +\paragraph{Bilateral generating functions} +\begin{multline} +\sum_{n=0}^\iy\frac{n!}{(2\la)_n}\,r^n\,C_n^\la(x)\,C_n^\la(y) +=\frac1{(1-2rxy+r^2)^\la}\,\hyp21{\thalf\la,\thalf(\la+1)}{\la+\thalf} +{\frac{4r^2(1-x^2)(1-y^2)}{(1-2rxy+r^2)^2}}\\ +(r\in(-1,1),\;x,y\in[-1,1]). +\label{66} +\end{multline} +For the proof put $\be:=\al$ in \eqref{58}, then use \eqref{63} and \eqref{65}. +The Poisson kernel for Gegenbauer polynomials can be derived in a similar way +from \eqref{59}, or alternatively by applying the operator +$r^{-\la+1}\frac d{dr}\circ r^\la$ to both sides of \eqref{66}: +\begin{multline} +\sum_{n=0}^\iy\frac{\la+n}\la\,\frac{n!}{(2\la)_n}\,r^n\,C_n^\la(x)\,C_n^\la(y) +=\frac{1-r^2}{(1-2rxy+r^2)^{\la+1}}\\ +\times\hyp21{\thalf(\la+1),\thalf(\la+2)}{\la+\thalf} +{\frac{4r^2(1-x^2)(1-y^2)}{(1-2rxy+r^2)^2}}\qquad +(r\in(-1,1),\;x,y\in[-1,1]). +\label{67} +\end{multline} +Formula \eqref{67} was obtained by Gasper \& Rahman \myciteKLS{234}{(4.4)} +as a limit case of their formula for the Poisson kernel for continuous +$q$-ultraspherical polynomials. +% +\paragraph{Trigonometric expansions} +By \mycite{DLMF}{(18.5.11), (15.8.1)}: +\begin{align} +C_n^{\la}(\cos\tha) +&=\sum_{k=0}^n\frac{(\la)_k(\la)_{n-k}}{k!\,(n-k)!}\,e^{i(n-2k)\tha} +=e^{in\tha}\frac{(\la)_n}{n!}\, +\hyp21{-n,\la}{1-\la-n}{e^{-2i\tha}}\label{103}\\ +&=\frac{(\la)_n}{2^\la n!}\, +e^{-\half i\la\pi}e^{i(n+\la)\tha}\,(\sin\tha)^{-\la}\, +\hyp21{\la,1-\la}{1-\la-n}{\frac{i e^{-i\tha}}{2\sin\tha}}\label{104}\\ +&=\frac{(\la)_n}{n!}\,\sum_{k=0}^\iy\frac{(\la)_k(1-\la)_k}{(1-\la-n)_k k!}\, +\frac{\cos((n-k+\la)\tha+\thalf(k-\la)\pi)}{(2\sin\tha)^{k+\la}}\,.\label{105} +\end{align} +In \eqref{104} and \eqref{105} we require that +$\tfrac16\pi<\tha<\tfrac56\pi$. Then the convergence is absolute for $\la>\thalf$ +and conditional for $0<\la\le\thalf$. + +By \mycite{DLMF}{(14.13.1), (14.3.21), (15.8.1)]}: +\begin{align} +C_n^\la(\cos\tha)&=\frac{2\Ga(\la+\thalf)}{\pi^\half\Ga(\la+1)}\, +\frac{(2\la)_n}{(\la+1)_n}\,(\sin\tha)^{1-2\la}\, +\sum_{k=0}^\iy\frac{(1-\la)_k(n+1)_k}{(n+\la+1)_k k!}\, +\sin\big((2k+n+1)\tha\big) +\label{7}\\ +&=\frac{2\Ga(\la+\thalf)}{\pi^\half\Ga(\la+1)}\, +\frac{(2\la)_n}{(\la+1)_n}\,(\sin\tha)^{1-2\la}\, +\Im\!\!\left(e^{i(n+1)\tha}\,\hyp21{1-\la,n+1}{n+\la+1}{e^{2i\tha}}\right)\nonumber\\ +&=\frac{2^\la\Ga(\la+\thalf)}{\pi^\half\Ga(\la+1)}\, +\frac{(2\la)_n}{(\la+1)_n}\,(\sin\tha)^{-\la}\, +\Re\!\!\left(e^{-\thalf i\la\pi}e^{i(n+\la)\tha}\, +\hyp21{\la,1-\la}{1+\la+n}{\frac{e^{i\tha}}{2i\sin\tha}}\right)\nonumber\\ +&=\frac{2^{2\la}\Ga(\la+\thalf)}{\pi^\half\Ga(\la+1)}\,\frac{(2\la)_n}{(\la+1)_n}\, +\sum_{k=0}^\iy\frac{(\la)_k(1-\la)_k}{(1+\la+n)_k k!}\, +\frac{\cos((n+k+\la)\tha-\thalf(k+\la)\pi)}{(2\sin\tha)^{k+\la}}\,. +\label{106} +\end{align} +We require that $0<\tha<\pi$ in \eqref{7} and $\tfrac16\pi<\tha<\tfrac56\pi$ in +\eqref{106} The convergence is absolute for $\la>\thalf$ and conditional for +$0<\la\le\thalf$. +For $\la\in\Zpos$ the above series terminate after the term with +$k=\la-1$. +Formulas \eqref{7} and \eqref{106} are also given in +\mycite{Sz}{(4.9.22), (4.9.25)}. +% +\paragraph{Fourier transform} +\begin{equation} +\frac{\Ga(\la+1)}{\Ga(\la+\thalf)\,\Ga(\thalf)}\, +\int_{-1}^1 \frac{C_n^\la(y)}{C_n^\la(1)}\,(1-y^2)^{\la-\half}\, +e^{ixy}\,dy +=i^n\,2^\la\,\Ga(\la+1)\,x^{-\la}\,J_{\la+n}(x). +\label{8} +\end{equation} +See \mycite{DLMF}{(18.17.17) and (18.17.18)}. +% +\paragraph{Laplace transforms} +\begin{equation} +\frac2{n!\,\Ga(\la)}\, +\int_0^\iy H_n(tx)\,t^{n+2\la-1}\,e^{-t^2}\,dt=C_n^\la(x). +\label{56} +\end{equation} +See Nielsen \cite[p.48, (4) with p.47, (1) and p.28, (10)]{K4} (1918) +or Feldheim \cite[(28)]{K3} (1942). +\begin{equation} +\frac2{\Ga(\la+\thalf)}\,\int_0^1 \frac{C_n^\la(t)}{C_n^\la(1)}\, +(1-t^2)^{\la-\half}\,t^{-1}\,(x/t)^{n+2\la+1}\,e^{-x^2/t^2}\,dt +=2^{-n}\,H_n(x)\,e^{-x^2}\quad(\la>-\thalf). +\label{46} +\end{equation} +Use Askey \& Fitch \cite[(3.29)]{K2} for $\al=\pm\thalf$ together with +\eqref{48}, \eqref{51}, \eqref{52}, \eqref{54} and \eqref{55}. +\paragraph{Addition formula} (see \mycite{AAR}{(9.8.5$'$)]}) +\begin{multline} +R_n^{(\al,\al)}\big(xy+(1-x^2)^\half(1-y^2)^\half t\big) +=\sum_{k=0}^n \frac{(-1)^k(-n)_k\,(n+2\al+1)_k}{2^{2k}((\al+1)_k)^2}\\ +\times(1-x^2)^{k/2} R_{n-k}^{(\al+k,\al+k)}(x)\,(1-y^2)^{k/2} R_{n-k}^{(\al+k,\al+k)}(y)\, +\om_k^{(\al-\half,\al-\half)}\,R_k^{(\al-\half,\al-\half)}(t), +\label{108} +\end{multline} +where +\[ +R_n^{(\al,\be)}(x):=P_n^{(\al,\be)}(x)/P_n^{(\al,\be)}(1),\quad +\om_n^{(\al,\be)}:=\frac{\int_{-1}^1 (1-x)^\al(1+x)^\be\,dx} +{\int_{-1}^1 (R_n^{(\al,\be)}(x))^2\,(1-x)^\al(1+x)^\be\,dx}\,. +\] +xP_n^{(\lambda-\frac{1}{2},\frac{1}{2})}(2x^2-1).$$ + +\subsection*{References} +\cite{Abram}, \cite{Ahmed+86}, \cite{AndrewsAskeyRoy}, \cite{Area+I}, \cite{Askey67}, \cite{Askey74}, \cite{Askey75}, +\cite{Askey89I}, \cite{AskeyFitch}, \cite{AskeyKoornRahman}, \cite{Berg}, \cite{BilodeauI}, +\cite{BojanovNikolov}, \cite{Brafman51}, \cite{Brafman57I}, \cite{Brown}, +\cite{BustozIsmail82}, \cite{BustozIsmail83}, \cite{BustozIsmail89}, \cite{BustozSavage79}, +\cite{BustozSavage80}, \cite{Carlitz61II}, \cite{Chihara78}, \cite{Common}, \cite{Danese}, +\cite{Dette94}, \cite{DijksmaKoorn}, \cite{DilcherStolarsky}, \cite{Dimitrov96}, +\cite{Dimitrov2003}, \cite{Doha2002II}, \cite{Driver}, \cite{ElbertLaforgia86I}, +\cite{ElbertLaforgia86II}, \cite{ElbertLaforgia90}, \cite{Erdelyi+}, \cite{Exton96}, +\cite{Gasper69}, \cite{Gasper72II}, \cite{Gasper85}, \cite{Grad}, \cite{Ismail74}, +\cite{Ismail2005II}, \cite{Koekoek2000}, \cite{Koorn88}, \cite{Laforgia}, \cite{LewanowiczI}, +\cite{Lorch}, \cite{Mathai}, \cite{Nagel}, \cite{Nikiforov+}, \cite{NikiforovUvarov}, +\cite{RahmanShah}, \cite{Rainville}, \cite{Reimer}, \cite{Sartoretto}, \cite{Srivastava71}, +\cite{Szego75}, \cite{Temme}, \cite{Viswanathan}, \cite{Zayed}. + +\subsection{Chebyshev}\index{Chebyshev polynomials} + +\par + +\subsection*{Hypergeometric representation} +The Chebyshev polynomials of the first kind can be obtained from the Jacobi +polynomials by taking $\alpha=\beta=-\frac{1}{2}$: +\begin{equation} +\label{DefChebyshevI} +T_n(x)=\frac{P_n^{(-\frac{1}{2},-\frac{1}{2})}(x)}{P_n^{(-\frac{1}{2},-\frac{1}{2})}(1)} +=\hyp{2}{1}{-n,n}{\frac{1}{2}}{\frac{1-x}{2}} +\end{equation} +and the Chebyshev polynomials of the second kind can be obtained from the +Jacobi polynomials by taking $\alpha=\beta=\frac{1}{2}$: +\begin{equation} +\label{DefChebyshevII} +U_n(x)=(n+1)\frac{P_n^{(\frac{1}{2},\frac{1}{2})}(x)}{P_n^{(\frac{1}{2},\frac{1}{2})}(1)} +=(n+1)\,\hyp{2}{1}{-n,n+2}{\frac{3}{2}}{\frac{1-x}{2}}. +\end{equation} + +\subsection*{Orthogonality relation} +\begin{equation} +\label{OrtChebyshevI} +\int_{-1}^1(1-x^2)^{-\frac{1}{2}}T_m(x)T_n(x)\,dx= +\left\{\begin{array}{ll} +\displaystyle\frac{\cpi}{2}\,\delta_{mn}, & n\neq 0\\[5mm] +\cpi\,\delta_{mn}, & n=0. +\end{array}\right. +\end{equation} + +\begin{equation} +\label{OrtChebyshevII} +\int_{-1}^1(1-x^2)^{\frac{1}{2}}U_m(x)U_n(x)\,dx=\frac{\cpi}{2}\,\delta_{mn}. +\end{equation} + +\subsection*{Recurrence relations} +\begin{equation} +\label{RecChebyshevI} +2xT_n(x)=T_{n+1}(x)+T_{n-1}(x),\quad T_{0}(x)=1\quad\textrm{and}\quad T_1(x)=x. +\end{equation} + +\begin{equation} +\label{RecChebyshevII} +2xU_n(x)=U_{n+1}(x)+U_{n-1}(x),\quad U_{0}(x)=1\quad\textrm{and}\quad U_1(x)=2x. +\end{equation} + +\subsection*{Normalized recurrence relations} +\begin{equation} +\label{NormRecChebyshevI} +xp_n(x)=p_{n+1}(x)+\frac{1}{4}p_{n-1}(x), +\end{equation} +where +$$T_1(x)=p_1(x)=x\quad\textrm{and}\quad T_n(x)=2^np_n(x),\quad n\neq 1.$$ + +\begin{equation} +\label{NormRecChebyshevII} +xp_n(x)=p_{n+1}(x)+\frac{1}{4}p_{n-1}(x), +\end{equation} +where +$$U_n(x)=2^np_n(x).$$ + +\subsection*{Differential equations} +\begin{equation} +\label{dvChebyshevI} +(1-x^2)y''(x)-xy'(x)+n^2y(x)=0,\quad y(x)=T_n(x). +\end{equation} + +\begin{equation} +\label{dvChebyshevII} +(1-x^2)y''(x)-3xy'(x)+n(n+2)y(x)=0,\quad y(x)=U_n(x). +\end{equation} + +\subsection*{Forward shift operator} +\begin{equation} +\label{shift1Chebyshev} +\frac{d}{dx}T_n(x)=nU_{n-1}(x). +\end{equation} + +\subsection*{Backward shift operator} +\begin{equation} +\label{shift2ChebyshevI} +(1-x^2)\frac{d}{dx}U_n(x)-xU_n(x)=-(n+1)T_{n+1}(x) +\end{equation} +or equivalently +\begin{equation} +\label{shift2ChebyshevII} +\frac{d}{dx}\left[\left(1-x^2\right)^{\frac{1}{2}}U_n(x)\right] +=-(n+1)\left(1-x^2\right)^{-\frac{1}{2}}T_{n+1}(x). +\end{equation} + +\subsection*{Rodrigues-type formulas} +\begin{equation} +\label{RodChebyshevI} +(1-x^2)^{-\frac{1}{2}}T_n(x)=\frac{(-1)^n}{(\frac{1}{2})_n2^n} +\left(\frac{d}{dx}\right)^n\left[(1-x^2)^{n-\frac{1}{2}}\right]. +\end{equation} + +\begin{equation} +\label{RodChebyshevII} +(1-x^2)^{\frac{1}{2}}U_n(x)=\frac{(n+1)(-1)^n}{(\frac{3}{2})_n2^n} +\left(\frac{d}{dx}\right)^n\left[(1-x^2)^{n+\frac{1}{2}}\right]. +\end{equation} + +\subsection*{Generating functions} +\begin{equation} +\label{GenChebyshevI1} +\frac{1-xt}{1-2xt+t^2}=\sum_{n=0}^{\infty}T_n(x)t^n. +\end{equation} + +\begin{equation} +\label{GenChebyshevI2} +R^{-1}\sqrt{\frac{1}{2}(1+R-xt)}=\sum_{n=0}^{\infty} +\frac{\left(\frac{1}{2}\right)_n}{n!}T_n(x)t^n,\quad R=\sqrt{1-2xt+t^2}. +\end{equation} + +\begin{equation} +\label{GenChebyshevI3} +\hyp{0}{1}{-}{\frac{1}{2}}{\frac{(x-1)t}{2}}\, +\hyp{0}{1}{-}{\frac{1}{2}}{\frac{(x+1)t}{2}}= +\sum_{n=0}^{\infty}\frac{T_n(x)}{\left(\frac{1}{2}\right)_nn!}t^n. +\end{equation} + +\begin{equation} +\label{GenChebyshevI4} +\e^{xt}\,\hyp{0}{1}{-}{\frac{1}{2}}{\frac{(x^2-1)t^2}{4}}= +\sum_{n=0}^{\infty}\frac{T_n(x)}{n!}t^n. +\end{equation} + +\begin{eqnarray} +\label{GenChebyshevI5} +& &\hyp{2}{1}{\gamma,-\gamma}{\frac{1}{2}}{\frac{1-R-t}{2}}\, +\hyp{2}{1}{\gamma,-\gamma}{\frac{1}{2}}{\frac{1-R+t}{2}}\nonumber\\ +& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n(-\gamma)_n}{\left(\frac{1}{2}\right)_nn!}T_n(x)t^n, +\quad R=\sqrt{1-2xt+t^2},\quad\textrm{$\gamma$ arbitrary}. +\end{eqnarray} + +\begin{eqnarray} +\label{GenChebyshevI6} +& &(1-xt)^{-\gamma}\,\hyp{2}{1}{\frac{1}{2}\gamma,\frac{1}{2}\gamma+\frac{1}{2}}{\frac{1}{2}} +{\frac{(x^2-1)t^2}{(1-xt)^2}}\nonumber\\ +& &=\sum_{n=0}^{\infty}\frac{(\gamma)_n}{n!}T_n(x)t^n,\quad\textrm{$\gamma$ arbitrary}. +\end{eqnarray} + +\begin{equation} +\label{GenChebyshevII1} +\frac{1}{1-2xt+t^2}=\sum_{n=0}^{\infty}U_n(x)t^n. +\end{equation} + +\begin{equation} +\label{GenChebyshevII2} +\frac{1}{R\sqrt{\frac{1}{2}(1+R-xt)}}=\sum_{n=0}^{\infty} +\frac{\left(\frac{3}{2}\right)_n}{(n+1)!}U_n(x)t^n,\quad R=\sqrt{1-2xt+t^2}. +\end{equation} + +\begin{equation} +\label{GenChebyshevII3} +\hyp{0}{1}{-}{\frac{3}{2}}{\frac{(x-1)t}{2}}\, +\hyp{0}{1}{-}{\frac{3}{2}}{\frac{(x+1)t}{2}}= +\sum_{n=0}^{\infty}\frac{U_n(x)}{\left(\frac{3}{2}\right)_n(n+1)!}t^n. +\end{equation} + +\begin{equation} +\label{GenChebyshevII4} +\e^{xt}\,\hyp{0}{1}{-}{\frac{3}{2}}{\frac{(x^2-1)t^2}{4}}= +\sum_{n=0}^{\infty}\frac{U_n(x)}{(n+1)!}t^n. +\end{equation} + +\begin{eqnarray} +\label{GenChebyshevII5} +& &\hyp{2}{1}{\gamma,2-\gamma}{\frac{3}{2}}{\frac{1-R-t}{2}}\, +\hyp{2}{1}{\gamma,2-\gamma}{\frac{3}{2}}{\frac{1-R+t}{2}}\nonumber\\ +& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n(2-\gamma)_n}{\left(\frac{3}{2}\right)_n(n+1)!}U_n(x)t^n, +\quad R=\sqrt{1-2xt+t^2},\quad\textrm{$\gamma$ arbitrary}. +\end{eqnarray} + +\begin{eqnarray} +\label{GenChebyshevII6} +& &(1-xt)^{-\gamma}\,\hyp{2}{1}{\frac{1}{2}\gamma,\frac{1}{2}\gamma+\frac{1}{2}}{\frac{3}{2}} +{\frac{(x^2-1)t^2}{(1-xt)^2}}\nonumber\\ +& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n}{(n+1)!}U_n(x)t^n,\quad\textrm{$\gamma$ arbitrary}. +\end{eqnarray} + +\subsection*{Remarks} +The Chebyshev polynomials can also be written as: +$$T_n(x)=\cos(n\theta),\quad x=\cos\theta$$ +and +$$U_n(x)=\frac{\sin (n+1)\theta}{\sin\theta},\quad x=\cos\theta.$$ +Further we have +$$U_n(x)=C_n^{(1)}(x)$$ +where $C_n^{(\lambda)}(x)$ denotes the Gegenbauer (or ultraspherical) +\subsection*{9.8.2 Chebyshev} +\label{sec9.8.2} +In addition to the Chebyshev polynomials $T_n$ of the first kind (9.8.35) +and $U_n$ of the second kind (9.8.36), +\begin{align} +T_n(x)&:=\frac{P_n^{(-\half,-\half)}(x)}{P_n^{(-\half,-\half)}(1)} +=\cos(n\tha),\quad x=\cos\tha,\\ +U_n(x)&:=(n+1)\,\frac{P_n^{(\half,\half)}(x)}{P_n^{(\half,\half)}(1)} +=\frac{\sin((n+1)\tha)}{\sin\tha}\,,\quad x=\cos\tha, +\end{align} +we have Chebyshev polynomials $V_n$ {\em of the third kind} +and $W_n$ {\em of the fourth kind}, +\begin{align} +V_n(x)&:=\frac{P_n^{(-\half,\half)}(x)}{P_n^{(-\half,\half)}(1)} +=\frac{\cos((n+\thalf)\tha)}{\cos(\thalf\tha)}\,,\quad x=\cos\tha,\\ +W_n(x)&:=(2n+1)\,\frac{P_n^{(\half,-\half)}(x)}{P_n^{(\half,-\half)}(1)} +=\frac{\sin((n+\thalf)\tha)}{\sin(\thalf\tha)}\,,\quad x=\cos\tha, +\end{align} +see \cite[Section 1.2.3]{K20}. Then there is the symmetry +\begin{equation} +V_n(-x)=(-1)^n W_n(x). +\label{140} +\end{equation} + +The names of Chebyshev polynomials of the third and fourth kind +and the notation $V_n(x)$ are due to Gautschi \cite{K21}. +The notation $W_n(x)$ was first used by Mason \cite{K22}. +Names and notations for Chebyshev polynomials of the third and fourth +kind are interchanged in \mycite{AAR}{Remark 2.5.3} and +\mycite{DLMF}{Table 18.3.1}. +polynomial given by (\ref{DefGegenbauer}) in the preceding subsection. + +\subsection*{References} +\cite{Abram}, \cite{AskeyFitch}, \cite{AskeyGasperHarris}, \cite{AskeyIsmail76}, +\cite{Bavinck95}, \cite{Chihara78}, \cite{Danese}, \cite{DilcherStolarsky}, +\cite{Erdelyi+}, \cite{Grad}, \cite{HartmannStephan}, \cite{Ismail2005II}, \cite{Koekoek2000}, +\cite{Luke}, \cite{Mathai}, \cite{Nikiforov+}, \cite{NikiforovUvarov}, \cite{Rainville}, +\cite{Rayes+}, \cite{Rivlin}, \cite{SainteViennot}, \cite{Szego75}, \cite{Temme}, +\cite{Wilson70I}, \cite{Zayed}, \cite{Zhang}, \cite{ZhangWang}. + +\subsection{Legendre / Spherical}\index{Legendre polynomials} +\index{Spherical polynomials} + +\par + +\subsection*{Hypergeometric representation} +The Legendre (or spherical) polynomials are Jacobi polynomials with $\alpha=\beta=0$: +\begin{equation} +\label{DefLegendre} +P_n(x)=P_n^{(0,0)}(x)=\hyp{2}{1}{-n,n+1}{1}{\frac{1-x}{2}}. +\end{equation} + +\subsection*{Orthogonality relation} +\begin{equation} +\label{OrtLegendre} +\int_{-1}^1P_m(x)P_n(x)\,dx=\frac{2}{2n+1}\,\delta_{mn}. +\end{equation} + +\subsection*{Recurrence relation} +\begin{equation} +\label{RecLegendre} +(2n+1)xP_n(x)=(n+1)P_{n+1}(x)+nP_{n-1}(x). +\end{equation} + +\subsection*{Normalized recurrence relation} +\begin{equation} +\label{NormRecLegendre} +xp_n(x)=p_{n+1}(x)+\frac{n^2}{(2n-1)(2n+1)}p_{n-1}(x), +\end{equation} +where +$$P_n(x)=\binom{2n}{n}\frac{1}{2^n}p_n(x).$$ + +\subsection*{Differential equation} +\begin{equation} +\label{dvLegendre} +(1-x^2)y''(x)-2xy'(x)+n(n+1)y(x)=0,\quad y(x)=P_n(x). +\end{equation} + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodLegendre} +P_n(x)=\frac{(-1)^n}{2^nn!}\left(\frac{d}{dx}\right)^n\left[(1-x^2)^n\right]. +\end{equation} + +\subsection*{Generating functions} +\begin{equation} +\label{GenLegendre1} +\frac{1}{\sqrt{1-2xt+t^2}}=\sum_{n=0}^{\infty}P_n(x)t^n. +\end{equation} + +\begin{equation} +\label{GenLegendre2} +\hyp{0}{1}{-}{1}{\frac{(x-1)t}{2}}\,\hyp{0}{1}{-}{1}{\frac{(x+1)t}{2}}= +\sum_{n=0}^{\infty}\frac{P_n(x)}{(n!)^2}t^n. +\end{equation} + +\begin{equation} +\label{GenLegendre3} +\e^{xt}\,\hyp{0}{1}{-}{1}{\frac{(x^2-1)t^2}{4}}= +\sum_{n=0}^{\infty}\frac{P_n(x)}{n!}t^n. +\end{equation} + +\begin{eqnarray} +\label{GenLegendre4} +& &\hyp{2}{1}{\gamma,1-\gamma}{1}{\frac{1-R-t}{2}}\, +\hyp{2}{1}{\gamma,1-\gamma}{1}{\frac{1-R+t}{2}}\nonumber\\ +& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n(1-\gamma)_n}{(n!)^2}P_n(x)t^n, +\quad R=\sqrt{1-2xt+t^2},\quad\textrm{$\gamma$ arbitrary}. +\end{eqnarray} + +\begin{eqnarray} +\label{GenLegendre5} +& &(1-xt)^{-\gamma}\,\hyp{2}{1}{\frac{1}{2}\gamma,\frac{1}{2}\gamma+\frac{1}{2}}{1} +{\frac{(x^2-1)t^2}{(1-xt)^2}}\nonumber\\ +& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n}{n!}P_n(x)t^n,\quad\textrm{$\gamma$ arbitrary}. +\subsection*{9.9 Pseudo Jacobi (or Routh-Romanovski)} +\label{sec9.9} +In this section in \mycite{KLS} the pseudo Jacobi polynomial $P_n(x;\nu,N)$ in (9.9.1) +is considered +for $N\in\ZZ_{\ge0}$ and $n=0,1,\ldots,n$. However, we can more generally take +$-\thalf\be>0\;\;{\rm or}\;\;\al<\be<0). +\] +Then $P_n$ can be expressed as a Meixner polynomial: +\[ +P_n(x)=(-k_2(\al\be)^{-1})_n\,\be^n\, +M_n\left(-\,\frac{x+k_2\al^{-1}}{\al-\be},-k_2(\al\be)^{-1},\be\al^{-1}\right). +\] + +In 1938 Gottlieb \cite[\S2]{K1} introduces polynomials $l_n$ ``of Laguerre type'' +which turn out to be special Meixner polynomials: +$l_n(x)=e^{-n\la} M_n(x;1,e^{-\la})$. +% +\paragraph{Uniqueness of orthogonality measure} +The coefficient of $p_{n-1}(x)$ in (9.10.4) behaves as $O(n^2)$ as $n\to\iy$. +Hence \eqref{93} holds, by which the orthogonality measure is unique. +=\frac{2^n}{(n-2N-1)_n}y_n(x;-2N-2).$$ + +\subsection*{References} +\cite{Askey87}, \cite{BorodinOlshanski}, \cite{Lesky96}. + + +\section{Meixner}\index{Meixner polynomials} + +\par\setcounter{equation}{0} + +\subsection*{Hypergeometric representation} +\begin{equation} +\label{DefMeixner} +M_n(x;\beta,c)=\hyp{2}{1}{-n,-x}{\beta}{1-\frac{1}{c}}. +\end{equation} + +\subsection*{Orthogonality relation} +\begin{equation} +\label{OrtMeixner} +\sum_{x=0}^{\infty}\frac{(\beta)_x}{x!}c^xM_m(x;\beta,c)M_n(x;\beta,c) +{}=\frac{c^{-n}n!}{(\beta)_n(1-c)^{\beta}}\,\delta_{mn} +% \constraint{ +% $\beta > 0$ & +% $0 < c < 1$ } +\end{equation} + +\subsection*{Recurrence relation} +\begin{eqnarray} +\label{RecMeixner} +(c-1)xM_n(x;\beta,c)&=&c(n+\beta)M_{n+1}(x;\beta,c)\nonumber\\ +& &{}\mathindent{}-\left[n+(n+\beta)c\right]M_n(x;\beta,c)+nM_{n-1}(x;\beta,c). +\end{eqnarray} + +\subsection*{Normalized recurrence relation} +\begin{equation} +\label{NormRecMeixner} +xp_n(x)=p_{n+1}(x)+\frac{n+(n+\beta)c}{1-c}p_n(x)+ +\frac{n(n+\beta-1)c}{(1-c)^2}p_{n-1}(x), +\end{equation} +where +$$M_n(x;\beta,c)=\frac{1}{(\beta)_n}\left(\frac{c-1}{c}\right)^np_n(x).$$ + +\subsection*{Difference equation} +\begin{equation} +\label{dvMeixner} +n(c-1)y(x)=c(x+\beta)y(x+1)-\left[x+(x+\beta)c\right]y(x)+xy(x-1), +\end{equation} +where +$$y(x)=M_n(x;\beta,c).$$ + +\subsection*{Forward shift operator} +\begin{equation} +\label{shift1MeixnerI} +M_n(x+1;\beta,c)-M_n(x;\beta,c)= +\frac{n}{\beta}\left(\frac{c-1}{c}\right)M_{n-1}(x;\beta+1,c) +\end{equation} +or equivalently +\begin{equation} +\label{shift1MeixnerII} +\Delta M_n(x;\beta,c)=\frac{n}{\beta}\left(\frac{c-1}{c}\right)M_{n-1}(x;\beta+1,c). +\end{equation} + +\subsection*{Backward shift operator} +\begin{equation} +\label{shift2MeixnerI} +c(\beta+x-1)M_n(x;\beta,c)-xM_n(x-1;\beta,c)=c(\beta-1)M_{n+1}(x;\beta-1,c) +\end{equation} +or equivalently +\begin{equation} +\label{shift2MeixnerII} +\nabla\left[\frac{(\beta)_xc^x}{x!}M_n(x;\beta,c)\right]= +\frac{(\beta-1)_xc^x}{x!}M_{n+1}(x;\beta-1,c). +\end{equation} + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodMeixner} +\frac{(\beta)_xc^x}{x!}M_n(x;\beta,c)=\nabla^n\left[\frac{(\beta+n)_xc^x}{x!}\right]. +\end{equation} + +\subsection*{Generating functions} +\begin{equation} +\label{GenMeixner1} +\left(1-\frac{t}{c}\right)^x(1-t)^{-x-\beta}= +\sum_{n=0}^{\infty}\frac{(\beta)_n}{n!}M_n(x;\beta,c)t^n. +\end{equation} + +\begin{equation} +\label{GenMeixner2} +\e^t\,\hyp{1}{1}{-x}{\beta}{\left(\frac{1-c}{c}\right)t}= +\sum_{n=0}^{\infty}\frac{M_n(x;\beta,c)}{n!}t^n. +\end{equation} + +\begin{equation} +\label{GenMeixner3} +(1-t)^{-\gamma}\,\hyp{2}{1}{\gamma,-x}{\beta}{\frac{(1-c)t}{c(1-t)}} +=\sum_{n=0}^{\infty}\frac{(\gamma)_n}{n!}M_n(x;\beta,c)t^n, +\quad\textrm{$\gamma$ arbitrary}. +\end{equation} + +\subsection*{Limit relations} + +\subsubsection*{Hahn $\rightarrow$ Meixner} +If we take $\alpha=b-1$, $\beta=N(1-c)c^{-1}$ in the definition (\ref{DefHahn}) +of the Hahn polynomials and let $N\rightarrow\infty$ we find the Meixner polynomials: +$$\lim_{N\rightarrow\infty} +Q_n(x;b-1,N(1-c)c^{-1},N)=M_n(x;b,c).$$ + +\subsubsection*{Dual Hahn $\rightarrow$ Meixner} +To obtain the Meixner polynomials from the dual Hahn polynomials we have to take +$\gamma=\beta-1$ and $\delta=N(1-c)c^{-1}$ in the definition (\ref{DefDualHahn}) of +the dual Hahn polynomials and let $N\rightarrow\infty$: +$$\lim_{N\rightarrow\infty} +R_n(\lambda(x);\beta-1,N(1-c)c^{-1},N)=M_n(x;\beta,c).$$ + +\subsubsection*{Meixner $\rightarrow$ Laguerre} +The Laguerre polynomials given by (\ref{DefLaguerre}) are obtained from the Meixner polynomials +if we take $\beta=\alpha+1$ and $x\rightarrow (1-c)^{-1}x$ and let $c\rightarrow 1$: +\begin{equation} +\lim_{c\rightarrow 1} +M_n((1-c)^{-1}x;\alpha+1,c)=\frac{L_n^{(\alpha)}(x)}{L_n^{(\alpha)}(0)}. +\end{equation} + +\subsubsection*{Meixner $\rightarrow$ Charlier} +The Charlier polynomials given by (\ref{DefCharlier}) are obtained from the Meixner polynomials +if we take $c=(a+\beta)^{-1}a$ and let $\beta\rightarrow\infty$: +\begin{equation} +\lim_{\beta\rightarrow\infty} +M_n(x;\beta,(a+\beta)^{-1}a)=C_n(x;a). +\end{equation} + +\subsection*{Remarks} +The Meixner polynomials are related to the Jacobi polynomials given by (\ref{DefJacobi}) +in the following way: +$$\frac{(\beta)_n}{n!}M_n(x;\beta,c)=P_n^{(\beta-1,-n-\beta-x)}((2-c)c^{-1}).$$ + +\noindent +The Meixner polynomials are also related to the Krawtchouk polynomials given by +(\ref{DefKrawtchouk}) in the following way: +\subsection*{9.11 Krawtchouk} +\label{sec9.11} +% +\paragraph{Special values} +By (9.11.1) and the binomial formula: +\begin{equation} +K_n(0;p,N)=1,\qquad +K_n(N;p,N)=(1-p^{-1})^n. +\label{9} +\end{equation} +The self-duality (p.240, Remarks, first formula) +\begin{equation} +K_n(x;p,N)=K_x(n;p,N)\qquad (n,x\in \{0,1,\ldots,N\}) +\label{147} +\end{equation} +combined with \eqref{9} yields: +\begin{equation} +K_N(x;p,N)=(1-p^{-1})^x\qquad(x\in\{0,1,\ldots,N\}). +\label{148} +\end{equation} +% +\paragraph{Symmetry} +By the orthogonality relation (9.11.2): +\begin{equation} +\frac{K_n(N-x;p,N)}{K_n(N;p,N)}=K_n(x;1-p,N). +\label{10} +\end{equation} +By \eqref{10} and \eqref{147} we have also +\begin{equation} +\frac{K_{N-n}(x;p,N)}{K_N(x;p,N)}=K_n(x;1-p,N) +\qquad(n,x\in\{0,1,\ldots,N\}), +\label{149} +\end{equation} +and, by \eqref{149}, \eqref{10} and \eqref{9}, +\begin{equation} +K_{N-n}(N-x;p,N)=\left(\frac p{p-1}\right)^{n+x-N}K_n(x;p,N) +\qquad(n,x\in\{0,1,\ldots,N\}). +\label{150} +\end{equation} +A particular case of \eqref{10} is: +\begin{equation} +K_n(N-x;\thalf,N)=(-1)^n K_n(x;\thalf,N). +\label{11} +\end{equation} +Hence +\begin{equation} +K_{2m+1}(N;\thalf,2N)=0. +\label{12} +\end{equation} +From (9.11.11): +\begin{equation} +K_{2m}(N;\thalf,2N)=\frac{(\thalf)_m}{(-N+\thalf)_m}\,. +\label{13} +\end{equation} +% +\paragraph{Quadratic transformations} +\begin{align} +K_{2m}(x+N;\thalf,2N)&=\frac{(\thalf)_m}{(-N+\thalf)_m}\, +R_m(x^2;-\thalf,-\thalf,N), +\label{31}\\ +K_{2m+1}(x+N;\thalf,2N)&=-\,\frac{(\tfrac32)_m}{N\,(-N+\thalf)_m}\, +x\,R_m(x^2-1;\thalf,\thalf,N-1), +\label{33}\\ +K_{2m}(x+N+1;\thalf,2N+1)&=\frac{(\tfrac12)_m}{(-N-\thalf)_m}\, +R_m(x(x+1);-\thalf,\thalf,N), +\label{32}\\ +K_{2m+1}(x+N+1;\thalf,2N+1)&=\frac{(\tfrac32)_m}{(-N-\thalf)_{m+1}}\, +(x+\thalf)\,R_m(x(x+1);\thalf,-\thalf,N), +\label{34} +\end{align} +where $R_m$ is a dual Hahn polynomial (9.6.1). For the proofs use +(9.6.2), (9.11.2), (9.6.4) and (9.11.4). +% +\paragraph{Generating functions} +\begin{multline} +\sum_{x=0}^N\binom Nx K_m(x;p,N)K_n(x;q,N)z^x\\ +=\left(\frac{p-z+pz}p\right)^m +\left(\frac{q-z+qz}q\right)^n +(1+z)^{N-m-n} +K_m\left(n;-\,\frac{(p-z+pz)(q-z+qz)}z,N\right). +\label{107} +\end{multline} +This follows immediately from Rosengren \cite[(3.5)]{K8}, which goes back +to Meixner \cite{K9}. +$$K_n(x;p,N)=M_n(x;-N,(p-1)^{-1}p).$$ + +\subsection*{References} +\cite{Allaway76}, \cite{NAlSalam66}, \cite{AlSalam90}, \cite{AlSalamChihara76}, +\cite{AlSalamIsmail76}, \cite{Alvarez+}, \cite{AndrewsAskey85}, \cite{Area+II}, \cite{Askey75}, +\cite{Askey89I}, \cite{Askey2005}, \cite{AskeyGasper77}, \cite{AskeyIsmail76}, +\cite{AskeyWilson85}, \cite{AtakRahmanSuslov}, \cite{AtakSuslov88}, \cite{Bavinck98}, +\cite{BavinckHaeringen}, \cite{Campigotto+}, \cite{Chihara78}, \cite{Cooper+}, +\cite{Erdelyi+}, \cite{FoataLabelle}, \cite{Gabutti}, \cite{GabuttiMathis}, \cite{Gasper73I}, +\cite{Gasper74}, \cite{HoareRahman}, \cite{Ismail2005II}, \cite{IsmailLetVal88}, +\cite{IsmailLi}, \cite{IsmailMuldoon}, \cite{IsmailStanton97}, \cite{JinWong}, +\cite{Karlin58}, \cite{Koekoek2000}, \cite{Koorn88}, \cite{LabelleYehI}, \cite{LabelleYehII}, +\cite{Lesky89}, \cite{Lesky94I}, \cite{Lesky95II}, \cite{LewanowiczII}, \cite{Meixner}, +\cite{Nikiforov+}, \cite{NikiforovUvarov}, \cite{Rahman78I}, \cite{ValentAssche}, +\cite{Viennot}, \cite{Zarzo+}, \cite{Zeng90}. + + +\section{Krawtchouk}\index{Krawtchouk polynomials} + +\par\setcounter{equation}{0} + +\subsection*{Hypergeometric representation} +\begin{equation} +\label{DefKrawtchouk} +K_n(x;p,N)=\hyp{2}{1}{-n,-x}{-N}{\frac{1}{p}},\quad n=0,1,2,\ldots,N. +\end{equation} + +\subsection*{Orthogonality relation} +\begin{eqnarray} +\label{OrtKrawtchouk} +& &\sum_{x=0}^N\binom{N}{x}p^x(1-p)^{N-x} K_m(x;p,N)K_n(x;p,N)\nonumber\\ +& &{}=\frac{(-1)^nn!}{(-N)_n}\left(\frac{1-p}{p}\right)^n\,\delta_{mn},\quad0 < p < 1. +\end{eqnarray} + +\subsection*{Recurrence relation} +\begin{eqnarray} +\label{RecKrawtchouk} +-xK_n(x;p,N)&=&p(N-n)K_{n+1}(x;p,N)\nonumber\\ +& &{}\mathindent{}-\left[p(N-n)+n(1-p)\right]K_n(x;p,N)\nonumber\\ +& &{}\mathindent\mathindent{}+n(1-p)K_{n-1}(x;p,N). +\end{eqnarray} + +\subsection*{Normalized recurrence relation} +\begin{eqnarray} +\label{NormRecKrawtchouk} +xp_n(x)&=&p_{n+1}(x)+\left[p(N-n)+n(1-p)\right]p_n(x)\nonumber\\ +& &{}\mathindent{}+np(1-p)(N+1-n)p_{n-1}(x), +\end{eqnarray} +where +$$K_n(x;p,N)=\frac{1}{(-N)_np^n}p_n(x).$$ + +\subsection*{Difference equation} +\begin{eqnarray} +\label{dvKrawtchouk} +-ny(x)&=&p(N-x)y(x+1)\nonumber\\ +& &{}\mathindent{}-\left[p(N-x)+x(1-p)\right]y(x)+x(1-p)y(x-1), +\end{eqnarray} +where +$$y(x)=K_n(x;p,N).$$ + +\subsection*{Forward shift operator} +\begin{equation} +\label{shift1KrawtchoukI} +K_n(x+1;p,N)-K_n(x;p,N)=-\frac{n}{Np}K_{n-1}(x;p,N-1) +\end{equation} +or equivalently +\begin{equation} +\label{shift1KrawtchoukII} +\Delta K_n(x;p,N)=-\frac{n}{Np}K_{n-1}(x;p,N-1). +\end{equation} + +\subsection*{Backward shift operator} +\begin{eqnarray} +\label{shift2KrawtchoukI} +& &(N+1-x)K_n(x;p,N)-x\left(\frac{1-p}{p}\right)K_n(x-1;p,N)\nonumber\\ +& &{}=(N+1)K_{n+1}(x;p,N+1) +\end{eqnarray} +or equivalently +\begin{equation} +\label{shift2KrawtchoukII} +\nabla\left[\binom{N}{x}\left(\frac{p}{1-p}\right)^xK_n(x;p,N)\right]= +\binom{N+1}{x}\left(\frac{p}{1-p}\right)^xK_{n+1}(x;p,N+1). +\end{equation} + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodKrawtchouk} +\binom{N}{x}\left(\frac{p}{1-p}\right)^xK_n(x;p,N)= +\nabla^n\left[\binom{N-n}{x}\left(\frac{p}{1-p}\right)^x\right]. +\end{equation} + +\subsection*{Generating functions} +For $x=0,1,2,\ldots,N$ we have +\begin{equation} +\label{GenKrawtchouk1} +\left(1-\frac{(1-p)}{p}t\right)^x(1+t)^{N-x}= +\sum_{n=0}^N\binom{N}{n}K_n(x;p,N)t^n. +\end{equation} + +\begin{equation} +\label{GenKrawtchouk2} +\left[\expe^t\,\hyp{1}{1}{-x}{-N}{-\frac{t}{p}}\right]_N= +\sum_{n=0}^N\frac{K_n(x;p,N)}{n!}t^n. +\end{equation} + +\begin{eqnarray} +\label{GenKrawtchouk3} +& &\left[(1-t)^{-\gamma}\,\hyp{2}{1}{\gamma,-x}{-N}{\frac{t}{p(t-1)}}\right]_N\nonumber\\ +& &{}=\sum_{n=0}^N\frac{(\gamma)_n}{n!}K_n(x;p,N)t^n, +\quad\textrm{$\gamma$ arbitrary}. +\end{eqnarray} + +\subsection*{Limit relations} + +\subsubsection*{Hahn $\rightarrow$ Krawtchouk} +If we take $\alpha=pt$ and $\beta=(1-p)t$ in the definition (\ref{DefHahn}) of the Hahn +polynomials and let $t\rightarrow\infty$ we obtain the Krawtchouk polynomials: +$$\lim_{t\rightarrow\infty}Q_n(x;pt,(1-p)t,N)=K_n(x;p,N).$$ + +\subsubsection*{Dual Hahn $\rightarrow$ Krawtchouk} +The Krawtchouk polynomials follow from the dual Hahn polynomials given by +(\ref{DefDualHahn}) if we set $\gamma=pt$, $\delta=(1-p)t$ and let $t\rightarrow\infty$: +$$\lim_{t\rightarrow\infty}R_n(\lambda(x);pt,(1-p)t,N)=K_n(x;p,N).$$ + +\subsubsection*{Krawtchouk $\rightarrow$ Charlier} +The Charlier polynomials given by (\ref{DefCharlier}) can be found from the Krawtchouk +polynomials by taking $p=N^{-1}a$ and letting $N\rightarrow\infty$: +\begin{equation} +\lim_{N\rightarrow\infty}K_n(x;N^{-1}a,N)=C_n(x;a). +\end{equation} + +\subsubsection*{Krawtchouk $\rightarrow$ Hermite} +The Hermite polynomials given by (\ref{DefHermite}) follow from the Krawtchouk polynomials +by setting $x\rightarrow pN+x\sqrt{2p(1-p)N}$ and then letting $N\rightarrow\infty$: +\begin{equation} +\lim_{N\rightarrow\infty} +\sqrt{\binom{N}{n}}K_n(pN+x\sqrt{2p(1-p)N};p,N) +=\frac{\displaystyle (-1)^nH_n(x)}{\displaystyle\sqrt{2^nn!\left(\frac{p}{1-p}\right)^n}}. +\end{equation} + +\subsection*{Remarks} +The Krawtchouk polynomials are self-dual, which means that +$$K_n(x;p,N)=K_x(n;p,N),\quad n,x\in\{0,1,2,\ldots,N\}.$$ +By using this relation we easily obtain the so-called dual orthogonality +relation from the orthogonality relation (\ref{OrtKrawtchouk}): +$$\sum_{n=0}^N\binom{N}{n}p^n(1-p)^{N-n} K_n(x;p,N)K_n(y;p,N)= +\frac{\displaystyle\left(\frac{1-p}{p}\right)^x}{\dbinom{N}{x}}\delta_{xy},$$ +where $0 < p < 1$ and $x,y\in\{0,1,2,\ldots,N\}$. + +\noindent +The Krawtchouk polynomials are related to the Meixner polynomials given by (\ref{DefMeixner}) +in the following way: +\subsection*{9.12 Laguerre} +\label{sec9.12} +\paragraph{Notation} +Here the Laguerre polynomial is denoted by $L_n^\al$ instead of +$L_n^{(\al)}$. +% +\paragraph{Hypergeometric representation} +\begin{align} +L_n^\al(x)&= +\frac{(\al+1)_n}{n!}\,\hyp11{-n}{\al+1}x +\label{182}\\ +&=\frac{(-x)^n}{n!} \hyp20{-n,-n-\al}-{-\,\frac1x} +\label{183}\\ +&=\frac{(-x)^n}{n!}\,C_n(n+\al;x), +\label{184} +\end{align} +where $C_n$ in \eqref{184} is a +\hyperref[sec9.14]{Charlier polynomial}. +Formula \eqref{182} is (9.12.1). Then \eqref{183} follows by reversal +of summation. Finally \eqref{184} follows by \eqref{183} and \eqref{179}. +It is also the remark on top of p.244 in \mycite{KLS}, and it is essentially +\myciteKLS{416}{(2.7.10)}. +% +\paragraph{Uniqueness of orthogonality measure} +The coefficient of $p_{n-1}(x)$ in (9.12.4) behaves as $O(n^2)$ as $n\to\iy$. +Hence \eqref{93} holds, by which the orthogonality measure is unique. +% +\paragraph{Special value} +\begin{equation} +L_n^{\al}(0)=\frac{(\al+1)_n}{n!}\,. +\label{53} +\end{equation} +Use (9.12.1) or see \mycite{DLMF}{18.6.1)}. +% +\paragraph{Quadratic transformations} +\begin{align} +H_{2n}(x)&=(-1)^n\,2^{2n}\,n!\,L_n^{-1/2}(x^2), +\label{54}\\ +H_{2n+1}(x)&=(-1)^n\,2^{2n+1}\,n!\,x\,L_n^{1/2}(x^2). +\label{55} +\end{align} +See p.244, Remarks, last two formulas. +Or see \mycite{DLMF}{(18.7.19), (18.7.20)}. +% +\paragraph{Fourier transform} +\begin{equation} +\frac1{\Ga(\al+1)}\,\int_0^\iy \frac{L_n^\al(y)}{L_n^\al(0)}\, +e^{-y}\,y^\al\,e^{ixy}\,dy= +i^n\,\frac{y^n}{(iy+1)^{n+\al+1}}\,, +\label{14} +\end{equation} +see \mycite{DLMF}{(18.17.34)}. +% +\paragraph{Differentiation formulas} +Each differentiation formula is given in two equivalent forms. +\begin{equation} +\frac d{dx}\left(x^\al L_n^\al(x)\right)= +(n+\al)\,x^{\al-1} L_n^{\al-1}(x),\qquad +\left(x\frac d{dx}+\al\right)L_n^\al(x)= +(n+\al)\,L_n^{\al-1}(x). +\label{76} +\end{equation} +% +\begin{equation} +\frac d{dx}\left(e^{-x} L_n^\al(x)\right)= +-e^{-x} L_n^{\al+1}(x),\qquad +\left(\frac d{dx}-1\right)L_n^\al(x)= +-L_n^{\al+1}(x). +\label{77} +\end{equation} +% +Formulas \eqref{76} and \eqref{77} follow from +\mycite{DLMF}{(13.3.18), (13.3.20)} +together with (9.12.1). +% +\paragraph{Generalized Hermite polynomials} +See \myciteKLS{146}{p.156}, \cite[Section 1.5.1]{K26}. +These are defined by +\begin{equation} +H_{2m}^\mu(x):=\const L_m^{\mu-\half}(x^2),\qquad +H_{2m+1}^\mu(x):=\const x\,L_m^{\mu+\half}(x^2). +\label{78} +\end{equation} +Then for $\mu>-\thalf$ we have orthogonality relation +\begin{equation} +\int_{-\iy}^{\iy} H_m^\mu(x)\,H_n^\mu(x)\,|x|^{2\mu}e^{-x^2}\,dx +=0\qquad(m\ne n). +\label{79} +\end{equation} +Let the Dunkl operator $T_\mu$ be defined by \eqref{72}. +If we choose the constants in \eqref{78} as +\begin{equation} +H_{2m}^\mu(x)=\frac{(-1)^m(2m)!}{(\mu+\thalf)_m}\,L_m^{\mu-\half}(x^2),\qquad +H_{2m+1}^\mu(x)=\frac{(-1)^m(2m+1)!}{(\mu+\thalf)_{m+1}}\, + x\,L_m^{\mu+\half}(x^2) + \label{80} +\end{equation} +then (see \cite[(1.6)]{K5}) +\begin{equation} +T_\mu H_n^\mu=2n\,H_{n-1}^\mu. +\label{81} +\end{equation} +Formula \eqref{81} with \eqref{80} substituted gives rise to two +differentiation formulas involving Laguerre polynomials which are equivalent to +(9.12.6) and \eqref{76}. + +Composition of \eqref{81} with itself gives +\[ +T_\mu^2 H_n^\mu=4n(n-1)\,H_{n-2}^\mu, +\] +which is equivalent to the composition of (9.12.6) and \eqref{76}: +\begin{equation} +\left(\frac{d^2}{dx^2}+\frac{2\al+1}x\,\frac d{dx}\right)L_n^\al(x^2) +=-4(n+\al)\,L_{n-1}^\al(x^2). +\label{82} +\end{equation} +$$K_n(x;p,N)=M_n(x;-N,(p-1)^{-1}p).$$ + +\subsection*{References} +\cite{AlSalam90}, \cite{AndrewsAskey85}, \cite{Area+II}, \cite{Askey75}, \cite{Askey89I}, +\cite{AskeyGasper77}, \cite{AskeyWilson85}, \cite{AtakRahmanSuslov}, \cite{Campigotto+}, +\cite{LChiharaStanton}, \cite{Chihara78}, \cite{Dette95}, \cite{Dominici}, \cite{Dunkl76}, +\cite{Dunkl84}, \cite{DunklRamirez}, \cite{Erdelyi+}, \cite{FeinsilverSchott}, +\cite{Gasper73I}, \cite{Gasper74}, \cite{HoareRahman}, \cite{Ismail2005II}, \cite{Karlin58}, +\cite{Koorn82}, \cite{Koorn88}, \cite{LabelleYehI}, \cite{LabelleYehII}, \cite{Lesky62}, +\cite{Lesky89}, \cite{Lesky94I}, \cite{Lesky95II}, \cite{LewanowiczII}, \cite{Nikiforov+}, +\cite{NikiforovUvarov}, \cite{Qiu}, \cite{Rahman78I}, \cite{Rahman79}, \cite{Stanton84}, +\cite{Stanton90}, \cite{Szego75}, \cite{Zarzo+}, \cite{Zeng90}. + + +\section{Laguerre}\index{Laguerre polynomials} + +\par\setcounter{equation}{0} + +\subsection*{Hypergeometric representation} +\begin{equation} +\label{DefLaguerre} +L_n^{(\alpha)}(x)=\frac{(\alpha+1)_n}{n!}\,\hyp{1}{1}{-n}{\alpha+1}{x}. +\end{equation} + +\subsection*{Orthogonality relation} +\begin{equation} +\label{OrtLaguerre} +\int_0^{\infty}\expe^{-x}x^{\alpha}L_m^{(\alpha)}(x)L_n^{(\alpha)}(x)\,dx= +\frac{\Gamma(n+\alpha+1)}{n!}\,\delta_{mn},\quad\alpha > -1. +\end{equation} + +\subsection*{Recurrence relation} +\begin{equation} +\label{RecLaguerre} +(n+1)L_{n+1}^{(\alpha)}(x)-(2n+\alpha+1-x)L_n^{(\alpha)}(x)+(n+\alpha)L_{n-1}^{(\alpha)}(x)=0. +\end{equation} + +\subsection*{Normalized recurrence relation} +\begin{equation} +\label{NormRecLaguerre} +xp_n(x)=p_{n+1}(x)+(2n+\alpha+1)p_n(x)+n(n+\alpha)p_{n-1}(x), +\end{equation} +where +$$L_n^{(\alpha)}(x)=\frac{(-1)^n}{n!}p_n(x).$$ + +\subsection*{Differential equation} +\begin{equation} +\label{dvLaguerre} +xy''(x)+(\alpha+1-x)y'(x)+ny(x)=0,\quad y(x)=L_n^{(\alpha)}(x). +\end{equation} + +\newpage + +\subsection*{Forward shift operator} +\begin{equation} +\label{shift1Laguerre} +\frac{d}{dx}L_n^{(\alpha)}(x)=-L_{n-1}^{(\alpha+1)}(x). +\end{equation} + +\subsection*{Backward shift operator} +\begin{equation} +\label{shift2LaguerreI} +x\frac{d}{dx}L_n^{(\alpha)}(x)+(\alpha-x)L_n^{(\alpha)}(x)=(n+1)L_{n+1}^{(\alpha-1)}(x) +\end{equation} +or equivalently +\begin{equation} +\label{shift2LaguerreII} +\frac{d}{dx}\left[\expe^{-x}x^{\alpha}L_n^{(\alpha)}(x)\right]=(n+1)\expe^{-x}x^{\alpha-1}L^{(\alpha-1)}_{n+1}(x). +\end{equation} + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodLaguerre} +\e^{-x}x^{\alpha}L_n^{(\alpha)}(x)=\frac{1}{n!}\left(\frac{d}{dx}\right)^n\left[\expe^{-x}x^{n+\alpha}\right]. +\end{equation} + +\subsection*{Generating functions} +\begin{equation} +\label{GenLaguerre1} +(1-t)^{-\alpha-1}\exp\left(\frac{xt}{t-1}\right)= +\sum_{n=0}^{\infty}L_n^{(\alpha)}(x)t^n. +\end{equation} + +\begin{equation} +\label{GenLaguerre2} +\e^t\,\hyp{0}{1}{-}{\alpha+1}{-xt} +=\sum_{n=0}^{\infty}\frac{L_n^{(\alpha)}(x)}{(\alpha+1)_n}t^n. +\end{equation} + +\begin{equation} +\label{GenLaguerre3} +(1-t)^{-\gamma}\,\hyp{1}{1}{\gamma}{\alpha+1}{\frac{xt}{t-1}} +=\sum_{n=0}^{\infty}\frac{(\gamma)_n}{(\alpha+1)_n}L_n^{(\alpha)}(x)t^n, +\quad\textrm{$\gamma$ arbitrary}. +\end{equation} + +\subsection*{Limit relations} + +\subsubsection*{Meixner-Pollaczek $\rightarrow$ Laguerre} +The Laguerre polynomials can be obtained from the Meixner-Pollaczek polynomials given by +(\ref{DefMP}) by the substitution $\lambda=\frac{1}{2}(\alpha+1)$, +$x\rightarrow -\frac{1}{2}\phi^{-1}x$ and the limit $\phi\rightarrow 0$: +$$\lim_{\phi\rightarrow 0} +P_n^{(\frac{1}{2}\alpha+\frac{1}{2})}(-\textstyle\frac{1}{2}\phi^{-1}x;\phi)=L_n^{(\alpha)}(x).$$ + +\subsubsection*{Jacobi $\rightarrow$ Laguerre} +The Laguerre polynomials are obtained from the Jacobi polynomials given by (\ref{DefJacobi}) +if we set $x\rightarrow 1-2\beta^{-1}x$ and then take the limit $\beta\rightarrow\infty$: +$$\lim_{\beta\rightarrow\infty} +P_n^{(\alpha,\beta)}(1-2\beta^{-1}x)=L_n^{(\alpha)}(x).$$ + +\subsubsection*{Meixner $\rightarrow$ Laguerre} +If we take $\beta=\alpha+1$ and $x\rightarrow (1-c)^{-1}x$ in the definition +(\ref{DefMeixner}) of the Meixner polynomials and let $c\rightarrow 1$ we obtain +the Laguerre polynomials: +$$\lim_{c\rightarrow 1} +M_n((1-c)^{-1}x;\alpha+1,c)=\frac{L_n^{(\alpha)}(x)}{L_n^{(\alpha)}(0)}.$$ + +\subsubsection*{Laguerre $\rightarrow$ Hermite} +The Hermite polynomials given by (\ref{DefHermite}) can be obtained from the Laguerre +polynomials by taking the limit $\alpha\rightarrow\infty$ in the following way: +\begin{equation} +\lim_{\alpha\rightarrow\infty} +\left(\frac{2}{\alpha}\right)^{\frac{1}{2}n} +L_n^{(\alpha)}((2\alpha)^{\frac{1}{2}}x+\alpha)=\frac{(-1)^n}{n!}H_n(x). +\end{equation} + +\subsection*{Remarks} +The definition (\ref{DefLaguerre}) of the Laguerre polynomials can also be +written as: +$$L_n^{(\alpha)}(x)=\frac{1}{n!}\sum_{k=0}^n\frac{(-n)_k}{k!}(\alpha+k+1)_{n-k}x^k.$$ +In this way the Laguerre polynomials can also be seen as polynomials in the parameter $\alpha$. +Therefore they can be defined for all $\alpha$. + +\noindent +The Laguerre polynomials are related to the Bessel polynomials given by (\ref{DefBessel}) +in the following way: +$$L_n^{(\alpha)}(x)=\frac{(-x)^n}{n!}y_n(2x^{-1};-2n-\alpha-1).$$ + +\noindent +The Laguerre polynomials are related to the Charlier polynomials given by (\ref{DefCharlier}) +in the following way: +$$\frac{(-a)^n}{n!}C_n(x;a)=L_n^{(x-n)}(a).$$ + +\noindent +The Laguerre polynomials and the Hermite polynomials given by (\ref{DefHermite}) are also +connected by the following quadratic transformations: +$$H_{2n}(x)=(-1)^nn!\,2^{2n}L_n^{(-\frac{1}{2})}(x^2)$$ +and +$$H_{2n+1}(x)=(-1)^nn!\,2^{2n+1}xL_n^{(\frac{1}{2})}(x^2).$$ + +\noindent +In combinatorics the Laguerre polynomials with $\alpha=0$ are often called Rook +\subsection*{9.14 Charlier} +\label{sec9.14} +% +\paragraph{Hypergeometric representation} +\begin{align} +C_n(x;a)&=\hyp20{-n,-x}-{-\,\frac1a} +\label{179}\\ +&=\frac{(-x)_n}{a^n} \hyp11{-n}{x-n+1}a +\label{180}\\ +&=\frac{n!}{(-a)^n}\,L_n^{x-n}(a), +\label{181} +\end{align} +where $L_n^\al(x)$ is a +\hyperref[sec9.12]{Laguerre polynomial}. +Formula \eqref{179} is (9.14.1). Then \eqref{180} follows by reversal +of the summation. Finally \eqref{181} follows by \eqref{180} and +(9.12.1). It is also the Remark on p.249 of \mycite{KLS}, and it +was earlier given in \myciteKLS{416}{(2.7.10)}. +% +\paragraph{Uniqueness of orthogonality measure} +The coefficient of $p_{n-1}(x)$ in (9.14.4) behaves as $O(n)$ as $n\to\iy$. +Hence \eqref{93} holds, by which the orthogonality measure is unique. +polynomials. + +\subsection*{References} +\cite{Abdul}, \cite{Abram}, \cite{Ahmed+82}, \cite{Allaway76}, \cite{Allaway91}, +\cite{NAlSalam66}, \cite{AlSalam64}, \cite{AlSalam90}, \cite{AlSalamChihara72}, +\cite{AlSalamChihara76}, \cite{AndrewsAskey85}, \cite{AndrewsAskeyRoy}, \cite{Askey68}, +\cite{Askey75}, \cite{Askey89I}, \cite{Askey2005}, \cite{AskeyGasper76}, \cite{AskeyGasper77}, +\cite{AskeyIsmail76}, \cite{AskeyIsmailKoorn}, \cite{AskeyWilson85}, \cite{Barrett}, +\cite{Bavinck96}, \cite{Brafman51}, \cite{Brafman57II}, \cite{Brenke}, \cite{Brown}, +\cite{BustozSavage79}, \cite{BustozSavage80}, \cite{Carlitz60}, \cite{Carlitz61I}, +\cite{Carlitz61II}, \cite{Carlitz62}, \cite{Carlitz68}, \cite{ChenSrivastava}, +\cite{ChenIsmail91}, \cite{Chihara68II}, \cite{Chihara78}, \cite{Cohen}, \cite{Cooper+}, +\cite{Dattoli2001}, \cite{DetteStudden92}, \cite{DetteStudden95}, \cite{Dimitrov2003}, +\cite{Doha2003II}, \cite{ElbertLaforgia87I}, \cite{Erdelyi}, \cite{Erdelyi+}, \cite{Exton98}, +\cite{Faldey}, \cite{Gasper73II}, \cite{Gasper77}, \cite{Gatteschi2002}, \cite{Gawronski87}, +\cite{Gawronski93}, \cite{Gillis}, \cite{GillisIsmailOffer}, \cite{Godoy+}, \cite{Grad}, +\cite{Hajmirzaahmad95}, \cite{Ismail74}, \cite{Ismail77}, \cite{Ismail2005II}, +\cite{IsmailLetVal88}, \cite{IsmailLi}, \cite{IsmailMassonRahman}, \cite{IsmailMuldoon}, +\cite{IsmailStanton97}, \cite{IsmailTamhankar}, \cite{Jackson}, \cite{Karlin58}, +\cite{Kochneff95}, \cite{Kochneff97II}, \cite{Koekoek2000}, \cite{Koorn77I}, \cite{Koorn78}, +\cite{Koorn85}, \cite{Koorn88}, \cite{Krasikov2003}, \cite{Krasikov2005}, +\cite{KuijlaarsMcLaughlin}, \cite{KwonLittle}, \cite{LabelleYehI}, \cite{LabelleYehII}, +\cite{LabelleYehIII}, \cite{Lee97I}, \cite{Lee97II}, \cite{Lee97III}, \cite{Lesky95II}, +\cite{Lesky96}, \cite{LewanowiczI}, \cite{LopezTemme2004}, \cite{Mathai}, \cite{Meixner}, +\cite{Nikiforov+}, \cite{NikiforovUvarov}, \cite{Olver}, \cite{PerezPinar}, \cite{Pittaluga}, +\cite{Rainville}, \cite{SainteViennot}, \cite{Srivastava66}, \cite{Srivastava69I}, +\cite{Srivastava69II}, \cite{Srivastava70}, \cite{Srivastava71}, \cite{Szego75}, \cite{Temme}, +\cite{Trickovic}, \cite{Trivedi}, \cite{Viennot}. + + +\section{Bessel}\index{Bessel polynomials} + +\par\setcounter{equation}{0} + +\subsection*{Hypergeometric representation} +\begin{eqnarray} +\label{DefBessel} +y_n(x;a)&=&\hyp{2}{0}{-n,n+a+1}{-}{-\frac{x}{2}}\\ +&=&(n+a+1)_n\left(\frac{x}{2}\right)^n\,\hyp{1}{1}{-n}{-2n-a}{\frac{2}{x}}, +\quad n=0,1,2,\ldots,N.\nonumber +\end{eqnarray} + +\subsection*{Orthogonality relation} +\begin{eqnarray} +\label{OrtBessel} +& &\int_0^{\infty}x^a\e^{-\frac{2}{x}}y_m(x;a)y_n(x;a)\,dx\nonumber\\ +& &=-\frac{2^{a+1}}{2n+a+1}\Gamma(-n-a)n!\,\delta_{mn},\quad a<-2N-1. +\end{eqnarray} + +\subsection*{Recurrence relation} +\begin{eqnarray} +\label{RecBessel} +& &2(n+a+1)(2n+a)y_{n+1}(x;a)\nonumber\\ +& &{}=(2n+a+1)\left[2a+(2n+a)(2n+a+2)x\right]y_n(x;a)\nonumber\\ +& &{}\mathindent{}+2n(2n+a+2)y_{n-1}(x;a). +\end{eqnarray} + +\subsection*{Normalized recurrence relation} +\begin{eqnarray} +\label{NormRecBessel} +xp_n(x)&=&p_{n+1}(x)-\frac{2a}{(2n+a)(2n+a+2)}p_n(x)\nonumber\\ +& &{}\mathindent{}-\frac{4n(n+a)}{(2n+a-1)(2n+a)^2(2n+a+1)}p_{n-1}(x), +\end{eqnarray} +where +$$y_n(x;a)=\frac{(n+a+1)_n}{2^n}p_n(x).$$ + +\subsection*{Differential equation} +\begin{equation} +\label{dvBessel} +x^2y''(x)+\left[(a+2)x+2\right]y'(x)-n(n+a+1)y(x)=0,\quad y(x)=y_n(x;a). +\end{equation} + +\subsection*{Forward shift operator} +\begin{equation} +\label{shift1Bessel} +\frac{d}{dx}y_n(x;a)=\frac{n(n+a+1)}{2}y_{n-1}(x;a+2). +\end{equation} + +\subsection*{Backward shift operator} +\begin{equation} +\label{shift2BesselI} +x^2\frac{d}{dx}y_n(x;a)+(ax+2)y_n(x;a)=2y_{n+1}(x;a-2) +\end{equation} +or equivalently +\begin{equation} +\label{shift2BesselII} +\frac{d}{dx}\left[x^a\expe^{-\frac{2}{x}}y_n(x;a)\right]=2x^{a-2}\expe^{-\frac{2}{x}}y_{n+1}(x;a-2). +\end{equation} + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodBessel} +y_n(x;a)=2^{-n}x^{-a}\expe^{\frac{2}{x}}D^n\left(x^{2n+a}\expe^{-\frac{2}{x}}\right). +\end{equation} + +\subsection*{Generating function} +\begin{equation} +\label{GenBessel} +\left(1-2xt\right)^{-\frac{1}{2}}\left(\frac{2}{1+\sqrt{1-2xt}}\right)^a +\exp\left(\frac{2t}{1+\sqrt{1-2xt}}\right)=\sum_{n=0}^{\infty}y_n(x;a)\frac{t^n}{n!}. +\end{equation} + +\subsection*{Limit relation} + +\subsubsection*{Jacobi $\rightarrow$ Bessel} +If we take $\beta=a-\alpha$ in the definition (\ref{DefJacobi}) of the Jacobi polynomials +and let $\alpha\rightarrow -\infty$ we find the Bessel polynomials: +$$\lim_{\alpha\rightarrow -\infty} +\frac{P_n^{(\alpha,a-\alpha)}(1+\alpha x)}{P_n^{(\alpha,a-\alpha)}(1)}=y_n(x;a).$$ + +\subsection*{Remarks} +The following notations are also used for the Bessel polynomials: +$$y_n(x;a,b)=y_n(2b^{-1}x;a)\quad\textrm{and}\quad\theta_n(x;a,b)=x^ny_n(x^{-1};a,b).$$ +However, the Bessel polynomials essentially depend on only one parameter. + +\noindent +The Bessel polynomials are related to the Laguerre polynomials given by (\ref{DefLaguerre}) +in the following way: +$$L_n^{(\alpha)}(x)=\frac{(-x)^n}{n!}y_n(2x^{-1};-2n-\alpha-1).$$ + +\noindent +The special case $a=-2N-2$ of the Bessel polynomials can be obtained from the pseudo Jacobi +polynomials by setting $x\rightarrow\nu x$ in the definition (\ref{DefPseudoJacobi}) of the +pseudo Jacobi polynomials and taking the limit $\nu\rightarrow\infty$ in the following way: +$$\lim\limits_{\nu\rightarrow\infty}\frac{P_n(\nu x;\nu,N)}{\nu^n} +\subsection*{9.15 Hermite} +\label{sec9.15} +% +\paragraph{Uniqueness of orthogonality measure} +The coefficient of $p_{n-1}(x)$ in (9.15.4) behaves as $O(n)$ as $n\to\iy$. +Hence \eqref{93} holds, by which the orthogonality measure is unique. +% +\paragraph{Fourier transforms} +\begin{equation} +\frac1{\sqrt{2\pi}}\,\int_{-\iy}^\iy H_n(y)\,e^{-\half y^2}\,e^{ixy}\,dy= +i^n\,H_n(x)\,e^{-\half x^2}, +\label{15} +\end{equation} +see \mycite{AAR}{(6.1.15) and Exercise 6.11}. +\begin{equation} +\frac1{\sqrt\pi}\,\int_{-\iy}^\iy H_n(y)\,e^{-y^2}\,e^{ixy}\,dy= +i^n\,x^n\,e^{-\frac14 x^2}, +\label{16} +\end{equation} +see \mycite{DLMF}{(18.17.35)}. +\begin{equation} +\frac{i^n}{2\sqrt\pi}\,\int_{-\iy}^\iy y^n\,e^{-\frac14 y^2}\,e^{-ixy}\,dy= +H_n(x)\,e^{-x^2}, +\label{17} +\end{equation} +see \mycite{AAR}{(6.1.4)}. +=\frac{2^n}{(n-2N-1)_n}y_n(x;-2N-2).$$ + +\subsection*{References} +\cite{Andrade}, \cite{BergVignat}, \cite{Carlitz57I}, \cite{Dattoli2003}, +\cite{DohaAhmed2004}, \cite{DohaAhmed2006}, \cite{Grosswald}, \cite{Ismail2005II}, +\cite{KrallFrink}, \cite{Lesky98}, \cite{NikiforovUvarov}. + + +\section{Charlier}\index{Charlier polynomials} + +\par\setcounter{equation}{0} + +\subsection*{Hypergeometric representation} +\begin{equation} +\label{DefCharlier} +C_n(x;a)=\hyp{2}{0}{-n,-x}{-}{-\frac{1}{a}}. +\end{equation} + +\subsection*{Orthogonality relation} +\begin{equation} +\label{OrtCharlier} +\sum_{x=0}^{\infty}\frac{a^x}{x!}C_m(x;a)C_n(x;a)= +a^{-n}\expe^an!\,\delta_{mn},\quad a > 0. +\end{equation} + +\subsection*{Recurrence relation} +\begin{equation} +\label{RecCharlier} +-xC_n(x;a)=aC_{n+1}(x;a)-(n+a)C_n(x;a)+nC_{n-1}(x;a). +\end{equation} + +\subsection*{Normalized recurrence relation} +\begin{equation} +\label{NormRecCharlier} +xp_n(x)=p_{n+1}(x)+(n+a)p_n(x)+nap_{n-1}(x), +\end{equation} +where +$$C_n(x;a)=\left(-\frac{1}{a}\right)^np_n(x).$$ + +\subsection*{Difference equation} +\begin{equation} +\label{dvCharlier} +-ny(x)=ay(x+1)-(x+a)y(x)+xy(x-1),\quad y(x)=C_n(x;a). +\end{equation} + +\subsection*{Forward shift operator} +\begin{equation} +\label{shift1CharlierI} +C_n(x+1;a)-C_n(x;a)=-\frac{n}{a}C_{n-1}(x;a) +\end{equation} +or equivalently +\begin{equation} +\label{shift1CharlierII} +\Delta C_n(x;a)=-\frac{n}{a}C_{n-1}(x;a). +\end{equation} + +\subsection*{Backward shift operator} +\begin{equation} +\label{shift2CharlierI} +C_n(x;a)-\frac{x}{a}C_n(x-1;a)=C_{n+1}(x;a) +\end{equation} +or equivalently +\begin{equation} +\label{shift2CharlierII} +\nabla\left[\frac{a^x}{x!}C_n(x;a)\right]=\frac{a^x}{x!}C_{n+1}(x;a). +\end{equation} + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodCharlier} +\frac{a^x}{x!}C_n(x;a)=\nabla^n\left[\frac{a^x}{x!}\right]. +\end{equation} + +\subsection*{Generating function} +\begin{equation} +\label{GenCharlier} +\e^t\left(1-\frac{t}{a}\right)^x=\sum_{n=0}^{\infty}\frac{C_n(x;a)}{n!}t^n. +\end{equation} + +\subsection*{Limit relations} + +\subsubsection*{Meixner $\rightarrow$ Charlier} +If we take $c=(a+\beta)^{-1}a$ in the definition (\ref{DefMeixner}) of the Meixner polynomials +and let $\beta\rightarrow\infty$ we find the Charlier polynomials: +$$\lim_{\beta\rightarrow\infty}M_n(x;\beta,(a+\beta)^{-1}a)=C_n(x;a).$$ + +\subsubsection*{Krawtchouk $\rightarrow$ Charlier} +The Charlier polynomials can be found from the Krawtchouk polynomials given by +(\ref{DefKrawtchouk}) by taking $p=N^{-1}a$ and letting $N\rightarrow\infty$: +$$\lim_{N\rightarrow\infty}K_n(x;N^{-1}a,N)=C_n(x;a).$$ + +\subsubsection*{Charlier $\rightarrow$ Hermite} +The Hermite polynomials given by (\ref{DefHermite}) are obtained from the Charlier polynomials +if we set $x\rightarrow (2a)^{1/2}x+a$ and let $a\rightarrow\infty$. In fact we have +\begin{equation} +\lim_{a\rightarrow\infty} +(2a)^{\frac{1}{2}n}C_n((2a)^{\frac{1}{2}}x+a;a)=(-1)^nH_n(x). +\end{equation} + +\subsection*{Remark} +The Charlier polynomials are related to the Laguerre polynomials given by (\ref{DefLaguerre}) +in the following way: +\subsection*{14.1 Askey-Wilson} +\label{sec14.1} +% +\paragraph{Symmetry} +The Askey-Wilson polynomials $p_n(x;a,b,c,d\,|\,q)$ are \,|\,symmetric +in $a,b,c,d$. +\sLP +This follows from the orthogonality relation (14.1.2) +together with the value of its coefficient of $x^n$ given in (14.1.5b). +Alternatively, combine (14.1.1) with \mycite{GR}{(III.15)}.\\ +As a consequence, it is sufficient to give generating function (14.1.13). Then the generating +functions (14.1.14), (14.1.15) will follow by symmetry in the parameters. +% +\paragraph{Basic hypergeometric representation} +In addition to (14.1.1) we have (in notation \eqref{111}): +\begin{multline} +p_n(\cos\tha;a,b,c,d\,|\, q) +=\frac{(ae^{-i\tha},be^{-i\tha},ce^{-i\tha},de^{-i\tha};q)_n} +{(e^{-2i\tha};q)_n}\,e^{in\tha}\\ +\times {}_8W_7\big(q^{-n}e^{2i\tha};ae^{i\tha},be^{i\tha}, +ce^{i\tha},de^{i\tha},q^{-n};q,q^{2-n}/(abcd)\big). +\label{113} +\end{multline} +This follows from (14.1.1) by combining (III.15) and (III.19) in +\mycite{GR}. +It is also given in \myciteKLS{513}{(4.2)}, but be aware for some slight errors. +The symmetry in $a,b,c,d$ is evident from \eqref{113}. +% +\paragraph{Special value} +\begin{equation} +p_n\big(\thalf(a+a^{-1});a,b,c,d\,|\, q\big)=a^{-n}\,(ab,ac,ad;q)_n\,, +\label{40} +\end{equation} +and similarly for arguments $\thalf(b+b^{-1})$, $\thalf(c+c^{-1})$ and +$\thalf(d+d^{-1})$ by symmetry of $p_n$ in $a,b,c,d$. +% +\paragraph{Trivial symmetry} +\begin{equation} +p_n(-x;a,b,c,d\,|\, q)=(-1)^n p_n(x;-a,-b,-c,-d\,|\, q). +\label{41} +\end{equation} +Both \eqref{40} and \eqref{41} are obtained from (14.1.1). +% +\paragraph{Re: (14.1.5)} +Let +\begin{equation} +p_n(x):=\frac{p_n(x;a,b,c,d\,|\, q)}{2^n(abcdq^{n-1};q)_n}=x^n+\wt k_n x^{n-1} ++\cdots\;. +\label{18} +\end{equation} +Then +\begin{equation} +\wt k_n=-\frac{(1-q^n)(a+b+c+d-(abc+abd+acd+bcd)q^{n-1})} +{2(1-q)(1-abcdq^{2n-2})}\,. +\label{19} +\end{equation} +This follows because $\tilde k_n-\tilde k_{n+1}$ equals the coefficient +$\thalf\bigl(a+a^{-1}-(A_n+C_n)\bigr)$ of $p_n(x)$ in (14.1.5). +% +\paragraph{Generating functions} +Rahman \myciteKLS{449}{(4.1), (4.9)} gives: +\begin{align} +&\sum_{n=0}^\iy \frac{(abcdq^{-1};q)_n a^n}{(ab,ac,ad,q;q)_n}\,t^n\, +p_n(\cos\tha;a,b,c,d\,|\,q) +\nonumber\\ +&=\frac{(abcdtq^{-1};q)_\iy}{(t;q)_\iy}\, +\qhyp65{(abcdq^{-1})^\half,-(abcdq^{-1})^\half,(abcd)^\half, +-(abcd)^\half,a e^{i\tha},a e^{-i\tha}} +{ab,ac,ad,abcdtq^{-1},qt^{-1}}{q,q} +\nonumber\\ +&+\frac{(abcdq^{-1},abt,act,adt,ae^{i\tha},ae^{-i\tha};q)_\iy} +{(ab,ac,ad,t^{-1},ate^{i\tha},ate^{-i\tha};q)_\iy} +\nonumber\\ +&\times\qhyp65{t(abcdq^{-1})^\half,-t(abcdq^{-1})^\half,t(abcd)^\half, +-t(abcd)^\half,at e^{i\tha},at e^{-i\tha}} +{abt,act,adt,abcdt^2q^{-1},qt}{q,q}\quad(|t|<1). +\label{185} +\end{align} +In the limit \eqref{109} the first term on the \RHS\ of \eqref{185} +tends to the \LHS\ of (9.1.15), while the second term tends formally +to 0. The special case $ad=bc$ of \eqref{185} was earlier given in +\myciteKLS{236}{(4.1), (4.6)}. +% +\paragraph{Limit relations}\quad\sLP +{\bf Askey-Wilson $\longrightarrow$ Wilson}\\ +Instead of (14.1.21) we can keep a polynomial of degree $n$ while the limit is approached: +\begin{equation} +\lim_{q\to1}\frac{p_n(1-\thalf x(1-q)^2;q^a,q^b,q^c,q^d\,|\, q)}{(1-q)^{3n}} +=W_n(x;a,b,c,d). +\label{109} +\end{equation} +For the proof first derive the corresponding limit for the monic polynomials by comparing +(14.1.5) with (9.4.4). +\bLP +{\bf Askey-Wilson $\longrightarrow$ Continuous Hahn}\\ +Instead of (14.4.15) we can keep a polynomial of degree $n$ while the limit is approached: +\begin{multline} +\lim_{q\uparrow1} +\frac{p_n\big(\cos\phi-x(1-q)\sin\phi;q^a e^{i\phi},q^b e^{i\phi},q^{\overline a} e^{-i\phi}, +q^{\overline b} e^{-i\phi}\,|\, q\big)} +{(1-q)^{2n}}\\ +=(-2\sin\phi)^n\,n!\,p_n(x;a,b,\overline a,\overline b)\qquad +(0<\phi<\pi). +\label{177} +\end{multline} +Here the \RHS\ has a continuous Hahn polynomial (9.4.1). +For the proof first derive the corresponding limit for the monic polynomials by comparing +(14.1.5) with (9.1.5). +In fact, define the monic polynomial +\[ +\wt p_n(x):= +\frac{p_n\big(\cos\phi-x(1-q)\sin\phi;q^a e^{i\phi},q^b e^{i\phi},q^{\overline a} e^{-i\phi}, +q^{\overline b} e^{-i\phi}\,|\, q\big)} +{(-2(1-q)\sin\phi)^n\,(abcdq^{n-1};q)_n}\,. +\] +Then it follows from (14.1.5) that +\begin{equation*} +x\,\wt p_n(x)=\wt p_{n+1}(x) ++\frac{(1-q^a)e^{i\phi}+(1-q^{-a})e^{-i\phi}+\wt A_n+\wt C_n}{2(1-q)\sin\phi}\,\wt p_n(x)\\ ++\frac{\wt A_{n-1} \wt C_n}{(1-q)^2 \sin^2\phi}\,\wt p_{n-1}(x), +\end{equation*} +where $\wt A_n$ and $\wt C_n$ are as given after (14.1.3) with $a,b,c,d$ replaced by +$q^a e^{i\phi},q^b e^{i\phi},q^{\overline a} e^{-i\phi},q^{\overline b} e^{-i\phi}$. +Then the recurrence equation for $\wt p_n(x)$ tends for $q\uparrow 1$ to +the recurrence equation (9.4.4) with $c=\overline a$, $d=\overline b$. +\bLP +{\bf Askey-Wilson $\longrightarrow$ Meixner-Pollaczek}\\ +Instead of (14.9.15) we can keep a polynomial of degree $n$ while the limit is approached: +\begin{equation} +\lim_{q\uparrow1} +\frac{p_n\big(\cos\phi-x(1-q)\sin\phi; +q^\la e^{i\phi},0,q^\la e^{-i\phi},0\,|\, q\big)}{(1-q)^n} +=n!\,P_n^{(\la)}(x;\pi-\phi)\quad +(0<\phi<\pi). +\label{178} +\end{equation} +Here the \RHS\ has a Meixner-Pollaczek polynomial (9.7.1). +For the proof first derive the corresponding limit for the monic polynomials by comparing +(14.1.5) with (9.7.4). +In fact, define the monic polynomial +\[ +\wt p_n(x):= +\frac{p_n\big(\cos\phi-x(1-q)\sin\phi; +q^\la e^{i\phi},0,q^\la e^{-i\phi},0\,|\, q\big)}{(-2(1-q)\sin\phi)^n}\,. +\] +Then it follows from (14.1.5) that +\begin{equation*} +x\,\wt p_n(x)=\wt p_{n+1}(x) ++\frac{(1-q^\la)e^{i\phi}+(1-q^{-\la})e^{-i\phi}+\wt A_n+\wt C_n} +{2(1-q)\sin\phi}\,\wt p_n(x)\\ ++\frac{\wt A_{n-1} \wt C_n}{(1-q)^2 \sin^2\phi}\,\wt p_{n-1}(x), +\end{equation*} +where $\wt A_n$ and $\wt C_n$ are as given after (14.1.3) with $a,b,c,d$ replaced by +$q^\la e^{i\phi},0,q^\la e^{-i\phi},0$. +Then the recurrence equation for $\wt p_n(x)$ tends for $q\uparrow 1$ to +the recurrence equation (9.7.4). +% +\paragraph{References} +See also Koornwinder \cite{K7}. +$$\frac{(-a)^n}{n!}C_n(x;a)=L_n^{(x-n)}(a).$$ + +\subsection*{References} +\cite{Allaway76}, \cite{NAlSalam66}, \cite{AlSalam90}, \cite{AlSalamChihara76}, +\cite{AlSalamIsmail76}, \cite{AndrewsAskey85}, \cite{Area+II}, \cite{Askey75}, +\cite{AskeyGasper77}, \cite{AskeyWilson85}, \cite{AtakRahmanSuslov}, \cite{Bavinck98}, +\cite{BavinckKoekoek}, \cite{Chihara78}, \cite{Chihara79}, \cite{Dunkl76}, \cite{Erdelyi+}, +\cite{Gasper73I}, \cite{Gasper74}, \cite{Goh}, \cite{HoareRahman}, \cite{IsmailLetVal88}, +\cite{Koekoek2000}, \cite{Koorn88}, \cite{Krasikov2002}, \cite{LabelleYehI}, +\cite{LabelleYehII}, \cite{LabelleYehIII}, \cite{Lesky62}, \cite{Lesky89}, \cite{Lesky94I}, +\cite{Lesky95II}, \cite{LewanowiczII}, \cite{LopezTemme2004}, \cite{Meixner}, \cite{Nikiforov+}, +\cite{NikiforovUvarov}, \cite{Szafraniec}, \cite{Szego75}, \cite{Viennot}, \cite{Zarzo+}, +\cite{Zeng90}. + + +\section{Hermite}\index{Hermite polynomials} + +\par\setcounter{equation}{0} + +\subsection*{Hypergeometric representation} +\begin{equation} +\label{DefHermite} +H_n(x)=(2x)^n\,\hyp{2}{0}{-n/2,-(n-1)/2}{-}{-\frac{1}{x^2}}. +\end{equation} + +\subsection*{Orthogonality relation} +\begin{equation} +\label{OrtHermite} +\frac{1}{\sqrt{\cpi}}\int_{-\infty}^{\infty}\expe^{-x^2}H_m(x)H_n(x)\,dx +=2^nn!\,\delta_{mn}. +\end{equation} + +\subsection*{Recurrence relation} +\begin{equation} +\label{RecHermite} +H_{n+1}(x)-2xH_n(x)+2nH_{n-1}(x)=0. +\end{equation} + +\subsection*{Normalized recurrence relation} +\begin{equation} +\label{NormRecHermite} +xp_n(x)=p_{n+1}(x)+\frac{n}{2}p_{n-1}(x), +\end{equation} +where +$$H_n(x)=2^np_n(x).$$ + +\subsection*{Differential equation} +\begin{equation} +\label{dvHermite} +y''(x)-2xy'(x)+2ny(x)=0,\quad y(x)=H_n(x). +\end{equation} + +\subsection*{Forward shift operator} +\begin{equation} +\label{shift1Hermite} +\frac{d}{dx}H_n(x)=2nH_{n-1}(x). +\end{equation} + +\subsection*{Backward shift operator} +\begin{equation} +\label{shift2HermiteI} +\frac{d}{dx}H_n(x)-2xH_n(x)=-H_{n+1}(x) +\end{equation} +or equivalently +\begin{equation} +\label{shift2HermiteII} +\frac{d}{dx}\left[\expe^{-x^2}H_n(x)\right]=-\expe^{-x^2}H_{n+1}(x). +\end{equation} + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodHermite} +\e^{-x^2}H_n(x)=(-1)^n\left(\frac{d}{dx}\right)^n\left[\expe^{-x^2}\right]. +\end{equation} + +\subsection*{Generating functions} +\begin{equation} +\label{GenHermite1} +\exp\left(2xt-t^2\right)=\sum_{n=0}^{\infty}\frac{H_n(x)}{n!}t^n. +\end{equation} + +\begin{equation} +\label{GenHermite2} +\left\{\begin{array}{l} +\displaystyle\expe^t\cos(2x\sqrt{t})=\sum_{n=0}^{\infty} +\frac{(-1)^n}{(2n)!}H_{2n}(x)t^n\\[5mm] +\displaystyle\frac{\expe^t}{\sqrt{t}}\sin(2x\sqrt{t})=\sum_{n=0}^{\infty} +\frac{(-1)^n}{(2n+1)!}H_{2n+1}(x)t^n. +\end{array}\right. +\end{equation} + +\begin{equation} +\label{GenHermite3} +\left\{\begin{array}{l} +\displaystyle \expe^{-t^2}\cosh(2xt)=\sum_{n=0}^{\infty} +\frac{H_{2n}(x)}{(2n)!}t^{2n}\\[5mm] +\displaystyle\expe^{-t^2}\sinh(2xt)=\sum_{n=0}^{\infty} +\frac{H_{2n+1}(x)}{(2n+1)!}t^{2n+1}. +\end{array}\right. +\end{equation} + +\begin{equation} +\label{GenHermite4} +\left\{\begin{array}{l} +\displaystyle (1+t^2)^{-\gamma}\,\hyp{1}{1}{\gamma}{\frac{1}{2}}{\frac{x^2t^2}{1+t^2}}= +\sum_{n=0}^{\infty}\frac{(\gamma)_n}{(2n)!}H_{2n}(x)t^{2n}\\[5mm] +\displaystyle\frac{xt}{\sqrt{1+t^2}}\, +\hyp{1}{1}{\gamma+\frac{1}{2}}{\frac{3}{2}}{\frac{x^2t^2}{1+t^2}} +=\sum_{n=0}^{\infty}\frac{(\gamma+\frac{1}{2})_n}{(2n+1)!}H_{2n+1}(x)t^{2n+1} +\end{array}\right. +\end{equation} +with $\gamma$ arbitrary. + +\begin{equation} +\label{GenHermite5} +\frac{1+2xt+4t^2}{(1+4t^2)^{\frac{3}{2}}}\exp\left(\frac{4x^2t^2}{1+4t^2}\right) +=\sum_{n=0}^{\infty}\frac{H_n(x)}{\lfloor n/2\rfloor\,!}t^n, +\end{equation} +where $\lfloor n/2\rfloor$ denotes the largest integer smaller than or equal to $n/2$. + +\subsection*{Limit relations} + +\subsubsection*{Meixner-Pollaczek $\rightarrow$ Hermite} +If we take $x\rightarrow (\sin\phi)^{-1}(x\sqrt{\lambda}-\lambda\cos\phi)$ +in the definition (\ref{DefMP}) of the Meixner-Pollaczek polynomials and +then let $\lambda\rightarrow\infty$ we obtain the Hermite polynomials: +$$\lim_{\lambda\rightarrow\infty} +\lambda^{-\frac{1}{2}n}P_n^{(\lambda)} +((\sin\phi)^{-1}(x\sqrt{\lambda}-\lambda\cos\phi);\phi)=\frac{H_n(x)}{n!}.$$ + +\subsubsection*{Jacobi $\rightarrow$ Hermite} +The Hermite polynomials follow from the Jacobi polynomials given by (\ref{DefJacobi}) by +taking $\beta=\alpha$ and letting $\alpha\rightarrow\infty$ in the following way: +$$\lim_{\alpha\rightarrow\infty} +\alpha^{-\frac{1}{2}n}P_n^{(\alpha,\alpha)}(\alpha^{-\frac{1}{2}}x)=\frac{H_n(x)}{2^nn!}.$$ + +\subsubsection*{Gegenbauer / Ultraspherical $\rightarrow$ Hermite} +The Hermite polynomials follow from the Gegenbauer (or ultraspherical) polynomials given by +(\ref{DefGegenbauer}) by taking $\lambda=\alpha+\frac{1}{2}$ and letting +$\alpha\rightarrow\infty$ in the following way: +$$\lim_{\alpha\rightarrow\infty} +\alpha^{-\frac{1}{2}n}C_n^{(\alpha+\frac{1}{2})}(\alpha^{-\frac{1}{2}}x)=\frac{H_n(x)}{n!}.$$ + +\subsubsection*{Krawtchouk $\rightarrow$ Hermite} +The Hermite polynomials follow from the Krawtchouk polynomials given by (\ref{DefKrawtchouk}) +by setting $x\rightarrow pN+x\sqrt{2p(1-p)N}$ and then letting $N\rightarrow\infty$: +$$\lim_{N\rightarrow\infty} +\sqrt{\binom{N}{n}}K_n(pN+x\sqrt{2p(1-p)N};p,N) +=\frac{\displaystyle (-1)^nH_n(x)}{\displaystyle\sqrt{2^nn!\left(\frac{p}{1-p}\right)^n}}.$$ + +\subsubsection*{Laguerre $\rightarrow$ Hermite} +The Hermite polynomials can be obtained from the Laguerre polynomials given by +(\ref{DefLaguerre}) by taking the limit $\alpha\rightarrow\infty$ in the following way: +$$\lim_{\alpha\rightarrow\infty} +\left(\frac{2}{\alpha}\right)^{\frac{1}{2}n} +L_n^{(\alpha)}((2\alpha)^{\frac{1}{2}}x+\alpha)=\frac{(-1)^n}{n!}H_n(x).$$ + +\subsubsection*{Charlier $\rightarrow$ Hermite} +If we set $x\rightarrow (2a)^{1/2}x+a$ in the definition (\ref{DefCharlier}) +of the Charlier polynomials and let $a\rightarrow\infty$ we find the Hermite +polynomials. In fact we have +$$\lim_{a\rightarrow\infty} +(2a)^{\frac{1}{2}n}C_n((2a)^{\frac{1}{2}}x+a;a)=(-1)^nH_n(x).$$ + +\subsection*{Remarks} +The Hermite polynomials can also be written as: +$$\frac{H_n(x)}{n!}=\sum_{k=0}^{\lfloor n/2\rfloor} +\frac{(-1)^k(2x)^{n-2k}}{k!\,(n-2k)!},$$ +where $\lfloor n/2\rfloor$ denotes the largest integer smaller than or equal to $n/2$. + +\noindent +The Hermite polynomials and the Laguerre polynomials given by (\ref{DefLaguerre}) are also +connected by the following quadratic transformations: +$$H_{2n}(x)=(-1)^nn!\,2^{2n}L_n^{(-\frac{1}{2})}(x^2)$$ +and +\subsection*{14.2 $q$-Racah} +\label{sec14.2} +\paragraph{Symmetry} +\begin{equation} +R_n(x;\al,\be,q^{-N-1},\de\,|\, q) +=\frac{(\be q,\al\de^{-1}q;q)_n}{(\al q,\be\de q;q)_n}\,\de^n\, +R_n(\de^{-1}x;\be,\al,q^{-N-1},\de^{-1}\,|\, q). +\label{84} +\end{equation} +This follows from (14.2.1) combined with \mycite{GR}{(III.15)}. +\sLP +In particular, +\begin{equation} +R_n(x;\al,\be,q^{-N-1},-1\,|\, q) +=\frac{(\be q,-\al q;q)_n}{(\al q,-\be q;q)_n}\,(-1)^n\, +R_n(-x;\be,\al,q^{-N-1},-1\,|\, q), +\label{85} +\end{equation} +and +\begin{equation} +R_n(x;\al,\al,q^{-N-1},-1\,|\, q) +=(-1)^n\,R_n(-x;\al,\al,q^{-N-1},-1\,|\, q), +\label{86} +\end{equation} + +\paragraph{Trivial symmetry} +Clearly from (14.2.1): +\begin{equation} +R_n(x;\al,\be,\ga,\de\,|\, q)=R_n(x;\be\de,\al\de^{-1},\ga,\de\,|\, q) +=R_n(x;\ga,\al\be\ga^{-1},\al,\ga\de\al^{-1}\,|\, q). +\label{83} +\end{equation} +For $\al=q^{-N-1}$ this shows that the three cases +$\al q=q^{-N}$ or $\be\de q=q^{-N}$ or $\ga q=q^{-N}$ of (14.2.1) +are not essentially different. +% +\paragraph{Duality} +It follows from (14.2.1) that +\begin{equation} +R_n(q^{-y}+\ga\de q^{y+1};q^{-N-1},\be,\ga,\de\,|\, q) +=R_y(q^{-n}+\be q^{n-N};\ga,\de,q^{-N-1},\be\,|\, q)\quad +(n,y=0,1,\ldots,N). +\end{equation} +$$H_{2n+1}(x)=(-1)^nn!\,2^{2n+1}xL_n^{(\frac{1}{2})}(x^2).$$ + +\subsection*{14.3 Continuous dual $q$-Hahn} +\label{sec14.3} +The continuous dual $q$-Hahn polynomials are the special case $d=0$ of the +Askey-Wilson polynomials: +\[ +p_n(x;a,b,c\,|\, q):=p_n(x;a,b,c,0\,|\, q). +\] +Hence all formulas in \S14.3 are specializations for $d=0$ of formulas in \S14.1. +\subsection*{References} +\cite{Abram}, \cite{NAlSalam66}, \cite{AlSalam90}, \cite{AlSalamChihara72}, +\cite{AlSalamChihara76}, \cite{AndrewsAskey85}, \cite{AndrewsAskeyRoy}, \cite{Area+I}, +\cite{Askey68}, \cite{Askey75}, \cite{Askey89I}, \cite{AskeyGasper76}, \cite{AskeyWilson85}, +\cite{Azor}, \cite{Berg}, \cite{BilodeauII}, \cite{Brafman51}, \cite{Brafman57II}, +\cite{Brenke}, \cite{CarlitzSrivastava}, \cite{ChenSrivastava}, \cite{Chihara78}, +\cite{Cohen}, \cite{Danese}, \cite{DetteStudden92}, \cite{DetteStudden95}, \cite{Doha2004I}, +\cite{Erdelyi+}, \cite{Faldey}, \cite{Gawronski87}, \cite{Gawronski93}, \cite{Gillis+}, +\cite{Godoy+}, \cite{Grad}, \cite{Ismail74}, \cite{Ismail2005II}, \cite{IsmailStanton97}, +\cite{Koekoek2000}, \cite{Koorn88}, \cite{Krasikov2004}, \cite{Kwon+}, \cite{LabelleYehIII}, +\cite{Lesky95II}, \cite{Lesky96}, \cite{LewanowiczI}, \cite{LopezTemme99}, \cite{Mathai}, +\cite{Meixner}, \cite{Nikiforov+}, \cite{NikiforovUvarov}, \cite{Olver}, \cite{Pittaluga}, +\cite{Rainville}, \cite{SainteViennot}, \cite{Srivastava71}, \cite{SrivastavaMathur}, +\cite{Szego75}, \cite{Temme}, \cite{TemmeLopez2000}, \cite{Trickovic}, \cite{Viennot}, +\cite{Weisner59}, \cite{Wyman}. + +\end{document} diff --git a/KLSadd_insertion/updated9.tex b/KLSadd_insertion/updated9.tex new file mode 100644 index 0000000..e1b34b5 --- /dev/null +++ b/KLSadd_insertion/updated9.tex @@ -0,0 +1,4162 @@ +\documentclass[envcountchap,graybox]{svmono} + +\addtolength{\textwidth}{1mm} + +\usepackage{amsmath,amssymb} + +\usepackage{amsfonts} +%\usepackage{breqn} +\usepackage{DLMFmath} +\usepackage{DRMFfcns} + +\usepackage{mathptmx} +\usepackage{helvet} +\usepackage{courier} + +\usepackage{makeidx} +\usepackage{graphicx} + +\usepackage{multicol} +\usepackage[bottom]{footmisc} + +\makeindex + +\def\bibname{Bibliography} +\def\refname{Bibliography} + +\def\theequation{\thesection.\arabic{equation}} + +\smartqed + +\let\corollary=\undefined +\spnewtheorem{corollary}[theorem]{Corollary}{\bfseries}{\itshape} + +\newcounter{rom} + +\newcommand{\hyp}[5]{\mbox{}_{#1}{F}_{#2} +\left(\genfrac{}{}{0pt}{}{#3}{#4}\,;\,#5\right)} + +\newcommand{\qhyp}[5]{\mbox{}_{#1}{\phi}_{#2} +\left(\genfrac{}{}{0pt}{}{#3}{#4}\,;\,q,\,#5\right)} + +\newcommand{\mathindent}{\hspace{7.5mm}} + +\newcommand{\e}{\textrm{e}} + +\renewcommand{\E}{\textrm{E}} + +\renewcommand{\textfraction}{-1} + +\renewcommand{\Gamma}{\varGamma} + +\renewcommand{\leftlegendglue}{\hfil} + +\settowidth{\tocchpnum}{14\enspace} +\settowidth{\tocsecnum}{14.30\enspace} +\settowidth{\tocsubsecnum}{14.12.1\enspace} + +\makeatletter +\def\cleartoversopage{\clearpage\if@twoside\ifodd\c@page + \hbox{}\newpage\if@twocolumn\hbox{}\newpage\fi + \else\fi\fi} + +\newcommand{\clearemptyversopage}{ + \clearpage{\pagestyle{empty}\cleartoversopage}} +\makeatother + +\oddsidemargin -1.5cm +\topmargin -2.0cm +\textwidth 16.3cm +\textheight 25cm + +\begin{document} + +\author{Roelof Koekoek\\[2.5mm]Peter A. Lesky\\[2.5mm]Ren\'e F. Swarttouw} +\title{Hypergeometric orthogonal polynomials and their\\$q$-analogues} +\subtitle{-- Monograph --} +\maketitle + +\frontmatter + +\large + +\addtocounter{chapter}{8} +\pagenumbering{roman} +\chapter{Hypergeometric orthogonal polynomials} +\label{HyperOrtPol} + + +In this chapter we deal with all families of hypergeometric orthogonal polynomials +appearing in the Askey scheme on page~\pageref{scheme}. For each family of orthogonal +polynomials we state the most important properties such as a representation as a +hypergeometric function, orthogonality relation(s), the three-term recurrence relation, +the second-order differential or difference equation, the forward shift (or degree lowering) +and backward shift (or degree raising) operator, a Rodrigues-type formula and some generating +functions. In each case we use the notation which seems to be most common in the literature. +Moreover, in each case we mention the connection between various families by stating the +appropriate limit relations. See also \cite{Terwilliger2006} for an algebraic approach of +this Askey scheme and \cite{TemmeLopez2001} for a view from asymptotic analysis. +For notations the reader is referred to chapter~\ref{Definitions}. + +\section{Wilson}\index{Wilson polynomials} + +\par + +\subsection*{Hypergeometric representation} +\begin{eqnarray} +\label{DefWilson} +& &\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\nonumber\\ +& &{}=\hyp{4}{3}{-n,n+a+b+c+d-1,a+ix,a-ix}{a+b,a+c,a+d}{1}. +\end{eqnarray} + +\newpage + +\subsection*{Orthogonality relation} +If $Re(a,b,c,d)>0$ and non-real parameters occur in conjugate pairs, then +\begin{eqnarray} +\label{OrthIWilson} +& &\frac{1}{2\pi}\int_0^{\infty} +\left|\frac{\Gamma(a+ix)\Gamma(b+ix)\Gamma(c+ix)\Gamma(d+ix)}{\Gamma(2ix)}\right|^2\nonumber\\ +& &{}\mathindent\times W_m(x^2;a,b,c,d)W_n(x^2;a,b,c,d)\,dx\nonumber\\ +& &{}=\frac{\Gamma(n+a+b)\cdots\Gamma(n+c+d)}{\Gamma(2n+a+b+c+d)}(n+a+b+c+d-1)_nn!\,\delta_{mn}, +\end{eqnarray} +where +\begin{eqnarray*} +& &\Gamma(n+a+b)\cdots\Gamma(n+c+d)\\ +& &{}=\Gamma(n+a+b)\Gamma(n+a+c)\Gamma(n+a+d)\Gamma(n+b+c)\Gamma(n+b+d)\Gamma(n+c+d). +\end{eqnarray*} +If $a<0$ and $a+b$, $a+c$, $a+d$ are positive or a pair of complex conjugates +occur with positive real parts, then +\begin{eqnarray} +\label{OrtIIWilson} +& &\frac{1}{2\pi}\int_0^{\infty}\left|\frac{\Gamma(a+ix)\Gamma(b+ix)\Gamma(c+ix)\Gamma(d+ix)}{\Gamma(2ix)}\right|^2\nonumber\\ +& &{}\mathindent\mathindent\times W_m(x^2;a,b,c,d)W_n(x^2;a,b,c,d)\,dx\nonumber\\ +& &{}\mathindent{}+\frac{\Gamma(a+b)\Gamma(a+c)\Gamma(a+d)\Gamma(b-a)\Gamma(c-a)\Gamma(d-a)}{\Gamma(-2a)}\nonumber\\ +& &{}\mathindent\mathindent{}\times\sum_{\begin{array}{c} +{\scriptstyle k=0,1,2\ldots}\\{\scriptstyle a+k<0}\end{array}} +\frac{(2a)_k(a+1)_k(a+b)_k(a+c)_k(a+d)_k}{(a)_k(a-b+1)_k(a-c+1)_k(a-d+1)_kk!}\nonumber\\ +& &{}\mathindent\mathindent\mathindent\mathindent{}\times W_m(-(a+k)^2;a,b,c,d)W_n(-(a+k)^2;a,b,c,d)\nonumber\\ +& &{}=\frac{\Gamma(n+a+b)\cdots\Gamma(n+c+d)}{\Gamma(2n+a+b+c+d)}(n+a+b+c+d-1)_nn!\,\delta_{mn}. +\end{eqnarray} + +\subsection*{Recurrence relation} +\begin{equation} +\label{RecWilson} +-\left(a^2+x^2\right){\tilde{W}}_n(x^2) +=A_n{\tilde{W}}_{n+1}(x^2)-\left(A_n+C_n\right){\tilde{W}}_n(x^2)+C_n{\tilde{W}}_{n-1}(x^2), +\end{equation} +where +$${\tilde{W}}_n(x^2):={\tilde{W}}_n(x^2;a,b,c,d)=\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}$$ +and +$$\left\{\begin{array}{l} +\displaystyle A_n=\frac{(n+a+b+c+d-1)(n+a+b)(n+a+c)(n+a+d)}{(2n+a+b+c+d-1)(2n+a+b+c+d)}\\[5mm] +\displaystyle C_n=\frac{n(n+b+c-1)(n+b+d-1)(n+c+d-1)}{(2n+a+b+c+d-2)(2n+a+b+c+d-1)}. +\end{array}\right.$$ + +\subsection*{Normalized recurrence relation} +\begin{equation} +\label{NormRecWilson} +xp_n(x)=p_{n+1}(x)+(A_n+C_n-a^2)p_n(x)+A_{n-1}C_np_{n-1}(x), +\end{equation} +where +$$W_n(x^2;a,b,c,d)=(-1)^n(n+a+b+c+d-1)_np_n(x^2).$$ + +\subsection*{Difference equation} +\begin{eqnarray} +\label{dvWilson} +& &n(n+a+b+c+d-1)y(x)\nonumber\\ +& &{}=B(x)y(x+i)-\left[B(x)+D(x)\right]y(x)+D(x)y(x-i), +\end{eqnarray} +where +$$y(x)=W_n(x^2;a,b,c,d)$$ +and +$$\left\{\begin{array}{l} +\displaystyle B(x)=\frac{(a-ix)(b-ix)(c-ix)(d-ix)}{2ix(2ix-1)}\\[5mm] +\displaystyle D(x)=\frac{(a+ix)(b+ix)(c+ix)(d+ix)}{2ix(2ix+1)}. +\end{array}\right.$$ + +\subsection*{Forward shift operator} +\begin{eqnarray} +\label{shift1WilsonI} +& &W_n((x+\textstyle\frac{1}{2}i)^2;a,b,c,d) +-W_n((x-\textstyle\frac{1}{2}i)^2;a,b,c,d)\nonumber\\ +& &{}=-2inx(n+a+b+c+d-1)W_{n-1}(x^2;a+\textstyle\frac{1}{2},b+\textstyle\frac{1}{2}, +c+\textstyle\frac{1}{2},d+\textstyle\frac{1}{2}) +\end{eqnarray} +or equivalently +\begin{eqnarray} +\label{shift1WilsonII} +& &\frac{\delta W_n(x^2;a,b,c,d)}{\delta x^2}\nonumber\\ +& &{}=-n(n+a+b+c+d-1)W_{n-1}(x^2;a+\textstyle\frac{1}{2},b+\textstyle\frac{1}{2}, +c+\textstyle\frac{1}{2},d+\textstyle\frac{1}{2}). +\end{eqnarray} + +\newpage + +\subsection*{Backward shift operator} +\begin{eqnarray} +\label{shift2WilsonI} +& &(a-\textstyle\frac{1}{2}-ix)(b-\textstyle\frac{1}{2}-ix) +(c-\textstyle\frac{1}{2}-ix)(d-\textstyle\frac{1}{2}-ix) +W_n((x+\textstyle\frac{1}{2}i)^2;a,b,c,d)\nonumber\\ +& &{}\mathindent{}-(a-\textstyle\frac{1}{2}+ix)(b-\textstyle\frac{1}{2}+ix) +(c-\textstyle\frac{1}{2}+ix)(d-\textstyle\frac{1}{2}+ix) +W_n((x-\textstyle\frac{1}{2}i)^2;a,b,c,d)\nonumber\\ +& &{}=-2ix W_{n+1}(x^2;a-\textstyle\frac{1}{2},b-\textstyle\frac{1}{2}, +c-\textstyle\frac{1}{2},d-\textstyle\frac{1}{2}) +\end{eqnarray} +or equivalently +\begin{eqnarray} +\label{shift2WilsonII} +& &\frac{\delta\left[\omega(x;a,b,c,d)W_n(x^2;a,b,c,d)\right]}{\delta x^2}\nonumber\\ +& &{}=\omega(x;a-\textstyle\frac{1}{2},b-\textstyle\frac{1}{2}, +c-\textstyle\frac{1}{2},d-\textstyle\frac{1}{2}) +W_{n+1}(x^2;a-\textstyle\frac{1}{2},b-\textstyle\frac{1}{2},c-\textstyle\frac{1}{2},d-\textstyle\frac{1}{2}), +\end{eqnarray} +where +$$\omega(x;a,b,c,d):=\frac{1}{2ix}\left|\frac{\Gamma(a+ix)\Gamma(b+ix)\Gamma(c+ix)\Gamma(d+ix)}{\Gamma(2ix)}\right|^2.$$ + +\subsection*{Rodrigues-type formula} +\begin{eqnarray} +\label{RodWilson} +& &\omega(x;a,b,c,d)W_n(x^2;a,b,c,d)\nonumber\\ +& &{}=\left(\frac{\delta}{\delta x^2}\right)^n +\left[\omega(x;a+\textstyle\frac{1}{2}n,b+\textstyle\frac{1}{2}n, +c+\textstyle\frac{1}{2}n,d+\textstyle\frac{1}{2}n)\right]. +\end{eqnarray} + +\subsection*{Generating functions} +\begin{equation} +\label{GenWilson1} +\hyp{2}{1}{a+ix,b+ix}{a+b}{t}\,\hyp{2}{1}{c-ix,d-ix}{c+d}{t}= +\sum_{n=0}^{\infty}\frac{W_n(x^2;a,b,c,d)t^n}{(a+b)_n(c+d)_nn!}. +\end{equation} + +\begin{equation} +\label{GenWilson2} +\hyp{2}{1}{a+ix,c+ix}{a+c}{t}\,\hyp{2}{1}{b-ix,d-ix}{b+d}{t}= +\sum_{n=0}^{\infty}\frac{W_n(x^2;a,b,c,d)t^n}{(a+c)_n(b+d)_nn!}. +\end{equation} + +\begin{equation} +\label{GenWilson3} +\hyp{2}{1}{a+ix,d+ix}{a+d}{t}\,\hyp{2}{1}{b-ix,c-ix}{b+c}{t}= +\sum_{n=0}^{\infty}\frac{W_n(x^2;a,b,c,d)t^n}{(a+d)_n(b+c)_nn!}. +\end{equation} + +\begin{eqnarray} +\label{GenWilson4} +& &(1-t)^{1-a-b-c-d}\nonumber\\ +& &{}\mathindent\times\hyp{4}{3}{\frac{1}{2}(a+b+c+d-1),\frac{1}{2}(a+b+c+d),a+ix,a-ix} +{a+b,a+c,a+d}{-\frac{4t}{(1-t)^2}}\nonumber\\ +& &{}=\sum_{n=0}^{\infty}\frac{(a+b+c+d-1)_n}{(a+b)_n(a+c)_n(a+d)_nn!}W_n(x^2;a,b,c,d)t^n. +\end{eqnarray} + +\subsection*{Limit relations} + +\subsubsection*{Wilson $\rightarrow$ Continuous dual Hahn} +The continuous dual Hahn polynomials given by (\ref{DefContDualHahn}) can be found from the +Wilson polynomials by dividing by $(a+d)_n$ and letting $d\rightarrow\infty$: +\begin{equation} +\lim_{d\rightarrow\infty}\frac{W_n(x^2;a,b,c,d)}{(a+d)_n}=S_n(x^2;a,b,c). +\end{equation} + +\subsubsection*{Wilson $\rightarrow$ Continuous Hahn} +The continuous Hahn polynomials given by (\ref{DefContHahn}) are obtained from the Wilson +polynomials by the substitutions $a\rightarrow a-it$, $b\rightarrow b-it$, $c\rightarrow c+it$, +$d\rightarrow d+it$ and $x\rightarrow x+t$ and the limit $t\rightarrow\infty$ in the following +way: +\begin{equation} +\lim_{t\rightarrow\infty} +\frac{W_n((x+t)^2;a-it,b-it,c+it,d+it)}{(-2t)^nn!}=p_n(x;a,b,c,d). +\end{equation} + +\subsubsection*{Wilson $\rightarrow$ Jacobi} +The Jacobi polynomials given by (\ref{DefJacobi}) can be found from the Wilson +polynomials by substituting $a=b=\frac{1}{2}(\alpha+1)$, $c=\frac{1}{2}(\beta+1)+it$, +$d=\frac{1}{2}(\beta+1)-it$ and $x\rightarrow t\sqrt{\frac{1}{2}(1-x)}$ in the +definition (\ref{DefWilson}) of the Wilson polynomials and taking the limit +$t\rightarrow\infty$. In fact we have +\begin{eqnarray} +& &\lim_{t\rightarrow\infty}\frac{W_n(\frac{1}{2}(1-x)t^2;\frac{1}{2}(\alpha+1), +\frac{1}{2}(\alpha+1),\frac{1}{2}(\beta+1)+it,\frac{1}{2}(\beta+1)-it)}{t^{2n}n!}\nonumber\\ +& &{}=P_n^{(\alpha,\beta)}(x). +\end{eqnarray} + +\subsection*{Remarks} +Note that for $k0$, $c=\bar{a}$ and $d=\bar{b}$, then +\begin{eqnarray} +\label{OrtContHahn} +& &\frac{1}{2\pi}\int_{-\infty}^{\infty}\Gamma(a+ix)\Gamma(b+ix)\Gamma(c-ix)\Gamma(d-ix) +p_m(x;a,b,c,d)p_n(x;a,b,c,d)\,dx\nonumber\\ +& &{}=\frac{\Gamma(n+a+c)\Gamma(n+a+d)\Gamma(n+b+c)\Gamma(n+b+d)} +{(2n+a+b+c+d-1)\Gamma(n+a+b+c+d-1)n!}\,\delta_{mn}. +\end{eqnarray} + +\subsection*{Recurrence relation} +\begin{equation} +\label{RecContHahn} +(a+ix)\tilde{p}_n(x)=A_n\tilde{p}_{n+1}(x)-\left(A_n+C_n\right)\tilde{p}_n(x)+C_n\tilde{p}_{n-1}(x), +\end{equation} +where +$$\tilde{p}_n(x):=\tilde{p}_n(x;a,b,c,d)=\frac{n!}{i^n(a+c)_n(a+d)_n}p_n(x;a,b,c,d)$$ +and +$$\left\{\begin{array}{l} +\displaystyle A_n=-\frac{(n+a+b+c+d-1)(n+a+c)(n+a+d)}{(2n+a+b+c+d-1)(2n+a+b+c+d)}\\[5mm] +\displaystyle C_n=\frac{n(n+b+c-1)(n+b+d-1)}{(2n+a+b+c+d-2)(2n+a+b+c+d-1)}. +\end{array}\right.$$ + +\subsection*{Normalized recurrence relation} +\begin{equation} +\label{NormRecContHahn} +xp_n(x)=p_{n+1}(x)+i(A_n+C_n+a)p_n(x)-A_{n-1}C_np_{n-1}(x), +\end{equation} +where +$$p_n(x;a,b,c,d)=\frac{(n+a+b+c+d-1)_n}{n!}p_n(x).$$ + +\subsection*{Difference equation} +\begin{eqnarray} +\label{dvContHahn} +& &n(n+a+b+c+d-1)y(x)\nonumber\\ +& &{}=B(x)y(x+i)-\left[B(x)+D(x)\right]y(x)+D(x)y(x-i), +\end{eqnarray} +where +$$y(x)=p_n(x;a,b,c,d)$$ +and +$$\left\{\begin{array}{l} +\displaystyle B(x)=(c-ix)(d-ix)\\[5mm] +\displaystyle D(x)=(a+ix)(b+ix). +\end{array}\right.$$ + +\subsection*{Forward shift operator} +\begin{eqnarray} +\label{shift1ContHahnI} +& &p_n(x+\textstyle\frac{1}{2}i;a,b,c,d)-p_n(x-\textstyle\frac{1}{2}i;a,b,c,d)\nonumber\\ +& &{}=i(n+a+b+c+d-1)p_{n-1}(x;a+\textstyle\frac{1}{2}, +b+\textstyle\frac{1}{2},c+\textstyle\frac{1}{2},d+\textstyle\frac{1}{2}) +\end{eqnarray} +or equivalently +\begin{equation} +\label{shift1ContHahnII} +\frac{\delta p_n(x;a,b,c,d)}{\delta x}=(n+a+b+c+d-1) +p_{n-1}(x;a+\textstyle\frac{1}{2},b+\textstyle\frac{1}{2}, +c+\textstyle\frac{1}{2},d+\textstyle\frac{1}{2}). +\end{equation} + +\subsection*{Backward shift operator} +\begin{eqnarray} +\label{shift2ContHahnI} +& &(c-\textstyle\frac{1}{2}-ix)(d-\textstyle\frac{1}{2}-ix) +p_n(x+\textstyle\frac{1}{2}i;a,b,c,d)\nonumber\\ +& &{}\mathindent{}-(a-\textstyle\frac{1}{2}+ix)(b-\textstyle\frac{1}{2}+ix) +p_n(x-\textstyle\frac{1}{2}i;a,b,c,d)\nonumber\\ +& &{}=\frac{n+1}{i}p_{n+1}(x;a-\textstyle\frac{1}{2}, +b-\textstyle\frac{1}{2},c-\textstyle\frac{1}{2},d-\textstyle\frac{1}{2}) +\end{eqnarray} +or equivalently +\begin{eqnarray} +\label{shift2ContHahnII} +& &\frac{\delta\left[\omega(x;a,b,c,d)p_n(x;a,b,c,d)\right]}{\delta x}\nonumber\\ +& &{}=-(n+1)\omega(x;a-\textstyle\frac{1}{2},b-\textstyle\frac{1}{2}, +c-\textstyle\frac{1}{2},d-\textstyle\frac{1}{2})\nonumber\\ +& &{}\mathindent\times p_{n+1}(x;a-\textstyle\frac{1}{2},b-\textstyle\frac{1}{2},c-\textstyle\frac{1}{2},d-\textstyle\frac{1}{2}), +\end{eqnarray} +where +$$\omega(x;a,b,c,d)=\Gamma(a+ix)\Gamma(b+ix)\Gamma(c-ix)\Gamma(d-ix).$$ + +\subsection*{Rodrigues-type formula} +\begin{eqnarray} +\label{RodContHahn} +& &\omega(x;a,b,c,d)p_n(x;a,b,c,d)\nonumber\\ +& &{}=\frac{(-1)^n}{n!}\left(\frac{\delta}{\delta x}\right)^n +\left[\omega(x;a+\textstyle\frac{1}{2}n,b+\textstyle\frac{1}{2}n, +c+\textstyle\frac{1}{2}n,d+\textstyle\frac{1}{2}n)\right]. +\end{eqnarray} + +\subsection*{Generating functions} +\begin{equation} +\label{GenContHahn1} +\hyp{1}{1}{a+ix}{a+c}{-it}\,\hyp{1}{1}{d-ix}{b+d}{it}= +\sum_{n=0}^{\infty}\frac{p_n(x;a,b,c,d)}{(a+c)_n(b+d)_n}t^n. +\end{equation} + +\begin{equation} +\label{GenContHahn2} +\hyp{1}{1}{a+ix}{a+d}{-it}\,\hyp{1}{1}{c-ix}{b+c}{it}= +\sum_{n=0}^{\infty}\frac{p_n(x;a,b,c,d)}{(a+d)_n(b+c)_n}t^n. +\end{equation} + +\begin{eqnarray} +\label{GenContHahn3} +& &(1-t)^{1-a-b-c-d}\,\hyp{3}{2}{\frac{1}{2}(a+b+c+d-1),\frac{1}{2}(a+b+c+d),a+ix} +{a+c,a+d}{-\frac{4t}{(1-t)^2}}\nonumber\\ +& &{}=\sum_{n=0}^{\infty} +\frac{(a+b+c+d-1)_n}{(a+c)_n(a+d)_ni^n}p_n(x;a,b,c,d)t^n. +\end{eqnarray} + +\subsection*{Limit relations} + +\subsubsection*{Wilson $\rightarrow$ Continuous Hahn} +The continuous Hahn polynomials are obtained from the Wilson polynomials given by +(\ref{DefWilson}) by the substitution $a\rightarrow a-it$, $b\rightarrow b-it$, +$c\rightarrow c+it$, $d\rightarrow d+it$ and $x\rightarrow x+t$ and the limit +$t\rightarrow\infty$ in the following way: +$$\lim_{t\rightarrow\infty} +\frac{W_n((x+t)^2;a-it,b-it,c+it,d+it)}{(-2t)^nn!}=p_n(x;a,b,c,d).$$ + +\subsubsection*{Continuous Hahn $\rightarrow$ Meixner-Pollaczek} +The Meixner-Pollaczek polynomials given by (\ref{DefMP}) can be obtained from the +continuous Hahn polynomials by setting $x\rightarrow x+t$, $a=\lambda-it$, $c=\lambda+it$ +and $b=d=t\tan\phi$ and taking the limit $t\rightarrow\infty$: +\begin{equation} +\lim_{t\rightarrow\infty}\frac{p_n(x+t;\lambda-it,t\tan\phi,\lambda+it,t\tan\phi)}{t^n} +=\frac{P_n^{(\lambda)}(x;\phi)}{(\cos\phi)^n}. +\end{equation} + +\subsubsection*{Continuous Hahn $\rightarrow$ Jacobi} +The Jacobi polynomials given by (\ref{DefJacobi}) follow from the continuous Hahn polynomials +by the substitution $x\rightarrow \frac{1}{2}xt$, $a=\frac{1}{2}(\alpha+1-it)$, +$b=\frac{1}{2}(\beta+1+it)$, $c=\frac{1}{2}(\alpha+1+it)$ and $d=\frac{1}{2}(\beta+1-it)$, +division by $t^n$ and the limit $t\rightarrow\infty$: +\begin{eqnarray} +& &\lim_{t\rightarrow\infty} +\frac{p_n(\frac{1}{2}xt;\frac{1}{2}(\alpha+1-it),\frac{1}{2}(\beta+1+it), +\frac{1}{2}(\alpha+1+it),\frac{1}{2}(\beta+1-it))}{t^n}\nonumber\\ +& &{}=P_n^{(\alpha,\beta)}(x). +\end{eqnarray} + +\subsubsection*{Continuous Hahn $\rightarrow$ Pseudo Jacobi} +The pseudo Jacobi polynomials given by (\ref{DefPseudoJacobi}) follow from the continuous +Hahn polynomials by the substitution $x\rightarrow xt$, $a=\frac{1}{2}(-N+i\nu-2t)$, +$b=\frac{1}{2}(-N-i\nu+2t)$, $c=\frac{1}{2}(-N-i\nu-2t)$ and $d=\frac{1}{2}(-N+i\nu+2t)$, +division by $t^n$ and the limit $t\rightarrow\infty$: +\begin{eqnarray} +& &\lim_{t\rightarrow\infty}\frac{p_n(xt;\frac{1}{2}(-N+i\nu-2t),\frac{1}{2}(-N-i\nu+2t), +\frac{1}{2}(-N+i\nu-2t),\frac{1}{2}(-N-i\nu+2t))}{t^n}\nonumber\\ +& &{}=\frac{(n-2N-1)_n}{n!}P_n(x;\nu,N). +\end{eqnarray} + +\subsection*{Remark} +Since we have for $k0$ and $(c,d)=(\overline a,\overline b)$ or $(\overline b,\overline a)$.\\ +Thus, under these assumptions, the continuous Hahn polynomial +$p_n(x;a,b,c,d)$ +is symmetric in $a,b$ and in $c,d$. +This follows from the orthogonality relation (9.4.2) +together with the value of its coefficient of $x^n$ given in (9.4.4b).\\ +As a consequence, it is sufficient to give generating function (9.4.11). Then the generating +function (9.4.12) will follow by symmetry in the parameters. +% +\paragraph{Uniqueness of orthogonality measure} +The coefficient of $p_{n-1}(x)$ in (9.4.4) behaves as $O(n^2)$ as $n\to\iy$. +Hence \eqref{93} holds, by which the orthogonality measure is unique. +% +\paragraph{Special cases} +In the following special case there is a reduction to +Meixner-Pollaczek: +\begin{equation} +p_n(x;a,a+\thalf,a,a+\thalf)= +\frac{(2a)_n (2a+\thalf)_n}{(4a)_n}\,P_n^{(2a)}(2x;\thalf\pi). +\end{equation} +See \myciteKLS{342}{(2.6)} (note that in \myciteKLS{342}{(2.3)} the +Meixner-Pollaczek polynonmials are defined different from (9.7.1), +without a constant factor in front). + +For $0-1$ and $\beta>-1$, or for $\alpha<-N$ and $\beta<-N$, we have +\begin{eqnarray} +\label{OrtHahn} +& &\sum_{x=0}^N\binom{\alpha +x}{x}\binom{\beta+N-x}{N-x}Q_m(x;\alpha,\beta,N)Q_n(x;\alpha,\beta,N)\nonumber\\ +& &{}=\frac{(-1)^n(n+\alpha+\beta+1)_{N+1}(\beta+1)_nn!}{(2n+\alpha+\beta+1)(\alpha+1)_n(-N)_nN!}\,\delta_{mn}. +\end{eqnarray} + +\subsection*{Recurrence relation} +\begin{equation} +\label{RecHahn} +-xQ_n(x)=A_nQ_{n+1}(x)-\left(A_n+C_n\right)Q_n(x)+C_nQ_{n-1}(x), +\end{equation} +where +$$Q_n(x):=Q_n(x;\alpha,\beta,N)$$ +and +$$\left\{\begin{array}{l} +\displaystyle A_n=\frac{(n+\alpha+\beta+1)(n+\alpha+1)(N-n)}{(2n+\alpha+\beta+1)(2n+\alpha+\beta+2)}\\[5mm] +\displaystyle C_n=\frac{n(n+\alpha+\beta+N+1)(n+\beta)}{(2n+\alpha+\beta)(2n+\alpha+\beta+1)}. +\end{array}\right.$$ + +\subsection*{Normalized recurrence relation} +\begin{equation} +\label{NormRecHahn} +xp_n(x)=p_{n+1}(x)+\left(A_n+C_n\right)p_n(x)+A_{n-1}C_np_{n-1}(x), +\end{equation} +where +$$Q_n(x;\alpha,\beta,N)=\frac{(n+\alpha+\beta+1)_n}{(\alpha+1)_n(-N)_n}p_n(x).$$ + +\subsection*{Difference equation} +\begin{equation} +\label{dvHahn} +n(n+\alpha+\beta+1)y(x)=B(x)y(x+1)-\left[B(x)+D(x)\right]y(x)+D(x)y(x-1), +\end{equation} +where +$$y(x)=Q_n(x;\alpha,\beta,N)$$ +and +$$\left\{\begin{array}{l} +\displaystyle B(x)=(x+\alpha+1)(x-N)\\[5mm] +\displaystyle D(x)=x(x-\beta-N-1). +\end{array}\right.$$ + +\subsection*{Forward shift operator} +\begin{eqnarray} +\label{shift1HahnI} +& &Q_n(x+1;\alpha,\beta,N)-Q_n(x;\alpha,\beta,N)\nonumber\\ +& &{}=-\frac{n(n+\alpha+\beta+1)}{(\alpha+1)N}Q_{n-1}(x;\alpha+1,\beta+1,N-1) +\end{eqnarray} +or equivalently +\begin{equation} +\label{shift1HahnII} +\Delta Q_n(x;\alpha,\beta,N)=-\frac{n(n+\alpha+\beta+1)}{(\alpha+1)N}Q_{n-1}(x;\alpha+1,\beta+1,N-1). +\end{equation} + +\subsection*{Backward shift operator} +\begin{eqnarray} +\label{shift2HahnI} +& &(x+\alpha)(N+1-x)Q_n(x;\alpha,\beta,N)-x(\beta+N+1-x)Q_n(x-1;\alpha,\beta,N)\nonumber\\ +& &{}=\alpha(N+1)Q_{n+1}(x;\alpha-1,\beta-1,N+1) +\end{eqnarray} +or equivalently +\begin{eqnarray} +\label{shift2HahnII} +& &\nabla\left[\omega(x;\alpha,\beta,N)Q_n(x;\alpha,\beta,N)\right]\nonumber\\ +& &{}=\frac{N+1}{\beta}\omega(x;\alpha-1,\beta-1,N+1)Q_{n+1}(x;\alpha-1,\beta-1,N+1), +\end{eqnarray} +where +$$\omega(x;\alpha,\beta,N)=\binom{\alpha+x}{x}\binom{\beta+N-x}{N-x}.$$ + +\subsection*{Rodrigues-type formula} +\begin{eqnarray} +\label{RodHahn} +& &\omega(x;\alpha,\beta,N)Q_n(x;\alpha,\beta,N)\nonumber\\ +& &{}=\frac{(-1)^n(\beta+1)_n}{(-N)_n}\nabla^n\left[\omega(x;\alpha+n,\beta+n,N-n)\right]. +\end{eqnarray} + +\subsection*{Generating functions} +For $x=0,1,2,\ldots,N$ we have +\begin{equation} +\label{GenHahn1} +\hyp{1}{1}{-x}{\alpha+1}{-t}\,\hyp{1}{1}{x-N}{\beta+1}{t} +=\sum_{n=0}^N\frac{(-N)_n}{(\beta+1)_nn!}Q_n(x;\alpha,\beta,N)t^n. +\end{equation} + +\begin{eqnarray} +\label{GenHahn2} +& &\hyp{2}{0}{-x,-x+\beta+N+1}{-}{-t}\,\hyp{2}{0}{x-N,x+\alpha+1}{-}{t}\nonumber\\ +& &{}=\sum_{n=0}^N\frac{(-N)_n(\alpha+1)_n}{n!}Q_n(x;\alpha,\beta,N)t^n. +\end{eqnarray} + +\begin{eqnarray} +\label{GenHahn3} +& &\left[(1-t)^{-\alpha-\beta-1}\,\hyp{3}{2}{\frac{1}{2}(\alpha+\beta+1),\frac{1}{2}(\alpha+\beta+2),-x} +{\alpha+1,-N}{-\frac{4t}{(1-t)^2}}\right]_N\nonumber\\ +& &{}=\sum_{n=0}^N\frac{(\alpha+\beta+1)_n}{n!}Q_n(x;\alpha,\beta,N)t^n. +\end{eqnarray} + +\subsection*{Limit relations} + +\subsubsection*{Racah $\rightarrow$ Hahn} +If we take $\gamma+1=-N$ and let $\delta\rightarrow\infty$ in the definition (\ref{DefRacah}) +of the Racah polynomials, we obtain the Hahn polynomials. Hence +$$\lim_{\delta\rightarrow\infty} +R_n(\lambda(x);\alpha,\beta,-N-1,\delta)=Q_n(x;\alpha,\beta,N).$$ +And if we take $\delta=-\beta-N-1$ and let $\gamma\rightarrow\infty$ in the definition (\ref{DefRacah}) +of the Racah polynomials, we also obtain the Hahn polynomials: +$$\lim_{\gamma\rightarrow\infty} +R_n(\lambda(x);\alpha,\beta,\gamma,-\beta-N-1)=Q_n(x;\alpha,\beta,N).$$ +Another way to do this is to take $\alpha+1=-N$ and $\beta\rightarrow\beta+\gamma+N+1$ in +the definition (\ref{DefRacah}) of the Racah polynomials and then take the limit +$\delta\rightarrow\infty$. In that case we obtain the Hahn polynomials in the following way: +$$\lim_{\delta\rightarrow\infty} +R_n(\lambda(x);-N-1,\beta+\gamma+N+1,\gamma,\delta)=Q_n(x;\gamma,\beta,N).$$ + +\subsubsection*{Hahn $\rightarrow$ Jacobi} +To find the Jacobi polynomials given by (\ref{DefJacobi}) from the Hahn polynomials we take +$x\rightarrow Nx$ and let $N\rightarrow\infty$. In fact we have +\begin{equation} +\lim_{N\rightarrow\infty} +Q_n(Nx;\alpha,\beta,N)=\frac{P_n^{(\alpha,\beta)}(1-2x)}{P_n^{(\alpha,\beta)}(1)}. +\end{equation} + +\subsubsection*{Hahn $\rightarrow$ Meixner} +The Meixner polynomials given by (\ref{DefMeixner}) can be obtained from the Hahn polynomials +by taking $\alpha=b-1$, $\beta=N(1-c)c^{-1}$ and letting $N\rightarrow\infty$: +\begin{equation} +\lim_{N\rightarrow\infty} +Q_n(x;b-1,N(1-c)c^{-1},N)=M_n(x;b,c). +\end{equation} + +\subsubsection*{Hahn $\rightarrow$ Krawtchouk} +The Krawtchouk polynomials given by (\ref{DefKrawtchouk}) are obtained from the Hahn polynomials +if we take $\alpha=pt$ and $\beta=(1-p)t$ and let $t\rightarrow\infty$: +\begin{equation} +\lim_{t\rightarrow\infty}Q_n(x;pt,(1-p)t,N)=K_n(x;p,N). +\end{equation} + +\subsection*{Remark} +If we interchange the role of $x$ and $n$ in (\ref{DefHahn}) we obtain the +dual Hahn polynomials given by (\ref{DefDualHahn}). + +\noindent +Since +$$Q_n(x;\alpha,\beta,N)=R_x(\lambda(n);\alpha,\beta,N)$$ +we obtain the dual orthogonality relation for the Hahn polynomials from the +orthogonality relation (\ref{OrtDualHahn}) of the dual Hahn polynomials: +\begin{eqnarray*} +& &\sum_{n=0}^N\frac{(2n+\alpha+\beta+1)(\alpha+1)_n(-N)_nN!}{(-1)^n(n+\alpha+\beta+1)_{N+1}(\beta+1)_nn!} +Q_n(x;\alpha,\beta,N)Q_n(y;\alpha,\beta,N)\\ +& &{}=\frac{\delta_{xy}}{\dbinom{\alpha+x}{x}\dbinom{\beta+N-x}{N-x}},\quad x,y \in \{0,1,2,\ldots,N\}. +\subsection*{9.5 Hahn} +\label{sec9.5} +% +\paragraph{Special values} +\begin{equation} +Q_n(0;\al,\be,N)=1,\quad +Q_n(N;\al,\be,N)=\frac{(-1)^n(\be+1)_n}{(\al+1)_n}\,. +\label{95} +\end{equation} +Use (9.5.1) and compare with (9.8.1) and \eqref{50}. + +From (9.5.3) and \eqref{1} it follows that +\begin{equation} +Q_{2n}(N;\al,\al,2N)=\frac{(\thalf)_n(N+\al+1)_n}{(-N+\thalf)_n(\al+1)_n}\,. +\label{30} +\end{equation} +From (9.5.1) and \mycite{DLMF}{(15.4.24)} it follows that +\begin{equation} +Q_N(x;\al,\be,N)=\frac{(-N-\be)_x}{(\al+1)_x}\qquad(x=0,1,\ldots,N). +\label{44} +\end{equation} +% +\paragraph{Symmetries} +By the orthogonality relation (9.5.2): +\begin{equation} +\frac{Q_n(N-x;\al,\be,N)}{Q_n(N;\al,\be,N)}=Q_n(x;\be,\al,N), +\label{96} +\end{equation} +It follows from \eqref{97} and \eqref{45} that +\begin{equation} +\frac{Q_{N-n}(x;\al,\be,N)}{Q_N(x;\al,\be,N)} +=Q_n(x;-N-\be-1,-N-\al-1,N) +\qquad(x=0,1,\ldots,N). +\label{100} +\end{equation} +% +\paragraph{Duality} +The Remark on p.208 gives the duality between Hahn and dual Hahn polynomials: +% +\begin{equation} +Q_n(x;\al,\be,N)=R_x(n(n+\al+\be+1);\al,\be,N)\quad(n,x\in\{0,1,\ldots N\}). +\label{45} +\end{equation} +\end{eqnarray*} + +\subsection*{References} +\cite{AlSalam90}, \cite{AndrewsAskey85}, \cite{Area+II}, \cite{Askey75}, +\cite{Askey89I}, \cite{Askey2005}, \cite{AskeyGasper77}, \cite{AskeyWilson85}, +\cite{AtakRahmanSuslov}, \cite{AtakSuslov88}, \cite{Chihara78}, +\cite{Ciesielski}, \cite{Cooper+}, \cite{Dette95}, \cite{Dunkl76}, +\cite{Dunkl78I}, \cite{Gasper73I}, \cite{Gasper74}, \cite{HoareRahman}, +\cite{Ismail77}, \cite{Ismail2005II}, \cite{Karlin61}, \cite{Koorn81}, \cite{Koorn88}, +\cite{LabelleYehI}, \cite{LabelleYehII}, \cite{Laine}, \cite{Lesky62}, +\cite{Lesky88}, \cite{Lesky89}, \cite{Lesky94I}, \cite{Lesky95II}, +\cite{LewanowiczII}, \cite{Neuman}, \cite{Nikiforov+}, \cite{NikiforovUvarov}, +\cite{Rahman76III}, \cite{Rahman78I}, \cite{Rahman78II}, \cite{Rahman81III}, +\cite{Sablonniere}, \cite{Stanton84}, \cite{Stanton90}, \cite{Wilson80}, \cite{Wilson70II}, +\cite{Zarzo+}. + + +\section{Dual Hahn}\index{Dual Hahn polynomials}\index{Hahn polynomials!Dual} + +\par\setcounter{equation}{0} + +\subsection*{Hypergeometric representation} +\begin{equation} +\label{DefDualHahn} +R_n(\lambda(x);\gamma,\delta,N)= +\hyp{3}{2}{-n,-x,x+\gamma+\delta+1}{\gamma+1,-N}{1},\quad n=0,1,2,\ldots,N, +\end{equation} +where +$$\lambda(x)=x(x+\gamma+\delta+1).$$ + +\subsection*{Orthogonality relation} +For $\gamma>-1$ and $\delta>-1$, or for $\gamma<-N$ and $\delta<-N$, we have +\begin{eqnarray} +\label{OrtDualHahn} +& &\sum_{x=0}^N\frac{(2x+\gamma+\delta+1)(\gamma+1)_x(-N)_xN!}{(-1)^x(x+\gamma+\delta+1)_{N+1}(\delta+1)_xx!} +R_m(\lambda(x);\gamma,\delta,N)R_n(\lambda(x);\gamma,\delta,N)\nonumber\\ +& &{}=\frac{\,\delta_{mn}}{\dbinom{\gamma+n}{n}\dbinom{\delta+N-n}{N-n}}. +\end{eqnarray} + +\subsection*{Recurrence relation} +\begin{equation} +\label{RecDualHahn} +\lambda(x)R_n(\lambda(x)) +=A_nR_{n+1}(\lambda(x))-\left(A_n+C_n\right)R_n(\lambda(x))+C_nR_{n-1}(\lambda(x)), +\end{equation} +where +$$R_n(\lambda(x)):=R_n(\lambda(x);\gamma,\delta,N)$$ +and +$$\left\{\begin{array}{l} +\displaystyle A_n=(n+\gamma+1)(n-N)\\[5mm] +\displaystyle C_n=n(n-\delta-N-1). +\end{array}\right.$$ + +\subsection*{Normalized recurrence relation} +\begin{equation} +\label{NormRecDualHahn} +xp_n(x)=p_{n+1}(x)-(A_n+C_n)p_n(x)+A_{n-1}C_np_{n-1}(x), +\end{equation} +where +$$R_n(\lambda(x);\gamma,\delta,N)=\frac{1}{(\gamma+1)_n(-N)_n}p_n(\lambda(x)).$$ + +\subsection*{Difference equation} +\begin{equation} +\label{dvDualHahn} +-ny(x)=B(x)y(x+1)-\left[B(x)+D(x)\right]y(x)+D(x)y(x-1), +\end{equation} +where +$$y(x)=R_n(\lambda(x);\gamma,\delta,N)$$ +and +$$\left\{\begin{array}{l} +\displaystyle B(x)=\frac{(x+\gamma+1)(x+\gamma+\delta+1)(N-x)}{(2x+\gamma+\delta+1)(2x+\gamma+\delta+2)}\\[5mm] +\displaystyle D(x)=\frac{x(x+\gamma+\delta+N+1)(x+\delta)}{(2x+\gamma+\delta)(2x+\gamma+\delta+1)}. +\end{array}\right.$$ + +\subsection*{Forward shift operator} +\begin{eqnarray} +\label{shift1DualHahnI} +& &R_n(\lambda(x+1);\gamma,\delta,N)-R_n(\lambda(x);\gamma,\delta,N)\nonumber\\ +& &{}=-\frac{n(2x+\gamma+\delta+2)}{(\gamma+1)N}R_{n-1}(\lambda(x);\gamma+1,\delta,N-1) +\end{eqnarray} +or equivalently +\begin{equation} +\label{shift1DualHahnII} +\frac{\Delta R_n(\lambda(x);\gamma,\delta,N)}{\Delta\lambda(x)}= +-\frac{n}{(\gamma+1)N}R_{n-1}(\lambda(x);\gamma+1,\delta,N-1). +\end{equation} + +\subsection*{Backward shift operator} +\begin{eqnarray} +\label{shift2DualHahnI} +& &(x+\gamma)(x+\gamma+\delta)(N+1-x)R_n(\lambda(x);\gamma,\delta,N)\nonumber\\ +& &{}\mathindent{}-x(x+\gamma+\delta+N+1)(x+\delta)R_n(\lambda(x-1);\gamma,\delta,N)\nonumber\\ +& &{}=\gamma(N+1)(2x+\gamma+\delta)R_{n+1}(\lambda(x);\gamma-1,\delta,N+1) +\end{eqnarray} +or equivalently +\begin{eqnarray} +\label{shift2DualHahnII} +& &\frac{\nabla\left[\omega(x;\gamma,\delta,N)R_n(\lambda(x);\gamma,\delta,N)\right]}{\nabla\lambda(x)}\nonumber\\ +& &{}=\frac{1}{\gamma+\delta}\omega(x;\gamma-1,\delta,N+1)R_{n+1}(\lambda(x);\gamma-1,\delta,N+1), +\end{eqnarray} +where +$$\omega(x;\gamma,\delta,N)=\frac{(-1)^x(\gamma+1)_x(\gamma+\delta+1)_x(-N)_x} +{(\gamma+\delta+N+2)_x(\delta+1)_xx!}.$$ + +\newpage + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodDualHahn} +\omega(x;\gamma,\delta,N)R_n(\lambda(x);\gamma,\delta,N) +=(\gamma+\delta+1)_n\left(\nabla_{\lambda}\right)^n\left[\omega(x;\gamma+n,\delta,N-n)\right], +\end{equation} +where +$$\nabla_{\lambda}:=\frac{\nabla}{\nabla\lambda(x)}.$$ + +\subsection*{Generating functions} +For $x=0,1,2,\ldots,N$ we have +\begin{equation} +\label{GenDualHahn1} +(1-t)^{N-x}\,\hyp{2}{1}{-x,-x-\delta}{\gamma+1}{t} +=\sum_{n=0}^N\frac{(-N)_n}{n!}R_n(\lambda(x);\gamma,\delta,N)t^n. +\end{equation} + +\begin{eqnarray} +\label{GenDualHahn2} +& &(1-t)^x\,\hyp{2}{1}{x-N,x+\gamma+1}{-\delta-N}{t}\nonumber\\ +& &=\sum_{n=0}^N\frac{(\gamma+1)_n(-N)_n}{(-\delta-N)_nn!}R_n(\lambda(x);\gamma,\delta,N)t^n. +\end{eqnarray} + +\begin{equation} +\label{GenDualHahn3} +\left[\expe^t\,\hyp{2}{2}{-x,x+\gamma+\delta+1}{\gamma+1,-N}{-t}\right]_N=\sum_{n=0}^N +\frac{R_n(\lambda(x);\gamma,\delta,N)}{n!}t^n. +\end{equation} + +\begin{eqnarray} +\label{GenDualHahn4} +& &\left[(1-t)^{-\epsilon}\,\hyp{3}{2}{\epsilon,-x,x+\gamma+\delta+1}{\gamma+1,-N}{\frac{t}{t-1}}\right]_N\nonumber\\ +& &{}=\sum_{n=0}^N\frac{(\epsilon)_n}{n!}R_n(\lambda(x);\gamma,\delta,N)t^n, +\quad\textrm{$\epsilon$ arbitrary}. +\end{eqnarray} + +\subsection*{Limit relations} + +\subsubsection*{Racah $\rightarrow$ Dual Hahn} +If we take $\alpha+1=-N$ and let $\beta\rightarrow\infty$ in the definition (\ref{DefRacah}) +of the Racah polynomials, then we obtain the dual Hahn polynomials: +$$\lim_{\beta\rightarrow\infty} +R_n(\lambda(x);-N-1,\beta,\gamma,\delta)=R_n(\lambda(x);\gamma,\delta,N).$$ +And if we take $\beta=-\delta-N-1$ and let $\alpha\rightarrow\infty$ in the definition (\ref{DefRacah}) +of the Racah polynomials, then we also obtain the dual Hahn polynomials: +$$\lim_{\alpha\rightarrow\infty} +R_n(\lambda(x);\alpha,-\delta-N-1,\gamma,\delta)=R_n(\lambda(x);\gamma,\delta,N).$$ +Finally, if we take $\gamma+1=-N$ and $\delta\rightarrow\alpha+\delta+N+1$ in the definition +(\ref{DefRacah}) of the Racah polynomials and take the limit +$\beta\rightarrow\infty$ we find the dual Hahn polynomials in the following way: +$$\lim_{\beta\rightarrow\infty} +R_n(\lambda(x);\alpha,\beta,-N-1,\alpha+\delta+N+1)=R_n(\lambda(x);\alpha,\delta,N).$$ + +\subsubsection*{Dual Hahn $\rightarrow$ Meixner} +The Meixner polynomials given by (\ref{DefMeixner}) are obtained from the dual Hahn polynomials +if we take $\gamma=\beta-1$ and $\delta=N(1-c)c^{-1}$ and let $N\rightarrow\infty$: +\begin{equation} +\lim_{N\rightarrow\infty} +R_n(\lambda(x);\beta-1,N(1-c)c^{-1},N)=M_n(x;\beta,c). +\end{equation} + +\subsubsection*{Dual Hahn $\rightarrow$ Krawtchouk} +The Krawtchouk polynomials given by (\ref{DefKrawtchouk}) can be obtained from the dual +Hahn polynomials by setting $\gamma=pt$, $\delta=(1-p)t$ and letting $t\rightarrow\infty$: +\begin{equation} +\lim_{t\rightarrow\infty}R_n(\lambda(x);pt,(1-p)t,N)=K_n(x;p,N). +\end{equation} + +\subsection*{Remark} +If we interchange the role of $x$ and $n$ in the definition (\ref{DefDualHahn}) +of the dual Hahn polynomials we obtain the Hahn polynomials given by (\ref{DefHahn}). + +\noindent +Since +$$R_n(\lambda(x);\gamma,\delta,N)=Q_x(n;\gamma,\delta,N)$$ +we obtain the dual orthogonality relation for the dual Hahn polynomials +from the orthogonality relation (\ref{OrtHahn}) for the Hahn polynomials: +\begin{eqnarray*} +& &\sum_{n=0}^N\binom{\gamma+n}{n}\binom{\delta+N-n}{N-n} R_n(\lambda(x);\gamma,\delta,N)R_n(\lambda(y);\gamma,\delta,N)\\ +& &{}=\frac{(-1)^x(x+\gamma+\delta+1)_{N+1}(\delta+1)_xx!} +{(2x+\gamma+\delta+1)(\gamma+1)_x(-N)_xN!}\delta_{xy},\quad x,y \in \{0,1,2,\ldots,N\}. +\subsection*{9.6 Dual Hahn} +\label{sec9.6} +% +\paragraph{Special values} +By \eqref{44} and \eqref{45} we have +\begin{equation} +R_n(N(N+\ga+\de+1);\ga,\de,N)=\frac{(-N-\de)_n}{(\ga+1)_n}\,. +\label{47} +\end{equation} +It follows from \eqref{95} and \eqref{45} that +\begin{equation} +R_N(x(x+\ga+\de+1);\ga,\de,N) +=\frac{(-1)^x(\de+1)_x}{(\ga+1)_x}\qquad(x=0,1,\ldots,N). +\label{101} +\end{equation} +% +\paragraph{Symmetries} +Write the weight in (9.6.2) as +\begin{equation} +w_x(\al,\be,N):=N!\,\frac{2x+\ga+\de+1}{(x+\ga+\de+1)_{N+1}}\, +\frac{(\ga+1)_x}{(\de+1)_x}\,\binom Nx. +\label{98} +\end{equation} +Then +\begin{equation} +(\de+1)_N\,w_{N-x}(\ga,\de,N)= +(-\ga-N)_N\,w_x(-\de-N-1,-\ga-N-1,N). +\label{99} +\end{equation} +Hence, by (9.6.2), +\begin{equation} +\frac{R_n((N-x)(N-x+\ga+\de+1);\ga,\de,N)}{R_n(N(N+\ga+\de+1);\ga,\de,N)} +=R_n(x(x-2N-\ga-\de-1);-N-\de-1,-N-\ga-1,N). +\label{97} +\end{equation} +Alternatively, \eqref{97} follows from (9.6.1) and +\mycite{DLMF}{(16.4.11)}. + +It follows from \eqref{96} and \eqref{45} that +\begin{equation} +\frac{R_{N-n}(x(x+\ga+\de+1);\ga,\de,N)} +{R_N(x(x+\ga+\de+1);\ga,\de,N)} +=R_n(x(x+\ga+\de+1);\de,\ga,N)\qquad(x=0,1,\ldots,N). +\label{102} +\end{equation} +% +\paragraph{Re: (9.6.11).} +The generating function (9.6.11) can be written in a more conceptual way as +\begin{equation} +(1-t)^x\,\hyp21{x-N,x+\ga+1}{-\de-N}t=\frac{N!}{(\de+1)_N}\, +\sum_{n=0}^N \om_n\,R_n(\la(x);\ga,\de,N)\,t^n, +\label{2} +\end{equation} +where +\begin{equation} +\om_n:=\binom{\ga+n}n \binom{\de+N-n}{N-n}, +\label{3} +\end{equation} +i.e., the denominator on the \RHS\ of (9.6.2). +By the duality between Hahn polynomials and dual Hahn polynomials (see \eqref{45}) the above generating function can be rewritten in +terms of Hahn polynomials: +\begin{equation} +(1-t)^n\,\hyp21{n-N,n+\al+1}{-\be-N}t=\frac{N!}{(\be+1)_N}\, +\sum_{x=0}^N w_x\,Q_n(x;\al,\be,N)\,t^x, +\label{4} +\end{equation} +where +\begin{equation} +w_x:=\binom{\al+x}x \binom{\be+N-x}{N-x}, +\label{5} +\end{equation} +i.e., the weight occurring in the orthogonality relation (9.5.2) +for Hahn polynomials. +\paragraph{Re: (9.6.15).} +There should be a closing bracket before the equality sign. +\end{eqnarray*} + +\subsection*{References} +\cite{Askey2005}, \cite{AskeyWilson85}, \cite{AtakRahmanSuslov}, \cite{AtakSuslov88}, +\cite{Ismail2005II}, \cite{Karlin61}, \cite{Koorn81}, \cite{Koorn88}, \cite{Lesky93}, +\cite{Lesky94I}, \cite{Lesky95I}, \cite{Lesky95II}, \cite{Nikiforov+}, \cite{NikiforovUvarov}, +\cite{Rahman81II}, \cite{Stanton84}, \cite{Wilson80}. + + +\section{Meixner-Pollaczek}\index{Meixner-Pollaczek polynomials} + +\par\setcounter{equation}{0} + +\subsection*{Hypergeometric representation} +\begin{equation} +\label{DefMP} +P_n^{(\lambda)}(x;\phi)=\frac{(2\lambda)_n}{n!}\expe^{in\phi}\, +\hyp{2}{1}{-n,\lambda+ix}{2\lambda}{1-\expe^{-2i\phi}}. +\end{equation} + +\subsection*{Orthogonality relation} +\begin{equation} +\label{OrtMP} +\frac{1}{2\pi}\int_{-\infty}^{\infty} +\e^{(2\phi-\pi)x}\left|\Gamma(\lambda+ix)\right|^2 +P_m^{(\lambda)}(x;\phi)P_n^{(\lambda)}(x;\phi)\,dx +{}=\frac{\Gamma(n+2\lambda)}{(2\sin\phi)^{2\lambda}n!}\,\delta_{mn}, +% \constraint{ +% $\lambda > 0$ & +% $0 < \phi < \pi$ } +\end{equation} + +\subsection*{Recurrence relation} +\begin{eqnarray} +\label{RecMP} +& &(n+1)P_{n+1}^{(\lambda)}(x;\phi)-2\left[x\sin\phi+(n+\lambda)\cos\phi\right] +P_n^{(\lambda)}(x;\phi)\nonumber\\ +& &{}\mathindent{}+(n+2\lambda-1)P_{n-1}^{(\lambda)}(x;\phi)=0. +\end{eqnarray} + +\subsection*{Normalized recurrence relation} +\begin{equation} +\label{NormRecMP} +xp_n(x)=p_{n+1}(x)-\left(\frac{n+\lambda}{\tan\phi}\right)p_n(x)+ +\frac{n(n+2\lambda-1)}{4\sin^2\phi}p_{n-1}(x), +\end{equation} +where +$$P_n^{(\lambda)}(x;\phi)=\frac{(2\sin\phi)^n}{n!}p_n(x).$$ + +\subsection*{Difference equation} +\begin{eqnarray} +\label{dvMP} +& &\e^{i\phi}(\lambda-ix)y(x+i)+2i\left[x\cos\phi-(n+\lambda)\sin\phi\right]y(x)\nonumber\\ +& &{}\mathindent{}-\e^{-i\phi}(\lambda+ix)y(x-i)=0,\quad y(x)=P_n^{(\lambda)}(x;\phi). +\end{eqnarray} + +\subsection*{Forward shift operator} +\begin{equation} +\label{shift1MPI} +P_n^{(\lambda)}(x+\textstyle\frac{1}{2}i;\phi)- +P_n^{(\lambda)}(x-\textstyle\frac{1}{2}i;\phi)=(\expe^{i\phi}-\expe^{-i\phi}) +P_{n-1}^{(\lambda+\frac{1}{2})}(x;\phi) +\end{equation} +or equivalently +\begin{equation} +\label{shift1MPII} +\frac{\delta P_n^{(\lambda)}(x;\phi)}{\delta x} +=2\sin\phi\,P_{n-1}^{(\lambda+\frac{1}{2})}(x;\phi). +\end{equation} + +\subsection*{Backward shift operator} +\begin{eqnarray} +\label{shift2MPI} +& &\e^{i\phi}(\lambda-\textstyle\frac{1}{2}-ix) +P_n^{(\lambda)}(x+\textstyle\frac{1}{2}i;\phi)+ +\e^{-i\phi}(\lambda-\textstyle\frac{1}{2}+ix) +P_n^{(\lambda)}(x-\textstyle\frac{1}{2}i;\phi)\nonumber\\ +& &{}=(n+1)P_{n+1}^{(\lambda-\frac{1}{2})}(x;\phi) +\end{eqnarray} +or equivalently +\begin{equation} +\label{shift2MPII} +\frac{\delta\left[\omega(x;\lambda,\phi)P_n^{(\lambda)}(x;\phi)\right]}{\delta x}= +-(n+1)\omega(x;\lambda-\textstyle\frac{1}{2},\phi) +P_{n+1}^{(\lambda-\frac{1}{2})}(x;\phi), +\end{equation} +where +$$\omega(x;\lambda,\phi)=\Gamma(\lambda+ix)\Gamma(\lambda-ix)\e^{(2\phi-\pi)x}.$$ + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodMP} +\omega(x;\lambda,\phi)P_n^{(\lambda)}(x;\phi)=\frac{(-1)^n}{n!} +\left(\frac{\delta}{\delta x}\right)^n\left[\omega(x;\lambda+\textstyle\frac{1}{2}n,\phi)\right]. +\end{equation} + +\newpage + +\subsection*{Generating functions} +\begin{equation} +\label{GenMP1} +(1-\expe^{i\phi}t)^{-\lambda+ix}(1-\expe^{-i\phi}t)^{-\lambda-ix}= +\sum_{n=0}^{\infty}P_n^{(\lambda)}(x;\phi)t^n. +\end{equation} + +\begin{equation} +\label{GenMP2} +\e^t\,\hyp{1}{1}{\lambda+ix}{2\lambda}{(\expe^{-2i\phi}-1)t}= +\sum_{n=0}^{\infty}\frac{P_n^{(\lambda)}(x;\phi)}{(2\lambda)_n\expe^{in\phi}}t^n. +\end{equation} + +\begin{eqnarray} +\label{GenMP3} +& &(1-t)^{-\gamma}\,\hyp{2}{1}{\gamma,\lambda+ix}{2\lambda}{\frac{(1-\e^{-2i\phi})t}{t-1}}\nonumber\\ +& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n}{(2\lambda)_n}\frac{P_n^{(\lambda)}(x;\phi)}{\e^{in\phi}}t^n, +\quad\textrm{$\gamma$ arbitrary}. +\end{eqnarray} + +\subsection*{Limit relations} + +\subsubsection*{Continuous dual Hahn $\rightarrow$ Meixner-Pollaczek} +The Meixner-Pollaczek polynomials can be obtained from the continuous dual Hahn polynomials +given by (\ref{DefContDualHahn}) by the substitutions $x\rightarrow x-t$, $a=\lambda+it$, +$b=\lambda-it$ and $c=t\cot\phi$ and the limit $t\rightarrow\infty$: +$$\lim_{t\rightarrow\infty}\frac{S_n((x-t)^2;\lambda+it,\lambda-it,t\cot\phi)}{t^nn!} +=\frac{P_n^{(\lambda)}(x;\phi)}{(\sin\phi)^n}.$$ + +\subsubsection*{Continuous Hahn $\rightarrow$ Meixner-Pollaczek} +By setting $x\rightarrow x+t$, $a=\lambda-it$, $c=\lambda+it$ and $b=d=t\tan\phi$ in the +definition (\ref{DefContHahn}) of the continuous Hahn polynomials and taking the limit +$t\rightarrow\infty$ we obtain the Meixner-Pollaczek polynomials: +$$\lim_{t\rightarrow\infty}\frac{p_n(x+t;\lambda-it,t\tan\phi,\lambda+it,t\tan\phi)}{t^nn!} +=\frac{P_n^{(\lambda)}(x;\phi)}{(\cos\phi)^n}.$$ + +\newpage + +\subsubsection*{Meixner-Pollaczek $\rightarrow$ Laguerre} +The Laguerre polynomials given by (\ref{DefLaguerre}) can be obtained from the +Meixner-Pollaczek polynomials by the substitution $\lambda=\frac{1}{2}(\alpha+1)$, +$x\rightarrow -\frac{1}{2}\phi^{-1}x$ and the limit $\phi\rightarrow 0$: +\begin{equation} +\lim_{\phi\rightarrow 0} +P_n^{(\frac{1}{2}\alpha+\frac{1}{2})}(-\textstyle\frac{1}{2}\phi^{-1}x;\phi)=L_n^{(\alpha)}(x). +\end{equation} + +\subsubsection*{Meixner-Pollaczek $\rightarrow$ Hermite} +The Hermite polynomials given by (\ref{DefHermite}) are obtained from the Meixner-Pollaczek +polynomials if we substitute $x\rightarrow (\sin\phi)^{-1}(x\sqrt{\lambda}-\lambda\cos\phi)$ +and then let $\lambda\rightarrow\infty$: +\begin{equation} +\lim_{\lambda\rightarrow\infty} +\lambda^{-\frac{1}{2}n}P_n^{(\lambda)} +((\sin\phi)^{-1}(x\sqrt{\lambda}-\lambda\cos\phi);\phi)=\frac{H_n(x)}{n!}. +\end{equation} + +\subsection*{Remark} +Since we have for $k-1$ and $\beta>-1$ we have +\begin{eqnarray} +\label{OrtJacobi1} +& &\int_{-1}^1(1-x)^{\alpha}(1+x)^{\beta}P_m^{(\alpha,\beta)}(x)P_n^{(\alpha,\beta)}(x)\,dx\nonumber\\ +& &{}=\frac{2^{\alpha+\beta+1}}{2n+\alpha+\beta+1}\frac{\Gamma(n+\alpha+1)\Gamma(n+\beta+1)}{\Gamma(n+\alpha+\beta+1)n!}\,\delta_{mn}. +\end{eqnarray} +For $\alpha+\beta<-2N-1$, $\beta>-1$ and $m,n\in\{0,1,2,\ldots,N\}$ we also have +\begin{eqnarray} +\label{OrtJacobi2} +& &\int_1^{\infty}(x+1)^{\alpha}(x-1)^{\beta}P_m^{(\alpha,\beta)}(-x)P_n^{(\alpha,\beta)}(-x)\,dx\nonumber\\ +& &{}=-\frac{2^{\alpha+\beta+1}}{2n+\alpha+\beta+1}\frac{\Gamma(-n-\alpha-\beta)\Gamma(n+\alpha+\beta+1)}{\Gamma(-n-\alpha)n!}\,\delta_{mn}. +\end{eqnarray} + +\subsection*{Recurrence relation} +\begin{eqnarray} +\label{RecJacobi} +xP_n^{(\alpha,\beta)}(x)&=&\frac{2(n+1)(n+\alpha+\beta+1)}{(2n+\alpha+\beta+1)(2n+\alpha+\beta+2)}P_{n+1}^{(\alpha,\beta)}(x)\nonumber\\ +& &{}\mathindent{}+\frac{\beta^2-\alpha^2}{(2n+\alpha+\beta)(2n+\alpha+\beta+2)}P_n^{(\alpha,\beta)}(x)\nonumber\\ +& &{}\mathindent\mathindent{}+\frac{2(n+\alpha)(n+\beta)}{(2n+\alpha+\beta)(2n+\alpha+\beta+1)}P_{n-1}^{(\alpha,\beta)}(x). +\end{eqnarray} + +\subsection*{Normalized recurrence relation} +\begin{eqnarray} +\label{NormRecJacobi} +xp_n(x)&=&p_{n+1}(x)+\frac{\beta^2-\alpha^2}{(2n+\alpha+\beta)(2n+\alpha+\beta+2)}p_n(x)\nonumber\\ +& &{}\mathindent{}+\frac{4n(n+\alpha)(n+\beta)(n+\alpha+\beta)} +{(2n+\alpha+\beta-1)(2n+\alpha+\beta)^2(2n+\alpha+\beta+1)}p_{n-1}(x) +\end{eqnarray} +where +$$P_n^{(\alpha,\beta)}(x)=\frac{(n+\alpha+\beta+1)_n}{2^nn!}p_n(x).$$ + +\subsection*{Differential equation} +\begin{eqnarray} +\label{dvJacobi} +& &(1-x^2)y''(x)+\left[\beta-\alpha-(\alpha+\beta+2)x\right]y'(x)\nonumber\\ +& &{}\mathindent{}+n(n+\alpha+\beta+1)y(x)=0,\quad y(x)=P_n^{(\alpha,\beta)}(x). +\end{eqnarray} + +\subsection*{Forward shift operator} +\begin{equation} +\label{shift1Jacobi} +\frac{d}{dx}P_n^{(\alpha,\beta)}(x)=\frac{n+\alpha+\beta+1}{2}P_{n-1}^{(\alpha+1,\beta+1)}(x). +\end{equation} + +\subsection*{Backward shift operator} +\begin{eqnarray} +\label{shift2JacobiI} +& &(1-x^2)\frac{d}{dx}P_n^{(\alpha,\beta)}(x)+ +\left[(\beta-\alpha)-(\alpha+\beta)x\right]P_n^{(\alpha,\beta)}(x)\nonumber\\ +& &{}=-2(n+1)P_{n+1}^{(\alpha-1,\beta-1)}(x) +\end{eqnarray} +or equivalently +\begin{eqnarray} +\label{shift2JacobiII} +& &\frac{d}{dx}\left[(1-x)^\alpha(1+x)^\beta P_n^{(\alpha,\beta)}(x)\right]\nonumber\\ +& &{}=-2(n+1)(1-x)^{\alpha-1}(1+x)^{\beta-1}P_{n+1}^{(\alpha-1,\beta-1)}(x). +\end{eqnarray} + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodJacobi} +(1-x)^{\alpha}(1+x)^{\beta}P_n^{(\alpha,\beta)}(x)= +\frac{(-1)^n}{2^nn!}\left(\frac{d}{dx}\right)^n +\left[(1-x)^{n+\alpha}(1+x)^{n+\beta}\right]. +\end{equation} + +\subsection*{Generating functions} +\begin{equation} +\label{GenJacobi1} +\frac{2^{\alpha+\beta}}{R(1+R-t)^{\alpha}(1+R+t)^{\beta}}= +\sum_{n=0}^{\infty}P_n^{(\alpha,\beta)}(x)t^n,\quad R=\sqrt{1-2xt+t^2}. +\end{equation} + +\begin{eqnarray} +\label{GenJacobi2} +& &\hyp{0}{1}{-}{\alpha+1}{\frac{(x-1)t}{2}}\,\hyp{0}{1}{-}{\beta+1}{\frac{(x+1)t}{2}}\nonumber\\ +& &{}=\sum_{n=0}^{\infty}\frac{P_n^{(\alpha,\beta)}(x)}{(\alpha+1)_n(\beta+1)_n}t^n. +\end{eqnarray} + +\begin{eqnarray} +\label{GenJacobi3} +& &(1-t)^{-\alpha-\beta-1}\,\hyp{2}{1}{\frac{1}{2}(\alpha+\beta+1),\frac{1}{2}(\alpha+\beta+2)} +{\alpha+1}{\frac{2(x-1)t}{(1-t)^2}}\nonumber\\ +& &{}=\sum_{n=0}^{\infty}\frac{(\alpha+\beta+1)_n}{(\alpha+1)_n}P_n^{(\alpha,\beta)}(x)t^n. +\end{eqnarray} + +\begin{eqnarray} +\label{GenJacobi4} +& &(1+t)^{-\alpha-\beta-1}\,\hyp{2}{1}{\frac{1}{2}(\alpha+\beta+1),\frac{1}{2}(\alpha+\beta+2)} +{\beta+1}{\frac{2(x+1)t}{(1+t)^2}}\nonumber\\ +& &{}=\sum_{n=0}^{\infty}\frac{(\alpha+\beta+1)_n}{(\beta+1)_n}P_n^{(\alpha,\beta)}(x)t^n. +\end{eqnarray} + +\begin{eqnarray} +\label{GenJacobi5} +& &\hyp{2}{1}{\gamma,\alpha+\beta+1-\gamma}{\alpha+1}{\frac{1-R-t}{2}}\, +\hyp{2}{1}{\gamma,\alpha+\beta+1-\gamma}{\beta+1}{\frac{1-R+t}{2}}\nonumber\\ +& &{}=\sum_{n=0}^{\infty} +\frac{(\gamma)_n(\alpha+\beta+1-\gamma)_n}{(\alpha+1)_n(\beta+1)_n}P_n^{(\alpha,\beta)}(x)t^n, +\quad R=\sqrt{1-2xt+t^2} +\end{eqnarray} +with $\gamma$ arbitrary. + +\subsection*{Limit relations} + +\subsubsection*{Wilson $\rightarrow$ Jacobi} +The Jacobi polynomials can be found from the Wilson polynomials given by (\ref{DefWilson}) by +substituting $a=b=\frac{1}{2}(\alpha+1)$, $c=\frac{1}{2}(\beta+1)+it$, +$d=\frac{1}{2}(\beta+1)-it$ and $x\rightarrow t\sqrt{\frac{1}{2}(1-x)}$ in the +definition (\ref{DefWilson}) of the Wilson polynomials and taking the limit +$t\rightarrow\infty$. In fact we have +$$\lim_{t\rightarrow\infty} +\frac{W_n(\frac{1}{2}(1-x)t^2;\frac{1}{2}(\alpha+1), +\frac{1}{2}(\alpha+1),\frac{1}{2}(\beta+1)+it,\frac{1}{2}(\beta+1)-it)} +{t^{2n}n!}=P_n^{(\alpha,\beta)}(x).$$ + +\subsubsection*{Continuous Hahn $\rightarrow$ Jacobi} +The Jacobi polynomials follow from the continuous Hahn polynomials given by (\ref{DefContHahn}) +by using the substitution $x\rightarrow \frac{1}{2}xt$, $a=\frac{1}{2}(\alpha+1-it)$, +$b=\frac{1}{2}(\beta+1+it)$, $c=\frac{1}{2}(\alpha+1+it)$ and $d=\frac{1}{2}(\beta+1-it)$ +in (\ref{DefContHahn}), division by $t^n$ and the limit $t\rightarrow\infty$: +$$\lim_{t\rightarrow\infty} +\frac{p_n(\frac{1}{2}xt;\frac{1}{2}(\alpha+1-it),\frac{1}{2}(\beta+1+it), +\frac{1}{2}(\alpha+1+it),\frac{1}{2}(\beta+1-it))}{t^n}=P_n^{(\alpha,\beta)}(x).$$ + +\subsubsection*{Hahn $\rightarrow$ Jacobi} +To find the Jacobi polynomials from the Hahn polynomials given by (\ref{DefHahn}) we take +$x\rightarrow Nx$ in (\ref{DefHahn}) and let $N\rightarrow\infty$. In fact we have +$$\lim_{N\rightarrow\infty} +Q_n(Nx;\alpha,\beta,N)=\frac{P_n^{(\alpha,\beta)}(1-2x)}{P_n^{(\alpha,\beta)}(1)}.$$ + +\subsubsection*{Jacobi $\rightarrow$ Laguerre} +The Laguerre polynomials given by (\ref{DefLaguerre}) can be obtained from the Jacobi +polynomials by setting $x\rightarrow 1-2\beta^{-1}x$ and then the limit $\beta\rightarrow\infty$: +\begin{equation} +\lim_{\beta\rightarrow\infty} +P_n^{(\alpha,\beta)}(1-2\beta^{-1}x)=L_n^{(\alpha)}(x). +\end{equation} + +\subsubsection*{Jacobi $\rightarrow$ Bessel} +The Bessel polynomials given by (\ref{DefBessel}) are obtained from the Jacobi polynomials +if we take $\beta=a-\alpha$ and let $\alpha\rightarrow-\infty$: +\begin{equation} +\lim_{\alpha\rightarrow-\infty} +\frac{P_n^{(\alpha,a-\alpha)}(1+\alpha x)}{P_n^{(\alpha,a-\alpha)}(1)}=y_n(x;a). +\end{equation} + +\subsubsection*{Jacobi $\rightarrow$ Hermite} +The Hermite polynomials given by (\ref{DefHermite}) follow from the Jacobi polynomials +by taking $\beta=\alpha$ and letting $\alpha\rightarrow\infty$ in the following way: +\begin{equation} +\lim_{\alpha\rightarrow\infty} +\alpha^{-\frac{1}{2}n}P_n^{(\alpha,\alpha)}(\alpha^{-\frac{1}{2}}x)=\frac{H_n(x)}{2^nn!}. +\end{equation} + +\subsection*{Remarks} +The definition (\ref{DefJacobi}) of the Jacobi polynomials can also be written as: +$$P_n^{(\alpha,\beta)}(x)=\frac{1}{n!}\sum_{k=0}^n\frac{(-n)_k}{k!} +(n+\alpha+\beta+1)_k(\alpha+k+1)_{n-k}\left(\frac{1-x}{2}\right)^k.$$ +In this way the Jacobi polynomials can also be seen as polynomials in the parameters $\alpha$ +and $\beta$. Therefore they can be defined for all $\alpha$ and $\beta$. Then we have the +following connection with the Meixner polynomials given by (\ref{DefMeixner}): +$$\frac{(\beta)_n}{n!}M_n(x;\beta,c)=P_n^{(\beta-1,-n-\beta-x)}((2-c)c^{-1}).$$ + +\noindent +The Jacobi polynomials are related to the pseudo Jacobi polynomials defined by +(\ref{DefPseudoJacobi}) in the following way: +$$P_n(x;\nu,N)=\frac{(-2i)^nn!}{(n-2N-1)_n}P_n^{(-N-1+i\nu,-N-1-i\nu)}(ix).$$ + +\noindent +The Jacobi polynomials are also related to the Gegenbauer (or ultraspherical) polynomials +given by (\ref{DefGegenbauer}) by the quadratic transformations: +$$C_{2n}^{(\lambda)}(x)=\frac{(\lambda)_n}{(\frac{1}{2})_n} +P_n^{(\lambda-\frac{1}{2},-\frac{1}{2})}(2x^2-1)$$ +and +$$C_{2n+1}^{(\lambda)}(x)=\frac{(\lambda)_{n+1}}{(\frac{1}{2})_{n+1}} +\subsection*{9.8 Jacobi} +\label{sec9.8} +% +\paragraph{Orthogonality relation} +Write the \RHS\ of (9.8.2) as $h_n\,\de_{m,n}$. Then +\begin{equation} +\begin{split} +&\frac{h_n}{h_0}= +\frac{n+\al+\be+1}{2n+\al+\be+1}\, +\frac{(\al+1)_n(\be+1)_n}{(\al+\be+2)_n\,n!}\,,\quad +h_0=\frac{2^{\al+\be+1}\Ga(\al+1)\Ga(\be+1)}{\Ga(\al+\be+2)}\,,\sLP +&\frac{h_n}{h_0\,(P_n^{(\al,\be)}(1))^2}= +\frac{n+\al+\be+1}{2n+\al+\be+1}\, +\frac{(\be+1)_n\,n!}{(\al+1)_n\,(\al+\be+2)_n}\,. +\end{split} +\label{60} +\end{equation} + +In (9.8.3) the numerator factor $\Ga(n+\al+\be+1)$ in the last line should be +$\Ga(\be+1)$. When thus corrected, (9.8.3) can be rewritten as: +\begin{equation} +\begin{split} +&\int_1^\iy P_m^{(\al,\be)}(x)\,P_n^{(\al,\be)}(x)\,(x-1)^\al (x+1)^\be\,dx=h_n\,\de_{m,n}\,,\\ +&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad-1-\be>\al>-1,\quad m,n<-\thalf(\al+\be+1),\\ +&\frac{h_n}{h_0}= +\frac{n+\al+\be+1}{2n+\al+\be+1}\, +\frac{(\al+1)_n(\be+1)_n}{(\al+\be+2)_n\,n!}\,,\quad +h_0=\frac{2^{\al+\be+1}\Ga(\al+1)\Ga(-\al-\be-1)}{\Ga(-\be)}\,. +\end{split} +\label{122} +\end{equation} + +% +\paragraph{Symmetry} +\begin{equation} +P_n^{(\al,\be)}(-x)=(-1)^n\,P_n^{(\be,\al)}(x). +\label{48} +\end{equation} +Use (9.8.2) and (9.8.5b) or see \mycite{DLMF}{Table 18.6.1}. +% +\paragraph{Special values} +\begin{equation} +P_n^{(\al,\be)}(1)=\frac{(\al+1)_n}{n!}\,,\quad +P_n^{(\al,\be)}(-1)=\frac{(-1)^n(\be+1)_n}{n!}\,,\quad +\frac{P_n^{(\al,\be)}(-1)}{P_n^{(\al,\be)}(1)}=\frac{(-1)^n(\be+1)_n}{(\al+1)_n}\,. +\label{50} +\end{equation} +Use (9.8.1) and \eqref{48} or see \mycite{DLMF}{Table 18.6.1}. +% +\paragraph{Generating functions} +Formula (9.8.15) was first obtained by Brafman \myciteKLS{109}. +% +\paragraph{Bilateral generating functions} +For $0\le r<1$ and $x,y\in[-1,1]$ we have in terms of $F_4$ (see~\eqref{62}): +\begin{align} +&\sum_{n=0}^\iy\frac{(\al+\be+1)_n\,n!}{(\al+1)_n(\be+1)_n}\,r^n\, +P_n^{(\al,\be)}(x)\,P_n^{(\al,\be)}(y) +=\frac1{(1+r)^{\al+\be+1}} +\nonumber\\ +&\qquad\quad\times F_4\Big(\thalf(\al+\be+1),\thalf(\al+\be+2);\al+1,\be+1; +\frac{r(1-x)(1-y)}{(1+r)^2},\frac{r(1+x)(1+y)}{(1+r)^2}\Big), +\label{58}\sLP +&\sum_{n=0}^\iy\frac{2n+\al+\be+1}{n+\al+\be+1} +\frac{(\al+\be+2)_n\,n!}{(\al+1)_n(\be+1)_n}\,r^n\, +P_n^{(\al,\be)}(x)\,P_n^{(\al,\be)}(y) +=\frac{1-r}{(1+r)^{\al+\be+2}}\nonumber\\ +&\qquad\quad\times F_4\Big(\thalf(\al+\be+2),\thalf(\al+\be+3);\al+1,\be+1; +\frac{r(1-x)(1-y)}{(1+r)^2},\frac{r(1+x)(1+y)}{(1+r)^2}\Big). +\label{59} +\end{align} +Formulas \eqref{58} and \eqref{59} were first +given by Bailey \myciteKLS{91}{(2.1), (2.3)}. +See Stanton \myciteKLS{485} for a shorter proof. +(However, in the second line of +\myciteKLS{485}{(1)} $z$ and $Z$ should be interchanged.)$\;$ +As observed in Bailey \myciteKLS{91}{p.10}, \eqref{59} follows +from \eqref{58} +by applying the operator $r^{-\half(\al+\be-1)}\,\frac d{dr}\circ r^{\half(\al+\be+1)}$ +to both sides of \eqref{58}. +In view of \eqref{60}, formula \eqref{59} is the Poisson kernel for Jacobi +polynomials. The \RHS\ of \eqref{59} makes clear that this kernel is positive. +See also the discussion in Askey \myciteKLS{46}{following (2.32)}. +% +\paragraph{Quadratic transformations} +\begin{align} +\frac{C_{2n}^{(\al+\half)}(x)}{C_{2n}^{(\al+\half)}(1)} +=\frac{P_{2n}^{(\al,\al)}(x)}{P_{2n}^{(\al,\al)}(1)} +&=\frac{P_n^{(\al,-\half)}(2x^2-1)}{P_n^{(\al,-\half)}(1)}\,, +\label{51}\\ +\frac{C_{2n+1}^{(\al+\half)}(x)}{C_{2n+1}^{(\al+\half)}(1)} +=\frac{P_{2n+1}^{(\al,\al)}(x)}{P_{2n+1}^{(\al,\al)}(1)} +&=\frac{x\,P_n^{(\al,\half)}(2x^2-1)}{P_n^{(\al,\half)}(1)}\,. +\label{52} +\end{align} +See p.221, Remarks, last two formulas together with \eqref{50} and \eqref{49}. +Or see \mycite{DLMF}{(18.7.13), (18.7.14)}. +% +\paragraph{Differentiation formulas} +Each differentiation formula is given in two equivalent forms. +\begin{equation} +\begin{split} +\frac d{dx}\left((1-x)^\al P_n^{(\al,\be)}(x)\right)&= +-(n+\al)\,(1-x)^{\al-1} P_n^{(\al-1,\be+1)}(x),\\ +\left((1-x)\frac d{dx}-\al\right)P_n^{(\al,\be)}(x)&= +-(n+\al)\,P_n^{(\al-1,\be+1)}(x). +\end{split} +\label{68} +\end{equation} +% +\begin{equation} +\begin{split} +\frac d{dx}\left((1+x)^\be P_n^{(\al,\be)}(x)\right)&= +(n+\be)\,(1+x)^{\be-1} P_n^{(\al+1,\be-1)}(x),\\ +\left((1+x)\frac d{dx}+\be\right)P_n^{(\al,\be)}(x)&= +(n+\be)\,P_n^{(\al+1,\be-1)}(x). +\end{split} +\label{69} +\end{equation} +Formulas \eqref{68} and \eqref{69} follow from +\mycite{DLMF}{(15.5.4), (15.5.6)} +together with (9.8.1). They also follow from each other by \eqref{48}. +% +\paragraph{Generalized Gegenbauer polynomials} +These are defined by +\begin{equation} +S_{2m}^{(\al,\be)}(x):=\const P_m^{(\al,\be)}(2x^2-1),\qquad +S_{2m+1}^{(\al,\be)}(x):=\const x\,P_m^{(\al,\be+1)}(2x^2-1) +\label{70} +\end{equation} +in the notation of \myciteKLS{146}{p.156} +(see also \cite{K27}), while \cite[Section 1.5.2]{K26} +has $C_n^{(\la,\mu)}(x)=\const\allowbreak\times S_n^{(\la-\half,\mu-\half)}(x)$. +For $\al,\be>-1$ we have the orthogonality relation +\begin{equation} +\int_{-1}^1 S_m^{(\al,\be)}(x)\,S_n^{(\al,\be)}(x)\,|x|^{2\be+1}(1-x^2)^\al\,dx +=0\qquad(m\ne n). +\label{71} +\end{equation} +For $\be=\al-1$ generalized Gegenbauer polynomials are limit cases of +continuous $q$-ultraspherical polynomials, see \eqref{176}. + +If we define the {\em Dunkl operator} $T_\mu$ by +\begin{equation} +(T_\mu f)(x):=f'(x)+\mu\,\frac{f(x)-f(-x)}x +\label{72} +\end{equation} +and if we choose the constants in \eqref{70} as +\begin{equation} +S_{2m}^{(\al,\be)}(x)=\frac{(\al+\be+1)_m}{(\be+1)_m}\, P_m^{(\al,\be)}(2x^2-1),\quad +S_{2m+1}^{(\al,\be)}(x)=\frac{(\al+\be+1)_{m+1}}{(\be+1)_{m+1}}\, +x\,P_m^{(\al,\be+1)}(2x^2-1) +\label{73} +\end{equation} +then (see \cite[(1.6)]{K5}) +\begin{equation} +T_{\be+\half}S_n^{(\al,\be)}=2(\al+\be+1)\,S_{n-1}^{(\al+1,\be)}. +\label{74} +\end{equation} +Formula \eqref{74} with \eqref{73} substituted gives rise to two +differentiation formulas involving Jacobi polynomials which are equivalent to +(9.8.7) and \eqref{69}. + +Composition of \eqref{74} with itself gives +\[ +T_{\be+\half}^2S_n^{(\al,\be)}=4(\al+\be+1)(\al+\be+2)\,S_{n-2}^{(\al+2,\be)}, +\] +which is equivalent to the composition of (9.8.7) and \eqref{69}: +\begin{equation} +\left(\frac{d^2}{dx^2}+\frac{2\be+1}x\,\frac d{dx}\right)P_n^{(\al,\be)}(2x^2-1) +=4(n+\al+\be+1)(n+\be)\,P_{n-1}^{(\al+2,\be)}(2x^2-1). +\label{75} +\end{equation} +Formula \eqref{75} was also given in \myciteKLS{322}{(2.4)}. +xP_n^{(\lambda-\frac{1}{2},\frac{1}{2})}(2x^2-1).$$ + +\subsection*{References} +\cite{Abram}, \cite{Ahmed+82}, \cite{Allaway89}, \cite{NAlSalam66}, \cite{AlSalam64}, +\cite{AlSalamChihara72}, \cite{AndrewsAskey85}, \cite{AndrewsAskeyRoy}, \cite{Askey68}, +\cite{Askey70}, \cite{Askey72}, \cite{Askey73}, \cite{Askey74}, \cite{Askey75}, \cite{Askey78}, +\cite{Askey89I}, \cite{Askey2005}, \cite{AskeyFitch}, \cite{AskeyGasper71I}, \cite{AskeyGasper71II}, +\cite{AskeyGasper76}, \cite{AskeyWainger}, \cite{AskeyWilson85}, \cite{Bailey38}, \cite{Brafman51}, +\cite{Brown}, \cite{Carlitz61II}, \cite{Carlitz67}, \cite{ChenIsmail91}, \cite{Chihara78}, +\cite{Chow+}, \cite{Ciesielski}, \cite{Cooper+}, \cite{DetteStudden92}, \cite{DetteStudden95}, +\cite{DijksmaKoorn}, \cite{DimitrovRafaeli}, \cite{DimitrovRodrigues}, \cite{Doha2002I}, +\cite{Doha2003I}, \cite{Doha2004II}, \cite{Dunkl84}, \cite{ElbertLaforgia87II}, +\cite{ElbertLaforgiaRodono}, \cite{Erdelyi+}, \cite{Faldey}, \cite{FoataLeroux}, +\cite{Gasper69}, \cite{Gasper70I}, \cite{Gasper70II}, \cite{Gasper71I}, \cite{Gasper71II}, +\cite{Gasper72I}, \cite{Gasper72II}, \cite{Gasper73I}, \cite{Gasper73II}, \cite{Gasper74}, +\cite{Gasper77}, \cite{Gatteschi87}, \cite{Gautschi2008}, \cite{Gautschi2009I}, +\cite{Gautschi2009II}, \cite{GautschiLeopardi}, \cite{GawronskiShawyer}, \cite{Godoy+}, +\cite{Grad}, \cite{Hajmirzaahmad94}, \cite{HartmannStephan}, \cite{Horton}, \cite{Ismail74}, +\cite{Ismail77}, \cite{Ismail96}, \cite{Ismail2005II}, \cite{IsmailLi}, +\cite{IsmailMassonRahman}, \cite{Kochneff97I}, \cite{Koekoek99}, \cite{Koekoek2000}, +\cite{Koelink96II}, \cite{Koorn73}, \cite{Koorn74}, \cite{Koorn75}, \cite{Koorn77II}, +\cite{Koorn78}, \cite{Koorn72}, \cite{Koorn85}, \cite{Koorn88}, \cite{KuijlaarsMartinez}, +\cite{Kuijlaars+}, \cite{LabelleYehI}, \cite{LabelleYehII}, \cite{Laine}, \cite{Lesky95II}, +\cite{Lesky96}, \cite{Li96}, \cite{Li97}, \cite{LopezTemme2004}, \cite{Luke}, \cite{Martinez}, +\cite{Mathai}, \cite{Meijer}, \cite{Miller89}, \cite{MoakSaffVarga}, \cite{Nikiforov+}, +\cite{NikiforovUvarov}, \cite{Olver}, \cite{Prasad}, \cite{Rahman76I}, \cite{Rahman76II}, +\cite{Rahman77}, \cite{Rahman81I}, \cite{RahmanShah}, \cite{Rainville}, \cite{Rusev}, +\cite{Shi}, \cite{Srivastava69II}, \cite{Srivastava71}, \cite{Srivastava82}, +\cite{SrivastavaSinghal}, \cite{Stanton80I}, \cite{Stanton90}, \cite{Szego75}, \cite{Temme}, +\cite{Vertesi}, \cite{Wimp87}, \cite{WongZhang94}, \cite{WongZhang2006}, \cite{Zarzo+}, +\cite{Zayed}. + +\section*{Special cases} + +\subsection{Gegenbauer / Ultraspherical}\index{Gegenbauer polynomials} +\index{Ultraspherical polynomials} + +\par + +\subsection*{Hypergeometric representation} +The Gegenbauer (or ultraspherical) polynomials are Jacobi polynomials with +$\alpha=\beta=\lambda-\frac{1}{2}$ and another normalization: +\begin{eqnarray} +\label{DefGegenbauer} +C_n^{(\lambda)}(x)&=&\frac{(2\lambda)_n}{(\lambda+\frac{1}{2})_n} +P_n^{(\lambda-\frac{1}{2},\lambda-\frac{1}{2})}(x)\nonumber\\ +&=&\frac{(2\lambda)_n}{n!}\,\hyp{2}{1}{-n,n+2\lambda} +{\lambda+\frac{1}{2}}{\frac{1-x}{2}},\quad\lambda\neq 0. +\end{eqnarray} + +\subsection*{Orthogonality relation} +\begin{eqnarray} +\label{OrtGegenbauer} +& &\int_{-1}^1(1-x^2)^{\lambda-\frac{1}{2}}C_m^{(\lambda)}(x)C_n^{(\lambda)}(x)\,dx\nonumber\\ +& &{}=\frac{\pi\Gamma(n+2\lambda)2^{1-2\lambda}}{\left\{\Gamma(\lambda)\right\}^2(n+\lambda)n!}\,\delta_{mn}, +\quad\lambda>-\frac{1}{2}\quad\lambda\neq 0. +\end{eqnarray} + +\subsection*{Recurrence relation} +\begin{equation} +\label{RecGegenbauer} +2(n+\lambda)xC_n^{(\lambda)}(x)=(n+1)C_{n+1}^{(\lambda)}(x)+(n+2\lambda-1)C_{n-1}^{(\lambda)}(x). +\end{equation} + +\subsection*{Normalized recurrence relation} +\begin{equation} +\label{NormRecGegenbauer} +xp_n(x)=p_{n+1}(x)+\frac{n(n+2\lambda-1)}{4(n+\lambda-1)(n+\lambda)}p_{n-1}(x), +\end{equation} +where +$$C_n^{(\lambda)}(x)=\frac{2^n(\lambda)_n}{n!}p_n(x).$$ + +\subsection*{Differential equation} +\begin{equation} +\label{dvGegenbauer} +(1-x^2)y''(x)-(2\lambda+1)xy'(x)+n(n+2\lambda)y(x)=0,\quad y(x)=C_n^{(\lambda)}(x). +\end{equation} + +\subsection*{Forward shift operator} +\begin{equation} +\label{shift1Gegenbauer} +\frac{d}{dx}C_n^{(\lambda)}(x)=2\lambda C_{n-1}^{(\lambda+1)}(x). +\end{equation} + +\subsection*{Backward shift operator} +\begin{equation} +\label{shift2GegenbauerI} +(1-x^2)\frac{d}{dx}C_n^{(\lambda)}(x)+(1-2\lambda)xC_n^{(\lambda)}(x)= +-\frac{(n+1)(2\lambda+n-1)}{2(\lambda-1)} C_{n+1}^{(\lambda-1)}(x) +\end{equation} +or equivalently +\begin{eqnarray} +\label{shift2GegenbauerII} +& &\frac{d}{dx}\left[(1-x^2)^{\lambda-\textstyle\frac{1}{2}}C_n^{(\lambda)}(x)\right]\nonumber\\ +& &{}=-\frac{(n+1)(2\lambda+n-1)}{2(\lambda-1)}(1-x^2)^{\lambda-\textstyle\frac{3}{2}}C_{n+1}^{(\lambda-1)}(x). +\end{eqnarray} + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodGegenbauer} +(1-x^2)^{\lambda-\frac{1}{2}}C_n^{(\lambda)}(x)= +\frac{(2\lambda)_n(-1)^n}{(\lambda+\frac{1}{2})_n2^nn!}\left(\frac{d}{dx}\right)^n +\left[(1-x^2)^{\lambda+n-\frac{1}{2}}\right]. +\end{equation} + +\subsection*{Generating functions} +\begin{equation} +\label{GenGegenbauer1} +(1-2xt+t^2)^{-\lambda}=\sum_{n=0}^{\infty}C_n^{(\lambda)}(x)t^n. +\end{equation} + +\begin{equation} +\label{GenGegenbauer2} +R^{-1}\left(\frac{1+R-xt}{2}\right)^{\frac{1}{2}-\lambda}=\sum_{n=0}^{\infty} +\frac{(\lambda+\frac{1}{2})_n}{(2\lambda)_n}C_n^{(\lambda)}(x)t^n, +\quad R=\sqrt{1-2xt+t^2}. +\end{equation} + +\begin{equation} +\label{GenGegenbauer3} +\hyp{0}{1}{-}{\lambda+\frac{1}{2}}{\frac{(x-1)t}{2}}\, +\hyp{0}{1}{-}{\lambda+\frac{1}{2}}{\frac{(x+1)t}{2}} +=\sum_{n=0}^{\infty}\frac{C_n^{(\lambda)}(x)} +{(2\lambda)_n(\lambda+\frac{1}{2})_n}t^n. +\end{equation} + +\begin{equation} +\label{GenGegenbauer4} +\e^{xt}\,\hyp{0}{1}{-}{\lambda+\frac{1}{2}}{\frac{(x^2-1)t^2}{4}}= +\sum_{n=0}^{\infty}\frac{C_n^{(\lambda)}(x)}{(2\lambda)_n}t^n. +\end{equation} + +\begin{eqnarray} +\label{GenGegenbauer5} +& &\hyp{2}{1}{\gamma,2\lambda-\gamma}{\lambda+\frac{1}{2}}{\frac{1-R-t}{2}}\, +\hyp{2}{1}{\gamma,2\lambda-\gamma}{\lambda+\frac{1}{2}}{\frac{1-R+t}{2}}\nonumber\\ +& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n(2\lambda-\gamma)_n} +{(2\lambda)_n(\lambda+\frac{1}{2})_n}C_n^{(\lambda)}(x)t^n, +\quad R=\sqrt{1-2xt+t^2},\quad\textrm{$\gamma$ arbitrary}. +\end{eqnarray} + +\begin{eqnarray} +\label{GenGegenbauer6} +& &(1-xt)^{-\gamma}\,\hyp{2}{1}{\frac{1}{2}\gamma,\frac{1}{2}\gamma+\frac{1}{2}} +{\lambda+\frac{1}{2}}{\frac{(x^2-1)t^2}{(1-xt)^2}}\nonumber\\ +& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n}{(2\lambda)_n}C_n^{(\lambda)}(x)t^n, +\quad\textrm{$\gamma$ arbitrary}. +\end{eqnarray} + +\subsection*{Limit relation} + +\subsubsection*{Gegenbauer / Ultraspherical $\rightarrow$ Hermite} +The Hermite polynomials given by (\ref{DefHermite}) follow from the Gegenbauer (or +ultraspherical) polynomials by taking $\lambda=\alpha+\frac{1}{2}$ and letting +$\alpha\rightarrow\infty$ in the following way: +\begin{equation} +\lim_{\alpha\rightarrow\infty} +\alpha^{-\frac{1}{2}n}C_n^{(\alpha+\frac{1}{2})}(\alpha^{-\frac{1}{2}}x)=\frac{H_n(x)}{n!}. +\end{equation} + +\subsection*{Remarks} +The case $\lambda=0$ needs another normalization. In that case we have the +Chebyshev polynomials of the first kind described in the next subsection. + +\noindent +The Gegenbauer (or ultraspherical) polynomials are related to the Jacobi polynomials +given by (\ref{DefJacobi}) by the quadratic transformations: +$$C_{2n}^{(\lambda)}(x)=\frac{(\lambda)_n}{(\frac{1}{2})_n} +P_n^{(\lambda-\frac{1}{2},-\frac{1}{2})}(2x^2-1)$$ +and +$$C_{2n+1}^{(\lambda)}(x)=\frac{(\lambda)_{n+1}}{(\frac{1}{2})_{n+1}} +\subsection*{9.8.1 Gegenbauer / Ultraspherical} +\label{sec9.8.1} +% +\paragraph{Notation} +Here the Gegenbauer polynomial is denoted by $C_n^\la$ instead of $C_n^{(\la)}$. +% +\paragraph{Orthogonality relation} +Write the \RHS\ of (9.8.20) as $h_n\,\de_{m,n}$. Then +\begin{equation} +\frac{h_n}{h_0}= +\frac\la{\la+n}\,\frac{(2\la)_n}{n!}\,,\quad +h_0=\frac{\pi^\half\,\Ga(\la+\thalf)}{\Ga(\la+1)},\quad +\frac{h_n}{h_0\,(C_n^\la(1))^2}= +\frac\la{\la+n}\,\frac{n!}{(2\la)_n}\,. +\label{61} +\end{equation} +% +\paragraph{Hypergeometric representation} +Beside (9.8.19) we have also +\begin{equation} +C_n^\lambda(x)=\sum_{\ell=0}^{\lfloor n/2\rfloor}\frac{(-1)^{\ell}(\lambda)_{n-\ell}} +{\ell!\;(n-2\ell)!}\,(2x)^{n-2\ell} +=(2x)^{n}\,\frac{(\lambda)_{n}}{n!}\, +\hyp21{-\thalf n,-\thalf n+\thalf}{1-\la-n}{\frac1{x^2}}. +\label{57} +\end{equation} +See \mycite{DLMF}{(18.5.10)}. +% +\paragraph{Special value} +\begin{equation} +C_n^{\la}(1)=\frac{(2\la)_n}{n!}\,. +\label{49} +\end{equation} +Use (9.8.19) or see \mycite{DLMF}{Table 18.6.1}. +% +\paragraph{Expression in terms of Jacobi} +% +\begin{equation} +\frac{C_n^\la(x)}{C_n^\la(1)}= +\frac{P_n^{(\la-\half,\la-\half)}(x)}{P_n^{(\la-\half,\la-\half)}(1)}\,,\qquad +C_n^\la(x)=\frac{(2\la)_n}{(\la+\thalf)_n}\,P_n^{(\la-\half,\la-\half)}(x). +\label{65} +\end{equation} +% +\paragraph{Re: (9.8.21)} +By iteration of recurrence relation (9.8.21): +\begin{multline} +x^2 C_n^\la(x)= +\frac{(n+1)(n+2)}{4(n+\la)(n+\la+1)}\,C_{n+2}^\la(x)+ +\frac{n^2+2n\la+\la-1}{2(n+\la-1)(n+\la+1)}\,C_n^\la(x)\\ ++\frac{(n+2\la-1)(n+2\la-2)}{4(n+\la)(n+\la-1)}\,C_{n-2}^\la(x). +\label{6} +\end{multline} +% +\paragraph{Bilateral generating functions} +\begin{multline} +\sum_{n=0}^\iy\frac{n!}{(2\la)_n}\,r^n\,C_n^\la(x)\,C_n^\la(y) +=\frac1{(1-2rxy+r^2)^\la}\,\hyp21{\thalf\la,\thalf(\la+1)}{\la+\thalf} +{\frac{4r^2(1-x^2)(1-y^2)}{(1-2rxy+r^2)^2}}\\ +(r\in(-1,1),\;x,y\in[-1,1]). +\label{66} +\end{multline} +For the proof put $\be:=\al$ in \eqref{58}, then use \eqref{63} and \eqref{65}. +The Poisson kernel for Gegenbauer polynomials can be derived in a similar way +from \eqref{59}, or alternatively by applying the operator +$r^{-\la+1}\frac d{dr}\circ r^\la$ to both sides of \eqref{66}: +\begin{multline} +\sum_{n=0}^\iy\frac{\la+n}\la\,\frac{n!}{(2\la)_n}\,r^n\,C_n^\la(x)\,C_n^\la(y) +=\frac{1-r^2}{(1-2rxy+r^2)^{\la+1}}\\ +\times\hyp21{\thalf(\la+1),\thalf(\la+2)}{\la+\thalf} +{\frac{4r^2(1-x^2)(1-y^2)}{(1-2rxy+r^2)^2}}\qquad +(r\in(-1,1),\;x,y\in[-1,1]). +\label{67} +\end{multline} +Formula \eqref{67} was obtained by Gasper \& Rahman \myciteKLS{234}{(4.4)} +as a limit case of their formula for the Poisson kernel for continuous +$q$-ultraspherical polynomials. +% +\paragraph{Trigonometric expansions} +By \mycite{DLMF}{(18.5.11), (15.8.1)}: +\begin{align} +C_n^{\la}(\cos\tha) +&=\sum_{k=0}^n\frac{(\la)_k(\la)_{n-k}}{k!\,(n-k)!}\,e^{i(n-2k)\tha} +=e^{in\tha}\frac{(\la)_n}{n!}\, +\hyp21{-n,\la}{1-\la-n}{e^{-2i\tha}}\label{103}\\ +&=\frac{(\la)_n}{2^\la n!}\, +e^{-\half i\la\pi}e^{i(n+\la)\tha}\,(\sin\tha)^{-\la}\, +\hyp21{\la,1-\la}{1-\la-n}{\frac{i e^{-i\tha}}{2\sin\tha}}\label{104}\\ +&=\frac{(\la)_n}{n!}\,\sum_{k=0}^\iy\frac{(\la)_k(1-\la)_k}{(1-\la-n)_k k!}\, +\frac{\cos((n-k+\la)\tha+\thalf(k-\la)\pi)}{(2\sin\tha)^{k+\la}}\,.\label{105} +\end{align} +In \eqref{104} and \eqref{105} we require that +$\tfrac16\pi<\tha<\tfrac56\pi$. Then the convergence is absolute for $\la>\thalf$ +and conditional for $0<\la\le\thalf$. + +By \mycite{DLMF}{(14.13.1), (14.3.21), (15.8.1)]}: +\begin{align} +C_n^\la(\cos\tha)&=\frac{2\Ga(\la+\thalf)}{\pi^\half\Ga(\la+1)}\, +\frac{(2\la)_n}{(\la+1)_n}\,(\sin\tha)^{1-2\la}\, +\sum_{k=0}^\iy\frac{(1-\la)_k(n+1)_k}{(n+\la+1)_k k!}\, +\sin\big((2k+n+1)\tha\big) +\label{7}\\ +&=\frac{2\Ga(\la+\thalf)}{\pi^\half\Ga(\la+1)}\, +\frac{(2\la)_n}{(\la+1)_n}\,(\sin\tha)^{1-2\la}\, +\Im\!\!\left(e^{i(n+1)\tha}\,\hyp21{1-\la,n+1}{n+\la+1}{e^{2i\tha}}\right)\nonumber\\ +&=\frac{2^\la\Ga(\la+\thalf)}{\pi^\half\Ga(\la+1)}\, +\frac{(2\la)_n}{(\la+1)_n}\,(\sin\tha)^{-\la}\, +\Re\!\!\left(e^{-\thalf i\la\pi}e^{i(n+\la)\tha}\, +\hyp21{\la,1-\la}{1+\la+n}{\frac{e^{i\tha}}{2i\sin\tha}}\right)\nonumber\\ +&=\frac{2^{2\la}\Ga(\la+\thalf)}{\pi^\half\Ga(\la+1)}\,\frac{(2\la)_n}{(\la+1)_n}\, +\sum_{k=0}^\iy\frac{(\la)_k(1-\la)_k}{(1+\la+n)_k k!}\, +\frac{\cos((n+k+\la)\tha-\thalf(k+\la)\pi)}{(2\sin\tha)^{k+\la}}\,. +\label{106} +\end{align} +We require that $0<\tha<\pi$ in \eqref{7} and $\tfrac16\pi<\tha<\tfrac56\pi$ in +\eqref{106} The convergence is absolute for $\la>\thalf$ and conditional for +$0<\la\le\thalf$. +For $\la\in\Zpos$ the above series terminate after the term with +$k=\la-1$. +Formulas \eqref{7} and \eqref{106} are also given in +\mycite{Sz}{(4.9.22), (4.9.25)}. +% +\paragraph{Fourier transform} +\begin{equation} +\frac{\Ga(\la+1)}{\Ga(\la+\thalf)\,\Ga(\thalf)}\, +\int_{-1}^1 \frac{C_n^\la(y)}{C_n^\la(1)}\,(1-y^2)^{\la-\half}\, +e^{ixy}\,dy +=i^n\,2^\la\,\Ga(\la+1)\,x^{-\la}\,J_{\la+n}(x). +\label{8} +\end{equation} +See \mycite{DLMF}{(18.17.17) and (18.17.18)}. +% +\paragraph{Laplace transforms} +\begin{equation} +\frac2{n!\,\Ga(\la)}\, +\int_0^\iy H_n(tx)\,t^{n+2\la-1}\,e^{-t^2}\,dt=C_n^\la(x). +\label{56} +\end{equation} +See Nielsen \cite[p.48, (4) with p.47, (1) and p.28, (10)]{K4} (1918) +or Feldheim \cite[(28)]{K3} (1942). +\begin{equation} +\frac2{\Ga(\la+\thalf)}\,\int_0^1 \frac{C_n^\la(t)}{C_n^\la(1)}\, +(1-t^2)^{\la-\half}\,t^{-1}\,(x/t)^{n+2\la+1}\,e^{-x^2/t^2}\,dt +=2^{-n}\,H_n(x)\,e^{-x^2}\quad(\la>-\thalf). +\label{46} +\end{equation} +Use Askey \& Fitch \cite[(3.29)]{K2} for $\al=\pm\thalf$ together with +\eqref{48}, \eqref{51}, \eqref{52}, \eqref{54} and \eqref{55}. +\paragraph{Addition formula} (see \mycite{AAR}{(9.8.5$'$)]}) +\begin{multline} +R_n^{(\al,\al)}\big(xy+(1-x^2)^\half(1-y^2)^\half t\big) +=\sum_{k=0}^n \frac{(-1)^k(-n)_k\,(n+2\al+1)_k}{2^{2k}((\al+1)_k)^2}\\ +\times(1-x^2)^{k/2} R_{n-k}^{(\al+k,\al+k)}(x)\,(1-y^2)^{k/2} R_{n-k}^{(\al+k,\al+k)}(y)\, +\om_k^{(\al-\half,\al-\half)}\,R_k^{(\al-\half,\al-\half)}(t), +\label{108} +\end{multline} +where +\[ +R_n^{(\al,\be)}(x):=P_n^{(\al,\be)}(x)/P_n^{(\al,\be)}(1),\quad +\om_n^{(\al,\be)}:=\frac{\int_{-1}^1 (1-x)^\al(1+x)^\be\,dx} +{\int_{-1}^1 (R_n^{(\al,\be)}(x))^2\,(1-x)^\al(1+x)^\be\,dx}\,. +\] +xP_n^{(\lambda-\frac{1}{2},\frac{1}{2})}(2x^2-1).$$ + +\subsection*{References} +\cite{Abram}, \cite{Ahmed+86}, \cite{AndrewsAskeyRoy}, \cite{Area+I}, \cite{Askey67}, \cite{Askey74}, \cite{Askey75}, +\cite{Askey89I}, \cite{AskeyFitch}, \cite{AskeyKoornRahman}, \cite{Berg}, \cite{BilodeauI}, +\cite{BojanovNikolov}, \cite{Brafman51}, \cite{Brafman57I}, \cite{Brown}, +\cite{BustozIsmail82}, \cite{BustozIsmail83}, \cite{BustozIsmail89}, \cite{BustozSavage79}, +\cite{BustozSavage80}, \cite{Carlitz61II}, \cite{Chihara78}, \cite{Common}, \cite{Danese}, +\cite{Dette94}, \cite{DijksmaKoorn}, \cite{DilcherStolarsky}, \cite{Dimitrov96}, +\cite{Dimitrov2003}, \cite{Doha2002II}, \cite{Driver}, \cite{ElbertLaforgia86I}, +\cite{ElbertLaforgia86II}, \cite{ElbertLaforgia90}, \cite{Erdelyi+}, \cite{Exton96}, +\cite{Gasper69}, \cite{Gasper72II}, \cite{Gasper85}, \cite{Grad}, \cite{Ismail74}, +\cite{Ismail2005II}, \cite{Koekoek2000}, \cite{Koorn88}, \cite{Laforgia}, \cite{LewanowiczI}, +\cite{Lorch}, \cite{Mathai}, \cite{Nagel}, \cite{Nikiforov+}, \cite{NikiforovUvarov}, +\cite{RahmanShah}, \cite{Rainville}, \cite{Reimer}, \cite{Sartoretto}, \cite{Srivastava71}, +\cite{Szego75}, \cite{Temme}, \cite{Viswanathan}, \cite{Zayed}. + +\subsection{Chebyshev}\index{Chebyshev polynomials} + +\par + +\subsection*{Hypergeometric representation} +The Chebyshev polynomials of the first kind can be obtained from the Jacobi +polynomials by taking $\alpha=\beta=-\frac{1}{2}$: +\begin{equation} +\label{DefChebyshevI} +T_n(x)=\frac{P_n^{(-\frac{1}{2},-\frac{1}{2})}(x)}{P_n^{(-\frac{1}{2},-\frac{1}{2})}(1)} +=\hyp{2}{1}{-n,n}{\frac{1}{2}}{\frac{1-x}{2}} +\end{equation} +and the Chebyshev polynomials of the second kind can be obtained from the +Jacobi polynomials by taking $\alpha=\beta=\frac{1}{2}$: +\begin{equation} +\label{DefChebyshevII} +U_n(x)=(n+1)\frac{P_n^{(\frac{1}{2},\frac{1}{2})}(x)}{P_n^{(\frac{1}{2},\frac{1}{2})}(1)} +=(n+1)\,\hyp{2}{1}{-n,n+2}{\frac{3}{2}}{\frac{1-x}{2}}. +\end{equation} + +\subsection*{Orthogonality relation} +\begin{equation} +\label{OrtChebyshevI} +\int_{-1}^1(1-x^2)^{-\frac{1}{2}}T_m(x)T_n(x)\,dx= +\left\{\begin{array}{ll} +\displaystyle\frac{\cpi}{2}\,\delta_{mn}, & n\neq 0\\[5mm] +\cpi\,\delta_{mn}, & n=0. +\end{array}\right. +\end{equation} + +\begin{equation} +\label{OrtChebyshevII} +\int_{-1}^1(1-x^2)^{\frac{1}{2}}U_m(x)U_n(x)\,dx=\frac{\cpi}{2}\,\delta_{mn}. +\end{equation} + +\subsection*{Recurrence relations} +\begin{equation} +\label{RecChebyshevI} +2xT_n(x)=T_{n+1}(x)+T_{n-1}(x),\quad T_{0}(x)=1\quad\textrm{and}\quad T_1(x)=x. +\end{equation} + +\begin{equation} +\label{RecChebyshevII} +2xU_n(x)=U_{n+1}(x)+U_{n-1}(x),\quad U_{0}(x)=1\quad\textrm{and}\quad U_1(x)=2x. +\end{equation} + +\subsection*{Normalized recurrence relations} +\begin{equation} +\label{NormRecChebyshevI} +xp_n(x)=p_{n+1}(x)+\frac{1}{4}p_{n-1}(x), +\end{equation} +where +$$T_1(x)=p_1(x)=x\quad\textrm{and}\quad T_n(x)=2^np_n(x),\quad n\neq 1.$$ + +\begin{equation} +\label{NormRecChebyshevII} +xp_n(x)=p_{n+1}(x)+\frac{1}{4}p_{n-1}(x), +\end{equation} +where +$$U_n(x)=2^np_n(x).$$ + +\subsection*{Differential equations} +\begin{equation} +\label{dvChebyshevI} +(1-x^2)y''(x)-xy'(x)+n^2y(x)=0,\quad y(x)=T_n(x). +\end{equation} + +\begin{equation} +\label{dvChebyshevII} +(1-x^2)y''(x)-3xy'(x)+n(n+2)y(x)=0,\quad y(x)=U_n(x). +\end{equation} + +\subsection*{Forward shift operator} +\begin{equation} +\label{shift1Chebyshev} +\frac{d}{dx}T_n(x)=nU_{n-1}(x). +\end{equation} + +\subsection*{Backward shift operator} +\begin{equation} +\label{shift2ChebyshevI} +(1-x^2)\frac{d}{dx}U_n(x)-xU_n(x)=-(n+1)T_{n+1}(x) +\end{equation} +or equivalently +\begin{equation} +\label{shift2ChebyshevII} +\frac{d}{dx}\left[\left(1-x^2\right)^{\frac{1}{2}}U_n(x)\right] +=-(n+1)\left(1-x^2\right)^{-\frac{1}{2}}T_{n+1}(x). +\end{equation} + +\subsection*{Rodrigues-type formulas} +\begin{equation} +\label{RodChebyshevI} +(1-x^2)^{-\frac{1}{2}}T_n(x)=\frac{(-1)^n}{(\frac{1}{2})_n2^n} +\left(\frac{d}{dx}\right)^n\left[(1-x^2)^{n-\frac{1}{2}}\right]. +\end{equation} + +\begin{equation} +\label{RodChebyshevII} +(1-x^2)^{\frac{1}{2}}U_n(x)=\frac{(n+1)(-1)^n}{(\frac{3}{2})_n2^n} +\left(\frac{d}{dx}\right)^n\left[(1-x^2)^{n+\frac{1}{2}}\right]. +\end{equation} + +\subsection*{Generating functions} +\begin{equation} +\label{GenChebyshevI1} +\frac{1-xt}{1-2xt+t^2}=\sum_{n=0}^{\infty}T_n(x)t^n. +\end{equation} + +\begin{equation} +\label{GenChebyshevI2} +R^{-1}\sqrt{\frac{1}{2}(1+R-xt)}=\sum_{n=0}^{\infty} +\frac{\left(\frac{1}{2}\right)_n}{n!}T_n(x)t^n,\quad R=\sqrt{1-2xt+t^2}. +\end{equation} + +\begin{equation} +\label{GenChebyshevI3} +\hyp{0}{1}{-}{\frac{1}{2}}{\frac{(x-1)t}{2}}\, +\hyp{0}{1}{-}{\frac{1}{2}}{\frac{(x+1)t}{2}}= +\sum_{n=0}^{\infty}\frac{T_n(x)}{\left(\frac{1}{2}\right)_nn!}t^n. +\end{equation} + +\begin{equation} +\label{GenChebyshevI4} +\e^{xt}\,\hyp{0}{1}{-}{\frac{1}{2}}{\frac{(x^2-1)t^2}{4}}= +\sum_{n=0}^{\infty}\frac{T_n(x)}{n!}t^n. +\end{equation} + +\begin{eqnarray} +\label{GenChebyshevI5} +& &\hyp{2}{1}{\gamma,-\gamma}{\frac{1}{2}}{\frac{1-R-t}{2}}\, +\hyp{2}{1}{\gamma,-\gamma}{\frac{1}{2}}{\frac{1-R+t}{2}}\nonumber\\ +& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n(-\gamma)_n}{\left(\frac{1}{2}\right)_nn!}T_n(x)t^n, +\quad R=\sqrt{1-2xt+t^2},\quad\textrm{$\gamma$ arbitrary}. +\end{eqnarray} + +\begin{eqnarray} +\label{GenChebyshevI6} +& &(1-xt)^{-\gamma}\,\hyp{2}{1}{\frac{1}{2}\gamma,\frac{1}{2}\gamma+\frac{1}{2}}{\frac{1}{2}} +{\frac{(x^2-1)t^2}{(1-xt)^2}}\nonumber\\ +& &=\sum_{n=0}^{\infty}\frac{(\gamma)_n}{n!}T_n(x)t^n,\quad\textrm{$\gamma$ arbitrary}. +\end{eqnarray} + +\begin{equation} +\label{GenChebyshevII1} +\frac{1}{1-2xt+t^2}=\sum_{n=0}^{\infty}U_n(x)t^n. +\end{equation} + +\begin{equation} +\label{GenChebyshevII2} +\frac{1}{R\sqrt{\frac{1}{2}(1+R-xt)}}=\sum_{n=0}^{\infty} +\frac{\left(\frac{3}{2}\right)_n}{(n+1)!}U_n(x)t^n,\quad R=\sqrt{1-2xt+t^2}. +\end{equation} + +\begin{equation} +\label{GenChebyshevII3} +\hyp{0}{1}{-}{\frac{3}{2}}{\frac{(x-1)t}{2}}\, +\hyp{0}{1}{-}{\frac{3}{2}}{\frac{(x+1)t}{2}}= +\sum_{n=0}^{\infty}\frac{U_n(x)}{\left(\frac{3}{2}\right)_n(n+1)!}t^n. +\end{equation} + +\begin{equation} +\label{GenChebyshevII4} +\e^{xt}\,\hyp{0}{1}{-}{\frac{3}{2}}{\frac{(x^2-1)t^2}{4}}= +\sum_{n=0}^{\infty}\frac{U_n(x)}{(n+1)!}t^n. +\end{equation} + +\begin{eqnarray} +\label{GenChebyshevII5} +& &\hyp{2}{1}{\gamma,2-\gamma}{\frac{3}{2}}{\frac{1-R-t}{2}}\, +\hyp{2}{1}{\gamma,2-\gamma}{\frac{3}{2}}{\frac{1-R+t}{2}}\nonumber\\ +& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n(2-\gamma)_n}{\left(\frac{3}{2}\right)_n(n+1)!}U_n(x)t^n, +\quad R=\sqrt{1-2xt+t^2},\quad\textrm{$\gamma$ arbitrary}. +\end{eqnarray} + +\begin{eqnarray} +\label{GenChebyshevII6} +& &(1-xt)^{-\gamma}\,\hyp{2}{1}{\frac{1}{2}\gamma,\frac{1}{2}\gamma+\frac{1}{2}}{\frac{3}{2}} +{\frac{(x^2-1)t^2}{(1-xt)^2}}\nonumber\\ +& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n}{(n+1)!}U_n(x)t^n,\quad\textrm{$\gamma$ arbitrary}. +\end{eqnarray} + +\subsection*{Remarks} +The Chebyshev polynomials can also be written as: +$$T_n(x)=\cos(n\theta),\quad x=\cos\theta$$ +and +$$U_n(x)=\frac{\sin (n+1)\theta}{\sin\theta},\quad x=\cos\theta.$$ +Further we have +$$U_n(x)=C_n^{(1)}(x)$$ +where $C_n^{(\lambda)}(x)$ denotes the Gegenbauer (or ultraspherical) +\subsection*{9.8.2 Chebyshev} +\label{sec9.8.2} +In addition to the Chebyshev polynomials $T_n$ of the first kind (9.8.35) +and $U_n$ of the second kind (9.8.36), +\begin{align} +T_n(x)&:=\frac{P_n^{(-\half,-\half)}(x)}{P_n^{(-\half,-\half)}(1)} +=\cos(n\tha),\quad x=\cos\tha,\\ +U_n(x)&:=(n+1)\,\frac{P_n^{(\half,\half)}(x)}{P_n^{(\half,\half)}(1)} +=\frac{\sin((n+1)\tha)}{\sin\tha}\,,\quad x=\cos\tha, +\end{align} +we have Chebyshev polynomials $V_n$ {\em of the third kind} +and $W_n$ {\em of the fourth kind}, +\begin{align} +V_n(x)&:=\frac{P_n^{(-\half,\half)}(x)}{P_n^{(-\half,\half)}(1)} +=\frac{\cos((n+\thalf)\tha)}{\cos(\thalf\tha)}\,,\quad x=\cos\tha,\\ +W_n(x)&:=(2n+1)\,\frac{P_n^{(\half,-\half)}(x)}{P_n^{(\half,-\half)}(1)} +=\frac{\sin((n+\thalf)\tha)}{\sin(\thalf\tha)}\,,\quad x=\cos\tha, +\end{align} +see \cite[Section 1.2.3]{K20}. Then there is the symmetry +\begin{equation} +V_n(-x)=(-1)^n W_n(x). +\label{140} +\end{equation} + +The names of Chebyshev polynomials of the third and fourth kind +and the notation $V_n(x)$ are due to Gautschi \cite{K21}. +The notation $W_n(x)$ was first used by Mason \cite{K22}. +Names and notations for Chebyshev polynomials of the third and fourth +kind are interchanged in \mycite{AAR}{Remark 2.5.3} and +\mycite{DLMF}{Table 18.3.1}. +polynomial given by (\ref{DefGegenbauer}) in the preceding subsection. + +\subsection*{References} +\cite{Abram}, \cite{AskeyFitch}, \cite{AskeyGasperHarris}, \cite{AskeyIsmail76}, +\cite{Bavinck95}, \cite{Chihara78}, \cite{Danese}, \cite{DilcherStolarsky}, +\cite{Erdelyi+}, \cite{Grad}, \cite{HartmannStephan}, \cite{Ismail2005II}, \cite{Koekoek2000}, +\cite{Luke}, \cite{Mathai}, \cite{Nikiforov+}, \cite{NikiforovUvarov}, \cite{Rainville}, +\cite{Rayes+}, \cite{Rivlin}, \cite{SainteViennot}, \cite{Szego75}, \cite{Temme}, +\cite{Wilson70I}, \cite{Zayed}, \cite{Zhang}, \cite{ZhangWang}. + +\subsection{Legendre / Spherical}\index{Legendre polynomials} +\index{Spherical polynomials} + +\par + +\subsection*{Hypergeometric representation} +The Legendre (or spherical) polynomials are Jacobi polynomials with $\alpha=\beta=0$: +\begin{equation} +\label{DefLegendre} +P_n(x)=P_n^{(0,0)}(x)=\hyp{2}{1}{-n,n+1}{1}{\frac{1-x}{2}}. +\end{equation} + +\subsection*{Orthogonality relation} +\begin{equation} +\label{OrtLegendre} +\int_{-1}^1P_m(x)P_n(x)\,dx=\frac{2}{2n+1}\,\delta_{mn}. +\end{equation} + +\subsection*{Recurrence relation} +\begin{equation} +\label{RecLegendre} +(2n+1)xP_n(x)=(n+1)P_{n+1}(x)+nP_{n-1}(x). +\end{equation} + +\subsection*{Normalized recurrence relation} +\begin{equation} +\label{NormRecLegendre} +xp_n(x)=p_{n+1}(x)+\frac{n^2}{(2n-1)(2n+1)}p_{n-1}(x), +\end{equation} +where +$$P_n(x)=\binom{2n}{n}\frac{1}{2^n}p_n(x).$$ + +\subsection*{Differential equation} +\begin{equation} +\label{dvLegendre} +(1-x^2)y''(x)-2xy'(x)+n(n+1)y(x)=0,\quad y(x)=P_n(x). +\end{equation} + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodLegendre} +P_n(x)=\frac{(-1)^n}{2^nn!}\left(\frac{d}{dx}\right)^n\left[(1-x^2)^n\right]. +\end{equation} + +\subsection*{Generating functions} +\begin{equation} +\label{GenLegendre1} +\frac{1}{\sqrt{1-2xt+t^2}}=\sum_{n=0}^{\infty}P_n(x)t^n. +\end{equation} + +\begin{equation} +\label{GenLegendre2} +\hyp{0}{1}{-}{1}{\frac{(x-1)t}{2}}\,\hyp{0}{1}{-}{1}{\frac{(x+1)t}{2}}= +\sum_{n=0}^{\infty}\frac{P_n(x)}{(n!)^2}t^n. +\end{equation} + +\begin{equation} +\label{GenLegendre3} +\e^{xt}\,\hyp{0}{1}{-}{1}{\frac{(x^2-1)t^2}{4}}= +\sum_{n=0}^{\infty}\frac{P_n(x)}{n!}t^n. +\end{equation} + +\begin{eqnarray} +\label{GenLegendre4} +& &\hyp{2}{1}{\gamma,1-\gamma}{1}{\frac{1-R-t}{2}}\, +\hyp{2}{1}{\gamma,1-\gamma}{1}{\frac{1-R+t}{2}}\nonumber\\ +& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n(1-\gamma)_n}{(n!)^2}P_n(x)t^n, +\quad R=\sqrt{1-2xt+t^2},\quad\textrm{$\gamma$ arbitrary}. +\end{eqnarray} + +\begin{eqnarray} +\label{GenLegendre5} +& &(1-xt)^{-\gamma}\,\hyp{2}{1}{\frac{1}{2}\gamma,\frac{1}{2}\gamma+\frac{1}{2}}{1} +{\frac{(x^2-1)t^2}{(1-xt)^2}}\nonumber\\ +& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n}{n!}P_n(x)t^n,\quad\textrm{$\gamma$ arbitrary}. +\subsection*{9.9 Pseudo Jacobi (or Routh-Romanovski)} +\label{sec9.9} +In this section in \mycite{KLS} the pseudo Jacobi polynomial $P_n(x;\nu,N)$ in (9.9.1) +is considered +for $N\in\ZZ_{\ge0}$ and $n=0,1,\ldots,n$. However, we can more generally take +$-\thalf\be>0\;\;{\rm or}\;\;\al<\be<0). +\] +Then $P_n$ can be expressed as a Meixner polynomial: +\[ +P_n(x)=(-k_2(\al\be)^{-1})_n\,\be^n\, +M_n\left(-\,\frac{x+k_2\al^{-1}}{\al-\be},-k_2(\al\be)^{-1},\be\al^{-1}\right). +\] + +In 1938 Gottlieb \cite[\S2]{K1} introduces polynomials $l_n$ ``of Laguerre type'' +which turn out to be special Meixner polynomials: +$l_n(x)=e^{-n\la} M_n(x;1,e^{-\la})$. +% +\paragraph{Uniqueness of orthogonality measure} +The coefficient of $p_{n-1}(x)$ in (9.10.4) behaves as $O(n^2)$ as $n\to\iy$. +Hence \eqref{93} holds, by which the orthogonality measure is unique. +=\frac{2^n}{(n-2N-1)_n}y_n(x;-2N-2).$$ + +\subsection*{References} +\cite{Askey87}, \cite{BorodinOlshanski}, \cite{Lesky96}. + + +\section{Meixner}\index{Meixner polynomials} + +\par\setcounter{equation}{0} + +\subsection*{Hypergeometric representation} +\begin{equation} +\label{DefMeixner} +M_n(x;\beta,c)=\hyp{2}{1}{-n,-x}{\beta}{1-\frac{1}{c}}. +\end{equation} + +\subsection*{Orthogonality relation} +\begin{equation} +\label{OrtMeixner} +\sum_{x=0}^{\infty}\frac{(\beta)_x}{x!}c^xM_m(x;\beta,c)M_n(x;\beta,c) +{}=\frac{c^{-n}n!}{(\beta)_n(1-c)^{\beta}}\,\delta_{mn} +% \constraint{ +% $\beta > 0$ & +% $0 < c < 1$ } +\end{equation} + +\subsection*{Recurrence relation} +\begin{eqnarray} +\label{RecMeixner} +(c-1)xM_n(x;\beta,c)&=&c(n+\beta)M_{n+1}(x;\beta,c)\nonumber\\ +& &{}\mathindent{}-\left[n+(n+\beta)c\right]M_n(x;\beta,c)+nM_{n-1}(x;\beta,c). +\end{eqnarray} + +\subsection*{Normalized recurrence relation} +\begin{equation} +\label{NormRecMeixner} +xp_n(x)=p_{n+1}(x)+\frac{n+(n+\beta)c}{1-c}p_n(x)+ +\frac{n(n+\beta-1)c}{(1-c)^2}p_{n-1}(x), +\end{equation} +where +$$M_n(x;\beta,c)=\frac{1}{(\beta)_n}\left(\frac{c-1}{c}\right)^np_n(x).$$ + +\subsection*{Difference equation} +\begin{equation} +\label{dvMeixner} +n(c-1)y(x)=c(x+\beta)y(x+1)-\left[x+(x+\beta)c\right]y(x)+xy(x-1), +\end{equation} +where +$$y(x)=M_n(x;\beta,c).$$ + +\subsection*{Forward shift operator} +\begin{equation} +\label{shift1MeixnerI} +M_n(x+1;\beta,c)-M_n(x;\beta,c)= +\frac{n}{\beta}\left(\frac{c-1}{c}\right)M_{n-1}(x;\beta+1,c) +\end{equation} +or equivalently +\begin{equation} +\label{shift1MeixnerII} +\Delta M_n(x;\beta,c)=\frac{n}{\beta}\left(\frac{c-1}{c}\right)M_{n-1}(x;\beta+1,c). +\end{equation} + +\subsection*{Backward shift operator} +\begin{equation} +\label{shift2MeixnerI} +c(\beta+x-1)M_n(x;\beta,c)-xM_n(x-1;\beta,c)=c(\beta-1)M_{n+1}(x;\beta-1,c) +\end{equation} +or equivalently +\begin{equation} +\label{shift2MeixnerII} +\nabla\left[\frac{(\beta)_xc^x}{x!}M_n(x;\beta,c)\right]= +\frac{(\beta-1)_xc^x}{x!}M_{n+1}(x;\beta-1,c). +\end{equation} + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodMeixner} +\frac{(\beta)_xc^x}{x!}M_n(x;\beta,c)=\nabla^n\left[\frac{(\beta+n)_xc^x}{x!}\right]. +\end{equation} + +\subsection*{Generating functions} +\begin{equation} +\label{GenMeixner1} +\left(1-\frac{t}{c}\right)^x(1-t)^{-x-\beta}= +\sum_{n=0}^{\infty}\frac{(\beta)_n}{n!}M_n(x;\beta,c)t^n. +\end{equation} + +\begin{equation} +\label{GenMeixner2} +\e^t\,\hyp{1}{1}{-x}{\beta}{\left(\frac{1-c}{c}\right)t}= +\sum_{n=0}^{\infty}\frac{M_n(x;\beta,c)}{n!}t^n. +\end{equation} + +\begin{equation} +\label{GenMeixner3} +(1-t)^{-\gamma}\,\hyp{2}{1}{\gamma,-x}{\beta}{\frac{(1-c)t}{c(1-t)}} +=\sum_{n=0}^{\infty}\frac{(\gamma)_n}{n!}M_n(x;\beta,c)t^n, +\quad\textrm{$\gamma$ arbitrary}. +\end{equation} + +\subsection*{Limit relations} + +\subsubsection*{Hahn $\rightarrow$ Meixner} +If we take $\alpha=b-1$, $\beta=N(1-c)c^{-1}$ in the definition (\ref{DefHahn}) +of the Hahn polynomials and let $N\rightarrow\infty$ we find the Meixner polynomials: +$$\lim_{N\rightarrow\infty} +Q_n(x;b-1,N(1-c)c^{-1},N)=M_n(x;b,c).$$ + +\subsubsection*{Dual Hahn $\rightarrow$ Meixner} +To obtain the Meixner polynomials from the dual Hahn polynomials we have to take +$\gamma=\beta-1$ and $\delta=N(1-c)c^{-1}$ in the definition (\ref{DefDualHahn}) of +the dual Hahn polynomials and let $N\rightarrow\infty$: +$$\lim_{N\rightarrow\infty} +R_n(\lambda(x);\beta-1,N(1-c)c^{-1},N)=M_n(x;\beta,c).$$ + +\subsubsection*{Meixner $\rightarrow$ Laguerre} +The Laguerre polynomials given by (\ref{DefLaguerre}) are obtained from the Meixner polynomials +if we take $\beta=\alpha+1$ and $x\rightarrow (1-c)^{-1}x$ and let $c\rightarrow 1$: +\begin{equation} +\lim_{c\rightarrow 1} +M_n((1-c)^{-1}x;\alpha+1,c)=\frac{L_n^{(\alpha)}(x)}{L_n^{(\alpha)}(0)}. +\end{equation} + +\subsubsection*{Meixner $\rightarrow$ Charlier} +The Charlier polynomials given by (\ref{DefCharlier}) are obtained from the Meixner polynomials +if we take $c=(a+\beta)^{-1}a$ and let $\beta\rightarrow\infty$: +\begin{equation} +\lim_{\beta\rightarrow\infty} +M_n(x;\beta,(a+\beta)^{-1}a)=C_n(x;a). +\end{equation} + +\subsection*{Remarks} +The Meixner polynomials are related to the Jacobi polynomials given by (\ref{DefJacobi}) +in the following way: +$$\frac{(\beta)_n}{n!}M_n(x;\beta,c)=P_n^{(\beta-1,-n-\beta-x)}((2-c)c^{-1}).$$ + +\noindent +The Meixner polynomials are also related to the Krawtchouk polynomials given by +(\ref{DefKrawtchouk}) in the following way: +\subsection*{9.11 Krawtchouk} +\label{sec9.11} +% +\paragraph{Special values} +By (9.11.1) and the binomial formula: +\begin{equation} +K_n(0;p,N)=1,\qquad +K_n(N;p,N)=(1-p^{-1})^n. +\label{9} +\end{equation} +The self-duality (p.240, Remarks, first formula) +\begin{equation} +K_n(x;p,N)=K_x(n;p,N)\qquad (n,x\in \{0,1,\ldots,N\}) +\label{147} +\end{equation} +combined with \eqref{9} yields: +\begin{equation} +K_N(x;p,N)=(1-p^{-1})^x\qquad(x\in\{0,1,\ldots,N\}). +\label{148} +\end{equation} +% +\paragraph{Symmetry} +By the orthogonality relation (9.11.2): +\begin{equation} +\frac{K_n(N-x;p,N)}{K_n(N;p,N)}=K_n(x;1-p,N). +\label{10} +\end{equation} +By \eqref{10} and \eqref{147} we have also +\begin{equation} +\frac{K_{N-n}(x;p,N)}{K_N(x;p,N)}=K_n(x;1-p,N) +\qquad(n,x\in\{0,1,\ldots,N\}), +\label{149} +\end{equation} +and, by \eqref{149}, \eqref{10} and \eqref{9}, +\begin{equation} +K_{N-n}(N-x;p,N)=\left(\frac p{p-1}\right)^{n+x-N}K_n(x;p,N) +\qquad(n,x\in\{0,1,\ldots,N\}). +\label{150} +\end{equation} +A particular case of \eqref{10} is: +\begin{equation} +K_n(N-x;\thalf,N)=(-1)^n K_n(x;\thalf,N). +\label{11} +\end{equation} +Hence +\begin{equation} +K_{2m+1}(N;\thalf,2N)=0. +\label{12} +\end{equation} +From (9.11.11): +\begin{equation} +K_{2m}(N;\thalf,2N)=\frac{(\thalf)_m}{(-N+\thalf)_m}\,. +\label{13} +\end{equation} +% +\paragraph{Quadratic transformations} +\begin{align} +K_{2m}(x+N;\thalf,2N)&=\frac{(\thalf)_m}{(-N+\thalf)_m}\, +R_m(x^2;-\thalf,-\thalf,N), +\label{31}\\ +K_{2m+1}(x+N;\thalf,2N)&=-\,\frac{(\tfrac32)_m}{N\,(-N+\thalf)_m}\, +x\,R_m(x^2-1;\thalf,\thalf,N-1), +\label{33}\\ +K_{2m}(x+N+1;\thalf,2N+1)&=\frac{(\tfrac12)_m}{(-N-\thalf)_m}\, +R_m(x(x+1);-\thalf,\thalf,N), +\label{32}\\ +K_{2m+1}(x+N+1;\thalf,2N+1)&=\frac{(\tfrac32)_m}{(-N-\thalf)_{m+1}}\, +(x+\thalf)\,R_m(x(x+1);\thalf,-\thalf,N), +\label{34} +\end{align} +where $R_m$ is a dual Hahn polynomial (9.6.1). For the proofs use +(9.6.2), (9.11.2), (9.6.4) and (9.11.4). +% +\paragraph{Generating functions} +\begin{multline} +\sum_{x=0}^N\binom Nx K_m(x;p,N)K_n(x;q,N)z^x\\ +=\left(\frac{p-z+pz}p\right)^m +\left(\frac{q-z+qz}q\right)^n +(1+z)^{N-m-n} +K_m\left(n;-\,\frac{(p-z+pz)(q-z+qz)}z,N\right). +\label{107} +\end{multline} +This follows immediately from Rosengren \cite[(3.5)]{K8}, which goes back +to Meixner \cite{K9}. +$$K_n(x;p,N)=M_n(x;-N,(p-1)^{-1}p).$$ + +\subsection*{References} +\cite{Allaway76}, \cite{NAlSalam66}, \cite{AlSalam90}, \cite{AlSalamChihara76}, +\cite{AlSalamIsmail76}, \cite{Alvarez+}, \cite{AndrewsAskey85}, \cite{Area+II}, \cite{Askey75}, +\cite{Askey89I}, \cite{Askey2005}, \cite{AskeyGasper77}, \cite{AskeyIsmail76}, +\cite{AskeyWilson85}, \cite{AtakRahmanSuslov}, \cite{AtakSuslov88}, \cite{Bavinck98}, +\cite{BavinckHaeringen}, \cite{Campigotto+}, \cite{Chihara78}, \cite{Cooper+}, +\cite{Erdelyi+}, \cite{FoataLabelle}, \cite{Gabutti}, \cite{GabuttiMathis}, \cite{Gasper73I}, +\cite{Gasper74}, \cite{HoareRahman}, \cite{Ismail2005II}, \cite{IsmailLetVal88}, +\cite{IsmailLi}, \cite{IsmailMuldoon}, \cite{IsmailStanton97}, \cite{JinWong}, +\cite{Karlin58}, \cite{Koekoek2000}, \cite{Koorn88}, \cite{LabelleYehI}, \cite{LabelleYehII}, +\cite{Lesky89}, \cite{Lesky94I}, \cite{Lesky95II}, \cite{LewanowiczII}, \cite{Meixner}, +\cite{Nikiforov+}, \cite{NikiforovUvarov}, \cite{Rahman78I}, \cite{ValentAssche}, +\cite{Viennot}, \cite{Zarzo+}, \cite{Zeng90}. + + +\section{Krawtchouk}\index{Krawtchouk polynomials} + +\par\setcounter{equation}{0} + +\subsection*{Hypergeometric representation} +\begin{equation} +\label{DefKrawtchouk} +K_n(x;p,N)=\hyp{2}{1}{-n,-x}{-N}{\frac{1}{p}},\quad n=0,1,2,\ldots,N. +\end{equation} + +\subsection*{Orthogonality relation} +\begin{eqnarray} +\label{OrtKrawtchouk} +& &\sum_{x=0}^N\binom{N}{x}p^x(1-p)^{N-x} K_m(x;p,N)K_n(x;p,N)\nonumber\\ +& &{}=\frac{(-1)^nn!}{(-N)_n}\left(\frac{1-p}{p}\right)^n\,\delta_{mn},\quad0 < p < 1. +\end{eqnarray} + +\subsection*{Recurrence relation} +\begin{eqnarray} +\label{RecKrawtchouk} +-xK_n(x;p,N)&=&p(N-n)K_{n+1}(x;p,N)\nonumber\\ +& &{}\mathindent{}-\left[p(N-n)+n(1-p)\right]K_n(x;p,N)\nonumber\\ +& &{}\mathindent\mathindent{}+n(1-p)K_{n-1}(x;p,N). +\end{eqnarray} + +\subsection*{Normalized recurrence relation} +\begin{eqnarray} +\label{NormRecKrawtchouk} +xp_n(x)&=&p_{n+1}(x)+\left[p(N-n)+n(1-p)\right]p_n(x)\nonumber\\ +& &{}\mathindent{}+np(1-p)(N+1-n)p_{n-1}(x), +\end{eqnarray} +where +$$K_n(x;p,N)=\frac{1}{(-N)_np^n}p_n(x).$$ + +\subsection*{Difference equation} +\begin{eqnarray} +\label{dvKrawtchouk} +-ny(x)&=&p(N-x)y(x+1)\nonumber\\ +& &{}\mathindent{}-\left[p(N-x)+x(1-p)\right]y(x)+x(1-p)y(x-1), +\end{eqnarray} +where +$$y(x)=K_n(x;p,N).$$ + +\subsection*{Forward shift operator} +\begin{equation} +\label{shift1KrawtchoukI} +K_n(x+1;p,N)-K_n(x;p,N)=-\frac{n}{Np}K_{n-1}(x;p,N-1) +\end{equation} +or equivalently +\begin{equation} +\label{shift1KrawtchoukII} +\Delta K_n(x;p,N)=-\frac{n}{Np}K_{n-1}(x;p,N-1). +\end{equation} + +\subsection*{Backward shift operator} +\begin{eqnarray} +\label{shift2KrawtchoukI} +& &(N+1-x)K_n(x;p,N)-x\left(\frac{1-p}{p}\right)K_n(x-1;p,N)\nonumber\\ +& &{}=(N+1)K_{n+1}(x;p,N+1) +\end{eqnarray} +or equivalently +\begin{equation} +\label{shift2KrawtchoukII} +\nabla\left[\binom{N}{x}\left(\frac{p}{1-p}\right)^xK_n(x;p,N)\right]= +\binom{N+1}{x}\left(\frac{p}{1-p}\right)^xK_{n+1}(x;p,N+1). +\end{equation} + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodKrawtchouk} +\binom{N}{x}\left(\frac{p}{1-p}\right)^xK_n(x;p,N)= +\nabla^n\left[\binom{N-n}{x}\left(\frac{p}{1-p}\right)^x\right]. +\end{equation} + +\subsection*{Generating functions} +For $x=0,1,2,\ldots,N$ we have +\begin{equation} +\label{GenKrawtchouk1} +\left(1-\frac{(1-p)}{p}t\right)^x(1+t)^{N-x}= +\sum_{n=0}^N\binom{N}{n}K_n(x;p,N)t^n. +\end{equation} + +\begin{equation} +\label{GenKrawtchouk2} +\left[\expe^t\,\hyp{1}{1}{-x}{-N}{-\frac{t}{p}}\right]_N= +\sum_{n=0}^N\frac{K_n(x;p,N)}{n!}t^n. +\end{equation} + +\begin{eqnarray} +\label{GenKrawtchouk3} +& &\left[(1-t)^{-\gamma}\,\hyp{2}{1}{\gamma,-x}{-N}{\frac{t}{p(t-1)}}\right]_N\nonumber\\ +& &{}=\sum_{n=0}^N\frac{(\gamma)_n}{n!}K_n(x;p,N)t^n, +\quad\textrm{$\gamma$ arbitrary}. +\end{eqnarray} + +\subsection*{Limit relations} + +\subsubsection*{Hahn $\rightarrow$ Krawtchouk} +If we take $\alpha=pt$ and $\beta=(1-p)t$ in the definition (\ref{DefHahn}) of the Hahn +polynomials and let $t\rightarrow\infty$ we obtain the Krawtchouk polynomials: +$$\lim_{t\rightarrow\infty}Q_n(x;pt,(1-p)t,N)=K_n(x;p,N).$$ + +\subsubsection*{Dual Hahn $\rightarrow$ Krawtchouk} +The Krawtchouk polynomials follow from the dual Hahn polynomials given by +(\ref{DefDualHahn}) if we set $\gamma=pt$, $\delta=(1-p)t$ and let $t\rightarrow\infty$: +$$\lim_{t\rightarrow\infty}R_n(\lambda(x);pt,(1-p)t,N)=K_n(x;p,N).$$ + +\subsubsection*{Krawtchouk $\rightarrow$ Charlier} +The Charlier polynomials given by (\ref{DefCharlier}) can be found from the Krawtchouk +polynomials by taking $p=N^{-1}a$ and letting $N\rightarrow\infty$: +\begin{equation} +\lim_{N\rightarrow\infty}K_n(x;N^{-1}a,N)=C_n(x;a). +\end{equation} + +\subsubsection*{Krawtchouk $\rightarrow$ Hermite} +The Hermite polynomials given by (\ref{DefHermite}) follow from the Krawtchouk polynomials +by setting $x\rightarrow pN+x\sqrt{2p(1-p)N}$ and then letting $N\rightarrow\infty$: +\begin{equation} +\lim_{N\rightarrow\infty} +\sqrt{\binom{N}{n}}K_n(pN+x\sqrt{2p(1-p)N};p,N) +=\frac{\displaystyle (-1)^nH_n(x)}{\displaystyle\sqrt{2^nn!\left(\frac{p}{1-p}\right)^n}}. +\end{equation} + +\subsection*{Remarks} +The Krawtchouk polynomials are self-dual, which means that +$$K_n(x;p,N)=K_x(n;p,N),\quad n,x\in\{0,1,2,\ldots,N\}.$$ +By using this relation we easily obtain the so-called dual orthogonality +relation from the orthogonality relation (\ref{OrtKrawtchouk}): +$$\sum_{n=0}^N\binom{N}{n}p^n(1-p)^{N-n} K_n(x;p,N)K_n(y;p,N)= +\frac{\displaystyle\left(\frac{1-p}{p}\right)^x}{\dbinom{N}{x}}\delta_{xy},$$ +where $0 < p < 1$ and $x,y\in\{0,1,2,\ldots,N\}$. + +\noindent +The Krawtchouk polynomials are related to the Meixner polynomials given by (\ref{DefMeixner}) +in the following way: +\subsection*{9.12 Laguerre} +\label{sec9.12} +\paragraph{Notation} +Here the Laguerre polynomial is denoted by $L_n^\al$ instead of +$L_n^{(\al)}$. +% +\paragraph{Hypergeometric representation} +\begin{align} +L_n^\al(x)&= +\frac{(\al+1)_n}{n!}\,\hyp11{-n}{\al+1}x +\label{182}\\ +&=\frac{(-x)^n}{n!} \hyp20{-n,-n-\al}-{-\,\frac1x} +\label{183}\\ +&=\frac{(-x)^n}{n!}\,C_n(n+\al;x), +\label{184} +\end{align} +where $C_n$ in \eqref{184} is a +\hyperref[sec9.14]{Charlier polynomial}. +Formula \eqref{182} is (9.12.1). Then \eqref{183} follows by reversal +of summation. Finally \eqref{184} follows by \eqref{183} and \eqref{179}. +It is also the remark on top of p.244 in \mycite{KLS}, and it is essentially +\myciteKLS{416}{(2.7.10)}. +% +\paragraph{Uniqueness of orthogonality measure} +The coefficient of $p_{n-1}(x)$ in (9.12.4) behaves as $O(n^2)$ as $n\to\iy$. +Hence \eqref{93} holds, by which the orthogonality measure is unique. +% +\paragraph{Special value} +\begin{equation} +L_n^{\al}(0)=\frac{(\al+1)_n}{n!}\,. +\label{53} +\end{equation} +Use (9.12.1) or see \mycite{DLMF}{18.6.1)}. +% +\paragraph{Quadratic transformations} +\begin{align} +H_{2n}(x)&=(-1)^n\,2^{2n}\,n!\,L_n^{-1/2}(x^2), +\label{54}\\ +H_{2n+1}(x)&=(-1)^n\,2^{2n+1}\,n!\,x\,L_n^{1/2}(x^2). +\label{55} +\end{align} +See p.244, Remarks, last two formulas. +Or see \mycite{DLMF}{(18.7.19), (18.7.20)}. +% +\paragraph{Fourier transform} +\begin{equation} +\frac1{\Ga(\al+1)}\,\int_0^\iy \frac{L_n^\al(y)}{L_n^\al(0)}\, +e^{-y}\,y^\al\,e^{ixy}\,dy= +i^n\,\frac{y^n}{(iy+1)^{n+\al+1}}\,, +\label{14} +\end{equation} +see \mycite{DLMF}{(18.17.34)}. +% +\paragraph{Differentiation formulas} +Each differentiation formula is given in two equivalent forms. +\begin{equation} +\frac d{dx}\left(x^\al L_n^\al(x)\right)= +(n+\al)\,x^{\al-1} L_n^{\al-1}(x),\qquad +\left(x\frac d{dx}+\al\right)L_n^\al(x)= +(n+\al)\,L_n^{\al-1}(x). +\label{76} +\end{equation} +% +\begin{equation} +\frac d{dx}\left(e^{-x} L_n^\al(x)\right)= +-e^{-x} L_n^{\al+1}(x),\qquad +\left(\frac d{dx}-1\right)L_n^\al(x)= +-L_n^{\al+1}(x). +\label{77} +\end{equation} +% +Formulas \eqref{76} and \eqref{77} follow from +\mycite{DLMF}{(13.3.18), (13.3.20)} +together with (9.12.1). +% +\paragraph{Generalized Hermite polynomials} +See \myciteKLS{146}{p.156}, \cite[Section 1.5.1]{K26}. +These are defined by +\begin{equation} +H_{2m}^\mu(x):=\const L_m^{\mu-\half}(x^2),\qquad +H_{2m+1}^\mu(x):=\const x\,L_m^{\mu+\half}(x^2). +\label{78} +\end{equation} +Then for $\mu>-\thalf$ we have orthogonality relation +\begin{equation} +\int_{-\iy}^{\iy} H_m^\mu(x)\,H_n^\mu(x)\,|x|^{2\mu}e^{-x^2}\,dx +=0\qquad(m\ne n). +\label{79} +\end{equation} +Let the Dunkl operator $T_\mu$ be defined by \eqref{72}. +If we choose the constants in \eqref{78} as +\begin{equation} +H_{2m}^\mu(x)=\frac{(-1)^m(2m)!}{(\mu+\thalf)_m}\,L_m^{\mu-\half}(x^2),\qquad +H_{2m+1}^\mu(x)=\frac{(-1)^m(2m+1)!}{(\mu+\thalf)_{m+1}}\, + x\,L_m^{\mu+\half}(x^2) + \label{80} +\end{equation} +then (see \cite[(1.6)]{K5}) +\begin{equation} +T_\mu H_n^\mu=2n\,H_{n-1}^\mu. +\label{81} +\end{equation} +Formula \eqref{81} with \eqref{80} substituted gives rise to two +differentiation formulas involving Laguerre polynomials which are equivalent to +(9.12.6) and \eqref{76}. + +Composition of \eqref{81} with itself gives +\[ +T_\mu^2 H_n^\mu=4n(n-1)\,H_{n-2}^\mu, +\] +which is equivalent to the composition of (9.12.6) and \eqref{76}: +\begin{equation} +\left(\frac{d^2}{dx^2}+\frac{2\al+1}x\,\frac d{dx}\right)L_n^\al(x^2) +=-4(n+\al)\,L_{n-1}^\al(x^2). +\label{82} +\end{equation} +$$K_n(x;p,N)=M_n(x;-N,(p-1)^{-1}p).$$ + +\subsection*{References} +\cite{AlSalam90}, \cite{AndrewsAskey85}, \cite{Area+II}, \cite{Askey75}, \cite{Askey89I}, +\cite{AskeyGasper77}, \cite{AskeyWilson85}, \cite{AtakRahmanSuslov}, \cite{Campigotto+}, +\cite{LChiharaStanton}, \cite{Chihara78}, \cite{Dette95}, \cite{Dominici}, \cite{Dunkl76}, +\cite{Dunkl84}, \cite{DunklRamirez}, \cite{Erdelyi+}, \cite{FeinsilverSchott}, +\cite{Gasper73I}, \cite{Gasper74}, \cite{HoareRahman}, \cite{Ismail2005II}, \cite{Karlin58}, +\cite{Koorn82}, \cite{Koorn88}, \cite{LabelleYehI}, \cite{LabelleYehII}, \cite{Lesky62}, +\cite{Lesky89}, \cite{Lesky94I}, \cite{Lesky95II}, \cite{LewanowiczII}, \cite{Nikiforov+}, +\cite{NikiforovUvarov}, \cite{Qiu}, \cite{Rahman78I}, \cite{Rahman79}, \cite{Stanton84}, +\cite{Stanton90}, \cite{Szego75}, \cite{Zarzo+}, \cite{Zeng90}. + + +\section{Laguerre}\index{Laguerre polynomials} + +\par\setcounter{equation}{0} + +\subsection*{Hypergeometric representation} +\begin{equation} +\label{DefLaguerre} +L_n^{(\alpha)}(x)=\frac{(\alpha+1)_n}{n!}\,\hyp{1}{1}{-n}{\alpha+1}{x}. +\end{equation} + +\subsection*{Orthogonality relation} +\begin{equation} +\label{OrtLaguerre} +\int_0^{\infty}\expe^{-x}x^{\alpha}L_m^{(\alpha)}(x)L_n^{(\alpha)}(x)\,dx= +\frac{\Gamma(n+\alpha+1)}{n!}\,\delta_{mn},\quad\alpha > -1. +\end{equation} + +\subsection*{Recurrence relation} +\begin{equation} +\label{RecLaguerre} +(n+1)L_{n+1}^{(\alpha)}(x)-(2n+\alpha+1-x)L_n^{(\alpha)}(x)+(n+\alpha)L_{n-1}^{(\alpha)}(x)=0. +\end{equation} + +\subsection*{Normalized recurrence relation} +\begin{equation} +\label{NormRecLaguerre} +xp_n(x)=p_{n+1}(x)+(2n+\alpha+1)p_n(x)+n(n+\alpha)p_{n-1}(x), +\end{equation} +where +$$L_n^{(\alpha)}(x)=\frac{(-1)^n}{n!}p_n(x).$$ + +\subsection*{Differential equation} +\begin{equation} +\label{dvLaguerre} +xy''(x)+(\alpha+1-x)y'(x)+ny(x)=0,\quad y(x)=L_n^{(\alpha)}(x). +\end{equation} + +\newpage + +\subsection*{Forward shift operator} +\begin{equation} +\label{shift1Laguerre} +\frac{d}{dx}L_n^{(\alpha)}(x)=-L_{n-1}^{(\alpha+1)}(x). +\end{equation} + +\subsection*{Backward shift operator} +\begin{equation} +\label{shift2LaguerreI} +x\frac{d}{dx}L_n^{(\alpha)}(x)+(\alpha-x)L_n^{(\alpha)}(x)=(n+1)L_{n+1}^{(\alpha-1)}(x) +\end{equation} +or equivalently +\begin{equation} +\label{shift2LaguerreII} +\frac{d}{dx}\left[\expe^{-x}x^{\alpha}L_n^{(\alpha)}(x)\right]=(n+1)\expe^{-x}x^{\alpha-1}L^{(\alpha-1)}_{n+1}(x). +\end{equation} + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodLaguerre} +\e^{-x}x^{\alpha}L_n^{(\alpha)}(x)=\frac{1}{n!}\left(\frac{d}{dx}\right)^n\left[\expe^{-x}x^{n+\alpha}\right]. +\end{equation} + +\subsection*{Generating functions} +\begin{equation} +\label{GenLaguerre1} +(1-t)^{-\alpha-1}\exp\left(\frac{xt}{t-1}\right)= +\sum_{n=0}^{\infty}L_n^{(\alpha)}(x)t^n. +\end{equation} + +\begin{equation} +\label{GenLaguerre2} +\e^t\,\hyp{0}{1}{-}{\alpha+1}{-xt} +=\sum_{n=0}^{\infty}\frac{L_n^{(\alpha)}(x)}{(\alpha+1)_n}t^n. +\end{equation} + +\begin{equation} +\label{GenLaguerre3} +(1-t)^{-\gamma}\,\hyp{1}{1}{\gamma}{\alpha+1}{\frac{xt}{t-1}} +=\sum_{n=0}^{\infty}\frac{(\gamma)_n}{(\alpha+1)_n}L_n^{(\alpha)}(x)t^n, +\quad\textrm{$\gamma$ arbitrary}. +\end{equation} + +\subsection*{Limit relations} + +\subsubsection*{Meixner-Pollaczek $\rightarrow$ Laguerre} +The Laguerre polynomials can be obtained from the Meixner-Pollaczek polynomials given by +(\ref{DefMP}) by the substitution $\lambda=\frac{1}{2}(\alpha+1)$, +$x\rightarrow -\frac{1}{2}\phi^{-1}x$ and the limit $\phi\rightarrow 0$: +$$\lim_{\phi\rightarrow 0} +P_n^{(\frac{1}{2}\alpha+\frac{1}{2})}(-\textstyle\frac{1}{2}\phi^{-1}x;\phi)=L_n^{(\alpha)}(x).$$ + +\subsubsection*{Jacobi $\rightarrow$ Laguerre} +The Laguerre polynomials are obtained from the Jacobi polynomials given by (\ref{DefJacobi}) +if we set $x\rightarrow 1-2\beta^{-1}x$ and then take the limit $\beta\rightarrow\infty$: +$$\lim_{\beta\rightarrow\infty} +P_n^{(\alpha,\beta)}(1-2\beta^{-1}x)=L_n^{(\alpha)}(x).$$ + +\subsubsection*{Meixner $\rightarrow$ Laguerre} +If we take $\beta=\alpha+1$ and $x\rightarrow (1-c)^{-1}x$ in the definition +(\ref{DefMeixner}) of the Meixner polynomials and let $c\rightarrow 1$ we obtain +the Laguerre polynomials: +$$\lim_{c\rightarrow 1} +M_n((1-c)^{-1}x;\alpha+1,c)=\frac{L_n^{(\alpha)}(x)}{L_n^{(\alpha)}(0)}.$$ + +\subsubsection*{Laguerre $\rightarrow$ Hermite} +The Hermite polynomials given by (\ref{DefHermite}) can be obtained from the Laguerre +polynomials by taking the limit $\alpha\rightarrow\infty$ in the following way: +\begin{equation} +\lim_{\alpha\rightarrow\infty} +\left(\frac{2}{\alpha}\right)^{\frac{1}{2}n} +L_n^{(\alpha)}((2\alpha)^{\frac{1}{2}}x+\alpha)=\frac{(-1)^n}{n!}H_n(x). +\end{equation} + +\subsection*{Remarks} +The definition (\ref{DefLaguerre}) of the Laguerre polynomials can also be +written as: +$$L_n^{(\alpha)}(x)=\frac{1}{n!}\sum_{k=0}^n\frac{(-n)_k}{k!}(\alpha+k+1)_{n-k}x^k.$$ +In this way the Laguerre polynomials can also be seen as polynomials in the parameter $\alpha$. +Therefore they can be defined for all $\alpha$. + +\noindent +The Laguerre polynomials are related to the Bessel polynomials given by (\ref{DefBessel}) +in the following way: +$$L_n^{(\alpha)}(x)=\frac{(-x)^n}{n!}y_n(2x^{-1};-2n-\alpha-1).$$ + +\noindent +The Laguerre polynomials are related to the Charlier polynomials given by (\ref{DefCharlier}) +in the following way: +$$\frac{(-a)^n}{n!}C_n(x;a)=L_n^{(x-n)}(a).$$ + +\noindent +The Laguerre polynomials and the Hermite polynomials given by (\ref{DefHermite}) are also +connected by the following quadratic transformations: +$$H_{2n}(x)=(-1)^nn!\,2^{2n}L_n^{(-\frac{1}{2})}(x^2)$$ +and +$$H_{2n+1}(x)=(-1)^nn!\,2^{2n+1}xL_n^{(\frac{1}{2})}(x^2).$$ + +\noindent +In combinatorics the Laguerre polynomials with $\alpha=0$ are often called Rook +\subsection*{9.14 Charlier} +\label{sec9.14} +% +\paragraph{Hypergeometric representation} +\begin{align} +C_n(x;a)&=\hyp20{-n,-x}-{-\,\frac1a} +\label{179}\\ +&=\frac{(-x)_n}{a^n} \hyp11{-n}{x-n+1}a +\label{180}\\ +&=\frac{n!}{(-a)^n}\,L_n^{x-n}(a), +\label{181} +\end{align} +where $L_n^\al(x)$ is a +\hyperref[sec9.12]{Laguerre polynomial}. +Formula \eqref{179} is (9.14.1). Then \eqref{180} follows by reversal +of the summation. Finally \eqref{181} follows by \eqref{180} and +(9.12.1). It is also the Remark on p.249 of \mycite{KLS}, and it +was earlier given in \myciteKLS{416}{(2.7.10)}. +% +\paragraph{Uniqueness of orthogonality measure} +The coefficient of $p_{n-1}(x)$ in (9.14.4) behaves as $O(n)$ as $n\to\iy$. +Hence \eqref{93} holds, by which the orthogonality measure is unique. +polynomials. + +\subsection*{References} +\cite{Abdul}, \cite{Abram}, \cite{Ahmed+82}, \cite{Allaway76}, \cite{Allaway91}, +\cite{NAlSalam66}, \cite{AlSalam64}, \cite{AlSalam90}, \cite{AlSalamChihara72}, +\cite{AlSalamChihara76}, \cite{AndrewsAskey85}, \cite{AndrewsAskeyRoy}, \cite{Askey68}, +\cite{Askey75}, \cite{Askey89I}, \cite{Askey2005}, \cite{AskeyGasper76}, \cite{AskeyGasper77}, +\cite{AskeyIsmail76}, \cite{AskeyIsmailKoorn}, \cite{AskeyWilson85}, \cite{Barrett}, +\cite{Bavinck96}, \cite{Brafman51}, \cite{Brafman57II}, \cite{Brenke}, \cite{Brown}, +\cite{BustozSavage79}, \cite{BustozSavage80}, \cite{Carlitz60}, \cite{Carlitz61I}, +\cite{Carlitz61II}, \cite{Carlitz62}, \cite{Carlitz68}, \cite{ChenSrivastava}, +\cite{ChenIsmail91}, \cite{Chihara68II}, \cite{Chihara78}, \cite{Cohen}, \cite{Cooper+}, +\cite{Dattoli2001}, \cite{DetteStudden92}, \cite{DetteStudden95}, \cite{Dimitrov2003}, +\cite{Doha2003II}, \cite{ElbertLaforgia87I}, \cite{Erdelyi}, \cite{Erdelyi+}, \cite{Exton98}, +\cite{Faldey}, \cite{Gasper73II}, \cite{Gasper77}, \cite{Gatteschi2002}, \cite{Gawronski87}, +\cite{Gawronski93}, \cite{Gillis}, \cite{GillisIsmailOffer}, \cite{Godoy+}, \cite{Grad}, +\cite{Hajmirzaahmad95}, \cite{Ismail74}, \cite{Ismail77}, \cite{Ismail2005II}, +\cite{IsmailLetVal88}, \cite{IsmailLi}, \cite{IsmailMassonRahman}, \cite{IsmailMuldoon}, +\cite{IsmailStanton97}, \cite{IsmailTamhankar}, \cite{Jackson}, \cite{Karlin58}, +\cite{Kochneff95}, \cite{Kochneff97II}, \cite{Koekoek2000}, \cite{Koorn77I}, \cite{Koorn78}, +\cite{Koorn85}, \cite{Koorn88}, \cite{Krasikov2003}, \cite{Krasikov2005}, +\cite{KuijlaarsMcLaughlin}, \cite{KwonLittle}, \cite{LabelleYehI}, \cite{LabelleYehII}, +\cite{LabelleYehIII}, \cite{Lee97I}, \cite{Lee97II}, \cite{Lee97III}, \cite{Lesky95II}, +\cite{Lesky96}, \cite{LewanowiczI}, \cite{LopezTemme2004}, \cite{Mathai}, \cite{Meixner}, +\cite{Nikiforov+}, \cite{NikiforovUvarov}, \cite{Olver}, \cite{PerezPinar}, \cite{Pittaluga}, +\cite{Rainville}, \cite{SainteViennot}, \cite{Srivastava66}, \cite{Srivastava69I}, +\cite{Srivastava69II}, \cite{Srivastava70}, \cite{Srivastava71}, \cite{Szego75}, \cite{Temme}, +\cite{Trickovic}, \cite{Trivedi}, \cite{Viennot}. + + +\section{Bessel}\index{Bessel polynomials} + +\par\setcounter{equation}{0} + +\subsection*{Hypergeometric representation} +\begin{eqnarray} +\label{DefBessel} +y_n(x;a)&=&\hyp{2}{0}{-n,n+a+1}{-}{-\frac{x}{2}}\\ +&=&(n+a+1)_n\left(\frac{x}{2}\right)^n\,\hyp{1}{1}{-n}{-2n-a}{\frac{2}{x}}, +\quad n=0,1,2,\ldots,N.\nonumber +\end{eqnarray} + +\subsection*{Orthogonality relation} +\begin{eqnarray} +\label{OrtBessel} +& &\int_0^{\infty}x^a\e^{-\frac{2}{x}}y_m(x;a)y_n(x;a)\,dx\nonumber\\ +& &=-\frac{2^{a+1}}{2n+a+1}\Gamma(-n-a)n!\,\delta_{mn},\quad a<-2N-1. +\end{eqnarray} + +\subsection*{Recurrence relation} +\begin{eqnarray} +\label{RecBessel} +& &2(n+a+1)(2n+a)y_{n+1}(x;a)\nonumber\\ +& &{}=(2n+a+1)\left[2a+(2n+a)(2n+a+2)x\right]y_n(x;a)\nonumber\\ +& &{}\mathindent{}+2n(2n+a+2)y_{n-1}(x;a). +\end{eqnarray} + +\subsection*{Normalized recurrence relation} +\begin{eqnarray} +\label{NormRecBessel} +xp_n(x)&=&p_{n+1}(x)-\frac{2a}{(2n+a)(2n+a+2)}p_n(x)\nonumber\\ +& &{}\mathindent{}-\frac{4n(n+a)}{(2n+a-1)(2n+a)^2(2n+a+1)}p_{n-1}(x), +\end{eqnarray} +where +$$y_n(x;a)=\frac{(n+a+1)_n}{2^n}p_n(x).$$ + +\subsection*{Differential equation} +\begin{equation} +\label{dvBessel} +x^2y''(x)+\left[(a+2)x+2\right]y'(x)-n(n+a+1)y(x)=0,\quad y(x)=y_n(x;a). +\end{equation} + +\subsection*{Forward shift operator} +\begin{equation} +\label{shift1Bessel} +\frac{d}{dx}y_n(x;a)=\frac{n(n+a+1)}{2}y_{n-1}(x;a+2). +\end{equation} + +\subsection*{Backward shift operator} +\begin{equation} +\label{shift2BesselI} +x^2\frac{d}{dx}y_n(x;a)+(ax+2)y_n(x;a)=2y_{n+1}(x;a-2) +\end{equation} +or equivalently +\begin{equation} +\label{shift2BesselII} +\frac{d}{dx}\left[x^a\expe^{-\frac{2}{x}}y_n(x;a)\right]=2x^{a-2}\expe^{-\frac{2}{x}}y_{n+1}(x;a-2). +\end{equation} + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodBessel} +y_n(x;a)=2^{-n}x^{-a}\expe^{\frac{2}{x}}D^n\left(x^{2n+a}\expe^{-\frac{2}{x}}\right). +\end{equation} + +\subsection*{Generating function} +\begin{equation} +\label{GenBessel} +\left(1-2xt\right)^{-\frac{1}{2}}\left(\frac{2}{1+\sqrt{1-2xt}}\right)^a +\exp\left(\frac{2t}{1+\sqrt{1-2xt}}\right)=\sum_{n=0}^{\infty}y_n(x;a)\frac{t^n}{n!}. +\end{equation} + +\subsection*{Limit relation} + +\subsubsection*{Jacobi $\rightarrow$ Bessel} +If we take $\beta=a-\alpha$ in the definition (\ref{DefJacobi}) of the Jacobi polynomials +and let $\alpha\rightarrow -\infty$ we find the Bessel polynomials: +$$\lim_{\alpha\rightarrow -\infty} +\frac{P_n^{(\alpha,a-\alpha)}(1+\alpha x)}{P_n^{(\alpha,a-\alpha)}(1)}=y_n(x;a).$$ + +\subsection*{Remarks} +The following notations are also used for the Bessel polynomials: +$$y_n(x;a,b)=y_n(2b^{-1}x;a)\quad\textrm{and}\quad\theta_n(x;a,b)=x^ny_n(x^{-1};a,b).$$ +However, the Bessel polynomials essentially depend on only one parameter. + +\noindent +The Bessel polynomials are related to the Laguerre polynomials given by (\ref{DefLaguerre}) +in the following way: +$$L_n^{(\alpha)}(x)=\frac{(-x)^n}{n!}y_n(2x^{-1};-2n-\alpha-1).$$ + +\noindent +The special case $a=-2N-2$ of the Bessel polynomials can be obtained from the pseudo Jacobi +polynomials by setting $x\rightarrow\nu x$ in the definition (\ref{DefPseudoJacobi}) of the +pseudo Jacobi polynomials and taking the limit $\nu\rightarrow\infty$ in the following way: +$$\lim\limits_{\nu\rightarrow\infty}\frac{P_n(\nu x;\nu,N)}{\nu^n} +\subsection*{9.15 Hermite} +\label{sec9.15} +% +\paragraph{Uniqueness of orthogonality measure} +The coefficient of $p_{n-1}(x)$ in (9.15.4) behaves as $O(n)$ as $n\to\iy$. +Hence \eqref{93} holds, by which the orthogonality measure is unique. +% +\paragraph{Fourier transforms} +\begin{equation} +\frac1{\sqrt{2\pi}}\,\int_{-\iy}^\iy H_n(y)\,e^{-\half y^2}\,e^{ixy}\,dy= +i^n\,H_n(x)\,e^{-\half x^2}, +\label{15} +\end{equation} +see \mycite{AAR}{(6.1.15) and Exercise 6.11}. +\begin{equation} +\frac1{\sqrt\pi}\,\int_{-\iy}^\iy H_n(y)\,e^{-y^2}\,e^{ixy}\,dy= +i^n\,x^n\,e^{-\frac14 x^2}, +\label{16} +\end{equation} +see \mycite{DLMF}{(18.17.35)}. +\begin{equation} +\frac{i^n}{2\sqrt\pi}\,\int_{-\iy}^\iy y^n\,e^{-\frac14 y^2}\,e^{-ixy}\,dy= +H_n(x)\,e^{-x^2}, +\label{17} +\end{equation} +see \mycite{AAR}{(6.1.4)}. +=\frac{2^n}{(n-2N-1)_n}y_n(x;-2N-2).$$ + +\subsection*{References} +\cite{Andrade}, \cite{BergVignat}, \cite{Carlitz57I}, \cite{Dattoli2003}, +\cite{DohaAhmed2004}, \cite{DohaAhmed2006}, \cite{Grosswald}, \cite{Ismail2005II}, +\cite{KrallFrink}, \cite{Lesky98}, \cite{NikiforovUvarov}. + + +\section{Charlier}\index{Charlier polynomials} + +\par\setcounter{equation}{0} + +\subsection*{Hypergeometric representation} +\begin{equation} +\label{DefCharlier} +C_n(x;a)=\hyp{2}{0}{-n,-x}{-}{-\frac{1}{a}}. +\end{equation} + +\subsection*{Orthogonality relation} +\begin{equation} +\label{OrtCharlier} +\sum_{x=0}^{\infty}\frac{a^x}{x!}C_m(x;a)C_n(x;a)= +a^{-n}\expe^an!\,\delta_{mn},\quad a > 0. +\end{equation} + +\subsection*{Recurrence relation} +\begin{equation} +\label{RecCharlier} +-xC_n(x;a)=aC_{n+1}(x;a)-(n+a)C_n(x;a)+nC_{n-1}(x;a). +\end{equation} + +\subsection*{Normalized recurrence relation} +\begin{equation} +\label{NormRecCharlier} +xp_n(x)=p_{n+1}(x)+(n+a)p_n(x)+nap_{n-1}(x), +\end{equation} +where +$$C_n(x;a)=\left(-\frac{1}{a}\right)^np_n(x).$$ + +\subsection*{Difference equation} +\begin{equation} +\label{dvCharlier} +-ny(x)=ay(x+1)-(x+a)y(x)+xy(x-1),\quad y(x)=C_n(x;a). +\end{equation} + +\subsection*{Forward shift operator} +\begin{equation} +\label{shift1CharlierI} +C_n(x+1;a)-C_n(x;a)=-\frac{n}{a}C_{n-1}(x;a) +\end{equation} +or equivalently +\begin{equation} +\label{shift1CharlierII} +\Delta C_n(x;a)=-\frac{n}{a}C_{n-1}(x;a). +\end{equation} + +\subsection*{Backward shift operator} +\begin{equation} +\label{shift2CharlierI} +C_n(x;a)-\frac{x}{a}C_n(x-1;a)=C_{n+1}(x;a) +\end{equation} +or equivalently +\begin{equation} +\label{shift2CharlierII} +\nabla\left[\frac{a^x}{x!}C_n(x;a)\right]=\frac{a^x}{x!}C_{n+1}(x;a). +\end{equation} + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodCharlier} +\frac{a^x}{x!}C_n(x;a)=\nabla^n\left[\frac{a^x}{x!}\right]. +\end{equation} + +\subsection*{Generating function} +\begin{equation} +\label{GenCharlier} +\e^t\left(1-\frac{t}{a}\right)^x=\sum_{n=0}^{\infty}\frac{C_n(x;a)}{n!}t^n. +\end{equation} + +\subsection*{Limit relations} + +\subsubsection*{Meixner $\rightarrow$ Charlier} +If we take $c=(a+\beta)^{-1}a$ in the definition (\ref{DefMeixner}) of the Meixner polynomials +and let $\beta\rightarrow\infty$ we find the Charlier polynomials: +$$\lim_{\beta\rightarrow\infty}M_n(x;\beta,(a+\beta)^{-1}a)=C_n(x;a).$$ + +\subsubsection*{Krawtchouk $\rightarrow$ Charlier} +The Charlier polynomials can be found from the Krawtchouk polynomials given by +(\ref{DefKrawtchouk}) by taking $p=N^{-1}a$ and letting $N\rightarrow\infty$: +$$\lim_{N\rightarrow\infty}K_n(x;N^{-1}a,N)=C_n(x;a).$$ + +\subsubsection*{Charlier $\rightarrow$ Hermite} +The Hermite polynomials given by (\ref{DefHermite}) are obtained from the Charlier polynomials +if we set $x\rightarrow (2a)^{1/2}x+a$ and let $a\rightarrow\infty$. In fact we have +\begin{equation} +\lim_{a\rightarrow\infty} +(2a)^{\frac{1}{2}n}C_n((2a)^{\frac{1}{2}}x+a;a)=(-1)^nH_n(x). +\end{equation} + +\subsection*{Remark} +The Charlier polynomials are related to the Laguerre polynomials given by (\ref{DefLaguerre}) +in the following way: +\subsection*{14.1 Askey-Wilson} +\label{sec14.1} +% +\paragraph{Symmetry} +The Askey-Wilson polynomials $p_n(x;a,b,c,d\,|\,q)$ are \,|\,symmetric +in $a,b,c,d$. +\sLP +This follows from the orthogonality relation (14.1.2) +together with the value of its coefficient of $x^n$ given in (14.1.5b). +Alternatively, combine (14.1.1) with \mycite{GR}{(III.15)}.\\ +As a consequence, it is sufficient to give generating function (14.1.13). Then the generating +functions (14.1.14), (14.1.15) will follow by symmetry in the parameters. +% +\paragraph{Basic hypergeometric representation} +In addition to (14.1.1) we have (in notation \eqref{111}): +\begin{multline} +p_n(\cos\tha;a,b,c,d\,|\, q) +=\frac{(ae^{-i\tha},be^{-i\tha},ce^{-i\tha},de^{-i\tha};q)_n} +{(e^{-2i\tha};q)_n}\,e^{in\tha}\\ +\times {}_8W_7\big(q^{-n}e^{2i\tha};ae^{i\tha},be^{i\tha}, +ce^{i\tha},de^{i\tha},q^{-n};q,q^{2-n}/(abcd)\big). +\label{113} +\end{multline} +This follows from (14.1.1) by combining (III.15) and (III.19) in +\mycite{GR}. +It is also given in \myciteKLS{513}{(4.2)}, but be aware for some slight errors. +The symmetry in $a,b,c,d$ is evident from \eqref{113}. +% +\paragraph{Special value} +\begin{equation} +p_n\big(\thalf(a+a^{-1});a,b,c,d\,|\, q\big)=a^{-n}\,(ab,ac,ad;q)_n\,, +\label{40} +\end{equation} +and similarly for arguments $\thalf(b+b^{-1})$, $\thalf(c+c^{-1})$ and +$\thalf(d+d^{-1})$ by symmetry of $p_n$ in $a,b,c,d$. +% +\paragraph{Trivial symmetry} +\begin{equation} +p_n(-x;a,b,c,d\,|\, q)=(-1)^n p_n(x;-a,-b,-c,-d\,|\, q). +\label{41} +\end{equation} +Both \eqref{40} and \eqref{41} are obtained from (14.1.1). +% +\paragraph{Re: (14.1.5)} +Let +\begin{equation} +p_n(x):=\frac{p_n(x;a,b,c,d\,|\, q)}{2^n(abcdq^{n-1};q)_n}=x^n+\wt k_n x^{n-1} ++\cdots\;. +\label{18} +\end{equation} +Then +\begin{equation} +\wt k_n=-\frac{(1-q^n)(a+b+c+d-(abc+abd+acd+bcd)q^{n-1})} +{2(1-q)(1-abcdq^{2n-2})}\,. +\label{19} +\end{equation} +This follows because $\tilde k_n-\tilde k_{n+1}$ equals the coefficient +$\thalf\bigl(a+a^{-1}-(A_n+C_n)\bigr)$ of $p_n(x)$ in (14.1.5). +% +\paragraph{Generating functions} +Rahman \myciteKLS{449}{(4.1), (4.9)} gives: +\begin{align} +&\sum_{n=0}^\iy \frac{(abcdq^{-1};q)_n a^n}{(ab,ac,ad,q;q)_n}\,t^n\, +p_n(\cos\tha;a,b,c,d\,|\,q) +\nonumber\\ +&=\frac{(abcdtq^{-1};q)_\iy}{(t;q)_\iy}\, +\qhyp65{(abcdq^{-1})^\half,-(abcdq^{-1})^\half,(abcd)^\half, +-(abcd)^\half,a e^{i\tha},a e^{-i\tha}} +{ab,ac,ad,abcdtq^{-1},qt^{-1}}{q,q} +\nonumber\\ +&+\frac{(abcdq^{-1},abt,act,adt,ae^{i\tha},ae^{-i\tha};q)_\iy} +{(ab,ac,ad,t^{-1},ate^{i\tha},ate^{-i\tha};q)_\iy} +\nonumber\\ +&\times\qhyp65{t(abcdq^{-1})^\half,-t(abcdq^{-1})^\half,t(abcd)^\half, +-t(abcd)^\half,at e^{i\tha},at e^{-i\tha}} +{abt,act,adt,abcdt^2q^{-1},qt}{q,q}\quad(|t|<1). +\label{185} +\end{align} +In the limit \eqref{109} the first term on the \RHS\ of \eqref{185} +tends to the \LHS\ of (9.1.15), while the second term tends formally +to 0. The special case $ad=bc$ of \eqref{185} was earlier given in +\myciteKLS{236}{(4.1), (4.6)}. +% +\paragraph{Limit relations}\quad\sLP +{\bf Askey-Wilson $\longrightarrow$ Wilson}\\ +Instead of (14.1.21) we can keep a polynomial of degree $n$ while the limit is approached: +\begin{equation} +\lim_{q\to1}\frac{p_n(1-\thalf x(1-q)^2;q^a,q^b,q^c,q^d\,|\, q)}{(1-q)^{3n}} +=W_n(x;a,b,c,d). +\label{109} +\end{equation} +For the proof first derive the corresponding limit for the monic polynomials by comparing +(14.1.5) with (9.4.4). +\bLP +{\bf Askey-Wilson $\longrightarrow$ Continuous Hahn}\\ +Instead of (14.4.15) we can keep a polynomial of degree $n$ while the limit is approached: +\begin{multline} +\lim_{q\uparrow1} +\frac{p_n\big(\cos\phi-x(1-q)\sin\phi;q^a e^{i\phi},q^b e^{i\phi},q^{\overline a} e^{-i\phi}, +q^{\overline b} e^{-i\phi}\,|\, q\big)} +{(1-q)^{2n}}\\ +=(-2\sin\phi)^n\,n!\,p_n(x;a,b,\overline a,\overline b)\qquad +(0<\phi<\pi). +\label{177} +\end{multline} +Here the \RHS\ has a continuous Hahn polynomial (9.4.1). +For the proof first derive the corresponding limit for the monic polynomials by comparing +(14.1.5) with (9.1.5). +In fact, define the monic polynomial +\[ +\wt p_n(x):= +\frac{p_n\big(\cos\phi-x(1-q)\sin\phi;q^a e^{i\phi},q^b e^{i\phi},q^{\overline a} e^{-i\phi}, +q^{\overline b} e^{-i\phi}\,|\, q\big)} +{(-2(1-q)\sin\phi)^n\,(abcdq^{n-1};q)_n}\,. +\] +Then it follows from (14.1.5) that +\begin{equation*} +x\,\wt p_n(x)=\wt p_{n+1}(x) ++\frac{(1-q^a)e^{i\phi}+(1-q^{-a})e^{-i\phi}+\wt A_n+\wt C_n}{2(1-q)\sin\phi}\,\wt p_n(x)\\ ++\frac{\wt A_{n-1} \wt C_n}{(1-q)^2 \sin^2\phi}\,\wt p_{n-1}(x), +\end{equation*} +where $\wt A_n$ and $\wt C_n$ are as given after (14.1.3) with $a,b,c,d$ replaced by +$q^a e^{i\phi},q^b e^{i\phi},q^{\overline a} e^{-i\phi},q^{\overline b} e^{-i\phi}$. +Then the recurrence equation for $\wt p_n(x)$ tends for $q\uparrow 1$ to +the recurrence equation (9.4.4) with $c=\overline a$, $d=\overline b$. +\bLP +{\bf Askey-Wilson $\longrightarrow$ Meixner-Pollaczek}\\ +Instead of (14.9.15) we can keep a polynomial of degree $n$ while the limit is approached: +\begin{equation} +\lim_{q\uparrow1} +\frac{p_n\big(\cos\phi-x(1-q)\sin\phi; +q^\la e^{i\phi},0,q^\la e^{-i\phi},0\,|\, q\big)}{(1-q)^n} +=n!\,P_n^{(\la)}(x;\pi-\phi)\quad +(0<\phi<\pi). +\label{178} +\end{equation} +Here the \RHS\ has a Meixner-Pollaczek polynomial (9.7.1). +For the proof first derive the corresponding limit for the monic polynomials by comparing +(14.1.5) with (9.7.4). +In fact, define the monic polynomial +\[ +\wt p_n(x):= +\frac{p_n\big(\cos\phi-x(1-q)\sin\phi; +q^\la e^{i\phi},0,q^\la e^{-i\phi},0\,|\, q\big)}{(-2(1-q)\sin\phi)^n}\,. +\] +Then it follows from (14.1.5) that +\begin{equation*} +x\,\wt p_n(x)=\wt p_{n+1}(x) ++\frac{(1-q^\la)e^{i\phi}+(1-q^{-\la})e^{-i\phi}+\wt A_n+\wt C_n} +{2(1-q)\sin\phi}\,\wt p_n(x)\\ ++\frac{\wt A_{n-1} \wt C_n}{(1-q)^2 \sin^2\phi}\,\wt p_{n-1}(x), +\end{equation*} +where $\wt A_n$ and $\wt C_n$ are as given after (14.1.3) with $a,b,c,d$ replaced by +$q^\la e^{i\phi},0,q^\la e^{-i\phi},0$. +Then the recurrence equation for $\wt p_n(x)$ tends for $q\uparrow 1$ to +the recurrence equation (9.7.4). +% +\paragraph{References} +See also Koornwinder \cite{K7}. +$$\frac{(-a)^n}{n!}C_n(x;a)=L_n^{(x-n)}(a).$$ + +\subsection*{References} +\cite{Allaway76}, \cite{NAlSalam66}, \cite{AlSalam90}, \cite{AlSalamChihara76}, +\cite{AlSalamIsmail76}, \cite{AndrewsAskey85}, \cite{Area+II}, \cite{Askey75}, +\cite{AskeyGasper77}, \cite{AskeyWilson85}, \cite{AtakRahmanSuslov}, \cite{Bavinck98}, +\cite{BavinckKoekoek}, \cite{Chihara78}, \cite{Chihara79}, \cite{Dunkl76}, \cite{Erdelyi+}, +\cite{Gasper73I}, \cite{Gasper74}, \cite{Goh}, \cite{HoareRahman}, \cite{IsmailLetVal88}, +\cite{Koekoek2000}, \cite{Koorn88}, \cite{Krasikov2002}, \cite{LabelleYehI}, +\cite{LabelleYehII}, \cite{LabelleYehIII}, \cite{Lesky62}, \cite{Lesky89}, \cite{Lesky94I}, +\cite{Lesky95II}, \cite{LewanowiczII}, \cite{LopezTemme2004}, \cite{Meixner}, \cite{Nikiforov+}, +\cite{NikiforovUvarov}, \cite{Szafraniec}, \cite{Szego75}, \cite{Viennot}, \cite{Zarzo+}, +\cite{Zeng90}. + + +\section{Hermite}\index{Hermite polynomials} + +\par\setcounter{equation}{0} + +\subsection*{Hypergeometric representation} +\begin{equation} +\label{DefHermite} +H_n(x)=(2x)^n\,\hyp{2}{0}{-n/2,-(n-1)/2}{-}{-\frac{1}{x^2}}. +\end{equation} + +\subsection*{Orthogonality relation} +\begin{equation} +\label{OrtHermite} +\frac{1}{\sqrt{\cpi}}\int_{-\infty}^{\infty}\expe^{-x^2}H_m(x)H_n(x)\,dx +=2^nn!\,\delta_{mn}. +\end{equation} + +\subsection*{Recurrence relation} +\begin{equation} +\label{RecHermite} +H_{n+1}(x)-2xH_n(x)+2nH_{n-1}(x)=0. +\end{equation} + +\subsection*{Normalized recurrence relation} +\begin{equation} +\label{NormRecHermite} +xp_n(x)=p_{n+1}(x)+\frac{n}{2}p_{n-1}(x), +\end{equation} +where +$$H_n(x)=2^np_n(x).$$ + +\subsection*{Differential equation} +\begin{equation} +\label{dvHermite} +y''(x)-2xy'(x)+2ny(x)=0,\quad y(x)=H_n(x). +\end{equation} + +\subsection*{Forward shift operator} +\begin{equation} +\label{shift1Hermite} +\frac{d}{dx}H_n(x)=2nH_{n-1}(x). +\end{equation} + +\subsection*{Backward shift operator} +\begin{equation} +\label{shift2HermiteI} +\frac{d}{dx}H_n(x)-2xH_n(x)=-H_{n+1}(x) +\end{equation} +or equivalently +\begin{equation} +\label{shift2HermiteII} +\frac{d}{dx}\left[\expe^{-x^2}H_n(x)\right]=-\expe^{-x^2}H_{n+1}(x). +\end{equation} + +\subsection*{Rodrigues-type formula} +\begin{equation} +\label{RodHermite} +\e^{-x^2}H_n(x)=(-1)^n\left(\frac{d}{dx}\right)^n\left[\expe^{-x^2}\right]. +\end{equation} + +\subsection*{Generating functions} +\begin{equation} +\label{GenHermite1} +\exp\left(2xt-t^2\right)=\sum_{n=0}^{\infty}\frac{H_n(x)}{n!}t^n. +\end{equation} + +\begin{equation} +\label{GenHermite2} +\left\{\begin{array}{l} +\displaystyle\expe^t\cos(2x\sqrt{t})=\sum_{n=0}^{\infty} +\frac{(-1)^n}{(2n)!}H_{2n}(x)t^n\\[5mm] +\displaystyle\frac{\expe^t}{\sqrt{t}}\sin(2x\sqrt{t})=\sum_{n=0}^{\infty} +\frac{(-1)^n}{(2n+1)!}H_{2n+1}(x)t^n. +\end{array}\right. +\end{equation} + +\begin{equation} +\label{GenHermite3} +\left\{\begin{array}{l} +\displaystyle \expe^{-t^2}\cosh(2xt)=\sum_{n=0}^{\infty} +\frac{H_{2n}(x)}{(2n)!}t^{2n}\\[5mm] +\displaystyle\expe^{-t^2}\sinh(2xt)=\sum_{n=0}^{\infty} +\frac{H_{2n+1}(x)}{(2n+1)!}t^{2n+1}. +\end{array}\right. +\end{equation} + +\begin{equation} +\label{GenHermite4} +\left\{\begin{array}{l} +\displaystyle (1+t^2)^{-\gamma}\,\hyp{1}{1}{\gamma}{\frac{1}{2}}{\frac{x^2t^2}{1+t^2}}= +\sum_{n=0}^{\infty}\frac{(\gamma)_n}{(2n)!}H_{2n}(x)t^{2n}\\[5mm] +\displaystyle\frac{xt}{\sqrt{1+t^2}}\, +\hyp{1}{1}{\gamma+\frac{1}{2}}{\frac{3}{2}}{\frac{x^2t^2}{1+t^2}} +=\sum_{n=0}^{\infty}\frac{(\gamma+\frac{1}{2})_n}{(2n+1)!}H_{2n+1}(x)t^{2n+1} +\end{array}\right. +\end{equation} +with $\gamma$ arbitrary. + +\begin{equation} +\label{GenHermite5} +\frac{1+2xt+4t^2}{(1+4t^2)^{\frac{3}{2}}}\exp\left(\frac{4x^2t^2}{1+4t^2}\right) +=\sum_{n=0}^{\infty}\frac{H_n(x)}{\lfloor n/2\rfloor\,!}t^n, +\end{equation} +where $\lfloor n/2\rfloor$ denotes the largest integer smaller than or equal to $n/2$. + +\subsection*{Limit relations} + +\subsubsection*{Meixner-Pollaczek $\rightarrow$ Hermite} +If we take $x\rightarrow (\sin\phi)^{-1}(x\sqrt{\lambda}-\lambda\cos\phi)$ +in the definition (\ref{DefMP}) of the Meixner-Pollaczek polynomials and +then let $\lambda\rightarrow\infty$ we obtain the Hermite polynomials: +$$\lim_{\lambda\rightarrow\infty} +\lambda^{-\frac{1}{2}n}P_n^{(\lambda)} +((\sin\phi)^{-1}(x\sqrt{\lambda}-\lambda\cos\phi);\phi)=\frac{H_n(x)}{n!}.$$ + +\subsubsection*{Jacobi $\rightarrow$ Hermite} +The Hermite polynomials follow from the Jacobi polynomials given by (\ref{DefJacobi}) by +taking $\beta=\alpha$ and letting $\alpha\rightarrow\infty$ in the following way: +$$\lim_{\alpha\rightarrow\infty} +\alpha^{-\frac{1}{2}n}P_n^{(\alpha,\alpha)}(\alpha^{-\frac{1}{2}}x)=\frac{H_n(x)}{2^nn!}.$$ + +\subsubsection*{Gegenbauer / Ultraspherical $\rightarrow$ Hermite} +The Hermite polynomials follow from the Gegenbauer (or ultraspherical) polynomials given by +(\ref{DefGegenbauer}) by taking $\lambda=\alpha+\frac{1}{2}$ and letting +$\alpha\rightarrow\infty$ in the following way: +$$\lim_{\alpha\rightarrow\infty} +\alpha^{-\frac{1}{2}n}C_n^{(\alpha+\frac{1}{2})}(\alpha^{-\frac{1}{2}}x)=\frac{H_n(x)}{n!}.$$ + +\subsubsection*{Krawtchouk $\rightarrow$ Hermite} +The Hermite polynomials follow from the Krawtchouk polynomials given by (\ref{DefKrawtchouk}) +by setting $x\rightarrow pN+x\sqrt{2p(1-p)N}$ and then letting $N\rightarrow\infty$: +$$\lim_{N\rightarrow\infty} +\sqrt{\binom{N}{n}}K_n(pN+x\sqrt{2p(1-p)N};p,N) +=\frac{\displaystyle (-1)^nH_n(x)}{\displaystyle\sqrt{2^nn!\left(\frac{p}{1-p}\right)^n}}.$$ + +\subsubsection*{Laguerre $\rightarrow$ Hermite} +The Hermite polynomials can be obtained from the Laguerre polynomials given by +(\ref{DefLaguerre}) by taking the limit $\alpha\rightarrow\infty$ in the following way: +$$\lim_{\alpha\rightarrow\infty} +\left(\frac{2}{\alpha}\right)^{\frac{1}{2}n} +L_n^{(\alpha)}((2\alpha)^{\frac{1}{2}}x+\alpha)=\frac{(-1)^n}{n!}H_n(x).$$ + +\subsubsection*{Charlier $\rightarrow$ Hermite} +If we set $x\rightarrow (2a)^{1/2}x+a$ in the definition (\ref{DefCharlier}) +of the Charlier polynomials and let $a\rightarrow\infty$ we find the Hermite +polynomials. In fact we have +$$\lim_{a\rightarrow\infty} +(2a)^{\frac{1}{2}n}C_n((2a)^{\frac{1}{2}}x+a;a)=(-1)^nH_n(x).$$ + +\subsection*{Remarks} +The Hermite polynomials can also be written as: +$$\frac{H_n(x)}{n!}=\sum_{k=0}^{\lfloor n/2\rfloor} +\frac{(-1)^k(2x)^{n-2k}}{k!\,(n-2k)!},$$ +where $\lfloor n/2\rfloor$ denotes the largest integer smaller than or equal to $n/2$. + +\noindent +The Hermite polynomials and the Laguerre polynomials given by (\ref{DefLaguerre}) are also +connected by the following quadratic transformations: +$$H_{2n}(x)=(-1)^nn!\,2^{2n}L_n^{(-\frac{1}{2})}(x^2)$$ +and +\subsection*{14.2 $q$-Racah} +\label{sec14.2} +\paragraph{Symmetry} +\begin{equation} +R_n(x;\al,\be,q^{-N-1},\de\,|\, q) +=\frac{(\be q,\al\de^{-1}q;q)_n}{(\al q,\be\de q;q)_n}\,\de^n\, +R_n(\de^{-1}x;\be,\al,q^{-N-1},\de^{-1}\,|\, q). +\label{84} +\end{equation} +This follows from (14.2.1) combined with \mycite{GR}{(III.15)}. +\sLP +In particular, +\begin{equation} +R_n(x;\al,\be,q^{-N-1},-1\,|\, q) +=\frac{(\be q,-\al q;q)_n}{(\al q,-\be q;q)_n}\,(-1)^n\, +R_n(-x;\be,\al,q^{-N-1},-1\,|\, q), +\label{85} +\end{equation} +and +\begin{equation} +R_n(x;\al,\al,q^{-N-1},-1\,|\, q) +=(-1)^n\,R_n(-x;\al,\al,q^{-N-1},-1\,|\, q), +\label{86} +\end{equation} + +\paragraph{Trivial symmetry} +Clearly from (14.2.1): +\begin{equation} +R_n(x;\al,\be,\ga,\de\,|\, q)=R_n(x;\be\de,\al\de^{-1},\ga,\de\,|\, q) +=R_n(x;\ga,\al\be\ga^{-1},\al,\ga\de\al^{-1}\,|\, q). +\label{83} +\end{equation} +For $\al=q^{-N-1}$ this shows that the three cases +$\al q=q^{-N}$ or $\be\de q=q^{-N}$ or $\ga q=q^{-N}$ of (14.2.1) +are not essentially different. +% +\paragraph{Duality} +It follows from (14.2.1) that +\begin{equation} +R_n(q^{-y}+\ga\de q^{y+1};q^{-N-1},\be,\ga,\de\,|\, q) +=R_y(q^{-n}+\be q^{n-N};\ga,\de,q^{-N-1},\be\,|\, q)\quad +(n,y=0,1,\ldots,N). +\end{equation} +$$H_{2n+1}(x)=(-1)^nn!\,2^{2n+1}xL_n^{(\frac{1}{2})}(x^2).$$ + +\subsection*{14.3 Continuous dual $q$-Hahn} +\label{sec14.3} +The continuous dual $q$-Hahn polynomials are the special case $d=0$ of the +Askey-Wilson polynomials: +\[ +p_n(x;a,b,c\,|\, q):=p_n(x;a,b,c,0\,|\, q). +\] +Hence all formulas in \S14.3 are specializations for $d=0$ of formulas in \S14.1. +\subsection*{References} +\cite{Abram}, \cite{NAlSalam66}, \cite{AlSalam90}, \cite{AlSalamChihara72}, +\cite{AlSalamChihara76}, \cite{AndrewsAskey85}, \cite{AndrewsAskeyRoy}, \cite{Area+I}, +\cite{Askey68}, \cite{Askey75}, \cite{Askey89I}, \cite{AskeyGasper76}, \cite{AskeyWilson85}, +\cite{Azor}, \cite{Berg}, \cite{BilodeauII}, \cite{Brafman51}, \cite{Brafman57II}, +\cite{Brenke}, \cite{CarlitzSrivastava}, \cite{ChenSrivastava}, \cite{Chihara78}, +\cite{Cohen}, \cite{Danese}, \cite{DetteStudden92}, \cite{DetteStudden95}, \cite{Doha2004I}, +\cite{Erdelyi+}, \cite{Faldey}, \cite{Gawronski87}, \cite{Gawronski93}, \cite{Gillis+}, +\cite{Godoy+}, \cite{Grad}, \cite{Ismail74}, \cite{Ismail2005II}, \cite{IsmailStanton97}, +\cite{Koekoek2000}, \cite{Koorn88}, \cite{Krasikov2004}, \cite{Kwon+}, \cite{LabelleYehIII}, +\cite{Lesky95II}, \cite{Lesky96}, \cite{LewanowiczI}, \cite{LopezTemme99}, \cite{Mathai}, +\cite{Meixner}, \cite{Nikiforov+}, \cite{NikiforovUvarov}, \cite{Olver}, \cite{Pittaluga}, +\cite{Rainville}, \cite{SainteViennot}, \cite{Srivastava71}, \cite{SrivastavaMathur}, +\cite{Szego75}, \cite{Temme}, \cite{TemmeLopez2000}, \cite{Trickovic}, \cite{Viennot}, +\cite{Weisner59}, \cite{Wyman}. + +\end{document} From 98b3c2be56ed2b331c0dee6b12d1b5b2ccfd6a63 Mon Sep 17 00:00:00 2001 From: Rahul Shah Date: Wed, 17 Feb 2016 15:34:48 -0500 Subject: [PATCH 002/402] ignored chap files --- KLSadd_insertion/.gitignore | 1 + 1 file changed, 1 insertion(+) create mode 100644 KLSadd_insertion/.gitignore diff --git a/KLSadd_insertion/.gitignore b/KLSadd_insertion/.gitignore new file mode 100644 index 0000000..adb2870 --- /dev/null +++ b/KLSadd_insertion/.gitignore @@ -0,0 +1 @@ +*chap*.tex From 2005562bb72c5944d47f29b0c8b439837866eb91 Mon Sep 17 00:00:00 2001 From: Rahul Shah Date: Wed, 17 Feb 2016 15:36:59 -0500 Subject: [PATCH 003/402] removing chap files --- KLSadd_insertion/.gitignore | 1 - README.md | 3 +++ 2 files changed, 3 insertions(+), 1 deletion(-) delete mode 100644 KLSadd_insertion/.gitignore diff --git a/KLSadd_insertion/.gitignore b/KLSadd_insertion/.gitignore deleted file mode 100644 index adb2870..0000000 --- a/KLSadd_insertion/.gitignore +++ /dev/null @@ -1 +0,0 @@ -*chap*.tex diff --git a/README.md b/README.md index ee19c88..75cc520 100644 --- a/README.md +++ b/README.md @@ -27,4 +27,7 @@ To achieve the desired result, one should execute the progams in the following o ## BMP Seeding Project +## KLSadd Insertion Project + +Edits the DRMF chapter files to include rel From 944aa81aeb358fc23552f60286f427caf5da9249 Mon Sep 17 00:00:00 2001 From: Rahul Shah Date: Wed, 17 Feb 2016 15:45:07 -0500 Subject: [PATCH 004/402] new --- KLSadd_insertion/chap01.tex | 1542 ------- KLSadd_insertion/chap09.tex | 3037 ------------- KLSadd_insertion/chap14.tex | 5734 ------------------------ KLSadd_insertion/newtempchap09.tex | 3934 ---------------- KLSadd_insertion/newtempchap14.tex | 6695 ---------------------------- KLSadd_insertion/tempchap09.tex | 3926 ---------------- KLSadd_insertion/tempchap14.tex | 5734 ------------------------ 7 files changed, 30602 deletions(-) delete mode 100644 KLSadd_insertion/chap01.tex delete mode 100644 KLSadd_insertion/chap09.tex delete mode 100644 KLSadd_insertion/chap14.tex delete mode 100644 KLSadd_insertion/newtempchap09.tex delete mode 100644 KLSadd_insertion/newtempchap14.tex delete mode 100644 KLSadd_insertion/tempchap09.tex delete mode 100644 KLSadd_insertion/tempchap14.tex diff --git a/KLSadd_insertion/chap01.tex b/KLSadd_insertion/chap01.tex deleted file mode 100644 index d2b4c10..0000000 --- a/KLSadd_insertion/chap01.tex +++ /dev/null @@ -1,1542 +0,0 @@ -\documentclass[envcountchap,graybox]{svmono} - -\addtolength{\textwidth}{1mm} - -\usepackage{amsmath,amssymb} - -\usepackage{amsfonts} -%\usepackage{breqn} -\usepackage{DLMFmath} -\usepackage{DRMFfcns} - -\usepackage{mathptmx} -\usepackage{helvet} -\usepackage{courier} - -\usepackage{makeidx} -\usepackage{graphicx} - -\usepackage{multicol} -\usepackage[bottom]{footmisc} - -\makeindex - -\def\bibname{Bibliography} -\def\refname{Bibliography} - -\def\theequation{\thesection.\arabic{equation}} - -\smartqed - -\let\corollary=\undefined -\spnewtheorem{corollary}[theorem]{Corollary}{\bfseries}{\itshape} - -\newcounter{rom} - -\newcommand{\hyp}[5]{\mbox{}_{#1}{F}_{#2} -\left(\genfrac{}{}{0pt}{}{#3}{#4}\,;\,#5\right)} - -\newcommand{\qhyp}[5]{\mbox{}_{#1}{\phi}_{#2} -\left(\genfrac{}{}{0pt}{}{#3}{#4}\,;\,q,\,#5\right)} - -\newcommand{\mathindent}{\hspace{7.5mm}} - -\newcommand{\e}{\textrm{e}} - -\renewcommand{\E}{\textrm{E}} - -\renewcommand{\textfraction}{-1} - -\renewcommand{\Gamma}{\varGamma} - -\renewcommand{\leftlegendglue}{\hfil} - -\settowidth{\tocchpnum}{14\enspace} -\settowidth{\tocsecnum}{14.30\enspace} -\settowidth{\tocsubsecnum}{14.12.1\enspace} - -\makeatletter -\def\cleartoversopage{\clearpage\if@twoside\ifodd\c@page - \hbox{}\newpage\if@twocolumn\hbox{}\newpage\fi - \else\fi\fi} - -\newcommand{\clearemptyversopage}{ - \clearpage{\pagestyle{empty}\cleartoversopage}} -\makeatother - -\oddsidemargin -1.5cm -\topmargin -2.0cm -\textwidth 16.3cm -\textheight 25cm - -\begin{document} - -\author{Roelof Koekoek\\[2.5mm]Peter A. Lesky\\[2.5mm]Ren\'e F. Swarttouw} -\title{Hypergeometric orthogonal polynomials and their\\$q$-analogues} -\subtitle{-- Monograph --} -\maketitle - -\frontmatter - -\pagenumbering{roman} - -\large - -\chapter{Definitions and miscellaneous formulas} -\label{Definitions} - - -\section{Orthogonal polynomials}\index{Orthogonal polynomials} -\par\setcounter{equation}{0} -\label{orthogonal polynomials} - -A system of polynomials $\left\{p_n(x)\right\}_{n=0}^{\infty}$ with degree$[p_n(x)]=n$ for all -$n\in\{0,1,2,\ldots\}$ is called orthogonal on an interval $(a,b)$ with respect to the weight -function $w(x)\geq 0$ if -\begin{equation} -\label{orth1} -\int_a^bp_m(x)p_n(x)w(x)\,dx=0,\quad m\neq n,\quad m,n\in\{0,1,2,\ldots\}. -\end{equation} -Here $w(x)$ is continuous or piecewise continuous or integrable, and such that -\begin{equation} -0<\int_a^bx^{2n}w(x)\,dx<\infty -% \constraint{ $n\in\{0,1,2,\ldots\}$ } -\end{equation} -More generally, $w(x)\,dx$ may be replaced in this definition by a positive measure -$d\alpha(x)$, where $\alpha(x)$ is a bounded nondecreasing function on $[a,b]\cap\mathbb{R}$ -with an infinite number of points of increase, and such that -\begin{equation} -0<\int_a^bx^{2n}\,d\alpha(x)<\infty -% \constraint{ $n\in\{0,1,2,\ldots\}$ } -\end{equation} -If this function $\alpha(x)$ is constant between its (countably many) jump points then we have -the situation of positive weights $w_x$ on a countable subset $X$ of $\mathbb{R}$. Then the -system $\{p_n(x)\}_{n=0}^{\infty}$ is orthogonal on $X$ with respect to these weights as follows: -\begin{equation} -\label{orth2} -\sum_{x\in X}p_m(x)p_n(x)w_x=0,\quad m\neq n,\quad m,n\in\{0,1,2,\ldots\}. -\end{equation} -The case of weights $w_x$ ($x\in X$) on a finite set $X$ of $N+1$ points yields orthogonality -for a finite system of polynomials $\{p_n(x)\}_{n=0}^N$: -\begin{equation} -\label{orth3} -\sum_{x\in X}p_m(x)p_n(x)w_x=0,\quad m\neq n,\quad m,n\in\{0,1,2,\ldots,N\}. -\end{equation} -The orthogonality relations (\ref{orth1}), (\ref{orth2}) or (\ref{orth3}) determine the -polynomials $\{p_n(x)\}_{n=0}^{\infty}$ up to constant factors, which may be fixed by a -suitable normalization. We set: -\begin{equation} -\label{norm1} -\sigma_n=\int_a^b\left\{p_n(x)\right\}^2w(x)\,dx,\quad n=0,1,2,\ldots, -\end{equation} -\begin{equation} -\label{norm2} -\sigma_n=\sum_{x\in X}\left\{p_n(x)\right\}^2w_x,\quad n=0,1,2,\ldots -\end{equation} -or -\begin{equation} -\label{norm3} -\sigma_n=\sum_{x\in X}\left\{p_n(x)\right\}^2w_x,\quad n=0,1,2,\ldots,N, -\end{equation} -respectively and -\begin{equation} -\label{polynomial} -p_n(x)=k_nx^n+\,\textrm{lower order terms},\quad n=0,1,2,\ldots. -\end{equation} -Then we have the following normalizations: - -\begin{enumerate}\itemsep2.5mm -\item $\sigma_n=1$ for all $n=0,1,2,\ldots$. In that case the system of polynomials is called -orthonormal. If moreover $k_n>0$, for instance, then the polynomials are uniquely determined. -\item $k_n=1$ for all $n=0,1,2,\ldots$. In that case the system of polynomials is called -monic. Also in this case the polynomials are uniquely determined. -\end{enumerate} -Polynomials $\{p_n(x)\}_{n=0}^{\infty}$ with degree$[p_n(x)]=n$ for all $n\in\{0,1,2,\ldots\}$ -which are orthogonal with respect to a (piecewise) continuous or integrable weight function -$w(x)$ as in (\ref{orth1}) are called continuous orthogonal polynomials.\\ -Polynomials $\{p_n(x)\}_{n=0}^N$ with degree$[p_n(x)]=n$ for all $n\in\{0,1,2,\ldots,N\}$ and -possibly $N\rightarrow\infty$ which are orthogonal with respect to a countable subset $X$ of -$\mathbb{R}$ as in (\ref{orth2}) or (\ref{orth3}) are called discrete orthogonal polynomials.\\ -By using the Kronecker delta, defined by -\begin{equation} -\label{Kronecker} -\,\delta_{mn}:=\left\{\begin{array}{ll}0, &m\neq n,\\[5mm] -1, & m=n,\end{array}\right.\quad m,n\in\{0,1,2,\ldots\}, -\end{equation} -orthogonality relations can be written in the form -\begin{equation} -\label{orth4} -\int_a^bp_m(x)p_n(x)w(x)\,dx=\sigma_n\,\delta_{mn},\quad m,n\in\{0,1,2,\ldots\} -\end{equation} -or (with possibly $N\rightarrow\infty$) -\begin{equation} -\label{orth5} -\sum_{x\in X}p_m(x)p_n(x)w_x=\sigma_n\,\delta_{mn},\quad m,n\in\{0,1,2,\ldots,N\}, -\end{equation} -respectively. - -\section{The gamma and beta function} -\par\setcounter{equation}{0} -\label{gamma function} - -For $\textrm{Re}\,z>0$ the gamma function can be defined by the -gamma integral $\EulerGamma@{z}$ -\begin{equation} -\label{DefGamma}\index{Gamma function} -\Gamma(z):=\int_0^{\infty}t^{z-1}\e^{-t}\,dt,\quad\textrm{Re}\,z>0. -\end{equation} -This gamma function satisfies the well-known functional equation -\begin{equation} -\label{GammaFunctional} -\Gamma(z+1)=z\Gamma(z) -% \constraint{ $\Gamma(1)=1$ } -\end{equation} -which shows that $\Gamma(n+1)=n!$ for $n\in\{0,1,2,\ldots\}$. -For non-integral values of $z$, the gamma function also satisfies the reflection formula -\begin{equation} -\label{GammaReflection} -\Gamma(z)\Gamma(1-z)=\frac{\pi}{\sin(\pi z)},\quad z\notin\mathbb{Z}. -\end{equation} -Hence we have $\Gamma(1/2)=\sqrt{\pi}$, which implies that -\begin{equation} -\label{IntExp} -\int_{-\infty}^{\infty}\e^{-x^2}\,dx=2\int_0^{\infty}\e^{-x^2}\,dx -=\int_0^{\infty}t^{-1/2}\e^{-t}\,dt=\Gamma(1/2)=\sqrt{\pi}. -\end{equation} -More general we have -\begin{equation} -\label{IntHermite} -\int_{-\infty}^{\infty}\e^{-\alpha^2x^2-2\beta x}\,dx -=\sqrt{\frac{\pi}{\alpha^2}}\,\e^{\beta^2/\alpha^2}, -% \constraint{ -% $\alpha,\beta\in\Real$ & -% $\alpha\neq 0$ } -\end{equation} -Further we have Legendre's duplication formula -\begin{equation} -\label{GammaDuplication} -\Gamma(z)\Gamma(z+1/2)=2^{1-2z}\sqrt{\pi}\,\Gamma(2z) -% \constraint{ -% $z\in\Complex$ & -% $2z\neq 0,-1,-2,\ldots$ } -\end{equation} -and Stirling's asymptotic formula -\begin{equation} -\label{GammaAsymptotic} -\Gamma(z)\sim\sqrt{2\pi}\,z^{z-1/2}\e^{-z},\quad\textrm{Re}\,z\rightarrow\infty. -\end{equation} -For $z=x+iy$ with $x,y\in\mathbb{R}$ we also have -\begin{equation} -\label{GammaAsymptotic2} -\Gamma(x+iy)\sim\sqrt{2\pi}\,|y|^{x-1/2}\e^{-|y|\pi/2},\quad|y|\rightarrow\infty -\end{equation} -and for the ratio of two gamma functions we have the asymptotic -formula -\begin{equation} -\label{GammaAsymptotic3} -\frac{\Gamma(z+a)}{\Gamma(z+b)}\sim z^{a-b},\quad -% \constraint{ -% $a,b\in\Complex$ & -% $|z|\rightarrow\infty$ } -\end{equation} - -For $\textrm{Re}\,x>0$ and $\textrm{Re}\,y>0$ the beta function can be defined -by the integral -\begin{equation} -\label{DefBeta}\index{Beta function} -\textrm{B}(x,y):=\int_0^1t^{x-1}(1-t)^{y-1}\,dt -% \constraint{ -% $\textrm{Re}\,x>0$ & -% $\textrm{Re}\,y>0$ } -\end{equation} -The connection between the beta function and the gamma function is -given by the relation -\begin{equation} -\label{BetaGamma} -\textrm{B}(x,y)=\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)} -% \constraint{ -% $\textrm{Re}\,x>0$ & -% $\textrm{Re}\,y>0$ } -\end{equation} -There is another beta integral due to Cauchy, which can be written as -\begin{equation} -\label{Cauchy} -\frac{1}{2\pi}\int_{-\infty}^{\infty}\frac{dt}{(r+it)^{\rho}(s-it)^{\sigma}} -=\frac{(r+s)^{1-\rho-\sigma}\Gamma(\rho+\sigma-1)}{\Gamma(\rho)\Gamma(\sigma)} -\end{equation} -for $\textrm{Re}\,r>0$, $\textrm{Re}\,s>0$ and $\textrm{Re}(\rho+\sigma)>1$. - -\section{The shifted factorial and binomial coefficients} -\par\setcounter{equation}{0} -\label{pochhammer symbol} - -The shifted factorial -- or Pochhammer symbol -- is defined by -\begin{equation} -\label{Pochhammer}\index{Shifted factorial} -(a)_0:=1\quad\textrm{and}\quad (a)_k:=\prod_{n=1}^k(a+n-1) -% \constraint{ $k=1,2,3,\ldots$ } -\end{equation} -This can be seen as a generalization of the factorial since -$$(1)_n=n!,\quad n=0,1,2,\ldots.$$ - -The binomial coefficient can be defined by -\begin{equation} -\label{binomialcoeff}\index{Binomial coefficient} -\binom{\alpha}{\beta}:=\frac{\Gamma(\alpha+1)}{\Gamma(\beta+1)\Gamma(\alpha-\beta+1)}. -\end{equation} -For integer values of the parameter $\beta$ we have -$$\binom{\alpha}{k}:=\frac{(-\alpha)_k}{k!}(-1)^k,\quad k=0,1,2,\ldots$$ -and when the parameter $\alpha$ is an integer too, we have -\begin{equation} -\binom{n}{k}:=\frac{n!}{k!\,(n-k)!} -% \constraint{ -% $k=0,1,2,\ldots,n$ & -% $n=0,1,2,\ldots$ } -\end{equation} -The latter formula can be used to show that -$$\binom{2n}{n}=\frac{\left(\frac{1}{2}\right)_n}{n!}4^n,\quad n=0,1,2,\ldots.$$ - -\section{Hypergeometric functions} -\par\setcounter{equation}{0} -\label{hypergeometric functions} - -The hypergeometric function $\mbox{}_rF_s$ is defined by the series -\begin{equation} -\label{DefHyp}\index{Hypergeometric function} -\hyp{r}{s}{a_1,\ldots,a_r}{b_1,\ldots,b_s}{z}:= -\sum\limits_{k=0}^{\infty}\frac{(a_1,\ldots,a_r)_k}{(b_1,\ldots,b_s)_k} -\frac{z^k}{k!}, -\end{equation} -where -$$(a_1,\ldots,a_r)_k:=(a_1)_k\cdots(a_r)_k.$$ -Of course, the parameters must be such that the denominator factors in the -terms of the series are never zero. When one of the numerator parameters -$a_j$ equals $-n$, where $n$ is a nonnegative integer, this hypergeometric -function is a polynomial in $z$. Otherwise the radius of convergence $\rho$ of -the hypergeometric series is given by -$$\rho=\left\{\begin{array}{ll} -\displaystyle \infty & \quad\textrm{if}\quad r < s+1\\[5mm] -\displaystyle 1 & \quad\textrm{if}\quad r = s+1\\[5mm] -\displaystyle 0 & \quad\textrm{if}\quad r > s+1.\end{array}\right.$$ - -A hypergeometric series of the form (\ref{DefHyp}) is called -balanced\index{Hypergeometric function!Balanced} or -Saalsch\"{u}tzian\index{Hypergeometric function!Saalsch\"{u}tzian} -if $r=s+1$, $z=1$ and $a_1+a_2+ \ldots +a_{s+1}+1=b_1+b_2+ \ldots +b_s$. - -Many limit relations between hypergeometric orthogonal polynomials are based on the observations that -\begin{equation} -\label{hypsub} -\hyp{r}{s}{a_1,\ldots,a_{r-1},\mu}{b_1,\ldots,b_{s-1},\mu}{z} -=\hyp{r-1}{s-1}{a_1,\ldots,a_{r-1}}{b_1,\ldots,b_{s-1}}{z}, -\end{equation} -\begin{equation} -\label{hyplim1} -\lim\limits_{\lambda\rightarrow\infty} -\hyp{r}{s}{a_1,\ldots,a_{r-1},\lambda a_r}{b_1,\ldots,b_s}{\frac{z}{\lambda}} -=\hyp{r-1}{s}{a_1,\ldots,a_{r-1}}{b_1,\ldots,b_s}{a_rz}, -\end{equation} -\begin{equation} -\label{hyplim2} -\lim\limits_{\lambda\rightarrow\infty} -\hyp{r}{s}{a_1,\ldots,a_r}{b_1,\ldots,b_{s-1},\lambda b_s}{\lambda z}= -\hyp{r}{s-1}{a_1,\ldots,a_r}{b_1,\ldots,b_{s-1}}{\frac{z}{b_s}} -\end{equation} -and -\begin{equation} -\label{hyplim3} -\lim\limits_{\lambda\rightarrow\infty} -\hyp{r}{s}{a_1,\ldots,a_{r-1},\lambda a_r}{b_1,\ldots,b_{s-1},\lambda b_s}{z} -=\hyp{r-1}{s-1}{a_1,\ldots,a_{r-1}}{b_1,\ldots,b_{s-1}}{\frac{a_rz}{b_s}}. -\end{equation} - -Mostly, the left-hand side of (\ref{hypsub}) occurs as a limit case where some numerator -parameter and some denominator parameter tend to the same value. - -All families of discrete orthogonal polynomials $\{P_n(x)\}_{n=0}^N$ are -defined for $n=0,1,2,\ldots,N$, where $N$ is a positive integer. In these -cases something like (\ref{hypsub}) occurs in the hypergeometric representation when $n=N$. -In these cases we have to be aware of the fact that we still have a polynomial (in that case of -degree $N$). For instance, if we take $n=N$ in the hypergeometric representation (\ref{DefHahn}) -of the Hahn polynomials, we have -$$Q_N(x;\alpha,\beta,N)=\sum_{k=0}^N\frac{(N+\alpha+\beta+1)_k(-x)_k}{(\alpha+1)_kk!}.$$ -So these cases must be understood by continuity. - -In cases of discrete orthogonal polynomials, we need a special notation for -some of the generating functions. We define -$$\left[f(t)\right]_N:=\sum_{k=0}^N\frac{f^{(k)}(0)}{k!}t^k,$$ -for every function $f$ for which $f^{(k)}(0)$, $k=0,1,2,\ldots,N$ exists. As -an example of the use of this $N$th partial sum of a power series in $t$ we -remark that the generating function (\ref{GenKrawtchouk2}) for the -Krawtchouk polynomials must be understood as follows: the $N$th partial sum -of -$$\e^t\,\hyp{1}{1}{-x}{-N}{-\frac{t}{p}}= -\sum_{k=0}^{\infty}\frac{t^k}{k!}\sum_{m=0}^x\frac{(-x)_m}{(-N)_mm!} -\left(-\frac{t}{p}\right)^m$$ -equals -$$\sum_{n=0}^N\frac{K_n(x;p,N)}{n!}t^n$$ -for $x=0,1,2,\ldots,N$. - -The classical exponential function $\e^z$ and the trigonometric functions -$\cos z$ and $\sin z$ can be expressed in terms of hypergeometric functions -as -\begin{equation} -\label{exp} -\e^z=\hyp{0}{0}{-}{-}{z}, -\end{equation} -\begin{equation} -\label{cos} -\cos z=\hyp{0}{1}{-}{\frac{1}{2}}{-\frac{z^2}{4}} -\end{equation} -and -\begin{equation} -\label{sin} -\sin z=z\,\hyp{0}{1}{-}{\frac{3}{2}}{-\frac{z^2}{4}}. -\end{equation} - -Further we have the well-known Bessel function of the first kind $J_{\nu}(z)$, which can be -defined by -\begin{equation} -\label{Bessel}\index{Bessel function} -J_{\nu}(z):=\frac{\left(\frac{1}{2}z\right)^{\nu}}{\Gamma(\nu+1)}\, -\hyp{0}{1}{-}{\nu+1}{-\frac{z^2}{4}}. -\end{equation} - -\section{The binomial theorem and other summation formulas} -\par\setcounter{equation}{0} -\label{summation formulas} - -One of the most important summation formulas for hypergeometric series is -given by the binomial theorem: -\begin{equation} -\label{binomial}\index{Binomial theorem}\index{Summation formula!Binomial theorem} -\hyp{1}{0}{a}{-}{z}=\sum_{n=0}^{\infty}\frac{(a)_n}{n!}z^n=(1-z)^{-a}, -% \constraint{ $\quad |z|<1$ } -\end{equation} -which is a generalization of Newton's binomium -\begin{equation} -\label{Newton} -\hyp{1}{0}{-n}{-}{z}=\sum_{k=0}^n\frac{(-n)_k}{k!}z^k -=\sum_{k=0}^n\binom{n}{k}(-z)^k=(1-z)^n,\quad n=0,1,2,\ldots. -\end{equation} - -We also have Gauss's summation formula -\begin{equation} -\label{Gauss}\index{Gauss summation formula}\index{Summation formula!Gauss} -\hyp{2}{1}{a,b}{c}{1}= -\frac{\Gamma(c)\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)} -% \constraint{ $\textrm{Re}(c-a-b)>0$ } -\end{equation} -and the Vandermonde or Chu-Vandermonde summation formula -\begin{equation} -\label{ChuVandermonde} -\index{Vandermonde summation formula}\index{Summation formula!Vandermonde} -\index{Chu-Vandermonde summation formula}\index{Summation formula!Chu-Vandermonde} -\hyp{2}{1}{-n,b}{c}{1}=\frac{(c-b)_n}{(c)_n},\quad n=0,1,2,\ldots. -\end{equation} - -On the next level we have the summation formula -\begin{equation} -\label{PfaffSaalschutz} -\index{Saalsch\"{u}tz summation formula}\index{Summation formula!Saalsch\"{u}tz} -\index{Pfaff-Saalsch\"{u}tz summation formula}\index{Summation formula!Pfaff-Saalsch\"{u}tz} -\hyp{3}{2}{-n,a,b}{c,1+a+b-c-n}{1}=\frac{(c-a)_n(c-b)_n}{(c)_n(c-a-b)_n}, -\quad n=0,1,2,\ldots, -\end{equation} -which is called the Saalsch\"{u}tz or Pfaff-Saalsch\"{u}tz summation formula. - -For a very-well-poised ${}_5F_4$ we have the summation formula -\begin{eqnarray} -\label{wellpoised}\index{Summation formula!for a very-well-poised ${}_5F_4$} -& &\hyp{5}{4}{1+a/2,a,b,c,d}{a/2,1+a-b,1+a-c,1+a-d}{1}\nonumber\\ -& &{}=\frac{\Gamma(1+a-b)\Gamma(1+a-c)\Gamma(1+a-d)\Gamma(1+a-b-c-d)} -{\Gamma(1+a)\Gamma(1+a-b-c)\Gamma(1+a-b-d)\Gamma(1+a-c-d)}. -\end{eqnarray} -The limit case $d\rightarrow -\infty$ leads to -\begin{equation} -\label{wellpoisedlimit} -\hyp{4}{3}{1+a/2,a,b,c}{a/2,1+a-b,1+a-c}{-1} -=\frac{\Gamma(1+a-b)\Gamma(1+a-c)}{\Gamma(1+a)\Gamma(1+a-b-c)}. -\end{equation} - -Finally, we mention Dougall's bilateral sum -\begin{equation} -\label{Dougall1}\index{Dougall's bilateral sum}\index{Summation formula!Dougall} -\sum_{n=-\infty}^{\infty}\frac{\Gamma(n+a)\Gamma(n+b)}{\Gamma(n+c)\Gamma(n+d)} -=\frac{\Gamma(a)\Gamma(1-a)\Gamma(b)\Gamma(1-b)\Gamma(c+d-a-b-1)} -{\Gamma(c-a)\Gamma(d-a)\Gamma(c-b)\Gamma(d-b)}, -\end{equation} -which holds for $\textrm{Re}(a+b)+1<\textrm{Re}(c+d)$.\\ -By using -$$\Gamma(n+a)\Gamma(n+b)=\frac{\Gamma(a)\Gamma(1-a)\Gamma(b)\Gamma(1-b)} -{\Gamma(1-a-n)\Gamma(1-b-n)},\quad n\in\mathbb{Z},$$ -Dougall's bilateral sum can also be written in the form -\begin{eqnarray} -\label{Dougall2} -& &\sum_{n=-\infty}^{\infty}\frac{1}{\Gamma(n+c)\Gamma(n+d)\Gamma(1-a-n)\Gamma(1-b-n)}\nonumber\\ -& &{}=\frac{\Gamma(c+d-a-b-1)}{\Gamma(c-a)\Gamma(d-a)\Gamma(c-b)\Gamma(d-b)} -\end{eqnarray} -for $\textrm{Re}(a+b)+1<\textrm{Re}(c+d)$. - -\section{Some integrals} -\par\setcounter{equation}{0} -\label{integals1} - -For the ${}_2F_1$ hypergeometric function we have Euler's integral representation -\begin{equation} -\label{Eulerint}\index{Euler's integral representation for a ${}_2F_1$} -\index{Hypergeometric function!Euler's integral representation} -\hyp{2}{1}{a,b}{c}{z}=\frac{\Gamma(c)}{\Gamma(b)\Gamma(c-b)} -\int_0^1t^{b-1}(1-t)^{c-b-1}(1-zt)^{-a}\,dt, -\end{equation} -where $\textrm{Re}\,c>\textrm{Re}\,b>0$, which holds for $z\in\mathbb{C}\setminus(1,\infty)$. -Here it is understood that $\arg t=\arg(1-t)=0$ and that $(1-zt)^{-a}$ has its -principal value. - -We also have Barnes' integral representation -\begin{equation} -\label{Barnes}\index{Barnes' integral representation} -\index{Hypergeometric function!Barnes' integral representation} -\frac{\Gamma(a)\Gamma(b)}{\Gamma(c)}\,\hyp{2}{1}{a,b}{c}{z}=\frac{1}{2\pi i} -\int_{-i\infty}^{i\infty}\frac{\Gamma(a+s)\Gamma(b+s)\Gamma(-s)}{\Gamma(c+s)}(-z)^s\,ds -\end{equation} -for $|z|<1$ with $\arg(-z)<\pi$, where the path of integration is deformed -if necessary, to separate the decreasing poles $s=-a-n$ and $s=-b-n$ from the -increasing poles $s=n$ for $n\in\{0,1,2,\ldots\}$. Such a path always exists if -$a,b\notin\{\ldots,-3,-2,-1\}$. - -Secondly, we have the Mellin-Barnes integral or Barnes' first lemma -\begin{eqnarray} -\label{Mellin-Barnes1}\index{Mellin-Barnes integral}\index{Barnes' first lemma} -& &\frac{1}{2\pi i}\int_{-i\infty}^{i\infty}\Gamma(a+s)\Gamma(b+s)\Gamma(c-s)\Gamma(d-s)\,ds\nonumber\\ -& &{}=\frac{\Gamma(a+c)\Gamma(a+d)\Gamma(b+c)\Gamma(b+d)}{\Gamma(a+b+c+d)} -\end{eqnarray} -for $\textrm{Re}(a+b+c+d)<1$, where the contour must also be taken in a such a -way that the increasing poles and the decreasing poles remain separate. By using -analytic continuation, one can avoid the condition $\textrm{Re}(a+b+c+d)<1$. The -Barnes integral (\ref{Mellin-Barnes1}) can be seen as a continuous analogue of -Gauss' summation formula (\ref{Gauss}). - -We also have a continuous analogue of the Pfaff-Saalsch\"utz summation -formula (\ref{PfaffSaalschutz}) given by the integral, which is also -called Barnes' second lemma\index{Barnes' second lemma}, -\begin{eqnarray} -\label{Mellin-Barnes2} -& &\frac{1}{2\pi i}\int_{-i\infty}^{i\infty} -\frac{\Gamma(a+s)\Gamma(b+s)\Gamma(c+s)\Gamma(1-d-s)\Gamma(-s)}{\Gamma(e+s)}\,ds\nonumber\\ -& &{}=\frac{\Gamma(a)\Gamma(b)\Gamma(c)\Gamma(1+a-d)\Gamma(1+b-d)\Gamma(1+c-d)} -{\Gamma(e-a)\Gamma(e-b)\Gamma(e-c)} -\end{eqnarray} -with $d+e=a+b+c+1$, where the contour must be taken in a such a way that -the increasing poles and the decreasing poles stay separated. - -A continuous analogue of the summation formula (\ref{wellpoised}) for a very-well-poised -${}_5F_4$ is given by Bailey's integral (see also \cite{Bailey35}) -\begin{eqnarray} -\label{Bailey1}\index{Bailey's integral} -& &\frac{1}{2\pi i}\int_{-i\infty}^{i\infty} -\frac{\Gamma(1+a/2+s)\Gamma(a+s)\Gamma(b+s)\Gamma(c+s)\Gamma(d+s)\Gamma(b-a-s)\Gamma(-s)} -{\Gamma(a/2+s)\Gamma(1+a-c+s)\Gamma(1+a-d+s)}\,ds\nonumber\\ -& &{}=\frac{\Gamma(b)\Gamma(c)\Gamma(d)\Gamma(b+c-a)\Gamma(b+d-a)} -{2\,\Gamma(1+a-c-d)\Gamma(b+c+d-a)} -\end{eqnarray} -which can also be written in a more symmetric form -\begin{eqnarray} -\label{Bailey2} -& &\frac{1}{2\pi i}\int_{-i\infty}^{i\infty} -\frac{\Gamma(a+s)\Gamma(b+s)\Gamma(c+s)\Gamma(d+s)}{\Gamma(a+2s)}\nonumber\\ -& &{}\mathindent\times\frac{\Gamma(-s)\Gamma(b-a-s)\Gamma(c-a-s)\Gamma(d-a-s)}{\Gamma(-a-2s)}\,ds\nonumber\\ -& &{}=\frac{2\,\Gamma(b)\Gamma(c)\Gamma(d)\Gamma(b+c-a)\Gamma(b+d-a)\Gamma(c+d-a)}{\Gamma(b+c+d-a)} -\end{eqnarray} -and also in the following form due to Wilson (see also \cite{Wilson80}) -\begin{eqnarray} -\label{Bailey3} -& &\frac{1}{2\pi i}\int_{-i\infty}^{i\infty} -\frac{\Gamma(a+s)\Gamma(b+s)\Gamma(c+s)\Gamma(d+s)\Gamma(a-s)\Gamma(b-s)\Gamma(c-s)\Gamma(d-s)} -{\Gamma(2s)\Gamma(-2s)}\,ds\nonumber\\ -& &{}=\frac{2\,\Gamma(a+b)\Gamma(a+c)\Gamma(a+d)\Gamma(b+c)\Gamma(b+d)\Gamma(c+d)}{\Gamma(a+b+c+d)}, -\end{eqnarray} -where the contours must be taken in a such a way that the increasing poles and the -decreasing poles remain separate. - -We also mention the integral -\begin{equation} -\label{Slater} -\frac{1}{2\pi i}\int_{-i\infty}^{i\infty}\Gamma(a+s)\Gamma(b-s)z^{-s}\,ds -=\Gamma(a+b)\frac{z^a}{(1+z)^{a+b}}. -\end{equation} -Again the contour must be taken in such a way that the increasing poles of $\Gamma(b-s)$ and -the decreasing poles of $\Gamma(a+s)$ remain separate. - -The integrals (\ref{Mellin-Barnes1}) through (\ref{Bailey3}) are all special cases of a more -general formula (id est formula (4.5.1.2) in \cite{Slater}), which has more interesting and -useful special cases such as -\begin{equation} -\label{Slater1} -\frac{1}{2\pi i}\int_{-i\infty}^{i\infty}\frac{\Gamma(a+s)\Gamma(b-s)}{\Gamma(c+s)\Gamma(d-s)}\,ds -=\frac{\Gamma(a+b)\Gamma(c+d-a-b-1)}{\Gamma(c+d-1)\Gamma(c-a)\Gamma(d-b)}, -\end{equation} -where the contour must be taken in a such a way that the increasing poles and the decreasing poles -remain separate and -\begin{eqnarray} -\label{Slater2} -& &\frac{1}{2\pi i}\int_{-i\infty}^{i\infty} -\frac{\Gamma(a+s)\Gamma(b+s)\Gamma(c+s)\Gamma(-s)\Gamma(b-a-s)\Gamma(c-a-s)} -{\Gamma(a+2s)\Gamma(-a-2s)}\,ds\nonumber\\ -& &{}=\frac{1}{2}\Gamma(b)\Gamma(c)\Gamma(b+c-a), -\end{eqnarray} -where the contour must be taken in a such a way that the increasing poles and the decreasing poles -remain separate. - -If $a,b,c,d$ are positive or $b=\overline{a}$ and/or $d=\overline{c}$ and the real parts are -positive, Wilson's integral (\ref{Bailey3}) can be written in the form -\begin{eqnarray} -\label{Wilson1}\index{Wilson's integral} -& &\frac{1}{2\pi}\int_0^{\infty} -\left|\frac{\Gamma(a+ix)\Gamma(b+ix)\Gamma(c+ix)\Gamma(d+ix)}{\Gamma(2ix)}\right|^2\,dx\nonumber\\ -& &{}=\frac{\Gamma(a+b)\Gamma(a+c)\Gamma(a+d)\Gamma(b+c)\Gamma(b+d)\Gamma(c+d)}{\Gamma(a+b+c+d)}. -\end{eqnarray} -The limit case $d\rightarrow\infty$ leads to -\begin{equation} -\label{Wilson2} -\frac{1}{2\pi}\int_0^{\infty} -\left|\frac{\Gamma(a+ix)\Gamma(b+ix)\Gamma(c+ix)}{\Gamma(2ix)}\right|^2\,dx -=\Gamma(a+b)\Gamma(a+c)\Gamma(b+c). -\end{equation} - -\section{Transformation formulas for generalized hypergeometric functions} -\par\setcounter{equation}{0} -\label{transformation formulas} - -In this section we list a number of transformation formulas which can be used to -transform hypergeometric representations and other formulas into equivalent -but different forms. - -First of all we have Euler's transformation formula: -\begin{equation} -\label{Eulertrans}\index{Euler's transformation formula} -\index{Transformation formula!Euler} -\hyp{2}{1}{a,b}{c}{z}=(1-z)^{c-a-b}\,\hyp{2}{1}{c-a,c-b}{c}{z}. -\end{equation} - -Another transformation formula for the ${}_2F_1$ series, which is also due to Euler, is -\begin{equation} -\label{PfaffKummer}\index{Pfaff-Kummer transformation formula} -\index{Transformation formula!Pfaff-Kummer} -\hyp{2}{1}{a,b}{c}{z}=(1-z)^{-a}\,\hyp{2}{1}{a,c-b}{c}{\frac{z}{z-1}}. -\end{equation} -This transformation formula is also known as the Pfaff or Pfaff-Kummer transformation formula. - -As a limit case of the Pfaff-Kummer transformation formula we have Kummer's transformation -formula for the confluent hypergeometric series: -\begin{equation} -\label{Kummer}\index{Kummer's transformation formula}\index{Transformation formula!Kummer} -\hyp{1}{1}{a}{c}{z}=\e^z\,\hyp{1}{1}{c-a}{c}{-z}. -\end{equation} - -If we reverse the order of summation in a terminating ${}_1F_1$ series, we -obtain a ${}_2F_0$ series; in fact we have -\begin{equation} -\label{reverse1} -\hyp{1}{1}{-n}{a}{x}=\frac{(-x)^n}{(a)_n}\,\hyp{2}{0}{-n,-a-n+1}{-}{-\frac{1}{x}}, -\quad n=0,1,2,\ldots. -\end{equation} -If we apply this technique to a terminating ${}_2F_1$ series, we find -\begin{equation} -\label{reverse2} -\hyp{2}{1}{-n,b}{c}{x}=\frac{(b)_n}{(c)_n}(-x)^n\,\hyp{2}{1}{-n,-c-n+1}{-b-n+1}{\frac{1}{x}}, -\quad n=0,1,2,\ldots. -\end{equation} - -On the next level we have Whipple's transformation formula for a terminating balanced -${}_4F_3$ series: -\begin{eqnarray} -\label{Whipple}\index{Whipple's transformation formula}\index{Transformation formula!Whipple} -& &\hyp{4}{3}{-n,a,b,c}{d,e,f}{1}\nonumber\\ -& &{}=\frac{(e-a)_n(f-a)_n}{(e)_n(f)_n}\hyp{4}{3}{-n,a,d-b,d-c}{d,a-e-n+1,a-f-n+1}{1} -\end{eqnarray} -provided that $a+b+c+1=d+e+f+n$. - -\section{The $q$-shifted factorial} -\par\setcounter{equation}{0} -\label{qshifted factorial} - -The theory of $q$-analogues or $q$-extensions of classical formulas and -functions is based on the observation that -$$\lim\limits_{q\rightarrow 1}\frac{1-q^{\alpha}}{1-q}=\alpha.$$ -Therefore the number $(1-q^{\alpha})/(1-q)$ is sometimes called the -\emph{basic number} $[\alpha]$. For $q\neq 0$ and $q\neq 1$ we define -\begin{equation} -\label{basic} -[\alpha]:=\frac{1-q^{\alpha}}{1-q}, -\end{equation} -which implies that -\begin{equation} -\label{basicdef} -[0]=0,\quad [n]=\frac{1-q^n}{1-q}=\sum_{k=0}^{n-1}q^k,\quad n=1,2,3,\ldots. -\end{equation} - -Now we can give a $q$-analogue of the Pochhammer symbol $(a)_k$ defined by -(\ref{Pochhammer}): -\begin{equation} -\label{qsh}\index{q-Shifted factorial@$q$-Shifted factorial} -(a;q)_0:=1\quad\textrm{and}\quad (a;q)_k:=\prod_{n=1}^k(1-aq^{n-1}) -% \constraint{ $k=1,2,3,\ldots$ } -\end{equation} -It is clear that -\begin{equation} -\label{limitqsh} -\lim\limits_{q\rightarrow 1}\frac{(q^{\alpha};q)_k}{(1-q)^k}=(\alpha)_k. -\end{equation} -The symbols $(a;q)_k$ are called $q$-shifted factorials. For negative subscripts we define -\begin{equation} -\label{qsh1} -(a;q)_{-k}:=\frac{1}{\displaystyle\prod_{n=1}^{k}(1-aq^{-n})},\quad a\neq q,q^2,q^3,\ldots,q^k, -\quad k=1,2,3,\ldots. -\end{equation} -Now we have -\begin{equation} -\label{qsh2} -(a;q)_{-n}=\frac{1}{(aq^{-n};q)_n}=\frac{(-qa^{-1})^n}{(qa^{-1};q)_n} -q^{\binom{n}{2}},\quad n=0,1,2,\ldots. -\end{equation} - -If we replace $q$ by $q^{-1}$, we obtain -\begin{equation} -\label{qsh3} -(a;q^{-1})_n=(a^{-1};q)_n(-a)^nq^{-\binom{n}{2}},\quad a\neq 0. -\end{equation} - -We can also define -$$(a;q)_{\infty}=\prod_{k=0}^{\infty}(1-aq^k),\quad 0<|q|<1.$$ -This implies that -\begin{equation} -\label{qsh4} -(a;q)_n=\frac{(a;q)_{\infty}}{(aq^n;q)_{\infty}},\quad 0<|q|<1, -\end{equation} -and, for any complex number $\lambda$, -\begin{equation} -\label{qsh5} -(a;q)_{\lambda}=\frac{(a;q)_{\infty}}{(aq^{\lambda};q)_{\infty}},\quad 0<|q|<1, -\end{equation} -where the principal value of $q^{\lambda}$ is taken. - -Finally, we list a number of transformation formulas for the $q$-shifted -factorials, where $k$ and $n$ are nonnegative integers: -\begin{equation} -\label{qsh6} -(a;q)_{n+k}=(a;q)_n(aq^n;q)_k, -\end{equation} -\begin{equation} -\label{qsh7} -\frac{(aq^n;q)_k}{(aq^k;q)_n}=\frac{(a;q)_k}{(a;q)_n}, -\end{equation} -\begin{equation} -\label{qsh8} -(aq^k;q)_{n-k}=\frac{(a;q)_n}{(a;q)_k},\quad k=0,1,2,\ldots,n, -\end{equation} -\begin{equation} -\label{qsh9} -(a;q)_n=(a^{-1}q^{1-n};q)_n(-a)^nq^{\binom{n}{2}},\quad a\neq 0, -\end{equation} -\begin{equation} -\label{qsh10} -(aq^{-n};q)_n=(a^{-1}q;q)_n(-a)^nq^{-n-\binom{n}{2}},\quad a\neq 0, -\end{equation} -\begin{equation} -\label{qsh11} -\frac{(aq^{-n};q)_n}{(bq^{-n};q)_n}=\frac{(a^{-1}q;q)_n}{(b^{-1}q;q)_n} -\left(\frac{a}{b}\right)^n -% \constraint{ -% $a\neq 0$ & -% $b\neq 0$ } -\end{equation} -\begin{equation} -\label{qsh12} -(a;q)_{n-k}=\frac{(a;q)_n}{(a^{-1}q^{1-n};q)_k}\left(-\frac{q}{a}\right)^k -q^{\binom{k}{2}-nk},\quad a\neq 0,\quad k=0,1,2,\ldots,n, -\end{equation} -\begin{eqnarray} -\label{qsh13} -& &\frac{(a;q)_{n-k}}{(b;q)_{n-k}}=\frac{(a;q)_n}{(b;q)_n} -\frac{(b^{-1}q^{1-n};q)_k}{(a^{-1}q^{1-n};q)_k}\left(\frac{b}{a}\right)^k,\nonumber\\ -& &{}\mathindent\mathindent\mathindent a\neq 0,\quad b\neq 0,\quad k=0,1,2,\ldots,n, -\end{eqnarray} -\begin{equation} -\label{qsh14} -(q^{-n};q)_k=\frac{(q;q)_n}{(q;q)_{n-k}}(-1)^kq^{\binom{k}{2}-nk}, -\quad k=0,1,2,\ldots,n, -\end{equation} -\begin{equation} -\label{qsh15} -(aq^{-n};q)_k=\frac{(a^{-1}q;q)_n}{(a^{-1}q^{1-k};q)_n}(a;q)_kq^{-nk}, -\quad a\neq 0, -\end{equation} -\begin{eqnarray} -\label{qsh16} -& &(aq^{-n};q)_{n-k}=\frac{(a^{-1}q;q)_n}{(a^{-1}q;q)_k} -\left(-\frac{a}{q}\right)^{n-k}q^{\binom{k}{2}-\binom{n}{2}},\nonumber\\ -& &{}\mathindent\mathindent\mathindent a\neq 0,\quad k=0,1,2,\ldots,n, -\end{eqnarray} -\begin{equation} -\label{qsh17} -(a;q)_{2n}=(a;q^2)_n(aq;q^2)_n, -\end{equation} -\begin{equation} -\label{qsh18} -(a^2;q^2)_n=(a;q)_n(-a;q)_n, -\end{equation} -\begin{equation} -\label{qsh19} -(a;q)_{\infty}=(a;q^2)_{\infty}(aq;q^2)_{\infty},\quad 0<|q|<1, -\end{equation} -\begin{equation} -\label{qsh20} -(a^2;q^2)_{\infty}=(a;q)_{\infty}(-a;q)_{\infty},\quad 0<|q|<1. -\end{equation} - -We remark that by using (\ref{qsh18}) we have -\begin{equation} -\label{qsh21} -\frac{1-a^2q^{2n}}{1-a^2}=\frac{(a^2q^2;q^2)_n}{(a^2;q^2)_n} -=\frac{(aq;q)_n(-aq;q)_n}{(a;q)_n(-a;q)_n}. -\end{equation} - -\section{The $q$-gamma function and $q$-binomial coefficients} -\par\setcounter{equation}{0} -\label{qgamma function} - -The $q$-gamma function is defined by -\begin{equation} -\label{DefqGamma}\index{q-Gamma function@$q$-Gamma function} -\Gamma_q(x):=\frac{(q;q)_{\infty}}{(q^x;q)_{\infty}}(1-q)^{1-x},\quad 01$ the $q$-gamma function can be defined by -\begin{equation} -\label{DefqGamma2} -\Gamma_q(x)=\frac{(q^{-1};q^{-1})_{\infty}}{(q^{-x};q^{-1})_{\infty}} -q^{\binom{x}{2}}(q-1)^{1-x},\quad q>1. -\end{equation} - -The $q$-binomial coefficient is defined by -\begin{equation} -\label{DefqBinomial1}\index{q-Binomial coefficient@$q$-Binomial coefficient} -\genfrac{[}{]}{0pt}{}{n}{k}_q:=\frac{(q;q)_n}{(q;q)_k(q;q)_{n-k}}=\genfrac{[}{]}{0pt}{}{n}{n-k}_q, -\quad k=0,1,2,\ldots,n, -\end{equation} -where $n$ denotes a nonnegative integer. - -This definition can be generalized in the following way. For arbitrary -complex $\alpha$ we have -\begin{equation} -\label{DefqBinomial2} -\genfrac{[}{]}{0pt}{}{\alpha}{k}_q:=\frac{(q^{-\alpha};q)_k}{(q;q)_k} -(-1)^kq^{k\alpha-\binom{k}{2}}. -\end{equation} -Or more general, for all complex $\alpha$ and $\beta$ and $0<|q|<1$ we have -\begin{equation} -\label{DefqBinomial3} -\genfrac{[}{]}{0pt}{}{\alpha}{\beta}_q:=\frac{\Gamma_q(\alpha+1)}{\Gamma_q(\beta+1)\Gamma_q(\alpha-\beta+1)} -=\frac{(q^{\beta+1};q)_{\infty}(q^{\alpha-\beta+1};q)_{\infty}} -{(q;q)_{\infty}(q^{\alpha+1};q)_{\infty}}. -\end{equation} - -For instance this implies that -$$\frac{(q^{\alpha+1};q)_n}{(q;q)_n}=\genfrac{[}{]}{0pt}{}{n+\alpha}{n}_q.$$ - -Note that -$$\lim\limits_{q\rightarrow 1}\genfrac{[}{]}{0pt}{}{\alpha}{\beta}_q -=\frac{\Gamma(\alpha+1)}{\Gamma(\beta+1)\Gamma(\alpha-\beta+1)}=\binom{\alpha}{\beta}.$$ - -Finally, we remark that -\begin{equation} -\label{qqn} -\frac{1}{(q;q)_n}=\sum_{k=0}^n\frac{q^k}{(q;q)_k},\quad n=0,1,2,\ldots, -\end{equation} -which can easily be shown by induction, and that -\begin{equation} -\label{aqn} -(a;q)_n=\sum_{k=0}^n\genfrac{[}{]}{0pt}{}{n}{k}_qq^{\binom{k}{2}}(-a)^k,\quad n=0,1,2,\ldots, -\end{equation} -which is a special case of (\ref{qbinomialn}). - -\section{Basic hypergeometric functions} -\par\setcounter{equation}{0}\label{hypergeometric} -\label{qhypergeometric functions} - -The basic hypergeometric or $q$-hypergeometric function $\mbox{}_r\phi_s$ is defined by the series -\begin{eqnarray} -\label{DefBasHyp}\index{Basic hypergeometric function} -& &\qhyp{r}{s}{a_1,\ldots,a_r}{b_1,\ldots,b_s}{z}\nonumber\\ -& &{}:=\sum\limits_{k=0}^{\infty}\frac{(a_1,\ldots,a_r;q)_k}{(b_1,\ldots,b_s;q)_k} -(-1)^{(1+s-r)k}q^{(1+s-r)\binom{k}{2}}\frac{z^k}{(q;q)_k}, -\end{eqnarray} -where -$$(a_1,\ldots,a_r;q)_k:=(a_1;q)_k\cdots(a_r;q)_k.$$ -Again we assume that the parameters are such that the denominator factors -in the terms of the series are never zero. If one of the numerator -parameters $a_j$ equals $q^{-n}$, where $n$ is a nonnegative integer, this -basic hypergeometric function is a polynomial in $z$. Otherwise the radius of -convergence $\rho$ of the basic hypergeometric series is given by -$$\rho=\left\{\begin{array}{ll} -\displaystyle \infty & \quad\textrm{if}\quad r < s+1\\[5mm] -\displaystyle 1 & \quad\textrm{if}\quad r = s+1\\[5mm] -\displaystyle 0 & \quad\textrm{if}\quad r > s+1.\end{array}\right.$$ - -The special case $r=s+1$ reads -$$\qhyp{s+1}{s}{a_1,\ldots,a_{s+1}}{b_1,\ldots,b_s}{z}= -\sum\limits_{k=0}^{\infty}\frac{(a_1,\ldots,a_{s+1};q)_k}{(b_1,\ldots,b_s;q)_k} -\frac{z^k}{(q;q)_k}.$$ -This basic hypergeometric series was first introduced by Heine in 1846; therefore it is -sometimes called Heine's\index{Heine's series} series. A basic hypergeometric series of -this form is called balanced\index{Basic hypergeometric function!Balanced} -or Saalsch\"{u}tzian\index{Basic hypergeometric function!Saalsch\"{u}tzian} if $z=q$ and -$a_1a_2 \cdots a_{s+1}q=b_1b_2 \cdots b_s$. - -The $q$-hypergeometric function is a $q$-analogue of the hypergeometric function -defined by (\ref{DefHyp}) since -$$\lim\limits_{q\rightarrow 1} -\qhyp{r}{s}{q^{a_1},\ldots,q^{a_r}}{q^{b_1},\ldots,q^{b_s}}{(q-1)^{1+s-r}z} -=\hyp{r}{s}{a_1,\ldots,a_r}{b_1,\ldots,b_s}{z}.$$ -This limit will be used frequently in chapter~\ref{BasicHyperOrtPol}. In all -cases the hypergeometric series involved is in fact a polynomial so that -convergence is guaranteed. - -In the sequel of this paragraph we also assume that each (basic) hypergeometric -function is in fact a polynomial. We remark that -$$\lim\limits_{a_r\rightarrow\infty} -\qhyp{r}{s}{a_1,\ldots,a_r}{b_1,\ldots,b_s}{\frac{z}{a_r}}= -\qhyp{r-1}{s}{a_1,\ldots,a_{r-1}}{b_1,\ldots,b_s}{z}.$$ -In fact this is the reason for the factors -$(-1)^{(1+s-r)k}q^{(1+s-r)\binom{k}{2}}$ in the definition (\ref{DefBasHyp}) -of the basic hypergeometric function. - -Many limit relations between basic hypergeometric orthogonal polynomials -are based on the observations that -\begin{equation} -\label{qhypsub} -\qhyp{r}{s}{a_1,\ldots,a_{r-1},\mu}{b_1,\ldots,b_{s-1},\mu}{z}= -\qhyp{r-1}{s-1}{a_1,\ldots,a_{r-1}}{b_1,\ldots,b_{s-1}}{z}, -\end{equation} -\begin{equation} -\label{qhyplim1} -\lim\limits_{\lambda\rightarrow\infty} -\qhyp{r}{s}{a_1,\ldots,a_{r-1},\lambda a_r}{b_1,\ldots,b_s}{\frac{z}{\lambda}} -=\qhyp{r-1}{s}{a_1,\ldots,a_{r-1}}{b_1,\ldots,b_s}{a_rz}, -\end{equation} -\begin{equation} -\label{qhyplim2} -\lim\limits_{\lambda\rightarrow\infty} -\qhyp{r}{s}{a_1,\ldots,a_r}{b_1,\ldots,b_{s-1},\lambda b_s}{\lambda z}= -\qhyp{r}{s-1}{a_1,\ldots,a_r}{b_1,\ldots,b_{s-1}}{\frac{z}{b_s}}, -\end{equation} -and -\begin{equation} -\label{qhyplim3} -\lim\limits_{\lambda\rightarrow\infty} -\qhyp{r}{s}{a_1,\ldots,a_{r-1},\lambda a_r}{b_1,\ldots,b_{s-1},\lambda b_s}{z} -=\qhyp{r-1}{s-1}{a_1,\ldots,a_{r-1}}{b_1,\ldots,b_{s-1}}{\frac{a_rz}{b_s}}. -\end{equation} - -Mostly, the left-hand side of (\ref{qhypsub}) occurs as a limit case when some numerator -parameter and some denominator parameter tend to the same value. - -All families of discrete orthogonal polynomials $\{P_n(x)\}_{n=0}^N$ are -defined for $n=0,1,2,\ldots,N$, where $N$ is a positive integer. In these -cases something like (\ref{qhypsub}) occurs in the basic hypergeometric representation -when $n=N$. In these cases we have to be aware of the fact that we still have a polynomial -(in that case of degree $N$). For instance, if we take $n=N$ in the basic hypergeometric -representation (\ref{DefqHahn}) of the $q$-Hahn polynomials, we have -$$Q_N(q^{-x};\alpha,\beta,N|q)=\sum_{k=0}^N -\frac{(\alpha\beta q^{N+1};q)_k(q^{-x};q)_k}{(\alpha q;q)_k(q;q)_k}q^k.$$ -So these cases must be understood by continuity. - -\section{The $q$-binomial theorem and other summation formulas} -\par\setcounter{equation}{0} -\label{qsummation formulas} - -A $q$-analogue of the binomial theorem (\ref{binomial}) is called the $q$-binomial theorem: -\begin{equation} -\label{qbinomial}\index{q-Binomial theorem@$q$-Binomial theorem} -\index{Summation formula!q-Binomial theorem@$q$-Binomial theorem} -\qhyp{1}{0}{a}{-}{z}=\sum_{n=0}^{\infty}\frac{(a;q)_n}{(q;q)_n}z^n= -\frac{(az;q)_{\infty}}{(z;q)_{\infty}},\quad 0<|q|<1,\quad |z|<1. -\end{equation} -For $a=q^{-n}$ with $n$ a nonnegative integer we find -\begin{equation} -\label{qbinomialn} -\qhyp{1}{0}{q^{-n}}{-}{z}=(zq^{-n};q)_n,\quad n=0,1,2,\ldots. -\end{equation} -In fact this is a $q$-analogue of Newton's binomium (\ref{Newton}). -Note that the case $a=0$ of (\ref{qbinomial}) is the limit case ($n\rightarrow\infty$) of (\ref{qqn}). - -Gauss's summation formula (\ref{Gauss}) and the Vandermonde or Chu-Vandermonde -summation formula (\ref{ChuVandermonde}) have the following $q$-analogues: -\begin{equation} -\label{qChuVandermonde1}\index{q-Gauss summation formula@$q$-Gauss summation formula} -\index{Summation formula!q-Gauss@$q$-Gauss} -\qhyp{2}{1}{a,b}{c}{\frac{c}{ab}}=\frac{(a^{-1}c,b^{-1}c;q)_{\infty}} -{(c,a^{-1}b^{-1}c;q)_{\infty}},\quad 0<|q|<1,\quad\left|\frac{c}{ab}\right|<1, -\end{equation} -\begin{equation} -\label{qChuVandermonde2}\index{q-Vandermonde summation formula@$q$-Vandermonde summation formula} -\index{Summation formula!q-Vandermonde@$q$-Vandermonde} -\index{q-Chu-Vandermonde summation formula@$q$-Chu-Vandermonde summation formula} -\index{Summation formula!q-Chu-Vandermonde@$q$-Chu-Vandermonde} -\qhyp{2}{1}{q^{-n},b}{c}{\frac{cq^n}{b}}=\frac{(b^{-1}c;q)_n}{(c;q)_n},\quad n=0,1,2,\ldots -\end{equation} -and -\begin{equation} -\label{qChuVandermonde3} -\qhyp{2}{1}{q^{-n},b}{c}{q}=\frac{(b^{-1}c;q)_n}{(c;q)_n}b^n,\quad n=0,1,2,\ldots. -\end{equation} -By taking the limit $b\rightarrow\infty$ in (\ref{qChuVandermonde1}) we also obtain a summation -formula for the ${}_1\phi_1$ series: -\begin{equation} -\label{sum1phi1}\index{Summation formula!for a ${}_1\phi_1$} -\qhyp{1}{1}{a}{c}{\frac{c}{a}}=\frac{(a^{-1}c;q)_{\infty}}{(c;q)_{\infty}},\quad 0<|q|<1. -\end{equation} -By taking the limit $c\rightarrow\infty$ in (\ref{qChuVandermonde2}) we obtain a summation -formula for a terminating ${}_2\phi_0$ series: -\begin{equation} -\label{sum2phi0}\index{Summation formula!for a terminating ${}_2\phi_0$} -\qhyp{2}{0}{q^{-n},b}{-}{\frac{q^n}{b}}=\frac{1}{b^n},\quad n=0,1,2,\ldots. -\end{equation} -By taking the limit $a\rightarrow\infty$ in (\ref{sum1phi1}) we obtain -\begin{equation} -\label{sum0phi1}\index{Summation formula!for a ${}_0\phi_1$} -\qhyp{0}{1}{-}{c}{c}=\frac{1}{(c;q)_{\infty}},\quad 0<|q|<1. -\end{equation} - -On the ${}_3\phi_2$ level we have the summation formula -\begin{equation} -\label{qPfaffSaalschutz} -\index{q-Saalschutz summation formula@$q$-Saalsch\"{u}tz summation formula} -\index{Summation formula!q-Saalschutz@$q$-Saalsch\"{u}tz} -\index{q-Pfaff-Saalschutz summation formula@$q$-Pfaff-Saalsch\"{u}tz summation formula} -\index{Summation formula!q-Pfaff-Saalschutz@$q$-Pfaff-Saalsch\"{u}tz} -\qhyp{3}{2}{q^{-n},a,b}{c,abc^{-1}q^{1-n}}{q}=\frac{(a^{-1}c,b^{-1}c;q)_n} -{(c,a^{-1}b^{-1}c;q)_n},\quad n=0,1,2,\ldots, -\end{equation} -which is a $q$-analogue of the Saalsch\"utz or Pfaff-Saalsch\"utz summation formula -(\ref{PfaffSaalschutz}). - -On the ${}_6\phi_5$ level we have the Jackson summation formula -\begin{eqnarray} -\label{qJackson1}\index{Jackson's summation formula}\index{Summation formula!Jackson} -\index{Summation formula!for a very-well-poised ${}_6\phi_5$} -& &\qhyp{6}{5}{q\sqrt{a},-q\sqrt{a},a,b,c,d}{\sqrt{a},-\sqrt{a},ab^{-1}q,ac^{-1}q,ad^{-1}q}{\frac{aq}{bcd}}\nonumber\\ -& &{}=\frac{(aq,ab^{-1}c^{-1}q,ab^{-1}d^{-1}q,ac^{-1}d^{-1}q;q)_{\infty}} -{(ab^{-1}q,ac^{-1}q,ad^{-1}q,ab^{-1}c^{-1}d^{-1}q;q)_{\infty}},\quad\left|\frac{aq}{bcd}\right|<1 -\end{eqnarray} -for a non-terminating very-well-poised ${}_6\phi_5$. We also have the summation formula -\begin{eqnarray} -\label{qJackson2} -& &\qhyp{6}{5}{q\sqrt{a},-q\sqrt{a},a,b,c,q^{-n}}{\sqrt{a},-\sqrt{a},ab^{-1}q,ac^{-1}q,aq^{n+1}}{\frac{aq^{n+1}}{bc}}\nonumber\\ -& &{}=\frac{(aq,ab^{-1}c^{-1}q;q)_n}{(ab^{-1}q,ac^{-1}q;q)_n},\quad n=0,1,2,\ldots -\end{eqnarray} -for a terminating very-well-poised ${}_6\phi_5$, which is also due to Jackson. -By taking the limit $c\rightarrow 0$ we obtain -\begin{eqnarray} -\label{qJackson3} -& &\qhyp{6}{4}{q\sqrt{a},-q\sqrt{a},a,b,0,q^{-n}}{\sqrt{a},-\sqrt{a},ab^{-1}q,aq^{n+1}}{\frac{q^n}{b}}\nonumber\\ -& &{}=\frac{(aq;q)_n}{(ab^{-1}q;q)_n}b^{-n},\quad n=0,1,2,\ldots. -\end{eqnarray} -If we take the limit $b\rightarrow 0$ we obtain -\begin{eqnarray} -\label{qJackson4} -& &\qhyp{6}{3}{q\sqrt{a},-q\sqrt{a},a,0,0,q^{-n}}{\sqrt{a},-\sqrt{a},aq^{n+1}}{\frac{q^{n-1}}{a}}\nonumber\\ -& &{}=(-1)^na^{-n}q^{-\binom{n+1}{2}}(aq;q)_n,\quad n=0,1,2,\ldots. -\end{eqnarray} - -\section{More integrals} -\par\setcounter{equation}{0} -\label{integrals2} - -In this section we list some $q$-extensions of the beta integral. -For instance, we will need the following integral: -\begin{equation} -\label{int1} -\int_0^{\infty}x^{c-1}\frac{(-ax,-bq/x;q)_{\infty}}{(-x,-q/x;q)_{\infty}}\,dx -=\frac{\pi}{\sin(\pi c)}\,\frac{(ab,q^c,q^{1-c};q)_{\infty}}{(bq^c,aq^{-c},q;q)_{\infty}}, -\end{equation} -which holds for $00$ and $\left|aq^{-c}\right|<1$. -If we set $b=1$ in (\ref{int1}), we find that -\begin{equation} -\label{int2} -\int_0^{\infty}x^{c-1}\frac{(-ax;q)_{\infty}}{(-x;q)_{\infty}}\,dx -=\frac{\pi}{\sin(\pi c)}\,\frac{(a,q^{1-c};q)_{\infty}}{(aq^{-c},q;q)_{\infty}} -\end{equation} -for $00$ and $\left|aq^{-c}\right|<1$, -and by taking the limit $c\rightarrow 1$ in (\ref{int1}), we obtain -\begin{equation} -\label{int3} -\int_0^{\infty}\frac{(-ax,-bq/x;q)_{\infty}}{(-x,-q/x;q)_{\infty}}\,dx -=-\ln q\,\frac{(ab,q;q)_{\infty}}{(bq,a/q;q)_{\infty}} -\end{equation} -for $00$ and non-real parameters occur in conjugate pairs, then -\begin{eqnarray} -\label{OrthIWilson} -& &\frac{1}{2\pi}\int_0^{\infty} -\left|\frac{\Gamma(a+ix)\Gamma(b+ix)\Gamma(c+ix)\Gamma(d+ix)}{\Gamma(2ix)}\right|^2\nonumber\\ -& &{}\mathindent\times W_m(x^2;a,b,c,d)W_n(x^2;a,b,c,d)\,dx\nonumber\\ -& &{}=\frac{\Gamma(n+a+b)\cdots\Gamma(n+c+d)}{\Gamma(2n+a+b+c+d)}(n+a+b+c+d-1)_nn!\,\delta_{mn}, -\end{eqnarray} -where -\begin{eqnarray*} -& &\Gamma(n+a+b)\cdots\Gamma(n+c+d)\\ -& &{}=\Gamma(n+a+b)\Gamma(n+a+c)\Gamma(n+a+d)\Gamma(n+b+c)\Gamma(n+b+d)\Gamma(n+c+d). -\end{eqnarray*} -If $a<0$ and $a+b$, $a+c$, $a+d$ are positive or a pair of complex conjugates -occur with positive real parts, then -\begin{eqnarray} -\label{OrtIIWilson} -& &\frac{1}{2\pi}\int_0^{\infty}\left|\frac{\Gamma(a+ix)\Gamma(b+ix)\Gamma(c+ix)\Gamma(d+ix)}{\Gamma(2ix)}\right|^2\nonumber\\ -& &{}\mathindent\mathindent\times W_m(x^2;a,b,c,d)W_n(x^2;a,b,c,d)\,dx\nonumber\\ -& &{}\mathindent{}+\frac{\Gamma(a+b)\Gamma(a+c)\Gamma(a+d)\Gamma(b-a)\Gamma(c-a)\Gamma(d-a)}{\Gamma(-2a)}\nonumber\\ -& &{}\mathindent\mathindent{}\times\sum_{\begin{array}{c} -{\scriptstyle k=0,1,2\ldots}\\{\scriptstyle a+k<0}\end{array}} -\frac{(2a)_k(a+1)_k(a+b)_k(a+c)_k(a+d)_k}{(a)_k(a-b+1)_k(a-c+1)_k(a-d+1)_kk!}\nonumber\\ -& &{}\mathindent\mathindent\mathindent\mathindent{}\times W_m(-(a+k)^2;a,b,c,d)W_n(-(a+k)^2;a,b,c,d)\nonumber\\ -& &{}=\frac{\Gamma(n+a+b)\cdots\Gamma(n+c+d)}{\Gamma(2n+a+b+c+d)}(n+a+b+c+d-1)_nn!\,\delta_{mn}. -\end{eqnarray} - -\subsection*{Recurrence relation} -\begin{equation} -\label{RecWilson} --\left(a^2+x^2\right){\tilde{W}}_n(x^2) -=A_n{\tilde{W}}_{n+1}(x^2)-\left(A_n+C_n\right){\tilde{W}}_n(x^2)+C_n{\tilde{W}}_{n-1}(x^2), -\end{equation} -where -$${\tilde{W}}_n(x^2):={\tilde{W}}_n(x^2;a,b,c,d)=\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}$$ -and -$$\left\{\begin{array}{l} -\displaystyle A_n=\frac{(n+a+b+c+d-1)(n+a+b)(n+a+c)(n+a+d)}{(2n+a+b+c+d-1)(2n+a+b+c+d)}\\[5mm] -\displaystyle C_n=\frac{n(n+b+c-1)(n+b+d-1)(n+c+d-1)}{(2n+a+b+c+d-2)(2n+a+b+c+d-1)}. -\end{array}\right.$$ - -\subsection*{Normalized recurrence relation} -\begin{equation} -\label{NormRecWilson} -xp_n(x)=p_{n+1}(x)+(A_n+C_n-a^2)p_n(x)+A_{n-1}C_np_{n-1}(x), -\end{equation} -where -$$W_n(x^2;a,b,c,d)=(-1)^n(n+a+b+c+d-1)_np_n(x^2).$$ - -\subsection*{Difference equation} -\begin{eqnarray} -\label{dvWilson} -& &n(n+a+b+c+d-1)y(x)\nonumber\\ -& &{}=B(x)y(x+i)-\left[B(x)+D(x)\right]y(x)+D(x)y(x-i), -\end{eqnarray} -where -$$y(x)=W_n(x^2;a,b,c,d)$$ -and -$$\left\{\begin{array}{l} -\displaystyle B(x)=\frac{(a-ix)(b-ix)(c-ix)(d-ix)}{2ix(2ix-1)}\\[5mm] -\displaystyle D(x)=\frac{(a+ix)(b+ix)(c+ix)(d+ix)}{2ix(2ix+1)}. -\end{array}\right.$$ - -\subsection*{Forward shift operator} -\begin{eqnarray} -\label{shift1WilsonI} -& &W_n((x+\textstyle\frac{1}{2}i)^2;a,b,c,d) --W_n((x-\textstyle\frac{1}{2}i)^2;a,b,c,d)\nonumber\\ -& &{}=-2inx(n+a+b+c+d-1)W_{n-1}(x^2;a+\textstyle\frac{1}{2},b+\textstyle\frac{1}{2}, -c+\textstyle\frac{1}{2},d+\textstyle\frac{1}{2}) -\end{eqnarray} -or equivalently -\begin{eqnarray} -\label{shift1WilsonII} -& &\frac{\delta W_n(x^2;a,b,c,d)}{\delta x^2}\nonumber\\ -& &{}=-n(n+a+b+c+d-1)W_{n-1}(x^2;a+\textstyle\frac{1}{2},b+\textstyle\frac{1}{2}, -c+\textstyle\frac{1}{2},d+\textstyle\frac{1}{2}). -\end{eqnarray} - -\newpage - -\subsection*{Backward shift operator} -\begin{eqnarray} -\label{shift2WilsonI} -& &(a-\textstyle\frac{1}{2}-ix)(b-\textstyle\frac{1}{2}-ix) -(c-\textstyle\frac{1}{2}-ix)(d-\textstyle\frac{1}{2}-ix) -W_n((x+\textstyle\frac{1}{2}i)^2;a,b,c,d)\nonumber\\ -& &{}\mathindent{}-(a-\textstyle\frac{1}{2}+ix)(b-\textstyle\frac{1}{2}+ix) -(c-\textstyle\frac{1}{2}+ix)(d-\textstyle\frac{1}{2}+ix) -W_n((x-\textstyle\frac{1}{2}i)^2;a,b,c,d)\nonumber\\ -& &{}=-2ix W_{n+1}(x^2;a-\textstyle\frac{1}{2},b-\textstyle\frac{1}{2}, -c-\textstyle\frac{1}{2},d-\textstyle\frac{1}{2}) -\end{eqnarray} -or equivalently -\begin{eqnarray} -\label{shift2WilsonII} -& &\frac{\delta\left[\omega(x;a,b,c,d)W_n(x^2;a,b,c,d)\right]}{\delta x^2}\nonumber\\ -& &{}=\omega(x;a-\textstyle\frac{1}{2},b-\textstyle\frac{1}{2}, -c-\textstyle\frac{1}{2},d-\textstyle\frac{1}{2}) -W_{n+1}(x^2;a-\textstyle\frac{1}{2},b-\textstyle\frac{1}{2},c-\textstyle\frac{1}{2},d-\textstyle\frac{1}{2}), -\end{eqnarray} -where -$$\omega(x;a,b,c,d):=\frac{1}{2ix}\left|\frac{\Gamma(a+ix)\Gamma(b+ix)\Gamma(c+ix)\Gamma(d+ix)}{\Gamma(2ix)}\right|^2.$$ - -\subsection*{Rodrigues-type formula} -\begin{eqnarray} -\label{RodWilson} -& &\omega(x;a,b,c,d)W_n(x^2;a,b,c,d)\nonumber\\ -& &{}=\left(\frac{\delta}{\delta x^2}\right)^n -\left[\omega(x;a+\textstyle\frac{1}{2}n,b+\textstyle\frac{1}{2}n, -c+\textstyle\frac{1}{2}n,d+\textstyle\frac{1}{2}n)\right]. -\end{eqnarray} - -\subsection*{Generating functions} -\begin{equation} -\label{GenWilson1} -\hyp{2}{1}{a+ix,b+ix}{a+b}{t}\,\hyp{2}{1}{c-ix,d-ix}{c+d}{t}= -\sum_{n=0}^{\infty}\frac{W_n(x^2;a,b,c,d)t^n}{(a+b)_n(c+d)_nn!}. -\end{equation} - -\begin{equation} -\label{GenWilson2} -\hyp{2}{1}{a+ix,c+ix}{a+c}{t}\,\hyp{2}{1}{b-ix,d-ix}{b+d}{t}= -\sum_{n=0}^{\infty}\frac{W_n(x^2;a,b,c,d)t^n}{(a+c)_n(b+d)_nn!}. -\end{equation} - -\begin{equation} -\label{GenWilson3} -\hyp{2}{1}{a+ix,d+ix}{a+d}{t}\,\hyp{2}{1}{b-ix,c-ix}{b+c}{t}= -\sum_{n=0}^{\infty}\frac{W_n(x^2;a,b,c,d)t^n}{(a+d)_n(b+c)_nn!}. -\end{equation} - -\begin{eqnarray} -\label{GenWilson4} -& &(1-t)^{1-a-b-c-d}\nonumber\\ -& &{}\mathindent\times\hyp{4}{3}{\frac{1}{2}(a+b+c+d-1),\frac{1}{2}(a+b+c+d),a+ix,a-ix} -{a+b,a+c,a+d}{-\frac{4t}{(1-t)^2}}\nonumber\\ -& &{}=\sum_{n=0}^{\infty}\frac{(a+b+c+d-1)_n}{(a+b)_n(a+c)_n(a+d)_nn!}W_n(x^2;a,b,c,d)t^n. -\end{eqnarray} - -\subsection*{Limit relations} - -\subsubsection*{Wilson $\rightarrow$ Continuous dual Hahn} -The continuous dual Hahn polynomials given by (\ref{DefContDualHahn}) can be found from the -Wilson polynomials by dividing by $(a+d)_n$ and letting $d\rightarrow\infty$: -\begin{equation} -\lim_{d\rightarrow\infty}\frac{W_n(x^2;a,b,c,d)}{(a+d)_n}=S_n(x^2;a,b,c). -\end{equation} - -\subsubsection*{Wilson $\rightarrow$ Continuous Hahn} -The continuous Hahn polynomials given by (\ref{DefContHahn}) are obtained from the Wilson -polynomials by the substitutions $a\rightarrow a-it$, $b\rightarrow b-it$, $c\rightarrow c+it$, -$d\rightarrow d+it$ and $x\rightarrow x+t$ and the limit $t\rightarrow\infty$ in the following -way: -\begin{equation} -\lim_{t\rightarrow\infty} -\frac{W_n((x+t)^2;a-it,b-it,c+it,d+it)}{(-2t)^nn!}=p_n(x;a,b,c,d). -\end{equation} - -\subsubsection*{Wilson $\rightarrow$ Jacobi} -The Jacobi polynomials given by (\ref{DefJacobi}) can be found from the Wilson -polynomials by substituting $a=b=\frac{1}{2}(\alpha+1)$, $c=\frac{1}{2}(\beta+1)+it$, -$d=\frac{1}{2}(\beta+1)-it$ and $x\rightarrow t\sqrt{\frac{1}{2}(1-x)}$ in the -definition (\ref{DefWilson}) of the Wilson polynomials and taking the limit -$t\rightarrow\infty$. In fact we have -\begin{eqnarray} -& &\lim_{t\rightarrow\infty}\frac{W_n(\frac{1}{2}(1-x)t^2;\frac{1}{2}(\alpha+1), -\frac{1}{2}(\alpha+1),\frac{1}{2}(\beta+1)+it,\frac{1}{2}(\beta+1)-it)}{t^{2n}n!}\nonumber\\ -& &{}=P_n^{(\alpha,\beta)}(x). -\end{eqnarray} - -\subsection*{Remarks} -Note that for $k0$, $c=\bar{a}$ and $d=\bar{b}$, then -\begin{eqnarray} -\label{OrtContHahn} -& &\frac{1}{2\pi}\int_{-\infty}^{\infty}\Gamma(a+ix)\Gamma(b+ix)\Gamma(c-ix)\Gamma(d-ix) -p_m(x;a,b,c,d)p_n(x;a,b,c,d)\,dx\nonumber\\ -& &{}=\frac{\Gamma(n+a+c)\Gamma(n+a+d)\Gamma(n+b+c)\Gamma(n+b+d)} -{(2n+a+b+c+d-1)\Gamma(n+a+b+c+d-1)n!}\,\delta_{mn}. -\end{eqnarray} - -\subsection*{Recurrence relation} -\begin{equation} -\label{RecContHahn} -(a+ix)\tilde{p}_n(x)=A_n\tilde{p}_{n+1}(x)-\left(A_n+C_n\right)\tilde{p}_n(x)+C_n\tilde{p}_{n-1}(x), -\end{equation} -where -$$\tilde{p}_n(x):=\tilde{p}_n(x;a,b,c,d)=\frac{n!}{i^n(a+c)_n(a+d)_n}p_n(x;a,b,c,d)$$ -and -$$\left\{\begin{array}{l} -\displaystyle A_n=-\frac{(n+a+b+c+d-1)(n+a+c)(n+a+d)}{(2n+a+b+c+d-1)(2n+a+b+c+d)}\\[5mm] -\displaystyle C_n=\frac{n(n+b+c-1)(n+b+d-1)}{(2n+a+b+c+d-2)(2n+a+b+c+d-1)}. -\end{array}\right.$$ - -\subsection*{Normalized recurrence relation} -\begin{equation} -\label{NormRecContHahn} -xp_n(x)=p_{n+1}(x)+i(A_n+C_n+a)p_n(x)-A_{n-1}C_np_{n-1}(x), -\end{equation} -where -$$p_n(x;a,b,c,d)=\frac{(n+a+b+c+d-1)_n}{n!}p_n(x).$$ - -\subsection*{Difference equation} -\begin{eqnarray} -\label{dvContHahn} -& &n(n+a+b+c+d-1)y(x)\nonumber\\ -& &{}=B(x)y(x+i)-\left[B(x)+D(x)\right]y(x)+D(x)y(x-i), -\end{eqnarray} -where -$$y(x)=p_n(x;a,b,c,d)$$ -and -$$\left\{\begin{array}{l} -\displaystyle B(x)=(c-ix)(d-ix)\\[5mm] -\displaystyle D(x)=(a+ix)(b+ix). -\end{array}\right.$$ - -\subsection*{Forward shift operator} -\begin{eqnarray} -\label{shift1ContHahnI} -& &p_n(x+\textstyle\frac{1}{2}i;a,b,c,d)-p_n(x-\textstyle\frac{1}{2}i;a,b,c,d)\nonumber\\ -& &{}=i(n+a+b+c+d-1)p_{n-1}(x;a+\textstyle\frac{1}{2}, -b+\textstyle\frac{1}{2},c+\textstyle\frac{1}{2},d+\textstyle\frac{1}{2}) -\end{eqnarray} -or equivalently -\begin{equation} -\label{shift1ContHahnII} -\frac{\delta p_n(x;a,b,c,d)}{\delta x}=(n+a+b+c+d-1) -p_{n-1}(x;a+\textstyle\frac{1}{2},b+\textstyle\frac{1}{2}, -c+\textstyle\frac{1}{2},d+\textstyle\frac{1}{2}). -\end{equation} - -\subsection*{Backward shift operator} -\begin{eqnarray} -\label{shift2ContHahnI} -& &(c-\textstyle\frac{1}{2}-ix)(d-\textstyle\frac{1}{2}-ix) -p_n(x+\textstyle\frac{1}{2}i;a,b,c,d)\nonumber\\ -& &{}\mathindent{}-(a-\textstyle\frac{1}{2}+ix)(b-\textstyle\frac{1}{2}+ix) -p_n(x-\textstyle\frac{1}{2}i;a,b,c,d)\nonumber\\ -& &{}=\frac{n+1}{i}p_{n+1}(x;a-\textstyle\frac{1}{2}, -b-\textstyle\frac{1}{2},c-\textstyle\frac{1}{2},d-\textstyle\frac{1}{2}) -\end{eqnarray} -or equivalently -\begin{eqnarray} -\label{shift2ContHahnII} -& &\frac{\delta\left[\omega(x;a,b,c,d)p_n(x;a,b,c,d)\right]}{\delta x}\nonumber\\ -& &{}=-(n+1)\omega(x;a-\textstyle\frac{1}{2},b-\textstyle\frac{1}{2}, -c-\textstyle\frac{1}{2},d-\textstyle\frac{1}{2})\nonumber\\ -& &{}\mathindent\times p_{n+1}(x;a-\textstyle\frac{1}{2},b-\textstyle\frac{1}{2},c-\textstyle\frac{1}{2},d-\textstyle\frac{1}{2}), -\end{eqnarray} -where -$$\omega(x;a,b,c,d)=\Gamma(a+ix)\Gamma(b+ix)\Gamma(c-ix)\Gamma(d-ix).$$ - -\subsection*{Rodrigues-type formula} -\begin{eqnarray} -\label{RodContHahn} -& &\omega(x;a,b,c,d)p_n(x;a,b,c,d)\nonumber\\ -& &{}=\frac{(-1)^n}{n!}\left(\frac{\delta}{\delta x}\right)^n -\left[\omega(x;a+\textstyle\frac{1}{2}n,b+\textstyle\frac{1}{2}n, -c+\textstyle\frac{1}{2}n,d+\textstyle\frac{1}{2}n)\right]. -\end{eqnarray} - -\subsection*{Generating functions} -\begin{equation} -\label{GenContHahn1} -\hyp{1}{1}{a+ix}{a+c}{-it}\,\hyp{1}{1}{d-ix}{b+d}{it}= -\sum_{n=0}^{\infty}\frac{p_n(x;a,b,c,d)}{(a+c)_n(b+d)_n}t^n. -\end{equation} - -\begin{equation} -\label{GenContHahn2} -\hyp{1}{1}{a+ix}{a+d}{-it}\,\hyp{1}{1}{c-ix}{b+c}{it}= -\sum_{n=0}^{\infty}\frac{p_n(x;a,b,c,d)}{(a+d)_n(b+c)_n}t^n. -\end{equation} - -\begin{eqnarray} -\label{GenContHahn3} -& &(1-t)^{1-a-b-c-d}\,\hyp{3}{2}{\frac{1}{2}(a+b+c+d-1),\frac{1}{2}(a+b+c+d),a+ix} -{a+c,a+d}{-\frac{4t}{(1-t)^2}}\nonumber\\ -& &{}=\sum_{n=0}^{\infty} -\frac{(a+b+c+d-1)_n}{(a+c)_n(a+d)_ni^n}p_n(x;a,b,c,d)t^n. -\end{eqnarray} - -\subsection*{Limit relations} - -\subsubsection*{Wilson $\rightarrow$ Continuous Hahn} -The continuous Hahn polynomials are obtained from the Wilson polynomials given by -(\ref{DefWilson}) by the substitution $a\rightarrow a-it$, $b\rightarrow b-it$, -$c\rightarrow c+it$, $d\rightarrow d+it$ and $x\rightarrow x+t$ and the limit -$t\rightarrow\infty$ in the following way: -$$\lim_{t\rightarrow\infty} -\frac{W_n((x+t)^2;a-it,b-it,c+it,d+it)}{(-2t)^nn!}=p_n(x;a,b,c,d).$$ - -\subsubsection*{Continuous Hahn $\rightarrow$ Meixner-Pollaczek} -The Meixner-Pollaczek polynomials given by (\ref{DefMP}) can be obtained from the -continuous Hahn polynomials by setting $x\rightarrow x+t$, $a=\lambda-it$, $c=\lambda+it$ -and $b=d=t\tan\phi$ and taking the limit $t\rightarrow\infty$: -\begin{equation} -\lim_{t\rightarrow\infty}\frac{p_n(x+t;\lambda-it,t\tan\phi,\lambda+it,t\tan\phi)}{t^n} -=\frac{P_n^{(\lambda)}(x;\phi)}{(\cos\phi)^n}. -\end{equation} - -\subsubsection*{Continuous Hahn $\rightarrow$ Jacobi} -The Jacobi polynomials given by (\ref{DefJacobi}) follow from the continuous Hahn polynomials -by the substitution $x\rightarrow \frac{1}{2}xt$, $a=\frac{1}{2}(\alpha+1-it)$, -$b=\frac{1}{2}(\beta+1+it)$, $c=\frac{1}{2}(\alpha+1+it)$ and $d=\frac{1}{2}(\beta+1-it)$, -division by $t^n$ and the limit $t\rightarrow\infty$: -\begin{eqnarray} -& &\lim_{t\rightarrow\infty} -\frac{p_n(\frac{1}{2}xt;\frac{1}{2}(\alpha+1-it),\frac{1}{2}(\beta+1+it), -\frac{1}{2}(\alpha+1+it),\frac{1}{2}(\beta+1-it))}{t^n}\nonumber\\ -& &{}=P_n^{(\alpha,\beta)}(x). -\end{eqnarray} - -\subsubsection*{Continuous Hahn $\rightarrow$ Pseudo Jacobi} -The pseudo Jacobi polynomials given by (\ref{DefPseudoJacobi}) follow from the continuous -Hahn polynomials by the substitution $x\rightarrow xt$, $a=\frac{1}{2}(-N+i\nu-2t)$, -$b=\frac{1}{2}(-N-i\nu+2t)$, $c=\frac{1}{2}(-N-i\nu-2t)$ and $d=\frac{1}{2}(-N+i\nu+2t)$, -division by $t^n$ and the limit $t\rightarrow\infty$: -\begin{eqnarray} -& &\lim_{t\rightarrow\infty}\frac{p_n(xt;\frac{1}{2}(-N+i\nu-2t),\frac{1}{2}(-N-i\nu+2t), -\frac{1}{2}(-N+i\nu-2t),\frac{1}{2}(-N-i\nu+2t))}{t^n}\nonumber\\ -& &{}=\frac{(n-2N-1)_n}{n!}P_n(x;\nu,N). -\end{eqnarray} - -\subsection*{Remark} -Since we have for $k-1$ and $\beta>-1$, or for $\alpha<-N$ and $\beta<-N$, we have -\begin{eqnarray} -\label{OrtHahn} -& &\sum_{x=0}^N\binom{\alpha +x}{x}\binom{\beta+N-x}{N-x}Q_m(x;\alpha,\beta,N)Q_n(x;\alpha,\beta,N)\nonumber\\ -& &{}=\frac{(-1)^n(n+\alpha+\beta+1)_{N+1}(\beta+1)_nn!}{(2n+\alpha+\beta+1)(\alpha+1)_n(-N)_nN!}\,\delta_{mn}. -\end{eqnarray} - -\subsection*{Recurrence relation} -\begin{equation} -\label{RecHahn} --xQ_n(x)=A_nQ_{n+1}(x)-\left(A_n+C_n\right)Q_n(x)+C_nQ_{n-1}(x), -\end{equation} -where -$$Q_n(x):=Q_n(x;\alpha,\beta,N)$$ -and -$$\left\{\begin{array}{l} -\displaystyle A_n=\frac{(n+\alpha+\beta+1)(n+\alpha+1)(N-n)}{(2n+\alpha+\beta+1)(2n+\alpha+\beta+2)}\\[5mm] -\displaystyle C_n=\frac{n(n+\alpha+\beta+N+1)(n+\beta)}{(2n+\alpha+\beta)(2n+\alpha+\beta+1)}. -\end{array}\right.$$ - -\subsection*{Normalized recurrence relation} -\begin{equation} -\label{NormRecHahn} -xp_n(x)=p_{n+1}(x)+\left(A_n+C_n\right)p_n(x)+A_{n-1}C_np_{n-1}(x), -\end{equation} -where -$$Q_n(x;\alpha,\beta,N)=\frac{(n+\alpha+\beta+1)_n}{(\alpha+1)_n(-N)_n}p_n(x).$$ - -\subsection*{Difference equation} -\begin{equation} -\label{dvHahn} -n(n+\alpha+\beta+1)y(x)=B(x)y(x+1)-\left[B(x)+D(x)\right]y(x)+D(x)y(x-1), -\end{equation} -where -$$y(x)=Q_n(x;\alpha,\beta,N)$$ -and -$$\left\{\begin{array}{l} -\displaystyle B(x)=(x+\alpha+1)(x-N)\\[5mm] -\displaystyle D(x)=x(x-\beta-N-1). -\end{array}\right.$$ - -\subsection*{Forward shift operator} -\begin{eqnarray} -\label{shift1HahnI} -& &Q_n(x+1;\alpha,\beta,N)-Q_n(x;\alpha,\beta,N)\nonumber\\ -& &{}=-\frac{n(n+\alpha+\beta+1)}{(\alpha+1)N}Q_{n-1}(x;\alpha+1,\beta+1,N-1) -\end{eqnarray} -or equivalently -\begin{equation} -\label{shift1HahnII} -\Delta Q_n(x;\alpha,\beta,N)=-\frac{n(n+\alpha+\beta+1)}{(\alpha+1)N}Q_{n-1}(x;\alpha+1,\beta+1,N-1). -\end{equation} - -\subsection*{Backward shift operator} -\begin{eqnarray} -\label{shift2HahnI} -& &(x+\alpha)(N+1-x)Q_n(x;\alpha,\beta,N)-x(\beta+N+1-x)Q_n(x-1;\alpha,\beta,N)\nonumber\\ -& &{}=\alpha(N+1)Q_{n+1}(x;\alpha-1,\beta-1,N+1) -\end{eqnarray} -or equivalently -\begin{eqnarray} -\label{shift2HahnII} -& &\nabla\left[\omega(x;\alpha,\beta,N)Q_n(x;\alpha,\beta,N)\right]\nonumber\\ -& &{}=\frac{N+1}{\beta}\omega(x;\alpha-1,\beta-1,N+1)Q_{n+1}(x;\alpha-1,\beta-1,N+1), -\end{eqnarray} -where -$$\omega(x;\alpha,\beta,N)=\binom{\alpha+x}{x}\binom{\beta+N-x}{N-x}.$$ - -\subsection*{Rodrigues-type formula} -\begin{eqnarray} -\label{RodHahn} -& &\omega(x;\alpha,\beta,N)Q_n(x;\alpha,\beta,N)\nonumber\\ -& &{}=\frac{(-1)^n(\beta+1)_n}{(-N)_n}\nabla^n\left[\omega(x;\alpha+n,\beta+n,N-n)\right]. -\end{eqnarray} - -\subsection*{Generating functions} -For $x=0,1,2,\ldots,N$ we have -\begin{equation} -\label{GenHahn1} -\hyp{1}{1}{-x}{\alpha+1}{-t}\,\hyp{1}{1}{x-N}{\beta+1}{t} -=\sum_{n=0}^N\frac{(-N)_n}{(\beta+1)_nn!}Q_n(x;\alpha,\beta,N)t^n. -\end{equation} - -\begin{eqnarray} -\label{GenHahn2} -& &\hyp{2}{0}{-x,-x+\beta+N+1}{-}{-t}\,\hyp{2}{0}{x-N,x+\alpha+1}{-}{t}\nonumber\\ -& &{}=\sum_{n=0}^N\frac{(-N)_n(\alpha+1)_n}{n!}Q_n(x;\alpha,\beta,N)t^n. -\end{eqnarray} - -\begin{eqnarray} -\label{GenHahn3} -& &\left[(1-t)^{-\alpha-\beta-1}\,\hyp{3}{2}{\frac{1}{2}(\alpha+\beta+1),\frac{1}{2}(\alpha+\beta+2),-x} -{\alpha+1,-N}{-\frac{4t}{(1-t)^2}}\right]_N\nonumber\\ -& &{}=\sum_{n=0}^N\frac{(\alpha+\beta+1)_n}{n!}Q_n(x;\alpha,\beta,N)t^n. -\end{eqnarray} - -\subsection*{Limit relations} - -\subsubsection*{Racah $\rightarrow$ Hahn} -If we take $\gamma+1=-N$ and let $\delta\rightarrow\infty$ in the definition (\ref{DefRacah}) -of the Racah polynomials, we obtain the Hahn polynomials. Hence -$$\lim_{\delta\rightarrow\infty} -R_n(\lambda(x);\alpha,\beta,-N-1,\delta)=Q_n(x;\alpha,\beta,N).$$ -And if we take $\delta=-\beta-N-1$ and let $\gamma\rightarrow\infty$ in the definition (\ref{DefRacah}) -of the Racah polynomials, we also obtain the Hahn polynomials: -$$\lim_{\gamma\rightarrow\infty} -R_n(\lambda(x);\alpha,\beta,\gamma,-\beta-N-1)=Q_n(x;\alpha,\beta,N).$$ -Another way to do this is to take $\alpha+1=-N$ and $\beta\rightarrow\beta+\gamma+N+1$ in -the definition (\ref{DefRacah}) of the Racah polynomials and then take the limit -$\delta\rightarrow\infty$. In that case we obtain the Hahn polynomials in the following way: -$$\lim_{\delta\rightarrow\infty} -R_n(\lambda(x);-N-1,\beta+\gamma+N+1,\gamma,\delta)=Q_n(x;\gamma,\beta,N).$$ - -\subsubsection*{Hahn $\rightarrow$ Jacobi} -To find the Jacobi polynomials given by (\ref{DefJacobi}) from the Hahn polynomials we take -$x\rightarrow Nx$ and let $N\rightarrow\infty$. In fact we have -\begin{equation} -\lim_{N\rightarrow\infty} -Q_n(Nx;\alpha,\beta,N)=\frac{P_n^{(\alpha,\beta)}(1-2x)}{P_n^{(\alpha,\beta)}(1)}. -\end{equation} - -\subsubsection*{Hahn $\rightarrow$ Meixner} -The Meixner polynomials given by (\ref{DefMeixner}) can be obtained from the Hahn polynomials -by taking $\alpha=b-1$, $\beta=N(1-c)c^{-1}$ and letting $N\rightarrow\infty$: -\begin{equation} -\lim_{N\rightarrow\infty} -Q_n(x;b-1,N(1-c)c^{-1},N)=M_n(x;b,c). -\end{equation} - -\subsubsection*{Hahn $\rightarrow$ Krawtchouk} -The Krawtchouk polynomials given by (\ref{DefKrawtchouk}) are obtained from the Hahn polynomials -if we take $\alpha=pt$ and $\beta=(1-p)t$ and let $t\rightarrow\infty$: -\begin{equation} -\lim_{t\rightarrow\infty}Q_n(x;pt,(1-p)t,N)=K_n(x;p,N). -\end{equation} - -\subsection*{Remark} -If we interchange the role of $x$ and $n$ in (\ref{DefHahn}) we obtain the -dual Hahn polynomials given by (\ref{DefDualHahn}). - -\noindent -Since -$$Q_n(x;\alpha,\beta,N)=R_x(\lambda(n);\alpha,\beta,N)$$ -we obtain the dual orthogonality relation for the Hahn polynomials from the -orthogonality relation (\ref{OrtDualHahn}) of the dual Hahn polynomials: -\begin{eqnarray*} -& &\sum_{n=0}^N\frac{(2n+\alpha+\beta+1)(\alpha+1)_n(-N)_nN!}{(-1)^n(n+\alpha+\beta+1)_{N+1}(\beta+1)_nn!} -Q_n(x;\alpha,\beta,N)Q_n(y;\alpha,\beta,N)\\ -& &{}=\frac{\delta_{xy}}{\dbinom{\alpha+x}{x}\dbinom{\beta+N-x}{N-x}},\quad x,y \in \{0,1,2,\ldots,N\}. -\end{eqnarray*} - -\subsection*{References} -\cite{AlSalam90}, \cite{AndrewsAskey85}, \cite{Area+II}, \cite{Askey75}, -\cite{Askey89I}, \cite{Askey2005}, \cite{AskeyGasper77}, \cite{AskeyWilson85}, -\cite{AtakRahmanSuslov}, \cite{AtakSuslov88}, \cite{Chihara78}, -\cite{Ciesielski}, \cite{Cooper+}, \cite{Dette95}, \cite{Dunkl76}, -\cite{Dunkl78I}, \cite{Gasper73I}, \cite{Gasper74}, \cite{HoareRahman}, -\cite{Ismail77}, \cite{Ismail2005II}, \cite{Karlin61}, \cite{Koorn81}, \cite{Koorn88}, -\cite{LabelleYehI}, \cite{LabelleYehII}, \cite{Laine}, \cite{Lesky62}, -\cite{Lesky88}, \cite{Lesky89}, \cite{Lesky94I}, \cite{Lesky95II}, -\cite{LewanowiczII}, \cite{Neuman}, \cite{Nikiforov+}, \cite{NikiforovUvarov}, -\cite{Rahman76III}, \cite{Rahman78I}, \cite{Rahman78II}, \cite{Rahman81III}, -\cite{Sablonniere}, \cite{Stanton84}, \cite{Stanton90}, \cite{Wilson80}, \cite{Wilson70II}, -\cite{Zarzo+}. - - -\section{Dual Hahn}\index{Dual Hahn polynomials}\index{Hahn polynomials!Dual} - -\par\setcounter{equation}{0} - -\subsection*{Hypergeometric representation} -\begin{equation} -\label{DefDualHahn} -R_n(\lambda(x);\gamma,\delta,N)= -\hyp{3}{2}{-n,-x,x+\gamma+\delta+1}{\gamma+1,-N}{1},\quad n=0,1,2,\ldots,N, -\end{equation} -where -$$\lambda(x)=x(x+\gamma+\delta+1).$$ - -\subsection*{Orthogonality relation} -For $\gamma>-1$ and $\delta>-1$, or for $\gamma<-N$ and $\delta<-N$, we have -\begin{eqnarray} -\label{OrtDualHahn} -& &\sum_{x=0}^N\frac{(2x+\gamma+\delta+1)(\gamma+1)_x(-N)_xN!}{(-1)^x(x+\gamma+\delta+1)_{N+1}(\delta+1)_xx!} -R_m(\lambda(x);\gamma,\delta,N)R_n(\lambda(x);\gamma,\delta,N)\nonumber\\ -& &{}=\frac{\,\delta_{mn}}{\dbinom{\gamma+n}{n}\dbinom{\delta+N-n}{N-n}}. -\end{eqnarray} - -\subsection*{Recurrence relation} -\begin{equation} -\label{RecDualHahn} -\lambda(x)R_n(\lambda(x)) -=A_nR_{n+1}(\lambda(x))-\left(A_n+C_n\right)R_n(\lambda(x))+C_nR_{n-1}(\lambda(x)), -\end{equation} -where -$$R_n(\lambda(x)):=R_n(\lambda(x);\gamma,\delta,N)$$ -and -$$\left\{\begin{array}{l} -\displaystyle A_n=(n+\gamma+1)(n-N)\\[5mm] -\displaystyle C_n=n(n-\delta-N-1). -\end{array}\right.$$ - -\subsection*{Normalized recurrence relation} -\begin{equation} -\label{NormRecDualHahn} -xp_n(x)=p_{n+1}(x)-(A_n+C_n)p_n(x)+A_{n-1}C_np_{n-1}(x), -\end{equation} -where -$$R_n(\lambda(x);\gamma,\delta,N)=\frac{1}{(\gamma+1)_n(-N)_n}p_n(\lambda(x)).$$ - -\subsection*{Difference equation} -\begin{equation} -\label{dvDualHahn} --ny(x)=B(x)y(x+1)-\left[B(x)+D(x)\right]y(x)+D(x)y(x-1), -\end{equation} -where -$$y(x)=R_n(\lambda(x);\gamma,\delta,N)$$ -and -$$\left\{\begin{array}{l} -\displaystyle B(x)=\frac{(x+\gamma+1)(x+\gamma+\delta+1)(N-x)}{(2x+\gamma+\delta+1)(2x+\gamma+\delta+2)}\\[5mm] -\displaystyle D(x)=\frac{x(x+\gamma+\delta+N+1)(x+\delta)}{(2x+\gamma+\delta)(2x+\gamma+\delta+1)}. -\end{array}\right.$$ - -\subsection*{Forward shift operator} -\begin{eqnarray} -\label{shift1DualHahnI} -& &R_n(\lambda(x+1);\gamma,\delta,N)-R_n(\lambda(x);\gamma,\delta,N)\nonumber\\ -& &{}=-\frac{n(2x+\gamma+\delta+2)}{(\gamma+1)N}R_{n-1}(\lambda(x);\gamma+1,\delta,N-1) -\end{eqnarray} -or equivalently -\begin{equation} -\label{shift1DualHahnII} -\frac{\Delta R_n(\lambda(x);\gamma,\delta,N)}{\Delta\lambda(x)}= --\frac{n}{(\gamma+1)N}R_{n-1}(\lambda(x);\gamma+1,\delta,N-1). -\end{equation} - -\subsection*{Backward shift operator} -\begin{eqnarray} -\label{shift2DualHahnI} -& &(x+\gamma)(x+\gamma+\delta)(N+1-x)R_n(\lambda(x);\gamma,\delta,N)\nonumber\\ -& &{}\mathindent{}-x(x+\gamma+\delta+N+1)(x+\delta)R_n(\lambda(x-1);\gamma,\delta,N)\nonumber\\ -& &{}=\gamma(N+1)(2x+\gamma+\delta)R_{n+1}(\lambda(x);\gamma-1,\delta,N+1) -\end{eqnarray} -or equivalently -\begin{eqnarray} -\label{shift2DualHahnII} -& &\frac{\nabla\left[\omega(x;\gamma,\delta,N)R_n(\lambda(x);\gamma,\delta,N)\right]}{\nabla\lambda(x)}\nonumber\\ -& &{}=\frac{1}{\gamma+\delta}\omega(x;\gamma-1,\delta,N+1)R_{n+1}(\lambda(x);\gamma-1,\delta,N+1), -\end{eqnarray} -where -$$\omega(x;\gamma,\delta,N)=\frac{(-1)^x(\gamma+1)_x(\gamma+\delta+1)_x(-N)_x} -{(\gamma+\delta+N+2)_x(\delta+1)_xx!}.$$ - -\newpage - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodDualHahn} -\omega(x;\gamma,\delta,N)R_n(\lambda(x);\gamma,\delta,N) -=(\gamma+\delta+1)_n\left(\nabla_{\lambda}\right)^n\left[\omega(x;\gamma+n,\delta,N-n)\right], -\end{equation} -where -$$\nabla_{\lambda}:=\frac{\nabla}{\nabla\lambda(x)}.$$ - -\subsection*{Generating functions} -For $x=0,1,2,\ldots,N$ we have -\begin{equation} -\label{GenDualHahn1} -(1-t)^{N-x}\,\hyp{2}{1}{-x,-x-\delta}{\gamma+1}{t} -=\sum_{n=0}^N\frac{(-N)_n}{n!}R_n(\lambda(x);\gamma,\delta,N)t^n. -\end{equation} - -\begin{eqnarray} -\label{GenDualHahn2} -& &(1-t)^x\,\hyp{2}{1}{x-N,x+\gamma+1}{-\delta-N}{t}\nonumber\\ -& &=\sum_{n=0}^N\frac{(\gamma+1)_n(-N)_n}{(-\delta-N)_nn!}R_n(\lambda(x);\gamma,\delta,N)t^n. -\end{eqnarray} - -\begin{equation} -\label{GenDualHahn3} -\left[\expe^t\,\hyp{2}{2}{-x,x+\gamma+\delta+1}{\gamma+1,-N}{-t}\right]_N=\sum_{n=0}^N -\frac{R_n(\lambda(x);\gamma,\delta,N)}{n!}t^n. -\end{equation} - -\begin{eqnarray} -\label{GenDualHahn4} -& &\left[(1-t)^{-\epsilon}\,\hyp{3}{2}{\epsilon,-x,x+\gamma+\delta+1}{\gamma+1,-N}{\frac{t}{t-1}}\right]_N\nonumber\\ -& &{}=\sum_{n=0}^N\frac{(\epsilon)_n}{n!}R_n(\lambda(x);\gamma,\delta,N)t^n, -\quad\textrm{$\epsilon$ arbitrary}. -\end{eqnarray} - -\subsection*{Limit relations} - -\subsubsection*{Racah $\rightarrow$ Dual Hahn} -If we take $\alpha+1=-N$ and let $\beta\rightarrow\infty$ in the definition (\ref{DefRacah}) -of the Racah polynomials, then we obtain the dual Hahn polynomials: -$$\lim_{\beta\rightarrow\infty} -R_n(\lambda(x);-N-1,\beta,\gamma,\delta)=R_n(\lambda(x);\gamma,\delta,N).$$ -And if we take $\beta=-\delta-N-1$ and let $\alpha\rightarrow\infty$ in the definition (\ref{DefRacah}) -of the Racah polynomials, then we also obtain the dual Hahn polynomials: -$$\lim_{\alpha\rightarrow\infty} -R_n(\lambda(x);\alpha,-\delta-N-1,\gamma,\delta)=R_n(\lambda(x);\gamma,\delta,N).$$ -Finally, if we take $\gamma+1=-N$ and $\delta\rightarrow\alpha+\delta+N+1$ in the definition -(\ref{DefRacah}) of the Racah polynomials and take the limit -$\beta\rightarrow\infty$ we find the dual Hahn polynomials in the following way: -$$\lim_{\beta\rightarrow\infty} -R_n(\lambda(x);\alpha,\beta,-N-1,\alpha+\delta+N+1)=R_n(\lambda(x);\alpha,\delta,N).$$ - -\subsubsection*{Dual Hahn $\rightarrow$ Meixner} -The Meixner polynomials given by (\ref{DefMeixner}) are obtained from the dual Hahn polynomials -if we take $\gamma=\beta-1$ and $\delta=N(1-c)c^{-1}$ and let $N\rightarrow\infty$: -\begin{equation} -\lim_{N\rightarrow\infty} -R_n(\lambda(x);\beta-1,N(1-c)c^{-1},N)=M_n(x;\beta,c). -\end{equation} - -\subsubsection*{Dual Hahn $\rightarrow$ Krawtchouk} -The Krawtchouk polynomials given by (\ref{DefKrawtchouk}) can be obtained from the dual -Hahn polynomials by setting $\gamma=pt$, $\delta=(1-p)t$ and letting $t\rightarrow\infty$: -\begin{equation} -\lim_{t\rightarrow\infty}R_n(\lambda(x);pt,(1-p)t,N)=K_n(x;p,N). -\end{equation} - -\subsection*{Remark} -If we interchange the role of $x$ and $n$ in the definition (\ref{DefDualHahn}) -of the dual Hahn polynomials we obtain the Hahn polynomials given by (\ref{DefHahn}). - -\noindent -Since -$$R_n(\lambda(x);\gamma,\delta,N)=Q_x(n;\gamma,\delta,N)$$ -we obtain the dual orthogonality relation for the dual Hahn polynomials -from the orthogonality relation (\ref{OrtHahn}) for the Hahn polynomials: -\begin{eqnarray*} -& &\sum_{n=0}^N\binom{\gamma+n}{n}\binom{\delta+N-n}{N-n} R_n(\lambda(x);\gamma,\delta,N)R_n(\lambda(y);\gamma,\delta,N)\\ -& &{}=\frac{(-1)^x(x+\gamma+\delta+1)_{N+1}(\delta+1)_xx!} -{(2x+\gamma+\delta+1)(\gamma+1)_x(-N)_xN!}\delta_{xy},\quad x,y \in \{0,1,2,\ldots,N\}. -\end{eqnarray*} - -\subsection*{References} -\cite{Askey2005}, \cite{AskeyWilson85}, \cite{AtakRahmanSuslov}, \cite{AtakSuslov88}, -\cite{Ismail2005II}, \cite{Karlin61}, \cite{Koorn81}, \cite{Koorn88}, \cite{Lesky93}, -\cite{Lesky94I}, \cite{Lesky95I}, \cite{Lesky95II}, \cite{Nikiforov+}, \cite{NikiforovUvarov}, -\cite{Rahman81II}, \cite{Stanton84}, \cite{Wilson80}. - - -\section{Meixner-Pollaczek}\index{Meixner-Pollaczek polynomials} - -\par\setcounter{equation}{0} - -\subsection*{Hypergeometric representation} -\begin{equation} -\label{DefMP} -P_n^{(\lambda)}(x;\phi)=\frac{(2\lambda)_n}{n!}\expe^{in\phi}\, -\hyp{2}{1}{-n,\lambda+ix}{2\lambda}{1-\expe^{-2i\phi}}. -\end{equation} - -\subsection*{Orthogonality relation} -\begin{equation} -\label{OrtMP} -\frac{1}{2\pi}\int_{-\infty}^{\infty} -\e^{(2\phi-\pi)x}\left|\Gamma(\lambda+ix)\right|^2 -P_m^{(\lambda)}(x;\phi)P_n^{(\lambda)}(x;\phi)\,dx -{}=\frac{\Gamma(n+2\lambda)}{(2\sin\phi)^{2\lambda}n!}\,\delta_{mn}, -% \constraint{ -% $\lambda > 0$ & -% $0 < \phi < \pi$ } -\end{equation} - -\subsection*{Recurrence relation} -\begin{eqnarray} -\label{RecMP} -& &(n+1)P_{n+1}^{(\lambda)}(x;\phi)-2\left[x\sin\phi+(n+\lambda)\cos\phi\right] -P_n^{(\lambda)}(x;\phi)\nonumber\\ -& &{}\mathindent{}+(n+2\lambda-1)P_{n-1}^{(\lambda)}(x;\phi)=0. -\end{eqnarray} - -\subsection*{Normalized recurrence relation} -\begin{equation} -\label{NormRecMP} -xp_n(x)=p_{n+1}(x)-\left(\frac{n+\lambda}{\tan\phi}\right)p_n(x)+ -\frac{n(n+2\lambda-1)}{4\sin^2\phi}p_{n-1}(x), -\end{equation} -where -$$P_n^{(\lambda)}(x;\phi)=\frac{(2\sin\phi)^n}{n!}p_n(x).$$ - -\subsection*{Difference equation} -\begin{eqnarray} -\label{dvMP} -& &\e^{i\phi}(\lambda-ix)y(x+i)+2i\left[x\cos\phi-(n+\lambda)\sin\phi\right]y(x)\nonumber\\ -& &{}\mathindent{}-\e^{-i\phi}(\lambda+ix)y(x-i)=0,\quad y(x)=P_n^{(\lambda)}(x;\phi). -\end{eqnarray} - -\subsection*{Forward shift operator} -\begin{equation} -\label{shift1MPI} -P_n^{(\lambda)}(x+\textstyle\frac{1}{2}i;\phi)- -P_n^{(\lambda)}(x-\textstyle\frac{1}{2}i;\phi)=(\expe^{i\phi}-\expe^{-i\phi}) -P_{n-1}^{(\lambda+\frac{1}{2})}(x;\phi) -\end{equation} -or equivalently -\begin{equation} -\label{shift1MPII} -\frac{\delta P_n^{(\lambda)}(x;\phi)}{\delta x} -=2\sin\phi\,P_{n-1}^{(\lambda+\frac{1}{2})}(x;\phi). -\end{equation} - -\subsection*{Backward shift operator} -\begin{eqnarray} -\label{shift2MPI} -& &\e^{i\phi}(\lambda-\textstyle\frac{1}{2}-ix) -P_n^{(\lambda)}(x+\textstyle\frac{1}{2}i;\phi)+ -\e^{-i\phi}(\lambda-\textstyle\frac{1}{2}+ix) -P_n^{(\lambda)}(x-\textstyle\frac{1}{2}i;\phi)\nonumber\\ -& &{}=(n+1)P_{n+1}^{(\lambda-\frac{1}{2})}(x;\phi) -\end{eqnarray} -or equivalently -\begin{equation} -\label{shift2MPII} -\frac{\delta\left[\omega(x;\lambda,\phi)P_n^{(\lambda)}(x;\phi)\right]}{\delta x}= --(n+1)\omega(x;\lambda-\textstyle\frac{1}{2},\phi) -P_{n+1}^{(\lambda-\frac{1}{2})}(x;\phi), -\end{equation} -where -$$\omega(x;\lambda,\phi)=\Gamma(\lambda+ix)\Gamma(\lambda-ix)\e^{(2\phi-\pi)x}.$$ - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodMP} -\omega(x;\lambda,\phi)P_n^{(\lambda)}(x;\phi)=\frac{(-1)^n}{n!} -\left(\frac{\delta}{\delta x}\right)^n\left[\omega(x;\lambda+\textstyle\frac{1}{2}n,\phi)\right]. -\end{equation} - -\newpage - -\subsection*{Generating functions} -\begin{equation} -\label{GenMP1} -(1-\expe^{i\phi}t)^{-\lambda+ix}(1-\expe^{-i\phi}t)^{-\lambda-ix}= -\sum_{n=0}^{\infty}P_n^{(\lambda)}(x;\phi)t^n. -\end{equation} - -\begin{equation} -\label{GenMP2} -\e^t\,\hyp{1}{1}{\lambda+ix}{2\lambda}{(\expe^{-2i\phi}-1)t}= -\sum_{n=0}^{\infty}\frac{P_n^{(\lambda)}(x;\phi)}{(2\lambda)_n\expe^{in\phi}}t^n. -\end{equation} - -\begin{eqnarray} -\label{GenMP3} -& &(1-t)^{-\gamma}\,\hyp{2}{1}{\gamma,\lambda+ix}{2\lambda}{\frac{(1-\e^{-2i\phi})t}{t-1}}\nonumber\\ -& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n}{(2\lambda)_n}\frac{P_n^{(\lambda)}(x;\phi)}{\e^{in\phi}}t^n, -\quad\textrm{$\gamma$ arbitrary}. -\end{eqnarray} - -\subsection*{Limit relations} - -\subsubsection*{Continuous dual Hahn $\rightarrow$ Meixner-Pollaczek} -The Meixner-Pollaczek polynomials can be obtained from the continuous dual Hahn polynomials -given by (\ref{DefContDualHahn}) by the substitutions $x\rightarrow x-t$, $a=\lambda+it$, -$b=\lambda-it$ and $c=t\cot\phi$ and the limit $t\rightarrow\infty$: -$$\lim_{t\rightarrow\infty}\frac{S_n((x-t)^2;\lambda+it,\lambda-it,t\cot\phi)}{t^nn!} -=\frac{P_n^{(\lambda)}(x;\phi)}{(\sin\phi)^n}.$$ - -\subsubsection*{Continuous Hahn $\rightarrow$ Meixner-Pollaczek} -By setting $x\rightarrow x+t$, $a=\lambda-it$, $c=\lambda+it$ and $b=d=t\tan\phi$ in the -definition (\ref{DefContHahn}) of the continuous Hahn polynomials and taking the limit -$t\rightarrow\infty$ we obtain the Meixner-Pollaczek polynomials: -$$\lim_{t\rightarrow\infty}\frac{p_n(x+t;\lambda-it,t\tan\phi,\lambda+it,t\tan\phi)}{t^nn!} -=\frac{P_n^{(\lambda)}(x;\phi)}{(\cos\phi)^n}.$$ - -\newpage - -\subsubsection*{Meixner-Pollaczek $\rightarrow$ Laguerre} -The Laguerre polynomials given by (\ref{DefLaguerre}) can be obtained from the -Meixner-Pollaczek polynomials by the substitution $\lambda=\frac{1}{2}(\alpha+1)$, -$x\rightarrow -\frac{1}{2}\phi^{-1}x$ and the limit $\phi\rightarrow 0$: -\begin{equation} -\lim_{\phi\rightarrow 0} -P_n^{(\frac{1}{2}\alpha+\frac{1}{2})}(-\textstyle\frac{1}{2}\phi^{-1}x;\phi)=L_n^{(\alpha)}(x). -\end{equation} - -\subsubsection*{Meixner-Pollaczek $\rightarrow$ Hermite} -The Hermite polynomials given by (\ref{DefHermite}) are obtained from the Meixner-Pollaczek -polynomials if we substitute $x\rightarrow (\sin\phi)^{-1}(x\sqrt{\lambda}-\lambda\cos\phi)$ -and then let $\lambda\rightarrow\infty$: -\begin{equation} -\lim_{\lambda\rightarrow\infty} -\lambda^{-\frac{1}{2}n}P_n^{(\lambda)} -((\sin\phi)^{-1}(x\sqrt{\lambda}-\lambda\cos\phi);\phi)=\frac{H_n(x)}{n!}. -\end{equation} - -\subsection*{Remark} -Since we have for $k-1$ and $\beta>-1$ we have -\begin{eqnarray} -\label{OrtJacobi1} -& &\int_{-1}^1(1-x)^{\alpha}(1+x)^{\beta}P_m^{(\alpha,\beta)}(x)P_n^{(\alpha,\beta)}(x)\,dx\nonumber\\ -& &{}=\frac{2^{\alpha+\beta+1}}{2n+\alpha+\beta+1}\frac{\Gamma(n+\alpha+1)\Gamma(n+\beta+1)}{\Gamma(n+\alpha+\beta+1)n!}\,\delta_{mn}. -\end{eqnarray} -For $\alpha+\beta<-2N-1$, $\beta>-1$ and $m,n\in\{0,1,2,\ldots,N\}$ we also have -\begin{eqnarray} -\label{OrtJacobi2} -& &\int_1^{\infty}(x+1)^{\alpha}(x-1)^{\beta}P_m^{(\alpha,\beta)}(-x)P_n^{(\alpha,\beta)}(-x)\,dx\nonumber\\ -& &{}=-\frac{2^{\alpha+\beta+1}}{2n+\alpha+\beta+1}\frac{\Gamma(-n-\alpha-\beta)\Gamma(n+\alpha+\beta+1)}{\Gamma(-n-\alpha)n!}\,\delta_{mn}. -\end{eqnarray} - -\subsection*{Recurrence relation} -\begin{eqnarray} -\label{RecJacobi} -xP_n^{(\alpha,\beta)}(x)&=&\frac{2(n+1)(n+\alpha+\beta+1)}{(2n+\alpha+\beta+1)(2n+\alpha+\beta+2)}P_{n+1}^{(\alpha,\beta)}(x)\nonumber\\ -& &{}\mathindent{}+\frac{\beta^2-\alpha^2}{(2n+\alpha+\beta)(2n+\alpha+\beta+2)}P_n^{(\alpha,\beta)}(x)\nonumber\\ -& &{}\mathindent\mathindent{}+\frac{2(n+\alpha)(n+\beta)}{(2n+\alpha+\beta)(2n+\alpha+\beta+1)}P_{n-1}^{(\alpha,\beta)}(x). -\end{eqnarray} - -\subsection*{Normalized recurrence relation} -\begin{eqnarray} -\label{NormRecJacobi} -xp_n(x)&=&p_{n+1}(x)+\frac{\beta^2-\alpha^2}{(2n+\alpha+\beta)(2n+\alpha+\beta+2)}p_n(x)\nonumber\\ -& &{}\mathindent{}+\frac{4n(n+\alpha)(n+\beta)(n+\alpha+\beta)} -{(2n+\alpha+\beta-1)(2n+\alpha+\beta)^2(2n+\alpha+\beta+1)}p_{n-1}(x) -\end{eqnarray} -where -$$P_n^{(\alpha,\beta)}(x)=\frac{(n+\alpha+\beta+1)_n}{2^nn!}p_n(x).$$ - -\subsection*{Differential equation} -\begin{eqnarray} -\label{dvJacobi} -& &(1-x^2)y''(x)+\left[\beta-\alpha-(\alpha+\beta+2)x\right]y'(x)\nonumber\\ -& &{}\mathindent{}+n(n+\alpha+\beta+1)y(x)=0,\quad y(x)=P_n^{(\alpha,\beta)}(x). -\end{eqnarray} - -\subsection*{Forward shift operator} -\begin{equation} -\label{shift1Jacobi} -\frac{d}{dx}P_n^{(\alpha,\beta)}(x)=\frac{n+\alpha+\beta+1}{2}P_{n-1}^{(\alpha+1,\beta+1)}(x). -\end{equation} - -\subsection*{Backward shift operator} -\begin{eqnarray} -\label{shift2JacobiI} -& &(1-x^2)\frac{d}{dx}P_n^{(\alpha,\beta)}(x)+ -\left[(\beta-\alpha)-(\alpha+\beta)x\right]P_n^{(\alpha,\beta)}(x)\nonumber\\ -& &{}=-2(n+1)P_{n+1}^{(\alpha-1,\beta-1)}(x) -\end{eqnarray} -or equivalently -\begin{eqnarray} -\label{shift2JacobiII} -& &\frac{d}{dx}\left[(1-x)^\alpha(1+x)^\beta P_n^{(\alpha,\beta)}(x)\right]\nonumber\\ -& &{}=-2(n+1)(1-x)^{\alpha-1}(1+x)^{\beta-1}P_{n+1}^{(\alpha-1,\beta-1)}(x). -\end{eqnarray} - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodJacobi} -(1-x)^{\alpha}(1+x)^{\beta}P_n^{(\alpha,\beta)}(x)= -\frac{(-1)^n}{2^nn!}\left(\frac{d}{dx}\right)^n -\left[(1-x)^{n+\alpha}(1+x)^{n+\beta}\right]. -\end{equation} - -\subsection*{Generating functions} -\begin{equation} -\label{GenJacobi1} -\frac{2^{\alpha+\beta}}{R(1+R-t)^{\alpha}(1+R+t)^{\beta}}= -\sum_{n=0}^{\infty}P_n^{(\alpha,\beta)}(x)t^n,\quad R=\sqrt{1-2xt+t^2}. -\end{equation} - -\begin{eqnarray} -\label{GenJacobi2} -& &\hyp{0}{1}{-}{\alpha+1}{\frac{(x-1)t}{2}}\,\hyp{0}{1}{-}{\beta+1}{\frac{(x+1)t}{2}}\nonumber\\ -& &{}=\sum_{n=0}^{\infty}\frac{P_n^{(\alpha,\beta)}(x)}{(\alpha+1)_n(\beta+1)_n}t^n. -\end{eqnarray} - -\begin{eqnarray} -\label{GenJacobi3} -& &(1-t)^{-\alpha-\beta-1}\,\hyp{2}{1}{\frac{1}{2}(\alpha+\beta+1),\frac{1}{2}(\alpha+\beta+2)} -{\alpha+1}{\frac{2(x-1)t}{(1-t)^2}}\nonumber\\ -& &{}=\sum_{n=0}^{\infty}\frac{(\alpha+\beta+1)_n}{(\alpha+1)_n}P_n^{(\alpha,\beta)}(x)t^n. -\end{eqnarray} - -\begin{eqnarray} -\label{GenJacobi4} -& &(1+t)^{-\alpha-\beta-1}\,\hyp{2}{1}{\frac{1}{2}(\alpha+\beta+1),\frac{1}{2}(\alpha+\beta+2)} -{\beta+1}{\frac{2(x+1)t}{(1+t)^2}}\nonumber\\ -& &{}=\sum_{n=0}^{\infty}\frac{(\alpha+\beta+1)_n}{(\beta+1)_n}P_n^{(\alpha,\beta)}(x)t^n. -\end{eqnarray} - -\begin{eqnarray} -\label{GenJacobi5} -& &\hyp{2}{1}{\gamma,\alpha+\beta+1-\gamma}{\alpha+1}{\frac{1-R-t}{2}}\, -\hyp{2}{1}{\gamma,\alpha+\beta+1-\gamma}{\beta+1}{\frac{1-R+t}{2}}\nonumber\\ -& &{}=\sum_{n=0}^{\infty} -\frac{(\gamma)_n(\alpha+\beta+1-\gamma)_n}{(\alpha+1)_n(\beta+1)_n}P_n^{(\alpha,\beta)}(x)t^n, -\quad R=\sqrt{1-2xt+t^2} -\end{eqnarray} -with $\gamma$ arbitrary. - -\subsection*{Limit relations} - -\subsubsection*{Wilson $\rightarrow$ Jacobi} -The Jacobi polynomials can be found from the Wilson polynomials given by (\ref{DefWilson}) by -substituting $a=b=\frac{1}{2}(\alpha+1)$, $c=\frac{1}{2}(\beta+1)+it$, -$d=\frac{1}{2}(\beta+1)-it$ and $x\rightarrow t\sqrt{\frac{1}{2}(1-x)}$ in the -definition (\ref{DefWilson}) of the Wilson polynomials and taking the limit -$t\rightarrow\infty$. In fact we have -$$\lim_{t\rightarrow\infty} -\frac{W_n(\frac{1}{2}(1-x)t^2;\frac{1}{2}(\alpha+1), -\frac{1}{2}(\alpha+1),\frac{1}{2}(\beta+1)+it,\frac{1}{2}(\beta+1)-it)} -{t^{2n}n!}=P_n^{(\alpha,\beta)}(x).$$ - -\subsubsection*{Continuous Hahn $\rightarrow$ Jacobi} -The Jacobi polynomials follow from the continuous Hahn polynomials given by (\ref{DefContHahn}) -by using the substitution $x\rightarrow \frac{1}{2}xt$, $a=\frac{1}{2}(\alpha+1-it)$, -$b=\frac{1}{2}(\beta+1+it)$, $c=\frac{1}{2}(\alpha+1+it)$ and $d=\frac{1}{2}(\beta+1-it)$ -in (\ref{DefContHahn}), division by $t^n$ and the limit $t\rightarrow\infty$: -$$\lim_{t\rightarrow\infty} -\frac{p_n(\frac{1}{2}xt;\frac{1}{2}(\alpha+1-it),\frac{1}{2}(\beta+1+it), -\frac{1}{2}(\alpha+1+it),\frac{1}{2}(\beta+1-it))}{t^n}=P_n^{(\alpha,\beta)}(x).$$ - -\subsubsection*{Hahn $\rightarrow$ Jacobi} -To find the Jacobi polynomials from the Hahn polynomials given by (\ref{DefHahn}) we take -$x\rightarrow Nx$ in (\ref{DefHahn}) and let $N\rightarrow\infty$. In fact we have -$$\lim_{N\rightarrow\infty} -Q_n(Nx;\alpha,\beta,N)=\frac{P_n^{(\alpha,\beta)}(1-2x)}{P_n^{(\alpha,\beta)}(1)}.$$ - -\subsubsection*{Jacobi $\rightarrow$ Laguerre} -The Laguerre polynomials given by (\ref{DefLaguerre}) can be obtained from the Jacobi -polynomials by setting $x\rightarrow 1-2\beta^{-1}x$ and then the limit $\beta\rightarrow\infty$: -\begin{equation} -\lim_{\beta\rightarrow\infty} -P_n^{(\alpha,\beta)}(1-2\beta^{-1}x)=L_n^{(\alpha)}(x). -\end{equation} - -\subsubsection*{Jacobi $\rightarrow$ Bessel} -The Bessel polynomials given by (\ref{DefBessel}) are obtained from the Jacobi polynomials -if we take $\beta=a-\alpha$ and let $\alpha\rightarrow-\infty$: -\begin{equation} -\lim_{\alpha\rightarrow-\infty} -\frac{P_n^{(\alpha,a-\alpha)}(1+\alpha x)}{P_n^{(\alpha,a-\alpha)}(1)}=y_n(x;a). -\end{equation} - -\subsubsection*{Jacobi $\rightarrow$ Hermite} -The Hermite polynomials given by (\ref{DefHermite}) follow from the Jacobi polynomials -by taking $\beta=\alpha$ and letting $\alpha\rightarrow\infty$ in the following way: -\begin{equation} -\lim_{\alpha\rightarrow\infty} -\alpha^{-\frac{1}{2}n}P_n^{(\alpha,\alpha)}(\alpha^{-\frac{1}{2}}x)=\frac{H_n(x)}{2^nn!}. -\end{equation} - -\subsection*{Remarks} -The definition (\ref{DefJacobi}) of the Jacobi polynomials can also be written as: -$$P_n^{(\alpha,\beta)}(x)=\frac{1}{n!}\sum_{k=0}^n\frac{(-n)_k}{k!} -(n+\alpha+\beta+1)_k(\alpha+k+1)_{n-k}\left(\frac{1-x}{2}\right)^k.$$ -In this way the Jacobi polynomials can also be seen as polynomials in the parameters $\alpha$ -and $\beta$. Therefore they can be defined for all $\alpha$ and $\beta$. Then we have the -following connection with the Meixner polynomials given by (\ref{DefMeixner}): -$$\frac{(\beta)_n}{n!}M_n(x;\beta,c)=P_n^{(\beta-1,-n-\beta-x)}((2-c)c^{-1}).$$ - -\noindent -The Jacobi polynomials are related to the pseudo Jacobi polynomials defined by -(\ref{DefPseudoJacobi}) in the following way: -$$P_n(x;\nu,N)=\frac{(-2i)^nn!}{(n-2N-1)_n}P_n^{(-N-1+i\nu,-N-1-i\nu)}(ix).$$ - -\noindent -The Jacobi polynomials are also related to the Gegenbauer (or ultraspherical) polynomials -given by (\ref{DefGegenbauer}) by the quadratic transformations: -$$C_{2n}^{(\lambda)}(x)=\frac{(\lambda)_n}{(\frac{1}{2})_n} -P_n^{(\lambda-\frac{1}{2},-\frac{1}{2})}(2x^2-1)$$ -and -$$C_{2n+1}^{(\lambda)}(x)=\frac{(\lambda)_{n+1}}{(\frac{1}{2})_{n+1}} -xP_n^{(\lambda-\frac{1}{2},\frac{1}{2})}(2x^2-1).$$ - -\subsection*{References} -\cite{Abram}, \cite{Ahmed+82}, \cite{Allaway89}, \cite{NAlSalam66}, \cite{AlSalam64}, -\cite{AlSalamChihara72}, \cite{AndrewsAskey85}, \cite{AndrewsAskeyRoy}, \cite{Askey68}, -\cite{Askey70}, \cite{Askey72}, \cite{Askey73}, \cite{Askey74}, \cite{Askey75}, \cite{Askey78}, -\cite{Askey89I}, \cite{Askey2005}, \cite{AskeyFitch}, \cite{AskeyGasper71I}, \cite{AskeyGasper71II}, -\cite{AskeyGasper76}, \cite{AskeyWainger}, \cite{AskeyWilson85}, \cite{Bailey38}, \cite{Brafman51}, -\cite{Brown}, \cite{Carlitz61II}, \cite{Carlitz67}, \cite{ChenIsmail91}, \cite{Chihara78}, -\cite{Chow+}, \cite{Ciesielski}, \cite{Cooper+}, \cite{DetteStudden92}, \cite{DetteStudden95}, -\cite{DijksmaKoorn}, \cite{DimitrovRafaeli}, \cite{DimitrovRodrigues}, \cite{Doha2002I}, -\cite{Doha2003I}, \cite{Doha2004II}, \cite{Dunkl84}, \cite{ElbertLaforgia87II}, -\cite{ElbertLaforgiaRodono}, \cite{Erdelyi+}, \cite{Faldey}, \cite{FoataLeroux}, -\cite{Gasper69}, \cite{Gasper70I}, \cite{Gasper70II}, \cite{Gasper71I}, \cite{Gasper71II}, -\cite{Gasper72I}, \cite{Gasper72II}, \cite{Gasper73I}, \cite{Gasper73II}, \cite{Gasper74}, -\cite{Gasper77}, \cite{Gatteschi87}, \cite{Gautschi2008}, \cite{Gautschi2009I}, -\cite{Gautschi2009II}, \cite{GautschiLeopardi}, \cite{GawronskiShawyer}, \cite{Godoy+}, -\cite{Grad}, \cite{Hajmirzaahmad94}, \cite{HartmannStephan}, \cite{Horton}, \cite{Ismail74}, -\cite{Ismail77}, \cite{Ismail96}, \cite{Ismail2005II}, \cite{IsmailLi}, -\cite{IsmailMassonRahman}, \cite{Kochneff97I}, \cite{Koekoek99}, \cite{Koekoek2000}, -\cite{Koelink96II}, \cite{Koorn73}, \cite{Koorn74}, \cite{Koorn75}, \cite{Koorn77II}, -\cite{Koorn78}, \cite{Koorn72}, \cite{Koorn85}, \cite{Koorn88}, \cite{KuijlaarsMartinez}, -\cite{Kuijlaars+}, \cite{LabelleYehI}, \cite{LabelleYehII}, \cite{Laine}, \cite{Lesky95II}, -\cite{Lesky96}, \cite{Li96}, \cite{Li97}, \cite{LopezTemme2004}, \cite{Luke}, \cite{Martinez}, -\cite{Mathai}, \cite{Meijer}, \cite{Miller89}, \cite{MoakSaffVarga}, \cite{Nikiforov+}, -\cite{NikiforovUvarov}, \cite{Olver}, \cite{Prasad}, \cite{Rahman76I}, \cite{Rahman76II}, -\cite{Rahman77}, \cite{Rahman81I}, \cite{RahmanShah}, \cite{Rainville}, \cite{Rusev}, -\cite{Shi}, \cite{Srivastava69II}, \cite{Srivastava71}, \cite{Srivastava82}, -\cite{SrivastavaSinghal}, \cite{Stanton80I}, \cite{Stanton90}, \cite{Szego75}, \cite{Temme}, -\cite{Vertesi}, \cite{Wimp87}, \cite{WongZhang94}, \cite{WongZhang2006}, \cite{Zarzo+}, -\cite{Zayed}. - -\section*{Special cases} - -\subsection{Gegenbauer / Ultraspherical}\index{Gegenbauer polynomials} -\index{Ultraspherical polynomials} - -\par - -\subsection*{Hypergeometric representation} -The Gegenbauer (or ultraspherical) polynomials are Jacobi polynomials with -$\alpha=\beta=\lambda-\frac{1}{2}$ and another normalization: -\begin{eqnarray} -\label{DefGegenbauer} -C_n^{(\lambda)}(x)&=&\frac{(2\lambda)_n}{(\lambda+\frac{1}{2})_n} -P_n^{(\lambda-\frac{1}{2},\lambda-\frac{1}{2})}(x)\nonumber\\ -&=&\frac{(2\lambda)_n}{n!}\,\hyp{2}{1}{-n,n+2\lambda} -{\lambda+\frac{1}{2}}{\frac{1-x}{2}},\quad\lambda\neq 0. -\end{eqnarray} - -\subsection*{Orthogonality relation} -\begin{eqnarray} -\label{OrtGegenbauer} -& &\int_{-1}^1(1-x^2)^{\lambda-\frac{1}{2}}C_m^{(\lambda)}(x)C_n^{(\lambda)}(x)\,dx\nonumber\\ -& &{}=\frac{\pi\Gamma(n+2\lambda)2^{1-2\lambda}}{\left\{\Gamma(\lambda)\right\}^2(n+\lambda)n!}\,\delta_{mn}, -\quad\lambda>-\frac{1}{2}\quad\lambda\neq 0. -\end{eqnarray} - -\subsection*{Recurrence relation} -\begin{equation} -\label{RecGegenbauer} -2(n+\lambda)xC_n^{(\lambda)}(x)=(n+1)C_{n+1}^{(\lambda)}(x)+(n+2\lambda-1)C_{n-1}^{(\lambda)}(x). -\end{equation} - -\subsection*{Normalized recurrence relation} -\begin{equation} -\label{NormRecGegenbauer} -xp_n(x)=p_{n+1}(x)+\frac{n(n+2\lambda-1)}{4(n+\lambda-1)(n+\lambda)}p_{n-1}(x), -\end{equation} -where -$$C_n^{(\lambda)}(x)=\frac{2^n(\lambda)_n}{n!}p_n(x).$$ - -\subsection*{Differential equation} -\begin{equation} -\label{dvGegenbauer} -(1-x^2)y''(x)-(2\lambda+1)xy'(x)+n(n+2\lambda)y(x)=0,\quad y(x)=C_n^{(\lambda)}(x). -\end{equation} - -\subsection*{Forward shift operator} -\begin{equation} -\label{shift1Gegenbauer} -\frac{d}{dx}C_n^{(\lambda)}(x)=2\lambda C_{n-1}^{(\lambda+1)}(x). -\end{equation} - -\subsection*{Backward shift operator} -\begin{equation} -\label{shift2GegenbauerI} -(1-x^2)\frac{d}{dx}C_n^{(\lambda)}(x)+(1-2\lambda)xC_n^{(\lambda)}(x)= --\frac{(n+1)(2\lambda+n-1)}{2(\lambda-1)} C_{n+1}^{(\lambda-1)}(x) -\end{equation} -or equivalently -\begin{eqnarray} -\label{shift2GegenbauerII} -& &\frac{d}{dx}\left[(1-x^2)^{\lambda-\textstyle\frac{1}{2}}C_n^{(\lambda)}(x)\right]\nonumber\\ -& &{}=-\frac{(n+1)(2\lambda+n-1)}{2(\lambda-1)}(1-x^2)^{\lambda-\textstyle\frac{3}{2}}C_{n+1}^{(\lambda-1)}(x). -\end{eqnarray} - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodGegenbauer} -(1-x^2)^{\lambda-\frac{1}{2}}C_n^{(\lambda)}(x)= -\frac{(2\lambda)_n(-1)^n}{(\lambda+\frac{1}{2})_n2^nn!}\left(\frac{d}{dx}\right)^n -\left[(1-x^2)^{\lambda+n-\frac{1}{2}}\right]. -\end{equation} - -\subsection*{Generating functions} -\begin{equation} -\label{GenGegenbauer1} -(1-2xt+t^2)^{-\lambda}=\sum_{n=0}^{\infty}C_n^{(\lambda)}(x)t^n. -\end{equation} - -\begin{equation} -\label{GenGegenbauer2} -R^{-1}\left(\frac{1+R-xt}{2}\right)^{\frac{1}{2}-\lambda}=\sum_{n=0}^{\infty} -\frac{(\lambda+\frac{1}{2})_n}{(2\lambda)_n}C_n^{(\lambda)}(x)t^n, -\quad R=\sqrt{1-2xt+t^2}. -\end{equation} - -\begin{equation} -\label{GenGegenbauer3} -\hyp{0}{1}{-}{\lambda+\frac{1}{2}}{\frac{(x-1)t}{2}}\, -\hyp{0}{1}{-}{\lambda+\frac{1}{2}}{\frac{(x+1)t}{2}} -=\sum_{n=0}^{\infty}\frac{C_n^{(\lambda)}(x)} -{(2\lambda)_n(\lambda+\frac{1}{2})_n}t^n. -\end{equation} - -\begin{equation} -\label{GenGegenbauer4} -\e^{xt}\,\hyp{0}{1}{-}{\lambda+\frac{1}{2}}{\frac{(x^2-1)t^2}{4}}= -\sum_{n=0}^{\infty}\frac{C_n^{(\lambda)}(x)}{(2\lambda)_n}t^n. -\end{equation} - -\begin{eqnarray} -\label{GenGegenbauer5} -& &\hyp{2}{1}{\gamma,2\lambda-\gamma}{\lambda+\frac{1}{2}}{\frac{1-R-t}{2}}\, -\hyp{2}{1}{\gamma,2\lambda-\gamma}{\lambda+\frac{1}{2}}{\frac{1-R+t}{2}}\nonumber\\ -& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n(2\lambda-\gamma)_n} -{(2\lambda)_n(\lambda+\frac{1}{2})_n}C_n^{(\lambda)}(x)t^n, -\quad R=\sqrt{1-2xt+t^2},\quad\textrm{$\gamma$ arbitrary}. -\end{eqnarray} - -\begin{eqnarray} -\label{GenGegenbauer6} -& &(1-xt)^{-\gamma}\,\hyp{2}{1}{\frac{1}{2}\gamma,\frac{1}{2}\gamma+\frac{1}{2}} -{\lambda+\frac{1}{2}}{\frac{(x^2-1)t^2}{(1-xt)^2}}\nonumber\\ -& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n}{(2\lambda)_n}C_n^{(\lambda)}(x)t^n, -\quad\textrm{$\gamma$ arbitrary}. -\end{eqnarray} - -\subsection*{Limit relation} - -\subsubsection*{Gegenbauer / Ultraspherical $\rightarrow$ Hermite} -The Hermite polynomials given by (\ref{DefHermite}) follow from the Gegenbauer (or -ultraspherical) polynomials by taking $\lambda=\alpha+\frac{1}{2}$ and letting -$\alpha\rightarrow\infty$ in the following way: -\begin{equation} -\lim_{\alpha\rightarrow\infty} -\alpha^{-\frac{1}{2}n}C_n^{(\alpha+\frac{1}{2})}(\alpha^{-\frac{1}{2}}x)=\frac{H_n(x)}{n!}. -\end{equation} - -\subsection*{Remarks} -The case $\lambda=0$ needs another normalization. In that case we have the -Chebyshev polynomials of the first kind described in the next subsection. - -\noindent -The Gegenbauer (or ultraspherical) polynomials are related to the Jacobi polynomials -given by (\ref{DefJacobi}) by the quadratic transformations: -$$C_{2n}^{(\lambda)}(x)=\frac{(\lambda)_n}{(\frac{1}{2})_n} -P_n^{(\lambda-\frac{1}{2},-\frac{1}{2})}(2x^2-1)$$ -and -$$C_{2n+1}^{(\lambda)}(x)=\frac{(\lambda)_{n+1}}{(\frac{1}{2})_{n+1}} -xP_n^{(\lambda-\frac{1}{2},\frac{1}{2})}(2x^2-1).$$ - -\subsection*{References} -\cite{Abram}, \cite{Ahmed+86}, \cite{AndrewsAskeyRoy}, \cite{Area+I}, \cite{Askey67}, \cite{Askey74}, \cite{Askey75}, -\cite{Askey89I}, \cite{AskeyFitch}, \cite{AskeyKoornRahman}, \cite{Berg}, \cite{BilodeauI}, -\cite{BojanovNikolov}, \cite{Brafman51}, \cite{Brafman57I}, \cite{Brown}, -\cite{BustozIsmail82}, \cite{BustozIsmail83}, \cite{BustozIsmail89}, \cite{BustozSavage79}, -\cite{BustozSavage80}, \cite{Carlitz61II}, \cite{Chihara78}, \cite{Common}, \cite{Danese}, -\cite{Dette94}, \cite{DijksmaKoorn}, \cite{DilcherStolarsky}, \cite{Dimitrov96}, -\cite{Dimitrov2003}, \cite{Doha2002II}, \cite{Driver}, \cite{ElbertLaforgia86I}, -\cite{ElbertLaforgia86II}, \cite{ElbertLaforgia90}, \cite{Erdelyi+}, \cite{Exton96}, -\cite{Gasper69}, \cite{Gasper72II}, \cite{Gasper85}, \cite{Grad}, \cite{Ismail74}, -\cite{Ismail2005II}, \cite{Koekoek2000}, \cite{Koorn88}, \cite{Laforgia}, \cite{LewanowiczI}, -\cite{Lorch}, \cite{Mathai}, \cite{Nagel}, \cite{Nikiforov+}, \cite{NikiforovUvarov}, -\cite{RahmanShah}, \cite{Rainville}, \cite{Reimer}, \cite{Sartoretto}, \cite{Srivastava71}, -\cite{Szego75}, \cite{Temme}, \cite{Viswanathan}, \cite{Zayed}. - -\subsection{Chebyshev}\index{Chebyshev polynomials} - -\par - -\subsection*{Hypergeometric representation} -The Chebyshev polynomials of the first kind can be obtained from the Jacobi -polynomials by taking $\alpha=\beta=-\frac{1}{2}$: -\begin{equation} -\label{DefChebyshevI} -T_n(x)=\frac{P_n^{(-\frac{1}{2},-\frac{1}{2})}(x)}{P_n^{(-\frac{1}{2},-\frac{1}{2})}(1)} -=\hyp{2}{1}{-n,n}{\frac{1}{2}}{\frac{1-x}{2}} -\end{equation} -and the Chebyshev polynomials of the second kind can be obtained from the -Jacobi polynomials by taking $\alpha=\beta=\frac{1}{2}$: -\begin{equation} -\label{DefChebyshevII} -U_n(x)=(n+1)\frac{P_n^{(\frac{1}{2},\frac{1}{2})}(x)}{P_n^{(\frac{1}{2},\frac{1}{2})}(1)} -=(n+1)\,\hyp{2}{1}{-n,n+2}{\frac{3}{2}}{\frac{1-x}{2}}. -\end{equation} - -\subsection*{Orthogonality relation} -\begin{equation} -\label{OrtChebyshevI} -\int_{-1}^1(1-x^2)^{-\frac{1}{2}}T_m(x)T_n(x)\,dx= -\left\{\begin{array}{ll} -\displaystyle\frac{\cpi}{2}\,\delta_{mn}, & n\neq 0\\[5mm] -\cpi\,\delta_{mn}, & n=0. -\end{array}\right. -\end{equation} - -\begin{equation} -\label{OrtChebyshevII} -\int_{-1}^1(1-x^2)^{\frac{1}{2}}U_m(x)U_n(x)\,dx=\frac{\cpi}{2}\,\delta_{mn}. -\end{equation} - -\subsection*{Recurrence relations} -\begin{equation} -\label{RecChebyshevI} -2xT_n(x)=T_{n+1}(x)+T_{n-1}(x),\quad T_{0}(x)=1\quad\textrm{and}\quad T_1(x)=x. -\end{equation} - -\begin{equation} -\label{RecChebyshevII} -2xU_n(x)=U_{n+1}(x)+U_{n-1}(x),\quad U_{0}(x)=1\quad\textrm{and}\quad U_1(x)=2x. -\end{equation} - -\subsection*{Normalized recurrence relations} -\begin{equation} -\label{NormRecChebyshevI} -xp_n(x)=p_{n+1}(x)+\frac{1}{4}p_{n-1}(x), -\end{equation} -where -$$T_1(x)=p_1(x)=x\quad\textrm{and}\quad T_n(x)=2^np_n(x),\quad n\neq 1.$$ - -\begin{equation} -\label{NormRecChebyshevII} -xp_n(x)=p_{n+1}(x)+\frac{1}{4}p_{n-1}(x), -\end{equation} -where -$$U_n(x)=2^np_n(x).$$ - -\subsection*{Differential equations} -\begin{equation} -\label{dvChebyshevI} -(1-x^2)y''(x)-xy'(x)+n^2y(x)=0,\quad y(x)=T_n(x). -\end{equation} - -\begin{equation} -\label{dvChebyshevII} -(1-x^2)y''(x)-3xy'(x)+n(n+2)y(x)=0,\quad y(x)=U_n(x). -\end{equation} - -\subsection*{Forward shift operator} -\begin{equation} -\label{shift1Chebyshev} -\frac{d}{dx}T_n(x)=nU_{n-1}(x). -\end{equation} - -\subsection*{Backward shift operator} -\begin{equation} -\label{shift2ChebyshevI} -(1-x^2)\frac{d}{dx}U_n(x)-xU_n(x)=-(n+1)T_{n+1}(x) -\end{equation} -or equivalently -\begin{equation} -\label{shift2ChebyshevII} -\frac{d}{dx}\left[\left(1-x^2\right)^{\frac{1}{2}}U_n(x)\right] -=-(n+1)\left(1-x^2\right)^{-\frac{1}{2}}T_{n+1}(x). -\end{equation} - -\subsection*{Rodrigues-type formulas} -\begin{equation} -\label{RodChebyshevI} -(1-x^2)^{-\frac{1}{2}}T_n(x)=\frac{(-1)^n}{(\frac{1}{2})_n2^n} -\left(\frac{d}{dx}\right)^n\left[(1-x^2)^{n-\frac{1}{2}}\right]. -\end{equation} - -\begin{equation} -\label{RodChebyshevII} -(1-x^2)^{\frac{1}{2}}U_n(x)=\frac{(n+1)(-1)^n}{(\frac{3}{2})_n2^n} -\left(\frac{d}{dx}\right)^n\left[(1-x^2)^{n+\frac{1}{2}}\right]. -\end{equation} - -\subsection*{Generating functions} -\begin{equation} -\label{GenChebyshevI1} -\frac{1-xt}{1-2xt+t^2}=\sum_{n=0}^{\infty}T_n(x)t^n. -\end{equation} - -\begin{equation} -\label{GenChebyshevI2} -R^{-1}\sqrt{\frac{1}{2}(1+R-xt)}=\sum_{n=0}^{\infty} -\frac{\left(\frac{1}{2}\right)_n}{n!}T_n(x)t^n,\quad R=\sqrt{1-2xt+t^2}. -\end{equation} - -\begin{equation} -\label{GenChebyshevI3} -\hyp{0}{1}{-}{\frac{1}{2}}{\frac{(x-1)t}{2}}\, -\hyp{0}{1}{-}{\frac{1}{2}}{\frac{(x+1)t}{2}}= -\sum_{n=0}^{\infty}\frac{T_n(x)}{\left(\frac{1}{2}\right)_nn!}t^n. -\end{equation} - -\begin{equation} -\label{GenChebyshevI4} -\e^{xt}\,\hyp{0}{1}{-}{\frac{1}{2}}{\frac{(x^2-1)t^2}{4}}= -\sum_{n=0}^{\infty}\frac{T_n(x)}{n!}t^n. -\end{equation} - -\begin{eqnarray} -\label{GenChebyshevI5} -& &\hyp{2}{1}{\gamma,-\gamma}{\frac{1}{2}}{\frac{1-R-t}{2}}\, -\hyp{2}{1}{\gamma,-\gamma}{\frac{1}{2}}{\frac{1-R+t}{2}}\nonumber\\ -& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n(-\gamma)_n}{\left(\frac{1}{2}\right)_nn!}T_n(x)t^n, -\quad R=\sqrt{1-2xt+t^2},\quad\textrm{$\gamma$ arbitrary}. -\end{eqnarray} - -\begin{eqnarray} -\label{GenChebyshevI6} -& &(1-xt)^{-\gamma}\,\hyp{2}{1}{\frac{1}{2}\gamma,\frac{1}{2}\gamma+\frac{1}{2}}{\frac{1}{2}} -{\frac{(x^2-1)t^2}{(1-xt)^2}}\nonumber\\ -& &=\sum_{n=0}^{\infty}\frac{(\gamma)_n}{n!}T_n(x)t^n,\quad\textrm{$\gamma$ arbitrary}. -\end{eqnarray} - -\begin{equation} -\label{GenChebyshevII1} -\frac{1}{1-2xt+t^2}=\sum_{n=0}^{\infty}U_n(x)t^n. -\end{equation} - -\begin{equation} -\label{GenChebyshevII2} -\frac{1}{R\sqrt{\frac{1}{2}(1+R-xt)}}=\sum_{n=0}^{\infty} -\frac{\left(\frac{3}{2}\right)_n}{(n+1)!}U_n(x)t^n,\quad R=\sqrt{1-2xt+t^2}. -\end{equation} - -\begin{equation} -\label{GenChebyshevII3} -\hyp{0}{1}{-}{\frac{3}{2}}{\frac{(x-1)t}{2}}\, -\hyp{0}{1}{-}{\frac{3}{2}}{\frac{(x+1)t}{2}}= -\sum_{n=0}^{\infty}\frac{U_n(x)}{\left(\frac{3}{2}\right)_n(n+1)!}t^n. -\end{equation} - -\begin{equation} -\label{GenChebyshevII4} -\e^{xt}\,\hyp{0}{1}{-}{\frac{3}{2}}{\frac{(x^2-1)t^2}{4}}= -\sum_{n=0}^{\infty}\frac{U_n(x)}{(n+1)!}t^n. -\end{equation} - -\begin{eqnarray} -\label{GenChebyshevII5} -& &\hyp{2}{1}{\gamma,2-\gamma}{\frac{3}{2}}{\frac{1-R-t}{2}}\, -\hyp{2}{1}{\gamma,2-\gamma}{\frac{3}{2}}{\frac{1-R+t}{2}}\nonumber\\ -& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n(2-\gamma)_n}{\left(\frac{3}{2}\right)_n(n+1)!}U_n(x)t^n, -\quad R=\sqrt{1-2xt+t^2},\quad\textrm{$\gamma$ arbitrary}. -\end{eqnarray} - -\begin{eqnarray} -\label{GenChebyshevII6} -& &(1-xt)^{-\gamma}\,\hyp{2}{1}{\frac{1}{2}\gamma,\frac{1}{2}\gamma+\frac{1}{2}}{\frac{3}{2}} -{\frac{(x^2-1)t^2}{(1-xt)^2}}\nonumber\\ -& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n}{(n+1)!}U_n(x)t^n,\quad\textrm{$\gamma$ arbitrary}. -\end{eqnarray} - -\subsection*{Remarks} -The Chebyshev polynomials can also be written as: -$$T_n(x)=\cos(n\theta),\quad x=\cos\theta$$ -and -$$U_n(x)=\frac{\sin (n+1)\theta}{\sin\theta},\quad x=\cos\theta.$$ -Further we have -$$U_n(x)=C_n^{(1)}(x)$$ -where $C_n^{(\lambda)}(x)$ denotes the Gegenbauer (or ultraspherical) -polynomial given by (\ref{DefGegenbauer}) in the preceding subsection. - -\subsection*{References} -\cite{Abram}, \cite{AskeyFitch}, \cite{AskeyGasperHarris}, \cite{AskeyIsmail76}, -\cite{Bavinck95}, \cite{Chihara78}, \cite{Danese}, \cite{DilcherStolarsky}, -\cite{Erdelyi+}, \cite{Grad}, \cite{HartmannStephan}, \cite{Ismail2005II}, \cite{Koekoek2000}, -\cite{Luke}, \cite{Mathai}, \cite{Nikiforov+}, \cite{NikiforovUvarov}, \cite{Rainville}, -\cite{Rayes+}, \cite{Rivlin}, \cite{SainteViennot}, \cite{Szego75}, \cite{Temme}, -\cite{Wilson70I}, \cite{Zayed}, \cite{Zhang}, \cite{ZhangWang}. - -\subsection{Legendre / Spherical}\index{Legendre polynomials} -\index{Spherical polynomials} - -\par - -\subsection*{Hypergeometric representation} -The Legendre (or spherical) polynomials are Jacobi polynomials with $\alpha=\beta=0$: -\begin{equation} -\label{DefLegendre} -P_n(x)=P_n^{(0,0)}(x)=\hyp{2}{1}{-n,n+1}{1}{\frac{1-x}{2}}. -\end{equation} - -\subsection*{Orthogonality relation} -\begin{equation} -\label{OrtLegendre} -\int_{-1}^1P_m(x)P_n(x)\,dx=\frac{2}{2n+1}\,\delta_{mn}. -\end{equation} - -\subsection*{Recurrence relation} -\begin{equation} -\label{RecLegendre} -(2n+1)xP_n(x)=(n+1)P_{n+1}(x)+nP_{n-1}(x). -\end{equation} - -\subsection*{Normalized recurrence relation} -\begin{equation} -\label{NormRecLegendre} -xp_n(x)=p_{n+1}(x)+\frac{n^2}{(2n-1)(2n+1)}p_{n-1}(x), -\end{equation} -where -$$P_n(x)=\binom{2n}{n}\frac{1}{2^n}p_n(x).$$ - -\subsection*{Differential equation} -\begin{equation} -\label{dvLegendre} -(1-x^2)y''(x)-2xy'(x)+n(n+1)y(x)=0,\quad y(x)=P_n(x). -\end{equation} - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodLegendre} -P_n(x)=\frac{(-1)^n}{2^nn!}\left(\frac{d}{dx}\right)^n\left[(1-x^2)^n\right]. -\end{equation} - -\subsection*{Generating functions} -\begin{equation} -\label{GenLegendre1} -\frac{1}{\sqrt{1-2xt+t^2}}=\sum_{n=0}^{\infty}P_n(x)t^n. -\end{equation} - -\begin{equation} -\label{GenLegendre2} -\hyp{0}{1}{-}{1}{\frac{(x-1)t}{2}}\,\hyp{0}{1}{-}{1}{\frac{(x+1)t}{2}}= -\sum_{n=0}^{\infty}\frac{P_n(x)}{(n!)^2}t^n. -\end{equation} - -\begin{equation} -\label{GenLegendre3} -\e^{xt}\,\hyp{0}{1}{-}{1}{\frac{(x^2-1)t^2}{4}}= -\sum_{n=0}^{\infty}\frac{P_n(x)}{n!}t^n. -\end{equation} - -\begin{eqnarray} -\label{GenLegendre4} -& &\hyp{2}{1}{\gamma,1-\gamma}{1}{\frac{1-R-t}{2}}\, -\hyp{2}{1}{\gamma,1-\gamma}{1}{\frac{1-R+t}{2}}\nonumber\\ -& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n(1-\gamma)_n}{(n!)^2}P_n(x)t^n, -\quad R=\sqrt{1-2xt+t^2},\quad\textrm{$\gamma$ arbitrary}. -\end{eqnarray} - -\begin{eqnarray} -\label{GenLegendre5} -& &(1-xt)^{-\gamma}\,\hyp{2}{1}{\frac{1}{2}\gamma,\frac{1}{2}\gamma+\frac{1}{2}}{1} -{\frac{(x^2-1)t^2}{(1-xt)^2}}\nonumber\\ -& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n}{n!}P_n(x)t^n,\quad\textrm{$\gamma$ arbitrary}. -\end{eqnarray} - -\subsection*{References} -\cite{Abram}, \cite{Alladi}, \cite{AlSalam90}, \cite{Bhonsle}, \cite{Brafman51}, -\cite{Carlitz57II}, \cite{Chihara78}, \cite{Danese}, \cite{Dattoli2001}, -\cite{DilcherStolarsky}, \cite{ElbertLaforgia94}, \cite{Erdelyi+}, \cite{Grad}, -\cite{Mathai}, \cite{Nikiforov+}, \cite{NikiforovUvarov}, \cite{Olver}, \cite{Rainville}, -\cite{Szego75}, \cite{Temme}, \cite{Zayed}. - -\newpage - -\section{Pseudo Jacobi}\index{Pseudo Jacobi polynomials}\index{Jacobi polynomials!Pseudo} - -\par\setcounter{equation}{0} - -\subsection*{Hypergeometric representation} -\begin{eqnarray} -\label{DefPseudoJacobi} -P_n(x;\nu,N)&=&\frac{(-2i)^n(-N+i\nu)_n}{(n-2N-1)_n}\,\hyp{2}{1}{-n,n-2N-1}{-N+i\nu}{\frac{1-ix}{2}}\\ -&=&(x+i)^n\,\hyp{2}{1}{-n,N+1-n-i\nu}{2N+2-2n}{\frac{2}{1-ix}},\quad n=0,1,2,\ldots,N.\nonumber -\end{eqnarray} - -\subsection*{Orthogonality relation} -\begin{eqnarray} -\label{OrtPseudoJacobi} -& &\frac{1}{2\pi}\int_{-\infty}^{\infty}(1+x^2)^{-N-1}\e^{2\nu\arctan x}P_m(x;\nu,N)P_n(x;\nu,N)\,dx\nonumber\\ -& &{}=\frac{\Gamma(2N+1-2n)\Gamma(2N+2-2n)2^{2n-2N-1}n!}{\Gamma(2N+2-n)\left|\Gamma(N+1-n+i\nu)\right|^2}\,\delta_{mn}. -\end{eqnarray} - -\subsection*{Recurrence relation} -\begin{eqnarray} -\label{RecPseudoJacobi} -xP_n(x;\nu,N)&=&P_{n+1}(x;\nu,N)+\frac{(N+1)\nu}{(n-N-1)(n-N)}P_n(x;\nu,N)\nonumber\\ -& &{}\mathindent{}-\frac{n(n-2N-2)}{(2n-2N-3)(n-N-1)^2(2n-2N-1)}\nonumber\\ -& &{}\mathindent\mathindent\times(n-N-1-i\nu)(n-N-1+i\nu)P_{n-1}(x;\nu,N). -\end{eqnarray} - -\subsection*{Normalized recurrence relation} -\begin{eqnarray} -\label{NormRecPseudoJacobi} -xp_n(x)&=&p_{n+1}(x)+\frac{(N+1)\nu}{(n-N-1)(n-N)}p_n(x)\nonumber\\ -& &{}\mathindent{}-\frac{n(n-2N-2)(n-N-1-i\nu)(n-N-1+i\nu)}{(2n-2N-3)(n-N-1)^2(2n-2N-1)}p_{n-1}(x), -\end{eqnarray} -where -$$P_n(x;\nu,N)=p_n(x).$$ - -\subsection*{Differential equation} -\begin{equation} -\label{dvPseudoJacobi} -(1+x^2)y''(x)+2\left(\nu-Nx\right)y'(x)-n(n-2N-1)y(x)=0, -\end{equation} -where -$$y(x)=P_n(x;\nu,N).$$ - -\subsection*{Forward shift operator} -\begin{equation} -\label{shift1PseudoJacobi} -\frac{d}{dx}P_n(x;\nu,N)=nP_{n-1}(x;\nu,N-1). -\end{equation} - -\subsection*{Backward shift operator} -\begin{eqnarray} -\label{shift2PseudoJacobiI} -& &(1+x^2)\frac{d}{dx}P_n(x;\nu,N)+2\left[\nu-(N+1)x\right]P_n(x;\nu,N)\nonumber\\ -& &{}=(n-2N-2)P_{n+1}(x;\nu,N+1) -\end{eqnarray} -or equivalently -\begin{eqnarray} -\label{shift2PseudoJacobiII} -& &\frac{d}{dx}\left[(1+x^2)^{-N-1}\e^{2\nu\arctan x}P_n(x;\nu,N)\right]\nonumber\\ -& &{}=(n-2N-2)(1+x^2)^{-N-2}\e^{2\nu\arctan x}P_{n+1}(x;\nu,N+1). -\end{eqnarray} - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodPseudoJacobi} -P_n(x;\nu,N)=\frac{(1+x^2)^{N+1}\expe^{-2\nu\arctan x}}{(n-2N-1)_n} -\left(\frac{d}{dx}\right)^n\left[(1+x^2)^{n-N-1}\expe^{2\nu\arctan x}\right]. -\end{equation} - -\subsection*{Generating function} -\begin{eqnarray} -\label{GenPseudoJacobi} -& &\left[\hyp{0}{1}{-}{-N+i\nu}{(x+i)t}\,\hyp{0}{1}{-}{-N-i\nu}{(x-i)t}\right]_N\nonumber\\ -& &{}=\sum_{n=0}^N\frac{(n-2N-1)_n}{(-N+i\nu)_n(-N-i\nu)_nn!}P_n(x;\nu,N)t^n. -\end{eqnarray} - -\subsection*{Limit relation} - -\subsubsection*{Continuous Hahn $\rightarrow$ Pseudo Jacobi} -The pseudo Jacobi polynomials follow from the continuous Hahn polynomials given by -(\ref{DefContHahn}) by the substitutions $x\rightarrow xt$, $a=\frac{1}{2}(-N+i\nu-2t)$, -$b=\frac{1}{2}(-N-i\nu+2t)$, $c=\frac{1}{2}(-N-i\nu-2t)$ and $d=\frac{1}{2}(-N+i\nu+2t)$, -division by $t^n$ and the limit $t\rightarrow\infty$: -\begin{eqnarray*} -& &\lim_{t\rightarrow\infty}\frac{p_n(xt;\frac{1}{2}(-N+i\nu-2t),\frac{1}{2}(-N-i\nu+2t), -\frac{1}{2}(-N+i\nu-2t),\frac{1}{2}(-N-i\nu+2t))}{t^n}\\ -& &{}=\frac{(n-2N-1)_n}{n!}P_n(x;\nu,N). -\end{eqnarray*} - -\subsection*{Remarks} -Since we have for $k 0$ & -% $0 < c < 1$ } -\end{equation} - -\subsection*{Recurrence relation} -\begin{eqnarray} -\label{RecMeixner} -(c-1)xM_n(x;\beta,c)&=&c(n+\beta)M_{n+1}(x;\beta,c)\nonumber\\ -& &{}\mathindent{}-\left[n+(n+\beta)c\right]M_n(x;\beta,c)+nM_{n-1}(x;\beta,c). -\end{eqnarray} - -\subsection*{Normalized recurrence relation} -\begin{equation} -\label{NormRecMeixner} -xp_n(x)=p_{n+1}(x)+\frac{n+(n+\beta)c}{1-c}p_n(x)+ -\frac{n(n+\beta-1)c}{(1-c)^2}p_{n-1}(x), -\end{equation} -where -$$M_n(x;\beta,c)=\frac{1}{(\beta)_n}\left(\frac{c-1}{c}\right)^np_n(x).$$ - -\subsection*{Difference equation} -\begin{equation} -\label{dvMeixner} -n(c-1)y(x)=c(x+\beta)y(x+1)-\left[x+(x+\beta)c\right]y(x)+xy(x-1), -\end{equation} -where -$$y(x)=M_n(x;\beta,c).$$ - -\subsection*{Forward shift operator} -\begin{equation} -\label{shift1MeixnerI} -M_n(x+1;\beta,c)-M_n(x;\beta,c)= -\frac{n}{\beta}\left(\frac{c-1}{c}\right)M_{n-1}(x;\beta+1,c) -\end{equation} -or equivalently -\begin{equation} -\label{shift1MeixnerII} -\Delta M_n(x;\beta,c)=\frac{n}{\beta}\left(\frac{c-1}{c}\right)M_{n-1}(x;\beta+1,c). -\end{equation} - -\subsection*{Backward shift operator} -\begin{equation} -\label{shift2MeixnerI} -c(\beta+x-1)M_n(x;\beta,c)-xM_n(x-1;\beta,c)=c(\beta-1)M_{n+1}(x;\beta-1,c) -\end{equation} -or equivalently -\begin{equation} -\label{shift2MeixnerII} -\nabla\left[\frac{(\beta)_xc^x}{x!}M_n(x;\beta,c)\right]= -\frac{(\beta-1)_xc^x}{x!}M_{n+1}(x;\beta-1,c). -\end{equation} - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodMeixner} -\frac{(\beta)_xc^x}{x!}M_n(x;\beta,c)=\nabla^n\left[\frac{(\beta+n)_xc^x}{x!}\right]. -\end{equation} - -\subsection*{Generating functions} -\begin{equation} -\label{GenMeixner1} -\left(1-\frac{t}{c}\right)^x(1-t)^{-x-\beta}= -\sum_{n=0}^{\infty}\frac{(\beta)_n}{n!}M_n(x;\beta,c)t^n. -\end{equation} - -\begin{equation} -\label{GenMeixner2} -\e^t\,\hyp{1}{1}{-x}{\beta}{\left(\frac{1-c}{c}\right)t}= -\sum_{n=0}^{\infty}\frac{M_n(x;\beta,c)}{n!}t^n. -\end{equation} - -\begin{equation} -\label{GenMeixner3} -(1-t)^{-\gamma}\,\hyp{2}{1}{\gamma,-x}{\beta}{\frac{(1-c)t}{c(1-t)}} -=\sum_{n=0}^{\infty}\frac{(\gamma)_n}{n!}M_n(x;\beta,c)t^n, -\quad\textrm{$\gamma$ arbitrary}. -\end{equation} - -\subsection*{Limit relations} - -\subsubsection*{Hahn $\rightarrow$ Meixner} -If we take $\alpha=b-1$, $\beta=N(1-c)c^{-1}$ in the definition (\ref{DefHahn}) -of the Hahn polynomials and let $N\rightarrow\infty$ we find the Meixner polynomials: -$$\lim_{N\rightarrow\infty} -Q_n(x;b-1,N(1-c)c^{-1},N)=M_n(x;b,c).$$ - -\subsubsection*{Dual Hahn $\rightarrow$ Meixner} -To obtain the Meixner polynomials from the dual Hahn polynomials we have to take -$\gamma=\beta-1$ and $\delta=N(1-c)c^{-1}$ in the definition (\ref{DefDualHahn}) of -the dual Hahn polynomials and let $N\rightarrow\infty$: -$$\lim_{N\rightarrow\infty} -R_n(\lambda(x);\beta-1,N(1-c)c^{-1},N)=M_n(x;\beta,c).$$ - -\subsubsection*{Meixner $\rightarrow$ Laguerre} -The Laguerre polynomials given by (\ref{DefLaguerre}) are obtained from the Meixner polynomials -if we take $\beta=\alpha+1$ and $x\rightarrow (1-c)^{-1}x$ and let $c\rightarrow 1$: -\begin{equation} -\lim_{c\rightarrow 1} -M_n((1-c)^{-1}x;\alpha+1,c)=\frac{L_n^{(\alpha)}(x)}{L_n^{(\alpha)}(0)}. -\end{equation} - -\subsubsection*{Meixner $\rightarrow$ Charlier} -The Charlier polynomials given by (\ref{DefCharlier}) are obtained from the Meixner polynomials -if we take $c=(a+\beta)^{-1}a$ and let $\beta\rightarrow\infty$: -\begin{equation} -\lim_{\beta\rightarrow\infty} -M_n(x;\beta,(a+\beta)^{-1}a)=C_n(x;a). -\end{equation} - -\subsection*{Remarks} -The Meixner polynomials are related to the Jacobi polynomials given by (\ref{DefJacobi}) -in the following way: -$$\frac{(\beta)_n}{n!}M_n(x;\beta,c)=P_n^{(\beta-1,-n-\beta-x)}((2-c)c^{-1}).$$ - -\noindent -The Meixner polynomials are also related to the Krawtchouk polynomials given by -(\ref{DefKrawtchouk}) in the following way: -$$K_n(x;p,N)=M_n(x;-N,(p-1)^{-1}p).$$ - -\subsection*{References} -\cite{Allaway76}, \cite{NAlSalam66}, \cite{AlSalam90}, \cite{AlSalamChihara76}, -\cite{AlSalamIsmail76}, \cite{Alvarez+}, \cite{AndrewsAskey85}, \cite{Area+II}, \cite{Askey75}, -\cite{Askey89I}, \cite{Askey2005}, \cite{AskeyGasper77}, \cite{AskeyIsmail76}, -\cite{AskeyWilson85}, \cite{AtakRahmanSuslov}, \cite{AtakSuslov88}, \cite{Bavinck98}, -\cite{BavinckHaeringen}, \cite{Campigotto+}, \cite{Chihara78}, \cite{Cooper+}, -\cite{Erdelyi+}, \cite{FoataLabelle}, \cite{Gabutti}, \cite{GabuttiMathis}, \cite{Gasper73I}, -\cite{Gasper74}, \cite{HoareRahman}, \cite{Ismail2005II}, \cite{IsmailLetVal88}, -\cite{IsmailLi}, \cite{IsmailMuldoon}, \cite{IsmailStanton97}, \cite{JinWong}, -\cite{Karlin58}, \cite{Koekoek2000}, \cite{Koorn88}, \cite{LabelleYehI}, \cite{LabelleYehII}, -\cite{Lesky89}, \cite{Lesky94I}, \cite{Lesky95II}, \cite{LewanowiczII}, \cite{Meixner}, -\cite{Nikiforov+}, \cite{NikiforovUvarov}, \cite{Rahman78I}, \cite{ValentAssche}, -\cite{Viennot}, \cite{Zarzo+}, \cite{Zeng90}. - - -\section{Krawtchouk}\index{Krawtchouk polynomials} - -\par\setcounter{equation}{0} - -\subsection*{Hypergeometric representation} -\begin{equation} -\label{DefKrawtchouk} -K_n(x;p,N)=\hyp{2}{1}{-n,-x}{-N}{\frac{1}{p}},\quad n=0,1,2,\ldots,N. -\end{equation} - -\subsection*{Orthogonality relation} -\begin{eqnarray} -\label{OrtKrawtchouk} -& &\sum_{x=0}^N\binom{N}{x}p^x(1-p)^{N-x} K_m(x;p,N)K_n(x;p,N)\nonumber\\ -& &{}=\frac{(-1)^nn!}{(-N)_n}\left(\frac{1-p}{p}\right)^n\,\delta_{mn},\quad0 < p < 1. -\end{eqnarray} - -\subsection*{Recurrence relation} -\begin{eqnarray} -\label{RecKrawtchouk} --xK_n(x;p,N)&=&p(N-n)K_{n+1}(x;p,N)\nonumber\\ -& &{}\mathindent{}-\left[p(N-n)+n(1-p)\right]K_n(x;p,N)\nonumber\\ -& &{}\mathindent\mathindent{}+n(1-p)K_{n-1}(x;p,N). -\end{eqnarray} - -\subsection*{Normalized recurrence relation} -\begin{eqnarray} -\label{NormRecKrawtchouk} -xp_n(x)&=&p_{n+1}(x)+\left[p(N-n)+n(1-p)\right]p_n(x)\nonumber\\ -& &{}\mathindent{}+np(1-p)(N+1-n)p_{n-1}(x), -\end{eqnarray} -where -$$K_n(x;p,N)=\frac{1}{(-N)_np^n}p_n(x).$$ - -\subsection*{Difference equation} -\begin{eqnarray} -\label{dvKrawtchouk} --ny(x)&=&p(N-x)y(x+1)\nonumber\\ -& &{}\mathindent{}-\left[p(N-x)+x(1-p)\right]y(x)+x(1-p)y(x-1), -\end{eqnarray} -where -$$y(x)=K_n(x;p,N).$$ - -\subsection*{Forward shift operator} -\begin{equation} -\label{shift1KrawtchoukI} -K_n(x+1;p,N)-K_n(x;p,N)=-\frac{n}{Np}K_{n-1}(x;p,N-1) -\end{equation} -or equivalently -\begin{equation} -\label{shift1KrawtchoukII} -\Delta K_n(x;p,N)=-\frac{n}{Np}K_{n-1}(x;p,N-1). -\end{equation} - -\subsection*{Backward shift operator} -\begin{eqnarray} -\label{shift2KrawtchoukI} -& &(N+1-x)K_n(x;p,N)-x\left(\frac{1-p}{p}\right)K_n(x-1;p,N)\nonumber\\ -& &{}=(N+1)K_{n+1}(x;p,N+1) -\end{eqnarray} -or equivalently -\begin{equation} -\label{shift2KrawtchoukII} -\nabla\left[\binom{N}{x}\left(\frac{p}{1-p}\right)^xK_n(x;p,N)\right]= -\binom{N+1}{x}\left(\frac{p}{1-p}\right)^xK_{n+1}(x;p,N+1). -\end{equation} - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodKrawtchouk} -\binom{N}{x}\left(\frac{p}{1-p}\right)^xK_n(x;p,N)= -\nabla^n\left[\binom{N-n}{x}\left(\frac{p}{1-p}\right)^x\right]. -\end{equation} - -\subsection*{Generating functions} -For $x=0,1,2,\ldots,N$ we have -\begin{equation} -\label{GenKrawtchouk1} -\left(1-\frac{(1-p)}{p}t\right)^x(1+t)^{N-x}= -\sum_{n=0}^N\binom{N}{n}K_n(x;p,N)t^n. -\end{equation} - -\begin{equation} -\label{GenKrawtchouk2} -\left[\expe^t\,\hyp{1}{1}{-x}{-N}{-\frac{t}{p}}\right]_N= -\sum_{n=0}^N\frac{K_n(x;p,N)}{n!}t^n. -\end{equation} - -\begin{eqnarray} -\label{GenKrawtchouk3} -& &\left[(1-t)^{-\gamma}\,\hyp{2}{1}{\gamma,-x}{-N}{\frac{t}{p(t-1)}}\right]_N\nonumber\\ -& &{}=\sum_{n=0}^N\frac{(\gamma)_n}{n!}K_n(x;p,N)t^n, -\quad\textrm{$\gamma$ arbitrary}. -\end{eqnarray} - -\subsection*{Limit relations} - -\subsubsection*{Hahn $\rightarrow$ Krawtchouk} -If we take $\alpha=pt$ and $\beta=(1-p)t$ in the definition (\ref{DefHahn}) of the Hahn -polynomials and let $t\rightarrow\infty$ we obtain the Krawtchouk polynomials: -$$\lim_{t\rightarrow\infty}Q_n(x;pt,(1-p)t,N)=K_n(x;p,N).$$ - -\subsubsection*{Dual Hahn $\rightarrow$ Krawtchouk} -The Krawtchouk polynomials follow from the dual Hahn polynomials given by -(\ref{DefDualHahn}) if we set $\gamma=pt$, $\delta=(1-p)t$ and let $t\rightarrow\infty$: -$$\lim_{t\rightarrow\infty}R_n(\lambda(x);pt,(1-p)t,N)=K_n(x;p,N).$$ - -\subsubsection*{Krawtchouk $\rightarrow$ Charlier} -The Charlier polynomials given by (\ref{DefCharlier}) can be found from the Krawtchouk -polynomials by taking $p=N^{-1}a$ and letting $N\rightarrow\infty$: -\begin{equation} -\lim_{N\rightarrow\infty}K_n(x;N^{-1}a,N)=C_n(x;a). -\end{equation} - -\subsubsection*{Krawtchouk $\rightarrow$ Hermite} -The Hermite polynomials given by (\ref{DefHermite}) follow from the Krawtchouk polynomials -by setting $x\rightarrow pN+x\sqrt{2p(1-p)N}$ and then letting $N\rightarrow\infty$: -\begin{equation} -\lim_{N\rightarrow\infty} -\sqrt{\binom{N}{n}}K_n(pN+x\sqrt{2p(1-p)N};p,N) -=\frac{\displaystyle (-1)^nH_n(x)}{\displaystyle\sqrt{2^nn!\left(\frac{p}{1-p}\right)^n}}. -\end{equation} - -\subsection*{Remarks} -The Krawtchouk polynomials are self-dual, which means that -$$K_n(x;p,N)=K_x(n;p,N),\quad n,x\in\{0,1,2,\ldots,N\}.$$ -By using this relation we easily obtain the so-called dual orthogonality -relation from the orthogonality relation (\ref{OrtKrawtchouk}): -$$\sum_{n=0}^N\binom{N}{n}p^n(1-p)^{N-n} K_n(x;p,N)K_n(y;p,N)= -\frac{\displaystyle\left(\frac{1-p}{p}\right)^x}{\dbinom{N}{x}}\delta_{xy},$$ -where $0 < p < 1$ and $x,y\in\{0,1,2,\ldots,N\}$. - -\noindent -The Krawtchouk polynomials are related to the Meixner polynomials given by (\ref{DefMeixner}) -in the following way: -$$K_n(x;p,N)=M_n(x;-N,(p-1)^{-1}p).$$ - -\subsection*{References} -\cite{AlSalam90}, \cite{AndrewsAskey85}, \cite{Area+II}, \cite{Askey75}, \cite{Askey89I}, -\cite{AskeyGasper77}, \cite{AskeyWilson85}, \cite{AtakRahmanSuslov}, \cite{Campigotto+}, -\cite{LChiharaStanton}, \cite{Chihara78}, \cite{Dette95}, \cite{Dominici}, \cite{Dunkl76}, -\cite{Dunkl84}, \cite{DunklRamirez}, \cite{Erdelyi+}, \cite{FeinsilverSchott}, -\cite{Gasper73I}, \cite{Gasper74}, \cite{HoareRahman}, \cite{Ismail2005II}, \cite{Karlin58}, -\cite{Koorn82}, \cite{Koorn88}, \cite{LabelleYehI}, \cite{LabelleYehII}, \cite{Lesky62}, -\cite{Lesky89}, \cite{Lesky94I}, \cite{Lesky95II}, \cite{LewanowiczII}, \cite{Nikiforov+}, -\cite{NikiforovUvarov}, \cite{Qiu}, \cite{Rahman78I}, \cite{Rahman79}, \cite{Stanton84}, -\cite{Stanton90}, \cite{Szego75}, \cite{Zarzo+}, \cite{Zeng90}. - - -\section{Laguerre}\index{Laguerre polynomials} - -\par\setcounter{equation}{0} - -\subsection*{Hypergeometric representation} -\begin{equation} -\label{DefLaguerre} -L_n^{(\alpha)}(x)=\frac{(\alpha+1)_n}{n!}\,\hyp{1}{1}{-n}{\alpha+1}{x}. -\end{equation} - -\subsection*{Orthogonality relation} -\begin{equation} -\label{OrtLaguerre} -\int_0^{\infty}\expe^{-x}x^{\alpha}L_m^{(\alpha)}(x)L_n^{(\alpha)}(x)\,dx= -\frac{\Gamma(n+\alpha+1)}{n!}\,\delta_{mn},\quad\alpha > -1. -\end{equation} - -\subsection*{Recurrence relation} -\begin{equation} -\label{RecLaguerre} -(n+1)L_{n+1}^{(\alpha)}(x)-(2n+\alpha+1-x)L_n^{(\alpha)}(x)+(n+\alpha)L_{n-1}^{(\alpha)}(x)=0. -\end{equation} - -\subsection*{Normalized recurrence relation} -\begin{equation} -\label{NormRecLaguerre} -xp_n(x)=p_{n+1}(x)+(2n+\alpha+1)p_n(x)+n(n+\alpha)p_{n-1}(x), -\end{equation} -where -$$L_n^{(\alpha)}(x)=\frac{(-1)^n}{n!}p_n(x).$$ - -\subsection*{Differential equation} -\begin{equation} -\label{dvLaguerre} -xy''(x)+(\alpha+1-x)y'(x)+ny(x)=0,\quad y(x)=L_n^{(\alpha)}(x). -\end{equation} - -\newpage - -\subsection*{Forward shift operator} -\begin{equation} -\label{shift1Laguerre} -\frac{d}{dx}L_n^{(\alpha)}(x)=-L_{n-1}^{(\alpha+1)}(x). -\end{equation} - -\subsection*{Backward shift operator} -\begin{equation} -\label{shift2LaguerreI} -x\frac{d}{dx}L_n^{(\alpha)}(x)+(\alpha-x)L_n^{(\alpha)}(x)=(n+1)L_{n+1}^{(\alpha-1)}(x) -\end{equation} -or equivalently -\begin{equation} -\label{shift2LaguerreII} -\frac{d}{dx}\left[\expe^{-x}x^{\alpha}L_n^{(\alpha)}(x)\right]=(n+1)\expe^{-x}x^{\alpha-1}L^{(\alpha-1)}_{n+1}(x). -\end{equation} - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodLaguerre} -\e^{-x}x^{\alpha}L_n^{(\alpha)}(x)=\frac{1}{n!}\left(\frac{d}{dx}\right)^n\left[\expe^{-x}x^{n+\alpha}\right]. -\end{equation} - -\subsection*{Generating functions} -\begin{equation} -\label{GenLaguerre1} -(1-t)^{-\alpha-1}\exp\left(\frac{xt}{t-1}\right)= -\sum_{n=0}^{\infty}L_n^{(\alpha)}(x)t^n. -\end{equation} - -\begin{equation} -\label{GenLaguerre2} -\e^t\,\hyp{0}{1}{-}{\alpha+1}{-xt} -=\sum_{n=0}^{\infty}\frac{L_n^{(\alpha)}(x)}{(\alpha+1)_n}t^n. -\end{equation} - -\begin{equation} -\label{GenLaguerre3} -(1-t)^{-\gamma}\,\hyp{1}{1}{\gamma}{\alpha+1}{\frac{xt}{t-1}} -=\sum_{n=0}^{\infty}\frac{(\gamma)_n}{(\alpha+1)_n}L_n^{(\alpha)}(x)t^n, -\quad\textrm{$\gamma$ arbitrary}. -\end{equation} - -\subsection*{Limit relations} - -\subsubsection*{Meixner-Pollaczek $\rightarrow$ Laguerre} -The Laguerre polynomials can be obtained from the Meixner-Pollaczek polynomials given by -(\ref{DefMP}) by the substitution $\lambda=\frac{1}{2}(\alpha+1)$, -$x\rightarrow -\frac{1}{2}\phi^{-1}x$ and the limit $\phi\rightarrow 0$: -$$\lim_{\phi\rightarrow 0} -P_n^{(\frac{1}{2}\alpha+\frac{1}{2})}(-\textstyle\frac{1}{2}\phi^{-1}x;\phi)=L_n^{(\alpha)}(x).$$ - -\subsubsection*{Jacobi $\rightarrow$ Laguerre} -The Laguerre polynomials are obtained from the Jacobi polynomials given by (\ref{DefJacobi}) -if we set $x\rightarrow 1-2\beta^{-1}x$ and then take the limit $\beta\rightarrow\infty$: -$$\lim_{\beta\rightarrow\infty} -P_n^{(\alpha,\beta)}(1-2\beta^{-1}x)=L_n^{(\alpha)}(x).$$ - -\subsubsection*{Meixner $\rightarrow$ Laguerre} -If we take $\beta=\alpha+1$ and $x\rightarrow (1-c)^{-1}x$ in the definition -(\ref{DefMeixner}) of the Meixner polynomials and let $c\rightarrow 1$ we obtain -the Laguerre polynomials: -$$\lim_{c\rightarrow 1} -M_n((1-c)^{-1}x;\alpha+1,c)=\frac{L_n^{(\alpha)}(x)}{L_n^{(\alpha)}(0)}.$$ - -\subsubsection*{Laguerre $\rightarrow$ Hermite} -The Hermite polynomials given by (\ref{DefHermite}) can be obtained from the Laguerre -polynomials by taking the limit $\alpha\rightarrow\infty$ in the following way: -\begin{equation} -\lim_{\alpha\rightarrow\infty} -\left(\frac{2}{\alpha}\right)^{\frac{1}{2}n} -L_n^{(\alpha)}((2\alpha)^{\frac{1}{2}}x+\alpha)=\frac{(-1)^n}{n!}H_n(x). -\end{equation} - -\subsection*{Remarks} -The definition (\ref{DefLaguerre}) of the Laguerre polynomials can also be -written as: -$$L_n^{(\alpha)}(x)=\frac{1}{n!}\sum_{k=0}^n\frac{(-n)_k}{k!}(\alpha+k+1)_{n-k}x^k.$$ -In this way the Laguerre polynomials can also be seen as polynomials in the parameter $\alpha$. -Therefore they can be defined for all $\alpha$. - -\noindent -The Laguerre polynomials are related to the Bessel polynomials given by (\ref{DefBessel}) -in the following way: -$$L_n^{(\alpha)}(x)=\frac{(-x)^n}{n!}y_n(2x^{-1};-2n-\alpha-1).$$ - -\noindent -The Laguerre polynomials are related to the Charlier polynomials given by (\ref{DefCharlier}) -in the following way: -$$\frac{(-a)^n}{n!}C_n(x;a)=L_n^{(x-n)}(a).$$ - -\noindent -The Laguerre polynomials and the Hermite polynomials given by (\ref{DefHermite}) are also -connected by the following quadratic transformations: -$$H_{2n}(x)=(-1)^nn!\,2^{2n}L_n^{(-\frac{1}{2})}(x^2)$$ -and -$$H_{2n+1}(x)=(-1)^nn!\,2^{2n+1}xL_n^{(\frac{1}{2})}(x^2).$$ - -\noindent -In combinatorics the Laguerre polynomials with $\alpha=0$ are often called Rook -polynomials. - -\subsection*{References} -\cite{Abdul}, \cite{Abram}, \cite{Ahmed+82}, \cite{Allaway76}, \cite{Allaway91}, -\cite{NAlSalam66}, \cite{AlSalam64}, \cite{AlSalam90}, \cite{AlSalamChihara72}, -\cite{AlSalamChihara76}, \cite{AndrewsAskey85}, \cite{AndrewsAskeyRoy}, \cite{Askey68}, -\cite{Askey75}, \cite{Askey89I}, \cite{Askey2005}, \cite{AskeyGasper76}, \cite{AskeyGasper77}, -\cite{AskeyIsmail76}, \cite{AskeyIsmailKoorn}, \cite{AskeyWilson85}, \cite{Barrett}, -\cite{Bavinck96}, \cite{Brafman51}, \cite{Brafman57II}, \cite{Brenke}, \cite{Brown}, -\cite{BustozSavage79}, \cite{BustozSavage80}, \cite{Carlitz60}, \cite{Carlitz61I}, -\cite{Carlitz61II}, \cite{Carlitz62}, \cite{Carlitz68}, \cite{ChenSrivastava}, -\cite{ChenIsmail91}, \cite{Chihara68II}, \cite{Chihara78}, \cite{Cohen}, \cite{Cooper+}, -\cite{Dattoli2001}, \cite{DetteStudden92}, \cite{DetteStudden95}, \cite{Dimitrov2003}, -\cite{Doha2003II}, \cite{ElbertLaforgia87I}, \cite{Erdelyi}, \cite{Erdelyi+}, \cite{Exton98}, -\cite{Faldey}, \cite{Gasper73II}, \cite{Gasper77}, \cite{Gatteschi2002}, \cite{Gawronski87}, -\cite{Gawronski93}, \cite{Gillis}, \cite{GillisIsmailOffer}, \cite{Godoy+}, \cite{Grad}, -\cite{Hajmirzaahmad95}, \cite{Ismail74}, \cite{Ismail77}, \cite{Ismail2005II}, -\cite{IsmailLetVal88}, \cite{IsmailLi}, \cite{IsmailMassonRahman}, \cite{IsmailMuldoon}, -\cite{IsmailStanton97}, \cite{IsmailTamhankar}, \cite{Jackson}, \cite{Karlin58}, -\cite{Kochneff95}, \cite{Kochneff97II}, \cite{Koekoek2000}, \cite{Koorn77I}, \cite{Koorn78}, -\cite{Koorn85}, \cite{Koorn88}, \cite{Krasikov2003}, \cite{Krasikov2005}, -\cite{KuijlaarsMcLaughlin}, \cite{KwonLittle}, \cite{LabelleYehI}, \cite{LabelleYehII}, -\cite{LabelleYehIII}, \cite{Lee97I}, \cite{Lee97II}, \cite{Lee97III}, \cite{Lesky95II}, -\cite{Lesky96}, \cite{LewanowiczI}, \cite{LopezTemme2004}, \cite{Mathai}, \cite{Meixner}, -\cite{Nikiforov+}, \cite{NikiforovUvarov}, \cite{Olver}, \cite{PerezPinar}, \cite{Pittaluga}, -\cite{Rainville}, \cite{SainteViennot}, \cite{Srivastava66}, \cite{Srivastava69I}, -\cite{Srivastava69II}, \cite{Srivastava70}, \cite{Srivastava71}, \cite{Szego75}, \cite{Temme}, -\cite{Trickovic}, \cite{Trivedi}, \cite{Viennot}. - - -\section{Bessel}\index{Bessel polynomials} - -\par\setcounter{equation}{0} - -\subsection*{Hypergeometric representation} -\begin{eqnarray} -\label{DefBessel} -y_n(x;a)&=&\hyp{2}{0}{-n,n+a+1}{-}{-\frac{x}{2}}\\ -&=&(n+a+1)_n\left(\frac{x}{2}\right)^n\,\hyp{1}{1}{-n}{-2n-a}{\frac{2}{x}}, -\quad n=0,1,2,\ldots,N.\nonumber -\end{eqnarray} - -\subsection*{Orthogonality relation} -\begin{eqnarray} -\label{OrtBessel} -& &\int_0^{\infty}x^a\e^{-\frac{2}{x}}y_m(x;a)y_n(x;a)\,dx\nonumber\\ -& &=-\frac{2^{a+1}}{2n+a+1}\Gamma(-n-a)n!\,\delta_{mn},\quad a<-2N-1. -\end{eqnarray} - -\subsection*{Recurrence relation} -\begin{eqnarray} -\label{RecBessel} -& &2(n+a+1)(2n+a)y_{n+1}(x;a)\nonumber\\ -& &{}=(2n+a+1)\left[2a+(2n+a)(2n+a+2)x\right]y_n(x;a)\nonumber\\ -& &{}\mathindent{}+2n(2n+a+2)y_{n-1}(x;a). -\end{eqnarray} - -\subsection*{Normalized recurrence relation} -\begin{eqnarray} -\label{NormRecBessel} -xp_n(x)&=&p_{n+1}(x)-\frac{2a}{(2n+a)(2n+a+2)}p_n(x)\nonumber\\ -& &{}\mathindent{}-\frac{4n(n+a)}{(2n+a-1)(2n+a)^2(2n+a+1)}p_{n-1}(x), -\end{eqnarray} -where -$$y_n(x;a)=\frac{(n+a+1)_n}{2^n}p_n(x).$$ - -\subsection*{Differential equation} -\begin{equation} -\label{dvBessel} -x^2y''(x)+\left[(a+2)x+2\right]y'(x)-n(n+a+1)y(x)=0,\quad y(x)=y_n(x;a). -\end{equation} - -\subsection*{Forward shift operator} -\begin{equation} -\label{shift1Bessel} -\frac{d}{dx}y_n(x;a)=\frac{n(n+a+1)}{2}y_{n-1}(x;a+2). -\end{equation} - -\subsection*{Backward shift operator} -\begin{equation} -\label{shift2BesselI} -x^2\frac{d}{dx}y_n(x;a)+(ax+2)y_n(x;a)=2y_{n+1}(x;a-2) -\end{equation} -or equivalently -\begin{equation} -\label{shift2BesselII} -\frac{d}{dx}\left[x^a\expe^{-\frac{2}{x}}y_n(x;a)\right]=2x^{a-2}\expe^{-\frac{2}{x}}y_{n+1}(x;a-2). -\end{equation} - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodBessel} -y_n(x;a)=2^{-n}x^{-a}\expe^{\frac{2}{x}}D^n\left(x^{2n+a}\expe^{-\frac{2}{x}}\right). -\end{equation} - -\subsection*{Generating function} -\begin{equation} -\label{GenBessel} -\left(1-2xt\right)^{-\frac{1}{2}}\left(\frac{2}{1+\sqrt{1-2xt}}\right)^a -\exp\left(\frac{2t}{1+\sqrt{1-2xt}}\right)=\sum_{n=0}^{\infty}y_n(x;a)\frac{t^n}{n!}. -\end{equation} - -\subsection*{Limit relation} - -\subsubsection*{Jacobi $\rightarrow$ Bessel} -If we take $\beta=a-\alpha$ in the definition (\ref{DefJacobi}) of the Jacobi polynomials -and let $\alpha\rightarrow -\infty$ we find the Bessel polynomials: -$$\lim_{\alpha\rightarrow -\infty} -\frac{P_n^{(\alpha,a-\alpha)}(1+\alpha x)}{P_n^{(\alpha,a-\alpha)}(1)}=y_n(x;a).$$ - -\subsection*{Remarks} -The following notations are also used for the Bessel polynomials: -$$y_n(x;a,b)=y_n(2b^{-1}x;a)\quad\textrm{and}\quad\theta_n(x;a,b)=x^ny_n(x^{-1};a,b).$$ -However, the Bessel polynomials essentially depend on only one parameter. - -\noindent -The Bessel polynomials are related to the Laguerre polynomials given by (\ref{DefLaguerre}) -in the following way: -$$L_n^{(\alpha)}(x)=\frac{(-x)^n}{n!}y_n(2x^{-1};-2n-\alpha-1).$$ - -\noindent -The special case $a=-2N-2$ of the Bessel polynomials can be obtained from the pseudo Jacobi -polynomials by setting $x\rightarrow\nu x$ in the definition (\ref{DefPseudoJacobi}) of the -pseudo Jacobi polynomials and taking the limit $\nu\rightarrow\infty$ in the following way: -$$\lim\limits_{\nu\rightarrow\infty}\frac{P_n(\nu x;\nu,N)}{\nu^n} -=\frac{2^n}{(n-2N-1)_n}y_n(x;-2N-2).$$ - -\subsection*{References} -\cite{Andrade}, \cite{BergVignat}, \cite{Carlitz57I}, \cite{Dattoli2003}, -\cite{DohaAhmed2004}, \cite{DohaAhmed2006}, \cite{Grosswald}, \cite{Ismail2005II}, -\cite{KrallFrink}, \cite{Lesky98}, \cite{NikiforovUvarov}. - - -\section{Charlier}\index{Charlier polynomials} - -\par\setcounter{equation}{0} - -\subsection*{Hypergeometric representation} -\begin{equation} -\label{DefCharlier} -C_n(x;a)=\hyp{2}{0}{-n,-x}{-}{-\frac{1}{a}}. -\end{equation} - -\subsection*{Orthogonality relation} -\begin{equation} -\label{OrtCharlier} -\sum_{x=0}^{\infty}\frac{a^x}{x!}C_m(x;a)C_n(x;a)= -a^{-n}\expe^an!\,\delta_{mn},\quad a > 0. -\end{equation} - -\subsection*{Recurrence relation} -\begin{equation} -\label{RecCharlier} --xC_n(x;a)=aC_{n+1}(x;a)-(n+a)C_n(x;a)+nC_{n-1}(x;a). -\end{equation} - -\subsection*{Normalized recurrence relation} -\begin{equation} -\label{NormRecCharlier} -xp_n(x)=p_{n+1}(x)+(n+a)p_n(x)+nap_{n-1}(x), -\end{equation} -where -$$C_n(x;a)=\left(-\frac{1}{a}\right)^np_n(x).$$ - -\subsection*{Difference equation} -\begin{equation} -\label{dvCharlier} --ny(x)=ay(x+1)-(x+a)y(x)+xy(x-1),\quad y(x)=C_n(x;a). -\end{equation} - -\subsection*{Forward shift operator} -\begin{equation} -\label{shift1CharlierI} -C_n(x+1;a)-C_n(x;a)=-\frac{n}{a}C_{n-1}(x;a) -\end{equation} -or equivalently -\begin{equation} -\label{shift1CharlierII} -\Delta C_n(x;a)=-\frac{n}{a}C_{n-1}(x;a). -\end{equation} - -\subsection*{Backward shift operator} -\begin{equation} -\label{shift2CharlierI} -C_n(x;a)-\frac{x}{a}C_n(x-1;a)=C_{n+1}(x;a) -\end{equation} -or equivalently -\begin{equation} -\label{shift2CharlierII} -\nabla\left[\frac{a^x}{x!}C_n(x;a)\right]=\frac{a^x}{x!}C_{n+1}(x;a). -\end{equation} - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodCharlier} -\frac{a^x}{x!}C_n(x;a)=\nabla^n\left[\frac{a^x}{x!}\right]. -\end{equation} - -\subsection*{Generating function} -\begin{equation} -\label{GenCharlier} -\e^t\left(1-\frac{t}{a}\right)^x=\sum_{n=0}^{\infty}\frac{C_n(x;a)}{n!}t^n. -\end{equation} - -\subsection*{Limit relations} - -\subsubsection*{Meixner $\rightarrow$ Charlier} -If we take $c=(a+\beta)^{-1}a$ in the definition (\ref{DefMeixner}) of the Meixner polynomials -and let $\beta\rightarrow\infty$ we find the Charlier polynomials: -$$\lim_{\beta\rightarrow\infty}M_n(x;\beta,(a+\beta)^{-1}a)=C_n(x;a).$$ - -\subsubsection*{Krawtchouk $\rightarrow$ Charlier} -The Charlier polynomials can be found from the Krawtchouk polynomials given by -(\ref{DefKrawtchouk}) by taking $p=N^{-1}a$ and letting $N\rightarrow\infty$: -$$\lim_{N\rightarrow\infty}K_n(x;N^{-1}a,N)=C_n(x;a).$$ - -\subsubsection*{Charlier $\rightarrow$ Hermite} -The Hermite polynomials given by (\ref{DefHermite}) are obtained from the Charlier polynomials -if we set $x\rightarrow (2a)^{1/2}x+a$ and let $a\rightarrow\infty$. In fact we have -\begin{equation} -\lim_{a\rightarrow\infty} -(2a)^{\frac{1}{2}n}C_n((2a)^{\frac{1}{2}}x+a;a)=(-1)^nH_n(x). -\end{equation} - -\subsection*{Remark} -The Charlier polynomials are related to the Laguerre polynomials given by (\ref{DefLaguerre}) -in the following way: -$$\frac{(-a)^n}{n!}C_n(x;a)=L_n^{(x-n)}(a).$$ - -\subsection*{References} -\cite{Allaway76}, \cite{NAlSalam66}, \cite{AlSalam90}, \cite{AlSalamChihara76}, -\cite{AlSalamIsmail76}, \cite{AndrewsAskey85}, \cite{Area+II}, \cite{Askey75}, -\cite{AskeyGasper77}, \cite{AskeyWilson85}, \cite{AtakRahmanSuslov}, \cite{Bavinck98}, -\cite{BavinckKoekoek}, \cite{Chihara78}, \cite{Chihara79}, \cite{Dunkl76}, \cite{Erdelyi+}, -\cite{Gasper73I}, \cite{Gasper74}, \cite{Goh}, \cite{HoareRahman}, \cite{IsmailLetVal88}, -\cite{Koekoek2000}, \cite{Koorn88}, \cite{Krasikov2002}, \cite{LabelleYehI}, -\cite{LabelleYehII}, \cite{LabelleYehIII}, \cite{Lesky62}, \cite{Lesky89}, \cite{Lesky94I}, -\cite{Lesky95II}, \cite{LewanowiczII}, \cite{LopezTemme2004}, \cite{Meixner}, \cite{Nikiforov+}, -\cite{NikiforovUvarov}, \cite{Szafraniec}, \cite{Szego75}, \cite{Viennot}, \cite{Zarzo+}, -\cite{Zeng90}. - - -\section{Hermite}\index{Hermite polynomials} - -\par\setcounter{equation}{0} - -\subsection*{Hypergeometric representation} -\begin{equation} -\label{DefHermite} -H_n(x)=(2x)^n\,\hyp{2}{0}{-n/2,-(n-1)/2}{-}{-\frac{1}{x^2}}. -\end{equation} - -\subsection*{Orthogonality relation} -\begin{equation} -\label{OrtHermite} -\frac{1}{\sqrt{\cpi}}\int_{-\infty}^{\infty}\expe^{-x^2}H_m(x)H_n(x)\,dx -=2^nn!\,\delta_{mn}. -\end{equation} - -\subsection*{Recurrence relation} -\begin{equation} -\label{RecHermite} -H_{n+1}(x)-2xH_n(x)+2nH_{n-1}(x)=0. -\end{equation} - -\subsection*{Normalized recurrence relation} -\begin{equation} -\label{NormRecHermite} -xp_n(x)=p_{n+1}(x)+\frac{n}{2}p_{n-1}(x), -\end{equation} -where -$$H_n(x)=2^np_n(x).$$ - -\subsection*{Differential equation} -\begin{equation} -\label{dvHermite} -y''(x)-2xy'(x)+2ny(x)=0,\quad y(x)=H_n(x). -\end{equation} - -\subsection*{Forward shift operator} -\begin{equation} -\label{shift1Hermite} -\frac{d}{dx}H_n(x)=2nH_{n-1}(x). -\end{equation} - -\subsection*{Backward shift operator} -\begin{equation} -\label{shift2HermiteI} -\frac{d}{dx}H_n(x)-2xH_n(x)=-H_{n+1}(x) -\end{equation} -or equivalently -\begin{equation} -\label{shift2HermiteII} -\frac{d}{dx}\left[\expe^{-x^2}H_n(x)\right]=-\expe^{-x^2}H_{n+1}(x). -\end{equation} - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodHermite} -\e^{-x^2}H_n(x)=(-1)^n\left(\frac{d}{dx}\right)^n\left[\expe^{-x^2}\right]. -\end{equation} - -\subsection*{Generating functions} -\begin{equation} -\label{GenHermite1} -\exp\left(2xt-t^2\right)=\sum_{n=0}^{\infty}\frac{H_n(x)}{n!}t^n. -\end{equation} - -\begin{equation} -\label{GenHermite2} -\left\{\begin{array}{l} -\displaystyle\expe^t\cos(2x\sqrt{t})=\sum_{n=0}^{\infty} -\frac{(-1)^n}{(2n)!}H_{2n}(x)t^n\\[5mm] -\displaystyle\frac{\expe^t}{\sqrt{t}}\sin(2x\sqrt{t})=\sum_{n=0}^{\infty} -\frac{(-1)^n}{(2n+1)!}H_{2n+1}(x)t^n. -\end{array}\right. -\end{equation} - -\begin{equation} -\label{GenHermite3} -\left\{\begin{array}{l} -\displaystyle \expe^{-t^2}\cosh(2xt)=\sum_{n=0}^{\infty} -\frac{H_{2n}(x)}{(2n)!}t^{2n}\\[5mm] -\displaystyle\expe^{-t^2}\sinh(2xt)=\sum_{n=0}^{\infty} -\frac{H_{2n+1}(x)}{(2n+1)!}t^{2n+1}. -\end{array}\right. -\end{equation} - -\begin{equation} -\label{GenHermite4} -\left\{\begin{array}{l} -\displaystyle (1+t^2)^{-\gamma}\,\hyp{1}{1}{\gamma}{\frac{1}{2}}{\frac{x^2t^2}{1+t^2}}= -\sum_{n=0}^{\infty}\frac{(\gamma)_n}{(2n)!}H_{2n}(x)t^{2n}\\[5mm] -\displaystyle\frac{xt}{\sqrt{1+t^2}}\, -\hyp{1}{1}{\gamma+\frac{1}{2}}{\frac{3}{2}}{\frac{x^2t^2}{1+t^2}} -=\sum_{n=0}^{\infty}\frac{(\gamma+\frac{1}{2})_n}{(2n+1)!}H_{2n+1}(x)t^{2n+1} -\end{array}\right. -\end{equation} -with $\gamma$ arbitrary. - -\begin{equation} -\label{GenHermite5} -\frac{1+2xt+4t^2}{(1+4t^2)^{\frac{3}{2}}}\exp\left(\frac{4x^2t^2}{1+4t^2}\right) -=\sum_{n=0}^{\infty}\frac{H_n(x)}{\lfloor n/2\rfloor\,!}t^n, -\end{equation} -where $\lfloor n/2\rfloor$ denotes the largest integer smaller than or equal to $n/2$. - -\subsection*{Limit relations} - -\subsubsection*{Meixner-Pollaczek $\rightarrow$ Hermite} -If we take $x\rightarrow (\sin\phi)^{-1}(x\sqrt{\lambda}-\lambda\cos\phi)$ -in the definition (\ref{DefMP}) of the Meixner-Pollaczek polynomials and -then let $\lambda\rightarrow\infty$ we obtain the Hermite polynomials: -$$\lim_{\lambda\rightarrow\infty} -\lambda^{-\frac{1}{2}n}P_n^{(\lambda)} -((\sin\phi)^{-1}(x\sqrt{\lambda}-\lambda\cos\phi);\phi)=\frac{H_n(x)}{n!}.$$ - -\subsubsection*{Jacobi $\rightarrow$ Hermite} -The Hermite polynomials follow from the Jacobi polynomials given by (\ref{DefJacobi}) by -taking $\beta=\alpha$ and letting $\alpha\rightarrow\infty$ in the following way: -$$\lim_{\alpha\rightarrow\infty} -\alpha^{-\frac{1}{2}n}P_n^{(\alpha,\alpha)}(\alpha^{-\frac{1}{2}}x)=\frac{H_n(x)}{2^nn!}.$$ - -\subsubsection*{Gegenbauer / Ultraspherical $\rightarrow$ Hermite} -The Hermite polynomials follow from the Gegenbauer (or ultraspherical) polynomials given by -(\ref{DefGegenbauer}) by taking $\lambda=\alpha+\frac{1}{2}$ and letting -$\alpha\rightarrow\infty$ in the following way: -$$\lim_{\alpha\rightarrow\infty} -\alpha^{-\frac{1}{2}n}C_n^{(\alpha+\frac{1}{2})}(\alpha^{-\frac{1}{2}}x)=\frac{H_n(x)}{n!}.$$ - -\subsubsection*{Krawtchouk $\rightarrow$ Hermite} -The Hermite polynomials follow from the Krawtchouk polynomials given by (\ref{DefKrawtchouk}) -by setting $x\rightarrow pN+x\sqrt{2p(1-p)N}$ and then letting $N\rightarrow\infty$: -$$\lim_{N\rightarrow\infty} -\sqrt{\binom{N}{n}}K_n(pN+x\sqrt{2p(1-p)N};p,N) -=\frac{\displaystyle (-1)^nH_n(x)}{\displaystyle\sqrt{2^nn!\left(\frac{p}{1-p}\right)^n}}.$$ - -\subsubsection*{Laguerre $\rightarrow$ Hermite} -The Hermite polynomials can be obtained from the Laguerre polynomials given by -(\ref{DefLaguerre}) by taking the limit $\alpha\rightarrow\infty$ in the following way: -$$\lim_{\alpha\rightarrow\infty} -\left(\frac{2}{\alpha}\right)^{\frac{1}{2}n} -L_n^{(\alpha)}((2\alpha)^{\frac{1}{2}}x+\alpha)=\frac{(-1)^n}{n!}H_n(x).$$ - -\subsubsection*{Charlier $\rightarrow$ Hermite} -If we set $x\rightarrow (2a)^{1/2}x+a$ in the definition (\ref{DefCharlier}) -of the Charlier polynomials and let $a\rightarrow\infty$ we find the Hermite -polynomials. In fact we have -$$\lim_{a\rightarrow\infty} -(2a)^{\frac{1}{2}n}C_n((2a)^{\frac{1}{2}}x+a;a)=(-1)^nH_n(x).$$ - -\subsection*{Remarks} -The Hermite polynomials can also be written as: -$$\frac{H_n(x)}{n!}=\sum_{k=0}^{\lfloor n/2\rfloor} -\frac{(-1)^k(2x)^{n-2k}}{k!\,(n-2k)!},$$ -where $\lfloor n/2\rfloor$ denotes the largest integer smaller than or equal to $n/2$. - -\noindent -The Hermite polynomials and the Laguerre polynomials given by (\ref{DefLaguerre}) are also -connected by the following quadratic transformations: -$$H_{2n}(x)=(-1)^nn!\,2^{2n}L_n^{(-\frac{1}{2})}(x^2)$$ -and -$$H_{2n+1}(x)=(-1)^nn!\,2^{2n+1}xL_n^{(\frac{1}{2})}(x^2).$$ - -\subsection*{References} -\cite{Abram}, \cite{NAlSalam66}, \cite{AlSalam90}, \cite{AlSalamChihara72}, -\cite{AlSalamChihara76}, \cite{AndrewsAskey85}, \cite{AndrewsAskeyRoy}, \cite{Area+I}, -\cite{Askey68}, \cite{Askey75}, \cite{Askey89I}, \cite{AskeyGasper76}, \cite{AskeyWilson85}, -\cite{Azor}, \cite{Berg}, \cite{BilodeauII}, \cite{Brafman51}, \cite{Brafman57II}, -\cite{Brenke}, \cite{CarlitzSrivastava}, \cite{ChenSrivastava}, \cite{Chihara78}, -\cite{Cohen}, \cite{Danese}, \cite{DetteStudden92}, \cite{DetteStudden95}, \cite{Doha2004I}, -\cite{Erdelyi+}, \cite{Faldey}, \cite{Gawronski87}, \cite{Gawronski93}, \cite{Gillis+}, -\cite{Godoy+}, \cite{Grad}, \cite{Ismail74}, \cite{Ismail2005II}, \cite{IsmailStanton97}, -\cite{Koekoek2000}, \cite{Koorn88}, \cite{Krasikov2004}, \cite{Kwon+}, \cite{LabelleYehIII}, -\cite{Lesky95II}, \cite{Lesky96}, \cite{LewanowiczI}, \cite{LopezTemme99}, \cite{Mathai}, -\cite{Meixner}, \cite{Nikiforov+}, \cite{NikiforovUvarov}, \cite{Olver}, \cite{Pittaluga}, -\cite{Rainville}, \cite{SainteViennot}, \cite{Srivastava71}, \cite{SrivastavaMathur}, -\cite{Szego75}, \cite{Temme}, \cite{TemmeLopez2000}, \cite{Trickovic}, \cite{Viennot}, -\cite{Weisner59}, \cite{Wyman}. - -\end{document} diff --git a/KLSadd_insertion/chap14.tex b/KLSadd_insertion/chap14.tex deleted file mode 100644 index 0b5f396..0000000 --- a/KLSadd_insertion/chap14.tex +++ /dev/null @@ -1,5734 +0,0 @@ -\documentclass[envcountchap,graybox]{svmono} - -\addtolength{\textwidth}{1mm} - -\usepackage{amsmath,amssymb} - -\usepackage{amsfonts} -%\usepackage{breqn} -\usepackage{DLMFmath} -\usepackage{DRMFfcns} - -\usepackage{mathptmx} -\usepackage{helvet} -\usepackage{courier} - -\usepackage{makeidx} -\usepackage{graphicx} - -\usepackage{multicol} -\usepackage[bottom]{footmisc} - -\makeindex - -\def\bibname{Bibliography} -\def\refname{Bibliography} - -\def\theequation{\thesection.\arabic{equation}} - -\smartqed - -\let\corollary=\undefined -\spnewtheorem{corollary}[theorem]{Corollary}{\bfseries}{\itshape} - -\newcounter{rom} - -\newcommand{\hyp}[5]{\mbox{}_{#1}{F}_{#2} -\left(\genfrac{}{}{0pt}{}{#3}{#4}\,;\,#5\right)} - -\newcommand{\qhyp}[5]{\mbox{}_{#1}{\phi}_{#2} -\left(\genfrac{}{}{0pt}{}{#3}{#4}\,;\,q,\,#5\right)} - -\newcommand{\qhypK}[5]{\,\mbox{}_{#1}\phi_{#2}\!\left( - \genfrac{}{}{0pt}{}{#3}{#4};#5\right)} - -\newcommand{\mathindent}{\hspace{7.5mm}} - -\newcommand{\e}{\textrm{e}} - -\renewcommand{\E}{\textrm{E}} - -\renewcommand{\textfraction}{-1} - -\renewcommand{\Gamma}{\varGamma} - -\renewcommand{\leftlegendglue}{\hfil} - -\settowidth{\tocchpnum}{14\enspace} -\settowidth{\tocsecnum}{14.30\enspace} -\settowidth{\tocsubsecnum}{14.12.1\enspace} - -\makeatletter -\def\cleartoversopage{\clearpage\if@twoside\ifodd\c@page - \hbox{}\newpage\if@twocolumn\hbox{}\newpage\fi - \else\fi\fi} - -\newcommand{\clearemptyversopage}{ - \clearpage{\pagestyle{empty}\cleartoversopage}} -\makeatother - -\oddsidemargin -1.5cm -\topmargin -2.0cm -\textwidth 16.3cm -\textheight 25cm - -\begin{document} - -\author{Roelof Koekoek\\[2.5mm]Peter A. Lesky\\[2.5mm]Ren\'e F. Swarttouw} -\title{Hypergeometric orthogonal polynomials and their\\$q$-analogues} -\subtitle{-- Monograph --} -\maketitle - -\frontmatter - -\large - -\pagenumbering{roman} -\addtocounter{chapter}{13} - -\chapter{Basic hypergeometric orthogonal polynomials} -\label{BasicHyperOrtPol} - - -In this chapter we deal with all families of basic hypergeometric orthogonal polynomials -appearing in the $q$-analogue of the Askey scheme on the pages~\pageref{qscheme1} and -\pageref{qscheme2}. For each family of orthogonal polynomials we state the most important -properties such as a representation as a basic hypergeometric function, orthogonality -relation(s), the three-term recurrence relation, the second-order $q$-difference equation, -the forward shift (or degree lowering) and backward shift (or degree raising) operator, a -Rodrigues-type formula and some generating functions. Throughout this chapter we assume -that $01$ and $b,c,d$ are real or one is real and the other two are complex conjugates,\\ -$\max(|b|,|c|,|d|)<1$ and the pairwise products of $a,b,c$ and $d$ have -absolute value less than $1$, then we have another orthogonality relation -given by: -\begin{eqnarray} -\label{OrtAskeyWilsonII} -& &\frac{1}{2\pi}\int_{-1}^1\frac{w(x)}{\sqrt{1-x^2}}p_m(x;a,b,c,d|q)p_n(x;a,b,c,d|q)\,dx\nonumber\\ -& &{}\mathindent{}+\sum_{\begin{array}{c}\scriptstyle k\\ \scriptstyle 11$ and $b$ and $c$ are real or complex conjugates, -$\max(|b|,|c|)<1$ and the pairwise products of $a,b$ and $c$ have -absolute value less than $1$, then we have another orthogonality relation -given by: -\begin{eqnarray} -\label{OrtContDualqHahnII} -& &\frac{1}{2\pi}\int_{-1}^1\frac{w(x)}{\sqrt{1-x^2}}p_m(x;a,b,c|q)p_n(x;a,b,c|q)\,dx\nonumber\\ -& &{}\mathindent{}+\sum_{\begin{array}{c}\scriptstyle k\\ \scriptstyle 10$) in the definition (\ref{DefBigqJacobi}) -of the big $q$-Jacobi polynomials and take the limit $c\rightarrow -\infty$ we -obtain the $q$-Meixner polynomials given by (\ref{DefqMeixner}): -\begin{equation} -\lim_{c\rightarrow -\infty}P_n(q^{-x};a,-a^{-1}cd^{-1},c;q)=M_n(q^{-x};a,d;q). -\end{equation} - -\subsubsection*{Big $q$-Jacobi $\rightarrow$ Jacobi} -If we set $c=0$, $a=q^{\alpha}$ and $b=q^{\beta}$ in the definition (\ref{DefBigqJacobi}) -of the big $q$-Jacobi polynomials and let $q\rightarrow 1$ we find the Jacobi -polynomials given by (\ref{DefJacobi}): -\begin{equation} -\lim_{q\rightarrow 1}P_n(x;q^{\alpha},q^{\beta},0;q)=\frac{P_n^{(\alpha,\beta)}(2x-1)}{P_n^{(\alpha,\beta)}(1)}. -\end{equation} -If we take $c=-q^{\gamma}$ for arbitrary real $\gamma$ instead of $c=0$ we -find -\begin{equation} -\lim_{q\rightarrow 1}P_n(x;q^{\alpha},q^{\beta},-q^{\gamma};q)=\frac{P_n^{(\alpha,\beta)}(x)}{P_n^{(\alpha,\beta)}(1)}. -\end{equation} - -\subsection*{Remarks} -The big $q$-Jacobi polynomials with $c=0$ and the little -$q$-Jacobi polynomials given by (\ref{DefLittleqJacobi}) are related -in the following way: -$$P_n(x;a,b,0;q)=\frac{(bq;q)_n}{(aq;q)_n}(-1)^na^nq^{n+\binom{n}{2}}p_n(a^{-1}q^{-1}x;b,a|q).$$ - -\noindent -Sometimes the big $q$-Jacobi polynomials are defined in terms of four -parameters instead of three. In fact the polynomials given by the definition -$$P_n(x;a,b,c,d;q)=\qhyp{3}{2}{q^{-n},abq^{n+1},ac^{-1}qx}{aq,-ac^{-1}dq}{q}$$ -are orthogonal on the interval $[-d,c]$ with respect to the weight function -$$\frac{(c^{-1}qx,-d^{-1}qx;q)_{\infty}}{(ac^{-1}qx,-bd^{-1}qx;q)_{\infty}}d_qx.$$ -These polynomials are not really different from those given by -(\ref{DefBigqJacobi}) since we have -$$P_n(x;a,b,c,d;q)=P_n(ac^{-1}qx;a,b,-ac^{-1}d;q)$$ -and -$$P_n(x;a,b,c;q)=P_n(x;a,b,aq,-cq;q).$$ - -\subsection*{References} -\cite{NAlSalam89}, \cite{AlSalam90}, \cite{AndrewsAskey85}, \cite{AtakKlimyk2004}, -\cite{AtakRahmanSuslov}, \cite{DattaGriffin}, \cite{FloreaniniVinetII}, -\cite{GasperRahman90}, \cite{GrunbaumHaine96}, \cite{Gupta92}, \cite{Hahn}, -\cite{Ismail86I}, \cite{IsmailStanton97}, \cite{IsmailWilson}, \cite{KalninsMiller88}, -\cite{KoelinkE}, \cite{Koorn90II}, \cite{Koorn93}, \cite{Koorn2007}, \cite{Miller89}, -\cite{Nikiforov+}, \cite{NoumiMimachi90II}, \cite{NoumiMimachi90III}, -\cite{NoumiMimachi91}, \cite{Spiridonov97}, \cite{SrivastavaJain90}. - - -\section*{Special case} - -\subsection{Big $q$-Legendre} -\index{Big q-Legendre polynomials@Big $q$-Legendre polynomials} -\index{q-Legendre polynomials@$q$-Legendre polynomials!Big} -\par - -\subsection*{Basic hypergeometric representation} The big $q$-Legendre polynomials are big $q$-Jacobi polynomials -with $a=b=1$: -\begin{equation} -\label{DefBigqLegendre} -P_n(x;c;q)=\qhyp{3}{2}{q^{-n},q^{n+1},x}{q,cq}{q}. -\end{equation} - -\subsection*{Orthogonality relation} -\begin{eqnarray} -\label{OrtBigqLegendre} -& &\int_{cq}^{q}P_m(x;c;q)P_n(x;c;q)\,d_qx\nonumber\\ -& &{}=q(1-c)\frac{(1-q)}{(1-q^{2n+1})} -\frac{(c^{-1}q;q)_n}{(cq;q)_n}(-cq^2)^nq^{\binom{n}{2}}\,\delta_{mn},\quad c<0. -\end{eqnarray} - -\subsection*{Recurrence relation} -\begin{eqnarray} -\label{RecBigqLegendre} -(x-1)P_n(x;c;q)&=&A_nP_{n+1}(x;c;q)-\left(A_n+C_n\right)P_n(x;c;q)\nonumber\\ -& &{}\mathindent{}+C_nP_{n-1}(x;c;q), -\end{eqnarray} -where -$$\left\{\begin{array}{l} -\displaystyle A_n=\frac{(1-q^{n+1})(1-cq^{n+1})}{(1+q^{n+1})(1-q^{2n+1})}\\ -\\ -\displaystyle C_n=-cq^{n+1}\frac{(1-q^n)(1-c^{-1}q^n)}{(1+q^n)(1-q^{2n+1})}. -\end{array}\right.$$ - -\subsection*{Normalized recurrence relation} -\begin{equation} -\label{NormRecBigqLegendre} -xp_n(x)=p_{n+1}(x)+\left[1-(A_n+C_n)\right]p_n(x)+A_{n-1}C_np_{n-1}(x), -\end{equation} -where -$$P_n(x;c;q)=\frac{(q^{n+1};q)_n}{(q,cq;q)_n}p_n(x).$$ - -\subsection*{$q$-Difference equation} -\begin{eqnarray} -\label{dvBigqLegendre} -& &q^{-n}(1-q^n)(1-q^{n+1})x^2y(x)\nonumber\\ -& &{}=B(x)y(qx)-\left[B(x)+D(x)\right]y(x)+D(x)y(q^{-1}x), -\end{eqnarray} -where -$$y(x)=P_n(x;c;q)$$ -and -$$\left\{\begin{array}{l}\displaystyle B(x)=q(x-1)(x-c)\\ -\\ -\displaystyle D(x)=(x-q)(x-cq).\end{array}\right.$$ - -\subsection*{Rodrigues-type formula} -\begin{eqnarray} -\label{RodBigqLegendre} -P_n(x;c;q)&=&\frac{c^nq^{n(n+1)}(1-q)^n}{(q,cq;q)_n} -\left(\mathcal{D}_q\right)^n\left[(q^{-n}x,c^{-1}q^{-n}x;q)_n\right]\\ -&=&\frac{(1-q)^n}{(q,cq;q)_n}\left(\mathcal{D}_q\right)^n -\left[(qx^{-1},cqx^{-1};q)_nx^{2n}\right].\nonumber -\end{eqnarray} - -\subsection*{Generating functions} -\begin{equation} -\label{GenBigqLegendre1} -\qhyp{2}{1}{qx^{-1},0}{q}{xt}\,\qhyp{1}{1}{c^{-1}x}{q}{cqt} -=\sum_{n=0}^{\infty}\frac{(cq;q)_n}{(q,q;q)_n}P_n(x;c;q)t^n. -\end{equation} - -\begin{equation} -\label{GenBigqLegendre2} -\qhyp{2}{1}{cqx^{-1},0}{cq}{xt}\,\qhyp{1}{1}{c^{-1}x}{c^{-1}q}{qt} -=\sum_{n=0}^{\infty}\frac{P_n(x;c;q)}{(c^{-1}q;q)_n}t^n. -\end{equation} - -\subsection*{Limit relations} - -\subsubsection*{Big $q$-Legendre $\rightarrow$ Legendre / Spherical} -If we set $c=0$ in the definition (\ref{DefBigqLegendre}) of the big -$q$-Legendre polynomials and let $q\rightarrow 1$ we simply obtain the Legendre -(or spherical) polynomials given by (\ref{DefLegendre}): -\begin{equation} -\lim_{q\rightarrow 1}P_n(x;0;q)=P_n(2x-1). -\end{equation} -If we take $c=-q^{\gamma}$ for arbitrary real $\gamma$ instead of $c=0$ we -find -\begin{equation} -\lim_{q\rightarrow 1}P_n(x;-q^{\gamma};q)=P_n(x). -\end{equation} - -\subsection*{References} -\cite{Koelink95I}, \cite{Koorn90II}. - - -\section{$q$-Hahn}\index{q-Hahn polynomials@$q$-Hahn polynomials} -\par\setcounter{equation}{0} - -\subsection*{Basic hypergeometric representation} -\begin{equation} -\label{DefqHahn} -Q_n(q^{-x};\alpha,\beta,N|q)=\qhyp{3}{2}{q^{-n},\alpha\beta q^{n+1},q^{-x}} -{\alpha q,q^{-N}}{q},\quad n=0,1,2,\ldots,N. -\end{equation} - -\subsection*{Orthogonality relation} -For $0<\alpha q<1$ and $0<\beta q<1$, or for $\alpha>q^{-N}$ and $\beta>q^{-N}$, we have -\begin{eqnarray} -\label{OrtqHahn} -& &\sum_{x=0}^N\frac{(\alpha q,q^{-N};q)_x}{(q,\beta^{-1}q^{-N};q)_x}(\alpha\beta q)^{-x} -Q_m(q^{-x};\alpha,\beta,N|q)Q_n(q^{-x};\alpha,\beta,N|q)\nonumber\\ -& &{}=\frac{(\alpha\beta q^2;q)_N}{(\beta q;q)_N(\alpha q)^N} -\frac{(q,\alpha\beta q^{N+2},\beta q;q)_n}{(\alpha q,\alpha\beta q,q^{-N};q)_n}\, -\frac{(1-\alpha\beta q)(-\alpha q)^n}{(1-\alpha\beta q^{2n+1})} -q^{\binom{n}{2}-Nn}\,\delta_{mn}. -\end{eqnarray} - -\newpage - -\subsection*{Recurrence relation} -\begin{eqnarray} -\label{RecqHahn} --\left(1-q^{-x}\right)Q_n(q^{-x})&=&A_nQ_{n+1}(q^{-x})-\left(A_n+C_n\right)Q_n(q^{-x})\nonumber\\ -& &{}\mathindent{}+C_nQ_{n-1}(q^{-x}), -\end{eqnarray} -where -$$Q_n(q^{-x}):=Q_n(q^{-x};\alpha,\beta,N|q)$$ -and -$$\left\{\begin{array}{l} -\displaystyle A_n=\frac{(1-q^{n-N})(1-\alpha q^{n+1})(1-\alpha\beta q^{n+1})}{(1-\alpha\beta q^{2n+1})(1-\alpha\beta q^{2n+2})}\\ -\\ -\displaystyle C_n=-\frac{\alpha q^{n-N}(1-q^n)(1-\alpha\beta q^{n+N+1})(1-\beta q^n)}{(1-\alpha\beta q^{2n})(1-\alpha\beta q^{2n+1})}. -\end{array}\right.$$ - -\subsection*{Normalized recurrence relation} -\begin{equation} -\label{NormRecqHahn} -xp_n(x)=p_{n+1}(x)+\left[1-(A_n+C_n)\right]p_n(x)+A_{n-1}C_np_{n-1}(x), -\end{equation} -where -$$Q_n(q^{-x};\alpha,\beta,N|q)= -\frac{(\alpha\beta q^{n+1};q)_n}{(\alpha q,q^{-N};q)_n}p_n(q^{-x}).$$ - -\subsection*{$q$-Difference equation} -\begin{eqnarray} -\label{dvqHahn} -& &q^{-n}(1-q^n)(1-\alpha\beta q^{n+1})y(x)\nonumber\\ -& &{}=B(x)y(x+1)-\left[B(x)+D(x)\right]y(x)+D(x)y(x-1), -\end{eqnarray} -where -$$y(x)=Q_n(q^{-x};\alpha,\beta,N|q)$$ -and -$$\left\{\begin{array}{l}\displaystyle B(x)=(1-q^{x-N})(1-\alpha q^{x+1})\\ -\\ -\displaystyle D(x)=\alpha q(1-q^x)(\beta-q^{x-N-1}).\end{array}\right.$$ - -\newpage - -\subsection*{Forward shift operator} -\begin{eqnarray} -\label{shift1qHahnI} -& &Q_n(q^{-x-1};\alpha,\beta,N|q)-Q_n(q^{-x};\alpha,\beta,N|q)\nonumber\\ -& &{}=\frac{q^{-n-x}(1-q^n)(1-\alpha\beta q^{n+1})}{(1-\alpha q)(1-q^{-N})} -Q_{n-1}(q^{-x};\alpha q,\beta q,N-1|q) -\end{eqnarray} -or equivalently -\begin{eqnarray} -\label{shift1qHahnII} -\frac{\Delta Q_n(q^{-x};\alpha,\beta,N|q)}{\Delta q^{-x}} -&=&\frac{q^{-n+1}(1-q^n)(1-\alpha\beta q^{n+1})}{(1-q)(1-\alpha q)(1-q^{-N})}\nonumber\\ -& &{}\mathindent{}\times Q_{n-1}(q^{-x};\alpha q,\beta q,N-1|q). -\end{eqnarray} - -\subsection*{Backward shift operator} -\begin{eqnarray} -\label{shift2qHahnI} -& &(1-\alpha q^x)(1-q^{x-N-1})Q_n(q^{-x};\alpha,\beta,N|q)\nonumber\\ -& &{}\mathindent{}-\alpha(1-q^x)(\beta-q^{x-N-1})Q_n(q^{-x+1};\alpha,\beta,N|q)\nonumber\\ -& &{}=q^x(1-\alpha)(1-q^{-N-1})Q_{n+1}(q^{-x};\alpha q^{-1},\beta q^{-1},N+1|q) -\end{eqnarray} -or equivalently -\begin{eqnarray} -\label{shift2qHahnII} -& &\frac{\nabla\left[w(x;\alpha,\beta,N|q)Q_n(q^{-x};\alpha,\beta,N|q)\right]}{\nabla q^{-x}}\nonumber\\ -& &{}=\frac{1}{1-q}w(x;\alpha q^{-1},\beta q^{-1},N+1|q)Q_{n+1}(q^{-x};\alpha q^{-1},\beta q^{-1},N+1|q), -\end{eqnarray} -where -$$w(x;\alpha,\beta,N|q)=\frac{(\alpha q,q^{-N};q)_x}{(q,\beta^{-1}q^{-N};q)_x}(\alpha\beta)^{-x}.$$ - -\subsection*{Rodrigues-type formula} -\begin{eqnarray} -\label{RodqHahn} -& &w(x;\alpha,\beta,N|q)Q_n(q^{-x};\alpha,\beta,N|q)\nonumber\\ -& &{}=(1-q)^n\left(\nabla_q\right)^n\left[w(x;\alpha q^n,\beta q^n,N-n|q)\right], -\end{eqnarray} -where -$$\nabla_q:=\frac{\nabla}{\nabla q^{-x}}.$$ - -\subsection*{Generating functions} For $x=0,1,2,\ldots,N$ we have -\begin{eqnarray} -\label{GenqHahn1} -& &\qhyp{1}{1}{q^{-x}}{\alpha q}{\alpha qt}\,\qhyp{2}{1}{q^{x-N},0}{\beta q}{q^{-x}t}\nonumber\\ -& &{}=\sum_{n=0}^N\frac{(q^{-N};q)_n}{(\beta q,q;q)_n}Q_n(q^{-x};\alpha,\beta,N|q)t^n. -\end{eqnarray} - -\begin{eqnarray} -\label{GenqHahn2} -& &\qhyp{2}{1}{q^{-x},\beta q^{N+1-x}}{0}{-\alpha q^{x-N+1}t}\, -\qhyp{2}{0}{q^{x-N},\alpha q^{x+1}}{-}{-q^{-x}t}\nonumber\\ -& &{}=\sum_{n=0}^N\frac{(\alpha q,q^{-N};q)_n}{(q;q)_n}q^{-\binom{n}{2}}Q_n(q^{-x};\alpha,\beta,N|q)t^n. -\end{eqnarray} - -\subsection*{Limit relations} - -\subsubsection*{$q$-Racah $\rightarrow$ $q$-Hahn} -The $q$-Hahn polynomials follow from the $q$-Racah polynomials by the substitution -$\delta=0$ and $\gamma q=q^{-N}$ in the definition (\ref{DefqRacah}) of the -$q$-Racah polynomials: -$$R_n(\mu(x);\alpha,\beta,q^{-N-1},0|q)=Q_n(q^{-x};\alpha,\beta,N|q).$$ -Another way to obtain the $q$-Hahn polynomials from the $q$-Racah -polynomials is by setting $\gamma=0$ and $\delta=\beta^{-1}q^{-N-1}$ in the definition -(\ref{DefqRacah}): -$$R_n(\mu(x);\alpha,\beta,0,\beta^{-1}q^{-N-1}|q)=Q_n(q^{-x};\alpha,\beta,N|q).$$ -And if we take $\alpha q=q^{-N}$, $\beta\rightarrow\beta\gamma q^{N+1}$ and $\delta=0$ in the -definition (\ref{DefqRacah}) of the $q$-Racah polynomials we find the -$q$-Hahn polynomials given by (\ref{DefqHahn}) in the following way: -$$R_n(\mu(x);q^{-N-1},\beta\gamma q^{N+1},\gamma,0|q)=Q_n(q^{-x};\gamma,\beta,N|q).$$ -Note that $\mu(x)=q^{-x}$ in each case. - -\subsubsection*{$q$-Hahn $\rightarrow$ Little $q$-Jacobi} -If we set $x\rightarrow N-x$ in the definition (\ref{DefqHahn}) of the $q$-Hahn -polynomials and take the limit $N\rightarrow\infty$ we find the little $q$-Jacobi polynomials: -\begin{equation} -\lim _{N\rightarrow\infty} Q_n(q^{x-N};\alpha,\beta,N|q)=p_n(q^x;\alpha,\beta|q), -\end{equation} -where $p_n(q^x;\alpha,\beta|q)$ is given by (\ref{DefLittleqJacobi}). - -\subsubsection*{$q$-Hahn $\rightarrow$ $q$-Meixner} -The $q$-Meixner polynomials given by (\ref{DefqMeixner}) can be obtained from the $q$-Hahn polynomials -by setting $\alpha=b$ and $\beta=-b^{-1}c^{-1}q^{-N-1}$ in the definition (\ref{DefqHahn}) of the -$q$-Hahn polynomials and letting $N\rightarrow\infty$: -\begin{equation} -\lim_{N\rightarrow\infty} -Q_n(q^{-x};b,-b^{-1}c^{-1}q^{-N-1},N|q)=M_n(q^{-x};b,c;q). -\end{equation} - -\subsubsection*{$q$-Hahn $\rightarrow$ Quantum $q$-Krawtchouk} -The quantum $q$-Krawtchouk polynomials given by (\ref{DefQuantumqKrawtchouk}) -simply follow from the $q$-Hahn polynomials by setting $\beta=p$ in the definition (\ref{DefqHahn}) of -the $q$-Hahn polynomials and taking the limit $\alpha\rightarrow\infty$: -\begin{equation} -\lim_{\alpha\rightarrow\infty}Q_n(q^{-x};\alpha,p,N|q)=K_n^{qtm}(q^{-x};p,N;q). -\end{equation} - -\subsubsection*{$q$-Hahn $\rightarrow$ $q$-Krawtchouk} -If we set $\beta=-\alpha^{-1}q^{-1}p$ in the definition (\ref{DefqHahn}) of the $q$-Hahn polynomials -and then let $\alpha\rightarrow 0$ we obtain the $q$-Krawtchouk polynomials given by -(\ref{DefqKrawtchouk}): -\begin{equation} -\lim_{\alpha\rightarrow 0} -Q_n(q^{-x};\alpha,-\alpha^{-1}q^{-1}p,N|q)=K_n(q^{-x};p,N;q). -\end{equation} - -\subsubsection*{$q$-Hahn $\rightarrow$ Affine $q$-Krawtchouk} -The affine $q$-Krawtchouk polynomials given by (\ref{DefAffqKrawtchouk}) -can be obtained from the $q$-Hahn polynomials by the substitution $\alpha=p$ and -$\beta=0$ in (\ref{DefqHahn}): -\begin{equation} -Q_n(q^{-x};p,0,N|q)=K_n^{Aff}(q^{-x};p,N;q). -\end{equation} - -\subsubsection*{$q$-Hahn $\rightarrow$ Hahn} -The Hahn polynomials given by (\ref{DefHahn}) simply follow from the $q$-Hahn -polynomials given by (\ref{DefqHahn}), after setting $\alpha\rightarrow q^{\alpha}$ -and $\beta\rightarrow q^{\beta}$, in the following way: -\begin{equation} -\lim_{q\rightarrow 1}Q_n(q^{-x};q^{\alpha},q^{\beta},N|q)=Q_n(x;\alpha,\beta,N). -\end{equation} - -\subsection*{Remark} -The $q$-Hahn polynomials given by (\ref{DefqHahn}) and the dual $q$-Hahn -polynomials given by (\ref{DefDualqHahn}) are related in the following way: -$$Q_n(q^{-x};\alpha,\beta,N|q)=R_x(\mu(n);\alpha,\beta,N|q),$$ -with -$$\mu(n)=q^{-n}+\alpha\beta q^{n+1}$$ -or -$$R_n(\mu(x);\gamma,\delta,N|q)=Q_x(q^{-n};\gamma,\delta,N|q),$$ -where -$$\mu(x)=q^{-x}+\gamma\delta q^{x+1}.$$ - -\subsection*{References} -\cite{AlSalam90}, \cite{AlvarezRonveaux}, \cite{AndrewsAskey85}, -\cite{AskeyWilson79}, \cite{AskeyWilson85}, \cite{AtakRahmanSuslov}, -\cite{Dunkl78II}, \cite{Fischer95}, \cite{GasperRahman84}, -\cite{GasperRahman90}, \cite{Hahn}, \cite{KalninsMiller88}, -\cite{Koelink96I}, \cite{KoelinkKoorn}, \cite{Koorn89III}, \cite{Koorn90II}, -\cite{Nikiforov+}, \cite{NoumiMimachi90II}, \cite{Rahman82}, -\cite{Stanton80II}, \cite{Stanton84}, \cite{Stanton90}. - - -\section{Dual $q$-Hahn}\index{Dual q-Hahn polynomials@Dual $q$-Hahn polynomials} -\index{q-Hahn polynomials@$q$-Hahn polynomials!Dual} -\par\setcounter{equation}{0} - -\subsection*{Basic hypergeometric representation} -\begin{equation} -\label{DefDualqHahn} -R_n(\mu(x);\gamma,\delta,N|q)= -\qhyp{3}{2}{q^{-n},q^{-x},\gamma\delta q^{x+1}}{\gamma q,q^{-N}}{q},\quad n=0,1,2,\ldots,N, -\end{equation} -where -$$\mu(x):=q^{-x}+\gamma\delta q^{x+1}.$$ - -\newpage - -\subsection*{Orthogonality relation} -For $0<\gamma q<1$ and $0<\delta q<1$, or for $\gamma>q^{-N}$ and $\delta>q^{-N}$, we have -\begin{eqnarray} -\label{OrtDualqHahn} -& &\sum_{x=0}^N\frac{(\gamma q,\gamma\delta q,q^{-N};q)_x}{(q,\gamma\delta q^{N+2},\delta q;q)_x} -\frac{(1-\gamma\delta q^{2x+1})}{(1-\gamma\delta q)(-\gamma q)^x}q^{Nx-\binom{x}{2}}\nonumber\\ -& &{}\mathindent{}\times R_m(\mu(x);\gamma,\delta,N|q)R_n(\mu(x);\gamma,\delta,N|q)\nonumber\\ -& &{}=\frac{(\gamma\delta q^2;q)_N}{(\delta q;q)_N}(\gamma q)^{-N} -\frac{(q,\delta^{-1}q^{-N};q)_n}{(\gamma q,q^{-N};q)_n}(\gamma\delta q)^n\,\delta_{mn}. -\end{eqnarray} - -\subsection*{Recurrence relation} -\begin{eqnarray} -\label{RecDualqHahn} -& &-\left(1-q^{-x}\right)\left(1-\gamma\delta q^{x+1}\right)R_n(\mu(x))\nonumber\\ -& &{}=A_nR_{n+1}(\mu(x))-\left(A_n+C_n\right)R_n(\mu(x))+C_nR_{n-1}(\mu(x)), -\end{eqnarray} -where -$$R_n(\mu(x)):=R_n(\mu(x);\gamma,\delta,N|q)$$ -and -$$\left\{\begin{array}{l} -\displaystyle A_n=\left(1-q^{n-N}\right)\left(1-\gamma q^{n+1}\right)\\ -\\ -\displaystyle C_n=\gamma q\left(1-q^n\right)\left(\delta -q^{n-N-1}\right). -\end{array}\right.$$ - -\subsection*{Normalized recurrence relation} -\begin{eqnarray} -\label{NormRecDualqHahn} -xp_n(x)&=&p_{n+1}(x)+\left[1+\gamma\delta q-(A_n+C_n)\right]p_n(x)\nonumber\\ -& &{}\mathindent{}+\gamma q(1-q^n)(1-\gamma q^n)\nonumber\\ -& &{}\mathindent\mathindent{}\times(1-q^{n-N-1})(\delta-q^{n-N-1})p_{n-1}(x), -\end{eqnarray} -where -$$R_n(\mu(x);\gamma,\delta,N|q)=\frac{1}{(\gamma q,q^{-N};q)_n}p_n(\mu(x)).$$ - -\subsection*{$q$-Difference equation} -\begin{equation} -\label{dvDualqHahn} -q^{-n}(1-q^n)y(x)=B(x)y(x+1)-\left[B(x)+D(x)\right]y(x) -+D(x)y(x-1), -\end{equation} -where -$$y(x)=R_n(\mu(x);\gamma,\delta,N|q)$$ -and -$$\left\{\begin{array}{l}\displaystyle B(x)=\frac{(1-q^{x-N})(1-\gamma q^{x+1})(1-\gamma\delta q^{x+1})} -{(1-\gamma\delta q^{2x+1})(1-\gamma\delta q^{2x+2})}\\ -\\ -\displaystyle D(x)=-\frac{\gamma q^{x-N}(1-q^x)(1-\gamma\delta q^{x+N+1})(1-\delta q^x)} -{(1-\gamma\delta q^{2x})(1-\gamma\delta q^{2x+1})}.\end{array}\right.$$ - -\subsection*{Forward shift operator} -\begin{eqnarray} -\label{shift1DualqHahnI} -& &R_n(\mu(x+1);\gamma,\delta,N|q)-R_n(\mu(x);\gamma,\delta,N|q)\nonumber\\ -& &{}=\frac{q^{-n-x}(1-q^n)(1-\gamma\delta q^{2x+2})}{(1-\gamma q)(1-q^{-N})} -R_{n-1}(\mu(x);\gamma q,\delta,N-1|q) -\end{eqnarray} -or equivalently -\begin{eqnarray} -\label{shift1DualqHahnII} -& &\frac{\Delta R_n(\mu(x);\gamma,\delta,N|q)}{\Delta\mu(x)}\nonumber\\ -& &{}=\frac{q^{-n+1}(1-q^n)}{(1-q)(1-\gamma q)(1-q^{-N})}R_{n-1}(\mu(x);\gamma q,\delta,N-1|q). -\end{eqnarray} - -\subsection*{Backward shift operator} -\begin{eqnarray} -\label{shift2DualqHahnI} -& &(1-\gamma q^x)(1-\gamma\delta q^x)(1-q^{x-N-1})R_n(\mu(x);\gamma,\delta,N|q)\nonumber\\ -& &{}\mathindent{}+\gamma q^{x-N-1}(1-q^x)(1-\gamma\delta q^{x+N+1})(1-\delta q^x)R_n(\mu(x-1);\gamma,\delta,N|q)\nonumber\\ -& &{}=q^x(1-\gamma)(1-q^{-N-1})(1-\gamma\delta q^{2x})R_{n+1}(\mu(x);\gamma q^{-1},\delta,N+1|q) -\end{eqnarray} -or equivalently -\begin{eqnarray} -\label{shift2DualqHahnII} -& &\frac{\nabla\left[w(x;\gamma,\delta,N|q)R_n(\mu(x);\gamma,\delta,N|q)\right]}{\nabla\mu(x)}\nonumber\\ -& &{}=\frac{1}{(1-q)(1-\gamma\delta)}w(x;\gamma q^{-1},\delta,N+1|q)\nonumber\\ -& &{}\mathindent{}\times R_{n+1}(\mu(x);\gamma q^{-1},\delta,N+1|q), -\end{eqnarray} -where -$$w(x;\gamma,\delta,N|q)=\frac{(\gamma q,\gamma\delta q,q^{-N};q)_x} -{(q,\gamma\delta q^{N+2},\delta q;q)_x}(-\gamma^{-1})^x q^{Nx-\binom{x}{2}}.$$ - -\subsection*{Rodrigues-type formula} -\begin{eqnarray} -\label{RodDualqHahn} -& &w(x;\gamma,\delta,N|q)R_n(\mu(x);\gamma,\delta,N|q)\nonumber\\ -& &{}=(1-q)^n(\gamma\delta q;q)_n -\left(\nabla_{\mu}\right)^n\left[w(x;\gamma q^n,\delta,N-n|q)\right], -\end{eqnarray} -where -$$\nabla_{\mu}:=\frac{\nabla}{\nabla\mu(x)}.$$ - -\subsection*{Generating functions} For $x=0,1,2,\ldots,N$ we have -\begin{eqnarray} -\label{GenDualqHahn1} -& &(q^{-N}t;q)_{N-x}\cdot\qhyp{2}{1}{q^{-x},\delta^{-1}q^{-x}}{\gamma q}{\gamma\delta q^{x+1}t}\nonumber\\ -& &{}=\sum_{n=0}^N\frac{(q^{-N};q)_n}{(q;q)_n}R_n(\mu(x);\gamma,\delta,N|q)t^n. -\end{eqnarray} - -\begin{eqnarray} -\label{GenDualqHahn2} -& &(\gamma\delta qt;q)_x\cdot\qhyp{2}{1}{q^{x-N},\gamma q^{x+1}}{\delta^{-1}q^{-N}}{q^{-x}t}\nonumber\\ -& &{}=\sum_{n=0}^N -\frac{(q^{-N},\gamma q;q)_n}{(\delta^{-1}q^{-N},q;q)_n}R_n(\mu(x);\gamma,\delta,N|q)t^n. -\end{eqnarray} - -\subsection*{Limit relations} - -\subsubsection*{$q$-Racah $\rightarrow$ Dual $q$-Hahn} -To obtain the dual $q$-Hahn polynomials from the $q$-Racah polynomials we have to -take $\beta=0$ and $\alpha q=q^{-N}$ in (\ref{DefqRacah}): -$$R_n(\mu(x);q^{-N-1},0,\gamma,\delta|q)=R_n(\mu(x);\gamma,\delta,N|q),$$ -with -$$\mu(x)=q^{-x}+\gamma\delta q^{x+1}.$$ -We may also take $\alpha=0$ and $\beta=\delta^{-1}q^{-N-1}$ in (\ref{DefqRacah}) to obtain -the dual $q$-Hahn polynomials from the $q$-Racah polynomials: -$$R_n(\mu(x);0,\delta^{-1}q^{-N-1},\gamma,\delta|q)=R_n(\mu(x);\gamma,\delta,N|q).$$ -And if we take $\gamma q=q^{-N}$, $\delta\rightarrow\alpha\delta q^{N+1}$ and $\beta=0$ in the -definition (\ref{DefqRacah}) of the $q$-Racah polynomials we find the dual -$q$-Hahn polynomials given by (\ref{DefDualqHahn}) in the following way: -$$R_n(\mu(x);\alpha,0,q^{-N-1},\alpha\delta q^{N+1}|q)=R_n({\tilde \mu}(x);\alpha,\delta,N|q),$$ -with -$${\tilde \mu}(x)=q^{-x}+\alpha\delta q^{x+1}.$$ - -\subsubsection*{Dual $q$-Hahn $\rightarrow$ Affine $q$-Krawtchouk} -The affine $q$-Krawtchouk polynomials given by (\ref{DefAffqKrawtchouk}) -can be obtained from the dual $q$-Hahn polynomials by the substitution $\gamma=p$ -and $\delta=0$ in (\ref{DefDualqHahn}): -\begin{equation} -R_n(\mu(x);p,0,N|q)=K_n^{Aff}(q^{-x};p,N;q). -\end{equation} -Note that $\mu(x)=q^{-x}$ in this case. - -\subsubsection*{Dual $q$-Hahn $\rightarrow$ Dual $q$-Krawtchouk} -The dual $q$-Krawtchouk polynomials given by (\ref{DefDualqKrawtchouk}) can -be obtained from the dual $q$-Hahn polynomials by setting $\delta=c\gamma^{-1}q^{-N-1}$ -in (\ref{DefDualqHahn}) and letting $\gamma\rightarrow 0$: -\begin{equation} -\lim_{\gamma\rightarrow 0} -R_n(\mu(x);\gamma,c\gamma^{-1}q^{-N-1},N|q)=K_n(\lambda(x);c,N|q). -\end{equation} - -\subsubsection*{Dual $q$-Hahn $\rightarrow$ Dual Hahn} -The dual Hahn polynomials given by (\ref{DefDualHahn}) follow from the dual $q$-Hahn -polynomials by simply taking the limit $q\rightarrow 1$ in the definition (\ref{DefDualqHahn}) -of the dual $q$-Hahn polynomials after applying the substitution -$\gamma\rightarrow q^{\gamma}$ and $\delta\rightarrow q^{\delta}$: -\begin{equation} -\lim_{q\rightarrow 1}R_n(\mu(x);q^{\gamma},q^{\delta},N|q)=R_n(\lambda(x);\gamma,\delta,N), -\end{equation} -where -$$\left\{\begin{array}{l} -\displaystyle\mu(x)=q^{-x}+q^{x+\gamma+\delta+1}\\[5mm] -\displaystyle\lambda(x)=x(x+\gamma+\delta+1). -\end{array}\right.$$ - -\subsection*{Remark} -The dual $q$-Hahn polynomials given by (\ref{DefDualqHahn}) and the -$q$-Hahn polynomials given by (\ref{DefqHahn}) are related in the following way: -$$Q_n(q^{-x};\alpha,\beta,N|q)=R_x(\mu(n);\alpha,\beta,N|q),$$ -with -$$\mu(n)=q^{-n}+\alpha\beta q^{n+1}$$ -or -$$R_n(\mu(x);\gamma,\delta,N|q)=Q_x(q^{-n};\gamma,\delta,N|q),$$ -where -$$\mu(x)=q^{-x}+\gamma\delta q^{x+1}.$$ - -\subsection*{References} -\cite{AlvarezSmirnov}, \cite{AndrewsAskey85}, \cite{AskeyWilson79}, -\cite{AskeyWilson85}, \cite{AtakRahmanSuslov}, \cite{GasperRahman90}, -\cite{KoelinkKoorn}, \cite{Nikiforov+}, \cite{Stanton84}. - - -\section{Al-Salam-Chihara}\index{Al-Salam-Chihara polynomials} -\par\setcounter{equation}{0} - -\subsection*{Basic hypergeometric representation} -\begin{eqnarray} -\label{DefAlSalamChihara} -Q_n(x;a,b|q)&=&\frac{(ab;q)_n}{a^n}\, -\qhyp{3}{2}{q^{-n},a\e^{i\theta},a\e^{-i\theta}}{ab,0}{q}\\ -&=&(a\e^{i\theta};q)_n\e^{-in\theta}\,\qhyp{2}{1}{q^{-n},b\e^{-i\theta}} -{a^{-1}q^{-n+1}\e^{-i\theta}}{a^{-1}q\e^{i\theta}}\nonumber\\ -&=&(b\e^{-i\theta};q)_n\e^{in\theta}\,\qhyp{2}{1}{q^{-n},a\e^{i\theta}} -{b^{-1}q^{-n+1}\e^{i\theta}}{b^{-1}q\e^{-i\theta}},\quad x=\cos\theta.\nonumber -\end{eqnarray} - -\subsection*{Orthogonality relation} -If $a$ and $b$ are real or complex conjugates and $\max(|a|,|b|)<1$, then we have the following orthogonality relation -\begin{equation} -\label{OrtAlSalamChiharaI} -\frac{1}{2\pi}\int_{-1}^1\frac{w(x)}{\sqrt{1-x^2}}Q_m(x;a,b|q)Q_n(x;a,b|q)\,dx -=\frac{\,\delta_{mn}}{(q^{n+1},abq^n;q)_{\infty}}, -\end{equation} -where -$$w(x):=w(x;a,b|q)=\left|\frac{(\e^{2i\theta};q)_{\infty}} -{(a\e^{i\theta},b\e^{i\theta};q)_{\infty}}\right|^2 -=\frac{h(x,1)h(x,-1)h(x,q^{\frac{1}{2}})h(x,-q^{\frac{1}{2}})}{h(x,a)h(x,b)},$$ -with -$$h(x,\alpha):=\prod_{k=0}^{\infty}\left(1-2\alpha xq^k+\alpha^2q^{2k}\right) -=\left(\alpha\e^{i\theta},\alpha\e^{-i\theta};q\right)_{\infty},\quad x=\cos\theta.$$ -If $a>1$ and $|ab|<1$, then we have another orthogonality relation given by: -\begin{eqnarray} -\label{OrtAlSalamChiharaII} -& &\frac{1}{2\pi}\int_{-1}^1\frac{w(x)}{\sqrt{1-x^2}}Q_m(x;a,b|q)Q_n(x;a,b|q)\,dx\nonumber\\ -& &{}\mathindent{}+\sum_{\begin{array}{c}\scriptstyle k\\ \scriptstyle 10. -\end{eqnarray} - -\subsection*{Recurrence relation} -\begin{eqnarray} -\label{RecqMeixner} -& &q^{2n+1}(1-q^{-x})M_n(q^{-x})\nonumber\\ -& &{}=c(1-bq^{n+1})M_{n+1}(q^{-x})\nonumber\\ -& &{}\mathindent{}-\left[c(1-bq^{n+1})+q(1-q^n)(c+q^n)\right]M_n(q^{-x})\nonumber\\ -& &{}\mathindent\mathindent{}+q(1-q^n)(c+q^n)M_{n-1}(q^{-x}), -\end{eqnarray} -where -$$M_n(q^{-x}):=M_n(q^{-x};b,c;q).$$ - -\subsection*{Normalized recurrence relation} -\begin{eqnarray} -\label{NormRecqMeixner} -xp_n(x)&=&p_{n+1}(x)+ -\left[1+q^{-2n-1}\left\{c(1-bq^{n+1})+q(1-q^n)(c+q^n)\right\}\right]p_n(x)\nonumber\\ -& &{}\mathindent{}+cq^{-4n+1}(1-q^n)(1-bq^n)(c+q^n)p_{n-1}(x), -\end{eqnarray} -where -$$M_n(q^{-x};b,c;q)=\frac{(-1)^nq^{n^2}}{(bq;q)_nc^n}p_n(q^{-x}).$$ - -\subsection*{$q$-Difference equation} -\begin{equation} -\label{dvqMeixner} --(1-q^n)y(x)=B(x)y(x+1)-\left[B(x)+D(x)\right]y(x)+D(x)y(x-1), -\end{equation} -where -$$y(x)=M_n(q^{-x};b,c;q)$$ -and -$$\left\{\begin{array}{l}\displaystyle B(x)=cq^x(1-bq^{x+1})\\ -\\ -\displaystyle D(x)=(1-q^x)(1+bcq^x).\end{array}\right.$$ - -\subsection*{Forward shift operator} -\begin{eqnarray} -\label{shift1qMeixnerI} -& &M_n(q^{-x-1};b,c;q)-M_n(q^{-x};b,c;q)\nonumber\\ -& &{}=-\frac{q^{-x}(1-q^n)}{c(1-bq)}M_{n-1}(q^{-x};bq,cq^{-1};q) -\end{eqnarray} -or equivalently -\begin{equation} -\label{shift1qMeixnerII} -\frac{\Delta M_n(q^{-x};b,c;q)}{\Delta q^{-x}} -=-\frac{q(1-q^n)}{c(1-q)(1-bq)}M_{n-1}(q^{-x};bq,cq^{-1};q). -\end{equation} - -\subsection*{Backward shift operator} -\begin{eqnarray} -\label{shift2qMeixnerI} -& &cq^x(1-bq^x)M_n(q^{-x};b,c;q)-(1-q^x)(1+bcq^x)M_n(q^{-x+1};b,c;q)\nonumber\\ -& &{}=cq^x(1-b)M_{n+1}(q^{-x};bq^{-1},cq;q) -\end{eqnarray} -or equivalently -\begin{eqnarray} -\label{shift2qMeixnerII} -& &\frac{\nabla\left[w(x;b,c;q)M_n(q^{-x};b,c;q)\right]}{\nabla q^{-x}}\nonumber\\ -& &{}=\frac{1}{1-q}w(x;bq^{-1},cq;q)M_{n+1}(q^{-x};bq^{-1},cq;q), -\end{eqnarray} -where -$$w(x;b,c;q)=\frac{(bq;q)_x}{(q,-bcq;q)_x}c^xq^{\binom{x+1}{2}}.$$ - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodqMeixner} -w(x;b,c;q)M_n(q^{-x};b,c;q)=(1-q)^n\left(\nabla_q\right)^n\left[w(x;bq^n,cq^{-n};q)\right], -\end{equation} -where -$$\nabla_q:=\frac{\nabla}{\nabla q^{-x}}.$$ - -\subsection*{Generating functions} -\begin{equation} -\label{GenqMeixner1} -\frac{1}{(t;q)_{\infty}}\,\qhyp{1}{1}{q^{-x}}{bq}{-c^{-1}qt} -=\sum_{n=0}^{\infty}\frac{M_n(q^{-x};b,c;q)}{(q;q)_n}t^n. -\end{equation} - -\begin{eqnarray} -\label{GenqMeixner2} -& &\frac{1}{(t;q)_{\infty}}\,\qhyp{1}{1}{-b^{-1}c^{-1}q^{-x}}{-c^{-1}q}{bqt}\nonumber\\ -& &{}=\sum_{n=0}^{\infty}\frac{(bq;q)_n}{(-c^{-1}q,q;q)_n}M_n(q^{-x};b,c;q)t^n. -\end{eqnarray} - -\subsection*{Limit relations} - -\subsubsection*{Big $q$-Jacobi $\rightarrow$ $q$-Meixner} -If we set $b=-a^{-1}cd^{-1}$ (with $d>0$) in the definition (\ref{DefBigqJacobi}) -of the big $q$-Jacobi polynomials and take the limit $c\rightarrow -\infty$ we -obtain the $q$-Meixner polynomials given by (\ref{DefqMeixner}): -\begin{equation} -\lim_{c\rightarrow -\infty}P_n(q^{-x};a,-a^{-1}cd^{-1},c;q)=M_n(q^{-x};a,d;q). -\end{equation} - -\subsubsection*{$q$-Hahn $\rightarrow$ $q$-Meixner} -The $q$-Meixner polynomials given by (\ref{DefqMeixner}) can be obtained from the $q$-Hahn polynomials -by setting $\alpha=b$ and $\beta=-b^{-1}c^{-1}q^{-N-1}$ in the definition (\ref{DefqHahn}) of the -$q$-Hahn polynomials and letting $N\rightarrow\infty$: -$$\lim_{N\rightarrow\infty}Q_n(q^{-x};b,-b^{-1}c^{-1}q^{-N-1},N|q)=M_n(q^{-x};b,c;q).$$ - -\subsubsection*{$q$-Meixner $\rightarrow$ $q$-Laguerre} -The $q$-Laguerre polynomials given by (\ref{DefqLaguerre}) can be obtained -from the $q$-Meixner polynomials given by (\ref{DefqMeixner}) by setting -$b=q^{\alpha}$ and $q^{-x}\rightarrow cq^{\alpha}x$ in the definition -(\ref{DefqMeixner}) of the $q$-Meixner polynomials and then taking the limit -$c\rightarrow\infty$: -\begin{equation} -\lim_{c\rightarrow\infty}M_n(cq^{\alpha}x;q^{\alpha},c;q)= -\frac{(q;q)_n}{(q^{\alpha+1};q)_n}L_n^{(\alpha)}(x;q). -\end{equation} - -\subsubsection*{$q$-Meixner $\rightarrow$ $q$-Charlier} -The $q$-Charlier polynomials given by (\ref{DefqCharlier}) can easily be -obtained from the $q$-Meixner given by (\ref{DefqMeixner}) by setting $b=0$ -in the definition (\ref{DefqMeixner}) of the $q$-Meixner polynomials: -\begin{equation} -M_n(x;0,c;q)=C_n(x;c;q). -\end{equation} - -\subsubsection*{$q$-Meixner $\rightarrow$ Al-Salam-Carlitz~II} -The Al-Salam-Carlitz~II polynomials given by (\ref{DefAlSalamCarlitzII}) -can be obtained from the $q$-Meixner polynomials given by (\ref{DefqMeixner}) -by setting $b=-ac^{-1}$ in the definition (\ref{DefqMeixner}) of the -$q$-Meixner polynomials and then taking the limit $c\rightarrow 0$: -\begin{equation} -\lim_{c\rightarrow 0}M_n(x;-ac^{-1},c;q)= -\left(-\frac{1}{a}\right)^nq^{\binom{n}{2}}V_n^{(a)}(x;q). -\end{equation} - -\subsubsection*{$q$-Meixner $\rightarrow$ Meixner} -To find the Meixner polynomials given by (\ref{DefMeixner}) from the $q$-Meixner polynomials given by -(\ref{DefqMeixner}) we set $b=q^{\beta-1}$ and $c\rightarrow (1-c)^{-1}c$ and let $q\rightarrow 1$: -\begin{equation} -\lim_{q\rightarrow 1}M_n(q^{-x};q^{\beta-1},(1-c)^{-1}c;q)=M_n(x;\beta,c). -\end{equation} - -\subsection*{Remarks} -The $q$-Meixner polynomials given by (\ref{DefqMeixner}) and the little -$q$-Jacobi polynomials given by (\ref{DefLittleqJacobi}) are related in the -following way: -$$M_n(q^{-x};b,c;q)=p_n(-c^{-1}q^n;b,b^{-1}q^{-n-x-1}|q).$$ - -\noindent -The $q$-Meixner polynomials and the quantum $q$-Krawtchouk polynomials -given by (\ref{DefQuantumqKrawtchouk}) are related in the following way: -$$K_n^{qtm}(q^{-x};p,N;q)=M_n(q^{-x};q^{-N-1},-p^{-1};q).$$ - -\subsection*{References} -\cite{AlSalam90}, \cite{AlSalamVerma82II}, \cite{AlSalamVerma88}, \cite{AlvarezRonveaux}, -\cite{AtakAtakKlimyk}, \cite{AtakRahmanSuslov}, \cite{Campigotto+}, \cite{GasperRahman90}, -\cite{Hahn}, \cite{Ismail2005I}, \cite{Nikiforov+}, \cite{Smirnov}. - - -\section{Quantum $q$-Krawtchouk} -\index{Quantum q-Krawtchouk polynomials@Quantum $q$-Krawtchouk polynomials} -\index{q-Krawtchouk polynomials@$q$-Krawtchouk polynomials!Quantum} -\par\setcounter{equation}{0} - -\subsection*{Basic hypergeometric representation} -\begin{equation} -\label{DefQuantumqKrawtchouk} -K_n^{qtm}(q^{-x};p,N;q)= -\qhyp{2}{1}{q^{-n},q^{-x}}{q^{-N}}{pq^{n+1}},\quad n=0,1,2,\ldots,N. -\end{equation} - -\subsection*{Orthogonality relation} -\begin{eqnarray} -\label{OrtQuantumqKrawtchouk} -& &\sum_{x=0}^N\frac{(pq;q)_{N-x}}{(q;q)_x(q;q)_{N-x}}(-1)^{N-x}q^{\binom{x}{2}} -K_m^{qtm}(q^{-x};p,N;q)K_n^{qtm}(q^{-x};p,N;q)\nonumber\\ -& &{}=\frac{(-1)^np^N(q;q)_{N-n}(q,pq;q)_n}{(q,q;q)_N} -q^{\binom{N+1}{2}-\binom{n+1}{2}+Nn}\,\delta_{mn},\quad p>q^{-N}. -\end{eqnarray} - -\subsection*{Recurrence relation} -\begin{eqnarray} -\label{RecQuantumqKrawtchouk} -& &-pq^{2n+1}(1-q^{-x})K_n^{qtm}(q^{-x})\nonumber\\ -& &{}=(1-q^{n-N})K_{n+1}^{qtm}(q^{-x})\nonumber\\ -& &{}\mathindent{}-\left[(1-q^{n-N})+q(1-q^n)(1-pq^n)\right]K_n^{qtm}(q^{-x})\nonumber\\ -& &{}\mathindent\mathindent{}+q(1-q^n)(1-pq^n)K_{n-1}^{qtm}(q^{-x}), -\end{eqnarray} -where -$$K_n^{qtm}(q^{-x}):=K_n^{qtm}(q^{-x};p,N;q).$$ - -\subsection*{Normalized recurrence relation} -\begin{eqnarray} -\label{NormRecQuantumqKrawtchouk} -xp_n(x)&=&p_{n+1}(x)+ -\left[1-p^{-1}q^{-2n-1}\left\{(1-q^{n-N})+q(1-q^n)(1-pq^n)\right\}\right]p_n(x)\nonumber\\ -& &{}\mathindent{}+p^{-2}q^{-4n+1}(1-q^n)(1-pq^n)(1-q^{n-N-1})p_{n-1}(x), -\end{eqnarray} -where -$$K_n^{qtm}(q^{-x};p,N;q)=\frac{p^nq^{n^2}}{(q^{-N};q)_n}p_n(q^{-x}).$$ - -\subsection*{$q$-Difference equation} -\begin{equation} -\label{dvQuantumqKrawtchouk} --p(1-q^n)y(x)=B(x)y(x+1)-\left[B(x)+D(x)\right]y(x)+D(x)y(x-1), -\end{equation} -where -$$y(x)=K_n^{qtm}(q^{-x};p,N;q)$$ -and -$$\left\{\begin{array}{l}\displaystyle B(x)=-q^x(1-q^{x-N})\\ -\\ -\displaystyle D(x)=(1-q^x)(p-q^{x-N-1}).\end{array}\right.$$ - -\subsection*{Forward shift operator} -\begin{eqnarray} -\label{shift1QuantumqKrawtchoukI} -& &K_n^{qtm}(q^{-x-1};p,N;q)-K_n^{qtm}(q^{-x};p,N;q)\nonumber\\ -& &{}=\frac{pq^{-x}(1-q^n)}{1-q^{-N}}K_{n-1}^{qtm}(q^{-x};pq,N-1;q) -\end{eqnarray} -or equivalently -\begin{equation} -\label{shift1QuantumqKrawtchoukII} -\frac{\Delta K_n^{qtm}(q^{-x};p,N;q)}{\Delta q^{-x}}= -\frac{pq(1-q^n)}{(1-q)(1-q^{-N})}K_{n-1}^{qtm}(q^{-x};pq,N-1;q). -\end{equation} - -\subsection*{Backward shift operator} -\begin{eqnarray} -\label{shift2QuantumqKrawtchoukI} -& &(1-q^{x-N-1})K_n^{qtm}(q^{-x};p,N;q)\nonumber\\ -& &{}\mathindent{}+q^{-x}(1-q^x)(p-q^{x-N-1})K_n^{qtm}(q^{-x+1};p,N;q)\nonumber\\ -& &{}=(1-q^{-N-1})K_{n+1}^{qtm}(q^{-x};pq^{-1},N+1;q) -\end{eqnarray} -or equivalently -\begin{eqnarray} -\label{shift2QuantumqKrawtchoukII} -& &\frac{\nabla\left[w(x;p,N;q)K_n^{qtm}(q^{-x};p,N;q)\right]}{\nabla q^{-x}}\nonumber\\ -& &{}=\frac{1}{1-q}w(x;pq^{-1},N+1;q)K_{n+1}^{qtm}(q^{-x};pq^{-1},N+1;q), -\end{eqnarray} -where -$$w(x;p,N;q)=\frac{(q^{-N};q)_x}{(q,p^{-1}q^{-N};q)_x}(-p)^{-x}q^{\binom{x+1}{2}}.$$ - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodQuantumqKrawtchouk} -w(x;p,N;q)K_n^{qtm}(q^{-x};p,N;q)=(1-q)^n\left(\nabla_q\right)^n\left[w(x;pq^n,N-n;q)\right], -\end{equation} -where -$$\nabla_q:=\frac{\nabla}{\nabla q^{-x}}.$$ - -\subsection*{Generating functions} For $x=0,1,2,\ldots,N$ we have -\begin{eqnarray} -\label{GenQuantumqKrawtchouk1} -& &(q^{x-N}t;q)_{N-x}\cdot\qhyp{2}{1}{q^{-x},pq^{N+1-x}}{0}{q^{x-N}t}\nonumber\\ -& &{}=\sum_{n=0}^N\frac{(q^{-N};q)_n}{(q;q)_n}K_n^{qtm}(q^{-x};p,N;q)t^n. -\end{eqnarray} - -\begin{eqnarray} -\label{GenQuantumqKrawtchouk2} -& &(q^{-x}t;q)_x\cdot\qhyp{2}{1}{q^{x-N},0}{pq}{q^{-x}t}\nonumber\\ -& &{}=\sum_{n=0}^N\frac{(q^{-N};q)_n}{(pq,q;q)_n}K_n^{qtm}(q^{-x};p,N;q)t^n. -\end{eqnarray} - -\subsection*{Limit relations} - -\subsubsection*{$q$-Hahn $\rightarrow$ Quantum $q$-Krawtchouk} -The quantum $q$-Krawtchouk polynomials given by (\ref{DefQuantumqKrawtchouk}) -simply follow from the $q$-Hahn polynomials by setting $\beta=p$ in the definition (\ref{DefqHahn}) of -the $q$-Hahn polynomials and taking the limit $\alpha\rightarrow\infty$: -$$\lim_{\alpha\rightarrow\infty}Q_n(q^{-x};\alpha,p,N|q)=K_n^{qtm}(q^{-x};p,N;q).$$ - -\subsubsection*{Quantum $q$-Krawtchouk $\rightarrow$ Al-Salam-Carlitz~II} -If we set $p=a^{-1}q^{-N-1}$ in the definition (\ref{DefQuantumqKrawtchouk}) -of the quantum $q$-Krawtchouk polynomials and let $N\rightarrow\infty$ we -obtain the Al-Salam-Carlitz~II polynomials given by -(\ref{DefAlSalamCarlitzII}). In fact we have -\begin{equation} -\lim_{N\rightarrow\infty}K_n^{qtm}(x;a^{-1}q^{-N-1},N;q)= -\left(-\frac{1}{a}\right)^nq^{\binom{n}{2}}V_n^{(a)}(x;q). -\end{equation} - -\subsubsection*{Quantum $q$-Krawtchouk $\rightarrow$ Krawtchouk} -The Krawtchouk polynomials given by (\ref{DefKrawtchouk}) easily follow from -the quantum $q$-Krawtchouk polynomials given by (\ref{DefQuantumqKrawtchouk}) -in the following way: -\begin{equation} -\lim_{q\rightarrow 1}K_n^{qtm}(q^{-x};p,N;q)=K_n(x;p^{-1},N). -\end{equation} - -\subsection*{Remarks} -The quantum $q$-Krawtchouk polynomials given by -(\ref{DefQuantumqKrawtchouk}) and the $q$-Meixner polynomials given by -(\ref{DefqMeixner}) are related in the following way: -$$K_n^{qtm}(q^{-x};p,N;q)=M_n(q^{-x};q^{-N-1},-p^{-1};q).$$ - -\noindent -The quantum $q$-Krawtchouk polynomials are related to the affine -$q$-Krawtchouk polynomials given by (\ref{DefAffqKrawtchouk}) -by the transformation $q\leftrightarrow q^{-1}$ in the following way: -$$K_n^{qtm}(q^x;p,N;q^{-1})=(p^{-1}q;q)_n\left(-\frac{p}{q}\right)^nq^{-\binom{n}{2}} -K_n^{Aff}(q^{x-N};p^{-1},N;q).$$ - -\subsection*{References} -\cite{GasperRahman90}, \cite{Koorn89III}, \cite{Koorn90II}, \cite{Smirnov}. - - -\section{$q$-Krawtchouk}\index{q-Krawtchouk polynomials@$q$-Krawtchouk polynomials} -\par\setcounter{equation}{0} - -\subsection*{Basic hypergeometric representation} For $n=0,1,2,\ldots,N$ we have -\begin{eqnarray} -\label{DefqKrawtchouk} -K_n(q^{-x};p,N;q)&=&\qhyp{3}{2}{q^{-n},q^{-x},-pq^n}{q^{-N},0}{q}\\ -&=&\frac{(q^{x-N};q)_n}{(q^{-N};q)_nq^{nx}}\, -\qhyp{2}{1}{q^{-n},q^{-x}}{q^{N-x-n+1}}{-pq^{n+N+1}}.\nonumber -\end{eqnarray} - -\subsection*{Orthogonality relation} -\begin{eqnarray} -\label{OrtqKrawtchouk} -& &\sum_{x=0}^N\frac{(q^{-N};q)_x}{(q;q)_x}(-p)^{-x}K_m(q^{-x};p,N;q)K_n(q^{-x};p,N;q)\nonumber\\ -& &{}=\frac{(q,-pq^{N+1};q)_n}{(-p,q^{-N};q)_n}\frac{(1+p)}{(1+pq^{2n})}\nonumber\\ -& &{}\mathindent{}\times (-pq;q)_Np^{-N}q^{-\binom{N+1}{2}} -\left(-pq^{-N}\right)^nq^{n^2}\,\delta_{mn},\quad p>0. -\end{eqnarray} - -\subsection*{Recurrence relation} -\begin{eqnarray} -\label{RecqKrawtchouk} --\left(1-q^{-x}\right)K_n(q^{-x})&=&A_nK_{n+1}(q^{-x})-\left(A_n+C_n\right)K_n(q^{-x})\nonumber\\ -& &{}\mathindent{}+C_nK_{n-1}(q^{-x}), -\end{eqnarray} -where -$$K_n(q^{-x}):=K_n(q^{-x};p,N;q)$$ -and -$$\left\{\begin{array}{l} -\displaystyle A_n=\frac{(1-q^{n-N})(1+pq^n)}{(1+pq^{2n})(1+pq^{2n+1})}\\ -\\ -\displaystyle C_n=-pq^{2n-N-1}\frac{(1+pq^{n+N})(1-q^n)}{(1+pq^{2n-1})(1+pq^{2n})}. -\end{array}\right.$$ - -\subsection*{Normalized recurrence relation} -\begin{equation} -\label{NormRecqKrawtchouk} -xp_n(x)=p_{n+1}(x)+\left[1-(A_n+C_n)\right]p_n(x)+A_{n-1}C_np_{n-1}(x), -\end{equation} -where -$$K_n(q^{-x};p,N;q)=\frac{(-pq^n;q)_n}{(q^{-N};q)_n}p_n(q^{-x}).$$ - -\subsection*{$q$-Difference equation} -\begin{eqnarray} -\label{dvqKrawtchouk} -& &q^{-n}(1-q^n)(1+pq^n)y(x)\nonumber\\ -& &{}=(1-q^{x-N})y(x+1)-\left[(1-q^{x-N})-p(1-q^x)\right]y(x)\nonumber\\ -& &{}\mathindent{}-p(1-q^x)y(x-1), -\end{eqnarray} -where -$$y(x)=K_n(q^{-x};p,N;q).$$ - -\subsection*{Forward shift operator} -\begin{eqnarray} -\label{shift1qKrawtchoukI} -& &K_n(q^{-x-1};p,N;q)-K_n(q^{-x};p,N;q)\nonumber\\ -& &{}=\frac{q^{-n-x}(1-q^n)(1+pq^n)}{1-q^{-N}}K_{n-1}(q^{-x};pq^2,N-1;q) -\end{eqnarray} -or equivalently -\begin{equation} -\label{shift1qKrawtchoukII} -\frac{\Delta K_n(q^{-x};p,N;q)}{\Delta q^{-x}}= -\frac{q^{-n+1}(1-q^n)(1+pq^n)}{(1-q)(1-q^{-N})}K_{n-1}(q^{-x};pq^2,N-1;q). -\end{equation} - -\subsection*{Backward shift operator} -\begin{eqnarray} -\label{shift2qKrawtchoukI} -& &(1-q^{x-N-1})K_n(q^{-x};p,N;q)+pq^{-1}(1-q^x)K_n(q^{-x+1};p,N;q)\nonumber\\ -& &{}=q^x(1-q^{-N-1})K_{n+1}(q^{-x};pq^{-2},N+1;q) -\end{eqnarray} -or equivalently -\begin{eqnarray} -\label{shift2qKrawtchoukII} -& &\frac{\nabla\left[w(x;p,N;q)K_n(q^{-x};p,N;q)\right]}{\nabla q^{-x}}\nonumber\\ -& &{}=\frac{1}{1-q}w(x;pq^{-2},N+1;q)K_{n+1}(q^{-x};pq^{-2},N+1;q), -\end{eqnarray} -where -$$w(x;p,N;q)=\frac{(q^{-N};q)_x}{(q;q)_x}\left(-\frac{q}{p}\right)^x.$$ - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodqKrawtchouk} -w(x;p,N;q)K_n(q^{-x};p,N;q)=(1-q)^n\left(\nabla_q\right)^n\left[w(x;pq^{2n},N-n;q)\right], -\end{equation} -where -$$\nabla_q:=\frac{\nabla}{\nabla q^{-x}}.$$ - -\subsection*{Generating function} For $x=0,1,2,\ldots,N$ we have -\begin{eqnarray} -\label{GenqKrawtchouk} -& &\qhyp{1}{1}{q^{-x}}{0}{pqt}\,\qhyp{2}{0}{q^{x-N},0}{-}{-q^{-x}t}\nonumber\\ -& &{}=\sum_{n=0}^N\frac{(q^{-N};q)_n}{(q;q)_n}q^{-\binom{n}{2}}K_n(q^{-x};p,N;q)t^n. -\end{eqnarray} - -\subsection*{Limit relations} - -\subsubsection*{$q$-Racah $\rightarrow$ $q$-Krawtchouk} -The $q$-Krawtchouk polynomials given by (\ref{DefqKrawtchouk}) can be obtained from -the $q$-Racah polynomials by setting $\alpha q=q^{-N}$, $\beta=-pq^N$ and -$\gamma=\delta=0$ in the definition (\ref{DefqRacah}) of the $q$-Racah polynomials: -$$R_n(q^{-x};q^{-N-1},-pq^N,0,0|q)=K_n(q^{-x};p,N;q).$$ -Note that $\mu(x)=q^{-x}$ in this case. - -\subsubsection*{$q$-Hahn $\rightarrow$ $q$-Krawtchouk} -If we set $\beta=-\alpha^{-1}q^{-1}p$ in the definition (\ref{DefqHahn}) of the $q$-Hahn polynomials -and then let $\alpha\rightarrow 0$ we obtain the $q$-Krawtchouk polynomials given by -(\ref{DefqKrawtchouk}): -$$\lim_{\alpha\rightarrow 0} -Q_n(q^{-x};\alpha,-\alpha^{-1}q^{-1}p,N|q)=K_n(q^{-x};p,N;q).$$ - -\subsubsection*{$q$-Krawtchouk $\rightarrow$ $q$-Bessel} -If we set $x\rightarrow N-x$ in the definition (\ref{DefqKrawtchouk}) of the -$q$-Krawtchouk polynomials and then take the limit $N\rightarrow\infty$ we -obtain the $q$-Bessel polynomials given by (\ref{DefqBessel}): -\begin{equation} -\lim_{N\rightarrow\infty}K_n(q^{x-N};p,N;q)=y_n(q^x;p;q). -\end{equation} - -\subsubsection*{$q$-Krawtchouk $\rightarrow$ $q$-Charlier} -By setting $p=a^{-1}q^{-N}$ in the definition (\ref{DefqKrawtchouk}) of the -$q$-Krawtchouk polynomials and then taking the limit $N\rightarrow\infty$ we -obtain the $q$-Charlier polynomials given by (\ref{DefqCharlier}): -\begin{equation} -\lim_{N\rightarrow\infty}K_n(q^{-x};a^{-1}q^{-N},N;q)=C_n(q^{-x};a;q). -\end{equation} - -\subsubsection*{$q$-Krawtchouk $\rightarrow$ Krawtchouk} -If we take the limit $q\rightarrow 1$ in the definition (\ref{DefqKrawtchouk}) of the $q$-Krawtchouk -polynomials we simply find the Krawtchouk polynomials given by (\ref{DefKrawtchouk}) in the -following way: -\begin{equation} -\lim_{q\rightarrow 1}K_n(q^{-x};p,N;q)=K_n(x;(p+1)^{-1},N). -\end{equation} - -\subsection*{Remark} -The $q$-Krawtchouk polynomials given by (\ref{DefqKrawtchouk}) and the -dual $q$-Krawtchouk polynomials given by (\ref{DefDualqKrawtchouk}) are -related in the following way: -$$K_n(q^{-x};p,N;q)=K_x(\lambda(n);-pq^N,N|q)$$ -with -$$\lambda(n)=q^{-n}-pq^n$$ -or -$$K_n(\lambda(x);c,N|q)=K_x(q^{-n};-cq^{-N},N;q)$$ -with -$$\lambda(x)=q^{-x}+cq^{x-N}.$$ - -\subsection*{References} -\cite{AlvarezRonveaux}, \cite{AskeyWilson79}, \cite{AtakRahmanSuslov}, -\cite{Campigotto+}, \cite{GasperRahman90}, \cite{Nikiforov+}, -\cite{NoumiMimachi91}, \cite{Stanton80III}, \cite{Stanton84}. - - -\newpage - -\section{Affine $q$-Krawtchouk} -\index{Affine q-Krawtchouk polynomials@Affine $q$-Krawtchouk polynomials} -\index{q-Krawtchouk polynomials@$q$-Krawtchouk polynomials!Affine} -\par\setcounter{equation}{0} - -\subsection*{Basic hypergeometric representation} -\begin{eqnarray} -\label{DefAffqKrawtchouk} -K_n^{Aff}(q^{-x};p,N;q)&=&\qhyp{3}{2}{q^{-n},0,q^{-x}}{pq,q^{-N}}{q}\\ -&=&\frac{(-pq)^nq^{\binom{n}{2}}}{(pq;q)_n}\, -\qhyp{2}{1}{q^{-n},q^{x-N}}{q^{-N}}{\frac{q^{-x}}{p}},\quad n=0,1,2,\ldots,N.\nonumber -\end{eqnarray} - -\subsection*{Orthogonality relation} -\begin{eqnarray} -\label{OrtAffqKrawtchouk} -& &\sum_{x=0}^N\frac{(pq;q)_x(q;q)_N}{(q;q)_x(q;q)_{N-x}}(pq)^{-x}K_m^{Aff}(q^{-x};p,N;q)K_n^{Aff}(q^{-x};p,N;q)\nonumber\\ -& &{}=(pq)^{n-N}\frac{(q;q)_n(q;q)_{N-n}}{(pq;q)_n(q;q)_N}\,\delta_{mn},\quad 01$, then we have another orthogonality relation given by: -\begin{eqnarray} -\label{OrtContBigqHermite2} -& &\frac{1}{2\pi}\int_{-1}^1\frac{w(x)}{\sqrt{1-x^2}}H_m(x;a|q)H_n(x;a|q)\,dx\nonumber\\ -& &{}\mathindent{}+\sum_{\begin{array}{c}\scriptstyle k\\ \scriptstyle 1-1 -\end{eqnarray} -and -\begin{eqnarray} -\label{OrtqLaguerre2} -& &\sum_{k=-\infty}^{\infty}\frac{q^{k\alpha+k}}{(-cq^k;q)_{\infty}}L_m^{(\alpha)}(cq^k;q)L_n^{(\alpha)}(cq^k;q)\nonumber\\ -& &{}=\frac{(q,-cq^{\alpha+1},-c^{-1}q^{-\alpha};q)_{\infty}} -{(q^{\alpha+1},-c,-c^{-1}q;q)_{\infty}}\frac{(q^{\alpha+1};q)_n}{(q;q)_nq^n}\,\delta_{mn}, -\quad\alpha>-1,\quad c>0. -\end{eqnarray} -For $c=1$ the latter orthogonality relation can also be written as -\begin{eqnarray} -\label{OrtqLaguerre3} -& &\int_0^{\infty}\frac{x^{\alpha}}{(-x;q)_{\infty}}L_m^{(\alpha)}(x;q)L_n^{(\alpha)}(x;q)\,d_qx\nonumber\\ -& &{}=\frac{1-q}{2}\,\frac{(q,-q^{\alpha+1},-q^{-\alpha};q)_{\infty}} -{(q^{\alpha+1},-q,-q;q)_{\infty}}\frac{(q^{\alpha+1};q)_n}{(q;q)_nq^n}\,\delta_{mn},\quad\alpha>-1. -\end{eqnarray} - -\newpage - -\subsection*{Recurrence relation} -\begin{eqnarray} -\label{RecqLaguerre} --q^{2n+\alpha+1}xL_n^{(\alpha)}(x;q)&=&(1-q^{n+1})L_{n+1}^{(\alpha)}(x;q)\nonumber\\ -& &{}\mathindent{}-\left[(1-q^{n+1})+q(1-q^{n+\alpha})\right]L_n^{(\alpha)}(x;q)\nonumber\\ -& &{}\mathindent\mathindent{}+q(1-q^{n+\alpha})L_{n-1}^{(\alpha)}(x;q). -\end{eqnarray} - -\subsection*{Normalized recurrence relation} -\begin{eqnarray} -\label{NormRecqLaguerre} -xp_n(x)&=&p_{n+1}(x)+q^{-2n-\alpha-1}\left[(1-q^{n+1})+q(1-q^{n+\alpha})\right]p_n(x)\nonumber\\ -& &{}\mathindent{}+q^{-4n-2\alpha+1}(1-q^n)(1-q^{n+\alpha})p_{n-1}(x), -\end{eqnarray} -where -$$L_n^{(\alpha)}(x;q)=\frac{(-1)^nq^{n(n+\alpha)}}{(q;q)_n}p_n(x).$$ - -\subsection*{$q$-Difference equation} -\begin{equation} -\label{dvqLaguerre} --q^{\alpha}(1-q^n)xy(x)=q^{\alpha}(1+x)y(qx)-\left[1+q^{\alpha}(1+x)\right]y(x)+y(q^{-1}x), -\end{equation} -where -$$y(x)=L_n^{(\alpha)}(x;q).$$ - -\subsection*{Forward shift operator} -\begin{equation} -\label{shift1qLaguerreI} -L_n^{(\alpha)}(x;q)-L_n^{(\alpha)}(qx;q)=-q^{\alpha+1}xL_{n-1}^{(\alpha+1)}(qx;q) -\end{equation} -or equivalently -\begin{equation} -\label{shift1qLaguerreII} -\mathcal{D}_qL_n^{(\alpha)}(x;q)=-\frac{q^{\alpha+1}}{1-q}L_{n-1}^{(\alpha+1)}(qx;q). -\end{equation} - -\subsection*{Backward shift operator} -\begin{equation} -\label{shift2qLaguerreI} -L_n^{(\alpha)}(x;q)-q^{\alpha}(1+x)L_n^{(\alpha)}(qx;q)=(1-q^{n+1})L_{n+1}^{(\alpha-1)}(x;q) -\end{equation} -or equivalently -\begin{equation} -\label{shift2qLaguerreII} -\mathcal{D}_q\left[w(x;\alpha;q)L_n^{(\alpha)}(x;q)\right]= -\frac{1-q^{n+1}}{1-q}w(x;\alpha-1;q)L_{n+1}^{(\alpha-1)}(x;q), -\end{equation} -where -$$w(x;\alpha;q)=\frac{x^{\alpha}}{(-x;q)_{\infty}}.$$ - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodqLaguerre} -w(x;\alpha;q)L_n^{(\alpha)}(x;q)= -\frac{(1-q)^n}{(q;q)_n}\left(\mathcal{D}_q\right)^n\left[w(x;\alpha+n;q)\right]. -\end{equation} - -\subsection*{Generating functions} -\begin{equation} -\label{GenqLaguerre1} -\frac{1}{(t;q)_{\infty}}\,\qhyp{1}{1}{-x}{0}{q^{\alpha+1}t} -=\sum_{n=0}^{\infty}L_n^{(\alpha)}(x;q)t^n. -\end{equation} - -\begin{equation} -\label{GenqLaguerre2} -\frac{1}{(t;q)_{\infty}}\,\qhyp{0}{1}{-}{q^{\alpha+1}}{-q^{\alpha+1}xt} -=\sum_{n=0}^{\infty}\frac{L_n^{(\alpha)}(x;q)}{(q^{\alpha+1};q)_n}t^n. -\end{equation} - -\begin{equation} -\label{GenqLaguerre3} -(t;q)_{\infty}\cdot\qhyp{0}{2}{-}{q^{\alpha+1},t}{-q^{\alpha+1}xt} -=\sum_{n=0}^{\infty}\frac{(-1)^nq^{\binom{n}{2}}}{(q^{\alpha+1};q)_n}L_n^{(\alpha)}(x;q)t^n. -\end{equation} - -\begin{eqnarray} -\label{GenqLaguerre4} -& &\frac{(\gamma t;q)_{\infty}}{(t;q)_{\infty}}\,\qhyp{1}{2}{\gamma}{q^{\alpha+1},\gamma t}{-q^{\alpha+1}xt}\nonumber\\ -& &{}=\sum_{n=0}^{\infty}\frac{(\gamma;q)_n}{(q^{\alpha+1};q)_n}L_n^{(\alpha)}(x;q)t^n, -\quad\textrm{$\gamma$ arbitrary}. -\end{eqnarray} - -\subsection*{Limit relations} - -\subsubsection*{Little $q$-Jacobi $\rightarrow$ $q$-Laguerre} -If we substitute $a=q^{\alpha}$ and $x\rightarrow -b^{-1}q^{-1}x$ in the definition -(\ref{DefLittleqJacobi}) of the little $q$-Jacobi polynomials and then take the limit -$b\rightarrow -\infty$ we find the $q$-Laguerre polynomials given by (\ref{DefqLaguerre}): -$$\lim_{b\rightarrow -\infty}p_n(-b^{-1}q^{-1}x;q^{\alpha},b|q)= -\frac{(q;q)_n}{(q^{\alpha+1};q)_n}L_n^{(\alpha)}(x;q).$$ - -\subsubsection*{$q$-Meixner $\rightarrow$ $q$-Laguerre} -The $q$-Laguerre polynomials given by (\ref{DefqLaguerre}) can be obtained -from the $q$-Meixner polynomials given by (\ref{DefqMeixner}) by setting -$b=q^{\alpha}$ and $q^{-x}\rightarrow cq^{\alpha}x$ in the definition -(\ref{DefqMeixner}) of the $q$-Meixner polynomials and then taking the limit -$c\rightarrow\infty$: -$$\lim_{c\rightarrow\infty}M_n(cq^{\alpha}x;q^{\alpha},c;q)= -\frac{(q;q)_n}{(q^{\alpha+1};q)_n}L_n^{(\alpha)}(x;q).$$ - -\subsubsection*{$q$-Laguerre $\rightarrow$ Stieltjes-Wigert} -If we set $x\rightarrow xq^{-\alpha}$ in the definition -(\ref{DefqLaguerre}) of the $q$-Laguerre polynomials and take the limit -$\alpha\rightarrow\infty$ we simply obtain the Stieltjes-Wigert polynomials given -by (\ref{DefStieltjesWigert}): -\begin{equation} -\lim_{\alpha\rightarrow\infty}L_n^{(\alpha)}\left(xq^{-\alpha};q\right)=S_n(x;q). -\end{equation} - -\subsubsection*{$q$-Laguerre $\rightarrow$ Laguerre / Charlier} -If we set $x\rightarrow (1-q)x$ in the definition (\ref{DefqLaguerre}) -of the $q$-Laguerre polynomials and take the limit $q\rightarrow 1$ we obtain -the Laguerre polynomials given by (\ref{DefLaguerre}): -\begin{equation} -\lim_{q\rightarrow 1}L_n^{(\alpha)}((1-q)x;q)=L_n^{(\alpha)}(x). -\end{equation} - -If we set $x\rightarrow -q^{-x}$ and $q^{\alpha}=a^{-1}(q-1)^{-1}$ (or -$\alpha=-(\ln q)^{-1}\ln (q-1)a$) in the definition (\ref{DefqLaguerre}) of the -$q$-Laguerre polynomials, multiply by $(q;q)_n$, and take the limit -$q\rightarrow 1$ we obtain the Charlier polynomials given by (\ref{DefCharlier}): -\begin{equation} -\lim_{q\rightarrow 1}\,(q;q)_nL_n^{(\alpha)}(-q^{-x};q)=C_n(x;a), -\end{equation} -where -$$q^{\alpha}=\frac{1}{a(q-1)}\quad\textrm{or}\quad\alpha=-\frac{\ln (q-1)a}{\ln q}.$$ - -\subsection*{Remarks} -The $q$-Laguerre polynomials are sometimes called the -generalized Stieltjes-Wigert polynomials. - -If we replace $q$ by $q^{-1}$ we obtain the little $q$-Laguerre -(or Wall) polynomials given by (\ref{DefLittleqLaguerre}) in the following -way: -$$L_n^{(\alpha)}(x;q^{-1})=\frac{(q^{\alpha+1};q)_n}{(q;q)_nq^{n\alpha}}p_n(-x;q^{\alpha}|q).$$ - -\noindent -The $q$-Laguerre polynomials given by (\ref{DefqLaguerre}) and the $q$-Bessel polynomials -given by (\ref{DefqBessel}) are related in the following way: -$$\frac{y_n(q^x;a;q)}{(q;q)_n}=L_n^{(x-n)}(aq^n;q).$$ - -\noindent -The $q$-Laguerre polynomials given by (\ref{DefqLaguerre}) and the -$q$-Charlier polynomials given by (\ref{DefqCharlier}) are related in the -following way: -$$\frac{C_n(-x;-q^{-\alpha};q)}{(q;q)_n}=L_n^{(\alpha)}(x;q).$$ - -\noindent -Since the Stieltjes and Hamburger moment problems corresponding to the -$q$-Laguerre polynomials are indeterminate there exist many different weight -functions. - -\subsection*{References} -\cite{NAlSalam89}, \cite{AlSalam90}, \cite{Askey86}, \cite{Askey89I}, \cite{AskeyWilson85}, -\cite{AtakAtakI}, \cite{ChenIsmailMuttalib}, \cite{Chihara68II}, \cite{Chihara78}, -\cite{Chihara79}, \cite{Chris}, \cite{Exton77}, \cite{GasperRahman90}, \cite{GrunbaumHaine96}, -\cite{Ismail2005I}, \cite{IsmailRahman98}, \cite{Jain95}, \cite{Moak}. - - -\section{$q$-Bessel} -\index{q-Bessel polynomials@$q$-Bessel polynomials} -\par\setcounter{equation}{0} - -\subsection*{Basic hypergeometric representation} -\begin{eqnarray} -\label{DefqBessel} -y_n(x;a;q)&=&\qhyp{2}{1}{q^{-n},-aq^n}{0}{qx}\\ -&=&(q^{-n+1}x;q)_n\cdot\qhyp{1}{1}{q^{-n}}{q^{-n+1}x}{-aq^{n+1}x}\nonumber\\ -&=&\left(-aq^nx\right)^n\cdot\qhyp{2}{1}{q^{-n},x^{-1}}{0}{-\frac{q^{-n+1}}{a}}.\nonumber -\end{eqnarray} - -\newpage - -\subsection*{Orthogonality relation} -\begin{eqnarray} -\label{OrtqBessel} -& &\sum_{k=0}^{\infty}\frac{a^k}{(q;q)_k}q^{\binom{k+1}{2}}y_m(q^k;a;q)y_n(q^k;a;q)\nonumber\\ -& &{}=(q;q)_n(-aq^n;q)_{\infty}\frac{a^nq^{\binom{n+1}{2}}}{(1+aq^{2n})}\,\delta_{mn},\quad a>0. -\end{eqnarray} - -\subsection*{Recurrence relation} -\begin{equation} -\label{RecqBessel} --xy_n(x;a;q)=A_ny_{n+1}(x;a;q)-(A_n+C_n)y_n(x;a;q)+C_ny_{n-1}(x;a;q), -\end{equation} -where -$$\left\{\begin{array}{l}\displaystyle A_n=q^n\frac{(1+aq^n)}{(1+aq^{2n})(1+aq^{2n+1})}\\ -\\ -\displaystyle C_n=aq^{2n-1}\frac{(1-q^n)}{(1+aq^{2n-1})(1+aq^{2n})}.\end{array}\right.$$ - -\subsection*{Normalized recurrence relation} -\begin{equation} -\label{NormRecqBessel} -xp_n(x)=p_{n+1}(x)+(A_n+C_n)p_n(x)+A_{n-1}C_np_{n-1}(x), -\end{equation} -where -$$y_n(x;a;q)=(-1)^nq^{-\binom{n}{2}}(-aq^n;q)_np_n(x).$$ - -\subsection*{$q$-Difference equation} -\begin{eqnarray} -\label{dvqBessel} -& &-q^{-n}(1-q^n)(1+aq^n)xy(x)\nonumber\\ -& &{}=axy(qx)-(ax+1-x)y(x)+(1-x)y(q^{-1}x), -\end{eqnarray} -where -$$y(x)=y_n(x;a;q).$$ - -\newpage - -\subsection*{Forward shift operator} -\begin{equation} -\label{shift1qBesselI} -y_n(x;a;q)-y_n(qx;a;q)=-q^{-n+1}(1-q^n)(1+aq^n)xy_{n-1}(x;aq^2;q) -\end{equation} -or equivalently -\begin{equation} -\label{shift1qBesselII} -\mathcal{D}_qy_n(x;a;q)=-\frac{q^{-n+1}(1-q^n)(1+aq^n)}{1-q}y_{n-1}(x;aq^2;q). -\end{equation} - -\subsection*{Backward shift operator} -\begin{equation} -\label{shift2qBesselI} -aq^{x-1}y_n(q^x;a;q)-(1-q^x)y_n(q^{x-1}x;a;q)=-y_{n+1}(q^x;aq^{-2};q) -\end{equation} -or equivalently -\begin{equation} -\label{shift2qBesselII} -\frac{\nabla\left[w(x;a;q)y_n(q^x;a;q)\right]}{\nabla q^x}= -\frac{q^2}{a(1-q)}w(x;aq^{-2};q)y_{n+1}(q^x;aq^{-2};q), -\end{equation} -where -$$w(x;a;q)=\frac{a^xq^{\binom{x}{2}}}{(q;q)_x}.$$ - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodqBessel} -w(x;a;q)y_n(q^x;a;q)=a^n(1-q)^nq^{n(n-1)}\left(\nabla_q\right)^n\left[w(x;aq^{2n};q)\right], -\end{equation} -where -$$\nabla_q:=\frac{\nabla}{\nabla q^x}.$$ - -\subsection*{Generating functions} -\begin{eqnarray} -\label{GenqBessel1} -& &\qhyp{0}{1}{-}{0}{-aq^{x+1}t}\,\qhyp{2}{0}{q^{-x},0}{-}{q^xt}\nonumber\\ -& &{}=\sum_{n=0}^{\infty}\frac{y_n(q^x;a;q)}{(q;q)_n}t^n,\quad x=0,1,2,\ldots. -\end{eqnarray} - -\begin{equation} -\label{GenqBessel2} -\frac{(t;q)_{\infty}}{(xt;q)_{\infty}}\,\qhyp{1}{3}{xt}{0,0,t}{-aqxt}= -\sum_{n=0}^{\infty}\frac{(-1)^nq^{\binom{n}{2}}}{(q;q)_n}y_n(x;a;q)t^n. -\end{equation} - -\subsection*{Limit relations} - -\subsubsection*{Little $q$-Jacobi $\rightarrow$ $q$-Bessel} -If we set $b\rightarrow -a^{-1}q^{-1}b$ in the definition (\ref{DefLittleqJacobi}) of -the little $q$-Jacobi polynomials and then take the limit $a\rightarrow 0$ we obtain -the $q$-Bessel polynomials given by (\ref{DefqBessel}): -$$\lim_{a\rightarrow 0}p_n(x;a,-a^{-1}q^{-1}b|q)=y_n(x;b;q).$$ - -\subsubsection*{$q$-Krawtchouk $\rightarrow$ $q$-Bessel} -If we set $x\rightarrow N-x$ in the definition (\ref{DefqKrawtchouk}) of the -$q$-Krawtchouk polynomials and then take the limit $N\rightarrow\infty$ we -obtain the $q$-Bessel polynomials given by (\ref{DefqBessel}): -$$\lim_{N\rightarrow\infty}K_n(q^{x-N};p,N;q)=y_n(q^x;p;q).$$ - -\subsubsection*{$q$-Bessel $\rightarrow$ Stieltjes-Wigert} -The Stieltjes-Wigert polynomials given by (\ref{DefStieltjesWigert}) can be obtained -from the $q$-Bessel polynomials by setting $x\rightarrow a^{-1}x$ in the definition -(\ref{DefqBessel}) of the $q$-Bessel polynomials and then taking the limit -$a\rightarrow\infty$. In fact we have -\begin{equation} -\lim_{a\rightarrow\infty}y_n(a^{-1}x;a;q)=(q;q)_nS_n(x;q). -\end{equation} - -\subsubsection*{$q$-Bessel $\rightarrow$ Bessel} -If we set $x\rightarrow -\frac{1}{2}(1-q)^{-1}x$ and $a\rightarrow -q^{a+1}$ in the definition -(\ref{DefqBessel}) of the $q$-Bessel polynomials and take the limit $q\rightarrow 1$ -we find the Bessel polynomials given by (\ref{DefBessel}): -\begin{equation} -\lim_{q\rightarrow 1}y_n(-\textstyle\frac{1}{2}(1-q)^{-1}x;-q^{a+1};q)=y_n(x;a). -\end{equation} - -\subsubsection*{$q$-Bessel $\rightarrow$ Charlier} -If we set $x\rightarrow q^x$ and $a\rightarrow a(1-q)$ in the definition (\ref{DefqBessel}) -of the $q$-Bessel polynomials and take the limit $q\rightarrow 1$ we find the Charlier -polynomials given by (\ref{DefCharlier}): -\begin{equation} -\lim_{q\rightarrow 1}\frac{y_n(q^x;a(1-q);q)}{(q-1)^n}=a^nC_n(x;a). -\end{equation} - -\subsection*{Remark} -In \cite{Koekoek94} and \cite{Koekoek98} these $q$-Bessel polynomials were called -\emph{alternative $q$-Charlier polynomials}. - -\noindent -The $q$-Bessel polynomials given by (\ref{DefqBessel}) and the $q$-Laguerre -polynomials given by (\ref{DefqLaguerre}) are related in the following way: -$$\frac{y_n(q^x;a;q)}{(q;q)_n}=L_n^{(x-n)}(aq^n;q).$$ - -\subsection*{Reference} -\cite{DattaGriffin}. - - -\section{$q$-Charlier}\index{q-Charlier polynomials@$q$-Charlier polynomials} -\par\setcounter{equation}{0} - -\subsection*{Basic hypergeometric representation} -\begin{eqnarray} -\label{DefqCharlier} -C_n(q^{-x};a;q)&=&\qhyp{2}{1}{q^{-n},q^{-x}}{0}{-\frac{q^{n+1}}{a}}\\ -&=&(-a^{-1}q;q)_n\cdot\qhyp{1}{1}{q^{-n}}{-a^{-1}q}{-\frac{q^{n+1-x}}{a}}.\nonumber -\end{eqnarray} - -\subsection*{Orthogonality relation} -\begin{eqnarray} -\label{OrtqCharlier} -& &\sum_{x=0}^{\infty}\frac{a^x}{(q;q)_x}q^{\binom{x}{2}}C_m(q^{-x};a;q)C_n(q^{-x};a;q)\nonumber\\ -& &{}=q^{-n}(-a;q)_{\infty}(-a^{-1}q,q;q)_n\,\delta_{mn},\quad a>0. -\end{eqnarray} - -\newpage - -\subsection*{Recurrence relation} -\begin{eqnarray} -\label{RecqCharlier} -& &q^{2n+1}(1-q^{-x})C_n(q^{-x})\nonumber\\ -& &{}=aC_{n+1}(q^{-x})-\left[a+q(1-q^n)(a+q^n)\right]C_n(q^{-x})\nonumber\\ -& &{}\mathindent{}+q(1-q^n)(a+q^n)C_{n-1}(q^{-x}), -\end{eqnarray} -where -$$C_n(q^{-x}):=C_n(q^{-x};a;q).$$ - -\subsection*{Normalized recurrence relation} -\begin{eqnarray} -\label{NormRecqCharlier} -xp_n(x)&=&p_{n+1}(x)+\left[1+q^{-2n-1}\left\{a+q(1-q^n)(a+q^n)\right\}\right]p_n(x)\nonumber\\ -& &{}\mathindent{}+aq^{-4n+1}(1-q^n)(a+q^n)p_{n-1}(x), -\end{eqnarray} -where -$$C_n(q^{-x};a;q)=\frac{(-1)^nq^{n^2}}{a^n}p_n(q^{-x}).$$ - -\subsection*{$q$-Difference equation} -\begin{equation} -\label{dvqCharlier} -q^ny(x)=aq^xy(x+1)-q^x(a-1)y(x)+(1-q^x)y(x-1), -\end{equation} -where -$$y(x)=C_n(q^{-x};a;q).$$ - -\subsection*{Forward shift operator} -\begin{equation} -\label{shift1qCharlierI} -C_n(q^{-x-1};a;q)-C_n(q^{-x};a;q)=-a^{-1}q^{-x}(1-q^n)C_{n-1}(q^{-x};aq^{-1};q) -\end{equation} -or equivalently -\begin{equation} -\label{shift1qCharlierII} -\frac{\Delta C_n(q^{-x};a;q)}{\Delta q^{-x}}=-\frac{q(1-q^n)}{a(1-q)}C_{n-1}(q^{-x};aq^{-1};q). -\end{equation} - -\newpage - -\subsection*{Backward shift operator} -\begin{equation} -\label{shift2qCharlierI} -C_n(q^{-x};a;q)-a^{-1}q^{-x}(1-q^x)C_n(q^{-x+1};a;q)=C_{n+1}(q^{-x};aq;q) -\end{equation} -or equivalently -\begin{equation} -\label{shift2qCharlierII} -\frac{\nabla\left[w(x;a;q)C_n(q^{-x};a;q)\right]}{\nabla q^{-x}} -=\frac{1}{1-q}w(x;aq;q)C_{n+1}(q^{-x};aq;q), -\end{equation} -where -$$w(x;a;q)=\frac{a^xq^{\binom{x+1}{2}}}{(q;q)_x}.$$ - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodqCharlier} -w(x;a;q)C_n(q^{-x};a;q)=(1-q)^n\left(\nabla_q\right)^n\left[w(x;aq^{-n};q)\right], -\end{equation} -where -$$\nabla_q:=\frac{\nabla}{\nabla q^{-x}}.$$ - -\subsection*{Generating functions} -\begin{equation} -\label{GenqCharlier1} -\frac{1}{(t;q)_{\infty}}\,\qhyp{1}{1}{q^{-x}}{0}{-a^{-1}qt} -=\sum_{n=0}^{\infty}\frac{C_n(q^{-x};a;q)}{(q;q)_n}t^n. -\end{equation} - -\begin{equation} -\label{GenqCharlier2} -\frac{1}{(t;q)_{\infty}}\,\qhyp{0}{1}{-}{-a^{-1}q}{-a^{-1}q^{-x+1}t} -=\sum_{n=0}^{\infty}\frac{C_n(q^{-x};a;q)}{(-a^{-1}q,q;q)_n}t^n. -\end{equation} - -\subsection*{Limit relations} - -\subsubsection*{$q$-Meixner $\rightarrow$ $q$-Charlier} -The $q$-Charlier polynomials given by (\ref{DefqCharlier}) can easily be -obtained from the $q$-Meixner given by (\ref{DefqMeixner}) by setting $b=0$ -in the definition (\ref{DefqMeixner}) of the $q$-Meixner polynomials: -\begin{equation} -M_n(x;0,c;q)=C_n(x;c;q). -\end{equation} - -\subsubsection*{$q$-Krawtchouk $\rightarrow$ $q$-Charlier} -By setting $p=a^{-1}q^{-N}$ in the definition (\ref{DefqKrawtchouk}) of the -$q$-Krawtchouk polynomials and then taking the limit $N\rightarrow\infty$ we -obtain the $q$-Charlier polynomials given by (\ref{DefqCharlier}): -$$\lim_{N\rightarrow\infty}K_n(q^{-x};a^{-1}q^{-N},N;q)=C_n(q^{-x};a;q).$$ - -\subsubsection*{$q$-Charlier $\rightarrow$ Stieltjes-Wigert} -If we set $q^{-x}\rightarrow ax$ in the definition (\ref{DefqCharlier}) of the -$q$-Charlier polynomials and take the limit $a\rightarrow\infty$ we obtain -the Stieltjes-Wigert polynomials given by (\ref{DefStieltjesWigert}) in the -following way: -\begin{equation} -\lim_{a\rightarrow\infty}C_n(ax;a;q)=(q;q)_nS_n(x;q). -\end{equation} - -\subsubsection*{$q$-Charlier $\rightarrow$ Charlier} -If we set $a\rightarrow a(1-q)$ in the definition (\ref{DefqCharlier}) -of the $q$-Charlier polynomials and take the limit $q\rightarrow 1$ we obtain the -Charlier polynomials given by (\ref{DefCharlier}): -\begin{equation} -\lim_{q\rightarrow 1}C_n(q^{-x};a(1-q);q)=C_n(x;a). -\end{equation} - -\subsection*{Remark} -The $q$-Charlier polynomials given by (\ref{DefqCharlier}) and the -$q$-Laguerre polynomials given by (\ref{DefqLaguerre}) are related in the -following way: -$$\frac{C_n(-x;-q^{-\alpha};q)}{(q;q)_n}=L_n^{(\alpha)}(x;q).$$ - -\subsection*{References} -\cite{AlvarezRonveaux}, \cite{AtakRahmanSuslov}, \cite{GasperRahman90}, -\cite{Hahn}, \cite{Koelink96III}, \cite{Nikiforov+}, \cite{Zeng95}. - - -\section{Al-Salam-Carlitz~I}\index{Al-Salam-Carlitz~I polynomials} -\par\setcounter{equation}{0} - -\subsection*{Basic hypergeometric representation} -\begin{equation} -\label{DefAlSalamCarlitzI} -U_n^{(a)}(x;q)=(-a)^nq^{\binom{n}{2}}\,\qhyp{2}{1}{q^{-n},x^{-1}}{0}{\frac{qx}{a}}. -\end{equation} - -\subsection*{Orthogonality relation} -\begin{eqnarray} -\label{OrtAlSalamCarlitzI} -& &\int_a^1(qx,a^{-1}qx;q)_{\infty}U_m^{(a)}(x;q)U_n^{(a)}(x;q)\,d_qx\nonumber\\ -& &{}=(-a)^n(1-q)(q;q)_n(q,a,a^{-1}q;q)_{\infty}q^{\binom{n}{2}}\,\delta_{mn},\quad a<0. -\end{eqnarray} - -\subsection*{Recurrence relation} -\begin{equation} -\label{RecAlSalamCarlitzI} -xU_n^{(a)}(x;q)=U_{n+1}^{(a)}(x;q)+(a+1)q^nU_n^{(a)}(x;q) --aq^{n-1}(1-q^n)U_{n-1}^{(a)}(x;q). -\end{equation} - -\subsection*{Normalized recurrence relation} -\begin{equation} -\label{NormRecAlSalamCarlitzI} -xp_n(x)=p_{n+1}(x)+(a+1)q^np_n(x)-aq^{n-1}(1-q^n)p_{n-1}(x), -\end{equation} -where -$$U_n^{(a)}(x;q)=p_n(x).$$ - -\subsection*{$q$-Difference equation} -\begin{eqnarray} -\label{dvAlSalamCarlitzI} -(1-q^n)x^2y(x)&=&aq^{n-1}y(qx)-\left[aq^{n-1}+q^n(1-x)(a-x)\right]y(x)\nonumber\\ -& &{}\mathindent{}+q^n(1-x)(a-x)y(q^{-1}x), -\end{eqnarray} -where -$$y(x)=U_n^{(a)}(x;q).$$ - -\subsection*{Forward shift operator} -\begin{equation} -\label{shift1AlSalamCarlitzI-I} -U_n^{(a)}(x;q)-U_n^{(a)}(qx;q)=(1-q^n)xU_{n-1}^{(a)}(x;q) -\end{equation} -or equivalently -\begin{equation} -\label{shift1AlSalamCarlitzI-II} -\mathcal{D}_qU_n^{(a)}(x;q)=\frac{1-q^n}{1-q}U_{n-1}^{(a)}(x;q). -\end{equation} - -\subsection*{Backward shift operator} -\begin{equation} -\label{shift2AlSalamCarlitzI-I} -aU_n^{(a)}(x;q)-(1-x)(a-x)U_n^{(a)}(q^{-1}x;q)=-q^{-n}xU_{n+1}^{(a)}(x;q) -\end{equation} -or equivalently -\begin{equation} -\label{shift2AlSalamCarlitzI-II} -\mathcal{D}_{q^{-1}}\left[w(x;a;q)U_n^{(a)}(x;q)\right]= -\frac{q^{-n+1}}{a(1-q)}w(x;a;q)U_{n+1}^{(a)}(x;q), -\end{equation} -where -$$w(x;a;q)=(qx,a^{-1}qx;q)_{\infty}.$$ - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodAlSalamCarlitzI} -w(x;a;q)U_n^{(a)}(x;q)=a^nq^{\frac{1}{2}n(n-3)}(1-q)^n -\left(\mathcal{D}_{q^{-1}}\right)^n\left[w(x;a;q)\right]. -\end{equation} - -\subsection*{Generating function} -\begin{equation} -\label{GenAlSalamCarlitzI} -\frac{(t,at;q)_{\infty}}{(xt;q)_{\infty}}= -\sum_{n=0}^{\infty}\frac{U_n^{(a)}(x;q)}{(q;q)_n}t^n. -\end{equation} - -\subsection*{Limit relations} - -\subsubsection*{Big $q$-Laguerre $\rightarrow$ Al-Salam-Carlitz~I} -If we set $x\rightarrow aqx$ and $b\rightarrow ab$ in the definition -(\ref{DefBigqLaguerre}) of the big $q$-Laguerre polynomials and take the -limit $a\rightarrow 0$ we obtain the Al-Salam-Carlitz~I polynomials given by -(\ref{DefAlSalamCarlitzI}): -$$\lim_{a\rightarrow 0}\frac{P_n(aqx;a,ab;q)}{a^n}=q^nU_n^{(b)}(x;q).$$ - -\subsubsection*{Dual $q$-Krawtchouk $\rightarrow$ Al-Salam-Carlitz~I} -If we set $c=a^{-1}$ in the definition (\ref{DefDualqKrawtchouk}) -of the dual $q$-Krawtchouk polynomials and take the limit -$N\rightarrow\infty$ we simply obtain the Al-Salam-Carlitz~I polynomials -given by (\ref{DefAlSalamCarlitzI}): -$$\lim_{N\rightarrow\infty}K_n(\lambda(x);a^{-1},N|q)= -\left(-\frac{1}{a}\right)^nq^{-\binom{n}{2}}U_n^{(a)}(q^x;q).$$ -Note that $\lambda(x)=q^{-x}+a^{-1}q^{x-N}$. - -\subsubsection*{Al-Salam-Carlitz~I $\rightarrow$ Discrete $q$-Hermite~I} -The discrete $q$-Hermite~I polynomials given by -(\ref{DefDiscreteqHermiteI}) can easily be obtained from the -Al-Salam-Carlitz~I polynomials given by (\ref{DefAlSalamCarlitzI}) by the -substitution $a=-1$: -\begin{equation} -U_n^{(-1)}(x;q)=h_n(x;q). -\end{equation} - -\subsubsection*{Al-Salam-Carlitz~I $\rightarrow$ Charlier / Hermite} -If we set $a\rightarrow a(q-1)$ and $x\rightarrow q^x$ in the definition -(\ref{DefAlSalamCarlitzI}) of the Al-Salam-Carlitz~I polynomials and take the -limit $q\rightarrow 1$ after dividing by $a^n(1-q)^n$ we obtain the Charlier -polynomials given by (\ref{DefCharlier}): -\begin{equation} -\lim_{q\rightarrow 1}\frac{U_n^{(a(q-1))}(q^x;q)}{(1-q)^n}=a^nC_n(x;a). -\end{equation} - -If we set $x\rightarrow x\sqrt{1-q^2}$ and $a\rightarrow a\sqrt{1-q^2}-1$ -in the definition (\ref{DefAlSalamCarlitzI}) of the Al-Salam-Carlitz~I -polynomials, divide by $(1-q^2)^{\frac{n}{2}}$, and let $q$ tend to $1$ we -obtain the Hermite polynomials given by (\ref{DefHermite}) with shifted -argument. In fact we have -\begin{equation} -\lim_{q\rightarrow 1}\frac{U_n^{(a\sqrt{1-q^2}-1)}(x\sqrt{1-q^2};q)} -{(1-q^2)^{\frac{n}{2}}}=\frac{H_n(x-a)}{2^n}. -\end{equation} - -\subsection*{Remark} -The Al-Salam-Carlitz~I polynomials are related to the Al-Salam-Carlitz~II -polynomials given by (\ref{DefAlSalamCarlitzII}) in the following way: -$$U_n^{(a)}(x;q^{-1})=V_n^{(a)}(x;q).$$ - -\subsection*{References} -\cite{AlSalam90}, \cite{AlSalamCarlitz}, \cite{AlSalamChihara76}, \cite{AskeySuslovII}, -\cite{AtakRahmanSuslov}, \cite{Chihara68II}, \cite{Chihara78}, \cite{DattaGriffin}, -\cite{Dehesa}, \cite{DohaAhmed2005}, \cite{GasperRahman90}, \cite{Ismail85I}, -\cite{IsmailMuldoon}, \cite{Kim}, \cite{Zeng95}. - - -\section{Al-Salam-Carlitz~II}\index{Al-Salam-Carlitz~II polynomials} -\par\setcounter{equation}{0} - -\subsection*{Basic hypergeometric representation} -\begin{equation} -\label{DefAlSalamCarlitzII} -V_n^{(a)}(x;q)= -(-a)^nq^{-\binom{n}{2}}\,\qhyp{2}{0}{q^{-n},x}{-}{\frac{q^n}{a}}. -\end{equation} - -\subsection*{Orthogonality relation} -\begin{eqnarray} -\label{OrtAlSalamCarlitzII} -& &\sum_{k=0}^{\infty}\frac{q^{k^2}a^k}{(q;q)_k(aq;q)_k} -V_m^{(a)}(q^{-k};q)V_n^{(a)}(q^{-k};q)\nonumber\\ -& &{}=\frac{(q;q)_na^n}{(aq;q)_{\infty}q^{n^2}}\,\delta_{mn},\quad 00,\quad\textrm{with}\quad -\gamma^2=-\frac{1}{2\ln q}.$$ - -\subsection*{References} -\cite{Askey86}, \cite{Askey89I}, \cite{AtakAtakIII}, \cite{Chihara70}, \cite{Chihara78}, -\cite{Dehesa}, \cite{Ismail2005I}, \cite{Nikiforov+}, \cite{Stieltjes}, \cite{Szego75}, -\cite{ValentAssche}, \cite{Wigert}. - - -\section{Discrete $q$-Hermite~I} -\index{Discrete q-Hermite~I polynomials@Discrete $q$-Hermite~I polynomials} -\index{q-Hermite~I polynomials@$q$-Hermite~I polynomials!Discrete} -\par\setcounter{equation}{0} - -\subsection*{Basic hypergeometric representation} -The discrete $q$-Hermite~I polynomials are Al-Salam-Carlitz~I polynomials -with $a=-1$: -\begin{eqnarray} -\label{DefDiscreteqHermiteI} -h_n(x;q)=U_n^{(-1)}(x;q)&=&q^{\binom{n}{2}}\,\qhyp{2}{1}{q^{-n},x^{-1}}{0}{-qx}\\ -&=&x^n\qhypK{2}{0}{q^{-n},q^{-n+1}}{0}{q^2,\frac{q^{2n-1}}{x^2}}\nonumber -\end{eqnarray} - -\subsection*{Orthogonality relation} -\begin{eqnarray} -\label{OrtDiscreteqHermiteI} -& &\int_{-1}^1(qx,-qx;q)_{\infty}h_m(x;q)h_n(x;q)\,d_qx\nonumber\\ -& &{}=(1-q)(q;q)_n(q,-1,-q;q)_{\infty}q^{\binom{n}{2}}\,\delta_{mn}. -\end{eqnarray} - -\subsection*{Recurrence relation} -\begin{equation} -\label{RecDiscreteqHermiteI} -xh_n(x;q)=h_{n+1}(x;q)+q^{n-1}(1-q^n)h_{n-1}(x;q). -\end{equation} - -\subsection*{Normalized recurrence relation} -\begin{equation} -\label{NormRecDiscreteqHermiteI} -xp_n(x)=p_{n+1}(x)+q^{n-1}(1-q^n)p_{n-1}(x), -\end{equation} -where -$$h_n(x;q)=p_n(x).$$ - -\subsection*{$q$-Difference equation} -\begin{equation} -\label{dvDiscreteqHermiteI} --q^{-n+1}x^2y(x)=y(qx)-(1+q)y(x)+q(1-x^2)y(q^{-1}x), -\end{equation} -where -$$y(x)=h_n(x;q).$$ - -\subsection*{Forward shift operator} -\begin{equation} -\label{shift1DiscreteqHermiteI-I} -h_n(x;q)-h_n(qx;q)=(1-q^n)xh_{n-1}(x;q) -\end{equation} -or equivalently -\begin{equation} -\label{shift1DiscreteqHermiteI-II} -\mathcal{D}_qh_n(x;q)=\frac{1-q^n}{1-q}h_{n-1}(x;q). -\end{equation} - -\subsection*{Backward shift operator} -\begin{equation} -\label{shift2DiscreteqHermiteI-I} -h_n(x;q)-(1-x^2)h_n(q^{-1}x;q)=q^{-n}xh_{n+1}(x;q) -\end{equation} -or equivalently -\begin{equation} -\label{shift2DiscreteqHermiteI-II} -\mathcal{D}_{q^{-1}}\left[w(x;q)h_n(x;q)\right] -=-\frac{q^{-n+1}}{1-q}w(x;q)h_{n+1}(x;q), -\end{equation} -where -$$w(x;q)=(qx,-qx;q)_{\infty}.$$ - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodDiscreteqHermiteI} -w(x;q)h_n(x;q)=(q-1)^nq^{\frac{1}{2}n(n-3)} -\left(\mathcal{D}_{q^{-1}}\right)^n\left[w(x;q)\right]. -\end{equation} - -\subsection*{Generating function} -\begin{equation} -\label{GenDiscreteqHermiteI} -\frac{(t^2;q^2)_{\infty}}{(xt;q)_{\infty}}= -\sum_{n=0}^{\infty}\frac{h_n(x;q)}{(q;q)_n}t^n. -\end{equation} - -\subsection*{Limit relations} - -\subsubsection*{Al-Salam-Carlitz~I $\rightarrow$ Discrete $q$-Hermite~I} -The discrete $q$-Hermite~I polynomials given by -(\ref{DefDiscreteqHermiteI}) can easily be obtained from the -Al-Salam-Carlitz~I polynomials given by (\ref{DefAlSalamCarlitzI}) by the -substitution $a=-1$: -$$U_n^{(-1)}(x;q)=h_n(x;q).$$ - -\subsubsection*{Discrete $q$-Hermite~I $\rightarrow$ Hermite} -The Hermite polynomials given by (\ref{DefHermite}) can be found -from the discrete $q$-Hermite~I polynomials given by -(\ref{DefDiscreteqHermiteI}) in the following way: -\begin{equation} -\lim_{q\rightarrow 1}\frac{h_n(x\sqrt{1-q^2};q)} -{(1-q^2)^{\frac{n}{2}}}=\frac{H_n(x)}{2^n}. -\end{equation} - -\subsection*{Remark} The discrete $q$-Hermite~I polynomials are related to the -discrete $q$-Hermite~II polynomials given by (\ref{DefDiscreteqHermiteII}) -in the following way: -$$h_n(ix;q^{-1})=i^n{\tilde h}_n(x;q).$$ - -\subsection*{References} -\cite{AlSalam90}, \cite{AlSalamCarlitz}, \cite{AtakRahmanSuslov}, -\cite{BergIsmail}, \cite{BustozIsmail82}, \cite{GasperRahman90}, \cite{Hahn}, -\cite{Koorn97}. - - -\newpage - -\section{Discrete $q$-Hermite~II} -\index{Discrete q-Hermite~II polynomials@Discrete $q$-Hermite~II polynomials} -\index{q-Hermite~II polynomials@$q$-Hermite~II polynomials!Discrete} -\par\setcounter{equation}{0} - -\subsection*{Basic hypergeometric representation} -The discrete $q$-Hermite~II polynomials are Al-Salam-Carlitz~II polynomials -with $a=-1$: -\begin{eqnarray} -\label{DefDiscreteqHermiteII} -{\tilde h}_n(x;q)=i^{-n}V_n^{(-1)}(ix;q) -&=&i^{-n}q^{-\binom{n}{2}}\,\qhyp{2}{0}{q^{-n},ix}{-}{-q^n}\\ -&=&x^n -\qhypK{2}{1}{q^{-n},q^{-n+1}}{0}{q^2,-\frac{q^2}{x^2}}\nonumber -\end{eqnarray} - -\subsection*{Orthogonality relation} -\begin{eqnarray} -\label{OrtDiscreteqHermiteIIa} -& &\sum_{k=-\infty}^{\infty}\left[{\tilde h}_m(cq^k;q) -{\tilde h}_n(cq^k;q)+{\tilde h}_m(-cq^k;q){\tilde h}_n(-cq^k;q)\right] -w(cq^k;q)q^k\nonumber\\ -& &{}=2\frac{(q^2,-c^2q,-c^{-2}q;q^2)_{\infty}}{(q,-c^2,-c^{-2}q^2;q^2)_{\infty}} -\frac{(q;q)_n}{q^{n^2}}\,\delta_{mn},\quad c>0, -\end{eqnarray} -where -$$w(x;q)=\frac{1}{(ix,-ix;q)_{\infty}}=\frac{1}{(-x^2;q^2)_{\infty}}.$$ -For $c=1$ this orthogonality relation can also be written as -\begin{equation} -\label{OrtDiscreteqHermiteIIb} -\int_{-\infty}^{\infty}\frac{{\tilde h}_m(x;q){\tilde h}_n(x;q)}{(-x^2;q^2)_{\infty}}\,d_qx -=\frac{\left(q^2,-q,-q;q^2\right)_{\infty}}{\left(q^3,-q^2,-q^2;q^2\right)_{\infty}} -\frac{(q;q)_n}{q^{n^2}}\,\delta_{mn}. -\end{equation} - -\subsection*{Recurrence relation} -\begin{equation} -\label{RecDiscreteqHermiteII} -x{\tilde h}_n(x;q)={\tilde h}_{n+1}(x;q)+q^{-2n+1}(1-q^n){\tilde h}_{n-1}(x;q). -\end{equation} - -\subsection*{Normalized recurrence relation} -\begin{equation} -\label{NormRecDiscreteqHermiteII} -xp_n(x)=p_{n+1}(x)+q^{-2n+1}(1-q^n)p_{n-1}(x), -\end{equation} -where -$${\tilde h}_n(x;q)=p_n(x).$$ - -\subsection*{$q$-Difference equation} -\begin{eqnarray} -\label{dvDiscreteqHermiteII} -& &-(1-q^n)x^2{\tilde h}_n(x;q)\nonumber\\ -& &{}=(1+x^2){\tilde h}_n(qx;q)-(1+x^2+q){\tilde h}_n(x;q)+q{\tilde h}_n(q^{-1}x;q). -\end{eqnarray} - -\subsection*{Forward shift operator} -\begin{equation} -\label{shift1DiscreteqHermiteII-I} -\tilde{h}_n(x;q)-\tilde{h}_n(qx;q)=q^{-n+1}(1-q^n)x\tilde{h}_{n-1}(qx;q) -\end{equation} -or equivalently -\begin{equation} -\label{shift1DiscreteqHermiteII-II} -\mathcal{D}_q\tilde{h}_n(x;q)=\frac{q^{-n+1}(1-q^n)}{1-q}\tilde{h}_{n-1}(qx;q). -\end{equation} - -\subsection*{Backward shift operator} -\begin{equation} -\label{shift2DiscreteqHermiteII-I} -\tilde{h}_n(x;q)-(1+x^2)\tilde{h}_n(qx;q)=-q^nx\tilde{h}_{n+1}(x;q) -\end{equation} -or equivalently -\begin{equation} -\label{shift2DiscreteqHermiteII-II} -\mathcal{D}_q\left[w(x;q)\tilde{h}_n(x;q)\right] -=-\frac{q^n}{1-q}w(x;q)\tilde{h}_{n+1}(x;q). -\end{equation} - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodDiscreteqHermiteII} -w(x;q)\tilde{h}_n(x;q)=(q-1)^nq^{-\binom{n}{2}} -\left(\mathcal{D}_q\right)^n\left[w(x;q)\right]. -\end{equation} - -\subsection*{Generating functions} -\begin{equation} -\label{GenDiscreteqHermiteII1} -\frac{(-xt;q)_{\infty}}{(-t^2;q^2)_{\infty}}=\sum_{n=0}^{\infty} -\frac{q^{\binom{n}{2}}}{(q;q)_n}{\tilde h}_n(x;q)t^n. -\end{equation} - -\begin{equation} -\label{GenDiscreteqHermiteII2} -(-it;q)_{\infty}\cdot\qhyp{1}{1}{ix}{-it}{it}=\sum_{n=0}^{\infty} -\frac{(-1)^nq^{n(n-1)}}{(q;q)_n}{\tilde h}_n(x;q)t^n. -\end{equation} - -\subsection*{Limit relations} - -\subsubsection*{Al-Salam-Carlitz~II $\rightarrow$ Discrete $q$-Hermite~II} -The discrete $q$-Hermite~II polynomials given by -(\ref{DefDiscreteqHermiteII}) follow from the Al-Salam-Carlitz~II -polynomials given by (\ref{DefAlSalamCarlitzII}) by the substitution $a=-1$ -in the following way: -$$i^{-n}V_n^{(-1)}(ix;q)={\tilde h}_n(x;q).$$ - -\subsubsection*{Discrete $q$-Hermite~II $\rightarrow$ Hermite} -The Hermite polynomials given by (\ref{DefHermite}) can be found -from the discrete $q$-Hermite~II polynomials given by -(\ref{DefDiscreteqHermiteII}) in the following way: -\begin{equation} -\lim_{q\rightarrow 1}\frac{{\tilde h}_n(x\sqrt{1-q^2};q)} -{(1-q^2)^{\frac{n}{2}}}=\frac{H_n(x)}{2^n}. -\end{equation} - -\subsection*{Remark} The discrete $q$-Hermite~II polynomials are related to the -discrete $q$-Hermite~I polynomials given by (\ref{DefDiscreteqHermiteI}) -in the following way: -$${\tilde h}_n(x;q^{-1})=i^{-n}h_n(ix;q).$$ - -\subsection*{References} -\cite{BergIsmail}, \cite{Koorn97}. - -\end{document} diff --git a/KLSadd_insertion/newtempchap09.tex b/KLSadd_insertion/newtempchap09.tex deleted file mode 100644 index 8186c8d..0000000 --- a/KLSadd_insertion/newtempchap09.tex +++ /dev/null @@ -1,3934 +0,0 @@ -\documentclass[envcountchap,graybox]{svmono} - -\addtolength{\textwidth}{1mm} - -\usepackage{amsmath,amssymb} - -\usepackage{amsfonts} -%\usepackage{breqn} -\usepackage{DLMFmath} -\usepackage{DRMFfcns} - -\usepackage{mathptmx} -\usepackage{helvet} -\usepackage{courier} - -\usepackage{makeidx} -\usepackage{graphicx} - -\usepackage{multicol} -\usepackage[bottom]{footmisc} - -\usepackage[pdftex]{hyperref} -\usepackage {xparse} -\usepackage{cite} -\makeindex - -\def\bibname{Bibliography} -\def\refname{Bibliography} - 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-\newcommand\sa{\smallskipamount} -\newcommand\sLP{\\[\sa]} -\newcommand\sPP{\\[\sa]\indent} -\newcommand\ba{\bigskipamount} -\newcommand\bLP{\\[\ba]} -\newcommand\CC{\mathbb{C}} -\newcommand\RR{\mathbb{R}} -\newcommand\ZZ{\mathbb{Z}} -\newcommand\al\alpha -\newcommand\be\beta -\newcommand\ga\gamma -\newcommand\de\delta -\newcommand\tha\theta -\newcommand\la\lambda -\newcommand\om\omega -\newcommand\Ga{\Gamma} -\newcommand\half{\frac12} -\newcommand\thalf{\tfrac12} -\newcommand\iy\infty -\newcommand\wt{\widetilde} -\newcommand\const{{\rm const.}\,} -\newcommand\Zpos{\ZZ_{>0}} -\newcommand\Znonneg{\ZZ_{\ge0}} -\newcommand{\hyp}[5]{\,\mbox{}_{#1}F_{#2}\!\left( - \genfrac{}{}{0pt}{}{#3}{#4};#5\right)} -\newcommand{\qhyp}[5]{\,\mbox{}_{#1}\phi_{#2}\!\left( - \genfrac{}{}{0pt}{}{#3}{#4};#5\right)} -\newcommand\LHS{left-hand side} -\newcommand\RHS{right-hand side} -\renewcommand\Re{{\rm Re}\,} -\renewcommand\Im{{\rm Im}\,} - -\NewDocumentCommand\mycite{m g}{% - \IfNoValueTF{#2} - {[\hyperlink{#1}{#1}]} - {[\hyperlink{#1}{#1}, #2]}% -} -\newcommand\mybibitem[1]{\bibitem[#1]{#1}\hypertarget{#1}{}} -\begin{document} - -\author{Roelof Koekoek\\[2.5mm]Peter A. Lesky\\[2.5mm]Ren\'e F. Swarttouw} -\title{Hypergeometric orthogonal polynomials and their\\$q$-analogues} -\subtitle{-- Monograph --} -\maketitle - -\frontmatter - -\large - -\addtocounter{chapter}{8} -\pagenumbering{roman} -\chapter{Hypergeometric orthogonal polynomials} -\label{HyperOrtPol} - - -In this chapter we deal with all families of hypergeometric orthogonal polynomials -appearing in the Askey scheme on page~\pageref{scheme}. For each family of orthogonal -polynomials we state the most important properties such as a representation as a -hypergeometric function, orthogonality relation(s), the three-term recurrence relation, -the second-order differential or difference equation, the forward shift (or degree lowering) -and backward shift (or degree raising) operator, a Rodrigues-type formula and some generating -functions. In each case we use the notation which seems to be most common in the literature. -Moreover, in each case we mention the connection between various families by stating the -appropriate limit relations. See also \cite{Terwilliger2006} for an algebraic approach of -this Askey scheme and \cite{TemmeLopez2001} for a view from asymptotic analysis. -For notations the reader is referred to chapter~\ref{Definitions}. - -\section{Wilson}\index{Wilson polynomials} - -\par - -\subsection*{Hypergeometric representation} -\begin{eqnarray} -\label{DefWilson} -& &\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\nonumber\\ -& &{}=\hyp{4}{3}{-n,n+a+b+c+d-1,a+ix,a-ix}{a+b,a+c,a+d}{1}. -\end{eqnarray} - -\newpage - -\subsection*{Orthogonality relation} -If $Re(a,b,c,d)>0$ and non-real parameters occur in conjugate pairs, then -\begin{eqnarray} -\label{OrthIWilson} -& &\frac{1}{2\pi}\int_0^{\infty} -\left|\frac{\Gamma(a+ix)\Gamma(b+ix)\Gamma(c+ix)\Gamma(d+ix)}{\Gamma(2ix)}\right|^2\nonumber\\ -& &{}\mathindent\times W_m(x^2;a,b,c,d)W_n(x^2;a,b,c,d)\,dx\nonumber\\ -& &{}=\frac{\Gamma(n+a+b)\cdots\Gamma(n+c+d)}{\Gamma(2n+a+b+c+d)}(n+a+b+c+d-1)_nn!\,\delta_{mn}, -\end{eqnarray} -where -\begin{eqnarray*} -& &\Gamma(n+a+b)\cdots\Gamma(n+c+d)\\ -& &{}=\Gamma(n+a+b)\Gamma(n+a+c)\Gamma(n+a+d)\Gamma(n+b+c)\Gamma(n+b+d)\Gamma(n+c+d). -\end{eqnarray*} -If $a<0$ and $a+b$, $a+c$, $a+d$ are positive or a pair of complex conjugates -occur with positive real parts, then -\begin{eqnarray} -\label{OrtIIWilson} -& &\frac{1}{2\pi}\int_0^{\infty}\left|\frac{\Gamma(a+ix)\Gamma(b+ix)\Gamma(c+ix)\Gamma(d+ix)}{\Gamma(2ix)}\right|^2\nonumber\\ -& &{}\mathindent\mathindent\times W_m(x^2;a,b,c,d)W_n(x^2;a,b,c,d)\,dx\nonumber\\ -& &{}\mathindent{}+\frac{\Gamma(a+b)\Gamma(a+c)\Gamma(a+d)\Gamma(b-a)\Gamma(c-a)\Gamma(d-a)}{\Gamma(-2a)}\nonumber\\ -& &{}\mathindent\mathindent{}\times\sum_{\begin{array}{c} -{\scriptstyle k=0,1,2\ldots}\\{\scriptstyle a+k<0}\end{array}} -\frac{(2a)_k(a+1)_k(a+b)_k(a+c)_k(a+d)_k}{(a)_k(a-b+1)_k(a-c+1)_k(a-d+1)_kk!}\nonumber\\ -& &{}\mathindent\mathindent\mathindent\mathindent{}\times W_m(-(a+k)^2;a,b,c,d)W_n(-(a+k)^2;a,b,c,d)\nonumber\\ -& &{}=\frac{\Gamma(n+a+b)\cdots\Gamma(n+c+d)}{\Gamma(2n+a+b+c+d)}(n+a+b+c+d-1)_nn!\,\delta_{mn}. -\end{eqnarray} - -\subsection*{Recurrence relation} -\begin{equation} -\label{RecWilson} --\left(a^2+x^2\right){\tilde{W}}_n(x^2) -=A_n{\tilde{W}}_{n+1}(x^2)-\left(A_n+C_n\right){\tilde{W}}_n(x^2)+C_n{\tilde{W}}_{n-1}(x^2), -\end{equation} -where -$${\tilde{W}}_n(x^2):={\tilde{W}}_n(x^2;a,b,c,d)=\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}$$ -and -$$\left\{\begin{array}{l} -\displaystyle A_n=\frac{(n+a+b+c+d-1)(n+a+b)(n+a+c)(n+a+d)}{(2n+a+b+c+d-1)(2n+a+b+c+d)}\\[5mm] -\displaystyle C_n=\frac{n(n+b+c-1)(n+b+d-1)(n+c+d-1)}{(2n+a+b+c+d-2)(2n+a+b+c+d-1)}. -\end{array}\right.$$ - -\subsection*{Normalized recurrence relation} -\begin{equation} -\label{NormRecWilson} -xp_n(x)=p_{n+1}(x)+(A_n+C_n-a^2)p_n(x)+A_{n-1}C_np_{n-1}(x), -\end{equation} -where -$$W_n(x^2;a,b,c,d)=(-1)^n(n+a+b+c+d-1)_np_n(x^2).$$ - -\subsection*{Difference equation} -\begin{eqnarray} -\label{dvWilson} -& &n(n+a+b+c+d-1)y(x)\nonumber\\ -& &{}=B(x)y(x+i)-\left[B(x)+D(x)\right]y(x)+D(x)y(x-i), -\end{eqnarray} -where -$$y(x)=W_n(x^2;a,b,c,d)$$ -and -$$\left\{\begin{array}{l} -\displaystyle B(x)=\frac{(a-ix)(b-ix)(c-ix)(d-ix)}{2ix(2ix-1)}\\[5mm] -\displaystyle D(x)=\frac{(a+ix)(b+ix)(c+ix)(d+ix)}{2ix(2ix+1)}. -\end{array}\right.$$ - -\subsection*{Forward shift operator} -\begin{eqnarray} -\label{shift1WilsonI} -& &W_n((x+\textstyle\frac{1}{2}i)^2;a,b,c,d) --W_n((x-\textstyle\frac{1}{2}i)^2;a,b,c,d)\nonumber\\ -& &{}=-2inx(n+a+b+c+d-1)W_{n-1}(x^2;a+\textstyle\frac{1}{2},b+\textstyle\frac{1}{2}, -c+\textstyle\frac{1}{2},d+\textstyle\frac{1}{2}) -\end{eqnarray} -or equivalently -\begin{eqnarray} -\label{shift1WilsonII} -& &\frac{\delta W_n(x^2;a,b,c,d)}{\delta x^2}\nonumber\\ -& &{}=-n(n+a+b+c+d-1)W_{n-1}(x^2;a+\textstyle\frac{1}{2},b+\textstyle\frac{1}{2}, -c+\textstyle\frac{1}{2},d+\textstyle\frac{1}{2}). -\end{eqnarray} - -\newpage - -\subsection*{Backward shift operator} -\begin{eqnarray} -\label{shift2WilsonI} -& &(a-\textstyle\frac{1}{2}-ix)(b-\textstyle\frac{1}{2}-ix) -(c-\textstyle\frac{1}{2}-ix)(d-\textstyle\frac{1}{2}-ix) -W_n((x+\textstyle\frac{1}{2}i)^2;a,b,c,d)\nonumber\\ -& &{}\mathindent{}-(a-\textstyle\frac{1}{2}+ix)(b-\textstyle\frac{1}{2}+ix) -(c-\textstyle\frac{1}{2}+ix)(d-\textstyle\frac{1}{2}+ix) -W_n((x-\textstyle\frac{1}{2}i)^2;a,b,c,d)\nonumber\\ -& &{}=-2ix W_{n+1}(x^2;a-\textstyle\frac{1}{2},b-\textstyle\frac{1}{2}, -c-\textstyle\frac{1}{2},d-\textstyle\frac{1}{2}) -\end{eqnarray} -or equivalently -\begin{eqnarray} -\label{shift2WilsonII} -& &\frac{\delta\left[\omega(x;a,b,c,d)W_n(x^2;a,b,c,d)\right]}{\delta x^2}\nonumber\\ -& &{}=\omega(x;a-\textstyle\frac{1}{2},b-\textstyle\frac{1}{2}, -c-\textstyle\frac{1}{2},d-\textstyle\frac{1}{2}) -W_{n+1}(x^2;a-\textstyle\frac{1}{2},b-\textstyle\frac{1}{2},c-\textstyle\frac{1}{2},d-\textstyle\frac{1}{2}), -\end{eqnarray} -where -$$\omega(x;a,b,c,d):=\frac{1}{2ix}\left|\frac{\Gamma(a+ix)\Gamma(b+ix)\Gamma(c+ix)\Gamma(d+ix)}{\Gamma(2ix)}\right|^2.$$ - -\subsection*{Rodrigues-type formula} -\begin{eqnarray} -\label{RodWilson} -& &\omega(x;a,b,c,d)W_n(x^2;a,b,c,d)\nonumber\\ -& &{}=\left(\frac{\delta}{\delta x^2}\right)^n -\left[\omega(x;a+\textstyle\frac{1}{2}n,b+\textstyle\frac{1}{2}n, -c+\textstyle\frac{1}{2}n,d+\textstyle\frac{1}{2}n)\right]. -\end{eqnarray} - -\subsection*{Generating functions} -\begin{equation} -\label{GenWilson1} -\hyp{2}{1}{a+ix,b+ix}{a+b}{t}\,\hyp{2}{1}{c-ix,d-ix}{c+d}{t}= -\sum_{n=0}^{\infty}\frac{W_n(x^2;a,b,c,d)t^n}{(a+b)_n(c+d)_nn!}. -\end{equation} - -\begin{equation} -\label{GenWilson2} -\hyp{2}{1}{a+ix,c+ix}{a+c}{t}\,\hyp{2}{1}{b-ix,d-ix}{b+d}{t}= -\sum_{n=0}^{\infty}\frac{W_n(x^2;a,b,c,d)t^n}{(a+c)_n(b+d)_nn!}. -\end{equation} - -\begin{equation} -\label{GenWilson3} -\hyp{2}{1}{a+ix,d+ix}{a+d}{t}\,\hyp{2}{1}{b-ix,c-ix}{b+c}{t}= -\sum_{n=0}^{\infty}\frac{W_n(x^2;a,b,c,d)t^n}{(a+d)_n(b+c)_nn!}. -\end{equation} - -\begin{eqnarray} -\label{GenWilson4} -& &(1-t)^{1-a-b-c-d}\nonumber\\ -& &{}\mathindent\times\hyp{4}{3}{\frac{1}{2}(a+b+c+d-1),\frac{1}{2}(a+b+c+d),a+ix,a-ix} -{a+b,a+c,a+d}{-\frac{4t}{(1-t)^2}}\nonumber\\ -& &{}=\sum_{n=0}^{\infty}\frac{(a+b+c+d-1)_n}{(a+b)_n(a+c)_n(a+d)_nn!}W_n(x^2;a,b,c,d)t^n. -\end{eqnarray} - -\subsection*{Limit relations} - -\subsubsection*{Wilson $\rightarrow$ Continuous dual Hahn} -The continuous dual Hahn polynomials given by (\ref{DefContDualHahn}) can be found from the -Wilson polynomials by dividing by $(a+d)_n$ and letting $d\rightarrow\infty$: -\begin{equation} -\lim_{d\rightarrow\infty}\frac{W_n(x^2;a,b,c,d)}{(a+d)_n}=S_n(x^2;a,b,c). -\end{equation} - -\subsubsection*{Wilson $\rightarrow$ Continuous Hahn} -The continuous Hahn polynomials given by (\ref{DefContHahn}) are obtained from the Wilson -polynomials by the substitutions $a\rightarrow a-it$, $b\rightarrow b-it$, $c\rightarrow c+it$, -$d\rightarrow d+it$ and $x\rightarrow x+t$ and the limit $t\rightarrow\infty$ in the following -way: -\begin{equation} -\lim_{t\rightarrow\infty} -\frac{W_n((x+t)^2;a-it,b-it,c+it,d+it)}{(-2t)^nn!}=p_n(x;a,b,c,d). -\end{equation} - -\subsubsection*{Wilson $\rightarrow$ Jacobi} -The Jacobi polynomials given by (\ref{DefJacobi}) can be found from the Wilson -polynomials by substituting $a=b=\frac{1}{2}(\alpha+1)$, $c=\frac{1}{2}(\beta+1)+it$, -$d=\frac{1}{2}(\beta+1)-it$ and $x\rightarrow t\sqrt{\frac{1}{2}(1-x)}$ in the -definition (\ref{DefWilson}) of the Wilson polynomials and taking the limit -$t\rightarrow\infty$. In fact we have -\begin{eqnarray} -& &\lim_{t\rightarrow\infty}\frac{W_n(\frac{1}{2}(1-x)t^2;\frac{1}{2}(\alpha+1), -\frac{1}{2}(\alpha+1),\frac{1}{2}(\beta+1)+it,\frac{1}{2}(\beta+1)-it)}{t^{2n}n!}\nonumber\\ -& &{}=P_n^{(\alpha,\beta)}(x). -\end{eqnarray} - -\subsection*{Remarks} -Note that for $k0$, $c=\bar{a}$ and $d=\bar{b}$, then -\begin{eqnarray} -\label{OrtContHahn} -& &\frac{1}{2\pi}\int_{-\infty}^{\infty}\Gamma(a+ix)\Gamma(b+ix)\Gamma(c-ix)\Gamma(d-ix) -p_m(x;a,b,c,d)p_n(x;a,b,c,d)\,dx\nonumber\\ -& &{}=\frac{\Gamma(n+a+c)\Gamma(n+a+d)\Gamma(n+b+c)\Gamma(n+b+d)} -{(2n+a+b+c+d-1)\Gamma(n+a+b+c+d-1)n!}\,\delta_{mn}. -\end{eqnarray} - -\subsection*{Recurrence relation} -\begin{equation} -\label{RecContHahn} -(a+ix)\tilde{p}_n(x)=A_n\tilde{p}_{n+1}(x)-\left(A_n+C_n\right)\tilde{p}_n(x)+C_n\tilde{p}_{n-1}(x), -\end{equation} -where -$$\tilde{p}_n(x):=\tilde{p}_n(x;a,b,c,d)=\frac{n!}{i^n(a+c)_n(a+d)_n}p_n(x;a,b,c,d)$$ -and -$$\left\{\begin{array}{l} -\displaystyle A_n=-\frac{(n+a+b+c+d-1)(n+a+c)(n+a+d)}{(2n+a+b+c+d-1)(2n+a+b+c+d)}\\[5mm] -\displaystyle C_n=\frac{n(n+b+c-1)(n+b+d-1)}{(2n+a+b+c+d-2)(2n+a+b+c+d-1)}. -\end{array}\right.$$ - -\subsection*{Normalized recurrence relation} -\begin{equation} -\label{NormRecContHahn} -xp_n(x)=p_{n+1}(x)+i(A_n+C_n+a)p_n(x)-A_{n-1}C_np_{n-1}(x), -\end{equation} -where -$$p_n(x;a,b,c,d)=\frac{(n+a+b+c+d-1)_n}{n!}p_n(x).$$ - -\subsection*{Difference equation} -\begin{eqnarray} -\label{dvContHahn} -& &n(n+a+b+c+d-1)y(x)\nonumber\\ -& &{}=B(x)y(x+i)-\left[B(x)+D(x)\right]y(x)+D(x)y(x-i), -\end{eqnarray} -where -$$y(x)=p_n(x;a,b,c,d)$$ -and -$$\left\{\begin{array}{l} -\displaystyle B(x)=(c-ix)(d-ix)\\[5mm] -\displaystyle D(x)=(a+ix)(b+ix). -\end{array}\right.$$ - -\subsection*{Forward shift operator} -\begin{eqnarray} -\label{shift1ContHahnI} -& &p_n(x+\textstyle\frac{1}{2}i;a,b,c,d)-p_n(x-\textstyle\frac{1}{2}i;a,b,c,d)\nonumber\\ -& &{}=i(n+a+b+c+d-1)p_{n-1}(x;a+\textstyle\frac{1}{2}, -b+\textstyle\frac{1}{2},c+\textstyle\frac{1}{2},d+\textstyle\frac{1}{2}) -\end{eqnarray} -or equivalently -\begin{equation} -\label{shift1ContHahnII} -\frac{\delta p_n(x;a,b,c,d)}{\delta x}=(n+a+b+c+d-1) -p_{n-1}(x;a+\textstyle\frac{1}{2},b+\textstyle\frac{1}{2}, -c+\textstyle\frac{1}{2},d+\textstyle\frac{1}{2}). -\end{equation} - -\subsection*{Backward shift operator} -\begin{eqnarray} -\label{shift2ContHahnI} -& &(c-\textstyle\frac{1}{2}-ix)(d-\textstyle\frac{1}{2}-ix) -p_n(x+\textstyle\frac{1}{2}i;a,b,c,d)\nonumber\\ -& &{}\mathindent{}-(a-\textstyle\frac{1}{2}+ix)(b-\textstyle\frac{1}{2}+ix) -p_n(x-\textstyle\frac{1}{2}i;a,b,c,d)\nonumber\\ -& &{}=\frac{n+1}{i}p_{n+1}(x;a-\textstyle\frac{1}{2}, -b-\textstyle\frac{1}{2},c-\textstyle\frac{1}{2},d-\textstyle\frac{1}{2}) -\end{eqnarray} -or equivalently -\begin{eqnarray} -\label{shift2ContHahnII} -& &\frac{\delta\left[\omega(x;a,b,c,d)p_n(x;a,b,c,d)\right]}{\delta x}\nonumber\\ -& &{}=-(n+1)\omega(x;a-\textstyle\frac{1}{2},b-\textstyle\frac{1}{2}, -c-\textstyle\frac{1}{2},d-\textstyle\frac{1}{2})\nonumber\\ -& &{}\mathindent\times p_{n+1}(x;a-\textstyle\frac{1}{2},b-\textstyle\frac{1}{2},c-\textstyle\frac{1}{2},d-\textstyle\frac{1}{2}), -\end{eqnarray} -where -$$\omega(x;a,b,c,d)=\Gamma(a+ix)\Gamma(b+ix)\Gamma(c-ix)\Gamma(d-ix).$$ - -\subsection*{Rodrigues-type formula} -\begin{eqnarray} -\label{RodContHahn} -& &\omega(x;a,b,c,d)p_n(x;a,b,c,d)\nonumber\\ -& &{}=\frac{(-1)^n}{n!}\left(\frac{\delta}{\delta x}\right)^n -\left[\omega(x;a+\textstyle\frac{1}{2}n,b+\textstyle\frac{1}{2}n, -c+\textstyle\frac{1}{2}n,d+\textstyle\frac{1}{2}n)\right]. -\end{eqnarray} - -\subsection*{Generating functions} -\begin{equation} -\label{GenContHahn1} -\hyp{1}{1}{a+ix}{a+c}{-it}\,\hyp{1}{1}{d-ix}{b+d}{it}= -\sum_{n=0}^{\infty}\frac{p_n(x;a,b,c,d)}{(a+c)_n(b+d)_n}t^n. -\end{equation} - -\begin{equation} -\label{GenContHahn2} -\hyp{1}{1}{a+ix}{a+d}{-it}\,\hyp{1}{1}{c-ix}{b+c}{it}= -\sum_{n=0}^{\infty}\frac{p_n(x;a,b,c,d)}{(a+d)_n(b+c)_n}t^n. -\end{equation} - -\begin{eqnarray} -\label{GenContHahn3} -& &(1-t)^{1-a-b-c-d}\,\hyp{3}{2}{\frac{1}{2}(a+b+c+d-1),\frac{1}{2}(a+b+c+d),a+ix} -{a+c,a+d}{-\frac{4t}{(1-t)^2}}\nonumber\\ -& &{}=\sum_{n=0}^{\infty} -\frac{(a+b+c+d-1)_n}{(a+c)_n(a+d)_ni^n}p_n(x;a,b,c,d)t^n. -\end{eqnarray} - -\subsection*{Limit relations} - -\subsubsection*{Wilson $\rightarrow$ Continuous Hahn} -The continuous Hahn polynomials are obtained from the Wilson polynomials given by -(\ref{DefWilson}) by the substitution $a\rightarrow a-it$, $b\rightarrow b-it$, -$c\rightarrow c+it$, $d\rightarrow d+it$ and $x\rightarrow x+t$ and the limit -$t\rightarrow\infty$ in the following way: -$$\lim_{t\rightarrow\infty} -\frac{W_n((x+t)^2;a-it,b-it,c+it,d+it)}{(-2t)^nn!}=p_n(x;a,b,c,d).$$ - -\subsubsection*{Continuous Hahn $\rightarrow$ Meixner-Pollaczek} -The Meixner-Pollaczek polynomials given by (\ref{DefMP}) can be obtained from the -continuous Hahn polynomials by setting $x\rightarrow x+t$, $a=\lambda-it$, $c=\lambda+it$ -and $b=d=t\tan\phi$ and taking the limit $t\rightarrow\infty$: -\begin{equation} -\lim_{t\rightarrow\infty}\frac{p_n(x+t;\lambda-it,t\tan\phi,\lambda+it,t\tan\phi)}{t^n} -=\frac{P_n^{(\lambda)}(x;\phi)}{(\cos\phi)^n}. -\end{equation} - -\subsubsection*{Continuous Hahn $\rightarrow$ Jacobi} -The Jacobi polynomials given by (\ref{DefJacobi}) follow from the continuous Hahn polynomials -by the substitution $x\rightarrow \frac{1}{2}xt$, $a=\frac{1}{2}(\alpha+1-it)$, -$b=\frac{1}{2}(\beta+1+it)$, $c=\frac{1}{2}(\alpha+1+it)$ and $d=\frac{1}{2}(\beta+1-it)$, -division by $t^n$ and the limit $t\rightarrow\infty$: -\begin{eqnarray} -& &\lim_{t\rightarrow\infty} -\frac{p_n(\frac{1}{2}xt;\frac{1}{2}(\alpha+1-it),\frac{1}{2}(\beta+1+it), -\frac{1}{2}(\alpha+1+it),\frac{1}{2}(\beta+1-it))}{t^n}\nonumber\\ -& &{}=P_n^{(\alpha,\beta)}(x). -\end{eqnarray} - -\subsubsection*{Continuous Hahn $\rightarrow$ Pseudo Jacobi} -The pseudo Jacobi polynomials given by (\ref{DefPseudoJacobi}) follow from the continuous -Hahn polynomials by the substitution $x\rightarrow xt$, $a=\frac{1}{2}(-N+i\nu-2t)$, -$b=\frac{1}{2}(-N-i\nu+2t)$, $c=\frac{1}{2}(-N-i\nu-2t)$ and $d=\frac{1}{2}(-N+i\nu+2t)$, -division by $t^n$ and the limit $t\rightarrow\infty$: -\begin{eqnarray} -& &\lim_{t\rightarrow\infty}\frac{p_n(xt;\frac{1}{2}(-N+i\nu-2t),\frac{1}{2}(-N-i\nu+2t), -\frac{1}{2}(-N+i\nu-2t),\frac{1}{2}(-N-i\nu+2t))}{t^n}\nonumber\\ -& &{}=\frac{(n-2N-1)_n}{n!}P_n(x;\nu,N). -\end{eqnarray} - -\subsection*{Remark} -Since we have for $k0$ and $(c,d)=(\overline a,\overline b)$ or $(\overline b,\overline a)$.\\ -Thus, under these assumptions, the continuous Hahn polynomial -$p_n(x;a,b,c,d)$ -is symmetric in $a,b$ and in $c,d$. -This follows from the orthogonality relation (9.4.2) -together with the value of its coefficient of $x^n$ given in (9.4.4b).\\ -As a consequence, it is sufficient to give generating function (9.4.11). Then the generating -function (9.4.12) will follow by symmetry in the parameters. -% -\paragraph{\large\bf KLSadd: Uniqueness of orthogonality measure}The coefficient of $p_{n-1}(x)$ in (9.4.4) behaves as $O(n^2)$ as $n\to\iy$. -Hence \eqref{93} holds, by which the orthogonality measure is unique. -% -\paragraph{\large\bf KLSadd: Special cases}In the following special case there is a reduction to -Meixner-Pollaczek: -\begin{equation} -p_n(x;a,a+\thalf,a,a+\thalf)= -\frac{(2a)_n (2a+\thalf)_n}{(4a)_n}\,P_n^{(2a)}(2x;\thalf\pi). -\end{equation} -See \myciteKLS{342}{(2.6)} (note that in \myciteKLS{342}{(2.3)} the -Meixner-Pollaczek polynonmials are defined different from (9.7.1), -without a constant factor in front). - -For $0-1$ and $\beta>-1$, or for $\alpha<-N$ and $\beta<-N$, we have -\begin{eqnarray} -\label{OrtHahn} -& &\sum_{x=0}^N\binom{\alpha +x}{x}\binom{\beta+N-x}{N-x}Q_m(x;\alpha,\beta,N)Q_n(x;\alpha,\beta,N)\nonumber\\ -& &{}=\frac{(-1)^n(n+\alpha+\beta+1)_{N+1}(\beta+1)_nn!}{(2n+\alpha+\beta+1)(\alpha+1)_n(-N)_nN!}\,\delta_{mn}. -\end{eqnarray} - -\subsection*{Recurrence relation} -\begin{equation} -\label{RecHahn} --xQ_n(x)=A_nQ_{n+1}(x)-\left(A_n+C_n\right)Q_n(x)+C_nQ_{n-1}(x), -\end{equation} -where -$$Q_n(x):=Q_n(x;\alpha,\beta,N)$$ -and -$$\left\{\begin{array}{l} -\displaystyle A_n=\frac{(n+\alpha+\beta+1)(n+\alpha+1)(N-n)}{(2n+\alpha+\beta+1)(2n+\alpha+\beta+2)}\\[5mm] -\displaystyle C_n=\frac{n(n+\alpha+\beta+N+1)(n+\beta)}{(2n+\alpha+\beta)(2n+\alpha+\beta+1)}. -\end{array}\right.$$ - -\subsection*{Normalized recurrence relation} -\begin{equation} -\label{NormRecHahn} -xp_n(x)=p_{n+1}(x)+\left(A_n+C_n\right)p_n(x)+A_{n-1}C_np_{n-1}(x), -\end{equation} -where -$$Q_n(x;\alpha,\beta,N)=\frac{(n+\alpha+\beta+1)_n}{(\alpha+1)_n(-N)_n}p_n(x).$$ - -\subsection*{Difference equation} -\begin{equation} -\label{dvHahn} -n(n+\alpha+\beta+1)y(x)=B(x)y(x+1)-\left[B(x)+D(x)\right]y(x)+D(x)y(x-1), -\end{equation} -where -$$y(x)=Q_n(x;\alpha,\beta,N)$$ -and -$$\left\{\begin{array}{l} -\displaystyle B(x)=(x+\alpha+1)(x-N)\\[5mm] -\displaystyle D(x)=x(x-\beta-N-1). -\end{array}\right.$$ - -\subsection*{Forward shift operator} -\begin{eqnarray} -\label{shift1HahnI} -& &Q_n(x+1;\alpha,\beta,N)-Q_n(x;\alpha,\beta,N)\nonumber\\ -& &{}=-\frac{n(n+\alpha+\beta+1)}{(\alpha+1)N}Q_{n-1}(x;\alpha+1,\beta+1,N-1) -\end{eqnarray} -or equivalently -\begin{equation} -\label{shift1HahnII} -\Delta Q_n(x;\alpha,\beta,N)=-\frac{n(n+\alpha+\beta+1)}{(\alpha+1)N}Q_{n-1}(x;\alpha+1,\beta+1,N-1). -\end{equation} - -\subsection*{Backward shift operator} -\begin{eqnarray} -\label{shift2HahnI} -& &(x+\alpha)(N+1-x)Q_n(x;\alpha,\beta,N)-x(\beta+N+1-x)Q_n(x-1;\alpha,\beta,N)\nonumber\\ -& &{}=\alpha(N+1)Q_{n+1}(x;\alpha-1,\beta-1,N+1) -\end{eqnarray} -or equivalently -\begin{eqnarray} -\label{shift2HahnII} -& &\nabla\left[\omega(x;\alpha,\beta,N)Q_n(x;\alpha,\beta,N)\right]\nonumber\\ -& &{}=\frac{N+1}{\beta}\omega(x;\alpha-1,\beta-1,N+1)Q_{n+1}(x;\alpha-1,\beta-1,N+1), -\end{eqnarray} -where -$$\omega(x;\alpha,\beta,N)=\binom{\alpha+x}{x}\binom{\beta+N-x}{N-x}.$$ - -\subsection*{Rodrigues-type formula} -\begin{eqnarray} -\label{RodHahn} -& &\omega(x;\alpha,\beta,N)Q_n(x;\alpha,\beta,N)\nonumber\\ -& &{}=\frac{(-1)^n(\beta+1)_n}{(-N)_n}\nabla^n\left[\omega(x;\alpha+n,\beta+n,N-n)\right]. -\end{eqnarray} - -\subsection*{Generating functions} -For $x=0,1,2,\ldots,N$ we have -\begin{equation} -\label{GenHahn1} -\hyp{1}{1}{-x}{\alpha+1}{-t}\,\hyp{1}{1}{x-N}{\beta+1}{t} -=\sum_{n=0}^N\frac{(-N)_n}{(\beta+1)_nn!}Q_n(x;\alpha,\beta,N)t^n. -\end{equation} - -\begin{eqnarray} -\label{GenHahn2} -& &\hyp{2}{0}{-x,-x+\beta+N+1}{-}{-t}\,\hyp{2}{0}{x-N,x+\alpha+1}{-}{t}\nonumber\\ -& &{}=\sum_{n=0}^N\frac{(-N)_n(\alpha+1)_n}{n!}Q_n(x;\alpha,\beta,N)t^n. -\end{eqnarray} - -\begin{eqnarray} -\label{GenHahn3} -& &\left[(1-t)^{-\alpha-\beta-1}\,\hyp{3}{2}{\frac{1}{2}(\alpha+\beta+1),\frac{1}{2}(\alpha+\beta+2),-x} -{\alpha+1,-N}{-\frac{4t}{(1-t)^2}}\right]_N\nonumber\\ -& &{}=\sum_{n=0}^N\frac{(\alpha+\beta+1)_n}{n!}Q_n(x;\alpha,\beta,N)t^n. -\end{eqnarray} - -\subsection*{Limit relations} - -\subsubsection*{Racah $\rightarrow$ Hahn} -If we take $\gamma+1=-N$ and let $\delta\rightarrow\infty$ in the definition (\ref{DefRacah}) -of the Racah polynomials, we obtain the Hahn polynomials. Hence -$$\lim_{\delta\rightarrow\infty} -R_n(\lambda(x);\alpha,\beta,-N-1,\delta)=Q_n(x;\alpha,\beta,N).$$ -And if we take $\delta=-\beta-N-1$ and let $\gamma\rightarrow\infty$ in the definition (\ref{DefRacah}) -of the Racah polynomials, we also obtain the Hahn polynomials: -$$\lim_{\gamma\rightarrow\infty} -R_n(\lambda(x);\alpha,\beta,\gamma,-\beta-N-1)=Q_n(x;\alpha,\beta,N).$$ -Another way to do this is to take $\alpha+1=-N$ and $\beta\rightarrow\beta+\gamma+N+1$ in -the definition (\ref{DefRacah}) of the Racah polynomials and then take the limit -$\delta\rightarrow\infty$. In that case we obtain the Hahn polynomials in the following way: -$$\lim_{\delta\rightarrow\infty} -R_n(\lambda(x);-N-1,\beta+\gamma+N+1,\gamma,\delta)=Q_n(x;\gamma,\beta,N).$$ - -\subsubsection*{Hahn $\rightarrow$ Jacobi} -To find the Jacobi polynomials given by (\ref{DefJacobi}) from the Hahn polynomials we take -$x\rightarrow Nx$ and let $N\rightarrow\infty$. In fact we have -\begin{equation} -\lim_{N\rightarrow\infty} -Q_n(Nx;\alpha,\beta,N)=\frac{P_n^{(\alpha,\beta)}(1-2x)}{P_n^{(\alpha,\beta)}(1)}. -\end{equation} - -\subsubsection*{Hahn $\rightarrow$ Meixner} -The Meixner polynomials given by (\ref{DefMeixner}) can be obtained from the Hahn polynomials -by taking $\alpha=b-1$, $\beta=N(1-c)c^{-1}$ and letting $N\rightarrow\infty$: -\begin{equation} -\lim_{N\rightarrow\infty} -Q_n(x;b-1,N(1-c)c^{-1},N)=M_n(x;b,c). -\end{equation} - -\subsubsection*{Hahn $\rightarrow$ Krawtchouk} -The Krawtchouk polynomials given by (\ref{DefKrawtchouk}) are obtained from the Hahn polynomials -if we take $\alpha=pt$ and $\beta=(1-p)t$ and let $t\rightarrow\infty$: -\begin{equation} -\lim_{t\rightarrow\infty}Q_n(x;pt,(1-p)t,N)=K_n(x;p,N). -\end{equation} - -\subsection*{Remark} -If we interchange the role of $x$ and $n$ in (\ref{DefHahn}) we obtain the -dual Hahn polynomials given by (\ref{DefDualHahn}). - -\noindent -Since -$$Q_n(x;\alpha,\beta,N)=R_x(\lambda(n);\alpha,\beta,N)$$ -we obtain the dual orthogonality relation for the Hahn polynomials from the -orthogonality relation (\ref{OrtDualHahn}) of the dual Hahn polynomials: -\begin{eqnarray*} -& &\sum_{n=0}^N\frac{(2n+\alpha+\beta+1)(\alpha+1)_n(-N)_nN!}{(-1)^n(n+\alpha+\beta+1)_{N+1}(\beta+1)_nn!} -Q_n(x;\alpha,\beta,N)Q_n(y;\alpha,\beta,N)\\ -& &{}=\frac{\delta_{xy}}{\dbinom{\alpha+x}{x}\dbinom{\beta+N-x}{N-x}},\quad x,y \in \{0,1,2,\ldots,N\}. -\end{eqnarray*} -% RS: add begin\label{sec9.5} -% -\paragraph{\large\bf KLSadd: Special values}\begin{equation} -Q_n(0;\al,\be,N)=1,\quad -Q_n(N;\al,\be,N)=\frac{(-1)^n(\be+1)_n}{(\al+1)_n}\,. -\label{95} -\end{equation} -Use (9.5.1) and compare with (9.8.1) and \eqref{50}. - -From (9.5.3) and \eqref{1} it follows that -\begin{equation} -Q_{2n}(N;\al,\al,2N)=\frac{(\thalf)_n(N+\al+1)_n}{(-N+\thalf)_n(\al+1)_n}\,. -\label{30} -\end{equation} -From (9.5.1) and \mycite{DLMF}{(15.4.24)} it follows that -\begin{equation} -Q_N(x;\al,\be,N)=\frac{(-N-\be)_x}{(\al+1)_x}\qquad(x=0,1,\ldots,N). -\label{44} -\end{equation} -% -\paragraph{\large\bf KLSadd: Symmetries}By the orthogonality relation (9.5.2): -\begin{equation} -\frac{Q_n(N-x;\al,\be,N)}{Q_n(N;\al,\be,N)}=Q_n(x;\be,\al,N), -\label{96} -\end{equation} -It follows from \eqref{97} and \eqref{45} that -\begin{equation} -\frac{Q_{N-n}(x;\al,\be,N)}{Q_N(x;\al,\be,N)} -=Q_n(x;-N-\be-1,-N-\al-1,N) -\qquad(x=0,1,\ldots,N). -\label{100} -\end{equation} -% -\paragraph{\large\bf KLSadd: Duality}The Remark on p.208 gives the duality between Hahn and dual Hahn polynomials: -% -\begin{equation} -Q_n(x;\al,\be,N)=R_x(n(n+\al+\be+1);\al,\be,N)\quad(n,x\in\{0,1,\ldots N\}). -\label{45} -\end{equation} -% -% RS: add end -\subsection*{References} -\cite{AlSalam90}, \cite{AndrewsAskey85}, \cite{Area+II}, \cite{Askey75}, -\cite{Askey89I}, \cite{Askey2005}, \cite{AskeyGasper77}, \cite{AskeyWilson85}, -\cite{AtakRahmanSuslov}, \cite{AtakSuslov88}, \cite{Chihara78}, -\cite{Ciesielski}, \cite{Cooper+}, \cite{Dette95}, \cite{Dunkl76}, -\cite{Dunkl78I}, \cite{Gasper73I}, \cite{Gasper74}, \cite{HoareRahman}, -\cite{Ismail77}, \cite{Ismail2005II}, \cite{Karlin61}, \cite{Koorn81}, \cite{Koorn88}, -\cite{LabelleYehI}, \cite{LabelleYehII}, \cite{Laine}, \cite{Lesky62}, -\cite{Lesky88}, \cite{Lesky89}, \cite{Lesky94I}, \cite{Lesky95II}, -\cite{LewanowiczII}, \cite{Neuman}, \cite{Nikiforov+}, \cite{NikiforovUvarov}, -\cite{Rahman76III}, \cite{Rahman78I}, \cite{Rahman78II}, \cite{Rahman81III}, -\cite{Sablonniere}, \cite{Stanton84}, \cite{Stanton90}, \cite{Wilson80}, \cite{Wilson70II}, -\cite{Zarzo+}. - - -\section{Dual Hahn}\index{Dual Hahn polynomials}\index{Hahn polynomials!Dual} - -\par\setcounter{equation}{0} - -\subsection*{Hypergeometric representation} -\begin{equation} -\label{DefDualHahn} -R_n(\lambda(x);\gamma,\delta,N)= -\hyp{3}{2}{-n,-x,x+\gamma+\delta+1}{\gamma+1,-N}{1},\quad n=0,1,2,\ldots,N, -\end{equation} -where -$$\lambda(x)=x(x+\gamma+\delta+1).$$ - -\subsection*{Orthogonality relation} -For $\gamma>-1$ and $\delta>-1$, or for $\gamma<-N$ and $\delta<-N$, we have -\begin{eqnarray} -\label{OrtDualHahn} -& &\sum_{x=0}^N\frac{(2x+\gamma+\delta+1)(\gamma+1)_x(-N)_xN!}{(-1)^x(x+\gamma+\delta+1)_{N+1}(\delta+1)_xx!} -R_m(\lambda(x);\gamma,\delta,N)R_n(\lambda(x);\gamma,\delta,N)\nonumber\\ -& &{}=\frac{\,\delta_{mn}}{\dbinom{\gamma+n}{n}\dbinom{\delta+N-n}{N-n}}. -\end{eqnarray} - -\subsection*{Recurrence relation} -\begin{equation} -\label{RecDualHahn} -\lambda(x)R_n(\lambda(x)) -=A_nR_{n+1}(\lambda(x))-\left(A_n+C_n\right)R_n(\lambda(x))+C_nR_{n-1}(\lambda(x)), -\end{equation} -where -$$R_n(\lambda(x)):=R_n(\lambda(x);\gamma,\delta,N)$$ -and -$$\left\{\begin{array}{l} -\displaystyle A_n=(n+\gamma+1)(n-N)\\[5mm] -\displaystyle C_n=n(n-\delta-N-1). -\end{array}\right.$$ - -\subsection*{Normalized recurrence relation} -\begin{equation} -\label{NormRecDualHahn} -xp_n(x)=p_{n+1}(x)-(A_n+C_n)p_n(x)+A_{n-1}C_np_{n-1}(x), -\end{equation} -where -$$R_n(\lambda(x);\gamma,\delta,N)=\frac{1}{(\gamma+1)_n(-N)_n}p_n(\lambda(x)).$$ - -\subsection*{Difference equation} -\begin{equation} -\label{dvDualHahn} --ny(x)=B(x)y(x+1)-\left[B(x)+D(x)\right]y(x)+D(x)y(x-1), -\end{equation} -where -$$y(x)=R_n(\lambda(x);\gamma,\delta,N)$$ -and -$$\left\{\begin{array}{l} -\displaystyle B(x)=\frac{(x+\gamma+1)(x+\gamma+\delta+1)(N-x)}{(2x+\gamma+\delta+1)(2x+\gamma+\delta+2)}\\[5mm] -\displaystyle D(x)=\frac{x(x+\gamma+\delta+N+1)(x+\delta)}{(2x+\gamma+\delta)(2x+\gamma+\delta+1)}. -\end{array}\right.$$ - -\subsection*{Forward shift operator} -\begin{eqnarray} -\label{shift1DualHahnI} -& &R_n(\lambda(x+1);\gamma,\delta,N)-R_n(\lambda(x);\gamma,\delta,N)\nonumber\\ -& &{}=-\frac{n(2x+\gamma+\delta+2)}{(\gamma+1)N}R_{n-1}(\lambda(x);\gamma+1,\delta,N-1) -\end{eqnarray} -or equivalently -\begin{equation} -\label{shift1DualHahnII} -\frac{\Delta R_n(\lambda(x);\gamma,\delta,N)}{\Delta\lambda(x)}= --\frac{n}{(\gamma+1)N}R_{n-1}(\lambda(x);\gamma+1,\delta,N-1). -\end{equation} - -\subsection*{Backward shift operator} -\begin{eqnarray} -\label{shift2DualHahnI} -& &(x+\gamma)(x+\gamma+\delta)(N+1-x)R_n(\lambda(x);\gamma,\delta,N)\nonumber\\ -& &{}\mathindent{}-x(x+\gamma+\delta+N+1)(x+\delta)R_n(\lambda(x-1);\gamma,\delta,N)\nonumber\\ -& &{}=\gamma(N+1)(2x+\gamma+\delta)R_{n+1}(\lambda(x);\gamma-1,\delta,N+1) -\end{eqnarray} -or equivalently -\begin{eqnarray} -\label{shift2DualHahnII} -& &\frac{\nabla\left[\omega(x;\gamma,\delta,N)R_n(\lambda(x);\gamma,\delta,N)\right]}{\nabla\lambda(x)}\nonumber\\ -& &{}=\frac{1}{\gamma+\delta}\omega(x;\gamma-1,\delta,N+1)R_{n+1}(\lambda(x);\gamma-1,\delta,N+1), -\end{eqnarray} -where -$$\omega(x;\gamma,\delta,N)=\frac{(-1)^x(\gamma+1)_x(\gamma+\delta+1)_x(-N)_x} -{(\gamma+\delta+N+2)_x(\delta+1)_xx!}.$$ - -\newpage - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodDualHahn} -\omega(x;\gamma,\delta,N)R_n(\lambda(x);\gamma,\delta,N) -=(\gamma+\delta+1)_n\left(\nabla_{\lambda}\right)^n\left[\omega(x;\gamma+n,\delta,N-n)\right], -\end{equation} -where -$$\nabla_{\lambda}:=\frac{\nabla}{\nabla\lambda(x)}.$$ - -\subsection*{Generating functions} -For $x=0,1,2,\ldots,N$ we have -\begin{equation} -\label{GenDualHahn1} -(1-t)^{N-x}\,\hyp{2}{1}{-x,-x-\delta}{\gamma+1}{t} -=\sum_{n=0}^N\frac{(-N)_n}{n!}R_n(\lambda(x);\gamma,\delta,N)t^n. -\end{equation} - -\begin{eqnarray} -\label{GenDualHahn2} -& &(1-t)^x\,\hyp{2}{1}{x-N,x+\gamma+1}{-\delta-N}{t}\nonumber\\ -& &=\sum_{n=0}^N\frac{(\gamma+1)_n(-N)_n}{(-\delta-N)_nn!}R_n(\lambda(x);\gamma,\delta,N)t^n. -\end{eqnarray} - -\begin{equation} -\label{GenDualHahn3} -\left[\expe^t\,\hyp{2}{2}{-x,x+\gamma+\delta+1}{\gamma+1,-N}{-t}\right]_N=\sum_{n=0}^N -\frac{R_n(\lambda(x);\gamma,\delta,N)}{n!}t^n. -\end{equation} - -\begin{eqnarray} -\label{GenDualHahn4} -& &\left[(1-t)^{-\epsilon}\,\hyp{3}{2}{\epsilon,-x,x+\gamma+\delta+1}{\gamma+1,-N}{\frac{t}{t-1}}\right]_N\nonumber\\ -& &{}=\sum_{n=0}^N\frac{(\epsilon)_n}{n!}R_n(\lambda(x);\gamma,\delta,N)t^n, -\quad\textrm{$\epsilon$ arbitrary}. -\end{eqnarray} - -\subsection*{Limit relations} - -\subsubsection*{Racah $\rightarrow$ Dual Hahn} -If we take $\alpha+1=-N$ and let $\beta\rightarrow\infty$ in the definition (\ref{DefRacah}) -of the Racah polynomials, then we obtain the dual Hahn polynomials: -$$\lim_{\beta\rightarrow\infty} -R_n(\lambda(x);-N-1,\beta,\gamma,\delta)=R_n(\lambda(x);\gamma,\delta,N).$$ -And if we take $\beta=-\delta-N-1$ and let $\alpha\rightarrow\infty$ in the definition (\ref{DefRacah}) -of the Racah polynomials, then we also obtain the dual Hahn polynomials: -$$\lim_{\alpha\rightarrow\infty} -R_n(\lambda(x);\alpha,-\delta-N-1,\gamma,\delta)=R_n(\lambda(x);\gamma,\delta,N).$$ -Finally, if we take $\gamma+1=-N$ and $\delta\rightarrow\alpha+\delta+N+1$ in the definition -(\ref{DefRacah}) of the Racah polynomials and take the limit -$\beta\rightarrow\infty$ we find the dual Hahn polynomials in the following way: -$$\lim_{\beta\rightarrow\infty} -R_n(\lambda(x);\alpha,\beta,-N-1,\alpha+\delta+N+1)=R_n(\lambda(x);\alpha,\delta,N).$$ - -\subsubsection*{Dual Hahn $\rightarrow$ Meixner} -The Meixner polynomials given by (\ref{DefMeixner}) are obtained from the dual Hahn polynomials -if we take $\gamma=\beta-1$ and $\delta=N(1-c)c^{-1}$ and let $N\rightarrow\infty$: -\begin{equation} -\lim_{N\rightarrow\infty} -R_n(\lambda(x);\beta-1,N(1-c)c^{-1},N)=M_n(x;\beta,c). -\end{equation} - -\subsubsection*{Dual Hahn $\rightarrow$ Krawtchouk} -The Krawtchouk polynomials given by (\ref{DefKrawtchouk}) can be obtained from the dual -Hahn polynomials by setting $\gamma=pt$, $\delta=(1-p)t$ and letting $t\rightarrow\infty$: -\begin{equation} -\lim_{t\rightarrow\infty}R_n(\lambda(x);pt,(1-p)t,N)=K_n(x;p,N). -\end{equation} - -\subsection*{Remark} -If we interchange the role of $x$ and $n$ in the definition (\ref{DefDualHahn}) -of the dual Hahn polynomials we obtain the Hahn polynomials given by (\ref{DefHahn}). - -\noindent -Since -$$R_n(\lambda(x);\gamma,\delta,N)=Q_x(n;\gamma,\delta,N)$$ -we obtain the dual orthogonality relation for the dual Hahn polynomials -from the orthogonality relation (\ref{OrtHahn}) for the Hahn polynomials: -\begin{eqnarray*} -& &\sum_{n=0}^N\binom{\gamma+n}{n}\binom{\delta+N-n}{N-n} R_n(\lambda(x);\gamma,\delta,N)R_n(\lambda(y);\gamma,\delta,N)\\ -& &{}=\frac{(-1)^x(x+\gamma+\delta+1)_{N+1}(\delta+1)_xx!} -{(2x+\gamma+\delta+1)(\gamma+1)_x(-N)_xN!}\delta_{xy},\quad x,y \in \{0,1,2,\ldots,N\}. -\end{eqnarray*} -% RS: add begin\label{sec9.6} -% -\paragraph{\large\bf KLSadd: Special values}By \eqref{44} and \eqref{45} we have -\begin{equation} -R_n(N(N+\ga+\de+1);\ga,\de,N)=\frac{(-N-\de)_n}{(\ga+1)_n}\,. -\label{47} -\end{equation} -It follows from \eqref{95} and \eqref{45} that -\begin{equation} -R_N(x(x+\ga+\de+1);\ga,\de,N) -=\frac{(-1)^x(\de+1)_x}{(\ga+1)_x}\qquad(x=0,1,\ldots,N). -\label{101} -\end{equation} -% -\paragraph{\large\bf KLSadd: Symmetries}Write the weight in (9.6.2) as -\begin{equation} -w_x(\al,\be,N):=N!\,\frac{2x+\ga+\de+1}{(x+\ga+\de+1)_{N+1}}\, -\frac{(\ga+1)_x}{(\de+1)_x}\,\binom Nx. -\label{98} -\end{equation} -Then -\begin{equation} -(\de+1)_N\,w_{N-x}(\ga,\de,N)= -(-\ga-N)_N\,w_x(-\de-N-1,-\ga-N-1,N). -\label{99} -\end{equation} -Hence, by (9.6.2), -\begin{equation} -\frac{R_n((N-x)(N-x+\ga+\de+1);\ga,\de,N)}{R_n(N(N+\ga+\de+1);\ga,\de,N)} -=R_n(x(x-2N-\ga-\de-1);-N-\de-1,-N-\ga-1,N). -\label{97} -\end{equation} -Alternatively, \eqref{97} follows from (9.6.1) and -\mycite{DLMF}{(16.4.11)}. - -It follows from \eqref{96} and \eqref{45} that -\begin{equation} -\frac{R_{N-n}(x(x+\ga+\de+1);\ga,\de,N)} -{R_N(x(x+\ga+\de+1);\ga,\de,N)} -=R_n(x(x+\ga+\de+1);\de,\ga,N)\qquad(x=0,1,\ldots,N). -\label{102} -\end{equation} -% -\paragraph{\large\bf KLSadd: Re: (9.6.11).}The generating function (9.6.11) can be written in a more conceptual way as -\begin{equation} -(1-t)^x\,\hyp21{x-N,x+\ga+1}{-\de-N}t=\frac{N!}{(\de+1)_N}\, -\sum_{n=0}^N \om_n\,R_n(\la(x);\ga,\de,N)\,t^n, -\label{2} -\end{equation} -where -\begin{equation} -\om_n:=\binom{\ga+n}n \binom{\de+N-n}{N-n}, -\label{3} -\end{equation} -i.e., the denominator on the \RHS\ of (9.6.2). -By the duality between Hahn polynomials and dual Hahn polynomials (see \eqref{45}) the above generating function can be rewritten in -terms of Hahn polynomials: -\begin{equation} -(1-t)^n\,\hyp21{n-N,n+\al+1}{-\be-N}t=\frac{N!}{(\be+1)_N}\, -\sum_{x=0}^N w_x\,Q_n(x;\al,\be,N)\,t^x, -\label{4} -\end{equation} -where -\begin{equation} -w_x:=\binom{\al+x}x \binom{\be+N-x}{N-x}, -\label{5} -\end{equation} -i.e., the weight occurring in the orthogonality relation (9.5.2) -for Hahn polynomials. -\paragraph{\large\bf KLSadd: Re: (9.6.15).}There should be a closing bracket before the equality sign. -% -% RS: add end -\subsection*{References} -\cite{Askey2005}, \cite{AskeyWilson85}, \cite{AtakRahmanSuslov}, \cite{AtakSuslov88}, -\cite{Ismail2005II}, \cite{Karlin61}, \cite{Koorn81}, \cite{Koorn88}, \cite{Lesky93}, -\cite{Lesky94I}, \cite{Lesky95I}, \cite{Lesky95II}, \cite{Nikiforov+}, \cite{NikiforovUvarov}, -\cite{Rahman81II}, \cite{Stanton84}, \cite{Wilson80}. - - -\section{Meixner-Pollaczek}\index{Meixner-Pollaczek polynomials} - -\par\setcounter{equation}{0} - -\subsection*{Hypergeometric representation} -\begin{equation} -\label{DefMP} -P_n^{(\lambda)}(x;\phi)=\frac{(2\lambda)_n}{n!}\expe^{in\phi}\, -\hyp{2}{1}{-n,\lambda+ix}{2\lambda}{1-\expe^{-2i\phi}}. -\end{equation} - -\subsection*{Orthogonality relation} -\begin{equation} -\label{OrtMP} -\frac{1}{2\pi}\int_{-\infty}^{\infty} -\e^{(2\phi-\pi)x}\left|\Gamma(\lambda+ix)\right|^2 -P_m^{(\lambda)}(x;\phi)P_n^{(\lambda)}(x;\phi)\,dx -{}=\frac{\Gamma(n+2\lambda)}{(2\sin\phi)^{2\lambda}n!}\,\delta_{mn}, -% \constraint{ -% $\lambda > 0$ & -% $0 < \phi < \pi$ } -\end{equation} - -\subsection*{Recurrence relation} -\begin{eqnarray} -\label{RecMP} -& &(n+1)P_{n+1}^{(\lambda)}(x;\phi)-2\left[x\sin\phi+(n+\lambda)\cos\phi\right] -P_n^{(\lambda)}(x;\phi)\nonumber\\ -& &{}\mathindent{}+(n+2\lambda-1)P_{n-1}^{(\lambda)}(x;\phi)=0. -\end{eqnarray} - -\subsection*{Normalized recurrence relation} -\begin{equation} -\label{NormRecMP} -xp_n(x)=p_{n+1}(x)-\left(\frac{n+\lambda}{\tan\phi}\right)p_n(x)+ -\frac{n(n+2\lambda-1)}{4\sin^2\phi}p_{n-1}(x), -\end{equation} -where -$$P_n^{(\lambda)}(x;\phi)=\frac{(2\sin\phi)^n}{n!}p_n(x).$$ - -\subsection*{Difference equation} -\begin{eqnarray} -\label{dvMP} -& &\e^{i\phi}(\lambda-ix)y(x+i)+2i\left[x\cos\phi-(n+\lambda)\sin\phi\right]y(x)\nonumber\\ -& &{}\mathindent{}-\e^{-i\phi}(\lambda+ix)y(x-i)=0,\quad y(x)=P_n^{(\lambda)}(x;\phi). -\end{eqnarray} - -\subsection*{Forward shift operator} -\begin{equation} -\label{shift1MPI} -P_n^{(\lambda)}(x+\textstyle\frac{1}{2}i;\phi)- -P_n^{(\lambda)}(x-\textstyle\frac{1}{2}i;\phi)=(\expe^{i\phi}-\expe^{-i\phi}) -P_{n-1}^{(\lambda+\frac{1}{2})}(x;\phi) -\end{equation} -or equivalently -\begin{equation} -\label{shift1MPII} -\frac{\delta P_n^{(\lambda)}(x;\phi)}{\delta x} -=2\sin\phi\,P_{n-1}^{(\lambda+\frac{1}{2})}(x;\phi). -\end{equation} - -\subsection*{Backward shift operator} -\begin{eqnarray} -\label{shift2MPI} -& &\e^{i\phi}(\lambda-\textstyle\frac{1}{2}-ix) -P_n^{(\lambda)}(x+\textstyle\frac{1}{2}i;\phi)+ -\e^{-i\phi}(\lambda-\textstyle\frac{1}{2}+ix) -P_n^{(\lambda)}(x-\textstyle\frac{1}{2}i;\phi)\nonumber\\ -& &{}=(n+1)P_{n+1}^{(\lambda-\frac{1}{2})}(x;\phi) -\end{eqnarray} -or equivalently -\begin{equation} -\label{shift2MPII} -\frac{\delta\left[\omega(x;\lambda,\phi)P_n^{(\lambda)}(x;\phi)\right]}{\delta x}= --(n+1)\omega(x;\lambda-\textstyle\frac{1}{2},\phi) -P_{n+1}^{(\lambda-\frac{1}{2})}(x;\phi), -\end{equation} -where -$$\omega(x;\lambda,\phi)=\Gamma(\lambda+ix)\Gamma(\lambda-ix)\e^{(2\phi-\pi)x}.$$ - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodMP} -\omega(x;\lambda,\phi)P_n^{(\lambda)}(x;\phi)=\frac{(-1)^n}{n!} -\left(\frac{\delta}{\delta x}\right)^n\left[\omega(x;\lambda+\textstyle\frac{1}{2}n,\phi)\right]. -\end{equation} - -\newpage - -\subsection*{Generating functions} -\begin{equation} -\label{GenMP1} -(1-\expe^{i\phi}t)^{-\lambda+ix}(1-\expe^{-i\phi}t)^{-\lambda-ix}= -\sum_{n=0}^{\infty}P_n^{(\lambda)}(x;\phi)t^n. -\end{equation} - -\begin{equation} -\label{GenMP2} -\e^t\,\hyp{1}{1}{\lambda+ix}{2\lambda}{(\expe^{-2i\phi}-1)t}= -\sum_{n=0}^{\infty}\frac{P_n^{(\lambda)}(x;\phi)}{(2\lambda)_n\expe^{in\phi}}t^n. -\end{equation} - -\begin{eqnarray} -\label{GenMP3} -& &(1-t)^{-\gamma}\,\hyp{2}{1}{\gamma,\lambda+ix}{2\lambda}{\frac{(1-\e^{-2i\phi})t}{t-1}}\nonumber\\ -& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n}{(2\lambda)_n}\frac{P_n^{(\lambda)}(x;\phi)}{\e^{in\phi}}t^n, -\quad\textrm{$\gamma$ arbitrary}. -\end{eqnarray} - -\subsection*{Limit relations} - -\subsubsection*{Continuous dual Hahn $\rightarrow$ Meixner-Pollaczek} -The Meixner-Pollaczek polynomials can be obtained from the continuous dual Hahn polynomials -given by (\ref{DefContDualHahn}) by the substitutions $x\rightarrow x-t$, $a=\lambda+it$, -$b=\lambda-it$ and $c=t\cot\phi$ and the limit $t\rightarrow\infty$: -$$\lim_{t\rightarrow\infty}\frac{S_n((x-t)^2;\lambda+it,\lambda-it,t\cot\phi)}{t^nn!} -=\frac{P_n^{(\lambda)}(x;\phi)}{(\sin\phi)^n}.$$ - -\subsubsection*{Continuous Hahn $\rightarrow$ Meixner-Pollaczek} -By setting $x\rightarrow x+t$, $a=\lambda-it$, $c=\lambda+it$ and $b=d=t\tan\phi$ in the -definition (\ref{DefContHahn}) of the continuous Hahn polynomials and taking the limit -$t\rightarrow\infty$ we obtain the Meixner-Pollaczek polynomials: -$$\lim_{t\rightarrow\infty}\frac{p_n(x+t;\lambda-it,t\tan\phi,\lambda+it,t\tan\phi)}{t^nn!} -=\frac{P_n^{(\lambda)}(x;\phi)}{(\cos\phi)^n}.$$ - -\newpage - -\subsubsection*{Meixner-Pollaczek $\rightarrow$ Laguerre} -The Laguerre polynomials given by (\ref{DefLaguerre}) can be obtained from the -Meixner-Pollaczek polynomials by the substitution $\lambda=\frac{1}{2}(\alpha+1)$, -$x\rightarrow -\frac{1}{2}\phi^{-1}x$ and the limit $\phi\rightarrow 0$: -\begin{equation} -\lim_{\phi\rightarrow 0} -P_n^{(\frac{1}{2}\alpha+\frac{1}{2})}(-\textstyle\frac{1}{2}\phi^{-1}x;\phi)=L_n^{(\alpha)}(x). -\end{equation} - -\subsubsection*{Meixner-Pollaczek $\rightarrow$ Hermite} -The Hermite polynomials given by (\ref{DefHermite}) are obtained from the Meixner-Pollaczek -polynomials if we substitute $x\rightarrow (\sin\phi)^{-1}(x\sqrt{\lambda}-\lambda\cos\phi)$ -and then let $\lambda\rightarrow\infty$: -\begin{equation} -\lim_{\lambda\rightarrow\infty} -\lambda^{-\frac{1}{2}n}P_n^{(\lambda)} -((\sin\phi)^{-1}(x\sqrt{\lambda}-\lambda\cos\phi);\phi)=\frac{H_n(x)}{n!}. -\end{equation} - -\subsection*{Remark} -Since we have for $k-1$ and $\beta>-1$ we have -\begin{eqnarray} -\label{OrtJacobi1} -& &\int_{-1}^1(1-x)^{\alpha}(1+x)^{\beta}P_m^{(\alpha,\beta)}(x)P_n^{(\alpha,\beta)}(x)\,dx\nonumber\\ -& &{}=\frac{2^{\alpha+\beta+1}}{2n+\alpha+\beta+1}\frac{\Gamma(n+\alpha+1)\Gamma(n+\beta+1)}{\Gamma(n+\alpha+\beta+1)n!}\,\delta_{mn}. -\end{eqnarray} -For $\alpha+\beta<-2N-1$, $\beta>-1$ and $m,n\in\{0,1,2,\ldots,N\}$ we also have -\begin{eqnarray} -\label{OrtJacobi2} -& &\int_1^{\infty}(x+1)^{\alpha}(x-1)^{\beta}P_m^{(\alpha,\beta)}(-x)P_n^{(\alpha,\beta)}(-x)\,dx\nonumber\\ -& &{}=-\frac{2^{\alpha+\beta+1}}{2n+\alpha+\beta+1}\frac{\Gamma(-n-\alpha-\beta)\Gamma(n+\alpha+\beta+1)}{\Gamma(-n-\alpha)n!}\,\delta_{mn}. -\end{eqnarray} - -\subsection*{Recurrence relation} -\begin{eqnarray} -\label{RecJacobi} -xP_n^{(\alpha,\beta)}(x)&=&\frac{2(n+1)(n+\alpha+\beta+1)}{(2n+\alpha+\beta+1)(2n+\alpha+\beta+2)}P_{n+1}^{(\alpha,\beta)}(x)\nonumber\\ -& &{}\mathindent{}+\frac{\beta^2-\alpha^2}{(2n+\alpha+\beta)(2n+\alpha+\beta+2)}P_n^{(\alpha,\beta)}(x)\nonumber\\ -& &{}\mathindent\mathindent{}+\frac{2(n+\alpha)(n+\beta)}{(2n+\alpha+\beta)(2n+\alpha+\beta+1)}P_{n-1}^{(\alpha,\beta)}(x). -\end{eqnarray} - -\subsection*{Normalized recurrence relation} -\begin{eqnarray} -\label{NormRecJacobi} -xp_n(x)&=&p_{n+1}(x)+\frac{\beta^2-\alpha^2}{(2n+\alpha+\beta)(2n+\alpha+\beta+2)}p_n(x)\nonumber\\ -& &{}\mathindent{}+\frac{4n(n+\alpha)(n+\beta)(n+\alpha+\beta)} -{(2n+\alpha+\beta-1)(2n+\alpha+\beta)^2(2n+\alpha+\beta+1)}p_{n-1}(x) -\end{eqnarray} -where -$$P_n^{(\alpha,\beta)}(x)=\frac{(n+\alpha+\beta+1)_n}{2^nn!}p_n(x).$$ - -\subsection*{Differential equation} -\begin{eqnarray} -\label{dvJacobi} -& &(1-x^2)y''(x)+\left[\beta-\alpha-(\alpha+\beta+2)x\right]y'(x)\nonumber\\ -& &{}\mathindent{}+n(n+\alpha+\beta+1)y(x)=0,\quad y(x)=P_n^{(\alpha,\beta)}(x). -\end{eqnarray} - -\subsection*{Forward shift operator} -\begin{equation} -\label{shift1Jacobi} -\frac{d}{dx}P_n^{(\alpha,\beta)}(x)=\frac{n+\alpha+\beta+1}{2}P_{n-1}^{(\alpha+1,\beta+1)}(x). -\end{equation} - -\subsection*{Backward shift operator} -\begin{eqnarray} -\label{shift2JacobiI} -& &(1-x^2)\frac{d}{dx}P_n^{(\alpha,\beta)}(x)+ -\left[(\beta-\alpha)-(\alpha+\beta)x\right]P_n^{(\alpha,\beta)}(x)\nonumber\\ -& &{}=-2(n+1)P_{n+1}^{(\alpha-1,\beta-1)}(x) -\end{eqnarray} -or equivalently -\begin{eqnarray} -\label{shift2JacobiII} -& &\frac{d}{dx}\left[(1-x)^\alpha(1+x)^\beta P_n^{(\alpha,\beta)}(x)\right]\nonumber\\ -& &{}=-2(n+1)(1-x)^{\alpha-1}(1+x)^{\beta-1}P_{n+1}^{(\alpha-1,\beta-1)}(x). -\end{eqnarray} - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodJacobi} -(1-x)^{\alpha}(1+x)^{\beta}P_n^{(\alpha,\beta)}(x)= -\frac{(-1)^n}{2^nn!}\left(\frac{d}{dx}\right)^n -\left[(1-x)^{n+\alpha}(1+x)^{n+\beta}\right]. -\end{equation} - -\subsection*{Generating functions} -\begin{equation} -\label{GenJacobi1} -\frac{2^{\alpha+\beta}}{R(1+R-t)^{\alpha}(1+R+t)^{\beta}}= -\sum_{n=0}^{\infty}P_n^{(\alpha,\beta)}(x)t^n,\quad R=\sqrt{1-2xt+t^2}. -\end{equation} - -\begin{eqnarray} -\label{GenJacobi2} -& &\hyp{0}{1}{-}{\alpha+1}{\frac{(x-1)t}{2}}\,\hyp{0}{1}{-}{\beta+1}{\frac{(x+1)t}{2}}\nonumber\\ -& &{}=\sum_{n=0}^{\infty}\frac{P_n^{(\alpha,\beta)}(x)}{(\alpha+1)_n(\beta+1)_n}t^n. -\end{eqnarray} - -\begin{eqnarray} -\label{GenJacobi3} -& &(1-t)^{-\alpha-\beta-1}\,\hyp{2}{1}{\frac{1}{2}(\alpha+\beta+1),\frac{1}{2}(\alpha+\beta+2)} -{\alpha+1}{\frac{2(x-1)t}{(1-t)^2}}\nonumber\\ -& &{}=\sum_{n=0}^{\infty}\frac{(\alpha+\beta+1)_n}{(\alpha+1)_n}P_n^{(\alpha,\beta)}(x)t^n. -\end{eqnarray} - -\begin{eqnarray} -\label{GenJacobi4} -& &(1+t)^{-\alpha-\beta-1}\,\hyp{2}{1}{\frac{1}{2}(\alpha+\beta+1),\frac{1}{2}(\alpha+\beta+2)} -{\beta+1}{\frac{2(x+1)t}{(1+t)^2}}\nonumber\\ -& &{}=\sum_{n=0}^{\infty}\frac{(\alpha+\beta+1)_n}{(\beta+1)_n}P_n^{(\alpha,\beta)}(x)t^n. -\end{eqnarray} - -\begin{eqnarray} -\label{GenJacobi5} -& &\hyp{2}{1}{\gamma,\alpha+\beta+1-\gamma}{\alpha+1}{\frac{1-R-t}{2}}\, -\hyp{2}{1}{\gamma,\alpha+\beta+1-\gamma}{\beta+1}{\frac{1-R+t}{2}}\nonumber\\ -& &{}=\sum_{n=0}^{\infty} -\frac{(\gamma)_n(\alpha+\beta+1-\gamma)_n}{(\alpha+1)_n(\beta+1)_n}P_n^{(\alpha,\beta)}(x)t^n, -\quad R=\sqrt{1-2xt+t^2} -\end{eqnarray} -with $\gamma$ arbitrary. - -\subsection*{Limit relations} - -\subsubsection*{Wilson $\rightarrow$ Jacobi} -The Jacobi polynomials can be found from the Wilson polynomials given by (\ref{DefWilson}) by -substituting $a=b=\frac{1}{2}(\alpha+1)$, $c=\frac{1}{2}(\beta+1)+it$, -$d=\frac{1}{2}(\beta+1)-it$ and $x\rightarrow t\sqrt{\frac{1}{2}(1-x)}$ in the -definition (\ref{DefWilson}) of the Wilson polynomials and taking the limit -$t\rightarrow\infty$. In fact we have -$$\lim_{t\rightarrow\infty} -\frac{W_n(\frac{1}{2}(1-x)t^2;\frac{1}{2}(\alpha+1), -\frac{1}{2}(\alpha+1),\frac{1}{2}(\beta+1)+it,\frac{1}{2}(\beta+1)-it)} -{t^{2n}n!}=P_n^{(\alpha,\beta)}(x).$$ - -\subsubsection*{Continuous Hahn $\rightarrow$ Jacobi} -The Jacobi polynomials follow from the continuous Hahn polynomials given by (\ref{DefContHahn}) -by using the substitution $x\rightarrow \frac{1}{2}xt$, $a=\frac{1}{2}(\alpha+1-it)$, -$b=\frac{1}{2}(\beta+1+it)$, $c=\frac{1}{2}(\alpha+1+it)$ and $d=\frac{1}{2}(\beta+1-it)$ -in (\ref{DefContHahn}), division by $t^n$ and the limit $t\rightarrow\infty$: -$$\lim_{t\rightarrow\infty} -\frac{p_n(\frac{1}{2}xt;\frac{1}{2}(\alpha+1-it),\frac{1}{2}(\beta+1+it), -\frac{1}{2}(\alpha+1+it),\frac{1}{2}(\beta+1-it))}{t^n}=P_n^{(\alpha,\beta)}(x).$$ - -\subsubsection*{Hahn $\rightarrow$ Jacobi} -To find the Jacobi polynomials from the Hahn polynomials given by (\ref{DefHahn}) we take -$x\rightarrow Nx$ in (\ref{DefHahn}) and let $N\rightarrow\infty$. In fact we have -$$\lim_{N\rightarrow\infty} -Q_n(Nx;\alpha,\beta,N)=\frac{P_n^{(\alpha,\beta)}(1-2x)}{P_n^{(\alpha,\beta)}(1)}.$$ - -\subsubsection*{Jacobi $\rightarrow$ Laguerre} -The Laguerre polynomials given by (\ref{DefLaguerre}) can be obtained from the Jacobi -polynomials by setting $x\rightarrow 1-2\beta^{-1}x$ and then the limit $\beta\rightarrow\infty$: -\begin{equation} -\lim_{\beta\rightarrow\infty} -P_n^{(\alpha,\beta)}(1-2\beta^{-1}x)=L_n^{(\alpha)}(x). -\end{equation} - -\subsubsection*{Jacobi $\rightarrow$ Bessel} -The Bessel polynomials given by (\ref{DefBessel}) are obtained from the Jacobi polynomials -if we take $\beta=a-\alpha$ and let $\alpha\rightarrow-\infty$: -\begin{equation} -\lim_{\alpha\rightarrow-\infty} -\frac{P_n^{(\alpha,a-\alpha)}(1+\alpha x)}{P_n^{(\alpha,a-\alpha)}(1)}=y_n(x;a). -\end{equation} - -\subsubsection*{Jacobi $\rightarrow$ Hermite} -The Hermite polynomials given by (\ref{DefHermite}) follow from the Jacobi polynomials -by taking $\beta=\alpha$ and letting $\alpha\rightarrow\infty$ in the following way: -\begin{equation} -\lim_{\alpha\rightarrow\infty} -\alpha^{-\frac{1}{2}n}P_n^{(\alpha,\alpha)}(\alpha^{-\frac{1}{2}}x)=\frac{H_n(x)}{2^nn!}. -\end{equation} - -\subsection*{Remarks} -The definition (\ref{DefJacobi}) of the Jacobi polynomials can also be written as: -$$P_n^{(\alpha,\beta)}(x)=\frac{1}{n!}\sum_{k=0}^n\frac{(-n)_k}{k!} -(n+\alpha+\beta+1)_k(\alpha+k+1)_{n-k}\left(\frac{1-x}{2}\right)^k.$$ -In this way the Jacobi polynomials can also be seen as polynomials in the parameters $\alpha$ -and $\beta$. Therefore they can be defined for all $\alpha$ and $\beta$. Then we have the -following connection with the Meixner polynomials given by (\ref{DefMeixner}): -$$\frac{(\beta)_n}{n!}M_n(x;\beta,c)=P_n^{(\beta-1,-n-\beta-x)}((2-c)c^{-1}).$$ - -\noindent -The Jacobi polynomials are related to the pseudo Jacobi polynomials defined by -(\ref{DefPseudoJacobi}) in the following way: -$$P_n(x;\nu,N)=\frac{(-2i)^nn!}{(n-2N-1)_n}P_n^{(-N-1+i\nu,-N-1-i\nu)}(ix).$$ - -\noindent -The Jacobi polynomials are also related to the Gegenbauer (or ultraspherical) polynomials -given by (\ref{DefGegenbauer}) by the quadratic transformations: -$$C_{2n}^{(\lambda)}(x)=\frac{(\lambda)_n}{(\frac{1}{2})_n} -P_n^{(\lambda-\frac{1}{2},-\frac{1}{2})}(2x^2-1)$$ -and -$$C_{2n+1}^{(\lambda)}(x)=\frac{(\lambda)_{n+1}}{(\frac{1}{2})_{n+1}} -xP_n^{(\lambda-\frac{1}{2},\frac{1}{2})}(2x^2-1).$$ -% RS: add begin\label{sec9.8} -% -\paragraph{\large\bf KLSadd: Orthogonality relation}Write the \RHS\ of (9.8.2) as $h_n\,\de_{m,n}$. Then -\begin{equation} -\begin{split} -&\frac{h_n}{h_0}= -\frac{n+\al+\be+1}{2n+\al+\be+1}\, -\frac{(\al+1)_n(\be+1)_n}{(\al+\be+2)_n\,n!}\,,\quad -h_0=\frac{2^{\al+\be+1}\Ga(\al+1)\Ga(\be+1)}{\Ga(\al+\be+2)}\,,\sLP -&\frac{h_n}{h_0\,(P_n^{(\al,\be)}(1))^2}= -\frac{n+\al+\be+1}{2n+\al+\be+1}\, -\frac{(\be+1)_n\,n!}{(\al+1)_n\,(\al+\be+2)_n}\,. -\end{split} -\label{60} -\end{equation} - -In (9.8.3) the numerator factor $\Ga(n+\al+\be+1)$ in the last line should be -$\Ga(\be+1)$. When thus corrected, (9.8.3) can be rewritten as: -\begin{equation} -\begin{split} -&\int_1^\iy P_m^{(\al,\be)}(x)\,P_n^{(\al,\be)}(x)\,(x-1)^\al (x+1)^\be\,dx=h_n\,\de_{m,n}\,,\\ -&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad-1-\be>\al>-1,\quad m,n<-\thalf(\al+\be+1),\\ -&\frac{h_n}{h_0}= -\frac{n+\al+\be+1}{2n+\al+\be+1}\, -\frac{(\al+1)_n(\be+1)_n}{(\al+\be+2)_n\,n!}\,,\quad -h_0=\frac{2^{\al+\be+1}\Ga(\al+1)\Ga(-\al-\be-1)}{\Ga(-\be)}\,. -\end{split} -\label{122} -\end{equation} - -% -\paragraph{\large\bf KLSadd: Symmetry}\begin{equation} -P_n^{(\al,\be)}(-x)=(-1)^n\,P_n^{(\be,\al)}(x). -\label{48} -\end{equation} -Use (9.8.2) and (9.8.5b) or see \mycite{DLMF}{Table 18.6.1}. -% -\paragraph{\large\bf KLSadd: Special values}\begin{equation} -P_n^{(\al,\be)}(1)=\frac{(\al+1)_n}{n!}\,,\quad -P_n^{(\al,\be)}(-1)=\frac{(-1)^n(\be+1)_n}{n!}\,,\quad -\frac{P_n^{(\al,\be)}(-1)}{P_n^{(\al,\be)}(1)}=\frac{(-1)^n(\be+1)_n}{(\al+1)_n}\,. -\label{50} -\end{equation} -Use (9.8.1) and \eqref{48} or see \mycite{DLMF}{Table 18.6.1}. -% -\paragraph{\large\bf KLSadd: Generating functions}Formula (9.8.15) was first obtained by Brafman \myciteKLS{109}. -% -\paragraph{\large\bf KLSadd: Bilateral generating functions}For $0\le r<1$ and $x,y\in[-1,1]$ we have in terms of $F_4$ (see~\eqref{62}): -\begin{align} -&\sum_{n=0}^\iy\frac{(\al+\be+1)_n\,n!}{(\al+1)_n(\be+1)_n}\,r^n\, -P_n^{(\al,\be)}(x)\,P_n^{(\al,\be)}(y) -=\frac1{(1+r)^{\al+\be+1}} -\nonumber\\ -&\qquad\quad\times F_4\Big(\thalf(\al+\be+1),\thalf(\al+\be+2);\al+1,\be+1; -\frac{r(1-x)(1-y)}{(1+r)^2},\frac{r(1+x)(1+y)}{(1+r)^2}\Big), -\label{58}\sLP -&\sum_{n=0}^\iy\frac{2n+\al+\be+1}{n+\al+\be+1} -\frac{(\al+\be+2)_n\,n!}{(\al+1)_n(\be+1)_n}\,r^n\, -P_n^{(\al,\be)}(x)\,P_n^{(\al,\be)}(y) -=\frac{1-r}{(1+r)^{\al+\be+2}}\nonumber\\ -&\qquad\quad\times F_4\Big(\thalf(\al+\be+2),\thalf(\al+\be+3);\al+1,\be+1; -\frac{r(1-x)(1-y)}{(1+r)^2},\frac{r(1+x)(1+y)}{(1+r)^2}\Big). -\label{59} -\end{align} -Formulas \eqref{58} and \eqref{59} were first -given by Bailey \myciteKLS{91}{(2.1), (2.3)}. -See Stanton \myciteKLS{485} for a shorter proof. -(However, in the second line of -\myciteKLS{485}{(1)} $z$ and $Z$ should be interchanged.)$\;$ -As observed in Bailey \myciteKLS{91}{p.10}, \eqref{59} follows -from \eqref{58} -by applying the operator $r^{-\half(\al+\be-1)}\,\frac d{dr}\circ r^{\half(\al+\be+1)}$ -to both sides of \eqref{58}. -In view of \eqref{60}, formula \eqref{59} is the Poisson kernel for Jacobi -polynomials. The \RHS\ of \eqref{59} makes clear that this kernel is positive. -See also the discussion in Askey \myciteKLS{46}{following (2.32)}. -% -\paragraph{\large\bf KLSadd: Quadratic transformations}\begin{align} -\frac{C_{2n}^{(\al+\half)}(x)}{C_{2n}^{(\al+\half)}(1)} -=\frac{P_{2n}^{(\al,\al)}(x)}{P_{2n}^{(\al,\al)}(1)} -&=\frac{P_n^{(\al,-\half)}(2x^2-1)}{P_n^{(\al,-\half)}(1)}\,, -\label{51}\\ -\frac{C_{2n+1}^{(\al+\half)}(x)}{C_{2n+1}^{(\al+\half)}(1)} -=\frac{P_{2n+1}^{(\al,\al)}(x)}{P_{2n+1}^{(\al,\al)}(1)} -&=\frac{x\,P_n^{(\al,\half)}(2x^2-1)}{P_n^{(\al,\half)}(1)}\,. -\label{52} -\end{align} -See p.221, Remarks, last two formulas together with \eqref{50} and \eqref{49}. -Or see \mycite{DLMF}{(18.7.13), (18.7.14)}. -% -\paragraph{\large\bf KLSadd: Differentiation formulas}Each differentiation formula is given in two equivalent forms. -\begin{equation} -\begin{split} -\frac d{dx}\left((1-x)^\al P_n^{(\al,\be)}(x)\right)&= --(n+\al)\,(1-x)^{\al-1} P_n^{(\al-1,\be+1)}(x),\\ -\left((1-x)\frac d{dx}-\al\right)P_n^{(\al,\be)}(x)&= --(n+\al)\,P_n^{(\al-1,\be+1)}(x). -\end{split} -\label{68} -\end{equation} -% -\begin{equation} -\begin{split} -\frac d{dx}\left((1+x)^\be P_n^{(\al,\be)}(x)\right)&= -(n+\be)\,(1+x)^{\be-1} P_n^{(\al+1,\be-1)}(x),\\ -\left((1+x)\frac d{dx}+\be\right)P_n^{(\al,\be)}(x)&= -(n+\be)\,P_n^{(\al+1,\be-1)}(x). -\end{split} -\label{69} -\end{equation} -Formulas \eqref{68} and \eqref{69} follow from -\mycite{DLMF}{(15.5.4), (15.5.6)} -together with (9.8.1). They also follow from each other by \eqref{48}. -% -\paragraph{\large\bf KLSadd: Generalized Gegenbauer polynomials}These are defined by -\begin{equation} -S_{2m}^{(\al,\be)}(x):=\const P_m^{(\al,\be)}(2x^2-1),\qquad -S_{2m+1}^{(\al,\be)}(x):=\const x\,P_m^{(\al,\be+1)}(2x^2-1) -\label{70} -\end{equation} -in the notation of \myciteKLS{146}{p.156} -(see also \cite{K27}), while \cite[Section 1.5.2]{K26} -has $C_n^{(\la,\mu)}(x)=\const\allowbreak\times S_n^{(\la-\half,\mu-\half)}(x)$. -For $\al,\be>-1$ we have the orthogonality relation -\begin{equation} -\int_{-1}^1 S_m^{(\al,\be)}(x)\,S_n^{(\al,\be)}(x)\,|x|^{2\be+1}(1-x^2)^\al\,dx -=0\qquad(m\ne n). -\label{71} -\end{equation} -For $\be=\al-1$ generalized Gegenbauer polynomials are limit cases of -continuous $q$-ultraspherical polynomials, see \eqref{176}. - -If we define the {\em Dunkl operator} $T_\mu$ by -\begin{equation} -(T_\mu f)(x):=f'(x)+\mu\,\frac{f(x)-f(-x)}x -\label{72} -\end{equation} -and if we choose the constants in \eqref{70} as -\begin{equation} -S_{2m}^{(\al,\be)}(x)=\frac{(\al+\be+1)_m}{(\be+1)_m}\, P_m^{(\al,\be)}(2x^2-1),\quad -S_{2m+1}^{(\al,\be)}(x)=\frac{(\al+\be+1)_{m+1}}{(\be+1)_{m+1}}\, -x\,P_m^{(\al,\be+1)}(2x^2-1) -\label{73} -\end{equation} -then (see \cite[(1.6)]{K5}) -\begin{equation} -T_{\be+\half}S_n^{(\al,\be)}=2(\al+\be+1)\,S_{n-1}^{(\al+1,\be)}. -\label{74} -\end{equation} -Formula \eqref{74} with \eqref{73} substituted gives rise to two -differentiation formulas involving Jacobi polynomials which are equivalent to -(9.8.7) and \eqref{69}. - -Composition of \eqref{74} with itself gives -\[ -T_{\be+\half}^2S_n^{(\al,\be)}=4(\al+\be+1)(\al+\be+2)\,S_{n-2}^{(\al+2,\be)}, -\] -which is equivalent to the composition of (9.8.7) and \eqref{69}: -\begin{equation} -\left(\frac{d^2}{dx^2}+\frac{2\be+1}x\,\frac d{dx}\right)P_n^{(\al,\be)}(2x^2-1) -=4(n+\al+\be+1)(n+\be)\,P_{n-1}^{(\al+2,\be)}(2x^2-1). -\label{75} -\end{equation} -Formula \eqref{75} was also given in \myciteKLS{322}{(2.4)}. -% -% RS: add end -\subsection*{References} -\cite{Abram}, \cite{Ahmed+82}, \cite{Allaway89}, \cite{NAlSalam66}, \cite{AlSalam64}, -\cite{AlSalamChihara72}, \cite{AndrewsAskey85}, \cite{AndrewsAskeyRoy}, \cite{Askey68}, -\cite{Askey70}, \cite{Askey72}, \cite{Askey73}, \cite{Askey74}, \cite{Askey75}, \cite{Askey78}, -\cite{Askey89I}, \cite{Askey2005}, \cite{AskeyFitch}, \cite{AskeyGasper71I}, \cite{AskeyGasper71II}, -\cite{AskeyGasper76}, \cite{AskeyWainger}, \cite{AskeyWilson85}, \cite{Bailey38}, \cite{Brafman51}, -\cite{Brown}, \cite{Carlitz61II}, \cite{Carlitz67}, \cite{ChenIsmail91}, \cite{Chihara78}, -\cite{Chow+}, \cite{Ciesielski}, \cite{Cooper+}, \cite{DetteStudden92}, \cite{DetteStudden95}, -\cite{DijksmaKoorn}, \cite{DimitrovRafaeli}, \cite{DimitrovRodrigues}, \cite{Doha2002I}, -\cite{Doha2003I}, \cite{Doha2004II}, \cite{Dunkl84}, \cite{ElbertLaforgia87II}, -\cite{ElbertLaforgiaRodono}, \cite{Erdelyi+}, \cite{Faldey}, \cite{FoataLeroux}, -\cite{Gasper69}, \cite{Gasper70I}, \cite{Gasper70II}, \cite{Gasper71I}, \cite{Gasper71II}, -\cite{Gasper72I}, \cite{Gasper72II}, \cite{Gasper73I}, \cite{Gasper73II}, \cite{Gasper74}, -\cite{Gasper77}, \cite{Gatteschi87}, \cite{Gautschi2008}, \cite{Gautschi2009I}, -\cite{Gautschi2009II}, \cite{GautschiLeopardi}, \cite{GawronskiShawyer}, \cite{Godoy+}, -\cite{Grad}, \cite{Hajmirzaahmad94}, \cite{HartmannStephan}, \cite{Horton}, \cite{Ismail74}, -\cite{Ismail77}, \cite{Ismail96}, \cite{Ismail2005II}, \cite{IsmailLi}, -\cite{IsmailMassonRahman}, \cite{Kochneff97I}, \cite{Koekoek99}, \cite{Koekoek2000}, -\cite{Koelink96II}, \cite{Koorn73}, \cite{Koorn74}, \cite{Koorn75}, \cite{Koorn77II}, -\cite{Koorn78}, \cite{Koorn72}, \cite{Koorn85}, \cite{Koorn88}, \cite{KuijlaarsMartinez}, -\cite{Kuijlaars+}, \cite{LabelleYehI}, \cite{LabelleYehII}, \cite{Laine}, \cite{Lesky95II}, -\cite{Lesky96}, \cite{Li96}, \cite{Li97}, \cite{LopezTemme2004}, \cite{Luke}, \cite{Martinez}, -\cite{Mathai}, \cite{Meijer}, \cite{Miller89}, \cite{MoakSaffVarga}, \cite{Nikiforov+}, -\cite{NikiforovUvarov}, \cite{Olver}, \cite{Prasad}, \cite{Rahman76I}, \cite{Rahman76II}, -\cite{Rahman77}, \cite{Rahman81I}, \cite{RahmanShah}, \cite{Rainville}, \cite{Rusev}, -\cite{Shi}, \cite{Srivastava69II}, \cite{Srivastava71}, \cite{Srivastava82}, -\cite{SrivastavaSinghal}, \cite{Stanton80I}, \cite{Stanton90}, \cite{Szego75}, \cite{Temme}, -\cite{Vertesi}, \cite{Wimp87}, \cite{WongZhang94}, \cite{WongZhang2006}, \cite{Zarzo+}, -\cite{Zayed}. - -\section*{Special cases} - -\subsection{Gegenbauer / Ultraspherical}\index{Gegenbauer polynomials} -\index{Ultraspherical polynomials} - -\par - -\subsection*{Hypergeometric representation} -The Gegenbauer (or ultraspherical) polynomials are Jacobi polynomials with -$\alpha=\beta=\lambda-\frac{1}{2}$ and another normalization: -\begin{eqnarray} -\label{DefGegenbauer} -C_n^{(\lambda)}(x)&=&\frac{(2\lambda)_n}{(\lambda+\frac{1}{2})_n} -P_n^{(\lambda-\frac{1}{2},\lambda-\frac{1}{2})}(x)\nonumber\\ -&=&\frac{(2\lambda)_n}{n!}\,\hyp{2}{1}{-n,n+2\lambda} -{\lambda+\frac{1}{2}}{\frac{1-x}{2}},\quad\lambda\neq 0. -\end{eqnarray} - -\subsection*{Orthogonality relation} -\begin{eqnarray} -\label{OrtGegenbauer} -& &\int_{-1}^1(1-x^2)^{\lambda-\frac{1}{2}}C_m^{(\lambda)}(x)C_n^{(\lambda)}(x)\,dx\nonumber\\ -& &{}=\frac{\pi\Gamma(n+2\lambda)2^{1-2\lambda}}{\left\{\Gamma(\lambda)\right\}^2(n+\lambda)n!}\,\delta_{mn}, -\quad\lambda>-\frac{1}{2}\quad\lambda\neq 0. -\end{eqnarray} - -\subsection*{Recurrence relation} -\begin{equation} -\label{RecGegenbauer} -2(n+\lambda)xC_n^{(\lambda)}(x)=(n+1)C_{n+1}^{(\lambda)}(x)+(n+2\lambda-1)C_{n-1}^{(\lambda)}(x). -\end{equation} - -\subsection*{Normalized recurrence relation} -\begin{equation} -\label{NormRecGegenbauer} -xp_n(x)=p_{n+1}(x)+\frac{n(n+2\lambda-1)}{4(n+\lambda-1)(n+\lambda)}p_{n-1}(x), -\end{equation} -where -$$C_n^{(\lambda)}(x)=\frac{2^n(\lambda)_n}{n!}p_n(x).$$ - -\subsection*{Differential equation} -\begin{equation} -\label{dvGegenbauer} -(1-x^2)y''(x)-(2\lambda+1)xy'(x)+n(n+2\lambda)y(x)=0,\quad y(x)=C_n^{(\lambda)}(x). -\end{equation} - -\subsection*{Forward shift operator} -\begin{equation} -\label{shift1Gegenbauer} -\frac{d}{dx}C_n^{(\lambda)}(x)=2\lambda C_{n-1}^{(\lambda+1)}(x). -\end{equation} - -\subsection*{Backward shift operator} -\begin{equation} -\label{shift2GegenbauerI} -(1-x^2)\frac{d}{dx}C_n^{(\lambda)}(x)+(1-2\lambda)xC_n^{(\lambda)}(x)= --\frac{(n+1)(2\lambda+n-1)}{2(\lambda-1)} C_{n+1}^{(\lambda-1)}(x) -\end{equation} -or equivalently -\begin{eqnarray} -\label{shift2GegenbauerII} -& &\frac{d}{dx}\left[(1-x^2)^{\lambda-\textstyle\frac{1}{2}}C_n^{(\lambda)}(x)\right]\nonumber\\ -& &{}=-\frac{(n+1)(2\lambda+n-1)}{2(\lambda-1)}(1-x^2)^{\lambda-\textstyle\frac{3}{2}}C_{n+1}^{(\lambda-1)}(x). -\end{eqnarray} - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodGegenbauer} -(1-x^2)^{\lambda-\frac{1}{2}}C_n^{(\lambda)}(x)= -\frac{(2\lambda)_n(-1)^n}{(\lambda+\frac{1}{2})_n2^nn!}\left(\frac{d}{dx}\right)^n -\left[(1-x^2)^{\lambda+n-\frac{1}{2}}\right]. -\end{equation} - -\subsection*{Generating functions} -\begin{equation} -\label{GenGegenbauer1} -(1-2xt+t^2)^{-\lambda}=\sum_{n=0}^{\infty}C_n^{(\lambda)}(x)t^n. -\end{equation} - -\begin{equation} -\label{GenGegenbauer2} -R^{-1}\left(\frac{1+R-xt}{2}\right)^{\frac{1}{2}-\lambda}=\sum_{n=0}^{\infty} -\frac{(\lambda+\frac{1}{2})_n}{(2\lambda)_n}C_n^{(\lambda)}(x)t^n, -\quad R=\sqrt{1-2xt+t^2}. -\end{equation} - -\begin{equation} -\label{GenGegenbauer3} -\hyp{0}{1}{-}{\lambda+\frac{1}{2}}{\frac{(x-1)t}{2}}\, -\hyp{0}{1}{-}{\lambda+\frac{1}{2}}{\frac{(x+1)t}{2}} -=\sum_{n=0}^{\infty}\frac{C_n^{(\lambda)}(x)} -{(2\lambda)_n(\lambda+\frac{1}{2})_n}t^n. -\end{equation} - -\begin{equation} -\label{GenGegenbauer4} -\e^{xt}\,\hyp{0}{1}{-}{\lambda+\frac{1}{2}}{\frac{(x^2-1)t^2}{4}}= -\sum_{n=0}^{\infty}\frac{C_n^{(\lambda)}(x)}{(2\lambda)_n}t^n. -\end{equation} - -\begin{eqnarray} -\label{GenGegenbauer5} -& &\hyp{2}{1}{\gamma,2\lambda-\gamma}{\lambda+\frac{1}{2}}{\frac{1-R-t}{2}}\, -\hyp{2}{1}{\gamma,2\lambda-\gamma}{\lambda+\frac{1}{2}}{\frac{1-R+t}{2}}\nonumber\\ -& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n(2\lambda-\gamma)_n} -{(2\lambda)_n(\lambda+\frac{1}{2})_n}C_n^{(\lambda)}(x)t^n, -\quad R=\sqrt{1-2xt+t^2},\quad\textrm{$\gamma$ arbitrary}. -\end{eqnarray} - -\begin{eqnarray} -\label{GenGegenbauer6} -& &(1-xt)^{-\gamma}\,\hyp{2}{1}{\frac{1}{2}\gamma,\frac{1}{2}\gamma+\frac{1}{2}} -{\lambda+\frac{1}{2}}{\frac{(x^2-1)t^2}{(1-xt)^2}}\nonumber\\ -& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n}{(2\lambda)_n}C_n^{(\lambda)}(x)t^n, -\quad\textrm{$\gamma$ arbitrary}. -\end{eqnarray} - -\subsection*{Limit relation} - -\subsubsection*{Gegenbauer / Ultraspherical $\rightarrow$ Hermite} -The Hermite polynomials given by (\ref{DefHermite}) follow from the Gegenbauer (or -ultraspherical) polynomials by taking $\lambda=\alpha+\frac{1}{2}$ and letting -$\alpha\rightarrow\infty$ in the following way: -\begin{equation} -\lim_{\alpha\rightarrow\infty} -\alpha^{-\frac{1}{2}n}C_n^{(\alpha+\frac{1}{2})}(\alpha^{-\frac{1}{2}}x)=\frac{H_n(x)}{n!}. -\end{equation} - -\subsection*{Remarks} -The case $\lambda=0$ needs another normalization. In that case we have the -Chebyshev polynomials of the first kind described in the next subsection. - -\noindent -The Gegenbauer (or ultraspherical) polynomials are related to the Jacobi polynomials -given by (\ref{DefJacobi}) by the quadratic transformations: -$$C_{2n}^{(\lambda)}(x)=\frac{(\lambda)_n}{(\frac{1}{2})_n} -P_n^{(\lambda-\frac{1}{2},-\frac{1}{2})}(2x^2-1)$$ -and -$$C_{2n+1}^{(\lambda)}(x)=\frac{(\lambda)_{n+1}}{(\frac{1}{2})_{n+1}} -xP_n^{(\lambda-\frac{1}{2},\frac{1}{2})}(2x^2-1).$$ -% RS: add begin\label{sec9.8.1} -% -\paragraph{\large\bf KLSadd: Notation}Here the Gegenbauer polynomial is denoted by $C_n^\la$ instead of $C_n^{(\la)}$. -% -\paragraph{\large\bf KLSadd: Orthogonality relation}Write the \RHS\ of (9.8.20) as $h_n\,\de_{m,n}$. Then -\begin{equation} -\frac{h_n}{h_0}= -\frac\la{\la+n}\,\frac{(2\la)_n}{n!}\,,\quad -h_0=\frac{\pi^\half\,\Ga(\la+\thalf)}{\Ga(\la+1)},\quad -\frac{h_n}{h_0\,(C_n^\la(1))^2}= -\frac\la{\la+n}\,\frac{n!}{(2\la)_n}\,. -\label{61} -\end{equation} -% -\paragraph{\large\bf KLSadd: Hypergeometric representation}Beside (9.8.19) we have also -\begin{equation} -C_n^\lambda(x)=\sum_{\ell=0}^{\lfloor n/2\rfloor}\frac{(-1)^{\ell}(\lambda)_{n-\ell}} -{\ell!\;(n-2\ell)!}\,(2x)^{n-2\ell} -=(2x)^{n}\,\frac{(\lambda)_{n}}{n!}\, -\hyp21{-\thalf n,-\thalf n+\thalf}{1-\la-n}{\frac1{x^2}}. -\label{57} -\end{equation} -See \mycite{DLMF}{(18.5.10)}. -% -\paragraph{\large\bf KLSadd: Special value}\begin{equation} -C_n^{\la}(1)=\frac{(2\la)_n}{n!}\,. -\label{49} -\end{equation} -Use (9.8.19) or see \mycite{DLMF}{Table 18.6.1}. -% -\paragraph{\large\bf KLSadd: Expression in terms of Jacobi}% -\begin{equation} -\frac{C_n^\la(x)}{C_n^\la(1)}= -\frac{P_n^{(\la-\half,\la-\half)}(x)}{P_n^{(\la-\half,\la-\half)}(1)}\,,\qquad -C_n^\la(x)=\frac{(2\la)_n}{(\la+\thalf)_n}\,P_n^{(\la-\half,\la-\half)}(x). -\label{65} -\end{equation} -% -\paragraph{\large\bf KLSadd: Re: (9.8.21)}By iteration of recurrence relation (9.8.21): -\begin{multline} -x^2 C_n^\la(x)= -\frac{(n+1)(n+2)}{4(n+\la)(n+\la+1)}\,C_{n+2}^\la(x)+ -\frac{n^2+2n\la+\la-1}{2(n+\la-1)(n+\la+1)}\,C_n^\la(x)\\ -+\frac{(n+2\la-1)(n+2\la-2)}{4(n+\la)(n+\la-1)}\,C_{n-2}^\la(x). -\label{6} -\end{multline} -% -\paragraph{\large\bf KLSadd: Bilateral generating functions}\begin{multline} -\sum_{n=0}^\iy\frac{n!}{(2\la)_n}\,r^n\,C_n^\la(x)\,C_n^\la(y) -=\frac1{(1-2rxy+r^2)^\la}\,\hyp21{\thalf\la,\thalf(\la+1)}{\la+\thalf} -{\frac{4r^2(1-x^2)(1-y^2)}{(1-2rxy+r^2)^2}}\\ -(r\in(-1,1),\;x,y\in[-1,1]). -\label{66} -\end{multline} -For the proof put $\be:=\al$ in \eqref{58}, then use \eqref{63} and \eqref{65}. -The Poisson kernel for Gegenbauer polynomials can be derived in a similar way -from \eqref{59}, or alternatively by applying the operator -$r^{-\la+1}\frac d{dr}\circ r^\la$ to both sides of \eqref{66}: -\begin{multline} -\sum_{n=0}^\iy\frac{\la+n}\la\,\frac{n!}{(2\la)_n}\,r^n\,C_n^\la(x)\,C_n^\la(y) -=\frac{1-r^2}{(1-2rxy+r^2)^{\la+1}}\\ -\times\hyp21{\thalf(\la+1),\thalf(\la+2)}{\la+\thalf} -{\frac{4r^2(1-x^2)(1-y^2)}{(1-2rxy+r^2)^2}}\qquad -(r\in(-1,1),\;x,y\in[-1,1]). -\label{67} -\end{multline} -Formula \eqref{67} was obtained by Gasper \& Rahman \myciteKLS{234}{(4.4)} -as a limit case of their formula for the Poisson kernel for continuous -$q$-ultraspherical polynomials. -% -\paragraph{\large\bf KLSadd: Trigonometric expansions}By \mycite{DLMF}{(18.5.11), (15.8.1)}: -\begin{align} -C_n^{\la}(\cos\tha) -&=\sum_{k=0}^n\frac{(\la)_k(\la)_{n-k}}{k!\,(n-k)!}\,e^{i(n-2k)\tha} -=e^{in\tha}\frac{(\la)_n}{n!}\, -\hyp21{-n,\la}{1-\la-n}{e^{-2i\tha}}\label{103}\\ -&=\frac{(\la)_n}{2^\la n!}\, -e^{-\half i\la\pi}e^{i(n+\la)\tha}\,(\sin\tha)^{-\la}\, -\hyp21{\la,1-\la}{1-\la-n}{\frac{i e^{-i\tha}}{2\sin\tha}}\label{104}\\ -&=\frac{(\la)_n}{n!}\,\sum_{k=0}^\iy\frac{(\la)_k(1-\la)_k}{(1-\la-n)_k k!}\, -\frac{\cos((n-k+\la)\tha+\thalf(k-\la)\pi)}{(2\sin\tha)^{k+\la}}\,.\label{105} -\end{align} -In \eqref{104} and \eqref{105} we require that -$\tfrac16\pi<\tha<\tfrac56\pi$. Then the convergence is absolute for $\la>\thalf$ -and conditional for $0<\la\le\thalf$. - -By \mycite{DLMF}{(14.13.1), (14.3.21), (15.8.1)]}: -\begin{align} -C_n^\la(\cos\tha)&=\frac{2\Ga(\la+\thalf)}{\pi^\half\Ga(\la+1)}\, -\frac{(2\la)_n}{(\la+1)_n}\,(\sin\tha)^{1-2\la}\, -\sum_{k=0}^\iy\frac{(1-\la)_k(n+1)_k}{(n+\la+1)_k k!}\, -\sin\big((2k+n+1)\tha\big) -\label{7}\\ -&=\frac{2\Ga(\la+\thalf)}{\pi^\half\Ga(\la+1)}\, -\frac{(2\la)_n}{(\la+1)_n}\,(\sin\tha)^{1-2\la}\, -\Im\!\!\left(e^{i(n+1)\tha}\,\hyp21{1-\la,n+1}{n+\la+1}{e^{2i\tha}}\right)\nonumber\\ -&=\frac{2^\la\Ga(\la+\thalf)}{\pi^\half\Ga(\la+1)}\, -\frac{(2\la)_n}{(\la+1)_n}\,(\sin\tha)^{-\la}\, -\Re\!\!\left(e^{-\thalf i\la\pi}e^{i(n+\la)\tha}\, -\hyp21{\la,1-\la}{1+\la+n}{\frac{e^{i\tha}}{2i\sin\tha}}\right)\nonumber\\ -&=\frac{2^{2\la}\Ga(\la+\thalf)}{\pi^\half\Ga(\la+1)}\,\frac{(2\la)_n}{(\la+1)_n}\, -\sum_{k=0}^\iy\frac{(\la)_k(1-\la)_k}{(1+\la+n)_k k!}\, -\frac{\cos((n+k+\la)\tha-\thalf(k+\la)\pi)}{(2\sin\tha)^{k+\la}}\,. -\label{106} -\end{align} -We require that $0<\tha<\pi$ in \eqref{7} and $\tfrac16\pi<\tha<\tfrac56\pi$ in -\eqref{106} The convergence is absolute for $\la>\thalf$ and conditional for -$0<\la\le\thalf$. -For $\la\in\Zpos$ the above series terminate after the term with -$k=\la-1$. -Formulas \eqref{7} and \eqref{106} are also given in -\mycite{Sz}{(4.9.22), (4.9.25)}. -% -\paragraph{\large\bf KLSadd: Fourier transform}\begin{equation} -\frac{\Ga(\la+1)}{\Ga(\la+\thalf)\,\Ga(\thalf)}\, -\int_{-1}^1 \frac{C_n^\la(y)}{C_n^\la(1)}\,(1-y^2)^{\la-\half}\, -e^{ixy}\,dy -=i^n\,2^\la\,\Ga(\la+1)\,x^{-\la}\,J_{\la+n}(x). -\label{8} -\end{equation} -See \mycite{DLMF}{(18.17.17) and (18.17.18)}. -% -\paragraph{\large\bf KLSadd: Laplace transforms}\begin{equation} -\frac2{n!\,\Ga(\la)}\, -\int_0^\iy H_n(tx)\,t^{n+2\la-1}\,e^{-t^2}\,dt=C_n^\la(x). -\label{56} -\end{equation} -See Nielsen \cite[p.48, (4) with p.47, (1) and p.28, (10)]{K4} (1918) -or Feldheim \cite[(28)]{K3} (1942). -\begin{equation} -\frac2{\Ga(\la+\thalf)}\,\int_0^1 \frac{C_n^\la(t)}{C_n^\la(1)}\, -(1-t^2)^{\la-\half}\,t^{-1}\,(x/t)^{n+2\la+1}\,e^{-x^2/t^2}\,dt -=2^{-n}\,H_n(x)\,e^{-x^2}\quad(\la>-\thalf). -\label{46} -\end{equation} -Use Askey \& Fitch \cite[(3.29)]{K2} for $\al=\pm\thalf$ together with -\eqref{48}, \eqref{51}, \eqref{52}, \eqref{54} and \eqref{55}. -\paragraph{\large\bf KLSadd: Addition formula}\begin{multline} -R_n^{(\al,\al)}\big(xy+(1-x^2)^\half(1-y^2)^\half t\big) -=\sum_{k=0}^n \frac{(-1)^k(-n)_k\,(n+2\al+1)_k}{2^{2k}((\al+1)_k)^2}\\ -\times(1-x^2)^{k/2} R_{n-k}^{(\al+k,\al+k)}(x)\,(1-y^2)^{k/2} R_{n-k}^{(\al+k,\al+k)}(y)\, -\om_k^{(\al-\half,\al-\half)}\,R_k^{(\al-\half,\al-\half)}(t), -\label{108} -\end{multline} -where -\[ -R_n^{(\al,\be)}(x):=P_n^{(\al,\be)}(x)/P_n^{(\al,\be)}(1),\quad -\om_n^{(\al,\be)}:=\frac{\int_{-1}^1 (1-x)^\al(1+x)^\be\,dx} -{\int_{-1}^1 (R_n^{(\al,\be)}(x))^2\,(1-x)^\al(1+x)^\be\,dx}\,. -\] -% -% RS: add end -\subsection*{References} -\cite{Abram}, \cite{Ahmed+86}, \cite{AndrewsAskeyRoy}, \cite{Area+I}, \cite{Askey67}, \cite{Askey74}, \cite{Askey75}, -\cite{Askey89I}, \cite{AskeyFitch}, \cite{AskeyKoornRahman}, \cite{Berg}, \cite{BilodeauI}, -\cite{BojanovNikolov}, \cite{Brafman51}, \cite{Brafman57I}, \cite{Brown}, -\cite{BustozIsmail82}, \cite{BustozIsmail83}, \cite{BustozIsmail89}, \cite{BustozSavage79}, -\cite{BustozSavage80}, \cite{Carlitz61II}, \cite{Chihara78}, \cite{Common}, \cite{Danese}, -\cite{Dette94}, \cite{DijksmaKoorn}, \cite{DilcherStolarsky}, \cite{Dimitrov96}, -\cite{Dimitrov2003}, \cite{Doha2002II}, \cite{Driver}, \cite{ElbertLaforgia86I}, -\cite{ElbertLaforgia86II}, \cite{ElbertLaforgia90}, \cite{Erdelyi+}, \cite{Exton96}, -\cite{Gasper69}, \cite{Gasper72II}, \cite{Gasper85}, \cite{Grad}, \cite{Ismail74}, -\cite{Ismail2005II}, \cite{Koekoek2000}, \cite{Koorn88}, \cite{Laforgia}, \cite{LewanowiczI}, -\cite{Lorch}, \cite{Mathai}, \cite{Nagel}, \cite{Nikiforov+}, \cite{NikiforovUvarov}, -\cite{RahmanShah}, \cite{Rainville}, \cite{Reimer}, \cite{Sartoretto}, \cite{Srivastava71}, -\cite{Szego75}, \cite{Temme}, \cite{Viswanathan}, \cite{Zayed}. - -\subsection{Chebyshev}\index{Chebyshev polynomials} - -\par - -\subsection*{Hypergeometric representation} -The Chebyshev polynomials of the first kind can be obtained from the Jacobi -polynomials by taking $\alpha=\beta=-\frac{1}{2}$: -\begin{equation} -\label{DefChebyshevI} -T_n(x)=\frac{P_n^{(-\frac{1}{2},-\frac{1}{2})}(x)}{P_n^{(-\frac{1}{2},-\frac{1}{2})}(1)} -=\hyp{2}{1}{-n,n}{\frac{1}{2}}{\frac{1-x}{2}} -\end{equation} -and the Chebyshev polynomials of the second kind can be obtained from the -Jacobi polynomials by taking $\alpha=\beta=\frac{1}{2}$: -\begin{equation} -\label{DefChebyshevII} -U_n(x)=(n+1)\frac{P_n^{(\frac{1}{2},\frac{1}{2})}(x)}{P_n^{(\frac{1}{2},\frac{1}{2})}(1)} -=(n+1)\,\hyp{2}{1}{-n,n+2}{\frac{3}{2}}{\frac{1-x}{2}}. -\end{equation} - -\subsection*{Orthogonality relation} -\begin{equation} -\label{OrtChebyshevI} -\int_{-1}^1(1-x^2)^{-\frac{1}{2}}T_m(x)T_n(x)\,dx= -\left\{\begin{array}{ll} -\displaystyle\frac{\cpi}{2}\,\delta_{mn}, & n\neq 0\\[5mm] -\cpi\,\delta_{mn}, & n=0. -\end{array}\right. -\end{equation} - -\begin{equation} -\label{OrtChebyshevII} -\int_{-1}^1(1-x^2)^{\frac{1}{2}}U_m(x)U_n(x)\,dx=\frac{\cpi}{2}\,\delta_{mn}. -\end{equation} - -\subsection*{Recurrence relations} -\begin{equation} -\label{RecChebyshevI} -2xT_n(x)=T_{n+1}(x)+T_{n-1}(x),\quad T_{0}(x)=1\quad\textrm{and}\quad T_1(x)=x. -\end{equation} - -\begin{equation} -\label{RecChebyshevII} -2xU_n(x)=U_{n+1}(x)+U_{n-1}(x),\quad U_{0}(x)=1\quad\textrm{and}\quad U_1(x)=2x. -\end{equation} - -\subsection*{Normalized recurrence relations} -\begin{equation} -\label{NormRecChebyshevI} -xp_n(x)=p_{n+1}(x)+\frac{1}{4}p_{n-1}(x), -\end{equation} -where -$$T_1(x)=p_1(x)=x\quad\textrm{and}\quad T_n(x)=2^np_n(x),\quad n\neq 1.$$ - -\begin{equation} -\label{NormRecChebyshevII} -xp_n(x)=p_{n+1}(x)+\frac{1}{4}p_{n-1}(x), -\end{equation} -where -$$U_n(x)=2^np_n(x).$$ - -\subsection*{Differential equations} -\begin{equation} -\label{dvChebyshevI} -(1-x^2)y''(x)-xy'(x)+n^2y(x)=0,\quad y(x)=T_n(x). -\end{equation} - -\begin{equation} -\label{dvChebyshevII} -(1-x^2)y''(x)-3xy'(x)+n(n+2)y(x)=0,\quad y(x)=U_n(x). -\end{equation} - -\subsection*{Forward shift operator} -\begin{equation} -\label{shift1Chebyshev} -\frac{d}{dx}T_n(x)=nU_{n-1}(x). -\end{equation} - -\subsection*{Backward shift operator} -\begin{equation} -\label{shift2ChebyshevI} -(1-x^2)\frac{d}{dx}U_n(x)-xU_n(x)=-(n+1)T_{n+1}(x) -\end{equation} -or equivalently -\begin{equation} -\label{shift2ChebyshevII} -\frac{d}{dx}\left[\left(1-x^2\right)^{\frac{1}{2}}U_n(x)\right] -=-(n+1)\left(1-x^2\right)^{-\frac{1}{2}}T_{n+1}(x). -\end{equation} - -\subsection*{Rodrigues-type formulas} -\begin{equation} -\label{RodChebyshevI} -(1-x^2)^{-\frac{1}{2}}T_n(x)=\frac{(-1)^n}{(\frac{1}{2})_n2^n} -\left(\frac{d}{dx}\right)^n\left[(1-x^2)^{n-\frac{1}{2}}\right]. -\end{equation} - -\begin{equation} -\label{RodChebyshevII} -(1-x^2)^{\frac{1}{2}}U_n(x)=\frac{(n+1)(-1)^n}{(\frac{3}{2})_n2^n} -\left(\frac{d}{dx}\right)^n\left[(1-x^2)^{n+\frac{1}{2}}\right]. -\end{equation} - -\subsection*{Generating functions} -\begin{equation} -\label{GenChebyshevI1} -\frac{1-xt}{1-2xt+t^2}=\sum_{n=0}^{\infty}T_n(x)t^n. -\end{equation} - -\begin{equation} -\label{GenChebyshevI2} -R^{-1}\sqrt{\frac{1}{2}(1+R-xt)}=\sum_{n=0}^{\infty} -\frac{\left(\frac{1}{2}\right)_n}{n!}T_n(x)t^n,\quad R=\sqrt{1-2xt+t^2}. -\end{equation} - -\begin{equation} -\label{GenChebyshevI3} -\hyp{0}{1}{-}{\frac{1}{2}}{\frac{(x-1)t}{2}}\, -\hyp{0}{1}{-}{\frac{1}{2}}{\frac{(x+1)t}{2}}= -\sum_{n=0}^{\infty}\frac{T_n(x)}{\left(\frac{1}{2}\right)_nn!}t^n. -\end{equation} - -\begin{equation} -\label{GenChebyshevI4} -\e^{xt}\,\hyp{0}{1}{-}{\frac{1}{2}}{\frac{(x^2-1)t^2}{4}}= -\sum_{n=0}^{\infty}\frac{T_n(x)}{n!}t^n. -\end{equation} - -\begin{eqnarray} -\label{GenChebyshevI5} -& &\hyp{2}{1}{\gamma,-\gamma}{\frac{1}{2}}{\frac{1-R-t}{2}}\, -\hyp{2}{1}{\gamma,-\gamma}{\frac{1}{2}}{\frac{1-R+t}{2}}\nonumber\\ -& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n(-\gamma)_n}{\left(\frac{1}{2}\right)_nn!}T_n(x)t^n, -\quad R=\sqrt{1-2xt+t^2},\quad\textrm{$\gamma$ arbitrary}. -\end{eqnarray} - -\begin{eqnarray} -\label{GenChebyshevI6} -& &(1-xt)^{-\gamma}\,\hyp{2}{1}{\frac{1}{2}\gamma,\frac{1}{2}\gamma+\frac{1}{2}}{\frac{1}{2}} -{\frac{(x^2-1)t^2}{(1-xt)^2}}\nonumber\\ -& &=\sum_{n=0}^{\infty}\frac{(\gamma)_n}{n!}T_n(x)t^n,\quad\textrm{$\gamma$ arbitrary}. -\end{eqnarray} - -\begin{equation} -\label{GenChebyshevII1} -\frac{1}{1-2xt+t^2}=\sum_{n=0}^{\infty}U_n(x)t^n. -\end{equation} - -\begin{equation} -\label{GenChebyshevII2} -\frac{1}{R\sqrt{\frac{1}{2}(1+R-xt)}}=\sum_{n=0}^{\infty} -\frac{\left(\frac{3}{2}\right)_n}{(n+1)!}U_n(x)t^n,\quad R=\sqrt{1-2xt+t^2}. -\end{equation} - -\begin{equation} -\label{GenChebyshevII3} -\hyp{0}{1}{-}{\frac{3}{2}}{\frac{(x-1)t}{2}}\, -\hyp{0}{1}{-}{\frac{3}{2}}{\frac{(x+1)t}{2}}= -\sum_{n=0}^{\infty}\frac{U_n(x)}{\left(\frac{3}{2}\right)_n(n+1)!}t^n. -\end{equation} - -\begin{equation} -\label{GenChebyshevII4} -\e^{xt}\,\hyp{0}{1}{-}{\frac{3}{2}}{\frac{(x^2-1)t^2}{4}}= -\sum_{n=0}^{\infty}\frac{U_n(x)}{(n+1)!}t^n. -\end{equation} - -\begin{eqnarray} -\label{GenChebyshevII5} -& &\hyp{2}{1}{\gamma,2-\gamma}{\frac{3}{2}}{\frac{1-R-t}{2}}\, -\hyp{2}{1}{\gamma,2-\gamma}{\frac{3}{2}}{\frac{1-R+t}{2}}\nonumber\\ -& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n(2-\gamma)_n}{\left(\frac{3}{2}\right)_n(n+1)!}U_n(x)t^n, -\quad R=\sqrt{1-2xt+t^2},\quad\textrm{$\gamma$ arbitrary}. -\end{eqnarray} - -\begin{eqnarray} -\label{GenChebyshevII6} -& &(1-xt)^{-\gamma}\,\hyp{2}{1}{\frac{1}{2}\gamma,\frac{1}{2}\gamma+\frac{1}{2}}{\frac{3}{2}} -{\frac{(x^2-1)t^2}{(1-xt)^2}}\nonumber\\ -& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n}{(n+1)!}U_n(x)t^n,\quad\textrm{$\gamma$ arbitrary}. -\end{eqnarray} - -\subsection*{Remarks} -The Chebyshev polynomials can also be written as: -$$T_n(x)=\cos(n\theta),\quad x=\cos\theta$$ -and -$$U_n(x)=\frac{\sin (n+1)\theta}{\sin\theta},\quad x=\cos\theta.$$ -Further we have -$$U_n(x)=C_n^{(1)}(x)$$ -where $C_n^{(\lambda)}(x)$ denotes the Gegenbauer (or ultraspherical) -polynomial given by (\ref{DefGegenbauer}) in the preceding subsection. -% RS: add begin\label{sec9.8.2} -In addition to the Chebyshev polynomials $T_n$ of the first kind (9.8.35) -and $U_n$ of the second kind (9.8.36), -\begin{align} -T_n(x)&:=\frac{P_n^{(-\half,-\half)}(x)}{P_n^{(-\half,-\half)}(1)} -=\cos(n\tha),\quad x=\cos\tha,\\ -U_n(x)&:=(n+1)\,\frac{P_n^{(\half,\half)}(x)}{P_n^{(\half,\half)}(1)} -=\frac{\sin((n+1)\tha)}{\sin\tha}\,,\quad x=\cos\tha, -\end{align} -we have Chebyshev polynomials $V_n$ {\em of the third kind} -and $W_n$ {\em of the fourth kind}, -\begin{align} -V_n(x)&:=\frac{P_n^{(-\half,\half)}(x)}{P_n^{(-\half,\half)}(1)} -=\frac{\cos((n+\thalf)\tha)}{\cos(\thalf\tha)}\,,\quad x=\cos\tha,\\ -W_n(x)&:=(2n+1)\,\frac{P_n^{(\half,-\half)}(x)}{P_n^{(\half,-\half)}(1)} -=\frac{\sin((n+\thalf)\tha)}{\sin(\thalf\tha)}\,,\quad x=\cos\tha, -\end{align} -see \cite[Section 1.2.3]{K20}. Then there is the symmetry -\begin{equation} -V_n(-x)=(-1)^n W_n(x). -\label{140} -\end{equation} - -The names of Chebyshev polynomials of the third and fourth kind -and the notation $V_n(x)$ are due to Gautschi \cite{K21}. -The notation $W_n(x)$ was first used by Mason \cite{K22}. -Names and notations for Chebyshev polynomials of the third and fourth -kind are interchanged in \mycite{AAR}{Remark 2.5.3} and -\mycite{DLMF}{Table 18.3.1}. -% -% RS: add end -\subsection*{References} -\cite{Abram}, \cite{AskeyFitch}, \cite{AskeyGasperHarris}, \cite{AskeyIsmail76}, -\cite{Bavinck95}, \cite{Chihara78}, \cite{Danese}, \cite{DilcherStolarsky}, -\cite{Erdelyi+}, \cite{Grad}, \cite{HartmannStephan}, \cite{Ismail2005II}, \cite{Koekoek2000}, -\cite{Luke}, \cite{Mathai}, \cite{Nikiforov+}, \cite{NikiforovUvarov}, \cite{Rainville}, -\cite{Rayes+}, \cite{Rivlin}, \cite{SainteViennot}, \cite{Szego75}, \cite{Temme}, -\cite{Wilson70I}, \cite{Zayed}, \cite{Zhang}, \cite{ZhangWang}. - -\subsection{Legendre / Spherical}\index{Legendre polynomials} -\index{Spherical polynomials} - -\par - -\subsection*{Hypergeometric representation} -The Legendre (or spherical) polynomials are Jacobi polynomials with $\alpha=\beta=0$: -\begin{equation} -\label{DefLegendre} -P_n(x)=P_n^{(0,0)}(x)=\hyp{2}{1}{-n,n+1}{1}{\frac{1-x}{2}}. -\end{equation} - -\subsection*{Orthogonality relation} -\begin{equation} -\label{OrtLegendre} -\int_{-1}^1P_m(x)P_n(x)\,dx=\frac{2}{2n+1}\,\delta_{mn}. -\end{equation} - -\subsection*{Recurrence relation} -\begin{equation} -\label{RecLegendre} -(2n+1)xP_n(x)=(n+1)P_{n+1}(x)+nP_{n-1}(x). -\end{equation} - -\subsection*{Normalized recurrence relation} -\begin{equation} -\label{NormRecLegendre} -xp_n(x)=p_{n+1}(x)+\frac{n^2}{(2n-1)(2n+1)}p_{n-1}(x), -\end{equation} -where -$$P_n(x)=\binom{2n}{n}\frac{1}{2^n}p_n(x).$$ - -\subsection*{Differential equation} -\begin{equation} -\label{dvLegendre} -(1-x^2)y''(x)-2xy'(x)+n(n+1)y(x)=0,\quad y(x)=P_n(x). -\end{equation} - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodLegendre} -P_n(x)=\frac{(-1)^n}{2^nn!}\left(\frac{d}{dx}\right)^n\left[(1-x^2)^n\right]. -\end{equation} - -\subsection*{Generating functions} -\begin{equation} -\label{GenLegendre1} -\frac{1}{\sqrt{1-2xt+t^2}}=\sum_{n=0}^{\infty}P_n(x)t^n. -\end{equation} - -\begin{equation} -\label{GenLegendre2} -\hyp{0}{1}{-}{1}{\frac{(x-1)t}{2}}\,\hyp{0}{1}{-}{1}{\frac{(x+1)t}{2}}= -\sum_{n=0}^{\infty}\frac{P_n(x)}{(n!)^2}t^n. -\end{equation} - -\begin{equation} -\label{GenLegendre3} -\e^{xt}\,\hyp{0}{1}{-}{1}{\frac{(x^2-1)t^2}{4}}= -\sum_{n=0}^{\infty}\frac{P_n(x)}{n!}t^n. -\end{equation} - -\begin{eqnarray} -\label{GenLegendre4} -& &\hyp{2}{1}{\gamma,1-\gamma}{1}{\frac{1-R-t}{2}}\, -\hyp{2}{1}{\gamma,1-\gamma}{1}{\frac{1-R+t}{2}}\nonumber\\ -& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n(1-\gamma)_n}{(n!)^2}P_n(x)t^n, -\quad R=\sqrt{1-2xt+t^2},\quad\textrm{$\gamma$ arbitrary}. -\end{eqnarray} - -\begin{eqnarray} -\label{GenLegendre5} -& &(1-xt)^{-\gamma}\,\hyp{2}{1}{\frac{1}{2}\gamma,\frac{1}{2}\gamma+\frac{1}{2}}{1} -{\frac{(x^2-1)t^2}{(1-xt)^2}}\nonumber\\ -& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n}{n!}P_n(x)t^n,\quad\textrm{$\gamma$ arbitrary}. -\end{eqnarray} - -\subsection*{References} -\cite{Abram}, \cite{Alladi}, \cite{AlSalam90}, \cite{Bhonsle}, \cite{Brafman51}, -\cite{Carlitz57II}, \cite{Chihara78}, \cite{Danese}, \cite{Dattoli2001}, -\cite{DilcherStolarsky}, \cite{ElbertLaforgia94}, \cite{Erdelyi+}, \cite{Grad}, -\cite{Mathai}, \cite{Nikiforov+}, \cite{NikiforovUvarov}, \cite{Olver}, \cite{Rainville}, -\cite{Szego75}, \cite{Temme}, \cite{Zayed}. - -\newpage - -\section{Pseudo Jacobi}\index{Pseudo Jacobi polynomials}\index{Jacobi polynomials!Pseudo} - -\par\setcounter{equation}{0} - -\subsection*{Hypergeometric representation} -\begin{eqnarray} -\label{DefPseudoJacobi} -P_n(x;\nu,N)&=&\frac{(-2i)^n(-N+i\nu)_n}{(n-2N-1)_n}\,\hyp{2}{1}{-n,n-2N-1}{-N+i\nu}{\frac{1-ix}{2}}\\ -&=&(x+i)^n\,\hyp{2}{1}{-n,N+1-n-i\nu}{2N+2-2n}{\frac{2}{1-ix}},\quad n=0,1,2,\ldots,N.\nonumber -\end{eqnarray} - -\subsection*{Orthogonality relation} -\begin{eqnarray} -\label{OrtPseudoJacobi} -& &\frac{1}{2\pi}\int_{-\infty}^{\infty}(1+x^2)^{-N-1}\e^{2\nu\arctan x}P_m(x;\nu,N)P_n(x;\nu,N)\,dx\nonumber\\ -& &{}=\frac{\Gamma(2N+1-2n)\Gamma(2N+2-2n)2^{2n-2N-1}n!}{\Gamma(2N+2-n)\left|\Gamma(N+1-n+i\nu)\right|^2}\,\delta_{mn}. -\end{eqnarray} - -\subsection*{Recurrence relation} -\begin{eqnarray} -\label{RecPseudoJacobi} -xP_n(x;\nu,N)&=&P_{n+1}(x;\nu,N)+\frac{(N+1)\nu}{(n-N-1)(n-N)}P_n(x;\nu,N)\nonumber\\ -& &{}\mathindent{}-\frac{n(n-2N-2)}{(2n-2N-3)(n-N-1)^2(2n-2N-1)}\nonumber\\ -& &{}\mathindent\mathindent\times(n-N-1-i\nu)(n-N-1+i\nu)P_{n-1}(x;\nu,N). -\end{eqnarray} - -\subsection*{Normalized recurrence relation} -\begin{eqnarray} -\label{NormRecPseudoJacobi} -xp_n(x)&=&p_{n+1}(x)+\frac{(N+1)\nu}{(n-N-1)(n-N)}p_n(x)\nonumber\\ -& &{}\mathindent{}-\frac{n(n-2N-2)(n-N-1-i\nu)(n-N-1+i\nu)}{(2n-2N-3)(n-N-1)^2(2n-2N-1)}p_{n-1}(x), -\end{eqnarray} -where -$$P_n(x;\nu,N)=p_n(x).$$ - -\subsection*{Differential equation} -\begin{equation} -\label{dvPseudoJacobi} -(1+x^2)y''(x)+2\left(\nu-Nx\right)y'(x)-n(n-2N-1)y(x)=0, -\end{equation} -where -$$y(x)=P_n(x;\nu,N).$$ - -\subsection*{Forward shift operator} -\begin{equation} -\label{shift1PseudoJacobi} -\frac{d}{dx}P_n(x;\nu,N)=nP_{n-1}(x;\nu,N-1). -\end{equation} - -\subsection*{Backward shift operator} -\begin{eqnarray} -\label{shift2PseudoJacobiI} -& &(1+x^2)\frac{d}{dx}P_n(x;\nu,N)+2\left[\nu-(N+1)x\right]P_n(x;\nu,N)\nonumber\\ -& &{}=(n-2N-2)P_{n+1}(x;\nu,N+1) -\end{eqnarray} -or equivalently -\begin{eqnarray} -\label{shift2PseudoJacobiII} -& &\frac{d}{dx}\left[(1+x^2)^{-N-1}\e^{2\nu\arctan x}P_n(x;\nu,N)\right]\nonumber\\ -& &{}=(n-2N-2)(1+x^2)^{-N-2}\e^{2\nu\arctan x}P_{n+1}(x;\nu,N+1). -\end{eqnarray} - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodPseudoJacobi} -P_n(x;\nu,N)=\frac{(1+x^2)^{N+1}\expe^{-2\nu\arctan x}}{(n-2N-1)_n} -\left(\frac{d}{dx}\right)^n\left[(1+x^2)^{n-N-1}\expe^{2\nu\arctan x}\right]. -\end{equation} - -\subsection*{Generating function} -\begin{eqnarray} -\label{GenPseudoJacobi} -& &\left[\hyp{0}{1}{-}{-N+i\nu}{(x+i)t}\,\hyp{0}{1}{-}{-N-i\nu}{(x-i)t}\right]_N\nonumber\\ -& &{}=\sum_{n=0}^N\frac{(n-2N-1)_n}{(-N+i\nu)_n(-N-i\nu)_nn!}P_n(x;\nu,N)t^n. -\end{eqnarray} - -\subsection*{Limit relation} - -\subsubsection*{Continuous Hahn $\rightarrow$ Pseudo Jacobi} -The pseudo Jacobi polynomials follow from the continuous Hahn polynomials given by -(\ref{DefContHahn}) by the substitutions $x\rightarrow xt$, $a=\frac{1}{2}(-N+i\nu-2t)$, -$b=\frac{1}{2}(-N-i\nu+2t)$, $c=\frac{1}{2}(-N-i\nu-2t)$ and $d=\frac{1}{2}(-N+i\nu+2t)$, -division by $t^n$ and the limit $t\rightarrow\infty$: -\begin{eqnarray*} -& &\lim_{t\rightarrow\infty}\frac{p_n(xt;\frac{1}{2}(-N+i\nu-2t),\frac{1}{2}(-N-i\nu+2t), -\frac{1}{2}(-N+i\nu-2t),\frac{1}{2}(-N-i\nu+2t))}{t^n}\\ -& &{}=\frac{(n-2N-1)_n}{n!}P_n(x;\nu,N). -\end{eqnarray*} - -\subsection*{Remarks} -Since we have for $k 0$ & -% $0 < c < 1$ } -\end{equation} - -\subsection*{Recurrence relation} -\begin{eqnarray} -\label{RecMeixner} -(c-1)xM_n(x;\beta,c)&=&c(n+\beta)M_{n+1}(x;\beta,c)\nonumber\\ -& &{}\mathindent{}-\left[n+(n+\beta)c\right]M_n(x;\beta,c)+nM_{n-1}(x;\beta,c). -\end{eqnarray} - -\subsection*{Normalized recurrence relation} -\begin{equation} -\label{NormRecMeixner} -xp_n(x)=p_{n+1}(x)+\frac{n+(n+\beta)c}{1-c}p_n(x)+ -\frac{n(n+\beta-1)c}{(1-c)^2}p_{n-1}(x), -\end{equation} -where -$$M_n(x;\beta,c)=\frac{1}{(\beta)_n}\left(\frac{c-1}{c}\right)^np_n(x).$$ - -\subsection*{Difference equation} -\begin{equation} -\label{dvMeixner} -n(c-1)y(x)=c(x+\beta)y(x+1)-\left[x+(x+\beta)c\right]y(x)+xy(x-1), -\end{equation} -where -$$y(x)=M_n(x;\beta,c).$$ - -\subsection*{Forward shift operator} -\begin{equation} -\label{shift1MeixnerI} -M_n(x+1;\beta,c)-M_n(x;\beta,c)= -\frac{n}{\beta}\left(\frac{c-1}{c}\right)M_{n-1}(x;\beta+1,c) -\end{equation} -or equivalently -\begin{equation} -\label{shift1MeixnerII} -\Delta M_n(x;\beta,c)=\frac{n}{\beta}\left(\frac{c-1}{c}\right)M_{n-1}(x;\beta+1,c). -\end{equation} - -\subsection*{Backward shift operator} -\begin{equation} -\label{shift2MeixnerI} -c(\beta+x-1)M_n(x;\beta,c)-xM_n(x-1;\beta,c)=c(\beta-1)M_{n+1}(x;\beta-1,c) -\end{equation} -or equivalently -\begin{equation} -\label{shift2MeixnerII} -\nabla\left[\frac{(\beta)_xc^x}{x!}M_n(x;\beta,c)\right]= -\frac{(\beta-1)_xc^x}{x!}M_{n+1}(x;\beta-1,c). -\end{equation} - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodMeixner} -\frac{(\beta)_xc^x}{x!}M_n(x;\beta,c)=\nabla^n\left[\frac{(\beta+n)_xc^x}{x!}\right]. -\end{equation} - -\subsection*{Generating functions} -\begin{equation} -\label{GenMeixner1} -\left(1-\frac{t}{c}\right)^x(1-t)^{-x-\beta}= -\sum_{n=0}^{\infty}\frac{(\beta)_n}{n!}M_n(x;\beta,c)t^n. -\end{equation} - -\begin{equation} -\label{GenMeixner2} -\e^t\,\hyp{1}{1}{-x}{\beta}{\left(\frac{1-c}{c}\right)t}= -\sum_{n=0}^{\infty}\frac{M_n(x;\beta,c)}{n!}t^n. -\end{equation} - -\begin{equation} -\label{GenMeixner3} -(1-t)^{-\gamma}\,\hyp{2}{1}{\gamma,-x}{\beta}{\frac{(1-c)t}{c(1-t)}} -=\sum_{n=0}^{\infty}\frac{(\gamma)_n}{n!}M_n(x;\beta,c)t^n, -\quad\textrm{$\gamma$ arbitrary}. -\end{equation} - -\subsection*{Limit relations} - -\subsubsection*{Hahn $\rightarrow$ Meixner} -If we take $\alpha=b-1$, $\beta=N(1-c)c^{-1}$ in the definition (\ref{DefHahn}) -of the Hahn polynomials and let $N\rightarrow\infty$ we find the Meixner polynomials: -$$\lim_{N\rightarrow\infty} -Q_n(x;b-1,N(1-c)c^{-1},N)=M_n(x;b,c).$$ - -\subsubsection*{Dual Hahn $\rightarrow$ Meixner} -To obtain the Meixner polynomials from the dual Hahn polynomials we have to take -$\gamma=\beta-1$ and $\delta=N(1-c)c^{-1}$ in the definition (\ref{DefDualHahn}) of -the dual Hahn polynomials and let $N\rightarrow\infty$: -$$\lim_{N\rightarrow\infty} -R_n(\lambda(x);\beta-1,N(1-c)c^{-1},N)=M_n(x;\beta,c).$$ - -\subsubsection*{Meixner $\rightarrow$ Laguerre} -The Laguerre polynomials given by (\ref{DefLaguerre}) are obtained from the Meixner polynomials -if we take $\beta=\alpha+1$ and $x\rightarrow (1-c)^{-1}x$ and let $c\rightarrow 1$: -\begin{equation} -\lim_{c\rightarrow 1} -M_n((1-c)^{-1}x;\alpha+1,c)=\frac{L_n^{(\alpha)}(x)}{L_n^{(\alpha)}(0)}. -\end{equation} - -\subsubsection*{Meixner $\rightarrow$ Charlier} -The Charlier polynomials given by (\ref{DefCharlier}) are obtained from the Meixner polynomials -if we take $c=(a+\beta)^{-1}a$ and let $\beta\rightarrow\infty$: -\begin{equation} -\lim_{\beta\rightarrow\infty} -M_n(x;\beta,(a+\beta)^{-1}a)=C_n(x;a). -\end{equation} - -\subsection*{Remarks} -The Meixner polynomials are related to the Jacobi polynomials given by (\ref{DefJacobi}) -in the following way: -$$\frac{(\beta)_n}{n!}M_n(x;\beta,c)=P_n^{(\beta-1,-n-\beta-x)}((2-c)c^{-1}).$$ - -\noindent -The Meixner polynomials are also related to the Krawtchouk polynomials given by -(\ref{DefKrawtchouk}) in the following way: -$$K_n(x;p,N)=M_n(x;-N,(p-1)^{-1}p).$$ -% RS: add begin\label{sec9.9} -In this section in \mycite{KLS} the pseudo Jacobi polynomial $P_n(x;\nu,N)$ in (9.9.1) -is considered -for $N\in\ZZ_{\ge0}$ and $n=0,1,\ldots,n$. However, we can more generally take -$-\thalf\be>0\;\;{\rm or}\;\;\al<\be<0). -\] -Then $P_n$ can be expressed as a Meixner polynomial: -\[ -P_n(x)=(-k_2(\al\be)^{-1})_n\,\be^n\, -M_n\left(-\,\frac{x+k_2\al^{-1}}{\al-\be},-k_2(\al\be)^{-1},\be\al^{-1}\right). -\] - -In 1938 Gottlieb \cite[\S2]{K1} introduces polynomials $l_n$ ``of Laguerre type'' -which turn out to be special Meixner polynomials: -$l_n(x)=e^{-n\la} M_n(x;1,e^{-\la})$. -% -\paragraph{\large\bf KLSadd: Uniqueness of orthogonality measure}The coefficient of $p_{n-1}(x)$ in (9.10.4) behaves as $O(n^2)$ as $n\to\iy$. -Hence \eqref{93} holds, by which the orthogonality measure is unique. -% -% RS: add end -\subsection*{References} -\cite{AlSalam90}, \cite{AndrewsAskey85}, \cite{Area+II}, \cite{Askey75}, \cite{Askey89I}, -\cite{AskeyGasper77}, \cite{AskeyWilson85}, \cite{AtakRahmanSuslov}, \cite{Campigotto+}, -\cite{LChiharaStanton}, \cite{Chihara78}, \cite{Dette95}, \cite{Dominici}, \cite{Dunkl76}, -\cite{Dunkl84}, \cite{DunklRamirez}, \cite{Erdelyi+}, \cite{FeinsilverSchott}, -\cite{Gasper73I}, \cite{Gasper74}, \cite{HoareRahman}, \cite{Ismail2005II}, \cite{Karlin58}, -\cite{Koorn82}, \cite{Koorn88}, \cite{LabelleYehI}, \cite{LabelleYehII}, \cite{Lesky62}, -\cite{Lesky89}, \cite{Lesky94I}, \cite{Lesky95II}, \cite{LewanowiczII}, \cite{Nikiforov+}, -\cite{NikiforovUvarov}, \cite{Qiu}, \cite{Rahman78I}, \cite{Rahman79}, \cite{Stanton84}, -\cite{Stanton90}, \cite{Szego75}, \cite{Zarzo+}, \cite{Zeng90}. - - -\section{Laguerre}\index{Laguerre polynomials} - -\par\setcounter{equation}{0} - -\subsection*{Hypergeometric representation} -\begin{equation} -\label{DefLaguerre} -L_n^{(\alpha)}(x)=\frac{(\alpha+1)_n}{n!}\,\hyp{1}{1}{-n}{\alpha+1}{x}. -\end{equation} - -\subsection*{Orthogonality relation} -\begin{equation} -\label{OrtLaguerre} -\int_0^{\infty}\expe^{-x}x^{\alpha}L_m^{(\alpha)}(x)L_n^{(\alpha)}(x)\,dx= -\frac{\Gamma(n+\alpha+1)}{n!}\,\delta_{mn},\quad\alpha > -1. -\end{equation} - -\subsection*{Recurrence relation} -\begin{equation} -\label{RecLaguerre} -(n+1)L_{n+1}^{(\alpha)}(x)-(2n+\alpha+1-x)L_n^{(\alpha)}(x)+(n+\alpha)L_{n-1}^{(\alpha)}(x)=0. -\end{equation} - -\subsection*{Normalized recurrence relation} -\begin{equation} -\label{NormRecLaguerre} -xp_n(x)=p_{n+1}(x)+(2n+\alpha+1)p_n(x)+n(n+\alpha)p_{n-1}(x), -\end{equation} -where -$$L_n^{(\alpha)}(x)=\frac{(-1)^n}{n!}p_n(x).$$ - -\subsection*{Differential equation} -\begin{equation} -\label{dvLaguerre} -xy''(x)+(\alpha+1-x)y'(x)+ny(x)=0,\quad y(x)=L_n^{(\alpha)}(x). -\end{equation} - -\newpage - -\subsection*{Forward shift operator} -\begin{equation} -\label{shift1Laguerre} -\frac{d}{dx}L_n^{(\alpha)}(x)=-L_{n-1}^{(\alpha+1)}(x). -\end{equation} - -\subsection*{Backward shift operator} -\begin{equation} -\label{shift2LaguerreI} -x\frac{d}{dx}L_n^{(\alpha)}(x)+(\alpha-x)L_n^{(\alpha)}(x)=(n+1)L_{n+1}^{(\alpha-1)}(x) -\end{equation} -or equivalently -\begin{equation} -\label{shift2LaguerreII} -\frac{d}{dx}\left[\expe^{-x}x^{\alpha}L_n^{(\alpha)}(x)\right]=(n+1)\expe^{-x}x^{\alpha-1}L^{(\alpha-1)}_{n+1}(x). -\end{equation} - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodLaguerre} -\e^{-x}x^{\alpha}L_n^{(\alpha)}(x)=\frac{1}{n!}\left(\frac{d}{dx}\right)^n\left[\expe^{-x}x^{n+\alpha}\right]. -\end{equation} - -\subsection*{Generating functions} -\begin{equation} -\label{GenLaguerre1} -(1-t)^{-\alpha-1}\exp\left(\frac{xt}{t-1}\right)= -\sum_{n=0}^{\infty}L_n^{(\alpha)}(x)t^n. -\end{equation} - -\begin{equation} -\label{GenLaguerre2} -\e^t\,\hyp{0}{1}{-}{\alpha+1}{-xt} -=\sum_{n=0}^{\infty}\frac{L_n^{(\alpha)}(x)}{(\alpha+1)_n}t^n. -\end{equation} - -\begin{equation} -\label{GenLaguerre3} -(1-t)^{-\gamma}\,\hyp{1}{1}{\gamma}{\alpha+1}{\frac{xt}{t-1}} -=\sum_{n=0}^{\infty}\frac{(\gamma)_n}{(\alpha+1)_n}L_n^{(\alpha)}(x)t^n, -\quad\textrm{$\gamma$ arbitrary}. -\end{equation} - -\subsection*{Limit relations} - -\subsubsection*{Meixner-Pollaczek $\rightarrow$ Laguerre} -The Laguerre polynomials can be obtained from the Meixner-Pollaczek polynomials given by -(\ref{DefMP}) by the substitution $\lambda=\frac{1}{2}(\alpha+1)$, -$x\rightarrow -\frac{1}{2}\phi^{-1}x$ and the limit $\phi\rightarrow 0$: -$$\lim_{\phi\rightarrow 0} -P_n^{(\frac{1}{2}\alpha+\frac{1}{2})}(-\textstyle\frac{1}{2}\phi^{-1}x;\phi)=L_n^{(\alpha)}(x).$$ - -\subsubsection*{Jacobi $\rightarrow$ Laguerre} -The Laguerre polynomials are obtained from the Jacobi polynomials given by (\ref{DefJacobi}) -if we set $x\rightarrow 1-2\beta^{-1}x$ and then take the limit $\beta\rightarrow\infty$: -$$\lim_{\beta\rightarrow\infty} -P_n^{(\alpha,\beta)}(1-2\beta^{-1}x)=L_n^{(\alpha)}(x).$$ - -\subsubsection*{Meixner $\rightarrow$ Laguerre} -If we take $\beta=\alpha+1$ and $x\rightarrow (1-c)^{-1}x$ in the definition -(\ref{DefMeixner}) of the Meixner polynomials and let $c\rightarrow 1$ we obtain -the Laguerre polynomials: -$$\lim_{c\rightarrow 1} -M_n((1-c)^{-1}x;\alpha+1,c)=\frac{L_n^{(\alpha)}(x)}{L_n^{(\alpha)}(0)}.$$ - -\subsubsection*{Laguerre $\rightarrow$ Hermite} -The Hermite polynomials given by (\ref{DefHermite}) can be obtained from the Laguerre -polynomials by taking the limit $\alpha\rightarrow\infty$ in the following way: -\begin{equation} -\lim_{\alpha\rightarrow\infty} -\left(\frac{2}{\alpha}\right)^{\frac{1}{2}n} -L_n^{(\alpha)}((2\alpha)^{\frac{1}{2}}x+\alpha)=\frac{(-1)^n}{n!}H_n(x). -\end{equation} - -\subsection*{Remarks} -The definition (\ref{DefLaguerre}) of the Laguerre polynomials can also be -written as: -$$L_n^{(\alpha)}(x)=\frac{1}{n!}\sum_{k=0}^n\frac{(-n)_k}{k!}(\alpha+k+1)_{n-k}x^k.$$ -In this way the Laguerre polynomials can also be seen as polynomials in the parameter $\alpha$. -Therefore they can be defined for all $\alpha$. - -\noindent -The Laguerre polynomials are related to the Bessel polynomials given by (\ref{DefBessel}) -in the following way: -$$L_n^{(\alpha)}(x)=\frac{(-x)^n}{n!}y_n(2x^{-1};-2n-\alpha-1).$$ - -\noindent -The Laguerre polynomials are related to the Charlier polynomials given by (\ref{DefCharlier}) -in the following way: -$$\frac{(-a)^n}{n!}C_n(x;a)=L_n^{(x-n)}(a).$$ - -\noindent -The Laguerre polynomials and the Hermite polynomials given by (\ref{DefHermite}) are also -connected by the following quadratic transformations: -$$H_{2n}(x)=(-1)^nn!\,2^{2n}L_n^{(-\frac{1}{2})}(x^2)$$ -and -$$H_{2n+1}(x)=(-1)^nn!\,2^{2n+1}xL_n^{(\frac{1}{2})}(x^2).$$ - -\noindent -In combinatorics the Laguerre polynomials with $\alpha=0$ are often called Rook -polynomials. -% RS: add begin\label{sec9.11} -% -\paragraph{\large\bf KLSadd: Special values}By (9.11.1) and the binomial formula: -\begin{equation} -K_n(0;p,N)=1,\qquad -K_n(N;p,N)=(1-p^{-1})^n. -\label{9} -\end{equation} -The self-duality (p.240, Remarks, first formula) -\begin{equation} -K_n(x;p,N)=K_x(n;p,N)\qquad (n,x\in \{0,1,\ldots,N\}) -\label{147} -\end{equation} -combined with \eqref{9} yields: -\begin{equation} -K_N(x;p,N)=(1-p^{-1})^x\qquad(x\in\{0,1,\ldots,N\}). -\label{148} -\end{equation} -% -\paragraph{\large\bf KLSadd: Symmetry}By the orthogonality relation (9.11.2): -\begin{equation} -\frac{K_n(N-x;p,N)}{K_n(N;p,N)}=K_n(x;1-p,N). -\label{10} -\end{equation} -By \eqref{10} and \eqref{147} we have also -\begin{equation} -\frac{K_{N-n}(x;p,N)}{K_N(x;p,N)}=K_n(x;1-p,N) -\qquad(n,x\in\{0,1,\ldots,N\}), -\label{149} -\end{equation} -and, by \eqref{149}, \eqref{10} and \eqref{9}, -\begin{equation} -K_{N-n}(N-x;p,N)=\left(\frac p{p-1}\right)^{n+x-N}K_n(x;p,N) -\qquad(n,x\in\{0,1,\ldots,N\}). -\label{150} -\end{equation} -A particular case of \eqref{10} is: -\begin{equation} -K_n(N-x;\thalf,N)=(-1)^n K_n(x;\thalf,N). -\label{11} -\end{equation} -Hence -\begin{equation} -K_{2m+1}(N;\thalf,2N)=0. -\label{12} -\end{equation} -From (9.11.11): -\begin{equation} -K_{2m}(N;\thalf,2N)=\frac{(\thalf)_m}{(-N+\thalf)_m}\,. -\label{13} -\end{equation} -% -\paragraph{\large\bf KLSadd: Quadratic transformations}\begin{align} -K_{2m}(x+N;\thalf,2N)&=\frac{(\thalf)_m}{(-N+\thalf)_m}\, -R_m(x^2;-\thalf,-\thalf,N), -\label{31}\\ -K_{2m+1}(x+N;\thalf,2N)&=-\,\frac{(\tfrac32)_m}{N\,(-N+\thalf)_m}\, -x\,R_m(x^2-1;\thalf,\thalf,N-1), -\label{33}\\ -K_{2m}(x+N+1;\thalf,2N+1)&=\frac{(\tfrac12)_m}{(-N-\thalf)_m}\, -R_m(x(x+1);-\thalf,\thalf,N), -\label{32}\\ -K_{2m+1}(x+N+1;\thalf,2N+1)&=\frac{(\tfrac32)_m}{(-N-\thalf)_{m+1}}\, -(x+\thalf)\,R_m(x(x+1);\thalf,-\thalf,N), -\label{34} -\end{align} -where $R_m$ is a dual Hahn polynomial (9.6.1). For the proofs use -(9.6.2), (9.11.2), (9.6.4) and (9.11.4). -% -\paragraph{\large\bf KLSadd: Generating functions}\begin{multline} -\sum_{x=0}^N\binom Nx K_m(x;p,N)K_n(x;q,N)z^x\\ -=\left(\frac{p-z+pz}p\right)^m -\left(\frac{q-z+qz}q\right)^n -(1+z)^{N-m-n} -K_m\left(n;-\,\frac{(p-z+pz)(q-z+qz)}z,N\right). -\label{107} -\end{multline} -This follows immediately from Rosengren \cite[(3.5)]{K8}, which goes back -to Meixner \cite{K9}. -% -% RS: add end -\subsection*{References} -\cite{Abdul}, \cite{Abram}, \cite{Ahmed+82}, \cite{Allaway76}, \cite{Allaway91}, -\cite{NAlSalam66}, \cite{AlSalam64}, \cite{AlSalam90}, \cite{AlSalamChihara72}, -\cite{AlSalamChihara76}, \cite{AndrewsAskey85}, \cite{AndrewsAskeyRoy}, \cite{Askey68}, -\cite{Askey75}, \cite{Askey89I}, \cite{Askey2005}, \cite{AskeyGasper76}, \cite{AskeyGasper77}, -\cite{AskeyIsmail76}, \cite{AskeyIsmailKoorn}, \cite{AskeyWilson85}, \cite{Barrett}, -\cite{Bavinck96}, \cite{Brafman51}, \cite{Brafman57II}, \cite{Brenke}, \cite{Brown}, -\cite{BustozSavage79}, \cite{BustozSavage80}, \cite{Carlitz60}, \cite{Carlitz61I}, -\cite{Carlitz61II}, \cite{Carlitz62}, \cite{Carlitz68}, \cite{ChenSrivastava}, -\cite{ChenIsmail91}, \cite{Chihara68II}, \cite{Chihara78}, \cite{Cohen}, \cite{Cooper+}, -\cite{Dattoli2001}, \cite{DetteStudden92}, \cite{DetteStudden95}, \cite{Dimitrov2003}, -\cite{Doha2003II}, \cite{ElbertLaforgia87I}, \cite{Erdelyi}, \cite{Erdelyi+}, \cite{Exton98}, -\cite{Faldey}, \cite{Gasper73II}, \cite{Gasper77}, \cite{Gatteschi2002}, \cite{Gawronski87}, -\cite{Gawronski93}, \cite{Gillis}, \cite{GillisIsmailOffer}, \cite{Godoy+}, \cite{Grad}, -\cite{Hajmirzaahmad95}, \cite{Ismail74}, \cite{Ismail77}, \cite{Ismail2005II}, -\cite{IsmailLetVal88}, \cite{IsmailLi}, \cite{IsmailMassonRahman}, \cite{IsmailMuldoon}, -\cite{IsmailStanton97}, \cite{IsmailTamhankar}, \cite{Jackson}, \cite{Karlin58}, -\cite{Kochneff95}, \cite{Kochneff97II}, \cite{Koekoek2000}, \cite{Koorn77I}, \cite{Koorn78}, -\cite{Koorn85}, \cite{Koorn88}, \cite{Krasikov2003}, \cite{Krasikov2005}, -\cite{KuijlaarsMcLaughlin}, \cite{KwonLittle}, \cite{LabelleYehI}, \cite{LabelleYehII}, -\cite{LabelleYehIII}, \cite{Lee97I}, \cite{Lee97II}, \cite{Lee97III}, \cite{Lesky95II}, -\cite{Lesky96}, \cite{LewanowiczI}, \cite{LopezTemme2004}, \cite{Mathai}, \cite{Meixner}, -\cite{Nikiforov+}, \cite{NikiforovUvarov}, \cite{Olver}, \cite{PerezPinar}, \cite{Pittaluga}, -\cite{Rainville}, \cite{SainteViennot}, \cite{Srivastava66}, \cite{Srivastava69I}, -\cite{Srivastava69II}, \cite{Srivastava70}, \cite{Srivastava71}, \cite{Szego75}, \cite{Temme}, -\cite{Trickovic}, \cite{Trivedi}, \cite{Viennot}. - - -\section{Bessel}\index{Bessel polynomials} - -\par\setcounter{equation}{0} - -\subsection*{Hypergeometric representation} -\begin{eqnarray} -\label{DefBessel} -y_n(x;a)&=&\hyp{2}{0}{-n,n+a+1}{-}{-\frac{x}{2}}\\ -&=&(n+a+1)_n\left(\frac{x}{2}\right)^n\,\hyp{1}{1}{-n}{-2n-a}{\frac{2}{x}}, -\quad n=0,1,2,\ldots,N.\nonumber -\end{eqnarray} - -\subsection*{Orthogonality relation} -\begin{eqnarray} -\label{OrtBessel} -& &\int_0^{\infty}x^a\e^{-\frac{2}{x}}y_m(x;a)y_n(x;a)\,dx\nonumber\\ -& &=-\frac{2^{a+1}}{2n+a+1}\Gamma(-n-a)n!\,\delta_{mn},\quad a<-2N-1. -\end{eqnarray} - -\subsection*{Recurrence relation} -\begin{eqnarray} -\label{RecBessel} -& &2(n+a+1)(2n+a)y_{n+1}(x;a)\nonumber\\ -& &{}=(2n+a+1)\left[2a+(2n+a)(2n+a+2)x\right]y_n(x;a)\nonumber\\ -& &{}\mathindent{}+2n(2n+a+2)y_{n-1}(x;a). -\end{eqnarray} - -\subsection*{Normalized recurrence relation} -\begin{eqnarray} -\label{NormRecBessel} -xp_n(x)&=&p_{n+1}(x)-\frac{2a}{(2n+a)(2n+a+2)}p_n(x)\nonumber\\ -& &{}\mathindent{}-\frac{4n(n+a)}{(2n+a-1)(2n+a)^2(2n+a+1)}p_{n-1}(x), -\end{eqnarray} -where -$$y_n(x;a)=\frac{(n+a+1)_n}{2^n}p_n(x).$$ - -\subsection*{Differential equation} -\begin{equation} -\label{dvBessel} -x^2y''(x)+\left[(a+2)x+2\right]y'(x)-n(n+a+1)y(x)=0,\quad y(x)=y_n(x;a). -\end{equation} - -\subsection*{Forward shift operator} -\begin{equation} -\label{shift1Bessel} -\frac{d}{dx}y_n(x;a)=\frac{n(n+a+1)}{2}y_{n-1}(x;a+2). -\end{equation} - -\subsection*{Backward shift operator} -\begin{equation} -\label{shift2BesselI} -x^2\frac{d}{dx}y_n(x;a)+(ax+2)y_n(x;a)=2y_{n+1}(x;a-2) -\end{equation} -or equivalently -\begin{equation} -\label{shift2BesselII} -\frac{d}{dx}\left[x^a\expe^{-\frac{2}{x}}y_n(x;a)\right]=2x^{a-2}\expe^{-\frac{2}{x}}y_{n+1}(x;a-2). -\end{equation} - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodBessel} -y_n(x;a)=2^{-n}x^{-a}\expe^{\frac{2}{x}}D^n\left(x^{2n+a}\expe^{-\frac{2}{x}}\right). -\end{equation} - -\subsection*{Generating function} -\begin{equation} -\label{GenBessel} -\left(1-2xt\right)^{-\frac{1}{2}}\left(\frac{2}{1+\sqrt{1-2xt}}\right)^a -\exp\left(\frac{2t}{1+\sqrt{1-2xt}}\right)=\sum_{n=0}^{\infty}y_n(x;a)\frac{t^n}{n!}. -\end{equation} - -\subsection*{Limit relation} - -\subsubsection*{Jacobi $\rightarrow$ Bessel} -If we take $\beta=a-\alpha$ in the definition (\ref{DefJacobi}) of the Jacobi polynomials -and let $\alpha\rightarrow -\infty$ we find the Bessel polynomials: -$$\lim_{\alpha\rightarrow -\infty} -\frac{P_n^{(\alpha,a-\alpha)}(1+\alpha x)}{P_n^{(\alpha,a-\alpha)}(1)}=y_n(x;a).$$ - -\subsection*{Remarks} -The following notations are also used for the Bessel polynomials: -$$y_n(x;a,b)=y_n(2b^{-1}x;a)\quad\textrm{and}\quad\theta_n(x;a,b)=x^ny_n(x^{-1};a,b).$$ -However, the Bessel polynomials essentially depend on only one parameter. - -\noindent -The Bessel polynomials are related to the Laguerre polynomials given by (\ref{DefLaguerre}) -in the following way: -$$L_n^{(\alpha)}(x)=\frac{(-x)^n}{n!}y_n(2x^{-1};-2n-\alpha-1).$$ - -\noindent -The special case $a=-2N-2$ of the Bessel polynomials can be obtained from the pseudo Jacobi -polynomials by setting $x\rightarrow\nu x$ in the definition (\ref{DefPseudoJacobi}) of the -pseudo Jacobi polynomials and taking the limit $\nu\rightarrow\infty$ in the following way: -$$\lim\limits_{\nu\rightarrow\infty}\frac{P_n(\nu x;\nu,N)}{\nu^n} -=\frac{2^n}{(n-2N-1)_n}y_n(x;-2N-2).$$ - -\subsection*{References} -\cite{Andrade}, \cite{BergVignat}, \cite{Carlitz57I}, \cite{Dattoli2003}, -\cite{DohaAhmed2004}, \cite{DohaAhmed2006}, \cite{Grosswald}, \cite{Ismail2005II}, -\cite{KrallFrink}, \cite{Lesky98}, \cite{NikiforovUvarov}. - - -\section{Charlier}\index{Charlier polynomials} - -\par\setcounter{equation}{0} - -\subsection*{Hypergeometric representation} -\begin{equation} -\label{DefCharlier} -C_n(x;a)=\hyp{2}{0}{-n,-x}{-}{-\frac{1}{a}}. -\end{equation} - -\subsection*{Orthogonality relation} -\begin{equation} -\label{OrtCharlier} -\sum_{x=0}^{\infty}\frac{a^x}{x!}C_m(x;a)C_n(x;a)= -a^{-n}\expe^an!\,\delta_{mn},\quad a > 0. -\end{equation} - -\subsection*{Recurrence relation} -\begin{equation} -\label{RecCharlier} --xC_n(x;a)=aC_{n+1}(x;a)-(n+a)C_n(x;a)+nC_{n-1}(x;a). -\end{equation} - -\subsection*{Normalized recurrence relation} -\begin{equation} -\label{NormRecCharlier} -xp_n(x)=p_{n+1}(x)+(n+a)p_n(x)+nap_{n-1}(x), -\end{equation} -where -$$C_n(x;a)=\left(-\frac{1}{a}\right)^np_n(x).$$ - -\subsection*{Difference equation} -\begin{equation} -\label{dvCharlier} --ny(x)=ay(x+1)-(x+a)y(x)+xy(x-1),\quad y(x)=C_n(x;a). -\end{equation} - -\subsection*{Forward shift operator} -\begin{equation} -\label{shift1CharlierI} -C_n(x+1;a)-C_n(x;a)=-\frac{n}{a}C_{n-1}(x;a) -\end{equation} -or equivalently -\begin{equation} -\label{shift1CharlierII} -\Delta C_n(x;a)=-\frac{n}{a}C_{n-1}(x;a). -\end{equation} - -\subsection*{Backward shift operator} -\begin{equation} -\label{shift2CharlierI} -C_n(x;a)-\frac{x}{a}C_n(x-1;a)=C_{n+1}(x;a) -\end{equation} -or equivalently -\begin{equation} -\label{shift2CharlierII} -\nabla\left[\frac{a^x}{x!}C_n(x;a)\right]=\frac{a^x}{x!}C_{n+1}(x;a). -\end{equation} - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodCharlier} -\frac{a^x}{x!}C_n(x;a)=\nabla^n\left[\frac{a^x}{x!}\right]. -\end{equation} - -\subsection*{Generating function} -\begin{equation} -\label{GenCharlier} -\e^t\left(1-\frac{t}{a}\right)^x=\sum_{n=0}^{\infty}\frac{C_n(x;a)}{n!}t^n. -\end{equation} - -\subsection*{Limit relations} - -\subsubsection*{Meixner $\rightarrow$ Charlier} -If we take $c=(a+\beta)^{-1}a$ in the definition (\ref{DefMeixner}) of the Meixner polynomials -and let $\beta\rightarrow\infty$ we find the Charlier polynomials: -$$\lim_{\beta\rightarrow\infty}M_n(x;\beta,(a+\beta)^{-1}a)=C_n(x;a).$$ - -\subsubsection*{Krawtchouk $\rightarrow$ Charlier} -The Charlier polynomials can be found from the Krawtchouk polynomials given by -(\ref{DefKrawtchouk}) by taking $p=N^{-1}a$ and letting $N\rightarrow\infty$: -$$\lim_{N\rightarrow\infty}K_n(x;N^{-1}a,N)=C_n(x;a).$$ - -\subsubsection*{Charlier $\rightarrow$ Hermite} -The Hermite polynomials given by (\ref{DefHermite}) are obtained from the Charlier polynomials -if we set $x\rightarrow (2a)^{1/2}x+a$ and let $a\rightarrow\infty$. In fact we have -\begin{equation} -\lim_{a\rightarrow\infty} -(2a)^{\frac{1}{2}n}C_n((2a)^{\frac{1}{2}}x+a;a)=(-1)^nH_n(x). -\end{equation} - -\subsection*{Remark} -The Charlier polynomials are related to the Laguerre polynomials given by (\ref{DefLaguerre}) -in the following way: -$$\frac{(-a)^n}{n!}C_n(x;a)=L_n^{(x-n)}(a).$$ -% RS: add begin\label{sec9.12} -\paragraph{\large\bf KLSadd: Notation}Here the Laguerre polynomial is denoted by $L_n^\al$ instead of -$L_n^{(\al)}$. -% -\paragraph{\large\bf KLSadd: Hypergeometric representation}\begin{align} -L_n^\al(x)&= -\frac{(\al+1)_n}{n!}\,\hyp11{-n}{\al+1}x -\label{182}\\ -&=\frac{(-x)^n}{n!} \hyp20{-n,-n-\al}-{-\,\frac1x} -\label{183}\\ -&=\frac{(-x)^n}{n!}\,C_n(n+\al;x), -\label{184} -\end{align} -where $C_n$ in \eqref{184} is a -\hyperref[sec9.14]{Charlier polynomial}. -Formula \eqref{182} is (9.12.1). Then \eqref{183} follows by reversal -of summation. Finally \eqref{184} follows by \eqref{183} and \eqref{179}. -It is also the remark on top of p.244 in \mycite{KLS}, and it is essentially -\myciteKLS{416}{(2.7.10)}. -% -\paragraph{\large\bf KLSadd: Uniqueness of orthogonality measure}The coefficient of $p_{n-1}(x)$ in (9.12.4) behaves as $O(n^2)$ as $n\to\iy$. -Hence \eqref{93} holds, by which the orthogonality measure is unique. -% -\paragraph{\large\bf KLSadd: Special value}\begin{equation} -L_n^{\al}(0)=\frac{(\al+1)_n}{n!}\,. -\label{53} -\end{equation} -Use (9.12.1) or see \mycite{DLMF}{18.6.1)}. -% -\paragraph{\large\bf KLSadd: Quadratic transformations}\begin{align} -H_{2n}(x)&=(-1)^n\,2^{2n}\,n!\,L_n^{-1/2}(x^2), -\label{54}\\ -H_{2n+1}(x)&=(-1)^n\,2^{2n+1}\,n!\,x\,L_n^{1/2}(x^2). -\label{55} -\end{align} -See p.244, Remarks, last two formulas. -Or see \mycite{DLMF}{(18.7.19), (18.7.20)}. -% -\paragraph{\large\bf KLSadd: Fourier transform}\begin{equation} -\frac1{\Ga(\al+1)}\,\int_0^\iy \frac{L_n^\al(y)}{L_n^\al(0)}\, -e^{-y}\,y^\al\,e^{ixy}\,dy= -i^n\,\frac{y^n}{(iy+1)^{n+\al+1}}\,, -\label{14} -\end{equation} -see \mycite{DLMF}{(18.17.34)}. -% -\paragraph{\large\bf KLSadd: Differentiation formulas}Each differentiation formula is given in two equivalent forms. -\begin{equation} -\frac d{dx}\left(x^\al L_n^\al(x)\right)= -(n+\al)\,x^{\al-1} L_n^{\al-1}(x),\qquad -\left(x\frac d{dx}+\al\right)L_n^\al(x)= -(n+\al)\,L_n^{\al-1}(x). -\label{76} -\end{equation} -% -\begin{equation} -\frac d{dx}\left(e^{-x} L_n^\al(x)\right)= --e^{-x} L_n^{\al+1}(x),\qquad -\left(\frac d{dx}-1\right)L_n^\al(x)= --L_n^{\al+1}(x). -\label{77} -\end{equation} -% -Formulas \eqref{76} and \eqref{77} follow from -\mycite{DLMF}{(13.3.18), (13.3.20)} -together with (9.12.1). -% -\paragraph{\large\bf KLSadd: Generalized Hermite polynomials}See \myciteKLS{146}{p.156}, \cite[Section 1.5.1]{K26}. -These are defined by -\begin{equation} -H_{2m}^\mu(x):=\const L_m^{\mu-\half}(x^2),\qquad -H_{2m+1}^\mu(x):=\const x\,L_m^{\mu+\half}(x^2). -\label{78} -\end{equation} -Then for $\mu>-\thalf$ we have orthogonality relation -\begin{equation} -\int_{-\iy}^{\iy} H_m^\mu(x)\,H_n^\mu(x)\,|x|^{2\mu}e^{-x^2}\,dx -=0\qquad(m\ne n). -\label{79} -\end{equation} -Let the Dunkl operator $T_\mu$ be defined by \eqref{72}. -If we choose the constants in \eqref{78} as -\begin{equation} -H_{2m}^\mu(x)=\frac{(-1)^m(2m)!}{(\mu+\thalf)_m}\,L_m^{\mu-\half}(x^2),\qquad -H_{2m+1}^\mu(x)=\frac{(-1)^m(2m+1)!}{(\mu+\thalf)_{m+1}}\, - x\,L_m^{\mu+\half}(x^2) - \label{80} -\end{equation} -then (see \cite[(1.6)]{K5}) -\begin{equation} -T_\mu H_n^\mu=2n\,H_{n-1}^\mu. -\label{81} -\end{equation} -Formula \eqref{81} with \eqref{80} substituted gives rise to two -differentiation formulas involving Laguerre polynomials which are equivalent to -(9.12.6) and \eqref{76}. - -Composition of \eqref{81} with itself gives -\[ -T_\mu^2 H_n^\mu=4n(n-1)\,H_{n-2}^\mu, -\] -which is equivalent to the composition of (9.12.6) and \eqref{76}: -\begin{equation} -\left(\frac{d^2}{dx^2}+\frac{2\al+1}x\,\frac d{dx}\right)L_n^\al(x^2) -=-4(n+\al)\,L_{n-1}^\al(x^2). -\label{82} -\end{equation} -% -% RS: add end -\subsection*{References} -\cite{Allaway76}, \cite{NAlSalam66}, \cite{AlSalam90}, \cite{AlSalamChihara76}, -\cite{AlSalamIsmail76}, \cite{AndrewsAskey85}, \cite{Area+II}, \cite{Askey75}, -\cite{AskeyGasper77}, \cite{AskeyWilson85}, \cite{AtakRahmanSuslov}, \cite{Bavinck98}, -\cite{BavinckKoekoek}, \cite{Chihara78}, \cite{Chihara79}, \cite{Dunkl76}, \cite{Erdelyi+}, -\cite{Gasper73I}, \cite{Gasper74}, \cite{Goh}, \cite{HoareRahman}, \cite{IsmailLetVal88}, -\cite{Koekoek2000}, \cite{Koorn88}, \cite{Krasikov2002}, \cite{LabelleYehI}, -\cite{LabelleYehII}, \cite{LabelleYehIII}, \cite{Lesky62}, \cite{Lesky89}, \cite{Lesky94I}, -\cite{Lesky95II}, \cite{LewanowiczII}, \cite{LopezTemme2004}, \cite{Meixner}, \cite{Nikiforov+}, -\cite{NikiforovUvarov}, \cite{Szafraniec}, \cite{Szego75}, \cite{Viennot}, \cite{Zarzo+}, -\cite{Zeng90}. - - -\section{Hermite}\index{Hermite polynomials} - -\par\setcounter{equation}{0} - -\subsection*{Hypergeometric representation} -\begin{equation} -\label{DefHermite} -H_n(x)=(2x)^n\,\hyp{2}{0}{-n/2,-(n-1)/2}{-}{-\frac{1}{x^2}}. -\end{equation} - -\subsection*{Orthogonality relation} -\begin{equation} -\label{OrtHermite} -\frac{1}{\sqrt{\cpi}}\int_{-\infty}^{\infty}\expe^{-x^2}H_m(x)H_n(x)\,dx -=2^nn!\,\delta_{mn}. -\end{equation} - -\subsection*{Recurrence relation} -\begin{equation} -\label{RecHermite} -H_{n+1}(x)-2xH_n(x)+2nH_{n-1}(x)=0. -\end{equation} - -\subsection*{Normalized recurrence relation} -\begin{equation} -\label{NormRecHermite} -xp_n(x)=p_{n+1}(x)+\frac{n}{2}p_{n-1}(x), -\end{equation} -where -$$H_n(x)=2^np_n(x).$$ - -\subsection*{Differential equation} -\begin{equation} -\label{dvHermite} -y''(x)-2xy'(x)+2ny(x)=0,\quad y(x)=H_n(x). -\end{equation} - -\subsection*{Forward shift operator} -\begin{equation} -\label{shift1Hermite} -\frac{d}{dx}H_n(x)=2nH_{n-1}(x). -\end{equation} - -\subsection*{Backward shift operator} -\begin{equation} -\label{shift2HermiteI} -\frac{d}{dx}H_n(x)-2xH_n(x)=-H_{n+1}(x) -\end{equation} -or equivalently -\begin{equation} -\label{shift2HermiteII} -\frac{d}{dx}\left[\expe^{-x^2}H_n(x)\right]=-\expe^{-x^2}H_{n+1}(x). -\end{equation} - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodHermite} -\e^{-x^2}H_n(x)=(-1)^n\left(\frac{d}{dx}\right)^n\left[\expe^{-x^2}\right]. -\end{equation} - -\subsection*{Generating functions} -\begin{equation} -\label{GenHermite1} -\exp\left(2xt-t^2\right)=\sum_{n=0}^{\infty}\frac{H_n(x)}{n!}t^n. -\end{equation} - -\begin{equation} -\label{GenHermite2} -\left\{\begin{array}{l} -\displaystyle\expe^t\cos(2x\sqrt{t})=\sum_{n=0}^{\infty} -\frac{(-1)^n}{(2n)!}H_{2n}(x)t^n\\[5mm] -\displaystyle\frac{\expe^t}{\sqrt{t}}\sin(2x\sqrt{t})=\sum_{n=0}^{\infty} -\frac{(-1)^n}{(2n+1)!}H_{2n+1}(x)t^n. -\end{array}\right. -\end{equation} - -\begin{equation} -\label{GenHermite3} -\left\{\begin{array}{l} -\displaystyle \expe^{-t^2}\cosh(2xt)=\sum_{n=0}^{\infty} -\frac{H_{2n}(x)}{(2n)!}t^{2n}\\[5mm] -\displaystyle\expe^{-t^2}\sinh(2xt)=\sum_{n=0}^{\infty} -\frac{H_{2n+1}(x)}{(2n+1)!}t^{2n+1}. -\end{array}\right. -\end{equation} - -\begin{equation} -\label{GenHermite4} -\left\{\begin{array}{l} -\displaystyle (1+t^2)^{-\gamma}\,\hyp{1}{1}{\gamma}{\frac{1}{2}}{\frac{x^2t^2}{1+t^2}}= -\sum_{n=0}^{\infty}\frac{(\gamma)_n}{(2n)!}H_{2n}(x)t^{2n}\\[5mm] -\displaystyle\frac{xt}{\sqrt{1+t^2}}\, -\hyp{1}{1}{\gamma+\frac{1}{2}}{\frac{3}{2}}{\frac{x^2t^2}{1+t^2}} -=\sum_{n=0}^{\infty}\frac{(\gamma+\frac{1}{2})_n}{(2n+1)!}H_{2n+1}(x)t^{2n+1} -\end{array}\right. -\end{equation} -with $\gamma$ arbitrary. - -\begin{equation} -\label{GenHermite5} -\frac{1+2xt+4t^2}{(1+4t^2)^{\frac{3}{2}}}\exp\left(\frac{4x^2t^2}{1+4t^2}\right) -=\sum_{n=0}^{\infty}\frac{H_n(x)}{\lfloor n/2\rfloor\,!}t^n, -\end{equation} -where $\lfloor n/2\rfloor$ denotes the largest integer smaller than or equal to $n/2$. - -\subsection*{Limit relations} - -\subsubsection*{Meixner-Pollaczek $\rightarrow$ Hermite} -If we take $x\rightarrow (\sin\phi)^{-1}(x\sqrt{\lambda}-\lambda\cos\phi)$ -in the definition (\ref{DefMP}) of the Meixner-Pollaczek polynomials and -then let $\lambda\rightarrow\infty$ we obtain the Hermite polynomials: -$$\lim_{\lambda\rightarrow\infty} -\lambda^{-\frac{1}{2}n}P_n^{(\lambda)} -((\sin\phi)^{-1}(x\sqrt{\lambda}-\lambda\cos\phi);\phi)=\frac{H_n(x)}{n!}.$$ - -\subsubsection*{Jacobi $\rightarrow$ Hermite} -The Hermite polynomials follow from the Jacobi polynomials given by (\ref{DefJacobi}) by -taking $\beta=\alpha$ and letting $\alpha\rightarrow\infty$ in the following way: -$$\lim_{\alpha\rightarrow\infty} -\alpha^{-\frac{1}{2}n}P_n^{(\alpha,\alpha)}(\alpha^{-\frac{1}{2}}x)=\frac{H_n(x)}{2^nn!}.$$ - -\subsubsection*{Gegenbauer / Ultraspherical $\rightarrow$ Hermite} -The Hermite polynomials follow from the Gegenbauer (or ultraspherical) polynomials given by -(\ref{DefGegenbauer}) by taking $\lambda=\alpha+\frac{1}{2}$ and letting -$\alpha\rightarrow\infty$ in the following way: -$$\lim_{\alpha\rightarrow\infty} -\alpha^{-\frac{1}{2}n}C_n^{(\alpha+\frac{1}{2})}(\alpha^{-\frac{1}{2}}x)=\frac{H_n(x)}{n!}.$$ - -\subsubsection*{Krawtchouk $\rightarrow$ Hermite} -The Hermite polynomials follow from the Krawtchouk polynomials given by (\ref{DefKrawtchouk}) -by setting $x\rightarrow pN+x\sqrt{2p(1-p)N}$ and then letting $N\rightarrow\infty$: -$$\lim_{N\rightarrow\infty} -\sqrt{\binom{N}{n}}K_n(pN+x\sqrt{2p(1-p)N};p,N) -=\frac{\displaystyle (-1)^nH_n(x)}{\displaystyle\sqrt{2^nn!\left(\frac{p}{1-p}\right)^n}}.$$ - -\subsubsection*{Laguerre $\rightarrow$ Hermite} -The Hermite polynomials can be obtained from the Laguerre polynomials given by -(\ref{DefLaguerre}) by taking the limit $\alpha\rightarrow\infty$ in the following way: -$$\lim_{\alpha\rightarrow\infty} -\left(\frac{2}{\alpha}\right)^{\frac{1}{2}n} -L_n^{(\alpha)}((2\alpha)^{\frac{1}{2}}x+\alpha)=\frac{(-1)^n}{n!}H_n(x).$$ - -\subsubsection*{Charlier $\rightarrow$ Hermite} -If we set $x\rightarrow (2a)^{1/2}x+a$ in the definition (\ref{DefCharlier}) -of the Charlier polynomials and let $a\rightarrow\infty$ we find the Hermite -polynomials. In fact we have -$$\lim_{a\rightarrow\infty} -(2a)^{\frac{1}{2}n}C_n((2a)^{\frac{1}{2}}x+a;a)=(-1)^nH_n(x).$$ - -\subsection*{Remarks} -The Hermite polynomials can also be written as: -$$\frac{H_n(x)}{n!}=\sum_{k=0}^{\lfloor n/2\rfloor} -\frac{(-1)^k(2x)^{n-2k}}{k!\,(n-2k)!},$$ -where $\lfloor n/2\rfloor$ denotes the largest integer smaller than or equal to $n/2$. - -\noindent -The Hermite polynomials and the Laguerre polynomials given by (\ref{DefLaguerre}) are also -connected by the following quadratic transformations: -$$H_{2n}(x)=(-1)^nn!\,2^{2n}L_n^{(-\frac{1}{2})}(x^2)$$ -and -$$H_{2n+1}(x)=(-1)^nn!\,2^{2n+1}xL_n^{(\frac{1}{2})}(x^2).$$ -% RS: add begin\label{sec9.14} -% -\paragraph{\large\bf KLSadd: Hypergeometric representation}\begin{align} -C_n(x;a)&=\hyp20{-n,-x}-{-\,\frac1a} -\label{179}\\ -&=\frac{(-x)_n}{a^n} \hyp11{-n}{x-n+1}a -\label{180}\\ -&=\frac{n!}{(-a)^n}\,L_n^{x-n}(a), -\label{181} -\end{align} -where $L_n^\al(x)$ is a -\hyperref[sec9.12]{Laguerre polynomial}. -Formula \eqref{179} is (9.14.1). Then \eqref{180} follows by reversal -of the summation. Finally \eqref{181} follows by \eqref{180} and -(9.12.1). It is also the Remark on p.249 of \mycite{KLS}, and it -was earlier given in \myciteKLS{416}{(2.7.10)}. -% -\paragraph{\large\bf KLSadd: Uniqueness of orthogonality measure}The coefficient of $p_{n-1}(x)$ in (9.14.4) behaves as $O(n)$ as $n\to\iy$. -Hence \eqref{93} holds, by which the orthogonality measure is unique. -% -% RS: add end -\subsection*{References} -\label{sec9.15} -% -\paragraph{\large\bf KLSadd: Uniqueness of orthogonality measure}The coefficient of $p_{n-1}(x)$ in (9.15.4) behaves as $O(n)$ as $n\to\iy$. -Hence \eqref{93} holds, by which the orthogonality measure is unique. -% -\paragraph{\large\bf KLSadd: Fourier transforms}\begin{equation} -\frac1{\sqrt{2\pi}}\,\int_{-\iy}^\iy H_n(y)\,e^{-\half y^2}\,e^{ixy}\,dy= -i^n\,H_n(x)\,e^{-\half x^2}, -\label{15} -\end{equation} -see \mycite{AAR}{(6.1.15) and Exercise 6.11}. -\begin{equation} -\frac1{\sqrt\pi}\,\int_{-\iy}^\iy H_n(y)\,e^{-y^2}\,e^{ixy}\,dy= -i^n\,x^n\,e^{-\frac14 x^2}, -\label{16} -\end{equation} -see \mycite{DLMF}{(18.17.35)}. -\begin{equation} -\frac{i^n}{2\sqrt\pi}\,\int_{-\iy}^\iy y^n\,e^{-\frac14 y^2}\,e^{-ixy}\,dy= -H_n(x)\,e^{-x^2}, -\label{17} -\end{equation} -see \mycite{AAR}{(6.1.4)}. -% -\cite{Abram}, \cite{NAlSalam66}, \cite{AlSalam90}, \cite{AlSalamChihara72}, -\cite{AlSalamChihara76}, \cite{AndrewsAskey85}, \cite{AndrewsAskeyRoy}, \cite{Area+I}, -\cite{Askey68}, \cite{Askey75}, \cite{Askey89I}, \cite{AskeyGasper76}, \cite{AskeyWilson85}, -\cite{Azor}, \cite{Berg}, \cite{BilodeauII}, \cite{Brafman51}, \cite{Brafman57II}, -\cite{Brenke}, \cite{CarlitzSrivastava}, \cite{ChenSrivastava}, \cite{Chihara78}, -\cite{Cohen}, \cite{Danese}, \cite{DetteStudden92}, \cite{DetteStudden95}, \cite{Doha2004I}, -\cite{Erdelyi+}, \cite{Faldey}, \cite{Gawronski87}, \cite{Gawronski93}, \cite{Gillis+}, -\cite{Godoy+}, \cite{Grad}, \cite{Ismail74}, \cite{Ismail2005II}, \cite{IsmailStanton97}, -\cite{Koekoek2000}, \cite{Koorn88}, \cite{Krasikov2004}, \cite{Kwon+}, \cite{LabelleYehIII}, -\cite{Lesky95II}, \cite{Lesky96}, \cite{LewanowiczI}, \cite{LopezTemme99}, \cite{Mathai}, -\cite{Meixner}, \cite{Nikiforov+}, \cite{NikiforovUvarov}, \cite{Olver}, \cite{Pittaluga}, -\cite{Rainville}, \cite{SainteViennot}, \cite{Srivastava71}, \cite{SrivastavaMathur}, -\cite{Szego75}, \cite{Temme}, \cite{TemmeLopez2000}, \cite{Trickovic}, \cite{Viennot}, -\cite{Weisner59}, \cite{Wyman}. - -\end{document} diff --git a/KLSadd_insertion/newtempchap14.tex b/KLSadd_insertion/newtempchap14.tex deleted file mode 100644 index 880a78e..0000000 --- a/KLSadd_insertion/newtempchap14.tex +++ /dev/null @@ -1,6695 +0,0 @@ -\documentclass[envcountchap,graybox]{svmono} - -\addtolength{\textwidth}{1mm} - -\usepackage{amsmath,amssymb} - -\usepackage{amsfonts} -%\usepackage{breqn} -\usepackage{DLMFmath} -\usepackage{DRMFfcns} - -\usepackage{mathptmx} -\usepackage{helvet} -\usepackage{courier} - -\usepackage{makeidx} -\usepackage{graphicx} - -\usepackage{multicol} -\usepackage[bottom]{footmisc} - -\usepackage[pdftex]{hyperref} -\usepackage {xparse} -\usepackage{cite} -\makeindex - -\def\bibname{Bibliography} -\def\refname{Bibliography} - -\def\theequation{\thesection.\arabic{equation}} - -\smartqed - -\let\corollary=\undefined -\spnewtheorem{corollary}[theorem]{Corollary}{\bfseries}{\itshape} - -\newcounter{rom} - -\newcommand{\hyp}[5]{\mbox{}_{#1}{F}_{#2} -\left(\genfrac{}{}{0pt}{}{#3}{#4}\,;\,#5\right)} - -\newcommand{\qhyp}[5]{\mbox{}_{#1}{\phi}_{#2} -\left(\genfrac{}{}{0pt}{}{#3}{#4}\,;\,q,\,#5\right)} - -\newcommand{\qhypK}[5]{\,\mbox{}_{#1}\phi_{#2}\!\left( - \genfrac{}{}{0pt}{}{#3}{#4};#5\right)} - -\newcommand{\mathindent}{\hspace{7.5mm}} - -\newcommand{\e}{\textrm{e}} - -\renewcommand{\E}{\textrm{E}} - -\renewcommand{\textfraction}{-1} - -\renewcommand{\Gamma}{\varGamma} - -\renewcommand{\leftlegendglue}{\hfil} - -\settowidth{\tocchpnum}{14\enspace} -\settowidth{\tocsecnum}{14.30\enspace} -\settowidth{\tocsubsecnum}{14.12.1\enspace} - -\makeatletter -\def\cleartoversopage{\clearpage\if@twoside\ifodd\c@page - \hbox{}\newpage\if@twocolumn\hbox{}\newpage\fi - \else\fi\fi} - -\newcommand{\clearemptyversopage}{ - \clearpage{\pagestyle{empty}\cleartoversopage}} -\makeatother - -\oddsidemargin -1.5cm -\topmargin -2.0cm -\textwidth 16.3cm -\newcommand\sa{\smallskipamount} -\newcommand\sLP{\\[\sa]} -\newcommand\sPP{\\[\sa]\indent} -\newcommand\ba{\bigskipamount} -\newcommand\bLP{\\[\ba]} -\newcommand\CC{\mathbb{C}} -\newcommand\RR{\mathbb{R}} -\newcommand\ZZ{\mathbb{Z}} -\newcommand\al\alpha -\newcommand\be\beta -\newcommand\ga\gamma -\newcommand\de\delta -\newcommand\tha\theta -\newcommand\la\lambda -\newcommand\om\omega -\newcommand\Ga{\Gamma} -\newcommand\half{\frac12} -\newcommand\thalf{\tfrac12} -\newcommand\iy\infty -\newcommand\wt{\widetilde} -\newcommand\const{{\rm const.}\,} -\newcommand\Zpos{\ZZ_{>0}} -\newcommand\Znonneg{\ZZ_{\ge0}} -\newcommand{\hyp}[5]{\,\mbox{}_{#1}F_{#2}\!\left( - \genfrac{}{}{0pt}{}{#3}{#4};#5\right)} -\newcommand{\qhyp}[5]{\,\mbox{}_{#1}\phi_{#2}\!\left( - \genfrac{}{}{0pt}{}{#3}{#4};#5\right)} -\newcommand\LHS{left-hand side} -\newcommand\RHS{right-hand side} -\renewcommand\Re{{\rm Re}\,} -\renewcommand\Im{{\rm Im}\,} - -\NewDocumentCommand\mycite{m g}{% - \IfNoValueTF{#2} - {[\hyperlink{#1}{#1}]} - {[\hyperlink{#1}{#1}, #2]}% -} -\newcommand\mybibitem[1]{\bibitem[#1]{#1}\hypertarget{#1}{}} -\textheight 25cm - -\begin{document} - -\author{Roelof Koekoek\\[2.5mm]Peter A. Lesky\\[2.5mm]Ren\'e F. Swarttouw} -\title{Hypergeometric orthogonal polynomials and their\\$q$-analogues} -\subtitle{-- Monograph --} -\maketitle - -\frontmatter - -\large - -\pagenumbering{roman} -\addtocounter{chapter}{13} - -\chapter{Basic hypergeometric orthogonal polynomials} -\label{BasicHyperOrtPol} - - -In this chapter we deal with all families of basic hypergeometric orthogonal polynomials -appearing in the $q$-analogue of the Askey scheme on the pages~\pageref{qscheme1} and -\pageref{qscheme2}. For each family of orthogonal polynomials we state the most important -properties such as a representation as a basic hypergeometric function, orthogonality -relation(s), the three-term recurrence relation, the second-order $q$-difference equation, -the forward shift (or degree lowering) and backward shift (or degree raising) operator, a -Rodrigues-type formula and some generating functions. Throughout this chapter we assume -that $01$ and $b,c,d$ are real or one is real and the other two are complex conjugates,\\ -$\max(|b|,|c|,|d|)<1$ and the pairwise products of $a,b,c$ and $d$ have -absolute value less than $1$, then we have another orthogonality relation -given by: -\begin{eqnarray} -\label{OrtAskeyWilsonII} -& &\frac{1}{2\pi}\int_{-1}^1\frac{w(x)}{\sqrt{1-x^2}}p_m(x;a,b,c,d|q)p_n(x;a,b,c,d|q)\,dx\nonumber\\ -& &{}\mathindent{}+\sum_{\begin{array}{c}\scriptstyle k\\ \scriptstyle 11$ and $b$ and $c$ are real or complex conjugates, -$\max(|b|,|c|)<1$ and the pairwise products of $a,b$ and $c$ have -absolute value less than $1$, then we have another orthogonality relation -given by: -\begin{eqnarray} -\label{OrtContDualqHahnII} -& &\frac{1}{2\pi}\int_{-1}^1\frac{w(x)}{\sqrt{1-x^2}}p_m(x;a,b,c|q)p_n(x;a,b,c|q)\,dx\nonumber\\ -& &{}\mathindent{}+\sum_{\begin{array}{c}\scriptstyle k\\ \scriptstyle 10$) in the definition (\ref{DefBigqJacobi}) -of the big $q$-Jacobi polynomials and take the limit $c\rightarrow -\infty$ we -obtain the $q$-Meixner polynomials given by (\ref{DefqMeixner}): -\begin{equation} -\lim_{c\rightarrow -\infty}P_n(q^{-x};a,-a^{-1}cd^{-1},c;q)=M_n(q^{-x};a,d;q). -\end{equation} - -\subsubsection*{Big $q$-Jacobi $\rightarrow$ Jacobi} -If we set $c=0$, $a=q^{\alpha}$ and $b=q^{\beta}$ in the definition (\ref{DefBigqJacobi}) -of the big $q$-Jacobi polynomials and let $q\rightarrow 1$ we find the Jacobi -polynomials given by (\ref{DefJacobi}): -\begin{equation} -\lim_{q\rightarrow 1}P_n(x;q^{\alpha},q^{\beta},0;q)=\frac{P_n^{(\alpha,\beta)}(2x-1)}{P_n^{(\alpha,\beta)}(1)}. -\end{equation} -If we take $c=-q^{\gamma}$ for arbitrary real $\gamma$ instead of $c=0$ we -find -\begin{equation} -\lim_{q\rightarrow 1}P_n(x;q^{\alpha},q^{\beta},-q^{\gamma};q)=\frac{P_n^{(\alpha,\beta)}(x)}{P_n^{(\alpha,\beta)}(1)}. -\end{equation} - -\subsection*{Remarks} -The big $q$-Jacobi polynomials with $c=0$ and the little -$q$-Jacobi polynomials given by (\ref{DefLittleqJacobi}) are related -in the following way: -$$P_n(x;a,b,0;q)=\frac{(bq;q)_n}{(aq;q)_n}(-1)^na^nq^{n+\binom{n}{2}}p_n(a^{-1}q^{-1}x;b,a|q).$$ - -\noindent -Sometimes the big $q$-Jacobi polynomials are defined in terms of four -parameters instead of three. In fact the polynomials given by the definition -$$P_n(x;a,b,c,d;q)=\qhyp{3}{2}{q^{-n},abq^{n+1},ac^{-1}qx}{aq,-ac^{-1}dq}{q}$$ -are orthogonal on the interval $[-d,c]$ with respect to the weight function -$$\frac{(c^{-1}qx,-d^{-1}qx;q)_{\infty}}{(ac^{-1}qx,-bd^{-1}qx;q)_{\infty}}d_qx.$$ -These polynomials are not really different from those given by -(\ref{DefBigqJacobi}) since we have -$$P_n(x;a,b,c,d;q)=P_n(ac^{-1}qx;a,b,-ac^{-1}d;q)$$ -and -$$P_n(x;a,b,c;q)=P_n(x;a,b,aq,-cq;q).$$ - RS: add begin\label{sec14.4} -The continuous $q$-Hahn polynomials are the special case -of Askey-Wilson polynomials with parameters -$a e^{i\phi},b e^{i\phi},a e^{-i\phi},b e^{-i\phi}$: -\[ -p_n(x;a,b,\phi\,|\, q):= -p_n(x;a e^{i\phi},b e^{i\phi},a e^{-i\phi},b e^{-i\phi}\,|\, q). -\] -In \myciteKLS{72}{(4.29)} and \mycite{GR}{(7.5.43)} -(who write $p_n(x;a,b\,|\,q)$, $x=\cos(\tha+\phi)$) -and in \mycite{KLS}{\S14.4} (who writes $p_n(x;a,b,c,d;q)$, -$x=\cos(\tha+\phi)$) -the parameter -dependence on $\phi$ is incorrectly omitted. - -Since all formulas in \S14.4 are specializations of formulas in \S14.1, -there is no real need to give these specializations explicitly. -In particular, the limit (14.4.15) is in fact a limit from Askey-Wilson to -continuous $q$-Hahn. See also \eqref{177}. -% - -\subsection*{References} -\cite{NAlSalam89}, \cite{AlSalam90}, \cite{AndrewsAskey85}, \cite{AtakKlimyk2004}, -\cite{AtakRahmanSuslov}, \cite{DattaGriffin}, \cite{FloreaniniVinetII}, -\cite{GasperRahman90}, \cite{GrunbaumHaine96}, \cite{Gupta92}, \cite{Hahn}, -\cite{Ismail86I}, \cite{IsmailStanton97}, \cite{IsmailWilson}, \cite{KalninsMiller88}, -\cite{KoelinkE}, \cite{Koorn90II}, \cite{Koorn93}, \cite{Koorn2007}, \cite{Miller89}, -\cite{Nikiforov+}, \cite{NoumiMimachi90II}, \cite{NoumiMimachi90III}, -\cite{NoumiMimachi91}, \cite{Spiridonov97}, \cite{SrivastavaJain90}. - - -\section*{Special case} - -\subsection{Big $q$-Legendre} -\index{Big q-Legendre polynomials@Big $q$-Legendre polynomials} -\index{q-Legendre polynomials@$q$-Legendre polynomials!Big} -\par - -\subsection*{Basic hypergeometric representation} The big $q$-Legendre polynomials are big $q$-Jacobi polynomials -with $a=b=1$: -\begin{equation} -\label{DefBigqLegendre} -P_n(x;c;q)=\qhyp{3}{2}{q^{-n},q^{n+1},x}{q,cq}{q}. -\end{equation} - -\subsection*{Orthogonality relation} -\begin{eqnarray} -\label{OrtBigqLegendre} -& &\int_{cq}^{q}P_m(x;c;q)P_n(x;c;q)\,d_qx\nonumber\\ -& &{}=q(1-c)\frac{(1-q)}{(1-q^{2n+1})} -\frac{(c^{-1}q;q)_n}{(cq;q)_n}(-cq^2)^nq^{\binom{n}{2}}\,\delta_{mn},\quad c<0. -\end{eqnarray} - -\subsection*{Recurrence relation} -\begin{eqnarray} -\label{RecBigqLegendre} -(x-1)P_n(x;c;q)&=&A_nP_{n+1}(x;c;q)-\left(A_n+C_n\right)P_n(x;c;q)\nonumber\\ -& &{}\mathindent{}+C_nP_{n-1}(x;c;q), -\end{eqnarray} -where -$$\left\{\begin{array}{l} -\displaystyle A_n=\frac{(1-q^{n+1})(1-cq^{n+1})}{(1+q^{n+1})(1-q^{2n+1})}\\ -\\ -\displaystyle C_n=-cq^{n+1}\frac{(1-q^n)(1-c^{-1}q^n)}{(1+q^n)(1-q^{2n+1})}. -\end{array}\right.$$ - -\subsection*{Normalized recurrence relation} -\begin{equation} -\label{NormRecBigqLegendre} -xp_n(x)=p_{n+1}(x)+\left[1-(A_n+C_n)\right]p_n(x)+A_{n-1}C_np_{n-1}(x), -\end{equation} -where -$$P_n(x;c;q)=\frac{(q^{n+1};q)_n}{(q,cq;q)_n}p_n(x).$$ - -\subsection*{$q$-Difference equation} -\begin{eqnarray} -\label{dvBigqLegendre} -& &q^{-n}(1-q^n)(1-q^{n+1})x^2y(x)\nonumber\\ -& &{}=B(x)y(qx)-\left[B(x)+D(x)\right]y(x)+D(x)y(q^{-1}x), -\end{eqnarray} -where -$$y(x)=P_n(x;c;q)$$ -and -$$\left\{\begin{array}{l}\displaystyle B(x)=q(x-1)(x-c)\\ -\\ -\displaystyle D(x)=(x-q)(x-cq).\end{array}\right.$$ - -\subsection*{Rodrigues-type formula} -\begin{eqnarray} -\label{RodBigqLegendre} -P_n(x;c;q)&=&\frac{c^nq^{n(n+1)}(1-q)^n}{(q,cq;q)_n} -\left(\mathcal{D}_q\right)^n\left[(q^{-n}x,c^{-1}q^{-n}x;q)_n\right]\\ -&=&\frac{(1-q)^n}{(q,cq;q)_n}\left(\mathcal{D}_q\right)^n -\left[(qx^{-1},cqx^{-1};q)_nx^{2n}\right].\nonumber -\end{eqnarray} - -\subsection*{Generating functions} -\begin{equation} -\label{GenBigqLegendre1} -\qhyp{2}{1}{qx^{-1},0}{q}{xt}\,\qhyp{1}{1}{c^{-1}x}{q}{cqt} -=\sum_{n=0}^{\infty}\frac{(cq;q)_n}{(q,q;q)_n}P_n(x;c;q)t^n. -\end{equation} - -\begin{equation} -\label{GenBigqLegendre2} -\qhyp{2}{1}{cqx^{-1},0}{cq}{xt}\,\qhyp{1}{1}{c^{-1}x}{c^{-1}q}{qt} -=\sum_{n=0}^{\infty}\frac{P_n(x;c;q)}{(c^{-1}q;q)_n}t^n. -\end{equation} - -\subsection*{Limit relations} - -\subsubsection*{Big $q$-Legendre $\rightarrow$ Legendre / Spherical} -If we set $c=0$ in the definition (\ref{DefBigqLegendre}) of the big -$q$-Legendre polynomials and let $q\rightarrow 1$ we simply obtain the Legendre -(or spherical) polynomials given by (\ref{DefLegendre}): -\begin{equation} -\lim_{q\rightarrow 1}P_n(x;0;q)=P_n(2x-1). -\end{equation} -If we take $c=-q^{\gamma}$ for arbitrary real $\gamma$ instead of $c=0$ we -find -\begin{equation} -\lim_{q\rightarrow 1}P_n(x;-q^{\gamma};q)=P_n(x). -\end{equation} - -\subsection*{References} -\cite{Koelink95I}, \cite{Koorn90II}. - - -\section{$q$-Hahn}\index{q-Hahn polynomials@$q$-Hahn polynomials} -\par\setcounter{equation}{0} - -\subsection*{Basic hypergeometric representation} -\begin{equation} -\label{DefqHahn} -Q_n(q^{-x};\alpha,\beta,N|q)=\qhyp{3}{2}{q^{-n},\alpha\beta q^{n+1},q^{-x}} -{\alpha q,q^{-N}}{q},\quad n=0,1,2,\ldots,N. -\end{equation} - -\subsection*{Orthogonality relation} -For $0<\alpha q<1$ and $0<\beta q<1$, or for $\alpha>q^{-N}$ and $\beta>q^{-N}$, we have -\begin{eqnarray} -\label{OrtqHahn} -& &\sum_{x=0}^N\frac{(\alpha q,q^{-N};q)_x}{(q,\beta^{-1}q^{-N};q)_x}(\alpha\beta q)^{-x} -Q_m(q^{-x};\alpha,\beta,N|q)Q_n(q^{-x};\alpha,\beta,N|q)\nonumber\\ -& &{}=\frac{(\alpha\beta q^2;q)_N}{(\beta q;q)_N(\alpha q)^N} -\frac{(q,\alpha\beta q^{N+2},\beta q;q)_n}{(\alpha q,\alpha\beta q,q^{-N};q)_n}\, -\frac{(1-\alpha\beta q)(-\alpha q)^n}{(1-\alpha\beta q^{2n+1})} -q^{\binom{n}{2}-Nn}\,\delta_{mn}. -\end{eqnarray} - -\newpage - -\subsection*{Recurrence relation} -\begin{eqnarray} -\label{RecqHahn} --\left(1-q^{-x}\right)Q_n(q^{-x})&=&A_nQ_{n+1}(q^{-x})-\left(A_n+C_n\right)Q_n(q^{-x})\nonumber\\ -& &{}\mathindent{}+C_nQ_{n-1}(q^{-x}), -\end{eqnarray} -where -$$Q_n(q^{-x}):=Q_n(q^{-x};\alpha,\beta,N|q)$$ -and -$$\left\{\begin{array}{l} -\displaystyle A_n=\frac{(1-q^{n-N})(1-\alpha q^{n+1})(1-\alpha\beta q^{n+1})}{(1-\alpha\beta q^{2n+1})(1-\alpha\beta q^{2n+2})}\\ -\\ -\displaystyle C_n=-\frac{\alpha q^{n-N}(1-q^n)(1-\alpha\beta q^{n+N+1})(1-\beta q^n)}{(1-\alpha\beta q^{2n})(1-\alpha\beta q^{2n+1})}. -\end{array}\right.$$ - -\subsection*{Normalized recurrence relation} -\begin{equation} -\label{NormRecqHahn} -xp_n(x)=p_{n+1}(x)+\left[1-(A_n+C_n)\right]p_n(x)+A_{n-1}C_np_{n-1}(x), -\end{equation} -where -$$Q_n(q^{-x};\alpha,\beta,N|q)= -\frac{(\alpha\beta q^{n+1};q)_n}{(\alpha q,q^{-N};q)_n}p_n(q^{-x}).$$ - -\subsection*{$q$-Difference equation} -\begin{eqnarray} -\label{dvqHahn} -& &q^{-n}(1-q^n)(1-\alpha\beta q^{n+1})y(x)\nonumber\\ -& &{}=B(x)y(x+1)-\left[B(x)+D(x)\right]y(x)+D(x)y(x-1), -\end{eqnarray} -where -$$y(x)=Q_n(q^{-x};\alpha,\beta,N|q)$$ -and -$$\left\{\begin{array}{l}\displaystyle B(x)=(1-q^{x-N})(1-\alpha q^{x+1})\\ -\\ -\displaystyle D(x)=\alpha q(1-q^x)(\beta-q^{x-N-1}).\end{array}\right.$$ - -\newpage - -\subsection*{Forward shift operator} -\begin{eqnarray} -\label{shift1qHahnI} -& &Q_n(q^{-x-1};\alpha,\beta,N|q)-Q_n(q^{-x};\alpha,\beta,N|q)\nonumber\\ -& &{}=\frac{q^{-n-x}(1-q^n)(1-\alpha\beta q^{n+1})}{(1-\alpha q)(1-q^{-N})} -Q_{n-1}(q^{-x};\alpha q,\beta q,N-1|q) -\end{eqnarray} -or equivalently -\begin{eqnarray} -\label{shift1qHahnII} -\frac{\Delta Q_n(q^{-x};\alpha,\beta,N|q)}{\Delta q^{-x}} -&=&\frac{q^{-n+1}(1-q^n)(1-\alpha\beta q^{n+1})}{(1-q)(1-\alpha q)(1-q^{-N})}\nonumber\\ -& &{}\mathindent{}\times Q_{n-1}(q^{-x};\alpha q,\beta q,N-1|q). -\end{eqnarray} - -\subsection*{Backward shift operator} -\begin{eqnarray} -\label{shift2qHahnI} -& &(1-\alpha q^x)(1-q^{x-N-1})Q_n(q^{-x};\alpha,\beta,N|q)\nonumber\\ -& &{}\mathindent{}-\alpha(1-q^x)(\beta-q^{x-N-1})Q_n(q^{-x+1};\alpha,\beta,N|q)\nonumber\\ -& &{}=q^x(1-\alpha)(1-q^{-N-1})Q_{n+1}(q^{-x};\alpha q^{-1},\beta q^{-1},N+1|q) -\end{eqnarray} -or equivalently -\begin{eqnarray} -\label{shift2qHahnII} -& &\frac{\nabla\left[w(x;\alpha,\beta,N|q)Q_n(q^{-x};\alpha,\beta,N|q)\right]}{\nabla q^{-x}}\nonumber\\ -& &{}=\frac{1}{1-q}w(x;\alpha q^{-1},\beta q^{-1},N+1|q)Q_{n+1}(q^{-x};\alpha q^{-1},\beta q^{-1},N+1|q), -\end{eqnarray} -where -$$w(x;\alpha,\beta,N|q)=\frac{(\alpha q,q^{-N};q)_x}{(q,\beta^{-1}q^{-N};q)_x}(\alpha\beta)^{-x}.$$ - -\subsection*{Rodrigues-type formula} -\begin{eqnarray} -\label{RodqHahn} -& &w(x;\alpha,\beta,N|q)Q_n(q^{-x};\alpha,\beta,N|q)\nonumber\\ -& &{}=(1-q)^n\left(\nabla_q\right)^n\left[w(x;\alpha q^n,\beta q^n,N-n|q)\right], -\end{eqnarray} -where -$$\nabla_q:=\frac{\nabla}{\nabla q^{-x}}.$$ - -\subsection*{Generating functions} For $x=0,1,2,\ldots,N$ we have -\begin{eqnarray} -\label{GenqHahn1} -& &\qhyp{1}{1}{q^{-x}}{\alpha q}{\alpha qt}\,\qhyp{2}{1}{q^{x-N},0}{\beta q}{q^{-x}t}\nonumber\\ -& &{}=\sum_{n=0}^N\frac{(q^{-N};q)_n}{(\beta q,q;q)_n}Q_n(q^{-x};\alpha,\beta,N|q)t^n. -\end{eqnarray} - -\begin{eqnarray} -\label{GenqHahn2} -& &\qhyp{2}{1}{q^{-x},\beta q^{N+1-x}}{0}{-\alpha q^{x-N+1}t}\, -\qhyp{2}{0}{q^{x-N},\alpha q^{x+1}}{-}{-q^{-x}t}\nonumber\\ -& &{}=\sum_{n=0}^N\frac{(\alpha q,q^{-N};q)_n}{(q;q)_n}q^{-\binom{n}{2}}Q_n(q^{-x};\alpha,\beta,N|q)t^n. -\end{eqnarray} - -\subsection*{Limit relations} - -\subsubsection*{$q$-Racah $\rightarrow$ $q$-Hahn} -The $q$-Hahn polynomials follow from the $q$-Racah polynomials by the substitution -$\delta=0$ and $\gamma q=q^{-N}$ in the definition (\ref{DefqRacah}) of the -$q$-Racah polynomials: -$$R_n(\mu(x);\alpha,\beta,q^{-N-1},0|q)=Q_n(q^{-x};\alpha,\beta,N|q).$$ -Another way to obtain the $q$-Hahn polynomials from the $q$-Racah -polynomials is by setting $\gamma=0$ and $\delta=\beta^{-1}q^{-N-1}$ in the definition -(\ref{DefqRacah}): -$$R_n(\mu(x);\alpha,\beta,0,\beta^{-1}q^{-N-1}|q)=Q_n(q^{-x};\alpha,\beta,N|q).$$ -And if we take $\alpha q=q^{-N}$, $\beta\rightarrow\beta\gamma q^{N+1}$ and $\delta=0$ in the -definition (\ref{DefqRacah}) of the $q$-Racah polynomials we find the -$q$-Hahn polynomials given by (\ref{DefqHahn}) in the following way: -$$R_n(\mu(x);q^{-N-1},\beta\gamma q^{N+1},\gamma,0|q)=Q_n(q^{-x};\gamma,\beta,N|q).$$ -Note that $\mu(x)=q^{-x}$ in each case. - -\subsubsection*{$q$-Hahn $\rightarrow$ Little $q$-Jacobi} -If we set $x\rightarrow N-x$ in the definition (\ref{DefqHahn}) of the $q$-Hahn -polynomials and take the limit $N\rightarrow\infty$ we find the little $q$-Jacobi polynomials: -\begin{equation} -\lim _{N\rightarrow\infty} Q_n(q^{x-N};\alpha,\beta,N|q)=p_n(q^x;\alpha,\beta|q), -\end{equation} -where $p_n(q^x;\alpha,\beta|q)$ is given by (\ref{DefLittleqJacobi}). - -\subsubsection*{$q$-Hahn $\rightarrow$ $q$-Meixner} -The $q$-Meixner polynomials given by (\ref{DefqMeixner}) can be obtained from the $q$-Hahn polynomials -by setting $\alpha=b$ and $\beta=-b^{-1}c^{-1}q^{-N-1}$ in the definition (\ref{DefqHahn}) of the -$q$-Hahn polynomials and letting $N\rightarrow\infty$: -\begin{equation} -\lim_{N\rightarrow\infty} -Q_n(q^{-x};b,-b^{-1}c^{-1}q^{-N-1},N|q)=M_n(q^{-x};b,c;q). -\end{equation} - -\subsubsection*{$q$-Hahn $\rightarrow$ Quantum $q$-Krawtchouk} -The quantum $q$-Krawtchouk polynomials given by (\ref{DefQuantumqKrawtchouk}) -simply follow from the $q$-Hahn polynomials by setting $\beta=p$ in the definition (\ref{DefqHahn}) of -the $q$-Hahn polynomials and taking the limit $\alpha\rightarrow\infty$: -\begin{equation} -\lim_{\alpha\rightarrow\infty}Q_n(q^{-x};\alpha,p,N|q)=K_n^{qtm}(q^{-x};p,N;q). -\end{equation} - -\subsubsection*{$q$-Hahn $\rightarrow$ $q$-Krawtchouk} -If we set $\beta=-\alpha^{-1}q^{-1}p$ in the definition (\ref{DefqHahn}) of the $q$-Hahn polynomials -and then let $\alpha\rightarrow 0$ we obtain the $q$-Krawtchouk polynomials given by -(\ref{DefqKrawtchouk}): -\begin{equation} -\lim_{\alpha\rightarrow 0} -Q_n(q^{-x};\alpha,-\alpha^{-1}q^{-1}p,N|q)=K_n(q^{-x};p,N;q). -\end{equation} - -\subsubsection*{$q$-Hahn $\rightarrow$ Affine $q$-Krawtchouk} -The affine $q$-Krawtchouk polynomials given by (\ref{DefAffqKrawtchouk}) -can be obtained from the $q$-Hahn polynomials by the substitution $\alpha=p$ and -$\beta=0$ in (\ref{DefqHahn}): -\begin{equation} -Q_n(q^{-x};p,0,N|q)=K_n^{Aff}(q^{-x};p,N;q). -\end{equation} - -\subsubsection*{$q$-Hahn $\rightarrow$ Hahn} -The Hahn polynomials given by (\ref{DefHahn}) simply follow from the $q$-Hahn -polynomials given by (\ref{DefqHahn}), after setting $\alpha\rightarrow q^{\alpha}$ -and $\beta\rightarrow q^{\beta}$, in the following way: -\begin{equation} -\lim_{q\rightarrow 1}Q_n(q^{-x};q^{\alpha},q^{\beta},N|q)=Q_n(x;\alpha,\beta,N). -\end{equation} - -\subsection*{Remark} -The $q$-Hahn polynomials given by (\ref{DefqHahn}) and the dual $q$-Hahn -polynomials given by (\ref{DefDualqHahn}) are related in the following way: -$$Q_n(q^{-x};\alpha,\beta,N|q)=R_x(\mu(n);\alpha,\beta,N|q),$$ -with -$$\mu(n)=q^{-n}+\alpha\beta q^{n+1}$$ -or -$$R_n(\mu(x);\gamma,\delta,N|q)=Q_x(q^{-n};\gamma,\delta,N|q),$$ -where -$$\mu(x)=q^{-x}+\gamma\delta q^{x+1}.$$ - -\subsection*{References} -\cite{AlSalam90}, \cite{AlvarezRonveaux}, \cite{AndrewsAskey85}, -\cite{AskeyWilson79}, \cite{AskeyWilson85}, \cite{AtakRahmanSuslov}, -\cite{Dunkl78II}, \cite{Fischer95}, \cite{GasperRahman84}, -\cite{GasperRahman90}, \cite{Hahn}, \cite{KalninsMiller88}, -\cite{Koelink96I}, \cite{KoelinkKoorn}, \cite{Koorn89III}, \cite{Koorn90II}, -\cite{Nikiforov+}, \cite{NoumiMimachi90II}, \cite{Rahman82}, -\cite{Stanton80II}, \cite{Stanton84}, \cite{Stanton90}. - - -\section{Dual $q$-Hahn}\index{Dual q-Hahn polynomials@Dual $q$-Hahn polynomials} -\index{q-Hahn polynomials@$q$-Hahn polynomials!Dual} -\par\setcounter{equation}{0} - -\subsection*{Basic hypergeometric representation} -\begin{equation} -\label{DefDualqHahn} -R_n(\mu(x);\gamma,\delta,N|q)= -\qhyp{3}{2}{q^{-n},q^{-x},\gamma\delta q^{x+1}}{\gamma q,q^{-N}}{q},\quad n=0,1,2,\ldots,N, -\end{equation} -where -$$\mu(x):=q^{-x}+\gamma\delta q^{x+1}.$$ - -\newpage - -\subsection*{Orthogonality relation} -For $0<\gamma q<1$ and $0<\delta q<1$, or for $\gamma>q^{-N}$ and $\delta>q^{-N}$, we have -\begin{eqnarray} -\label{OrtDualqHahn} -& &\sum_{x=0}^N\frac{(\gamma q,\gamma\delta q,q^{-N};q)_x}{(q,\gamma\delta q^{N+2},\delta q;q)_x} -\frac{(1-\gamma\delta q^{2x+1})}{(1-\gamma\delta q)(-\gamma q)^x}q^{Nx-\binom{x}{2}}\nonumber\\ -& &{}\mathindent{}\times R_m(\mu(x);\gamma,\delta,N|q)R_n(\mu(x);\gamma,\delta,N|q)\nonumber\\ -& &{}=\frac{(\gamma\delta q^2;q)_N}{(\delta q;q)_N}(\gamma q)^{-N} -\frac{(q,\delta^{-1}q^{-N};q)_n}{(\gamma q,q^{-N};q)_n}(\gamma\delta q)^n\,\delta_{mn}. -\end{eqnarray} - -\subsection*{Recurrence relation} -\begin{eqnarray} -\label{RecDualqHahn} -& &-\left(1-q^{-x}\right)\left(1-\gamma\delta q^{x+1}\right)R_n(\mu(x))\nonumber\\ -& &{}=A_nR_{n+1}(\mu(x))-\left(A_n+C_n\right)R_n(\mu(x))+C_nR_{n-1}(\mu(x)), -\end{eqnarray} -where -$$R_n(\mu(x)):=R_n(\mu(x);\gamma,\delta,N|q)$$ -and -$$\left\{\begin{array}{l} -\displaystyle A_n=\left(1-q^{n-N}\right)\left(1-\gamma q^{n+1}\right)\\ -\\ -\displaystyle C_n=\gamma q\left(1-q^n\right)\left(\delta -q^{n-N-1}\right). -\end{array}\right.$$ - -\subsection*{Normalized recurrence relation} -\begin{eqnarray} -\label{NormRecDualqHahn} -xp_n(x)&=&p_{n+1}(x)+\left[1+\gamma\delta q-(A_n+C_n)\right]p_n(x)\nonumber\\ -& &{}\mathindent{}+\gamma q(1-q^n)(1-\gamma q^n)\nonumber\\ -& &{}\mathindent\mathindent{}\times(1-q^{n-N-1})(\delta-q^{n-N-1})p_{n-1}(x), -\end{eqnarray} -where -$$R_n(\mu(x);\gamma,\delta,N|q)=\frac{1}{(\gamma q,q^{-N};q)_n}p_n(\mu(x)).$$ - -\subsection*{$q$-Difference equation} -\begin{equation} -\label{dvDualqHahn} -q^{-n}(1-q^n)y(x)=B(x)y(x+1)-\left[B(x)+D(x)\right]y(x) -+D(x)y(x-1), -\end{equation} -where -$$y(x)=R_n(\mu(x);\gamma,\delta,N|q)$$ -and -$$\left\{\begin{array}{l}\displaystyle B(x)=\frac{(1-q^{x-N})(1-\gamma q^{x+1})(1-\gamma\delta q^{x+1})} -{(1-\gamma\delta q^{2x+1})(1-\gamma\delta q^{2x+2})}\\ -\\ -\displaystyle D(x)=-\frac{\gamma q^{x-N}(1-q^x)(1-\gamma\delta q^{x+N+1})(1-\delta q^x)} -{(1-\gamma\delta q^{2x})(1-\gamma\delta q^{2x+1})}.\end{array}\right.$$ - -\subsection*{Forward shift operator} -\begin{eqnarray} -\label{shift1DualqHahnI} -& &R_n(\mu(x+1);\gamma,\delta,N|q)-R_n(\mu(x);\gamma,\delta,N|q)\nonumber\\ -& &{}=\frac{q^{-n-x}(1-q^n)(1-\gamma\delta q^{2x+2})}{(1-\gamma q)(1-q^{-N})} -R_{n-1}(\mu(x);\gamma q,\delta,N-1|q) -\end{eqnarray} -or equivalently -\begin{eqnarray} -\label{shift1DualqHahnII} -& &\frac{\Delta R_n(\mu(x);\gamma,\delta,N|q)}{\Delta\mu(x)}\nonumber\\ -& &{}=\frac{q^{-n+1}(1-q^n)}{(1-q)(1-\gamma q)(1-q^{-N})}R_{n-1}(\mu(x);\gamma q,\delta,N-1|q). -\end{eqnarray} - -\subsection*{Backward shift operator} -\begin{eqnarray} -\label{shift2DualqHahnI} -& &(1-\gamma q^x)(1-\gamma\delta q^x)(1-q^{x-N-1})R_n(\mu(x);\gamma,\delta,N|q)\nonumber\\ -& &{}\mathindent{}+\gamma q^{x-N-1}(1-q^x)(1-\gamma\delta q^{x+N+1})(1-\delta q^x)R_n(\mu(x-1);\gamma,\delta,N|q)\nonumber\\ -& &{}=q^x(1-\gamma)(1-q^{-N-1})(1-\gamma\delta q^{2x})R_{n+1}(\mu(x);\gamma q^{-1},\delta,N+1|q) -\end{eqnarray} -or equivalently -\begin{eqnarray} -\label{shift2DualqHahnII} -& &\frac{\nabla\left[w(x;\gamma,\delta,N|q)R_n(\mu(x);\gamma,\delta,N|q)\right]}{\nabla\mu(x)}\nonumber\\ -& &{}=\frac{1}{(1-q)(1-\gamma\delta)}w(x;\gamma q^{-1},\delta,N+1|q)\nonumber\\ -& &{}\mathindent{}\times R_{n+1}(\mu(x);\gamma q^{-1},\delta,N+1|q), -\end{eqnarray} -where -$$w(x;\gamma,\delta,N|q)=\frac{(\gamma q,\gamma\delta q,q^{-N};q)_x} -{(q,\gamma\delta q^{N+2},\delta q;q)_x}(-\gamma^{-1})^x q^{Nx-\binom{x}{2}}.$$ - -\subsection*{Rodrigues-type formula} -\begin{eqnarray} -\label{RodDualqHahn} -& &w(x;\gamma,\delta,N|q)R_n(\mu(x);\gamma,\delta,N|q)\nonumber\\ -& &{}=(1-q)^n(\gamma\delta q;q)_n -\left(\nabla_{\mu}\right)^n\left[w(x;\gamma q^n,\delta,N-n|q)\right], -\end{eqnarray} -where -$$\nabla_{\mu}:=\frac{\nabla}{\nabla\mu(x)}.$$ - -\subsection*{Generating functions} For $x=0,1,2,\ldots,N$ we have -\begin{eqnarray} -\label{GenDualqHahn1} -& &(q^{-N}t;q)_{N-x}\cdot\qhyp{2}{1}{q^{-x},\delta^{-1}q^{-x}}{\gamma q}{\gamma\delta q^{x+1}t}\nonumber\\ -& &{}=\sum_{n=0}^N\frac{(q^{-N};q)_n}{(q;q)_n}R_n(\mu(x);\gamma,\delta,N|q)t^n. -\end{eqnarray} - -\begin{eqnarray} -\label{GenDualqHahn2} -& &(\gamma\delta qt;q)_x\cdot\qhyp{2}{1}{q^{x-N},\gamma q^{x+1}}{\delta^{-1}q^{-N}}{q^{-x}t}\nonumber\\ -& &{}=\sum_{n=0}^N -\frac{(q^{-N},\gamma q;q)_n}{(\delta^{-1}q^{-N},q;q)_n}R_n(\mu(x);\gamma,\delta,N|q)t^n. -\end{eqnarray} - -\subsection*{Limit relations} - -\subsubsection*{$q$-Racah $\rightarrow$ Dual $q$-Hahn} -To obtain the dual $q$-Hahn polynomials from the $q$-Racah polynomials we have to -take $\beta=0$ and $\alpha q=q^{-N}$ in (\ref{DefqRacah}): -$$R_n(\mu(x);q^{-N-1},0,\gamma,\delta|q)=R_n(\mu(x);\gamma,\delta,N|q),$$ -with -$$\mu(x)=q^{-x}+\gamma\delta q^{x+1}.$$ -We may also take $\alpha=0$ and $\beta=\delta^{-1}q^{-N-1}$ in (\ref{DefqRacah}) to obtain -the dual $q$-Hahn polynomials from the $q$-Racah polynomials: -$$R_n(\mu(x);0,\delta^{-1}q^{-N-1},\gamma,\delta|q)=R_n(\mu(x);\gamma,\delta,N|q).$$ -And if we take $\gamma q=q^{-N}$, $\delta\rightarrow\alpha\delta q^{N+1}$ and $\beta=0$ in the -definition (\ref{DefqRacah}) of the $q$-Racah polynomials we find the dual -$q$-Hahn polynomials given by (\ref{DefDualqHahn}) in the following way: -$$R_n(\mu(x);\alpha,0,q^{-N-1},\alpha\delta q^{N+1}|q)=R_n({\tilde \mu}(x);\alpha,\delta,N|q),$$ -with -$${\tilde \mu}(x)=q^{-x}+\alpha\delta q^{x+1}.$$ - -\subsubsection*{Dual $q$-Hahn $\rightarrow$ Affine $q$-Krawtchouk} -The affine $q$-Krawtchouk polynomials given by (\ref{DefAffqKrawtchouk}) -can be obtained from the dual $q$-Hahn polynomials by the substitution $\gamma=p$ -and $\delta=0$ in (\ref{DefDualqHahn}): -\begin{equation} -R_n(\mu(x);p,0,N|q)=K_n^{Aff}(q^{-x};p,N;q). -\end{equation} -Note that $\mu(x)=q^{-x}$ in this case. - -\subsubsection*{Dual $q$-Hahn $\rightarrow$ Dual $q$-Krawtchouk} -The dual $q$-Krawtchouk polynomials given by (\ref{DefDualqKrawtchouk}) can -be obtained from the dual $q$-Hahn polynomials by setting $\delta=c\gamma^{-1}q^{-N-1}$ -in (\ref{DefDualqHahn}) and letting $\gamma\rightarrow 0$: -\begin{equation} -\lim_{\gamma\rightarrow 0} -R_n(\mu(x);\gamma,c\gamma^{-1}q^{-N-1},N|q)=K_n(\lambda(x);c,N|q). -\end{equation} - -\subsubsection*{Dual $q$-Hahn $\rightarrow$ Dual Hahn} -The dual Hahn polynomials given by (\ref{DefDualHahn}) follow from the dual $q$-Hahn -polynomials by simply taking the limit $q\rightarrow 1$ in the definition (\ref{DefDualqHahn}) -of the dual $q$-Hahn polynomials after applying the substitution -$\gamma\rightarrow q^{\gamma}$ and $\delta\rightarrow q^{\delta}$: -\begin{equation} -\lim_{q\rightarrow 1}R_n(\mu(x);q^{\gamma},q^{\delta},N|q)=R_n(\lambda(x);\gamma,\delta,N), -\end{equation} -where -$$\left\{\begin{array}{l} -\displaystyle\mu(x)=q^{-x}+q^{x+\gamma+\delta+1}\\[5mm] -\displaystyle\lambda(x)=x(x+\gamma+\delta+1). -\end{array}\right.$$ - -\subsection*{Remark} -The dual $q$-Hahn polynomials given by (\ref{DefDualqHahn}) and the -$q$-Hahn polynomials given by (\ref{DefqHahn}) are related in the following way: -$$Q_n(q^{-x};\alpha,\beta,N|q)=R_x(\mu(n);\alpha,\beta,N|q),$$ -with -$$\mu(n)=q^{-n}+\alpha\beta q^{n+1}$$ -or -$$R_n(\mu(x);\gamma,\delta,N|q)=Q_x(q^{-n};\gamma,\delta,N|q),$$ -where -$$\mu(x)=q^{-x}+\gamma\delta q^{x+1}.$$ - RS: add begin\label{sec14.5} -% -\paragraph{\large\bf KLSadd: Different notation}See p.442, Remarks: -\begin{equation} -P_n(x;a,b,c,d;q):=P_n(qac^{-1}x;a,b,-ac^{-1}d;q) -=\qhyp32{q^{-n},q^{n+1}ab,qac^{-1}x}{qa,-qac^{-1}d}{q,q}. -\label{123} -\end{equation} -Furthermore, -\begin{equation} -P_n(x;a,b,c,d;q)=P_n(\la x;a,b,\la c,\la d;q), -\label{141} -\end{equation} -\begin{equation} -P_n(x;a,b,c;q)=P_n(-q^{-1}c^{-1}x;a,b,-ac^{-1},1;q) -\label{142} -\end{equation} -% -\paragraph{\large\bf KLSadd: Orthogonality relation}(equivalent to (14.5.2), see also \cite[(2.42), (2.41), (2.36), (2.35)]{K17}). -Let $c,d>0$ and either $a\in (-c/(qd),1/q)$, $b\in(-d/(cq),1/q)$ or -$a/c=-\overline b/d\notin\RR$. Then -\begin{equation} -\int_{-d}^c P_m(x;a,b,c,d;q) P_n(x;a,b,c,d;q)\, -\frac{(qx/c,-qx/d;q)_\iy}{(qax/c,-qbx/d;q)_\iy}\,d_qx=h_n\,\de_{m,n}\,, -\label{124} -\end{equation} -where -\begin{equation} -\frac{h_n}{h_0}=q^{\half n(n-1)}\left(\frac{q^2a^2d}c\right)^n\, -\frac{1-qab}{1-q^{2n+1}ab}\, -\frac{(q,qb,-qbc/d;q)_n}{(qa,qab,-qad/c;q)_n} -\label{125} -\end{equation} -and -\begin{equation} -h_0=(1-q)c\,\frac{(q,-d/c,-qc/d,q^2ab;q)_\iy} -{(qa,qb,-qbc/d,-qad/c;q)_\iy}\,. -\label{126} -\end{equation} -% -\paragraph{\large\bf KLSadd: Other hypergeometric representation and asymptotics}\begin{align} -&P_n(x;a,b,c,d;q) -=\frac{(-qbd^{-1}x;q)_n}{(-q^{-n}a^{-1}cd^{-1};q)_n}\, -\qhyp32{q^{-n},q^{-n}b^{-1},cx^{-1}}{qa,-q^{-n}b^{-1}dx^{-1}}{q,q} -\label{138}\\ -&\qquad=(qac^{-1}x)^n\,\frac{(qb,cx^{-1};q)_n}{(qa,-qac^{-1}d;q)_n}\, -\qhyp32{q^{-n},q^{-n}a^{-1},-qbd^{-1}x}{qb,q^{1-n}c^{-1}x} -{q,-q^{n+1}ac^{-1}d} -\label{132}\\ -&\qquad=(qac^{-1}x)^n\,\frac{(qb,q;q)_n}{(-qac^{-1}d;q)_n}\, -\sum_{k=0}^n\frac{(cx^{-1};q)_{n-k}}{(q,qa;q)_{n-k}}\, -\frac{(-qbd^{-1}x;q)_k}{(qb,q;q)_k}\,(-1)^k q^{\half k(k-1)}(-dx^{-1})^k. -\label{133} -\end{align} -Formula \eqref{138} follows from \eqref{123} by -\mycite{GR}{(III.11)} and next \eqref{132} follows by series inversion -\mycite{GR}{Exercise 1.4(ii)}. -Formulas \eqref{138} and \eqref{133} are also given in -\mycite{Ism}{(18.4.28), (18.4.29)}. -It follows from \eqref{132} or \eqref{133} that -(see \myciteKLS{298}{(1.17)} or \mycite{Ism}{(18.4.31)}) -\begin{equation} -\lim_{n\to\iy}(qac^{-1}x)^{-n} P_n(x;a,b,c,d;q) -=\frac{(cx^{-1},-dx^{-1};q)_\iy}{(-qac^{-1}d,qa;q)_\iy}\,, -\label{134} -\end{equation} -uniformly for $x$ in compact subsets of $\CC\backslash\{0\}$. -(Exclusion of the spectral points $x=cq^m,dq^m$ ($m=0,1,2,\ldots$), -as was done in \myciteKLS{298} and \mycite{Ism}, is not necessary. However, -while \eqref{134} yields 0 at these points, a more refined asymptotics -at these points is given in \myciteKLS{298} and \mycite{Ism}.)$\;$ -For the proof of \eqref{134} use that -\begin{equation} -\lim_{n\to\iy}(qac^{-1}x)^{-n} P_n(x;a,b,c,d;q) -=\frac{(qb,cx^{-1};q)_n}{(qa,-qac^{-1}d;q)_n}\, -\qhyp11{-qbd^{-1}x}{qb}{q,-dx^{-1}}, -\label{135} -\end{equation} -which can be evaluated by \mycite{GR}{(II.5)}. -Formula \eqref{135} follows formally from \eqref{132}, and it follows rigorously, by -dominated convergence, from \eqref{133}. -% -\paragraph{\large\bf KLSadd: Symmetry}(see \cite[\S2.5]{K17}). -\begin{equation} -\frac{P_n(-x;a,b,c,d;q)}{P_n(-d/(qb);a,b,c,d;q)} -=P_n(x;b,a,d,c;q). -\end{equation} -% -\paragraph{\large\bf KLSadd: Special values}\begin{align} -P_n(c/(qa);a,b,c,d;q)&=1,\\ -P_n(-d/(qb);a,b,c,d;q)&=\left(-\,\frac{ad}{bc}\right)^n\, -\frac{(qb,-qbc/d;q)_n}{(qa,-qad/c;q)_n}\,,\\ -P_n(c;a,b,c,d;q)&= -q^{\half n(n+1)}\left(\frac{ad}c\right)^n -\frac{(-qbc/d;q)_n}{(-qad/c;q)_n}\,,\\ -P_n(-d;a,b,c,d;q)&=q^{\half n(n+1)} (-a)^n\,\frac{(qb;q)_n}{(qa;q)_n}\,. -\end{align} -% -\paragraph{\large\bf KLSadd: Quadratic transformations}(see \cite[(2.48), (2.49)]{K17} and \eqref{128}).\\ -These express big $q$-Jacobi polynomials $P_m(x;a,a,1,1;q)$ in terms of little -$q$-Jacobi polynomials (see \S14.12). -\begin{align} -P_{2n}(x;a,a,1,1;q)&=\frac{p_n(x^2;q^{-1},a^2;q^2)}{p_n((qa)^{-2};q^{-1},a^2;q^2)}\,, -\label{130}\\ -P_{2n+1}(x;a,a,1,1;q)&=\frac{qax\,p_n(x^2;q,a^2;q^2)}{p_n((qa)^{-2};q,a^2;q^2)}\,. -\label{131} -\end{align} -Hence, by (14.12.1), \mycite{GR}{Exercise 1.4(ii)} and \eqref{128}, -\begin{align} -P_n(x;a,a,1,1;q)&=\frac{(qa^2;q^2)_n}{(qa^2;q)_n}\,(qax)^n\, -\qhyp21{q^{-n},q^{-n+1}}{q^{-2n+1}a^{-2}}{q^2,(ax)^{-2}} -\label{136}\\ -&=\frac{(q;q)_n}{(qa^2;q)_n}\,(qa)^n\, -\sum_{k=0}^{[\half n]}(-1)^k q^{k(k-1)} -\frac{(qa^2;q^2)_{n-k}}{(q^2;q^2)_k\,(q;q)_{n-2k}}\,x^{n-2k}. -\label{137} -\end{align} -% -\paragraph{\large\bf KLSadd: $q$-Chebyshev polynomials}In \eqref{123}, with $c=d=1$, the cases $a=b=q^{-\half}$ and $a=b=q^\half$ can be considered -as $q$-analogues of the Chebyshev polynomials of the first and second kind, respectively -(\S9.8.2) because of the limit (14.5.17). The quadratic relations \eqref{130}, \eqref{131} -can also be specialized to these cases. The definition of the $q$-Chebyshev polynomials -may vary by normalization and by dilation of argument. They were considered in -\cite{K18}. -By \myciteKLS{24}{p.279} and \eqref{130}, \eqref{131}, the {\em Al-Salam-Ismail polynomials} -$U_n(x;a,b)$ ($q$-dependence suppressed) in the case $a=q$ can be expressed as -$q$-Chebyshev polynomials of the second kind: -\begin{equation*} -U_n(x,q,b)=(q^{-3} b)^{\half n}\,\frac{1-q^{n+1}}{1-q}\, -P_n(b^{-\half}x;q^\half,q^\half,1,1;q). -\end{equation*} -Similarly, by \cite[(5.4), (5.1), (5.3)]{K19} and \eqref{130}, \eqref{131}, Cigler's $q$-Chebyshev -polynomials $T_n(x,s,q)$ and $U_n(x,s,q)$ -can be expressed in terms of the $q$-Chebyshev cases of \eqref{123}: -\begin{align*} -T_n(x,s,q)&=(-s)^{\half n}\,P_n((-qs)^{-\half} x;q^{-\half},q^{-\half},1,1;q),\\ -U_n(x,s,q)&=(-q^{-2}s)^{\half n}\,\frac{1-q^{n+1}}{1-q}\, -P_n((-qs)^{-\half} x;q^{\half},q^{\half},1,1;q). -\end{align*} -% -\paragraph{\large\bf KLSadd: Limit to Discrete $q$-Hermite I}\begin{equation} -\lim_{a\to0} a^{-n}\,P_n(x;a,a,1,1;q)=q^n\,h_n(x;q). -\label{139} -\end{equation} -Here $h_n(x;q)$ is given by (14.28.1). -For the proof of \eqref{139} use \eqref{138}. -% -\paragraph{\large\bf KLSadd: Pseudo big $q$-Jacobi polynomials}Let $a,b,c,d\in\CC$, $z_+>0$, $z_-<0$ such that -$\tfrac{(ax,bx;q)_\iy}{(cx,dx;q)_\iy}>0$ for $x\in z_- q^\ZZ\cup z_+ q^\ZZ$. -Then $(ab)/(qcd)>0$. Assume that $(ab)/(qcd)<1$. -Let $N$ be the largest nonnegative integer such that $q^{2N}>(ab)/(qcd)$. -Then -\begin{multline} -\int_{z_- q^\ZZ\cup z_+ q^\ZZ}P_m(cx;c/b,d/a,c/a;q)\,P_n(cx;c/b,d/a,c/a;q)\, -\frac{(ax,bx;q)_\iy}{(cx,dx;q)_\iy}\,d_qx=h_n\de_{m,n}\\ -(m,n=0,1,\ldots,N), -\label{114} -\end{multline} -where -\begin{equation} -\frac{h_n}{h_0}=(-1)^n\left(\frac{c^2}{ab}\right)^n q^{\half n(n-1)} q^{2n}\, -\frac{(q,qd/a,qd/b;q)_n}{(qcd/(ab),qc/a,qc/b;q)_n}\, -\frac{1-qcd/(ab)}{1-q^{2n+1}cd/(ab)} -\label{115} -\end{equation} -and -\begin{equation} -h_0=\int_{z_- q^\ZZ\cup z_+ q^\ZZ}\frac{(ax,bx;q)_\iy}{(cx,dx;q)_\iy}\,d_qx -=(1-q)z_+\, -\frac{(q,a/c,a/d,b/c,b/d;q)_\iy}{(ab/(qcd);q)_\iy}\, -\frac{\tha(z_-/z_+,cdz_-z_+;q)}{\tha(cz_-,dz_-,cz_+,dz_+;q)}\,. -\label{116} -\end{equation} -See Groenevelt \& Koelink \cite[Prop.~2.2]{K14}. -Formula \eqref{116} was first given by Slater \cite[(5)]{K15} as an evaluation -of a sum of two ${}_2\psi_2$ series. -The same formula is given in Slater \myciteKLS{471}{(7.2.6)} and in -\mycite{GR}{Exercise 5.10}, but in both cases with the same slight error, -see \cite[2nd paragraph after Lemma 2.1]{K14} for correction. -The theta function is given by \eqref{117}. -Note that -\begin{equation} -P_n(cx;c/b,d/a,c/a;q)=P_n(-q^{-1}ax;c/b,d/a,-a/b,1;q). -\label{145} -\end{equation} - -In \cite{K29} the weights of the pseudo big $q$-Jacobi polynomials -occur in certain measures on the space of $N$-point configurations -on the so-called extended Gelfand-Tsetlin graph. -% -\subsubsection*{Limit relations} -\paragraph{\large\bf KLSadd: Pseudo big $q$-Jacobi $\longrightarrow$ Discrete Hermite II}\begin{equation} -\lim_{a\to\iy}i^n q^{\half n(n-1)} P_n(q^{-1}a^{-1}ix;a,a,1,1;q)= -\wt h_n(x;q). -\label{144} -\end{equation} -For the proof use \eqref{137} and \eqref{143}. -Note that $P_n(q^{-1}a^{-1}ix;a,a,1,1;q)$ is obtained from the -\RHS\ of \eqref{144} by replacing $a,b,c,d$ by $-ia^{-1},ia^{-1},i,-i$. -% -\paragraph{\large\bf KLSadd: Pseudo big $q$-Jacobi $\longrightarrow$ Pseudo Jacobi}\begin{equation} -\lim_{q\uparrow1}P_n(iq^{\half(-N-1+i\nu)}x;-q^{-N-1},-q^{-N-1},q^{-N+i\nu-1};q) -=\frac{P_n(x;\nu,N)}{P_n(-i;\nu,N)}\,. -\label{118} -\end{equation} -Here the big $q$-Jacobi polynomial on the \LHS\ equals -$P_n(cx;c/b,d/a,c/a;q)$ with\\ -$a=iq^{\half(N+1-i\nu)}$, $b=-iq^{\half(N+1+i\nu)}$, -$c=iq^{\half(-N-1+i\nu)}$, $d=-iq^{\half(-N-1-i\nu)}$. -% - -\subsection*{References} -\cite{AlvarezSmirnov}, \cite{AndrewsAskey85}, \cite{AskeyWilson79}, -\cite{AskeyWilson85}, \cite{AtakRahmanSuslov}, \cite{GasperRahman90}, -\cite{KoelinkKoorn}, \cite{Nikiforov+}, \cite{Stanton84}. - - -\section{Al-Salam-Chihara}\index{Al-Salam-Chihara polynomials} -\par\setcounter{equation}{0} - -\subsection*{Basic hypergeometric representation} -\begin{eqnarray} -\label{DefAlSalamChihara} -Q_n(x;a,b|q)&=&\frac{(ab;q)_n}{a^n}\, -\qhyp{3}{2}{q^{-n},a\e^{i\theta},a\e^{-i\theta}}{ab,0}{q}\\ -&=&(a\e^{i\theta};q)_n\e^{-in\theta}\,\qhyp{2}{1}{q^{-n},b\e^{-i\theta}} -{a^{-1}q^{-n+1}\e^{-i\theta}}{a^{-1}q\e^{i\theta}}\nonumber\\ -&=&(b\e^{-i\theta};q)_n\e^{in\theta}\,\qhyp{2}{1}{q^{-n},a\e^{i\theta}} -{b^{-1}q^{-n+1}\e^{i\theta}}{b^{-1}q\e^{-i\theta}},\quad x=\cos\theta.\nonumber -\end{eqnarray} - -\subsection*{Orthogonality relation} -If $a$ and $b$ are real or complex conjugates and $\max(|a|,|b|)<1$, then we have the following orthogonality relation -\begin{equation} -\label{OrtAlSalamChiharaI} -\frac{1}{2\pi}\int_{-1}^1\frac{w(x)}{\sqrt{1-x^2}}Q_m(x;a,b|q)Q_n(x;a,b|q)\,dx -=\frac{\,\delta_{mn}}{(q^{n+1},abq^n;q)_{\infty}}, -\end{equation} -where -$$w(x):=w(x;a,b|q)=\left|\frac{(\e^{2i\theta};q)_{\infty}} -{(a\e^{i\theta},b\e^{i\theta};q)_{\infty}}\right|^2 -=\frac{h(x,1)h(x,-1)h(x,q^{\frac{1}{2}})h(x,-q^{\frac{1}{2}})}{h(x,a)h(x,b)},$$ -with -$$h(x,\alpha):=\prod_{k=0}^{\infty}\left(1-2\alpha xq^k+\alpha^2q^{2k}\right) -=\left(\alpha\e^{i\theta},\alpha\e^{-i\theta};q\right)_{\infty},\quad x=\cos\theta.$$ -If $a>1$ and $|ab|<1$, then we have another orthogonality relation given by: -\begin{eqnarray} -\label{OrtAlSalamChiharaII} -& &\frac{1}{2\pi}\int_{-1}^1\frac{w(x)}{\sqrt{1-x^2}}Q_m(x;a,b|q)Q_n(x;a,b|q)\,dx\nonumber\\ -& &{}\mathindent{}+\sum_{\begin{array}{c}\scriptstyle k\\ \scriptstyle 1q^{-N}$;\quad -(iv) $\ga>q^{-N}$, $\de>q^{-N}$;\quad -(v) $01,\;qb0. -\end{eqnarray} - -\subsection*{Recurrence relation} -\begin{eqnarray} -\label{RecqMeixner} -& &q^{2n+1}(1-q^{-x})M_n(q^{-x})\nonumber\\ -& &{}=c(1-bq^{n+1})M_{n+1}(q^{-x})\nonumber\\ -& &{}\mathindent{}-\left[c(1-bq^{n+1})+q(1-q^n)(c+q^n)\right]M_n(q^{-x})\nonumber\\ -& &{}\mathindent\mathindent{}+q(1-q^n)(c+q^n)M_{n-1}(q^{-x}), -\end{eqnarray} -where -$$M_n(q^{-x}):=M_n(q^{-x};b,c;q).$$ - -\subsection*{Normalized recurrence relation} -\begin{eqnarray} -\label{NormRecqMeixner} -xp_n(x)&=&p_{n+1}(x)+ -\left[1+q^{-2n-1}\left\{c(1-bq^{n+1})+q(1-q^n)(c+q^n)\right\}\right]p_n(x)\nonumber\\ -& &{}\mathindent{}+cq^{-4n+1}(1-q^n)(1-bq^n)(c+q^n)p_{n-1}(x), -\end{eqnarray} -where -$$M_n(q^{-x};b,c;q)=\frac{(-1)^nq^{n^2}}{(bq;q)_nc^n}p_n(q^{-x}).$$ - -\subsection*{$q$-Difference equation} -\begin{equation} -\label{dvqMeixner} --(1-q^n)y(x)=B(x)y(x+1)-\left[B(x)+D(x)\right]y(x)+D(x)y(x-1), -\end{equation} -where -$$y(x)=M_n(q^{-x};b,c;q)$$ -and -$$\left\{\begin{array}{l}\displaystyle B(x)=cq^x(1-bq^{x+1})\\ -\\ -\displaystyle D(x)=(1-q^x)(1+bcq^x).\end{array}\right.$$ - -\subsection*{Forward shift operator} -\begin{eqnarray} -\label{shift1qMeixnerI} -& &M_n(q^{-x-1};b,c;q)-M_n(q^{-x};b,c;q)\nonumber\\ -& &{}=-\frac{q^{-x}(1-q^n)}{c(1-bq)}M_{n-1}(q^{-x};bq,cq^{-1};q) -\end{eqnarray} -or equivalently -\begin{equation} -\label{shift1qMeixnerII} -\frac{\Delta M_n(q^{-x};b,c;q)}{\Delta q^{-x}} -=-\frac{q(1-q^n)}{c(1-q)(1-bq)}M_{n-1}(q^{-x};bq,cq^{-1};q). -\end{equation} - -\subsection*{Backward shift operator} -\begin{eqnarray} -\label{shift2qMeixnerI} -& &cq^x(1-bq^x)M_n(q^{-x};b,c;q)-(1-q^x)(1+bcq^x)M_n(q^{-x+1};b,c;q)\nonumber\\ -& &{}=cq^x(1-b)M_{n+1}(q^{-x};bq^{-1},cq;q) -\end{eqnarray} -or equivalently -\begin{eqnarray} -\label{shift2qMeixnerII} -& &\frac{\nabla\left[w(x;b,c;q)M_n(q^{-x};b,c;q)\right]}{\nabla q^{-x}}\nonumber\\ -& &{}=\frac{1}{1-q}w(x;bq^{-1},cq;q)M_{n+1}(q^{-x};bq^{-1},cq;q), -\end{eqnarray} -where -$$w(x;b,c;q)=\frac{(bq;q)_x}{(q,-bcq;q)_x}c^xq^{\binom{x+1}{2}}.$$ - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodqMeixner} -w(x;b,c;q)M_n(q^{-x};b,c;q)=(1-q)^n\left(\nabla_q\right)^n\left[w(x;bq^n,cq^{-n};q)\right], -\end{equation} -where -$$\nabla_q:=\frac{\nabla}{\nabla q^{-x}}.$$ - -\subsection*{Generating functions} -\begin{equation} -\label{GenqMeixner1} -\frac{1}{(t;q)_{\infty}}\,\qhyp{1}{1}{q^{-x}}{bq}{-c^{-1}qt} -=\sum_{n=0}^{\infty}\frac{M_n(q^{-x};b,c;q)}{(q;q)_n}t^n. -\end{equation} - -\begin{eqnarray} -\label{GenqMeixner2} -& &\frac{1}{(t;q)_{\infty}}\,\qhyp{1}{1}{-b^{-1}c^{-1}q^{-x}}{-c^{-1}q}{bqt}\nonumber\\ -& &{}=\sum_{n=0}^{\infty}\frac{(bq;q)_n}{(-c^{-1}q,q;q)_n}M_n(q^{-x};b,c;q)t^n. -\end{eqnarray} - -\subsection*{Limit relations} - -\subsubsection*{Big $q$-Jacobi $\rightarrow$ $q$-Meixner} -If we set $b=-a^{-1}cd^{-1}$ (with $d>0$) in the definition (\ref{DefBigqJacobi}) -of the big $q$-Jacobi polynomials and take the limit $c\rightarrow -\infty$ we -obtain the $q$-Meixner polynomials given by (\ref{DefqMeixner}): -\begin{equation} -\lim_{c\rightarrow -\infty}P_n(q^{-x};a,-a^{-1}cd^{-1},c;q)=M_n(q^{-x};a,d;q). -\end{equation} - -\subsubsection*{$q$-Hahn $\rightarrow$ $q$-Meixner} -The $q$-Meixner polynomials given by (\ref{DefqMeixner}) can be obtained from the $q$-Hahn polynomials -by setting $\alpha=b$ and $\beta=-b^{-1}c^{-1}q^{-N-1}$ in the definition (\ref{DefqHahn}) of the -$q$-Hahn polynomials and letting $N\rightarrow\infty$: -$$\lim_{N\rightarrow\infty}Q_n(q^{-x};b,-b^{-1}c^{-1}q^{-N-1},N|q)=M_n(q^{-x};b,c;q).$$ - -\subsubsection*{$q$-Meixner $\rightarrow$ $q$-Laguerre} -The $q$-Laguerre polynomials given by (\ref{DefqLaguerre}) can be obtained -from the $q$-Meixner polynomials given by (\ref{DefqMeixner}) by setting -$b=q^{\alpha}$ and $q^{-x}\rightarrow cq^{\alpha}x$ in the definition -(\ref{DefqMeixner}) of the $q$-Meixner polynomials and then taking the limit -$c\rightarrow\infty$: -\begin{equation} -\lim_{c\rightarrow\infty}M_n(cq^{\alpha}x;q^{\alpha},c;q)= -\frac{(q;q)_n}{(q^{\alpha+1};q)_n}L_n^{(\alpha)}(x;q). -\end{equation} - -\subsubsection*{$q$-Meixner $\rightarrow$ $q$-Charlier} -The $q$-Charlier polynomials given by (\ref{DefqCharlier}) can easily be -obtained from the $q$-Meixner given by (\ref{DefqMeixner}) by setting $b=0$ -in the definition (\ref{DefqMeixner}) of the $q$-Meixner polynomials: -\begin{equation} -M_n(x;0,c;q)=C_n(x;c;q). -\end{equation} - -\subsubsection*{$q$-Meixner $\rightarrow$ Al-Salam-Carlitz~II} -The Al-Salam-Carlitz~II polynomials given by (\ref{DefAlSalamCarlitzII}) -can be obtained from the $q$-Meixner polynomials given by (\ref{DefqMeixner}) -by setting $b=-ac^{-1}$ in the definition (\ref{DefqMeixner}) of the -$q$-Meixner polynomials and then taking the limit $c\rightarrow 0$: -\begin{equation} -\lim_{c\rightarrow 0}M_n(x;-ac^{-1},c;q)= -\left(-\frac{1}{a}\right)^nq^{\binom{n}{2}}V_n^{(a)}(x;q). -\end{equation} - -\subsubsection*{$q$-Meixner $\rightarrow$ Meixner} -To find the Meixner polynomials given by (\ref{DefMeixner}) from the $q$-Meixner polynomials given by -(\ref{DefqMeixner}) we set $b=q^{\beta-1}$ and $c\rightarrow (1-c)^{-1}c$ and let $q\rightarrow 1$: -\begin{equation} -\lim_{q\rightarrow 1}M_n(q^{-x};q^{\beta-1},(1-c)^{-1}c;q)=M_n(x;\beta,c). -\end{equation} - -\subsection*{Remarks} -The $q$-Meixner polynomials given by (\ref{DefqMeixner}) and the little -$q$-Jacobi polynomials given by (\ref{DefLittleqJacobi}) are related in the -following way: -$$M_n(q^{-x};b,c;q)=p_n(-c^{-1}q^n;b,b^{-1}q^{-n-x-1}|q).$$ - -\noindent -The $q$-Meixner polynomials and the quantum $q$-Krawtchouk polynomials -given by (\ref{DefQuantumqKrawtchouk}) are related in the following way: -$$K_n^{qtm}(q^{-x};p,N;q)=M_n(q^{-x};q^{-N-1},-p^{-1};q).$$ - -\subsection*{References} -\cite{AlSalam90}, \cite{AlSalamVerma82II}, \cite{AlSalamVerma88}, \cite{AlvarezRonveaux}, -\cite{AtakAtakKlimyk}, \cite{AtakRahmanSuslov}, \cite{Campigotto+}, \cite{GasperRahman90}, -\cite{Hahn}, \cite{Ismail2005I}, \cite{Nikiforov+}, \cite{Smirnov}. - - -\section{Quantum $q$-Krawtchouk} -\index{Quantum q-Krawtchouk polynomials@Quantum $q$-Krawtchouk polynomials} -\index{q-Krawtchouk polynomials@$q$-Krawtchouk polynomials!Quantum} -\par\setcounter{equation}{0} - -\subsection*{Basic hypergeometric representation} -\begin{equation} -\label{DefQuantumqKrawtchouk} -K_n^{qtm}(q^{-x};p,N;q)= -\qhyp{2}{1}{q^{-n},q^{-x}}{q^{-N}}{pq^{n+1}},\quad n=0,1,2,\ldots,N. -\end{equation} - -\subsection*{Orthogonality relation} -\begin{eqnarray} -\label{OrtQuantumqKrawtchouk} -& &\sum_{x=0}^N\frac{(pq;q)_{N-x}}{(q;q)_x(q;q)_{N-x}}(-1)^{N-x}q^{\binom{x}{2}} -K_m^{qtm}(q^{-x};p,N;q)K_n^{qtm}(q^{-x};p,N;q)\nonumber\\ -& &{}=\frac{(-1)^np^N(q;q)_{N-n}(q,pq;q)_n}{(q,q;q)_N} -q^{\binom{N+1}{2}-\binom{n+1}{2}+Nn}\,\delta_{mn},\quad p>q^{-N}. -\end{eqnarray} - -\subsection*{Recurrence relation} -\begin{eqnarray} -\label{RecQuantumqKrawtchouk} -& &-pq^{2n+1}(1-q^{-x})K_n^{qtm}(q^{-x})\nonumber\\ -& &{}=(1-q^{n-N})K_{n+1}^{qtm}(q^{-x})\nonumber\\ -& &{}\mathindent{}-\left[(1-q^{n-N})+q(1-q^n)(1-pq^n)\right]K_n^{qtm}(q^{-x})\nonumber\\ -& &{}\mathindent\mathindent{}+q(1-q^n)(1-pq^n)K_{n-1}^{qtm}(q^{-x}), -\end{eqnarray} -where -$$K_n^{qtm}(q^{-x}):=K_n^{qtm}(q^{-x};p,N;q).$$ - -\subsection*{Normalized recurrence relation} -\begin{eqnarray} -\label{NormRecQuantumqKrawtchouk} -xp_n(x)&=&p_{n+1}(x)+ -\left[1-p^{-1}q^{-2n-1}\left\{(1-q^{n-N})+q(1-q^n)(1-pq^n)\right\}\right]p_n(x)\nonumber\\ -& &{}\mathindent{}+p^{-2}q^{-4n+1}(1-q^n)(1-pq^n)(1-q^{n-N-1})p_{n-1}(x), -\end{eqnarray} -where -$$K_n^{qtm}(q^{-x};p,N;q)=\frac{p^nq^{n^2}}{(q^{-N};q)_n}p_n(q^{-x}).$$ - -\subsection*{$q$-Difference equation} -\begin{equation} -\label{dvQuantumqKrawtchouk} --p(1-q^n)y(x)=B(x)y(x+1)-\left[B(x)+D(x)\right]y(x)+D(x)y(x-1), -\end{equation} -where -$$y(x)=K_n^{qtm}(q^{-x};p,N;q)$$ -and -$$\left\{\begin{array}{l}\displaystyle B(x)=-q^x(1-q^{x-N})\\ -\\ -\displaystyle D(x)=(1-q^x)(p-q^{x-N-1}).\end{array}\right.$$ - -\subsection*{Forward shift operator} -\begin{eqnarray} -\label{shift1QuantumqKrawtchoukI} -& &K_n^{qtm}(q^{-x-1};p,N;q)-K_n^{qtm}(q^{-x};p,N;q)\nonumber\\ -& &{}=\frac{pq^{-x}(1-q^n)}{1-q^{-N}}K_{n-1}^{qtm}(q^{-x};pq,N-1;q) -\end{eqnarray} -or equivalently -\begin{equation} -\label{shift1QuantumqKrawtchoukII} -\frac{\Delta K_n^{qtm}(q^{-x};p,N;q)}{\Delta q^{-x}}= -\frac{pq(1-q^n)}{(1-q)(1-q^{-N})}K_{n-1}^{qtm}(q^{-x};pq,N-1;q). -\end{equation} - -\subsection*{Backward shift operator} -\begin{eqnarray} -\label{shift2QuantumqKrawtchoukI} -& &(1-q^{x-N-1})K_n^{qtm}(q^{-x};p,N;q)\nonumber\\ -& &{}\mathindent{}+q^{-x}(1-q^x)(p-q^{x-N-1})K_n^{qtm}(q^{-x+1};p,N;q)\nonumber\\ -& &{}=(1-q^{-N-1})K_{n+1}^{qtm}(q^{-x};pq^{-1},N+1;q) -\end{eqnarray} -or equivalently -\begin{eqnarray} -\label{shift2QuantumqKrawtchoukII} -& &\frac{\nabla\left[w(x;p,N;q)K_n^{qtm}(q^{-x};p,N;q)\right]}{\nabla q^{-x}}\nonumber\\ -& &{}=\frac{1}{1-q}w(x;pq^{-1},N+1;q)K_{n+1}^{qtm}(q^{-x};pq^{-1},N+1;q), -\end{eqnarray} -where -$$w(x;p,N;q)=\frac{(q^{-N};q)_x}{(q,p^{-1}q^{-N};q)_x}(-p)^{-x}q^{\binom{x+1}{2}}.$$ - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodQuantumqKrawtchouk} -w(x;p,N;q)K_n^{qtm}(q^{-x};p,N;q)=(1-q)^n\left(\nabla_q\right)^n\left[w(x;pq^n,N-n;q)\right], -\end{equation} -where -$$\nabla_q:=\frac{\nabla}{\nabla q^{-x}}.$$ - -\subsection*{Generating functions} For $x=0,1,2,\ldots,N$ we have -\begin{eqnarray} -\label{GenQuantumqKrawtchouk1} -& &(q^{x-N}t;q)_{N-x}\cdot\qhyp{2}{1}{q^{-x},pq^{N+1-x}}{0}{q^{x-N}t}\nonumber\\ -& &{}=\sum_{n=0}^N\frac{(q^{-N};q)_n}{(q;q)_n}K_n^{qtm}(q^{-x};p,N;q)t^n. -\end{eqnarray} - -\begin{eqnarray} -\label{GenQuantumqKrawtchouk2} -& &(q^{-x}t;q)_x\cdot\qhyp{2}{1}{q^{x-N},0}{pq}{q^{-x}t}\nonumber\\ -& &{}=\sum_{n=0}^N\frac{(q^{-N};q)_n}{(pq,q;q)_n}K_n^{qtm}(q^{-x};p,N;q)t^n. -\end{eqnarray} - -\subsection*{Limit relations} - -\subsubsection*{$q$-Hahn $\rightarrow$ Quantum $q$-Krawtchouk} -The quantum $q$-Krawtchouk polynomials given by (\ref{DefQuantumqKrawtchouk}) -simply follow from the $q$-Hahn polynomials by setting $\beta=p$ in the definition (\ref{DefqHahn}) of -the $q$-Hahn polynomials and taking the limit $\alpha\rightarrow\infty$: -$$\lim_{\alpha\rightarrow\infty}Q_n(q^{-x};\alpha,p,N|q)=K_n^{qtm}(q^{-x};p,N;q).$$ - -\subsubsection*{Quantum $q$-Krawtchouk $\rightarrow$ Al-Salam-Carlitz~II} -If we set $p=a^{-1}q^{-N-1}$ in the definition (\ref{DefQuantumqKrawtchouk}) -of the quantum $q$-Krawtchouk polynomials and let $N\rightarrow\infty$ we -obtain the Al-Salam-Carlitz~II polynomials given by -(\ref{DefAlSalamCarlitzII}). In fact we have -\begin{equation} -\lim_{N\rightarrow\infty}K_n^{qtm}(x;a^{-1}q^{-N-1},N;q)= -\left(-\frac{1}{a}\right)^nq^{\binom{n}{2}}V_n^{(a)}(x;q). -\end{equation} - -\subsubsection*{Quantum $q$-Krawtchouk $\rightarrow$ Krawtchouk} -The Krawtchouk polynomials given by (\ref{DefKrawtchouk}) easily follow from -the quantum $q$-Krawtchouk polynomials given by (\ref{DefQuantumqKrawtchouk}) -in the following way: -\begin{equation} -\lim_{q\rightarrow 1}K_n^{qtm}(q^{-x};p,N;q)=K_n(x;p^{-1},N). -\end{equation} - -\subsection*{Remarks} -The quantum $q$-Krawtchouk polynomials given by -(\ref{DefQuantumqKrawtchouk}) and the $q$-Meixner polynomials given by -(\ref{DefqMeixner}) are related in the following way: -$$K_n^{qtm}(q^{-x};p,N;q)=M_n(q^{-x};q^{-N-1},-p^{-1};q).$$ - -\noindent -The quantum $q$-Krawtchouk polynomials are related to the affine -$q$-Krawtchouk polynomials given by (\ref{DefAffqKrawtchouk}) -by the transformation $q\leftrightarrow q^{-1}$ in the following way: -$$K_n^{qtm}(q^x;p,N;q^{-1})=(p^{-1}q;q)_n\left(-\frac{p}{q}\right)^nq^{-\binom{n}{2}} -K_n^{Aff}(q^{x-N};p^{-1},N;q).$$ - RS: add begin\label{sec14.12} -% -\paragraph{\large\bf KLSadd: Notation}Here the little $q$-Jacobi polynomial is denoted by -$p_n(x;a,b;q)$ instead of -$p_n(x;a,b\,|\, q)$. -% -\paragraph{\large\bf KLSadd: Special values}(see \cite[\S2.4]{K17}). -\begin{align} -p_n(0;a,b;q)&=1,\label{127}\\ -p_n(q^{-1}b^{-1};a,b;q)&=(-qb)^{-n}\,q^{-\half n(n-1)}\,\frac{(qb;q)_n}{(qa;q)_n}\,,\label{128}\\ -p_n(1;a,b;q)&=(-a)^n\,q^{\half n(n+1)}\,\frac{(qb;q)_n}{(qa;q)_n}\,.\label{129} -\end{align} -% - -\subsection*{References} -\cite{GasperRahman90}, \cite{Koorn89III}, \cite{Koorn90II}, \cite{Smirnov}. - - -\section{$q$-Krawtchouk}\index{q-Krawtchouk polynomials@$q$-Krawtchouk polynomials} -\par\setcounter{equation}{0} - -\subsection*{Basic hypergeometric representation} For $n=0,1,2,\ldots,N$ we have -\begin{eqnarray} -\label{DefqKrawtchouk} -K_n(q^{-x};p,N;q)&=&\qhyp{3}{2}{q^{-n},q^{-x},-pq^n}{q^{-N},0}{q}\\ -&=&\frac{(q^{x-N};q)_n}{(q^{-N};q)_nq^{nx}}\, -\qhyp{2}{1}{q^{-n},q^{-x}}{q^{N-x-n+1}}{-pq^{n+N+1}}.\nonumber -\end{eqnarray} - -\subsection*{Orthogonality relation} -\begin{eqnarray} -\label{OrtqKrawtchouk} -& &\sum_{x=0}^N\frac{(q^{-N};q)_x}{(q;q)_x}(-p)^{-x}K_m(q^{-x};p,N;q)K_n(q^{-x};p,N;q)\nonumber\\ -& &{}=\frac{(q,-pq^{N+1};q)_n}{(-p,q^{-N};q)_n}\frac{(1+p)}{(1+pq^{2n})}\nonumber\\ -& &{}\mathindent{}\times (-pq;q)_Np^{-N}q^{-\binom{N+1}{2}} -\left(-pq^{-N}\right)^nq^{n^2}\,\delta_{mn},\quad p>0. -\end{eqnarray} - -\subsection*{Recurrence relation} -\begin{eqnarray} -\label{RecqKrawtchouk} --\left(1-q^{-x}\right)K_n(q^{-x})&=&A_nK_{n+1}(q^{-x})-\left(A_n+C_n\right)K_n(q^{-x})\nonumber\\ -& &{}\mathindent{}+C_nK_{n-1}(q^{-x}), -\end{eqnarray} -where -$$K_n(q^{-x}):=K_n(q^{-x};p,N;q)$$ -and -$$\left\{\begin{array}{l} -\displaystyle A_n=\frac{(1-q^{n-N})(1+pq^n)}{(1+pq^{2n})(1+pq^{2n+1})}\\ -\\ -\displaystyle C_n=-pq^{2n-N-1}\frac{(1+pq^{n+N})(1-q^n)}{(1+pq^{2n-1})(1+pq^{2n})}. -\end{array}\right.$$ - -\subsection*{Normalized recurrence relation} -\begin{equation} -\label{NormRecqKrawtchouk} -xp_n(x)=p_{n+1}(x)+\left[1-(A_n+C_n)\right]p_n(x)+A_{n-1}C_np_{n-1}(x), -\end{equation} -where -$$K_n(q^{-x};p,N;q)=\frac{(-pq^n;q)_n}{(q^{-N};q)_n}p_n(q^{-x}).$$ - -\subsection*{$q$-Difference equation} -\begin{eqnarray} -\label{dvqKrawtchouk} -& &q^{-n}(1-q^n)(1+pq^n)y(x)\nonumber\\ -& &{}=(1-q^{x-N})y(x+1)-\left[(1-q^{x-N})-p(1-q^x)\right]y(x)\nonumber\\ -& &{}\mathindent{}-p(1-q^x)y(x-1), -\end{eqnarray} -where -$$y(x)=K_n(q^{-x};p,N;q).$$ - -\subsection*{Forward shift operator} -\begin{eqnarray} -\label{shift1qKrawtchoukI} -& &K_n(q^{-x-1};p,N;q)-K_n(q^{-x};p,N;q)\nonumber\\ -& &{}=\frac{q^{-n-x}(1-q^n)(1+pq^n)}{1-q^{-N}}K_{n-1}(q^{-x};pq^2,N-1;q) -\end{eqnarray} -or equivalently -\begin{equation} -\label{shift1qKrawtchoukII} -\frac{\Delta K_n(q^{-x};p,N;q)}{\Delta q^{-x}}= -\frac{q^{-n+1}(1-q^n)(1+pq^n)}{(1-q)(1-q^{-N})}K_{n-1}(q^{-x};pq^2,N-1;q). -\end{equation} - -\subsection*{Backward shift operator} -\begin{eqnarray} -\label{shift2qKrawtchoukI} -& &(1-q^{x-N-1})K_n(q^{-x};p,N;q)+pq^{-1}(1-q^x)K_n(q^{-x+1};p,N;q)\nonumber\\ -& &{}=q^x(1-q^{-N-1})K_{n+1}(q^{-x};pq^{-2},N+1;q) -\end{eqnarray} -or equivalently -\begin{eqnarray} -\label{shift2qKrawtchoukII} -& &\frac{\nabla\left[w(x;p,N;q)K_n(q^{-x};p,N;q)\right]}{\nabla q^{-x}}\nonumber\\ -& &{}=\frac{1}{1-q}w(x;pq^{-2},N+1;q)K_{n+1}(q^{-x};pq^{-2},N+1;q), -\end{eqnarray} -where -$$w(x;p,N;q)=\frac{(q^{-N};q)_x}{(q;q)_x}\left(-\frac{q}{p}\right)^x.$$ - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodqKrawtchouk} -w(x;p,N;q)K_n(q^{-x};p,N;q)=(1-q)^n\left(\nabla_q\right)^n\left[w(x;pq^{2n},N-n;q)\right], -\end{equation} -where -$$\nabla_q:=\frac{\nabla}{\nabla q^{-x}}.$$ - -\subsection*{Generating function} For $x=0,1,2,\ldots,N$ we have -\begin{eqnarray} -\label{GenqKrawtchouk} -& &\qhyp{1}{1}{q^{-x}}{0}{pqt}\,\qhyp{2}{0}{q^{x-N},0}{-}{-q^{-x}t}\nonumber\\ -& &{}=\sum_{n=0}^N\frac{(q^{-N};q)_n}{(q;q)_n}q^{-\binom{n}{2}}K_n(q^{-x};p,N;q)t^n. -\end{eqnarray} - -\subsection*{Limit relations} - -\subsubsection*{$q$-Racah $\rightarrow$ $q$-Krawtchouk} -The $q$-Krawtchouk polynomials given by (\ref{DefqKrawtchouk}) can be obtained from -the $q$-Racah polynomials by setting $\alpha q=q^{-N}$, $\beta=-pq^N$ and -$\gamma=\delta=0$ in the definition (\ref{DefqRacah}) of the $q$-Racah polynomials: -$$R_n(q^{-x};q^{-N-1},-pq^N,0,0|q)=K_n(q^{-x};p,N;q).$$ -Note that $\mu(x)=q^{-x}$ in this case. - -\subsubsection*{$q$-Hahn $\rightarrow$ $q$-Krawtchouk} -If we set $\beta=-\alpha^{-1}q^{-1}p$ in the definition (\ref{DefqHahn}) of the $q$-Hahn polynomials -and then let $\alpha\rightarrow 0$ we obtain the $q$-Krawtchouk polynomials given by -(\ref{DefqKrawtchouk}): -$$\lim_{\alpha\rightarrow 0} -Q_n(q^{-x};\alpha,-\alpha^{-1}q^{-1}p,N|q)=K_n(q^{-x};p,N;q).$$ - -\subsubsection*{$q$-Krawtchouk $\rightarrow$ $q$-Bessel} -If we set $x\rightarrow N-x$ in the definition (\ref{DefqKrawtchouk}) of the -$q$-Krawtchouk polynomials and then take the limit $N\rightarrow\infty$ we -obtain the $q$-Bessel polynomials given by (\ref{DefqBessel}): -\begin{equation} -\lim_{N\rightarrow\infty}K_n(q^{x-N};p,N;q)=y_n(q^x;p;q). -\end{equation} - -\subsubsection*{$q$-Krawtchouk $\rightarrow$ $q$-Charlier} -By setting $p=a^{-1}q^{-N}$ in the definition (\ref{DefqKrawtchouk}) of the -$q$-Krawtchouk polynomials and then taking the limit $N\rightarrow\infty$ we -obtain the $q$-Charlier polynomials given by (\ref{DefqCharlier}): -\begin{equation} -\lim_{N\rightarrow\infty}K_n(q^{-x};a^{-1}q^{-N},N;q)=C_n(q^{-x};a;q). -\end{equation} - -\subsubsection*{$q$-Krawtchouk $\rightarrow$ Krawtchouk} -If we take the limit $q\rightarrow 1$ in the definition (\ref{DefqKrawtchouk}) of the $q$-Krawtchouk -polynomials we simply find the Krawtchouk polynomials given by (\ref{DefKrawtchouk}) in the -following way: -\begin{equation} -\lim_{q\rightarrow 1}K_n(q^{-x};p,N;q)=K_n(x;(p+1)^{-1},N). -\end{equation} - -\subsection*{Remark} -The $q$-Krawtchouk polynomials given by (\ref{DefqKrawtchouk}) and the -dual $q$-Krawtchouk polynomials given by (\ref{DefDualqKrawtchouk}) are -related in the following way: -$$K_n(q^{-x};p,N;q)=K_x(\lambda(n);-pq^N,N|q)$$ -with -$$\lambda(n)=q^{-n}-pq^n$$ -or -$$K_n(\lambda(x);c,N|q)=K_x(q^{-n};-cq^{-N},N;q)$$ -with -$$\lambda(x)=q^{-x}+cq^{x-N}.$$ - -\subsection*{References} -\cite{AlvarezRonveaux}, \cite{AskeyWilson79}, \cite{AtakRahmanSuslov}, -\cite{Campigotto+}, \cite{GasperRahman90}, \cite{Nikiforov+}, -\cite{NoumiMimachi91}, \cite{Stanton80III}, \cite{Stanton84}. - - -\newpage - -\section{Affine $q$-Krawtchouk} -\index{Affine q-Krawtchouk polynomials@Affine $q$-Krawtchouk polynomials} -\index{q-Krawtchouk polynomials@$q$-Krawtchouk polynomials!Affine} -\par\setcounter{equation}{0} - -\subsection*{Basic hypergeometric representation} -\begin{eqnarray} -\label{DefAffqKrawtchouk} -K_n^{Aff}(q^{-x};p,N;q)&=&\qhyp{3}{2}{q^{-n},0,q^{-x}}{pq,q^{-N}}{q}\\ -&=&\frac{(-pq)^nq^{\binom{n}{2}}}{(pq;q)_n}\, -\qhyp{2}{1}{q^{-n},q^{x-N}}{q^{-N}}{\frac{q^{-x}}{p}},\quad n=0,1,2,\ldots,N.\nonumber -\end{eqnarray} - -\subsection*{Orthogonality relation} -\begin{eqnarray} -\label{OrtAffqKrawtchouk} -& &\sum_{x=0}^N\frac{(pq;q)_x(q;q)_N}{(q;q)_x(q;q)_{N-x}}(pq)^{-x}K_m^{Aff}(q^{-x};p,N;q)K_n^{Aff}(q^{-x};p,N;q)\nonumber\\ -& &{}=(pq)^{n-N}\frac{(q;q)_n(q;q)_{N-n}}{(pq;q)_n(q;q)_N}\,\delta_{mn},\quad 01$ -if $p>q^{-1}$. -% -\paragraph{\large\bf KLSadd: History}The origin of the name of the quantum $q$-Krawtchouk polynomials -is by their interpretation -as matrix elements of irreducible corepresentations of (the quantized -function algebra of) the quantum group $SU_q(2)$ considered -with respect to its quantum subgroup $U(1)$. The orthogonality -relation and dual orthogonality relation of these polynomials -are an expression of the unitarity of these corepresentations. -See for instance \myciteKLS{343}{Section 6}. -% - -\subsection*{References} -\cite{AtakRahmanSuslov}, \cite{LChiharaStanton}, \cite{Delsarte}, -\cite{DelsarteGoethals}, \cite{Dunkl78II}, \cite{FlorisKoelink}, -\cite{GasperRahman90}, \cite{Stanton84}. - - -\section{Dual $q$-Krawtchouk} -\index{Dual q-Krawtchouk polynomials@Dual $q$-Krawtchouk polynomials} -\index{q-Krawtchouk polynomials@$q$-Krawtchouk polynomials!Dual} -\par\setcounter{equation}{0} - -\subsection*{Basic hypergeometric representation} -\begin{eqnarray} -\label{DefDualqKrawtchouk} -K_n(\lambda(x);c,N|q)&=&\qhyp{3}{2}{q^{-n},q^{-x},cq^{x-N}}{q^{-N},0}{q}\\ -&=&\frac{(q^{x-N};q)_n}{(q^{-N};q)_nq^{nx}}\, -\qhyp{2}{1}{q^{-n},q^{-x}}{q^{N-x-n+1}}{cq^{x+1}},\quad n=0,1,2,\ldots,N,\nonumber -\end{eqnarray} -where -$$\lambda(x):=q^{-x}+cq^{x-N}.$$ - -\subsection*{Orthogonality relation} -\begin{eqnarray} -\label{OrtDualqKrawtchouk} -& &\sum_{x=0}^N\frac{(cq^{-N},q^{-N};q)_x}{(q,cq;q)_x} -\frac{(1-cq^{2x-N})}{(1-cq^{-N})}c^{-x}q^{x(2N-x)}K_m(\lambda(x))K_n(\lambda(x))\nonumber\\ -& &{}=(c^{-1};q)_N\frac{(q;q)_n}{(q^{-N};q)_n}(cq^{-N})^n\,\delta_{mn},\quad c<0, -\end{eqnarray} -where -$$K_n(\lambda(x)):=K_n(\lambda(x);c,N|q).$$ - -\newpage - -\subsection*{Recurrence relation} -\begin{eqnarray} -\label{RecDualqKrawtchouk} -& &-(1-q^{-x})(1-cq^{x-N})K_n(\lambda(x))\nonumber\\ -& &{}=(1-q^{n-N})K_{n+1}(\lambda(x))\nonumber\\ -& &{}\mathindent{}-\left[(1-q^{n-N})+cq^{-N}(1-q^n)\right]K_n(\lambda(x))\nonumber\\ -& &{}\mathindent\mathindent{}+cq^{-N}(1-q^n)K_{n-1}(\lambda(x)), -\end{eqnarray} -where -$$K_n(\lambda(x)):=K_n(\lambda(x);c,N|q).$$ - -\subsection*{Normalized recurrence relation} -\begin{eqnarray} -\label{NormRecDualqKrawtchouk} -xp_n(x)&=&p_{n+1}(x)+(1+c)q^{n-N}p_n(x)\nonumber\\ -& &{}\mathindent{}+cq^{-N}(1-q^n)(1-q^{n-N-1})p_{n-1}(x), -\end{eqnarray} -where -$$K_n(\lambda(x);c,N|q)=\frac{1}{(q^{-N};q)_n}p_n(\lambda(x)).$$ - -\subsection*{$q$-Difference equation} -\begin{equation} -\label{dvDualqKrawtchouk} -q^{-n}(1-q^n)y(x)=B(x)y(x+1)-\left[B(x)+D(x)\right]y(x)+D(x)y(x-1), -\end{equation} -where -$$y(x)=K_n(\lambda(x);c,N|q)$$ -and -$$\left\{\begin{array}{l}\displaystyle B(x)=\frac{(1-q^{x-N})(1-cq^{x-N})}{(1-cq^{2x-N})(1-cq^{2x-N+1})}\\ -\\ -\displaystyle D(x)=cq^{2x-2N-1}\frac{(1-q^x)(1-cq^x)}{(1-cq^{2x-N-1})(1-cq^{2x-N})}.\end{array}\right.$$ - -\newpage - -\subsection*{Forward shift operator} -\begin{eqnarray} -\label{shift1DualqKrawtchoukI} -& &K_n(\lambda(x+1);c,N|q)-K_n(\lambda(x);c,N|q)\nonumber\\ -& &{}=\frac{q^{-n-x}(1-q^n)(1-cq^{2x-N+1})}{1-q^{-N}} -K_{n-1}(\lambda(x);c,N-1|q) -\end{eqnarray} -or equivalently -\begin{equation} -\label{shift1DualqKrawtchoukII} -\frac{\Delta K_n(\lambda(x);c,N|q)}{\Delta\lambda(x)}= -\frac{q^{-n+1}(1-q^n)}{(1-q)(1-q^{-N})}K_{n-1}(\lambda(x);c,N-1|q). -\end{equation} - -\subsection*{Backward shift operator} -\begin{eqnarray} -\label{shift2DualqKrawtchoukI} -& &(1-q^{x-N-1})(1-cq^{x-N-1})K_n(\lambda(x);c,N|q)\nonumber\\ -& &{}\mathindent{}-cq^{2(x-N-1)}(1-q^x)(1-cq^x)K_n(\lambda(x-1);c,N|q)\nonumber\\ -& &{}=q^x(1-q^{-N-1})(1-cq^{2x-N-1})K_{n+1}(\lambda(x);c,N+1|q) -\end{eqnarray} -or equivalently -\begin{eqnarray} -\label{shift2DualqKrawtchoukII} -& &\frac{\nabla\left[w(x;c,N|q)K_n(\lambda(x);c,N|q)\right]}{\nabla\lambda(x)}\nonumber\\ -& &{}=\frac{1}{(1-q)(1-cq^{-N-1})}w(x;c,N+1|q)K_{n+1}(\lambda(x);c,N+1|q), -\end{eqnarray} -where -$$w(x;c,N|q)=\frac{(q^{-N},cq^{-N};q)_x}{(q,cq;q)_x}c^{-x}q^{2Nx-x(x-1)}.$$ - -\subsection*{Rodrigues-type formula} -\begin{eqnarray} -\label{RodDualqKrawtchouk} -& &w(x;c,N|q)K_n(\lambda(x);c,N|q)\nonumber\\ -& &{}=(1-q)^n(cq^{-N};q)_n\left(\nabla_{\lambda}\right)^n\left[w(x;c,N-n|q)\right], -\end{eqnarray} -where -$$\nabla_{\lambda}:=\frac{\nabla}{\nabla\lambda(x)}.$$ - -\subsection*{Generating function} For $x=0,1,2,\ldots,N$ we have -\begin{equation} -\label{GenDualqKrawtchouk} -(cq^{-N}t;q)_x\cdot (q^{-N}t;q)_{N-x}=\sum_{n=0}^N -\frac{(q^{-N};q)_n}{(q;q)_n}K_n(\lambda(x);c,N|q)t^n. -\end{equation} - -\subsection*{Limit relations} - -\subsubsection*{$q$-Racah $\rightarrow$ Dual $q$-Krawtchouk} -The dual $q$-Krawtchouk polynomials given by (\ref{DefDualqKrawtchouk}) easily -follow from the $q$-Racah polynomials given by (\ref{DefqRacah}) by using the -substitutions $\alpha=\beta=0$, $\gamma q=q^{-N}$ and $\delta=c$: -$$R_n(\mu(x);0,0,q^{-N-1},c|q)=K_n(\lambda(x);c,N|q).$$ -Note that -$$\mu(x)=\lambda(x)=q^{-x}+cq^{x-N}.$$ - -\subsubsection*{Dual $q$-Hahn $\rightarrow$ Dual $q$-Krawtchouk} -The dual $q$-Krawtchouk polynomials given by (\ref{DefDualqKrawtchouk}) can -be obtained from the dual $q$-Hahn polynomials by setting $\delta=c\gamma^{-1}q^{-N-1}$ -in (\ref{DefDualqHahn}) and letting $\gamma\rightarrow 0$: -$$\lim_{\gamma\rightarrow 0} -R_n(\mu(x);\gamma,c\gamma^{-1}q^{-N-1},N|q)=K_n(\lambda(x);c,N|q).$$ - -\subsubsection*{Dual $q$-Krawtchouk $\rightarrow$ Al-Salam-Carlitz~I} -If we set $c=a^{-1}$ in the definition (\ref{DefDualqKrawtchouk}) -of the dual $q$-Krawtchouk polynomials and take the limit -$N\rightarrow\infty$ we simply obtain the Al-Salam-Carlitz~I polynomials -given by (\ref{DefAlSalamCarlitzI}): -\begin{equation} -\lim_{N\rightarrow\infty}K_n(\lambda(x);a^{-1},N|q)= -\left(-\frac{1}{a}\right)^nq^{-\binom{n}{2}}U_n^{(a)}(q^x;q). -\end{equation} -Note that $\lambda(x)=q^{-x}+a^{-1}q^{x-N}$. - -\subsubsection*{Dual $q$-Krawtchouk $\rightarrow$ Krawtchouk} -If we set $c=1-p^{-1}$ in the definition (\ref{DefDualqKrawtchouk}) of the -dual $q$-Krawtchouk polynomials and take the limit $q\rightarrow 1$ we simply find -the Krawtchouk polynomials given by (\ref{DefKrawtchouk}): -\begin{equation} -\lim_{q\rightarrow 1}K_n(\lambda(x);1-p^{-1},N|q)=K_n(x;p,N). -\end{equation} - -\subsection*{Remark} -The dual $q$-Krawtchouk polynomials given by (\ref{DefDualqKrawtchouk}) -and the $q$-Krawtchouk polynomials given by (\ref{DefqKrawtchouk}) are -related in the following way: -$$K_n(q^{-x};p,N;q)=K_x(\lambda(n);-pq^N,N|q)$$ -with -$$\lambda(n)=q^{-n}-pq^n$$ -or -$$K_n(\lambda(x);c,N|q)=K_x(q^{-n};-cq^{-N},N;q)$$ -with -$$\lambda(x)=q^{-x}+cq^{x-N}.$$ - RS: add begin\label{sec14.16} -% -\paragraph{\large\bf KLSadd: $q$-Hypergeometric representation}For $n=0,1,\ldots,N$ -(see (14.16.1)): -\begin{align} -K_n^{\rm Aff}(y;p,N;q) -&=\frac1{(p^{-1}q^{-1};q^{-1})_n}\,\qhyp21{q^{-n},q^{-N}y^{-1}}{q^{-N}}{q,p^{-1}y} -\label{156}\\ -&=\qhyp32{q^{-n},y,0}{q^{-N},pq}{q,q}. -\label{157} -\end{align} -% -\paragraph{\large\bf KLSadd: Self-duality}By \eqref{157}: -\begin{equation} -K_n^{\rm Aff}(q^{-x};p,N;q)=K_x^{\rm Aff}(q^{-n};p,N;q)\qquad -(n,x\in\{0,1,\ldots,N\}). -\label{168} -\end{equation} -% -\paragraph{\large\bf KLSadd: Special values}By \eqref{156} and \mycite{GR}{(II.4)}: -\begin{equation} -K_n^{\rm Aff}(1;p,N;q)=1,\qquad -K_n^{\rm Aff}(q^{-N};p,N;q)=\frac1{((pq)^{-1};q^{-1})_n}\,. -\label{165} -\end{equation} -By \eqref{165} and \eqref{168} we have also -\begin{equation} -K_N^{\rm Aff}(q^{-x};p,N;q)=\frac1{((pq)^{-1};q^{-1})_x}\,. -\label{170} -\end{equation} -% -\paragraph{\large\bf KLSadd: Limit for $q\to1$ to Krawtchouk}\begin{align} -\lim_{q\to1} K_n^{\rm Aff}(1+(1-q)x;p,N;q)&=K_n(x;1-p,N), -\label{162}\\ -\lim_{q\to1} K_n^{\rm Aff}(q^{-x};p,N;q)&=K_n(x;1-p,N). -\label{166} -\end{align} -% -\paragraph{\large\bf KLSadd: A relation between quantum and affine $q$-Krawtchouk}By \eqref{152}, \eqref{156}, \eqref{165} and \eqref{168} -we have for $x\in\{0,1,\ldots,N\}$: -\begin{align} -K_{N-n}^{\rm qtm}(q^{-x};p^{-1}q^{-N-1},N;q) -&=\frac{K_x^{\rm Aff}(q^{-n};p,N;q)}{K_x^{\rm Aff}(q^{-N};p,N;q)} -\label{171}\\ -&=\frac{K_n^{\rm Aff}(q^{-x};p,N;q)}{K_N^{\rm Aff}(q^{-x};p,N;q)}\,. -\label{172} -\end{align} -Formula \eqref{171} is given in \cite[formula after (12)]{K24} -and \cite[(59)]{K25}. -In view of \eqref{164} and \eqref{166} -formula \eqref{172} has \eqref{149} as a limit case for -$q\to 1$. -% -\paragraph{\large\bf KLSadd: Affine $q^{-1}By \eqref{156}, \eqref{165}, -\eqref{154} and \eqref{152} (see also p.505, first formula): -\begin{align} -\frac{K_n^{\rm Aff}(y;p,N;q^{-1})}{K_n^{\rm Aff}(q^N;p,N;q^{-1})} -&=\qhyp21{q^{-n},q^{-N}y}{q^{-N}}{q,p^{-1}q^{n+1}} -\label{158}\\ -&=K_n^{\rm qtm}(q^{-N}y;p^{-1},N;q). -\label{159} -\end{align} -Formula \eqref{159} is equivalent to \eqref{160}. -Just as for \eqref{160}, it tends after suitable substitutions to -\eqref{10} as $q\to1$. - -The orthogonality relation (14.16.2) holds with positive weights for $q>1$ -if $01$, then we have another orthogonality relation given by: -\begin{eqnarray} -\label{OrtContBigqHermite2} -& &\frac{1}{2\pi}\int_{-1}^1\frac{w(x)}{\sqrt{1-x^2}}H_m(x;a|q)H_n(x;a|q)\,dx\nonumber\\ -& &{}\mathindent{}+\sum_{\begin{array}{c}\scriptstyle k\\ \scriptstyle 1-1 -\end{eqnarray} -and -\begin{eqnarray} -\label{OrtqLaguerre2} -& &\sum_{k=-\infty}^{\infty}\frac{q^{k\alpha+k}}{(-cq^k;q)_{\infty}}L_m^{(\alpha)}(cq^k;q)L_n^{(\alpha)}(cq^k;q)\nonumber\\ -& &{}=\frac{(q,-cq^{\alpha+1},-c^{-1}q^{-\alpha};q)_{\infty}} -{(q^{\alpha+1},-c,-c^{-1}q;q)_{\infty}}\frac{(q^{\alpha+1};q)_n}{(q;q)_nq^n}\,\delta_{mn}, -\quad\alpha>-1,\quad c>0. -\end{eqnarray} -For $c=1$ the latter orthogonality relation can also be written as -\begin{eqnarray} -\label{OrtqLaguerre3} -& &\int_0^{\infty}\frac{x^{\alpha}}{(-x;q)_{\infty}}L_m^{(\alpha)}(x;q)L_n^{(\alpha)}(x;q)\,d_qx\nonumber\\ -& &{}=\frac{1-q}{2}\,\frac{(q,-q^{\alpha+1},-q^{-\alpha};q)_{\infty}} -{(q^{\alpha+1},-q,-q;q)_{\infty}}\frac{(q^{\alpha+1};q)_n}{(q;q)_nq^n}\,\delta_{mn},\quad\alpha>-1. -\end{eqnarray} - -\newpage - -\subsection*{Recurrence relation} -\begin{eqnarray} -\label{RecqLaguerre} --q^{2n+\alpha+1}xL_n^{(\alpha)}(x;q)&=&(1-q^{n+1})L_{n+1}^{(\alpha)}(x;q)\nonumber\\ -& &{}\mathindent{}-\left[(1-q^{n+1})+q(1-q^{n+\alpha})\right]L_n^{(\alpha)}(x;q)\nonumber\\ -& &{}\mathindent\mathindent{}+q(1-q^{n+\alpha})L_{n-1}^{(\alpha)}(x;q). -\end{eqnarray} - -\subsection*{Normalized recurrence relation} -\begin{eqnarray} -\label{NormRecqLaguerre} -xp_n(x)&=&p_{n+1}(x)+q^{-2n-\alpha-1}\left[(1-q^{n+1})+q(1-q^{n+\alpha})\right]p_n(x)\nonumber\\ -& &{}\mathindent{}+q^{-4n-2\alpha+1}(1-q^n)(1-q^{n+\alpha})p_{n-1}(x), -\end{eqnarray} -where -$$L_n^{(\alpha)}(x;q)=\frac{(-1)^nq^{n(n+\alpha)}}{(q;q)_n}p_n(x).$$ - -\subsection*{$q$-Difference equation} -\begin{equation} -\label{dvqLaguerre} --q^{\alpha}(1-q^n)xy(x)=q^{\alpha}(1+x)y(qx)-\left[1+q^{\alpha}(1+x)\right]y(x)+y(q^{-1}x), -\end{equation} -where -$$y(x)=L_n^{(\alpha)}(x;q).$$ - -\subsection*{Forward shift operator} -\begin{equation} -\label{shift1qLaguerreI} -L_n^{(\alpha)}(x;q)-L_n^{(\alpha)}(qx;q)=-q^{\alpha+1}xL_{n-1}^{(\alpha+1)}(qx;q) -\end{equation} -or equivalently -\begin{equation} -\label{shift1qLaguerreII} -\mathcal{D}_qL_n^{(\alpha)}(x;q)=-\frac{q^{\alpha+1}}{1-q}L_{n-1}^{(\alpha+1)}(qx;q). -\end{equation} - -\subsection*{Backward shift operator} -\begin{equation} -\label{shift2qLaguerreI} -L_n^{(\alpha)}(x;q)-q^{\alpha}(1+x)L_n^{(\alpha)}(qx;q)=(1-q^{n+1})L_{n+1}^{(\alpha-1)}(x;q) -\end{equation} -or equivalently -\begin{equation} -\label{shift2qLaguerreII} -\mathcal{D}_q\left[w(x;\alpha;q)L_n^{(\alpha)}(x;q)\right]= -\frac{1-q^{n+1}}{1-q}w(x;\alpha-1;q)L_{n+1}^{(\alpha-1)}(x;q), -\end{equation} -where -$$w(x;\alpha;q)=\frac{x^{\alpha}}{(-x;q)_{\infty}}.$$ - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodqLaguerre} -w(x;\alpha;q)L_n^{(\alpha)}(x;q)= -\frac{(1-q)^n}{(q;q)_n}\left(\mathcal{D}_q\right)^n\left[w(x;\alpha+n;q)\right]. -\end{equation} - -\subsection*{Generating functions} -\begin{equation} -\label{GenqLaguerre1} -\frac{1}{(t;q)_{\infty}}\,\qhyp{1}{1}{-x}{0}{q^{\alpha+1}t} -=\sum_{n=0}^{\infty}L_n^{(\alpha)}(x;q)t^n. -\end{equation} - -\begin{equation} -\label{GenqLaguerre2} -\frac{1}{(t;q)_{\infty}}\,\qhyp{0}{1}{-}{q^{\alpha+1}}{-q^{\alpha+1}xt} -=\sum_{n=0}^{\infty}\frac{L_n^{(\alpha)}(x;q)}{(q^{\alpha+1};q)_n}t^n. -\end{equation} - -\begin{equation} -\label{GenqLaguerre3} -(t;q)_{\infty}\cdot\qhyp{0}{2}{-}{q^{\alpha+1},t}{-q^{\alpha+1}xt} -=\sum_{n=0}^{\infty}\frac{(-1)^nq^{\binom{n}{2}}}{(q^{\alpha+1};q)_n}L_n^{(\alpha)}(x;q)t^n. -\end{equation} - -\begin{eqnarray} -\label{GenqLaguerre4} -& &\frac{(\gamma t;q)_{\infty}}{(t;q)_{\infty}}\,\qhyp{1}{2}{\gamma}{q^{\alpha+1},\gamma t}{-q^{\alpha+1}xt}\nonumber\\ -& &{}=\sum_{n=0}^{\infty}\frac{(\gamma;q)_n}{(q^{\alpha+1};q)_n}L_n^{(\alpha)}(x;q)t^n, -\quad\textrm{$\gamma$ arbitrary}. -\end{eqnarray} - -\subsection*{Limit relations} - -\subsubsection*{Little $q$-Jacobi $\rightarrow$ $q$-Laguerre} -If we substitute $a=q^{\alpha}$ and $x\rightarrow -b^{-1}q^{-1}x$ in the definition -(\ref{DefLittleqJacobi}) of the little $q$-Jacobi polynomials and then take the limit -$b\rightarrow -\infty$ we find the $q$-Laguerre polynomials given by (\ref{DefqLaguerre}): -$$\lim_{b\rightarrow -\infty}p_n(-b^{-1}q^{-1}x;q^{\alpha},b|q)= -\frac{(q;q)_n}{(q^{\alpha+1};q)_n}L_n^{(\alpha)}(x;q).$$ - -\subsubsection*{$q$-Meixner $\rightarrow$ $q$-Laguerre} -The $q$-Laguerre polynomials given by (\ref{DefqLaguerre}) can be obtained -from the $q$-Meixner polynomials given by (\ref{DefqMeixner}) by setting -$b=q^{\alpha}$ and $q^{-x}\rightarrow cq^{\alpha}x$ in the definition -(\ref{DefqMeixner}) of the $q$-Meixner polynomials and then taking the limit -$c\rightarrow\infty$: -$$\lim_{c\rightarrow\infty}M_n(cq^{\alpha}x;q^{\alpha},c;q)= -\frac{(q;q)_n}{(q^{\alpha+1};q)_n}L_n^{(\alpha)}(x;q).$$ - -\subsubsection*{$q$-Laguerre $\rightarrow$ Stieltjes-Wigert} -If we set $x\rightarrow xq^{-\alpha}$ in the definition -(\ref{DefqLaguerre}) of the $q$-Laguerre polynomials and take the limit -$\alpha\rightarrow\infty$ we simply obtain the Stieltjes-Wigert polynomials given -by (\ref{DefStieltjesWigert}): -\begin{equation} -\lim_{\alpha\rightarrow\infty}L_n^{(\alpha)}\left(xq^{-\alpha};q\right)=S_n(x;q). -\end{equation} - -\subsubsection*{$q$-Laguerre $\rightarrow$ Laguerre / Charlier} -If we set $x\rightarrow (1-q)x$ in the definition (\ref{DefqLaguerre}) -of the $q$-Laguerre polynomials and take the limit $q\rightarrow 1$ we obtain -the Laguerre polynomials given by (\ref{DefLaguerre}): -\begin{equation} -\lim_{q\rightarrow 1}L_n^{(\alpha)}((1-q)x;q)=L_n^{(\alpha)}(x). -\end{equation} - -If we set $x\rightarrow -q^{-x}$ and $q^{\alpha}=a^{-1}(q-1)^{-1}$ (or -$\alpha=-(\ln q)^{-1}\ln (q-1)a$) in the definition (\ref{DefqLaguerre}) of the -$q$-Laguerre polynomials, multiply by $(q;q)_n$, and take the limit -$q\rightarrow 1$ we obtain the Charlier polynomials given by (\ref{DefCharlier}): -\begin{equation} -\lim_{q\rightarrow 1}\,(q;q)_nL_n^{(\alpha)}(-q^{-x};q)=C_n(x;a), -\end{equation} -where -$$q^{\alpha}=\frac{1}{a(q-1)}\quad\textrm{or}\quad\alpha=-\frac{\ln (q-1)a}{\ln q}.$$ - -\subsection*{Remarks} -The $q$-Laguerre polynomials are sometimes called the -generalized Stieltjes-Wigert polynomials. - -If we replace $q$ by $q^{-1}$ we obtain the little $q$-Laguerre -(or Wall) polynomials given by (\ref{DefLittleqLaguerre}) in the following -way: -$$L_n^{(\alpha)}(x;q^{-1})=\frac{(q^{\alpha+1};q)_n}{(q;q)_nq^{n\alpha}}p_n(-x;q^{\alpha}|q).$$ - -\noindent -The $q$-Laguerre polynomials given by (\ref{DefqLaguerre}) and the $q$-Bessel polynomials -given by (\ref{DefqBessel}) are related in the following way: -$$\frac{y_n(q^x;a;q)}{(q;q)_n}=L_n^{(x-n)}(aq^n;q).$$ - -\noindent -The $q$-Laguerre polynomials given by (\ref{DefqLaguerre}) and the -$q$-Charlier polynomials given by (\ref{DefqCharlier}) are related in the -following way: -$$\frac{C_n(-x;-q^{-\alpha};q)}{(q;q)_n}=L_n^{(\alpha)}(x;q).$$ - -\noindent -Since the Stieltjes and Hamburger moment problems corresponding to the -$q$-Laguerre polynomials are indeterminate there exist many different weight -functions. - RS: add begin\label{sec14.20} -% -\paragraph{\large\bf KLSadd: Notation}Here the little $q$-Laguerre polynomial is denoted by -$p_n(x;a;q)$ instead of -$p_n(x;a\,|\, q)$. -% -\paragraph{\large\bf KLSadd: Re: (14.20.11)}The \RHS\ of this generating function converges for $|xt|<1$. -We can rewrite the \LHS\ by use of the transformation -\begin{equation*} -\qhyp21{0,0}c{q,z}=\frac1{(z;q)_\iy}\,\qhyp01-c{q,cz}. -\end{equation*} -Then we obtain: -\begin{equation} -(t;q)_\iy\,\qhyp21{0,0}{aq}{q,xt} -=\sum_{n=0}^\iy\frac{(-1)^n\,q^{\half n(n-1)}}{(q;q)_n}\, -p_n(x;a;q)\,t^n\qquad(|xt|<1). -\label{35} -\end{equation} -% -\subsubsection*{Expansion of $x^n$} -Divide both sides of \eqref{35} by $(t;q)_\iy$. Then coefficients of the -same power of $t$ on both sides must be equal. We obtain: -\begin{equation} -x^n=(a;q)_n\,\sum_{k=0}^n \frac{(q^{-n};q)_k}{(q;q)_k}\,q^{nk}\,p_k(x;a;q). -\label{36} -\end{equation} -% -\subsubsection*{Quadratic transformations} -Little $q$-Laguerre polynomials $p_n(x;a;q)$ with $a=q^{\pm\half}$ are -related to discrete $q$-Hermite I polynomials $h_n(x;q)$: -\begin{align} -p_n(x^2;q^{-1};q^2)&= -\frac{(-1)^n q^{-n(n-1)}}{(q;q^2)_n}\,h_{2n}(x;q), -\label{28}\\ -xp_n(x^2;q;q^2)&= -\frac{(-1)^n q^{-n(n-1)}}{(q^3;q^2)_n}\,h_{2n+1}(x;q). -\label{29} -\end{align} -% - -\subsection*{References} -\cite{NAlSalam89}, \cite{AlSalam90}, \cite{Askey86}, \cite{Askey89I}, \cite{AskeyWilson85}, -\cite{AtakAtakI}, \cite{ChenIsmailMuttalib}, \cite{Chihara68II}, \cite{Chihara78}, -\cite{Chihara79}, \cite{Chris}, \cite{Exton77}, \cite{GasperRahman90}, \cite{GrunbaumHaine96}, -\cite{Ismail2005I}, \cite{IsmailRahman98}, \cite{Jain95}, \cite{Moak}. - - -\section{$q$-Bessel} -\index{q-Bessel polynomials@$q$-Bessel polynomials} -\par\setcounter{equation}{0} - -\subsection*{Basic hypergeometric representation} -\begin{eqnarray} -\label{DefqBessel} -y_n(x;a;q)&=&\qhyp{2}{1}{q^{-n},-aq^n}{0}{qx}\\ -&=&(q^{-n+1}x;q)_n\cdot\qhyp{1}{1}{q^{-n}}{q^{-n+1}x}{-aq^{n+1}x}\nonumber\\ -&=&\left(-aq^nx\right)^n\cdot\qhyp{2}{1}{q^{-n},x^{-1}}{0}{-\frac{q^{-n+1}}{a}}.\nonumber -\end{eqnarray} - -\newpage - -\subsection*{Orthogonality relation} -\begin{eqnarray} -\label{OrtqBessel} -& &\sum_{k=0}^{\infty}\frac{a^k}{(q;q)_k}q^{\binom{k+1}{2}}y_m(q^k;a;q)y_n(q^k;a;q)\nonumber\\ -& &{}=(q;q)_n(-aq^n;q)_{\infty}\frac{a^nq^{\binom{n+1}{2}}}{(1+aq^{2n})}\,\delta_{mn},\quad a>0. -\end{eqnarray} - -\subsection*{Recurrence relation} -\begin{equation} -\label{RecqBessel} --xy_n(x;a;q)=A_ny_{n+1}(x;a;q)-(A_n+C_n)y_n(x;a;q)+C_ny_{n-1}(x;a;q), -\end{equation} -where -$$\left\{\begin{array}{l}\displaystyle A_n=q^n\frac{(1+aq^n)}{(1+aq^{2n})(1+aq^{2n+1})}\\ -\\ -\displaystyle C_n=aq^{2n-1}\frac{(1-q^n)}{(1+aq^{2n-1})(1+aq^{2n})}.\end{array}\right.$$ - -\subsection*{Normalized recurrence relation} -\begin{equation} -\label{NormRecqBessel} -xp_n(x)=p_{n+1}(x)+(A_n+C_n)p_n(x)+A_{n-1}C_np_{n-1}(x), -\end{equation} -where -$$y_n(x;a;q)=(-1)^nq^{-\binom{n}{2}}(-aq^n;q)_np_n(x).$$ - -\subsection*{$q$-Difference equation} -\begin{eqnarray} -\label{dvqBessel} -& &-q^{-n}(1-q^n)(1+aq^n)xy(x)\nonumber\\ -& &{}=axy(qx)-(ax+1-x)y(x)+(1-x)y(q^{-1}x), -\end{eqnarray} -where -$$y(x)=y_n(x;a;q).$$ - -\newpage - -\subsection*{Forward shift operator} -\begin{equation} -\label{shift1qBesselI} -y_n(x;a;q)-y_n(qx;a;q)=-q^{-n+1}(1-q^n)(1+aq^n)xy_{n-1}(x;aq^2;q) -\end{equation} -or equivalently -\begin{equation} -\label{shift1qBesselII} -\mathcal{D}_qy_n(x;a;q)=-\frac{q^{-n+1}(1-q^n)(1+aq^n)}{1-q}y_{n-1}(x;aq^2;q). -\end{equation} - -\subsection*{Backward shift operator} -\begin{equation} -\label{shift2qBesselI} -aq^{x-1}y_n(q^x;a;q)-(1-q^x)y_n(q^{x-1}x;a;q)=-y_{n+1}(q^x;aq^{-2};q) -\end{equation} -or equivalently -\begin{equation} -\label{shift2qBesselII} -\frac{\nabla\left[w(x;a;q)y_n(q^x;a;q)\right]}{\nabla q^x}= -\frac{q^2}{a(1-q)}w(x;aq^{-2};q)y_{n+1}(q^x;aq^{-2};q), -\end{equation} -where -$$w(x;a;q)=\frac{a^xq^{\binom{x}{2}}}{(q;q)_x}.$$ - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodqBessel} -w(x;a;q)y_n(q^x;a;q)=a^n(1-q)^nq^{n(n-1)}\left(\nabla_q\right)^n\left[w(x;aq^{2n};q)\right], -\end{equation} -where -$$\nabla_q:=\frac{\nabla}{\nabla q^x}.$$ - -\subsection*{Generating functions} -\begin{eqnarray} -\label{GenqBessel1} -& &\qhyp{0}{1}{-}{0}{-aq^{x+1}t}\,\qhyp{2}{0}{q^{-x},0}{-}{q^xt}\nonumber\\ -& &{}=\sum_{n=0}^{\infty}\frac{y_n(q^x;a;q)}{(q;q)_n}t^n,\quad x=0,1,2,\ldots. -\end{eqnarray} - -\begin{equation} -\label{GenqBessel2} -\frac{(t;q)_{\infty}}{(xt;q)_{\infty}}\,\qhyp{1}{3}{xt}{0,0,t}{-aqxt}= -\sum_{n=0}^{\infty}\frac{(-1)^nq^{\binom{n}{2}}}{(q;q)_n}y_n(x;a;q)t^n. -\end{equation} - -\subsection*{Limit relations} - -\subsubsection*{Little $q$-Jacobi $\rightarrow$ $q$-Bessel} -If we set $b\rightarrow -a^{-1}q^{-1}b$ in the definition (\ref{DefLittleqJacobi}) of -the little $q$-Jacobi polynomials and then take the limit $a\rightarrow 0$ we obtain -the $q$-Bessel polynomials given by (\ref{DefqBessel}): -$$\lim_{a\rightarrow 0}p_n(x;a,-a^{-1}q^{-1}b|q)=y_n(x;b;q).$$ - -\subsubsection*{$q$-Krawtchouk $\rightarrow$ $q$-Bessel} -If we set $x\rightarrow N-x$ in the definition (\ref{DefqKrawtchouk}) of the -$q$-Krawtchouk polynomials and then take the limit $N\rightarrow\infty$ we -obtain the $q$-Bessel polynomials given by (\ref{DefqBessel}): -$$\lim_{N\rightarrow\infty}K_n(q^{x-N};p,N;q)=y_n(q^x;p;q).$$ - -\subsubsection*{$q$-Bessel $\rightarrow$ Stieltjes-Wigert} -The Stieltjes-Wigert polynomials given by (\ref{DefStieltjesWigert}) can be obtained -from the $q$-Bessel polynomials by setting $x\rightarrow a^{-1}x$ in the definition -(\ref{DefqBessel}) of the $q$-Bessel polynomials and then taking the limit -$a\rightarrow\infty$. In fact we have -\begin{equation} -\lim_{a\rightarrow\infty}y_n(a^{-1}x;a;q)=(q;q)_nS_n(x;q). -\end{equation} - -\subsubsection*{$q$-Bessel $\rightarrow$ Bessel} -If we set $x\rightarrow -\frac{1}{2}(1-q)^{-1}x$ and $a\rightarrow -q^{a+1}$ in the definition -(\ref{DefqBessel}) of the $q$-Bessel polynomials and take the limit $q\rightarrow 1$ -we find the Bessel polynomials given by (\ref{DefBessel}): -\begin{equation} -\lim_{q\rightarrow 1}y_n(-\textstyle\frac{1}{2}(1-q)^{-1}x;-q^{a+1};q)=y_n(x;a). -\end{equation} - -\subsubsection*{$q$-Bessel $\rightarrow$ Charlier} -If we set $x\rightarrow q^x$ and $a\rightarrow a(1-q)$ in the definition (\ref{DefqBessel}) -of the $q$-Bessel polynomials and take the limit $q\rightarrow 1$ we find the Charlier -polynomials given by (\ref{DefCharlier}): -\begin{equation} -\lim_{q\rightarrow 1}\frac{y_n(q^x;a(1-q);q)}{(q-1)^n}=a^nC_n(x;a). -\end{equation} - -\subsection*{Remark} -In \cite{Koekoek94} and \cite{Koekoek98} these $q$-Bessel polynomials were called -\emph{alternative $q$-Charlier polynomials}. - -\noindent -The $q$-Bessel polynomials given by (\ref{DefqBessel}) and the $q$-Laguerre -polynomials given by (\ref{DefqLaguerre}) are related in the following way: -$$\frac{y_n(q^x;a;q)}{(q;q)_n}=L_n^{(x-n)}(aq^n;q).$$ - -\subsection*{Reference} -\cite{DattaGriffin}. - - -\section{$q$-Charlier}\index{q-Charlier polynomials@$q$-Charlier polynomials} -\par\setcounter{equation}{0} - -\subsection*{Basic hypergeometric representation} -\begin{eqnarray} -\label{DefqCharlier} -C_n(q^{-x};a;q)&=&\qhyp{2}{1}{q^{-n},q^{-x}}{0}{-\frac{q^{n+1}}{a}}\\ -&=&(-a^{-1}q;q)_n\cdot\qhyp{1}{1}{q^{-n}}{-a^{-1}q}{-\frac{q^{n+1-x}}{a}}.\nonumber -\end{eqnarray} - -\subsection*{Orthogonality relation} -\begin{eqnarray} -\label{OrtqCharlier} -& &\sum_{x=0}^{\infty}\frac{a^x}{(q;q)_x}q^{\binom{x}{2}}C_m(q^{-x};a;q)C_n(q^{-x};a;q)\nonumber\\ -& &{}=q^{-n}(-a;q)_{\infty}(-a^{-1}q,q;q)_n\,\delta_{mn},\quad a>0. -\end{eqnarray} - -\newpage - -\subsection*{Recurrence relation} -\begin{eqnarray} -\label{RecqCharlier} -& &q^{2n+1}(1-q^{-x})C_n(q^{-x})\nonumber\\ -& &{}=aC_{n+1}(q^{-x})-\left[a+q(1-q^n)(a+q^n)\right]C_n(q^{-x})\nonumber\\ -& &{}\mathindent{}+q(1-q^n)(a+q^n)C_{n-1}(q^{-x}), -\end{eqnarray} -where -$$C_n(q^{-x}):=C_n(q^{-x};a;q).$$ - -\subsection*{Normalized recurrence relation} -\begin{eqnarray} -\label{NormRecqCharlier} -xp_n(x)&=&p_{n+1}(x)+\left[1+q^{-2n-1}\left\{a+q(1-q^n)(a+q^n)\right\}\right]p_n(x)\nonumber\\ -& &{}\mathindent{}+aq^{-4n+1}(1-q^n)(a+q^n)p_{n-1}(x), -\end{eqnarray} -where -$$C_n(q^{-x};a;q)=\frac{(-1)^nq^{n^2}}{a^n}p_n(q^{-x}).$$ - -\subsection*{$q$-Difference equation} -\begin{equation} -\label{dvqCharlier} -q^ny(x)=aq^xy(x+1)-q^x(a-1)y(x)+(1-q^x)y(x-1), -\end{equation} -where -$$y(x)=C_n(q^{-x};a;q).$$ - -\subsection*{Forward shift operator} -\begin{equation} -\label{shift1qCharlierI} -C_n(q^{-x-1};a;q)-C_n(q^{-x};a;q)=-a^{-1}q^{-x}(1-q^n)C_{n-1}(q^{-x};aq^{-1};q) -\end{equation} -or equivalently -\begin{equation} -\label{shift1qCharlierII} -\frac{\Delta C_n(q^{-x};a;q)}{\Delta q^{-x}}=-\frac{q(1-q^n)}{a(1-q)}C_{n-1}(q^{-x};aq^{-1};q). -\end{equation} - -\newpage - -\subsection*{Backward shift operator} -\begin{equation} -\label{shift2qCharlierI} -C_n(q^{-x};a;q)-a^{-1}q^{-x}(1-q^x)C_n(q^{-x+1};a;q)=C_{n+1}(q^{-x};aq;q) -\end{equation} -or equivalently -\begin{equation} -\label{shift2qCharlierII} -\frac{\nabla\left[w(x;a;q)C_n(q^{-x};a;q)\right]}{\nabla q^{-x}} -=\frac{1}{1-q}w(x;aq;q)C_{n+1}(q^{-x};aq;q), -\end{equation} -where -$$w(x;a;q)=\frac{a^xq^{\binom{x+1}{2}}}{(q;q)_x}.$$ - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodqCharlier} -w(x;a;q)C_n(q^{-x};a;q)=(1-q)^n\left(\nabla_q\right)^n\left[w(x;aq^{-n};q)\right], -\end{equation} -where -$$\nabla_q:=\frac{\nabla}{\nabla q^{-x}}.$$ - -\subsection*{Generating functions} -\begin{equation} -\label{GenqCharlier1} -\frac{1}{(t;q)_{\infty}}\,\qhyp{1}{1}{q^{-x}}{0}{-a^{-1}qt} -=\sum_{n=0}^{\infty}\frac{C_n(q^{-x};a;q)}{(q;q)_n}t^n. -\end{equation} - -\begin{equation} -\label{GenqCharlier2} -\frac{1}{(t;q)_{\infty}}\,\qhyp{0}{1}{-}{-a^{-1}q}{-a^{-1}q^{-x+1}t} -=\sum_{n=0}^{\infty}\frac{C_n(q^{-x};a;q)}{(-a^{-1}q,q;q)_n}t^n. -\end{equation} - -\subsection*{Limit relations} - -\subsubsection*{$q$-Meixner $\rightarrow$ $q$-Charlier} -The $q$-Charlier polynomials given by (\ref{DefqCharlier}) can easily be -obtained from the $q$-Meixner given by (\ref{DefqMeixner}) by setting $b=0$ -in the definition (\ref{DefqMeixner}) of the $q$-Meixner polynomials: -\begin{equation} -M_n(x;0,c;q)=C_n(x;c;q). -\end{equation} - -\subsubsection*{$q$-Krawtchouk $\rightarrow$ $q$-Charlier} -By setting $p=a^{-1}q^{-N}$ in the definition (\ref{DefqKrawtchouk}) of the -$q$-Krawtchouk polynomials and then taking the limit $N\rightarrow\infty$ we -obtain the $q$-Charlier polynomials given by (\ref{DefqCharlier}): -$$\lim_{N\rightarrow\infty}K_n(q^{-x};a^{-1}q^{-N},N;q)=C_n(q^{-x};a;q).$$ - -\subsubsection*{$q$-Charlier $\rightarrow$ Stieltjes-Wigert} -If we set $q^{-x}\rightarrow ax$ in the definition (\ref{DefqCharlier}) of the -$q$-Charlier polynomials and take the limit $a\rightarrow\infty$ we obtain -the Stieltjes-Wigert polynomials given by (\ref{DefStieltjesWigert}) in the -following way: -\begin{equation} -\lim_{a\rightarrow\infty}C_n(ax;a;q)=(q;q)_nS_n(x;q). -\end{equation} - -\subsubsection*{$q$-Charlier $\rightarrow$ Charlier} -If we set $a\rightarrow a(1-q)$ in the definition (\ref{DefqCharlier}) -of the $q$-Charlier polynomials and take the limit $q\rightarrow 1$ we obtain the -Charlier polynomials given by (\ref{DefCharlier}): -\begin{equation} -\lim_{q\rightarrow 1}C_n(q^{-x};a(1-q);q)=C_n(x;a). -\end{equation} - -\subsection*{Remark} -The $q$-Charlier polynomials given by (\ref{DefqCharlier}) and the -$q$-Laguerre polynomials given by (\ref{DefqLaguerre}) are related in the -following way: -$$\frac{C_n(-x;-q^{-\alpha};q)}{(q;q)_n}=L_n^{(\alpha)}(x;q).$$ - -\subsection*{References} -\cite{AlvarezRonveaux}, \cite{AtakRahmanSuslov}, \cite{GasperRahman90}, -\cite{Hahn}, \cite{Koelink96III}, \cite{Nikiforov+}, \cite{Zeng95}. - - -\section{Al-Salam-Carlitz~I}\index{Al-Salam-Carlitz~I polynomials} -\par\setcounter{equation}{0} - -\subsection*{Basic hypergeometric representation} -\begin{equation} -\label{DefAlSalamCarlitzI} -U_n^{(a)}(x;q)=(-a)^nq^{\binom{n}{2}}\,\qhyp{2}{1}{q^{-n},x^{-1}}{0}{\frac{qx}{a}}. -\end{equation} - -\subsection*{Orthogonality relation} -\begin{eqnarray} -\label{OrtAlSalamCarlitzI} -& &\int_a^1(qx,a^{-1}qx;q)_{\infty}U_m^{(a)}(x;q)U_n^{(a)}(x;q)\,d_qx\nonumber\\ -& &{}=(-a)^n(1-q)(q;q)_n(q,a,a^{-1}q;q)_{\infty}q^{\binom{n}{2}}\,\delta_{mn},\quad a<0. -\end{eqnarray} - -\subsection*{Recurrence relation} -\begin{equation} -\label{RecAlSalamCarlitzI} -xU_n^{(a)}(x;q)=U_{n+1}^{(a)}(x;q)+(a+1)q^nU_n^{(a)}(x;q) --aq^{n-1}(1-q^n)U_{n-1}^{(a)}(x;q). -\end{equation} - -\subsection*{Normalized recurrence relation} -\begin{equation} -\label{NormRecAlSalamCarlitzI} -xp_n(x)=p_{n+1}(x)+(a+1)q^np_n(x)-aq^{n-1}(1-q^n)p_{n-1}(x), -\end{equation} -where -$$U_n^{(a)}(x;q)=p_n(x).$$ - -\subsection*{$q$-Difference equation} -\begin{eqnarray} -\label{dvAlSalamCarlitzI} -(1-q^n)x^2y(x)&=&aq^{n-1}y(qx)-\left[aq^{n-1}+q^n(1-x)(a-x)\right]y(x)\nonumber\\ -& &{}\mathindent{}+q^n(1-x)(a-x)y(q^{-1}x), -\end{eqnarray} -where -$$y(x)=U_n^{(a)}(x;q).$$ - -\subsection*{Forward shift operator} -\begin{equation} -\label{shift1AlSalamCarlitzI-I} -U_n^{(a)}(x;q)-U_n^{(a)}(qx;q)=(1-q^n)xU_{n-1}^{(a)}(x;q) -\end{equation} -or equivalently -\begin{equation} -\label{shift1AlSalamCarlitzI-II} -\mathcal{D}_qU_n^{(a)}(x;q)=\frac{1-q^n}{1-q}U_{n-1}^{(a)}(x;q). -\end{equation} - -\subsection*{Backward shift operator} -\begin{equation} -\label{shift2AlSalamCarlitzI-I} -aU_n^{(a)}(x;q)-(1-x)(a-x)U_n^{(a)}(q^{-1}x;q)=-q^{-n}xU_{n+1}^{(a)}(x;q) -\end{equation} -or equivalently -\begin{equation} -\label{shift2AlSalamCarlitzI-II} -\mathcal{D}_{q^{-1}}\left[w(x;a;q)U_n^{(a)}(x;q)\right]= -\frac{q^{-n+1}}{a(1-q)}w(x;a;q)U_{n+1}^{(a)}(x;q), -\end{equation} -where -$$w(x;a;q)=(qx,a^{-1}qx;q)_{\infty}.$$ - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodAlSalamCarlitzI} -w(x;a;q)U_n^{(a)}(x;q)=a^nq^{\frac{1}{2}n(n-3)}(1-q)^n -\left(\mathcal{D}_{q^{-1}}\right)^n\left[w(x;a;q)\right]. -\end{equation} - -\subsection*{Generating function} -\begin{equation} -\label{GenAlSalamCarlitzI} -\frac{(t,at;q)_{\infty}}{(xt;q)_{\infty}}= -\sum_{n=0}^{\infty}\frac{U_n^{(a)}(x;q)}{(q;q)_n}t^n. -\end{equation} - -\subsection*{Limit relations} - -\subsubsection*{Big $q$-Laguerre $\rightarrow$ Al-Salam-Carlitz~I} -If we set $x\rightarrow aqx$ and $b\rightarrow ab$ in the definition -(\ref{DefBigqLaguerre}) of the big $q$-Laguerre polynomials and take the -limit $a\rightarrow 0$ we obtain the Al-Salam-Carlitz~I polynomials given by -(\ref{DefAlSalamCarlitzI}): -$$\lim_{a\rightarrow 0}\frac{P_n(aqx;a,ab;q)}{a^n}=q^nU_n^{(b)}(x;q).$$ - -\subsubsection*{Dual $q$-Krawtchouk $\rightarrow$ Al-Salam-Carlitz~I} -If we set $c=a^{-1}$ in the definition (\ref{DefDualqKrawtchouk}) -of the dual $q$-Krawtchouk polynomials and take the limit -$N\rightarrow\infty$ we simply obtain the Al-Salam-Carlitz~I polynomials -given by (\ref{DefAlSalamCarlitzI}): -$$\lim_{N\rightarrow\infty}K_n(\lambda(x);a^{-1},N|q)= -\left(-\frac{1}{a}\right)^nq^{-\binom{n}{2}}U_n^{(a)}(q^x;q).$$ -Note that $\lambda(x)=q^{-x}+a^{-1}q^{x-N}$. - -\subsubsection*{Al-Salam-Carlitz~I $\rightarrow$ Discrete $q$-Hermite~I} -The discrete $q$-Hermite~I polynomials given by -(\ref{DefDiscreteqHermiteI}) can easily be obtained from the -Al-Salam-Carlitz~I polynomials given by (\ref{DefAlSalamCarlitzI}) by the -substitution $a=-1$: -\begin{equation} -U_n^{(-1)}(x;q)=h_n(x;q). -\end{equation} - -\subsubsection*{Al-Salam-Carlitz~I $\rightarrow$ Charlier / Hermite} -If we set $a\rightarrow a(q-1)$ and $x\rightarrow q^x$ in the definition -(\ref{DefAlSalamCarlitzI}) of the Al-Salam-Carlitz~I polynomials and take the -limit $q\rightarrow 1$ after dividing by $a^n(1-q)^n$ we obtain the Charlier -polynomials given by (\ref{DefCharlier}): -\begin{equation} -\lim_{q\rightarrow 1}\frac{U_n^{(a(q-1))}(q^x;q)}{(1-q)^n}=a^nC_n(x;a). -\end{equation} - -If we set $x\rightarrow x\sqrt{1-q^2}$ and $a\rightarrow a\sqrt{1-q^2}-1$ -in the definition (\ref{DefAlSalamCarlitzI}) of the Al-Salam-Carlitz~I -polynomials, divide by $(1-q^2)^{\frac{n}{2}}$, and let $q$ tend to $1$ we -obtain the Hermite polynomials given by (\ref{DefHermite}) with shifted -argument. In fact we have -\begin{equation} -\lim_{q\rightarrow 1}\frac{U_n^{(a\sqrt{1-q^2}-1)}(x\sqrt{1-q^2};q)} -{(1-q^2)^{\frac{n}{2}}}=\frac{H_n(x-a)}{2^n}. -\end{equation} - -\subsection*{Remark} -The Al-Salam-Carlitz~I polynomials are related to the Al-Salam-Carlitz~II -polynomials given by (\ref{DefAlSalamCarlitzII}) in the following way: -$$U_n^{(a)}(x;q^{-1})=V_n^{(a)}(x;q).$$ - -\subsection*{References} -\cite{AlSalam90}, \cite{AlSalamCarlitz}, \cite{AlSalamChihara76}, \cite{AskeySuslovII}, -\cite{AtakRahmanSuslov}, \cite{Chihara68II}, \cite{Chihara78}, \cite{DattaGriffin}, -\cite{Dehesa}, \cite{DohaAhmed2005}, \cite{GasperRahman90}, \cite{Ismail85I}, -\cite{IsmailMuldoon}, \cite{Kim}, \cite{Zeng95}. - - -\section{Al-Salam-Carlitz~II}\index{Al-Salam-Carlitz~II polynomials} -\par\setcounter{equation}{0} - -\subsection*{Basic hypergeometric representation} -\begin{equation} -\label{DefAlSalamCarlitzII} -V_n^{(a)}(x;q)= -(-a)^nq^{-\binom{n}{2}}\,\qhyp{2}{0}{q^{-n},x}{-}{\frac{q^n}{a}}. -\end{equation} - -\subsection*{Orthogonality relation} -\begin{eqnarray} -\label{OrtAlSalamCarlitzII} -& &\sum_{k=0}^{\infty}\frac{q^{k^2}a^k}{(q;q)_k(aq;q)_k} -V_m^{(a)}(q^{-k};q)V_n^{(a)}(q^{-k};q)\nonumber\\ -& &{}=\frac{(q;q)_na^n}{(aq;q)_{\infty}q^{n^2}}\,\delta_{mn},\quad 00,\quad\textrm{with}\quad -\gamma^2=-\frac{1}{2\ln q}.$$ - RS: add begin\label{sec14.21} -% -\paragraph{\large\bf KLSadd: Notation}Here the $q$-Laguerre polynomial is denoted by $L_n^\al(x;q)$ instead of -$L_n^{(\al)}(x;q)$. -% -\subsubsection*{Orthogonality relation} -(14.21.2) can be rewritten with simplified \RHS: -\begin{equation} -\int_0^\iy L_m^{\al}(x;q)\,L_n^{\al}(x;q)\,\frac{x^\al}{(-x;q)_\iy}\,dx=h_n\,\de_{m,n} -\qquad(\al>-1) -\label{119} -\end{equation} -with -\begin{equation} -\frac{h_n}{h_0}=\frac{(q^{\al+1};q)_n}{(q;q)_n q^n},\qquad -h_0=-\,\frac{(q^{-\al};q)_\iy}{(q;q)_\iy}\,\frac\pi{\sin(\pi\al)}\,. -\label{120} -\end{equation} -The expression for $h_0$ (which is Askey's $q$-gamma evaluation -\cite[(4.2)]{K16}) -should be interpreted by continuity in $\al$ for -$\al\in\Znonneg$. -Explicitly we can write -\begin{equation} -h_n=q^{-\half\al(\al+1)}\,(q;q)_\al\,\log(q^{-1})\qquad(\al\in\Znonneg). -\label{121} -\end{equation} -% -\subsubsection*{Expansion of $x^n$} -\begin{equation} -x^n=q^{-\half n(n+2\al+1)}\,(q^{\al+1};q)_n\, -\sum_{k=0}^n\frac{(q^{-n};q)_k}{(q^{\al+1};q)_k}\,q^k\,L_k^\al(x;q). -\label{37} -\end{equation} -This follows from \eqref{36} by the equality given in the Remark at the end -of \S14.20. Alternatively, it can be derived in the same way as \eqref{36} -from the generating function (14.21.14). -% -\subsubsection*{Quadratic transformations} -$q$-Laguerre polynomials $L_n^\al(x;q)$ with $\al=\pm\half$ are -related to discrete $q$-Hermite II polynomials $\wt h_n(x;q)$: -\begin{align} -L_n^{-1/2}(x^2;q^2)&= -\frac{(-1)^n q^{2n^2-n}}{(q^2;q^2)_n}\,\wt h_{2n}(x;q), -\label{38}\\ -xL_n^{1/2}(x^2;q^2)&= -\frac{(-1)^n q^{2n^2+n}}{(q^2;q^2)_n}\,\wt h_{2n+1}(x;q). -\label{39} -\end{align} -These follows from \eqref{28} and \eqref{29}, respectively, by applying -the equalities given in the Remarks at the end of \S14.20 and \S14.28. -% - -\subsection*{References} -\cite{Askey86}, \cite{Askey89I}, \cite{AtakAtakIII}, \cite{Chihara70}, \cite{Chihara78}, -\cite{Dehesa}, \cite{Ismail2005I}, \cite{Nikiforov+}, \cite{Stieltjes}, \cite{Szego75}, -\cite{ValentAssche}, \cite{Wigert}. - - -\section{Discrete $q$-Hermite~I} -\index{Discrete q-Hermite~I polynomials@Discrete $q$-Hermite~I polynomials} -\index{q-Hermite~I polynomials@$q$-Hermite~I polynomials!Discrete} -\par\setcounter{equation}{0} - -\subsection*{Basic hypergeometric representation} -The discrete $q$-Hermite~I polynomials are Al-Salam-Carlitz~I polynomials -with $a=-1$: -\begin{eqnarray} -\label{DefDiscreteqHermiteI} -h_n(x;q)=U_n^{(-1)}(x;q)&=&q^{\binom{n}{2}}\,\qhyp{2}{1}{q^{-n},x^{-1}}{0}{-qx}\\ -&=&x^n\qhypK{2}{0}{q^{-n},q^{-n+1}}{0}{q^2,\frac{q^{2n-1}}{x^2}}\nonumber -\end{eqnarray} - -\subsection*{Orthogonality relation} -\begin{eqnarray} -\label{OrtDiscreteqHermiteI} -& &\int_{-1}^1(qx,-qx;q)_{\infty}h_m(x;q)h_n(x;q)\,d_qx\nonumber\\ -& &{}=(1-q)(q;q)_n(q,-1,-q;q)_{\infty}q^{\binom{n}{2}}\,\delta_{mn}. -\end{eqnarray} - -\subsection*{Recurrence relation} -\begin{equation} -\label{RecDiscreteqHermiteI} -xh_n(x;q)=h_{n+1}(x;q)+q^{n-1}(1-q^n)h_{n-1}(x;q). -\end{equation} - -\subsection*{Normalized recurrence relation} -\begin{equation} -\label{NormRecDiscreteqHermiteI} -xp_n(x)=p_{n+1}(x)+q^{n-1}(1-q^n)p_{n-1}(x), -\end{equation} -where -$$h_n(x;q)=p_n(x).$$ - -\subsection*{$q$-Difference equation} -\begin{equation} -\label{dvDiscreteqHermiteI} --q^{-n+1}x^2y(x)=y(qx)-(1+q)y(x)+q(1-x^2)y(q^{-1}x), -\end{equation} -where -$$y(x)=h_n(x;q).$$ - -\subsection*{Forward shift operator} -\begin{equation} -\label{shift1DiscreteqHermiteI-I} -h_n(x;q)-h_n(qx;q)=(1-q^n)xh_{n-1}(x;q) -\end{equation} -or equivalently -\begin{equation} -\label{shift1DiscreteqHermiteI-II} -\mathcal{D}_qh_n(x;q)=\frac{1-q^n}{1-q}h_{n-1}(x;q). -\end{equation} - -\subsection*{Backward shift operator} -\begin{equation} -\label{shift2DiscreteqHermiteI-I} -h_n(x;q)-(1-x^2)h_n(q^{-1}x;q)=q^{-n}xh_{n+1}(x;q) -\end{equation} -or equivalently -\begin{equation} -\label{shift2DiscreteqHermiteI-II} -\mathcal{D}_{q^{-1}}\left[w(x;q)h_n(x;q)\right] -=-\frac{q^{-n+1}}{1-q}w(x;q)h_{n+1}(x;q), -\end{equation} -where -$$w(x;q)=(qx,-qx;q)_{\infty}.$$ - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodDiscreteqHermiteI} -w(x;q)h_n(x;q)=(q-1)^nq^{\frac{1}{2}n(n-3)} -\left(\mathcal{D}_{q^{-1}}\right)^n\left[w(x;q)\right]. -\end{equation} - -\subsection*{Generating function} -\begin{equation} -\label{GenDiscreteqHermiteI} -\frac{(t^2;q^2)_{\infty}}{(xt;q)_{\infty}}= -\sum_{n=0}^{\infty}\frac{h_n(x;q)}{(q;q)_n}t^n. -\end{equation} - -\subsection*{Limit relations} - -\subsubsection*{Al-Salam-Carlitz~I $\rightarrow$ Discrete $q$-Hermite~I} -The discrete $q$-Hermite~I polynomials given by -(\ref{DefDiscreteqHermiteI}) can easily be obtained from the -Al-Salam-Carlitz~I polynomials given by (\ref{DefAlSalamCarlitzI}) by the -substitution $a=-1$: -$$U_n^{(-1)}(x;q)=h_n(x;q).$$ - -\subsubsection*{Discrete $q$-Hermite~I $\rightarrow$ Hermite} -The Hermite polynomials given by (\ref{DefHermite}) can be found -from the discrete $q$-Hermite~I polynomials given by -(\ref{DefDiscreteqHermiteI}) in the following way: -\begin{equation} -\lim_{q\rightarrow 1}\frac{h_n(x\sqrt{1-q^2};q)} -{(1-q^2)^{\frac{n}{2}}}=\frac{H_n(x)}{2^n}. -\end{equation} - -\subsection*{Remark} The discrete $q$-Hermite~I polynomials are related to the -discrete $q$-Hermite~II polynomials given by (\ref{DefDiscreteqHermiteII}) -in the following way: -$$h_n(ix;q^{-1})=i^n{\tilde h}_n(x;q).$$ - -\subsection*{References} -\cite{AlSalam90}, \cite{AlSalamCarlitz}, \cite{AtakRahmanSuslov}, -\cite{BergIsmail}, \cite{BustozIsmail82}, \cite{GasperRahman90}, \cite{Hahn}, -\cite{Koorn97}. - - -\newpage - -\section{Discrete $q$-Hermite~II} -\index{Discrete q-Hermite~II polynomials@Discrete $q$-Hermite~II polynomials} -\index{q-Hermite~II polynomials@$q$-Hermite~II polynomials!Discrete} -\par\setcounter{equation}{0} - -\subsection*{Basic hypergeometric representation} -The discrete $q$-Hermite~II polynomials are Al-Salam-Carlitz~II polynomials -with $a=-1$: -\begin{eqnarray} -\label{DefDiscreteqHermiteII} -{\tilde h}_n(x;q)=i^{-n}V_n^{(-1)}(ix;q) -&=&i^{-n}q^{-\binom{n}{2}}\,\qhyp{2}{0}{q^{-n},ix}{-}{-q^n}\\ -&=&x^n -\qhypK{2}{1}{q^{-n},q^{-n+1}}{0}{q^2,-\frac{q^2}{x^2}}\nonumber -\end{eqnarray} - -\subsection*{Orthogonality relation} -\begin{eqnarray} -\label{OrtDiscreteqHermiteIIa} -& &\sum_{k=-\infty}^{\infty}\left[{\tilde h}_m(cq^k;q) -{\tilde h}_n(cq^k;q)+{\tilde h}_m(-cq^k;q){\tilde h}_n(-cq^k;q)\right] -w(cq^k;q)q^k\nonumber\\ -& &{}=2\frac{(q^2,-c^2q,-c^{-2}q;q^2)_{\infty}}{(q,-c^2,-c^{-2}q^2;q^2)_{\infty}} -\frac{(q;q)_n}{q^{n^2}}\,\delta_{mn},\quad c>0, -\end{eqnarray} -where -$$w(x;q)=\frac{1}{(ix,-ix;q)_{\infty}}=\frac{1}{(-x^2;q^2)_{\infty}}.$$ -For $c=1$ this orthogonality relation can also be written as -\begin{equation} -\label{OrtDiscreteqHermiteIIb} -\int_{-\infty}^{\infty}\frac{{\tilde h}_m(x;q){\tilde h}_n(x;q)}{(-x^2;q^2)_{\infty}}\,d_qx -=\frac{\left(q^2,-q,-q;q^2\right)_{\infty}}{\left(q^3,-q^2,-q^2;q^2\right)_{\infty}} -\frac{(q;q)_n}{q^{n^2}}\,\delta_{mn}. -\end{equation} - -\subsection*{Recurrence relation} -\begin{equation} -\label{RecDiscreteqHermiteII} -x{\tilde h}_n(x;q)={\tilde h}_{n+1}(x;q)+q^{-2n+1}(1-q^n){\tilde h}_{n-1}(x;q). -\end{equation} - -\subsection*{Normalized recurrence relation} -\begin{equation} -\label{NormRecDiscreteqHermiteII} -xp_n(x)=p_{n+1}(x)+q^{-2n+1}(1-q^n)p_{n-1}(x), -\end{equation} -where -$${\tilde h}_n(x;q)=p_n(x).$$ - -\subsection*{$q$-Difference equation} -\begin{eqnarray} -\label{dvDiscreteqHermiteII} -& &-(1-q^n)x^2{\tilde h}_n(x;q)\nonumber\\ -& &{}=(1+x^2){\tilde h}_n(qx;q)-(1+x^2+q){\tilde h}_n(x;q)+q{\tilde h}_n(q^{-1}x;q). -\end{eqnarray} - -\subsection*{Forward shift operator} -\begin{equation} -\label{shift1DiscreteqHermiteII-I} -\tilde{h}_n(x;q)-\tilde{h}_n(qx;q)=q^{-n+1}(1-q^n)x\tilde{h}_{n-1}(qx;q) -\end{equation} -or equivalently -\begin{equation} -\label{shift1DiscreteqHermiteII-II} -\mathcal{D}_q\tilde{h}_n(x;q)=\frac{q^{-n+1}(1-q^n)}{1-q}\tilde{h}_{n-1}(qx;q). -\end{equation} - -\subsection*{Backward shift operator} -\begin{equation} -\label{shift2DiscreteqHermiteII-I} -\tilde{h}_n(x;q)-(1+x^2)\tilde{h}_n(qx;q)=-q^nx\tilde{h}_{n+1}(x;q) -\end{equation} -or equivalently -\begin{equation} -\label{shift2DiscreteqHermiteII-II} -\mathcal{D}_q\left[w(x;q)\tilde{h}_n(x;q)\right] -=-\frac{q^n}{1-q}w(x;q)\tilde{h}_{n+1}(x;q). -\end{equation} - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodDiscreteqHermiteII} -w(x;q)\tilde{h}_n(x;q)=(q-1)^nq^{-\binom{n}{2}} -\left(\mathcal{D}_q\right)^n\left[w(x;q)\right]. -\end{equation} - -\subsection*{Generating functions} -\begin{equation} -\label{GenDiscreteqHermiteII1} -\frac{(-xt;q)_{\infty}}{(-t^2;q^2)_{\infty}}=\sum_{n=0}^{\infty} -\frac{q^{\binom{n}{2}}}{(q;q)_n}{\tilde h}_n(x;q)t^n. -\end{equation} - -\begin{equation} -\label{GenDiscreteqHermiteII2} -(-it;q)_{\infty}\cdot\qhyp{1}{1}{ix}{-it}{it}=\sum_{n=0}^{\infty} -\frac{(-1)^nq^{n(n-1)}}{(q;q)_n}{\tilde h}_n(x;q)t^n. -\end{equation} - -\subsection*{Limit relations} - -\subsubsection*{Al-Salam-Carlitz~II $\rightarrow$ Discrete $q$-Hermite~II} -The discrete $q$-Hermite~II polynomials given by -(\ref{DefDiscreteqHermiteII}) follow from the Al-Salam-Carlitz~II -polynomials given by (\ref{DefAlSalamCarlitzII}) by the substitution $a=-1$ -in the following way: -$$i^{-n}V_n^{(-1)}(ix;q)={\tilde h}_n(x;q).$$ - -\subsubsection*{Discrete $q$-Hermite~II $\rightarrow$ Hermite} -The Hermite polynomials given by (\ref{DefHermite}) can be found -from the discrete $q$-Hermite~II polynomials given by -(\ref{DefDiscreteqHermiteII}) in the following way: -\begin{equation} -\lim_{q\rightarrow 1}\frac{{\tilde h}_n(x\sqrt{1-q^2};q)} -{(1-q^2)^{\frac{n}{2}}}=\frac{H_n(x)}{2^n}. -\end{equation} - -\subsection*{Remark} The discrete $q$-Hermite~II polynomials are related to the -discrete $q$-Hermite~I polynomials given by (\ref{DefDiscreteqHermiteI}) -in the following way: -$${\tilde h}_n(x;q^{-1})=i^{-n}h_n(ix;q).$$ - -\subsection*{References} -\cite{BergIsmail}, \cite{Koorn97}. - -\end{document} diff --git a/KLSadd_insertion/tempchap09.tex b/KLSadd_insertion/tempchap09.tex deleted file mode 100644 index 71f90cb..0000000 --- a/KLSadd_insertion/tempchap09.tex +++ /dev/null @@ -1,3926 +0,0 @@ -\documentclass[envcountchap,graybox]{svmono} - -\addtolength{\textwidth}{1mm} - -\usepackage{amsmath,amssymb} - -\usepackage{amsfonts} -%\usepackage{breqn} -\usepackage{DLMFmath} -\usepackage{DRMFfcns} - -\usepackage{mathptmx} -\usepackage{helvet} -\usepackage{courier} - -\usepackage{makeidx} -\usepackage{graphicx} - -\usepackage{multicol} -\usepackage[bottom]{footmisc} - -\makeindex - -\def\bibname{Bibliography} -\def\refname{Bibliography} - -\def\theequation{\thesection.\arabic{equation}} - -\smartqed - -\let\corollary=\undefined -\spnewtheorem{corollary}[theorem]{Corollary}{\bfseries}{\itshape} - -\newcounter{rom} - -\newcommand{\hyp}[5]{\mbox{}_{#1}{F}_{#2} -\left(\genfrac{}{}{0pt}{}{#3}{#4}\,;\,#5\right)} - -\newcommand{\qhyp}[5]{\mbox{}_{#1}{\phi}_{#2} -\left(\genfrac{}{}{0pt}{}{#3}{#4}\,;\,q,\,#5\right)} - -\newcommand{\mathindent}{\hspace{7.5mm}} - -\newcommand{\e}{\textrm{e}} - -\renewcommand{\E}{\textrm{E}} - -\renewcommand{\textfraction}{-1} - -\renewcommand{\Gamma}{\varGamma} - -\renewcommand{\leftlegendglue}{\hfil} - -\settowidth{\tocchpnum}{14\enspace} -\settowidth{\tocsecnum}{14.30\enspace} -\settowidth{\tocsubsecnum}{14.12.1\enspace} - -\makeatletter -\def\cleartoversopage{\clearpage\if@twoside\ifodd\c@page - \hbox{}\newpage\if@twocolumn\hbox{}\newpage\fi - \else\fi\fi} - -\newcommand{\clearemptyversopage}{ - \clearpage{\pagestyle{empty}\cleartoversopage}} -\makeatother - -\oddsidemargin -1.5cm -\topmargin -2.0cm -\textwidth 16.3cm -\textheight 25cm - -\begin{document} - -\author{Roelof Koekoek\\[2.5mm]Peter A. Lesky\\[2.5mm]Ren\'e F. Swarttouw} -\title{Hypergeometric orthogonal polynomials and their\\$q$-analogues} -\subtitle{-- Monograph --} -\maketitle - -\frontmatter - -\large - -\addtocounter{chapter}{8} -\pagenumbering{roman} -\chapter{Hypergeometric orthogonal polynomials} -\label{HyperOrtPol} - - -In this chapter we deal with all families of hypergeometric orthogonal polynomials -appearing in the Askey scheme on page~\pageref{scheme}. For each family of orthogonal -polynomials we state the most important properties such as a representation as a -hypergeometric function, orthogonality relation(s), the three-term recurrence relation, -the second-order differential or difference equation, the forward shift (or degree lowering) -and backward shift (or degree raising) operator, a Rodrigues-type formula and some generating -functions. In each case we use the notation which seems to be most common in the literature. -Moreover, in each case we mention the connection between various families by stating the -appropriate limit relations. See also \cite{Terwilliger2006} for an algebraic approach of -this Askey scheme and \cite{TemmeLopez2001} for a view from asymptotic analysis. -For notations the reader is referred to chapter~\ref{Definitions}. - -\section{Wilson}\index{Wilson polynomials} - -\par - -\subsection*{Hypergeometric representation} -\begin{eqnarray} -\label{DefWilson} -& &\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\nonumber\\ -& &{}=\hyp{4}{3}{-n,n+a+b+c+d-1,a+ix,a-ix}{a+b,a+c,a+d}{1}. -\end{eqnarray} - -\newpage - -\subsection*{Orthogonality relation} -If $Re(a,b,c,d)>0$ and non-real parameters occur in conjugate pairs, then -\begin{eqnarray} -\label{OrthIWilson} -& &\frac{1}{2\pi}\int_0^{\infty} -\left|\frac{\Gamma(a+ix)\Gamma(b+ix)\Gamma(c+ix)\Gamma(d+ix)}{\Gamma(2ix)}\right|^2\nonumber\\ -& &{}\mathindent\times W_m(x^2;a,b,c,d)W_n(x^2;a,b,c,d)\,dx\nonumber\\ -& &{}=\frac{\Gamma(n+a+b)\cdots\Gamma(n+c+d)}{\Gamma(2n+a+b+c+d)}(n+a+b+c+d-1)_nn!\,\delta_{mn}, -\end{eqnarray} -where -\begin{eqnarray*} -& &\Gamma(n+a+b)\cdots\Gamma(n+c+d)\\ -& &{}=\Gamma(n+a+b)\Gamma(n+a+c)\Gamma(n+a+d)\Gamma(n+b+c)\Gamma(n+b+d)\Gamma(n+c+d). -\end{eqnarray*} -If $a<0$ and $a+b$, $a+c$, $a+d$ are positive or a pair of complex conjugates -occur with positive real parts, then -\begin{eqnarray} -\label{OrtIIWilson} -& &\frac{1}{2\pi}\int_0^{\infty}\left|\frac{\Gamma(a+ix)\Gamma(b+ix)\Gamma(c+ix)\Gamma(d+ix)}{\Gamma(2ix)}\right|^2\nonumber\\ -& &{}\mathindent\mathindent\times W_m(x^2;a,b,c,d)W_n(x^2;a,b,c,d)\,dx\nonumber\\ -& &{}\mathindent{}+\frac{\Gamma(a+b)\Gamma(a+c)\Gamma(a+d)\Gamma(b-a)\Gamma(c-a)\Gamma(d-a)}{\Gamma(-2a)}\nonumber\\ -& &{}\mathindent\mathindent{}\times\sum_{\begin{array}{c} -{\scriptstyle k=0,1,2\ldots}\\{\scriptstyle a+k<0}\end{array}} -\frac{(2a)_k(a+1)_k(a+b)_k(a+c)_k(a+d)_k}{(a)_k(a-b+1)_k(a-c+1)_k(a-d+1)_kk!}\nonumber\\ -& &{}\mathindent\mathindent\mathindent\mathindent{}\times W_m(-(a+k)^2;a,b,c,d)W_n(-(a+k)^2;a,b,c,d)\nonumber\\ -& &{}=\frac{\Gamma(n+a+b)\cdots\Gamma(n+c+d)}{\Gamma(2n+a+b+c+d)}(n+a+b+c+d-1)_nn!\,\delta_{mn}. -\end{eqnarray} - -\subsection*{Recurrence relation} -\begin{equation} -\label{RecWilson} --\left(a^2+x^2\right){\tilde{W}}_n(x^2) -=A_n{\tilde{W}}_{n+1}(x^2)-\left(A_n+C_n\right){\tilde{W}}_n(x^2)+C_n{\tilde{W}}_{n-1}(x^2), -\end{equation} -where -$${\tilde{W}}_n(x^2):={\tilde{W}}_n(x^2;a,b,c,d)=\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}$$ -and -$$\left\{\begin{array}{l} -\displaystyle A_n=\frac{(n+a+b+c+d-1)(n+a+b)(n+a+c)(n+a+d)}{(2n+a+b+c+d-1)(2n+a+b+c+d)}\\[5mm] -\displaystyle C_n=\frac{n(n+b+c-1)(n+b+d-1)(n+c+d-1)}{(2n+a+b+c+d-2)(2n+a+b+c+d-1)}. -\end{array}\right.$$ - -\subsection*{Normalized recurrence relation} -\begin{equation} -\label{NormRecWilson} -xp_n(x)=p_{n+1}(x)+(A_n+C_n-a^2)p_n(x)+A_{n-1}C_np_{n-1}(x), -\end{equation} -where -$$W_n(x^2;a,b,c,d)=(-1)^n(n+a+b+c+d-1)_np_n(x^2).$$ - -\subsection*{Difference equation} -\begin{eqnarray} -\label{dvWilson} -& &n(n+a+b+c+d-1)y(x)\nonumber\\ -& &{}=B(x)y(x+i)-\left[B(x)+D(x)\right]y(x)+D(x)y(x-i), -\end{eqnarray} -where -$$y(x)=W_n(x^2;a,b,c,d)$$ -and -$$\left\{\begin{array}{l} -\displaystyle B(x)=\frac{(a-ix)(b-ix)(c-ix)(d-ix)}{2ix(2ix-1)}\\[5mm] -\displaystyle D(x)=\frac{(a+ix)(b+ix)(c+ix)(d+ix)}{2ix(2ix+1)}. -\end{array}\right.$$ - -\subsection*{Forward shift operator} -\begin{eqnarray} -\label{shift1WilsonI} -& &W_n((x+\textstyle\frac{1}{2}i)^2;a,b,c,d) --W_n((x-\textstyle\frac{1}{2}i)^2;a,b,c,d)\nonumber\\ -& &{}=-2inx(n+a+b+c+d-1)W_{n-1}(x^2;a+\textstyle\frac{1}{2},b+\textstyle\frac{1}{2}, -c+\textstyle\frac{1}{2},d+\textstyle\frac{1}{2}) -\end{eqnarray} -or equivalently -\begin{eqnarray} -\label{shift1WilsonII} -& &\frac{\delta W_n(x^2;a,b,c,d)}{\delta x^2}\nonumber\\ -& &{}=-n(n+a+b+c+d-1)W_{n-1}(x^2;a+\textstyle\frac{1}{2},b+\textstyle\frac{1}{2}, -c+\textstyle\frac{1}{2},d+\textstyle\frac{1}{2}). -\end{eqnarray} - -\newpage - -\subsection*{Backward shift operator} -\begin{eqnarray} -\label{shift2WilsonI} -& &(a-\textstyle\frac{1}{2}-ix)(b-\textstyle\frac{1}{2}-ix) -(c-\textstyle\frac{1}{2}-ix)(d-\textstyle\frac{1}{2}-ix) -W_n((x+\textstyle\frac{1}{2}i)^2;a,b,c,d)\nonumber\\ -& &{}\mathindent{}-(a-\textstyle\frac{1}{2}+ix)(b-\textstyle\frac{1}{2}+ix) -(c-\textstyle\frac{1}{2}+ix)(d-\textstyle\frac{1}{2}+ix) -W_n((x-\textstyle\frac{1}{2}i)^2;a,b,c,d)\nonumber\\ -& &{}=-2ix W_{n+1}(x^2;a-\textstyle\frac{1}{2},b-\textstyle\frac{1}{2}, -c-\textstyle\frac{1}{2},d-\textstyle\frac{1}{2}) -\end{eqnarray} -or equivalently -\begin{eqnarray} -\label{shift2WilsonII} -& &\frac{\delta\left[\omega(x;a,b,c,d)W_n(x^2;a,b,c,d)\right]}{\delta x^2}\nonumber\\ -& &{}=\omega(x;a-\textstyle\frac{1}{2},b-\textstyle\frac{1}{2}, -c-\textstyle\frac{1}{2},d-\textstyle\frac{1}{2}) -W_{n+1}(x^2;a-\textstyle\frac{1}{2},b-\textstyle\frac{1}{2},c-\textstyle\frac{1}{2},d-\textstyle\frac{1}{2}), -\end{eqnarray} -where -$$\omega(x;a,b,c,d):=\frac{1}{2ix}\left|\frac{\Gamma(a+ix)\Gamma(b+ix)\Gamma(c+ix)\Gamma(d+ix)}{\Gamma(2ix)}\right|^2.$$ - -\subsection*{Rodrigues-type formula} -\begin{eqnarray} -\label{RodWilson} -& &\omega(x;a,b,c,d)W_n(x^2;a,b,c,d)\nonumber\\ -& &{}=\left(\frac{\delta}{\delta x^2}\right)^n -\left[\omega(x;a+\textstyle\frac{1}{2}n,b+\textstyle\frac{1}{2}n, -c+\textstyle\frac{1}{2}n,d+\textstyle\frac{1}{2}n)\right]. -\end{eqnarray} - -\subsection*{Generating functions} -\begin{equation} -\label{GenWilson1} -\hyp{2}{1}{a+ix,b+ix}{a+b}{t}\,\hyp{2}{1}{c-ix,d-ix}{c+d}{t}= -\sum_{n=0}^{\infty}\frac{W_n(x^2;a,b,c,d)t^n}{(a+b)_n(c+d)_nn!}. -\end{equation} - -\begin{equation} -\label{GenWilson2} -\hyp{2}{1}{a+ix,c+ix}{a+c}{t}\,\hyp{2}{1}{b-ix,d-ix}{b+d}{t}= -\sum_{n=0}^{\infty}\frac{W_n(x^2;a,b,c,d)t^n}{(a+c)_n(b+d)_nn!}. -\end{equation} - -\begin{equation} -\label{GenWilson3} -\hyp{2}{1}{a+ix,d+ix}{a+d}{t}\,\hyp{2}{1}{b-ix,c-ix}{b+c}{t}= -\sum_{n=0}^{\infty}\frac{W_n(x^2;a,b,c,d)t^n}{(a+d)_n(b+c)_nn!}. -\end{equation} - -\begin{eqnarray} -\label{GenWilson4} -& &(1-t)^{1-a-b-c-d}\nonumber\\ -& &{}\mathindent\times\hyp{4}{3}{\frac{1}{2}(a+b+c+d-1),\frac{1}{2}(a+b+c+d),a+ix,a-ix} -{a+b,a+c,a+d}{-\frac{4t}{(1-t)^2}}\nonumber\\ -& &{}=\sum_{n=0}^{\infty}\frac{(a+b+c+d-1)_n}{(a+b)_n(a+c)_n(a+d)_nn!}W_n(x^2;a,b,c,d)t^n. -\end{eqnarray} - -\subsection*{Limit relations} - -\subsubsection*{Wilson $\rightarrow$ Continuous dual Hahn} -The continuous dual Hahn polynomials given by (\ref{DefContDualHahn}) can be found from the -Wilson polynomials by dividing by $(a+d)_n$ and letting $d\rightarrow\infty$: -\begin{equation} -\lim_{d\rightarrow\infty}\frac{W_n(x^2;a,b,c,d)}{(a+d)_n}=S_n(x^2;a,b,c). -\end{equation} - -\subsubsection*{Wilson $\rightarrow$ Continuous Hahn} -The continuous Hahn polynomials given by (\ref{DefContHahn}) are obtained from the Wilson -polynomials by the substitutions $a\rightarrow a-it$, $b\rightarrow b-it$, $c\rightarrow c+it$, -$d\rightarrow d+it$ and $x\rightarrow x+t$ and the limit $t\rightarrow\infty$ in the following -way: -\begin{equation} -\lim_{t\rightarrow\infty} -\frac{W_n((x+t)^2;a-it,b-it,c+it,d+it)}{(-2t)^nn!}=p_n(x;a,b,c,d). -\end{equation} - -\subsubsection*{Wilson $\rightarrow$ Jacobi} -The Jacobi polynomials given by (\ref{DefJacobi}) can be found from the Wilson -polynomials by substituting $a=b=\frac{1}{2}(\alpha+1)$, $c=\frac{1}{2}(\beta+1)+it$, -$d=\frac{1}{2}(\beta+1)-it$ and $x\rightarrow t\sqrt{\frac{1}{2}(1-x)}$ in the -definition (\ref{DefWilson}) of the Wilson polynomials and taking the limit -$t\rightarrow\infty$. In fact we have -\begin{eqnarray} -& &\lim_{t\rightarrow\infty}\frac{W_n(\frac{1}{2}(1-x)t^2;\frac{1}{2}(\alpha+1), -\frac{1}{2}(\alpha+1),\frac{1}{2}(\beta+1)+it,\frac{1}{2}(\beta+1)-it)}{t^{2n}n!}\nonumber\\ -& &{}=P_n^{(\alpha,\beta)}(x). -\end{eqnarray} - -\subsection*{Remarks} -Note that for $k0$, $c=\bar{a}$ and $d=\bar{b}$, then -\begin{eqnarray} -\label{OrtContHahn} -& &\frac{1}{2\pi}\int_{-\infty}^{\infty}\Gamma(a+ix)\Gamma(b+ix)\Gamma(c-ix)\Gamma(d-ix) -p_m(x;a,b,c,d)p_n(x;a,b,c,d)\,dx\nonumber\\ -& &{}=\frac{\Gamma(n+a+c)\Gamma(n+a+d)\Gamma(n+b+c)\Gamma(n+b+d)} -{(2n+a+b+c+d-1)\Gamma(n+a+b+c+d-1)n!}\,\delta_{mn}. -\end{eqnarray} - -\subsection*{Recurrence relation} -\begin{equation} -\label{RecContHahn} -(a+ix)\tilde{p}_n(x)=A_n\tilde{p}_{n+1}(x)-\left(A_n+C_n\right)\tilde{p}_n(x)+C_n\tilde{p}_{n-1}(x), -\end{equation} -where -$$\tilde{p}_n(x):=\tilde{p}_n(x;a,b,c,d)=\frac{n!}{i^n(a+c)_n(a+d)_n}p_n(x;a,b,c,d)$$ -and -$$\left\{\begin{array}{l} -\displaystyle A_n=-\frac{(n+a+b+c+d-1)(n+a+c)(n+a+d)}{(2n+a+b+c+d-1)(2n+a+b+c+d)}\\[5mm] -\displaystyle C_n=\frac{n(n+b+c-1)(n+b+d-1)}{(2n+a+b+c+d-2)(2n+a+b+c+d-1)}. -\end{array}\right.$$ - -\subsection*{Normalized recurrence relation} -\begin{equation} -\label{NormRecContHahn} -xp_n(x)=p_{n+1}(x)+i(A_n+C_n+a)p_n(x)-A_{n-1}C_np_{n-1}(x), -\end{equation} -where -$$p_n(x;a,b,c,d)=\frac{(n+a+b+c+d-1)_n}{n!}p_n(x).$$ - -\subsection*{Difference equation} -\begin{eqnarray} -\label{dvContHahn} -& &n(n+a+b+c+d-1)y(x)\nonumber\\ -& &{}=B(x)y(x+i)-\left[B(x)+D(x)\right]y(x)+D(x)y(x-i), -\end{eqnarray} -where -$$y(x)=p_n(x;a,b,c,d)$$ -and -$$\left\{\begin{array}{l} -\displaystyle B(x)=(c-ix)(d-ix)\\[5mm] -\displaystyle D(x)=(a+ix)(b+ix). -\end{array}\right.$$ - -\subsection*{Forward shift operator} -\begin{eqnarray} -\label{shift1ContHahnI} -& &p_n(x+\textstyle\frac{1}{2}i;a,b,c,d)-p_n(x-\textstyle\frac{1}{2}i;a,b,c,d)\nonumber\\ -& &{}=i(n+a+b+c+d-1)p_{n-1}(x;a+\textstyle\frac{1}{2}, -b+\textstyle\frac{1}{2},c+\textstyle\frac{1}{2},d+\textstyle\frac{1}{2}) -\end{eqnarray} -or equivalently -\begin{equation} -\label{shift1ContHahnII} -\frac{\delta p_n(x;a,b,c,d)}{\delta x}=(n+a+b+c+d-1) -p_{n-1}(x;a+\textstyle\frac{1}{2},b+\textstyle\frac{1}{2}, -c+\textstyle\frac{1}{2},d+\textstyle\frac{1}{2}). -\end{equation} - -\subsection*{Backward shift operator} -\begin{eqnarray} -\label{shift2ContHahnI} -& &(c-\textstyle\frac{1}{2}-ix)(d-\textstyle\frac{1}{2}-ix) -p_n(x+\textstyle\frac{1}{2}i;a,b,c,d)\nonumber\\ -& &{}\mathindent{}-(a-\textstyle\frac{1}{2}+ix)(b-\textstyle\frac{1}{2}+ix) -p_n(x-\textstyle\frac{1}{2}i;a,b,c,d)\nonumber\\ -& &{}=\frac{n+1}{i}p_{n+1}(x;a-\textstyle\frac{1}{2}, -b-\textstyle\frac{1}{2},c-\textstyle\frac{1}{2},d-\textstyle\frac{1}{2}) -\end{eqnarray} -or equivalently -\begin{eqnarray} -\label{shift2ContHahnII} -& &\frac{\delta\left[\omega(x;a,b,c,d)p_n(x;a,b,c,d)\right]}{\delta x}\nonumber\\ -& &{}=-(n+1)\omega(x;a-\textstyle\frac{1}{2},b-\textstyle\frac{1}{2}, -c-\textstyle\frac{1}{2},d-\textstyle\frac{1}{2})\nonumber\\ -& &{}\mathindent\times p_{n+1}(x;a-\textstyle\frac{1}{2},b-\textstyle\frac{1}{2},c-\textstyle\frac{1}{2},d-\textstyle\frac{1}{2}), -\end{eqnarray} -where -$$\omega(x;a,b,c,d)=\Gamma(a+ix)\Gamma(b+ix)\Gamma(c-ix)\Gamma(d-ix).$$ - -\subsection*{Rodrigues-type formula} -\begin{eqnarray} -\label{RodContHahn} -& &\omega(x;a,b,c,d)p_n(x;a,b,c,d)\nonumber\\ -& &{}=\frac{(-1)^n}{n!}\left(\frac{\delta}{\delta x}\right)^n -\left[\omega(x;a+\textstyle\frac{1}{2}n,b+\textstyle\frac{1}{2}n, -c+\textstyle\frac{1}{2}n,d+\textstyle\frac{1}{2}n)\right]. -\end{eqnarray} - -\subsection*{Generating functions} -\begin{equation} -\label{GenContHahn1} -\hyp{1}{1}{a+ix}{a+c}{-it}\,\hyp{1}{1}{d-ix}{b+d}{it}= -\sum_{n=0}^{\infty}\frac{p_n(x;a,b,c,d)}{(a+c)_n(b+d)_n}t^n. -\end{equation} - -\begin{equation} -\label{GenContHahn2} -\hyp{1}{1}{a+ix}{a+d}{-it}\,\hyp{1}{1}{c-ix}{b+c}{it}= -\sum_{n=0}^{\infty}\frac{p_n(x;a,b,c,d)}{(a+d)_n(b+c)_n}t^n. -\end{equation} - -\begin{eqnarray} -\label{GenContHahn3} -& &(1-t)^{1-a-b-c-d}\,\hyp{3}{2}{\frac{1}{2}(a+b+c+d-1),\frac{1}{2}(a+b+c+d),a+ix} -{a+c,a+d}{-\frac{4t}{(1-t)^2}}\nonumber\\ -& &{}=\sum_{n=0}^{\infty} -\frac{(a+b+c+d-1)_n}{(a+c)_n(a+d)_ni^n}p_n(x;a,b,c,d)t^n. -\end{eqnarray} - -\subsection*{Limit relations} - -\subsubsection*{Wilson $\rightarrow$ Continuous Hahn} -The continuous Hahn polynomials are obtained from the Wilson polynomials given by -(\ref{DefWilson}) by the substitution $a\rightarrow a-it$, $b\rightarrow b-it$, -$c\rightarrow c+it$, $d\rightarrow d+it$ and $x\rightarrow x+t$ and the limit -$t\rightarrow\infty$ in the following way: -$$\lim_{t\rightarrow\infty} -\frac{W_n((x+t)^2;a-it,b-it,c+it,d+it)}{(-2t)^nn!}=p_n(x;a,b,c,d).$$ - -\subsubsection*{Continuous Hahn $\rightarrow$ Meixner-Pollaczek} -The Meixner-Pollaczek polynomials given by (\ref{DefMP}) can be obtained from the -continuous Hahn polynomials by setting $x\rightarrow x+t$, $a=\lambda-it$, $c=\lambda+it$ -and $b=d=t\tan\phi$ and taking the limit $t\rightarrow\infty$: -\begin{equation} -\lim_{t\rightarrow\infty}\frac{p_n(x+t;\lambda-it,t\tan\phi,\lambda+it,t\tan\phi)}{t^n} -=\frac{P_n^{(\lambda)}(x;\phi)}{(\cos\phi)^n}. -\end{equation} - -\subsubsection*{Continuous Hahn $\rightarrow$ Jacobi} -The Jacobi polynomials given by (\ref{DefJacobi}) follow from the continuous Hahn polynomials -by the substitution $x\rightarrow \frac{1}{2}xt$, $a=\frac{1}{2}(\alpha+1-it)$, -$b=\frac{1}{2}(\beta+1+it)$, $c=\frac{1}{2}(\alpha+1+it)$ and $d=\frac{1}{2}(\beta+1-it)$, -division by $t^n$ and the limit $t\rightarrow\infty$: -\begin{eqnarray} -& &\lim_{t\rightarrow\infty} -\frac{p_n(\frac{1}{2}xt;\frac{1}{2}(\alpha+1-it),\frac{1}{2}(\beta+1+it), -\frac{1}{2}(\alpha+1+it),\frac{1}{2}(\beta+1-it))}{t^n}\nonumber\\ -& &{}=P_n^{(\alpha,\beta)}(x). -\end{eqnarray} - -\subsubsection*{Continuous Hahn $\rightarrow$ Pseudo Jacobi} -The pseudo Jacobi polynomials given by (\ref{DefPseudoJacobi}) follow from the continuous -Hahn polynomials by the substitution $x\rightarrow xt$, $a=\frac{1}{2}(-N+i\nu-2t)$, -$b=\frac{1}{2}(-N-i\nu+2t)$, $c=\frac{1}{2}(-N-i\nu-2t)$ and $d=\frac{1}{2}(-N+i\nu+2t)$, -division by $t^n$ and the limit $t\rightarrow\infty$: -\begin{eqnarray} -& &\lim_{t\rightarrow\infty}\frac{p_n(xt;\frac{1}{2}(-N+i\nu-2t),\frac{1}{2}(-N-i\nu+2t), -\frac{1}{2}(-N+i\nu-2t),\frac{1}{2}(-N-i\nu+2t))}{t^n}\nonumber\\ -& &{}=\frac{(n-2N-1)_n}{n!}P_n(x;\nu,N). -\end{eqnarray} - -\subsection*{Remark} -Since we have for $k0$ and $(c,d)=(\overline a,\overline b)$ or $(\overline b,\overline a)$.\\ -Thus, under these assumptions, the continuous Hahn polynomial -$p_n(x;a,b,c,d)$ -is symmetric in $a,b$ and in $c,d$. -This follows from the orthogonality relation (9.4.2) -together with the value of its coefficient of $x^n$ given in (9.4.4b).\\ -As a consequence, it is sufficient to give generating function (9.4.11). Then the generating -function (9.4.12) will follow by symmetry in the parameters. -% -\paragraph{Uniqueness of orthogonality measure} -The coefficient of $p_{n-1}(x)$ in (9.4.4) behaves as $O(n^2)$ as $n\to\iy$. -Hence \eqref{93} holds, by which the orthogonality measure is unique. -% -\paragraph{Special cases} -In the following special case there is a reduction to -Meixner-Pollaczek: -\begin{equation} -p_n(x;a,a+\thalf,a,a+\thalf)= -\frac{(2a)_n (2a+\thalf)_n}{(4a)_n}\,P_n^{(2a)}(2x;\thalf\pi). -\end{equation} -See \myciteKLS{342}{(2.6)} (note that in \myciteKLS{342}{(2.3)} the -Meixner-Pollaczek polynonmials are defined different from (9.7.1), -without a constant factor in front). - -For $0-1$ and $\beta>-1$, or for $\alpha<-N$ and $\beta<-N$, we have -\begin{eqnarray} -\label{OrtHahn} -& &\sum_{x=0}^N\binom{\alpha +x}{x}\binom{\beta+N-x}{N-x}Q_m(x;\alpha,\beta,N)Q_n(x;\alpha,\beta,N)\nonumber\\ -& &{}=\frac{(-1)^n(n+\alpha+\beta+1)_{N+1}(\beta+1)_nn!}{(2n+\alpha+\beta+1)(\alpha+1)_n(-N)_nN!}\,\delta_{mn}. -\end{eqnarray} - -\subsection*{Recurrence relation} -\begin{equation} -\label{RecHahn} --xQ_n(x)=A_nQ_{n+1}(x)-\left(A_n+C_n\right)Q_n(x)+C_nQ_{n-1}(x), -\end{equation} -where -$$Q_n(x):=Q_n(x;\alpha,\beta,N)$$ -and -$$\left\{\begin{array}{l} -\displaystyle A_n=\frac{(n+\alpha+\beta+1)(n+\alpha+1)(N-n)}{(2n+\alpha+\beta+1)(2n+\alpha+\beta+2)}\\[5mm] -\displaystyle C_n=\frac{n(n+\alpha+\beta+N+1)(n+\beta)}{(2n+\alpha+\beta)(2n+\alpha+\beta+1)}. -\end{array}\right.$$ - -\subsection*{Normalized recurrence relation} -\begin{equation} -\label{NormRecHahn} -xp_n(x)=p_{n+1}(x)+\left(A_n+C_n\right)p_n(x)+A_{n-1}C_np_{n-1}(x), -\end{equation} -where -$$Q_n(x;\alpha,\beta,N)=\frac{(n+\alpha+\beta+1)_n}{(\alpha+1)_n(-N)_n}p_n(x).$$ - -\subsection*{Difference equation} -\begin{equation} -\label{dvHahn} -n(n+\alpha+\beta+1)y(x)=B(x)y(x+1)-\left[B(x)+D(x)\right]y(x)+D(x)y(x-1), -\end{equation} -where -$$y(x)=Q_n(x;\alpha,\beta,N)$$ -and -$$\left\{\begin{array}{l} -\displaystyle B(x)=(x+\alpha+1)(x-N)\\[5mm] -\displaystyle D(x)=x(x-\beta-N-1). -\end{array}\right.$$ - -\subsection*{Forward shift operator} -\begin{eqnarray} -\label{shift1HahnI} -& &Q_n(x+1;\alpha,\beta,N)-Q_n(x;\alpha,\beta,N)\nonumber\\ -& &{}=-\frac{n(n+\alpha+\beta+1)}{(\alpha+1)N}Q_{n-1}(x;\alpha+1,\beta+1,N-1) -\end{eqnarray} -or equivalently -\begin{equation} -\label{shift1HahnII} -\Delta Q_n(x;\alpha,\beta,N)=-\frac{n(n+\alpha+\beta+1)}{(\alpha+1)N}Q_{n-1}(x;\alpha+1,\beta+1,N-1). -\end{equation} - -\subsection*{Backward shift operator} -\begin{eqnarray} -\label{shift2HahnI} -& &(x+\alpha)(N+1-x)Q_n(x;\alpha,\beta,N)-x(\beta+N+1-x)Q_n(x-1;\alpha,\beta,N)\nonumber\\ -& &{}=\alpha(N+1)Q_{n+1}(x;\alpha-1,\beta-1,N+1) -\end{eqnarray} -or equivalently -\begin{eqnarray} -\label{shift2HahnII} -& &\nabla\left[\omega(x;\alpha,\beta,N)Q_n(x;\alpha,\beta,N)\right]\nonumber\\ -& &{}=\frac{N+1}{\beta}\omega(x;\alpha-1,\beta-1,N+1)Q_{n+1}(x;\alpha-1,\beta-1,N+1), -\end{eqnarray} -where -$$\omega(x;\alpha,\beta,N)=\binom{\alpha+x}{x}\binom{\beta+N-x}{N-x}.$$ - -\subsection*{Rodrigues-type formula} -\begin{eqnarray} -\label{RodHahn} -& &\omega(x;\alpha,\beta,N)Q_n(x;\alpha,\beta,N)\nonumber\\ -& &{}=\frac{(-1)^n(\beta+1)_n}{(-N)_n}\nabla^n\left[\omega(x;\alpha+n,\beta+n,N-n)\right]. -\end{eqnarray} - -\subsection*{Generating functions} -For $x=0,1,2,\ldots,N$ we have -\begin{equation} -\label{GenHahn1} -\hyp{1}{1}{-x}{\alpha+1}{-t}\,\hyp{1}{1}{x-N}{\beta+1}{t} -=\sum_{n=0}^N\frac{(-N)_n}{(\beta+1)_nn!}Q_n(x;\alpha,\beta,N)t^n. -\end{equation} - -\begin{eqnarray} -\label{GenHahn2} -& &\hyp{2}{0}{-x,-x+\beta+N+1}{-}{-t}\,\hyp{2}{0}{x-N,x+\alpha+1}{-}{t}\nonumber\\ -& &{}=\sum_{n=0}^N\frac{(-N)_n(\alpha+1)_n}{n!}Q_n(x;\alpha,\beta,N)t^n. -\end{eqnarray} - -\begin{eqnarray} -\label{GenHahn3} -& &\left[(1-t)^{-\alpha-\beta-1}\,\hyp{3}{2}{\frac{1}{2}(\alpha+\beta+1),\frac{1}{2}(\alpha+\beta+2),-x} -{\alpha+1,-N}{-\frac{4t}{(1-t)^2}}\right]_N\nonumber\\ -& &{}=\sum_{n=0}^N\frac{(\alpha+\beta+1)_n}{n!}Q_n(x;\alpha,\beta,N)t^n. -\end{eqnarray} - -\subsection*{Limit relations} - -\subsubsection*{Racah $\rightarrow$ Hahn} -If we take $\gamma+1=-N$ and let $\delta\rightarrow\infty$ in the definition (\ref{DefRacah}) -of the Racah polynomials, we obtain the Hahn polynomials. Hence -$$\lim_{\delta\rightarrow\infty} -R_n(\lambda(x);\alpha,\beta,-N-1,\delta)=Q_n(x;\alpha,\beta,N).$$ -And if we take $\delta=-\beta-N-1$ and let $\gamma\rightarrow\infty$ in the definition (\ref{DefRacah}) -of the Racah polynomials, we also obtain the Hahn polynomials: -$$\lim_{\gamma\rightarrow\infty} -R_n(\lambda(x);\alpha,\beta,\gamma,-\beta-N-1)=Q_n(x;\alpha,\beta,N).$$ -Another way to do this is to take $\alpha+1=-N$ and $\beta\rightarrow\beta+\gamma+N+1$ in -the definition (\ref{DefRacah}) of the Racah polynomials and then take the limit -$\delta\rightarrow\infty$. In that case we obtain the Hahn polynomials in the following way: -$$\lim_{\delta\rightarrow\infty} -R_n(\lambda(x);-N-1,\beta+\gamma+N+1,\gamma,\delta)=Q_n(x;\gamma,\beta,N).$$ - -\subsubsection*{Hahn $\rightarrow$ Jacobi} -To find the Jacobi polynomials given by (\ref{DefJacobi}) from the Hahn polynomials we take -$x\rightarrow Nx$ and let $N\rightarrow\infty$. In fact we have -\begin{equation} -\lim_{N\rightarrow\infty} -Q_n(Nx;\alpha,\beta,N)=\frac{P_n^{(\alpha,\beta)}(1-2x)}{P_n^{(\alpha,\beta)}(1)}. -\end{equation} - -\subsubsection*{Hahn $\rightarrow$ Meixner} -The Meixner polynomials given by (\ref{DefMeixner}) can be obtained from the Hahn polynomials -by taking $\alpha=b-1$, $\beta=N(1-c)c^{-1}$ and letting $N\rightarrow\infty$: -\begin{equation} -\lim_{N\rightarrow\infty} -Q_n(x;b-1,N(1-c)c^{-1},N)=M_n(x;b,c). -\end{equation} - -\subsubsection*{Hahn $\rightarrow$ Krawtchouk} -The Krawtchouk polynomials given by (\ref{DefKrawtchouk}) are obtained from the Hahn polynomials -if we take $\alpha=pt$ and $\beta=(1-p)t$ and let $t\rightarrow\infty$: -\begin{equation} -\lim_{t\rightarrow\infty}Q_n(x;pt,(1-p)t,N)=K_n(x;p,N). -\end{equation} - -\subsection*{Remark} -If we interchange the role of $x$ and $n$ in (\ref{DefHahn}) we obtain the -dual Hahn polynomials given by (\ref{DefDualHahn}). - -\noindent -Since -$$Q_n(x;\alpha,\beta,N)=R_x(\lambda(n);\alpha,\beta,N)$$ -we obtain the dual orthogonality relation for the Hahn polynomials from the -orthogonality relation (\ref{OrtDualHahn}) of the dual Hahn polynomials: -\begin{eqnarray*} -& &\sum_{n=0}^N\frac{(2n+\alpha+\beta+1)(\alpha+1)_n(-N)_nN!}{(-1)^n(n+\alpha+\beta+1)_{N+1}(\beta+1)_nn!} -Q_n(x;\alpha,\beta,N)Q_n(y;\alpha,\beta,N)\\ -& &{}=\frac{\delta_{xy}}{\dbinom{\alpha+x}{x}\dbinom{\beta+N-x}{N-x}},\quad x,y \in \{0,1,2,\ldots,N\}. -\end{eqnarray*} - -\subsection*{References} -\label{sec9.5} -% -\paragraph{Special values} -\begin{equation} -Q_n(0;\al,\be,N)=1,\quad -Q_n(N;\al,\be,N)=\frac{(-1)^n(\be+1)_n}{(\al+1)_n}\,. -\label{95} -\end{equation} -Use (9.5.1) and compare with (9.8.1) and \eqref{50}. - -From (9.5.3) and \eqref{1} it follows that -\begin{equation} -Q_{2n}(N;\al,\al,2N)=\frac{(\thalf)_n(N+\al+1)_n}{(-N+\thalf)_n(\al+1)_n}\,. -\label{30} -\end{equation} -From (9.5.1) and \mycite{DLMF}{(15.4.24)} it follows that -\begin{equation} -Q_N(x;\al,\be,N)=\frac{(-N-\be)_x}{(\al+1)_x}\qquad(x=0,1,\ldots,N). -\label{44} -\end{equation} -% -\paragraph{Symmetries} -By the orthogonality relation (9.5.2): -\begin{equation} -\frac{Q_n(N-x;\al,\be,N)}{Q_n(N;\al,\be,N)}=Q_n(x;\be,\al,N), -\label{96} -\end{equation} -It follows from \eqref{97} and \eqref{45} that -\begin{equation} -\frac{Q_{N-n}(x;\al,\be,N)}{Q_N(x;\al,\be,N)} -=Q_n(x;-N-\be-1,-N-\al-1,N) -\qquad(x=0,1,\ldots,N). -\label{100} -\end{equation} -% -\paragraph{Duality} -The Remark on p.208 gives the duality between Hahn and dual Hahn polynomials: -% -\begin{equation} -Q_n(x;\al,\be,N)=R_x(n(n+\al+\be+1);\al,\be,N)\quad(n,x\in\{0,1,\ldots N\}). -\label{45} -\end{equation} -% -\cite{AlSalam90}, \cite{AndrewsAskey85}, \cite{Area+II}, \cite{Askey75}, -\cite{Askey89I}, \cite{Askey2005}, \cite{AskeyGasper77}, \cite{AskeyWilson85}, -\cite{AtakRahmanSuslov}, \cite{AtakSuslov88}, \cite{Chihara78}, -\cite{Ciesielski}, \cite{Cooper+}, \cite{Dette95}, \cite{Dunkl76}, -\cite{Dunkl78I}, \cite{Gasper73I}, \cite{Gasper74}, \cite{HoareRahman}, -\cite{Ismail77}, \cite{Ismail2005II}, \cite{Karlin61}, \cite{Koorn81}, \cite{Koorn88}, -\cite{LabelleYehI}, \cite{LabelleYehII}, \cite{Laine}, \cite{Lesky62}, -\cite{Lesky88}, \cite{Lesky89}, \cite{Lesky94I}, \cite{Lesky95II}, -\cite{LewanowiczII}, \cite{Neuman}, \cite{Nikiforov+}, \cite{NikiforovUvarov}, -\cite{Rahman76III}, \cite{Rahman78I}, \cite{Rahman78II}, \cite{Rahman81III}, -\cite{Sablonniere}, \cite{Stanton84}, \cite{Stanton90}, \cite{Wilson80}, \cite{Wilson70II}, -\cite{Zarzo+}. - - -\section{Dual Hahn}\index{Dual Hahn polynomials}\index{Hahn polynomials!Dual} - -\par\setcounter{equation}{0} - -\subsection*{Hypergeometric representation} -\begin{equation} -\label{DefDualHahn} -R_n(\lambda(x);\gamma,\delta,N)= -\hyp{3}{2}{-n,-x,x+\gamma+\delta+1}{\gamma+1,-N}{1},\quad n=0,1,2,\ldots,N, -\end{equation} -where -$$\lambda(x)=x(x+\gamma+\delta+1).$$ - -\subsection*{Orthogonality relation} -For $\gamma>-1$ and $\delta>-1$, or for $\gamma<-N$ and $\delta<-N$, we have -\begin{eqnarray} -\label{OrtDualHahn} -& &\sum_{x=0}^N\frac{(2x+\gamma+\delta+1)(\gamma+1)_x(-N)_xN!}{(-1)^x(x+\gamma+\delta+1)_{N+1}(\delta+1)_xx!} -R_m(\lambda(x);\gamma,\delta,N)R_n(\lambda(x);\gamma,\delta,N)\nonumber\\ -& &{}=\frac{\,\delta_{mn}}{\dbinom{\gamma+n}{n}\dbinom{\delta+N-n}{N-n}}. -\end{eqnarray} - -\subsection*{Recurrence relation} -\begin{equation} -\label{RecDualHahn} -\lambda(x)R_n(\lambda(x)) -=A_nR_{n+1}(\lambda(x))-\left(A_n+C_n\right)R_n(\lambda(x))+C_nR_{n-1}(\lambda(x)), -\end{equation} -where -$$R_n(\lambda(x)):=R_n(\lambda(x);\gamma,\delta,N)$$ -and -$$\left\{\begin{array}{l} -\displaystyle A_n=(n+\gamma+1)(n-N)\\[5mm] -\displaystyle C_n=n(n-\delta-N-1). -\end{array}\right.$$ - -\subsection*{Normalized recurrence relation} -\begin{equation} -\label{NormRecDualHahn} -xp_n(x)=p_{n+1}(x)-(A_n+C_n)p_n(x)+A_{n-1}C_np_{n-1}(x), -\end{equation} -where -$$R_n(\lambda(x);\gamma,\delta,N)=\frac{1}{(\gamma+1)_n(-N)_n}p_n(\lambda(x)).$$ - -\subsection*{Difference equation} -\begin{equation} -\label{dvDualHahn} --ny(x)=B(x)y(x+1)-\left[B(x)+D(x)\right]y(x)+D(x)y(x-1), -\end{equation} -where -$$y(x)=R_n(\lambda(x);\gamma,\delta,N)$$ -and -$$\left\{\begin{array}{l} -\displaystyle B(x)=\frac{(x+\gamma+1)(x+\gamma+\delta+1)(N-x)}{(2x+\gamma+\delta+1)(2x+\gamma+\delta+2)}\\[5mm] -\displaystyle D(x)=\frac{x(x+\gamma+\delta+N+1)(x+\delta)}{(2x+\gamma+\delta)(2x+\gamma+\delta+1)}. -\end{array}\right.$$ - -\subsection*{Forward shift operator} -\begin{eqnarray} -\label{shift1DualHahnI} -& &R_n(\lambda(x+1);\gamma,\delta,N)-R_n(\lambda(x);\gamma,\delta,N)\nonumber\\ -& &{}=-\frac{n(2x+\gamma+\delta+2)}{(\gamma+1)N}R_{n-1}(\lambda(x);\gamma+1,\delta,N-1) -\end{eqnarray} -or equivalently -\begin{equation} -\label{shift1DualHahnII} -\frac{\Delta R_n(\lambda(x);\gamma,\delta,N)}{\Delta\lambda(x)}= --\frac{n}{(\gamma+1)N}R_{n-1}(\lambda(x);\gamma+1,\delta,N-1). -\end{equation} - -\subsection*{Backward shift operator} -\begin{eqnarray} -\label{shift2DualHahnI} -& &(x+\gamma)(x+\gamma+\delta)(N+1-x)R_n(\lambda(x);\gamma,\delta,N)\nonumber\\ -& &{}\mathindent{}-x(x+\gamma+\delta+N+1)(x+\delta)R_n(\lambda(x-1);\gamma,\delta,N)\nonumber\\ -& &{}=\gamma(N+1)(2x+\gamma+\delta)R_{n+1}(\lambda(x);\gamma-1,\delta,N+1) -\end{eqnarray} -or equivalently -\begin{eqnarray} -\label{shift2DualHahnII} -& &\frac{\nabla\left[\omega(x;\gamma,\delta,N)R_n(\lambda(x);\gamma,\delta,N)\right]}{\nabla\lambda(x)}\nonumber\\ -& &{}=\frac{1}{\gamma+\delta}\omega(x;\gamma-1,\delta,N+1)R_{n+1}(\lambda(x);\gamma-1,\delta,N+1), -\end{eqnarray} -where -$$\omega(x;\gamma,\delta,N)=\frac{(-1)^x(\gamma+1)_x(\gamma+\delta+1)_x(-N)_x} -{(\gamma+\delta+N+2)_x(\delta+1)_xx!}.$$ - -\newpage - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodDualHahn} -\omega(x;\gamma,\delta,N)R_n(\lambda(x);\gamma,\delta,N) -=(\gamma+\delta+1)_n\left(\nabla_{\lambda}\right)^n\left[\omega(x;\gamma+n,\delta,N-n)\right], -\end{equation} -where -$$\nabla_{\lambda}:=\frac{\nabla}{\nabla\lambda(x)}.$$ - -\subsection*{Generating functions} -For $x=0,1,2,\ldots,N$ we have -\begin{equation} -\label{GenDualHahn1} -(1-t)^{N-x}\,\hyp{2}{1}{-x,-x-\delta}{\gamma+1}{t} -=\sum_{n=0}^N\frac{(-N)_n}{n!}R_n(\lambda(x);\gamma,\delta,N)t^n. -\end{equation} - -\begin{eqnarray} -\label{GenDualHahn2} -& &(1-t)^x\,\hyp{2}{1}{x-N,x+\gamma+1}{-\delta-N}{t}\nonumber\\ -& &=\sum_{n=0}^N\frac{(\gamma+1)_n(-N)_n}{(-\delta-N)_nn!}R_n(\lambda(x);\gamma,\delta,N)t^n. -\end{eqnarray} - -\begin{equation} -\label{GenDualHahn3} -\left[\expe^t\,\hyp{2}{2}{-x,x+\gamma+\delta+1}{\gamma+1,-N}{-t}\right]_N=\sum_{n=0}^N -\frac{R_n(\lambda(x);\gamma,\delta,N)}{n!}t^n. -\end{equation} - -\begin{eqnarray} -\label{GenDualHahn4} -& &\left[(1-t)^{-\epsilon}\,\hyp{3}{2}{\epsilon,-x,x+\gamma+\delta+1}{\gamma+1,-N}{\frac{t}{t-1}}\right]_N\nonumber\\ -& &{}=\sum_{n=0}^N\frac{(\epsilon)_n}{n!}R_n(\lambda(x);\gamma,\delta,N)t^n, -\quad\textrm{$\epsilon$ arbitrary}. -\end{eqnarray} - -\subsection*{Limit relations} - -\subsubsection*{Racah $\rightarrow$ Dual Hahn} -If we take $\alpha+1=-N$ and let $\beta\rightarrow\infty$ in the definition (\ref{DefRacah}) -of the Racah polynomials, then we obtain the dual Hahn polynomials: -$$\lim_{\beta\rightarrow\infty} -R_n(\lambda(x);-N-1,\beta,\gamma,\delta)=R_n(\lambda(x);\gamma,\delta,N).$$ -And if we take $\beta=-\delta-N-1$ and let $\alpha\rightarrow\infty$ in the definition (\ref{DefRacah}) -of the Racah polynomials, then we also obtain the dual Hahn polynomials: -$$\lim_{\alpha\rightarrow\infty} -R_n(\lambda(x);\alpha,-\delta-N-1,\gamma,\delta)=R_n(\lambda(x);\gamma,\delta,N).$$ -Finally, if we take $\gamma+1=-N$ and $\delta\rightarrow\alpha+\delta+N+1$ in the definition -(\ref{DefRacah}) of the Racah polynomials and take the limit -$\beta\rightarrow\infty$ we find the dual Hahn polynomials in the following way: -$$\lim_{\beta\rightarrow\infty} -R_n(\lambda(x);\alpha,\beta,-N-1,\alpha+\delta+N+1)=R_n(\lambda(x);\alpha,\delta,N).$$ - -\subsubsection*{Dual Hahn $\rightarrow$ Meixner} -The Meixner polynomials given by (\ref{DefMeixner}) are obtained from the dual Hahn polynomials -if we take $\gamma=\beta-1$ and $\delta=N(1-c)c^{-1}$ and let $N\rightarrow\infty$: -\begin{equation} -\lim_{N\rightarrow\infty} -R_n(\lambda(x);\beta-1,N(1-c)c^{-1},N)=M_n(x;\beta,c). -\end{equation} - -\subsubsection*{Dual Hahn $\rightarrow$ Krawtchouk} -The Krawtchouk polynomials given by (\ref{DefKrawtchouk}) can be obtained from the dual -Hahn polynomials by setting $\gamma=pt$, $\delta=(1-p)t$ and letting $t\rightarrow\infty$: -\begin{equation} -\lim_{t\rightarrow\infty}R_n(\lambda(x);pt,(1-p)t,N)=K_n(x;p,N). -\end{equation} - -\subsection*{Remark} -If we interchange the role of $x$ and $n$ in the definition (\ref{DefDualHahn}) -of the dual Hahn polynomials we obtain the Hahn polynomials given by (\ref{DefHahn}). - -\noindent -Since -$$R_n(\lambda(x);\gamma,\delta,N)=Q_x(n;\gamma,\delta,N)$$ -we obtain the dual orthogonality relation for the dual Hahn polynomials -from the orthogonality relation (\ref{OrtHahn}) for the Hahn polynomials: -\begin{eqnarray*} -& &\sum_{n=0}^N\binom{\gamma+n}{n}\binom{\delta+N-n}{N-n} R_n(\lambda(x);\gamma,\delta,N)R_n(\lambda(y);\gamma,\delta,N)\\ -& &{}=\frac{(-1)^x(x+\gamma+\delta+1)_{N+1}(\delta+1)_xx!} -{(2x+\gamma+\delta+1)(\gamma+1)_x(-N)_xN!}\delta_{xy},\quad x,y \in \{0,1,2,\ldots,N\}. -\end{eqnarray*} - -\subsection*{References} -\label{sec9.6} -% -\paragraph{Special values} -By \eqref{44} and \eqref{45} we have -\begin{equation} -R_n(N(N+\ga+\de+1);\ga,\de,N)=\frac{(-N-\de)_n}{(\ga+1)_n}\,. -\label{47} -\end{equation} -It follows from \eqref{95} and \eqref{45} that -\begin{equation} -R_N(x(x+\ga+\de+1);\ga,\de,N) -=\frac{(-1)^x(\de+1)_x}{(\ga+1)_x}\qquad(x=0,1,\ldots,N). -\label{101} -\end{equation} -% -\paragraph{Symmetries} -Write the weight in (9.6.2) as -\begin{equation} -w_x(\al,\be,N):=N!\,\frac{2x+\ga+\de+1}{(x+\ga+\de+1)_{N+1}}\, -\frac{(\ga+1)_x}{(\de+1)_x}\,\binom Nx. -\label{98} -\end{equation} -Then -\begin{equation} -(\de+1)_N\,w_{N-x}(\ga,\de,N)= -(-\ga-N)_N\,w_x(-\de-N-1,-\ga-N-1,N). -\label{99} -\end{equation} -Hence, by (9.6.2), -\begin{equation} -\frac{R_n((N-x)(N-x+\ga+\de+1);\ga,\de,N)}{R_n(N(N+\ga+\de+1);\ga,\de,N)} -=R_n(x(x-2N-\ga-\de-1);-N-\de-1,-N-\ga-1,N). -\label{97} -\end{equation} -Alternatively, \eqref{97} follows from (9.6.1) and -\mycite{DLMF}{(16.4.11)}. - -It follows from \eqref{96} and \eqref{45} that -\begin{equation} -\frac{R_{N-n}(x(x+\ga+\de+1);\ga,\de,N)} -{R_N(x(x+\ga+\de+1);\ga,\de,N)} -=R_n(x(x+\ga+\de+1);\de,\ga,N)\qquad(x=0,1,\ldots,N). -\label{102} -\end{equation} -% -\paragraph{Re: (9.6.11).} -The generating function (9.6.11) can be written in a more conceptual way as -\begin{equation} -(1-t)^x\,\hyp21{x-N,x+\ga+1}{-\de-N}t=\frac{N!}{(\de+1)_N}\, -\sum_{n=0}^N \om_n\,R_n(\la(x);\ga,\de,N)\,t^n, -\label{2} -\end{equation} -where -\begin{equation} -\om_n:=\binom{\ga+n}n \binom{\de+N-n}{N-n}, -\label{3} -\end{equation} -i.e., the denominator on the \RHS\ of (9.6.2). -By the duality between Hahn polynomials and dual Hahn polynomials (see \eqref{45}) the above generating function can be rewritten in -terms of Hahn polynomials: -\begin{equation} -(1-t)^n\,\hyp21{n-N,n+\al+1}{-\be-N}t=\frac{N!}{(\be+1)_N}\, -\sum_{x=0}^N w_x\,Q_n(x;\al,\be,N)\,t^x, -\label{4} -\end{equation} -where -\begin{equation} -w_x:=\binom{\al+x}x \binom{\be+N-x}{N-x}, -\label{5} -\end{equation} -i.e., the weight occurring in the orthogonality relation (9.5.2) -for Hahn polynomials. -\paragraph{Re: (9.6.15).} -There should be a closing bracket before the equality sign. -% -\cite{Askey2005}, \cite{AskeyWilson85}, \cite{AtakRahmanSuslov}, \cite{AtakSuslov88}, -\cite{Ismail2005II}, \cite{Karlin61}, \cite{Koorn81}, \cite{Koorn88}, \cite{Lesky93}, -\cite{Lesky94I}, \cite{Lesky95I}, \cite{Lesky95II}, \cite{Nikiforov+}, \cite{NikiforovUvarov}, -\cite{Rahman81II}, \cite{Stanton84}, \cite{Wilson80}. - - -\section{Meixner-Pollaczek}\index{Meixner-Pollaczek polynomials} - -\par\setcounter{equation}{0} - -\subsection*{Hypergeometric representation} -\begin{equation} -\label{DefMP} -P_n^{(\lambda)}(x;\phi)=\frac{(2\lambda)_n}{n!}\expe^{in\phi}\, -\hyp{2}{1}{-n,\lambda+ix}{2\lambda}{1-\expe^{-2i\phi}}. -\end{equation} - -\subsection*{Orthogonality relation} -\begin{equation} -\label{OrtMP} -\frac{1}{2\pi}\int_{-\infty}^{\infty} -\e^{(2\phi-\pi)x}\left|\Gamma(\lambda+ix)\right|^2 -P_m^{(\lambda)}(x;\phi)P_n^{(\lambda)}(x;\phi)\,dx -{}=\frac{\Gamma(n+2\lambda)}{(2\sin\phi)^{2\lambda}n!}\,\delta_{mn}, -% \constraint{ -% $\lambda > 0$ & -% $0 < \phi < \pi$ } -\end{equation} - -\subsection*{Recurrence relation} -\begin{eqnarray} -\label{RecMP} -& &(n+1)P_{n+1}^{(\lambda)}(x;\phi)-2\left[x\sin\phi+(n+\lambda)\cos\phi\right] -P_n^{(\lambda)}(x;\phi)\nonumber\\ -& &{}\mathindent{}+(n+2\lambda-1)P_{n-1}^{(\lambda)}(x;\phi)=0. -\end{eqnarray} - -\subsection*{Normalized recurrence relation} -\begin{equation} -\label{NormRecMP} -xp_n(x)=p_{n+1}(x)-\left(\frac{n+\lambda}{\tan\phi}\right)p_n(x)+ -\frac{n(n+2\lambda-1)}{4\sin^2\phi}p_{n-1}(x), -\end{equation} -where -$$P_n^{(\lambda)}(x;\phi)=\frac{(2\sin\phi)^n}{n!}p_n(x).$$ - -\subsection*{Difference equation} -\begin{eqnarray} -\label{dvMP} -& &\e^{i\phi}(\lambda-ix)y(x+i)+2i\left[x\cos\phi-(n+\lambda)\sin\phi\right]y(x)\nonumber\\ -& &{}\mathindent{}-\e^{-i\phi}(\lambda+ix)y(x-i)=0,\quad y(x)=P_n^{(\lambda)}(x;\phi). -\end{eqnarray} - -\subsection*{Forward shift operator} -\begin{equation} -\label{shift1MPI} -P_n^{(\lambda)}(x+\textstyle\frac{1}{2}i;\phi)- -P_n^{(\lambda)}(x-\textstyle\frac{1}{2}i;\phi)=(\expe^{i\phi}-\expe^{-i\phi}) -P_{n-1}^{(\lambda+\frac{1}{2})}(x;\phi) -\end{equation} -or equivalently -\begin{equation} -\label{shift1MPII} -\frac{\delta P_n^{(\lambda)}(x;\phi)}{\delta x} -=2\sin\phi\,P_{n-1}^{(\lambda+\frac{1}{2})}(x;\phi). -\end{equation} - -\subsection*{Backward shift operator} -\begin{eqnarray} -\label{shift2MPI} -& &\e^{i\phi}(\lambda-\textstyle\frac{1}{2}-ix) -P_n^{(\lambda)}(x+\textstyle\frac{1}{2}i;\phi)+ -\e^{-i\phi}(\lambda-\textstyle\frac{1}{2}+ix) -P_n^{(\lambda)}(x-\textstyle\frac{1}{2}i;\phi)\nonumber\\ -& &{}=(n+1)P_{n+1}^{(\lambda-\frac{1}{2})}(x;\phi) -\end{eqnarray} -or equivalently -\begin{equation} -\label{shift2MPII} -\frac{\delta\left[\omega(x;\lambda,\phi)P_n^{(\lambda)}(x;\phi)\right]}{\delta x}= --(n+1)\omega(x;\lambda-\textstyle\frac{1}{2},\phi) -P_{n+1}^{(\lambda-\frac{1}{2})}(x;\phi), -\end{equation} -where -$$\omega(x;\lambda,\phi)=\Gamma(\lambda+ix)\Gamma(\lambda-ix)\e^{(2\phi-\pi)x}.$$ - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodMP} -\omega(x;\lambda,\phi)P_n^{(\lambda)}(x;\phi)=\frac{(-1)^n}{n!} -\left(\frac{\delta}{\delta x}\right)^n\left[\omega(x;\lambda+\textstyle\frac{1}{2}n,\phi)\right]. -\end{equation} - -\newpage - -\subsection*{Generating functions} -\begin{equation} -\label{GenMP1} -(1-\expe^{i\phi}t)^{-\lambda+ix}(1-\expe^{-i\phi}t)^{-\lambda-ix}= -\sum_{n=0}^{\infty}P_n^{(\lambda)}(x;\phi)t^n. -\end{equation} - -\begin{equation} -\label{GenMP2} -\e^t\,\hyp{1}{1}{\lambda+ix}{2\lambda}{(\expe^{-2i\phi}-1)t}= -\sum_{n=0}^{\infty}\frac{P_n^{(\lambda)}(x;\phi)}{(2\lambda)_n\expe^{in\phi}}t^n. -\end{equation} - -\begin{eqnarray} -\label{GenMP3} -& &(1-t)^{-\gamma}\,\hyp{2}{1}{\gamma,\lambda+ix}{2\lambda}{\frac{(1-\e^{-2i\phi})t}{t-1}}\nonumber\\ -& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n}{(2\lambda)_n}\frac{P_n^{(\lambda)}(x;\phi)}{\e^{in\phi}}t^n, -\quad\textrm{$\gamma$ arbitrary}. -\end{eqnarray} - -\subsection*{Limit relations} - -\subsubsection*{Continuous dual Hahn $\rightarrow$ Meixner-Pollaczek} -The Meixner-Pollaczek polynomials can be obtained from the continuous dual Hahn polynomials -given by (\ref{DefContDualHahn}) by the substitutions $x\rightarrow x-t$, $a=\lambda+it$, -$b=\lambda-it$ and $c=t\cot\phi$ and the limit $t\rightarrow\infty$: -$$\lim_{t\rightarrow\infty}\frac{S_n((x-t)^2;\lambda+it,\lambda-it,t\cot\phi)}{t^nn!} -=\frac{P_n^{(\lambda)}(x;\phi)}{(\sin\phi)^n}.$$ - -\subsubsection*{Continuous Hahn $\rightarrow$ Meixner-Pollaczek} -By setting $x\rightarrow x+t$, $a=\lambda-it$, $c=\lambda+it$ and $b=d=t\tan\phi$ in the -definition (\ref{DefContHahn}) of the continuous Hahn polynomials and taking the limit -$t\rightarrow\infty$ we obtain the Meixner-Pollaczek polynomials: -$$\lim_{t\rightarrow\infty}\frac{p_n(x+t;\lambda-it,t\tan\phi,\lambda+it,t\tan\phi)}{t^nn!} -=\frac{P_n^{(\lambda)}(x;\phi)}{(\cos\phi)^n}.$$ - -\newpage - -\subsubsection*{Meixner-Pollaczek $\rightarrow$ Laguerre} -The Laguerre polynomials given by (\ref{DefLaguerre}) can be obtained from the -Meixner-Pollaczek polynomials by the substitution $\lambda=\frac{1}{2}(\alpha+1)$, -$x\rightarrow -\frac{1}{2}\phi^{-1}x$ and the limit $\phi\rightarrow 0$: -\begin{equation} -\lim_{\phi\rightarrow 0} -P_n^{(\frac{1}{2}\alpha+\frac{1}{2})}(-\textstyle\frac{1}{2}\phi^{-1}x;\phi)=L_n^{(\alpha)}(x). -\end{equation} - -\subsubsection*{Meixner-Pollaczek $\rightarrow$ Hermite} -The Hermite polynomials given by (\ref{DefHermite}) are obtained from the Meixner-Pollaczek -polynomials if we substitute $x\rightarrow (\sin\phi)^{-1}(x\sqrt{\lambda}-\lambda\cos\phi)$ -and then let $\lambda\rightarrow\infty$: -\begin{equation} -\lim_{\lambda\rightarrow\infty} -\lambda^{-\frac{1}{2}n}P_n^{(\lambda)} -((\sin\phi)^{-1}(x\sqrt{\lambda}-\lambda\cos\phi);\phi)=\frac{H_n(x)}{n!}. -\end{equation} - -\subsection*{Remark} -Since we have for $k-1$ and $\beta>-1$ we have -\begin{eqnarray} -\label{OrtJacobi1} -& &\int_{-1}^1(1-x)^{\alpha}(1+x)^{\beta}P_m^{(\alpha,\beta)}(x)P_n^{(\alpha,\beta)}(x)\,dx\nonumber\\ -& &{}=\frac{2^{\alpha+\beta+1}}{2n+\alpha+\beta+1}\frac{\Gamma(n+\alpha+1)\Gamma(n+\beta+1)}{\Gamma(n+\alpha+\beta+1)n!}\,\delta_{mn}. -\end{eqnarray} -For $\alpha+\beta<-2N-1$, $\beta>-1$ and $m,n\in\{0,1,2,\ldots,N\}$ we also have -\begin{eqnarray} -\label{OrtJacobi2} -& &\int_1^{\infty}(x+1)^{\alpha}(x-1)^{\beta}P_m^{(\alpha,\beta)}(-x)P_n^{(\alpha,\beta)}(-x)\,dx\nonumber\\ -& &{}=-\frac{2^{\alpha+\beta+1}}{2n+\alpha+\beta+1}\frac{\Gamma(-n-\alpha-\beta)\Gamma(n+\alpha+\beta+1)}{\Gamma(-n-\alpha)n!}\,\delta_{mn}. -\end{eqnarray} - -\subsection*{Recurrence relation} -\begin{eqnarray} -\label{RecJacobi} -xP_n^{(\alpha,\beta)}(x)&=&\frac{2(n+1)(n+\alpha+\beta+1)}{(2n+\alpha+\beta+1)(2n+\alpha+\beta+2)}P_{n+1}^{(\alpha,\beta)}(x)\nonumber\\ -& &{}\mathindent{}+\frac{\beta^2-\alpha^2}{(2n+\alpha+\beta)(2n+\alpha+\beta+2)}P_n^{(\alpha,\beta)}(x)\nonumber\\ -& &{}\mathindent\mathindent{}+\frac{2(n+\alpha)(n+\beta)}{(2n+\alpha+\beta)(2n+\alpha+\beta+1)}P_{n-1}^{(\alpha,\beta)}(x). -\end{eqnarray} - -\subsection*{Normalized recurrence relation} -\begin{eqnarray} -\label{NormRecJacobi} -xp_n(x)&=&p_{n+1}(x)+\frac{\beta^2-\alpha^2}{(2n+\alpha+\beta)(2n+\alpha+\beta+2)}p_n(x)\nonumber\\ -& &{}\mathindent{}+\frac{4n(n+\alpha)(n+\beta)(n+\alpha+\beta)} -{(2n+\alpha+\beta-1)(2n+\alpha+\beta)^2(2n+\alpha+\beta+1)}p_{n-1}(x) -\end{eqnarray} -where -$$P_n^{(\alpha,\beta)}(x)=\frac{(n+\alpha+\beta+1)_n}{2^nn!}p_n(x).$$ - -\subsection*{Differential equation} -\begin{eqnarray} -\label{dvJacobi} -& &(1-x^2)y''(x)+\left[\beta-\alpha-(\alpha+\beta+2)x\right]y'(x)\nonumber\\ -& &{}\mathindent{}+n(n+\alpha+\beta+1)y(x)=0,\quad y(x)=P_n^{(\alpha,\beta)}(x). -\end{eqnarray} - -\subsection*{Forward shift operator} -\begin{equation} -\label{shift1Jacobi} -\frac{d}{dx}P_n^{(\alpha,\beta)}(x)=\frac{n+\alpha+\beta+1}{2}P_{n-1}^{(\alpha+1,\beta+1)}(x). -\end{equation} - -\subsection*{Backward shift operator} -\begin{eqnarray} -\label{shift2JacobiI} -& &(1-x^2)\frac{d}{dx}P_n^{(\alpha,\beta)}(x)+ -\left[(\beta-\alpha)-(\alpha+\beta)x\right]P_n^{(\alpha,\beta)}(x)\nonumber\\ -& &{}=-2(n+1)P_{n+1}^{(\alpha-1,\beta-1)}(x) -\end{eqnarray} -or equivalently -\begin{eqnarray} -\label{shift2JacobiII} -& &\frac{d}{dx}\left[(1-x)^\alpha(1+x)^\beta P_n^{(\alpha,\beta)}(x)\right]\nonumber\\ -& &{}=-2(n+1)(1-x)^{\alpha-1}(1+x)^{\beta-1}P_{n+1}^{(\alpha-1,\beta-1)}(x). -\end{eqnarray} - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodJacobi} -(1-x)^{\alpha}(1+x)^{\beta}P_n^{(\alpha,\beta)}(x)= -\frac{(-1)^n}{2^nn!}\left(\frac{d}{dx}\right)^n -\left[(1-x)^{n+\alpha}(1+x)^{n+\beta}\right]. -\end{equation} - -\subsection*{Generating functions} -\begin{equation} -\label{GenJacobi1} -\frac{2^{\alpha+\beta}}{R(1+R-t)^{\alpha}(1+R+t)^{\beta}}= -\sum_{n=0}^{\infty}P_n^{(\alpha,\beta)}(x)t^n,\quad R=\sqrt{1-2xt+t^2}. -\end{equation} - -\begin{eqnarray} -\label{GenJacobi2} -& &\hyp{0}{1}{-}{\alpha+1}{\frac{(x-1)t}{2}}\,\hyp{0}{1}{-}{\beta+1}{\frac{(x+1)t}{2}}\nonumber\\ -& &{}=\sum_{n=0}^{\infty}\frac{P_n^{(\alpha,\beta)}(x)}{(\alpha+1)_n(\beta+1)_n}t^n. -\end{eqnarray} - -\begin{eqnarray} -\label{GenJacobi3} -& &(1-t)^{-\alpha-\beta-1}\,\hyp{2}{1}{\frac{1}{2}(\alpha+\beta+1),\frac{1}{2}(\alpha+\beta+2)} -{\alpha+1}{\frac{2(x-1)t}{(1-t)^2}}\nonumber\\ -& &{}=\sum_{n=0}^{\infty}\frac{(\alpha+\beta+1)_n}{(\alpha+1)_n}P_n^{(\alpha,\beta)}(x)t^n. -\end{eqnarray} - -\begin{eqnarray} -\label{GenJacobi4} -& &(1+t)^{-\alpha-\beta-1}\,\hyp{2}{1}{\frac{1}{2}(\alpha+\beta+1),\frac{1}{2}(\alpha+\beta+2)} -{\beta+1}{\frac{2(x+1)t}{(1+t)^2}}\nonumber\\ -& &{}=\sum_{n=0}^{\infty}\frac{(\alpha+\beta+1)_n}{(\beta+1)_n}P_n^{(\alpha,\beta)}(x)t^n. -\end{eqnarray} - -\begin{eqnarray} -\label{GenJacobi5} -& &\hyp{2}{1}{\gamma,\alpha+\beta+1-\gamma}{\alpha+1}{\frac{1-R-t}{2}}\, -\hyp{2}{1}{\gamma,\alpha+\beta+1-\gamma}{\beta+1}{\frac{1-R+t}{2}}\nonumber\\ -& &{}=\sum_{n=0}^{\infty} -\frac{(\gamma)_n(\alpha+\beta+1-\gamma)_n}{(\alpha+1)_n(\beta+1)_n}P_n^{(\alpha,\beta)}(x)t^n, -\quad R=\sqrt{1-2xt+t^2} -\end{eqnarray} -with $\gamma$ arbitrary. - -\subsection*{Limit relations} - -\subsubsection*{Wilson $\rightarrow$ Jacobi} -The Jacobi polynomials can be found from the Wilson polynomials given by (\ref{DefWilson}) by -substituting $a=b=\frac{1}{2}(\alpha+1)$, $c=\frac{1}{2}(\beta+1)+it$, -$d=\frac{1}{2}(\beta+1)-it$ and $x\rightarrow t\sqrt{\frac{1}{2}(1-x)}$ in the -definition (\ref{DefWilson}) of the Wilson polynomials and taking the limit -$t\rightarrow\infty$. In fact we have -$$\lim_{t\rightarrow\infty} -\frac{W_n(\frac{1}{2}(1-x)t^2;\frac{1}{2}(\alpha+1), -\frac{1}{2}(\alpha+1),\frac{1}{2}(\beta+1)+it,\frac{1}{2}(\beta+1)-it)} -{t^{2n}n!}=P_n^{(\alpha,\beta)}(x).$$ - -\subsubsection*{Continuous Hahn $\rightarrow$ Jacobi} -The Jacobi polynomials follow from the continuous Hahn polynomials given by (\ref{DefContHahn}) -by using the substitution $x\rightarrow \frac{1}{2}xt$, $a=\frac{1}{2}(\alpha+1-it)$, -$b=\frac{1}{2}(\beta+1+it)$, $c=\frac{1}{2}(\alpha+1+it)$ and $d=\frac{1}{2}(\beta+1-it)$ -in (\ref{DefContHahn}), division by $t^n$ and the limit $t\rightarrow\infty$: -$$\lim_{t\rightarrow\infty} -\frac{p_n(\frac{1}{2}xt;\frac{1}{2}(\alpha+1-it),\frac{1}{2}(\beta+1+it), -\frac{1}{2}(\alpha+1+it),\frac{1}{2}(\beta+1-it))}{t^n}=P_n^{(\alpha,\beta)}(x).$$ - -\subsubsection*{Hahn $\rightarrow$ Jacobi} -To find the Jacobi polynomials from the Hahn polynomials given by (\ref{DefHahn}) we take -$x\rightarrow Nx$ in (\ref{DefHahn}) and let $N\rightarrow\infty$. In fact we have -$$\lim_{N\rightarrow\infty} -Q_n(Nx;\alpha,\beta,N)=\frac{P_n^{(\alpha,\beta)}(1-2x)}{P_n^{(\alpha,\beta)}(1)}.$$ - -\subsubsection*{Jacobi $\rightarrow$ Laguerre} -The Laguerre polynomials given by (\ref{DefLaguerre}) can be obtained from the Jacobi -polynomials by setting $x\rightarrow 1-2\beta^{-1}x$ and then the limit $\beta\rightarrow\infty$: -\begin{equation} -\lim_{\beta\rightarrow\infty} -P_n^{(\alpha,\beta)}(1-2\beta^{-1}x)=L_n^{(\alpha)}(x). -\end{equation} - -\subsubsection*{Jacobi $\rightarrow$ Bessel} -The Bessel polynomials given by (\ref{DefBessel}) are obtained from the Jacobi polynomials -if we take $\beta=a-\alpha$ and let $\alpha\rightarrow-\infty$: -\begin{equation} -\lim_{\alpha\rightarrow-\infty} -\frac{P_n^{(\alpha,a-\alpha)}(1+\alpha x)}{P_n^{(\alpha,a-\alpha)}(1)}=y_n(x;a). -\end{equation} - -\subsubsection*{Jacobi $\rightarrow$ Hermite} -The Hermite polynomials given by (\ref{DefHermite}) follow from the Jacobi polynomials -by taking $\beta=\alpha$ and letting $\alpha\rightarrow\infty$ in the following way: -\begin{equation} -\lim_{\alpha\rightarrow\infty} -\alpha^{-\frac{1}{2}n}P_n^{(\alpha,\alpha)}(\alpha^{-\frac{1}{2}}x)=\frac{H_n(x)}{2^nn!}. -\end{equation} - -\subsection*{Remarks} -The definition (\ref{DefJacobi}) of the Jacobi polynomials can also be written as: -$$P_n^{(\alpha,\beta)}(x)=\frac{1}{n!}\sum_{k=0}^n\frac{(-n)_k}{k!} -(n+\alpha+\beta+1)_k(\alpha+k+1)_{n-k}\left(\frac{1-x}{2}\right)^k.$$ -In this way the Jacobi polynomials can also be seen as polynomials in the parameters $\alpha$ -and $\beta$. Therefore they can be defined for all $\alpha$ and $\beta$. Then we have the -following connection with the Meixner polynomials given by (\ref{DefMeixner}): -$$\frac{(\beta)_n}{n!}M_n(x;\beta,c)=P_n^{(\beta-1,-n-\beta-x)}((2-c)c^{-1}).$$ - -\noindent -The Jacobi polynomials are related to the pseudo Jacobi polynomials defined by -(\ref{DefPseudoJacobi}) in the following way: -$$P_n(x;\nu,N)=\frac{(-2i)^nn!}{(n-2N-1)_n}P_n^{(-N-1+i\nu,-N-1-i\nu)}(ix).$$ - -\noindent -The Jacobi polynomials are also related to the Gegenbauer (or ultraspherical) polynomials -given by (\ref{DefGegenbauer}) by the quadratic transformations: -$$C_{2n}^{(\lambda)}(x)=\frac{(\lambda)_n}{(\frac{1}{2})_n} -P_n^{(\lambda-\frac{1}{2},-\frac{1}{2})}(2x^2-1)$$ -and -$$C_{2n+1}^{(\lambda)}(x)=\frac{(\lambda)_{n+1}}{(\frac{1}{2})_{n+1}} -xP_n^{(\lambda-\frac{1}{2},\frac{1}{2})}(2x^2-1).$$ - -\subsection*{References} -\label{sec9.8} -% -\paragraph{Orthogonality relation} -Write the \RHS\ of (9.8.2) as $h_n\,\de_{m,n}$. Then -\begin{equation} -\begin{split} -&\frac{h_n}{h_0}= -\frac{n+\al+\be+1}{2n+\al+\be+1}\, -\frac{(\al+1)_n(\be+1)_n}{(\al+\be+2)_n\,n!}\,,\quad -h_0=\frac{2^{\al+\be+1}\Ga(\al+1)\Ga(\be+1)}{\Ga(\al+\be+2)}\,,\sLP -&\frac{h_n}{h_0\,(P_n^{(\al,\be)}(1))^2}= -\frac{n+\al+\be+1}{2n+\al+\be+1}\, -\frac{(\be+1)_n\,n!}{(\al+1)_n\,(\al+\be+2)_n}\,. -\end{split} -\label{60} -\end{equation} - -In (9.8.3) the numerator factor $\Ga(n+\al+\be+1)$ in the last line should be -$\Ga(\be+1)$. When thus corrected, (9.8.3) can be rewritten as: -\begin{equation} -\begin{split} -&\int_1^\iy P_m^{(\al,\be)}(x)\,P_n^{(\al,\be)}(x)\,(x-1)^\al (x+1)^\be\,dx=h_n\,\de_{m,n}\,,\\ -&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad-1-\be>\al>-1,\quad m,n<-\thalf(\al+\be+1),\\ -&\frac{h_n}{h_0}= -\frac{n+\al+\be+1}{2n+\al+\be+1}\, -\frac{(\al+1)_n(\be+1)_n}{(\al+\be+2)_n\,n!}\,,\quad -h_0=\frac{2^{\al+\be+1}\Ga(\al+1)\Ga(-\al-\be-1)}{\Ga(-\be)}\,. -\end{split} -\label{122} -\end{equation} - -% -\paragraph{Symmetry} -\begin{equation} -P_n^{(\al,\be)}(-x)=(-1)^n\,P_n^{(\be,\al)}(x). -\label{48} -\end{equation} -Use (9.8.2) and (9.8.5b) or see \mycite{DLMF}{Table 18.6.1}. -% -\paragraph{Special values} -\begin{equation} -P_n^{(\al,\be)}(1)=\frac{(\al+1)_n}{n!}\,,\quad -P_n^{(\al,\be)}(-1)=\frac{(-1)^n(\be+1)_n}{n!}\,,\quad -\frac{P_n^{(\al,\be)}(-1)}{P_n^{(\al,\be)}(1)}=\frac{(-1)^n(\be+1)_n}{(\al+1)_n}\,. -\label{50} -\end{equation} -Use (9.8.1) and \eqref{48} or see \mycite{DLMF}{Table 18.6.1}. -% -\paragraph{Generating functions} -Formula (9.8.15) was first obtained by Brafman \myciteKLS{109}. -% -\paragraph{Bilateral generating functions} -For $0\le r<1$ and $x,y\in[-1,1]$ we have in terms of $F_4$ (see~\eqref{62}): -\begin{align} -&\sum_{n=0}^\iy\frac{(\al+\be+1)_n\,n!}{(\al+1)_n(\be+1)_n}\,r^n\, -P_n^{(\al,\be)}(x)\,P_n^{(\al,\be)}(y) -=\frac1{(1+r)^{\al+\be+1}} -\nonumber\\ -&\qquad\quad\times F_4\Big(\thalf(\al+\be+1),\thalf(\al+\be+2);\al+1,\be+1; -\frac{r(1-x)(1-y)}{(1+r)^2},\frac{r(1+x)(1+y)}{(1+r)^2}\Big), -\label{58}\sLP -&\sum_{n=0}^\iy\frac{2n+\al+\be+1}{n+\al+\be+1} -\frac{(\al+\be+2)_n\,n!}{(\al+1)_n(\be+1)_n}\,r^n\, -P_n^{(\al,\be)}(x)\,P_n^{(\al,\be)}(y) -=\frac{1-r}{(1+r)^{\al+\be+2}}\nonumber\\ -&\qquad\quad\times F_4\Big(\thalf(\al+\be+2),\thalf(\al+\be+3);\al+1,\be+1; -\frac{r(1-x)(1-y)}{(1+r)^2},\frac{r(1+x)(1+y)}{(1+r)^2}\Big). -\label{59} -\end{align} -Formulas \eqref{58} and \eqref{59} were first -given by Bailey \myciteKLS{91}{(2.1), (2.3)}. -See Stanton \myciteKLS{485} for a shorter proof. -(However, in the second line of -\myciteKLS{485}{(1)} $z$ and $Z$ should be interchanged.)$\;$ -As observed in Bailey \myciteKLS{91}{p.10}, \eqref{59} follows -from \eqref{58} -by applying the operator $r^{-\half(\al+\be-1)}\,\frac d{dr}\circ r^{\half(\al+\be+1)}$ -to both sides of \eqref{58}. -In view of \eqref{60}, formula \eqref{59} is the Poisson kernel for Jacobi -polynomials. The \RHS\ of \eqref{59} makes clear that this kernel is positive. -See also the discussion in Askey \myciteKLS{46}{following (2.32)}. -% -\paragraph{Quadratic transformations} -\begin{align} -\frac{C_{2n}^{(\al+\half)}(x)}{C_{2n}^{(\al+\half)}(1)} -=\frac{P_{2n}^{(\al,\al)}(x)}{P_{2n}^{(\al,\al)}(1)} -&=\frac{P_n^{(\al,-\half)}(2x^2-1)}{P_n^{(\al,-\half)}(1)}\,, -\label{51}\\ -\frac{C_{2n+1}^{(\al+\half)}(x)}{C_{2n+1}^{(\al+\half)}(1)} -=\frac{P_{2n+1}^{(\al,\al)}(x)}{P_{2n+1}^{(\al,\al)}(1)} -&=\frac{x\,P_n^{(\al,\half)}(2x^2-1)}{P_n^{(\al,\half)}(1)}\,. -\label{52} -\end{align} -See p.221, Remarks, last two formulas together with \eqref{50} and \eqref{49}. -Or see \mycite{DLMF}{(18.7.13), (18.7.14)}. -% -\paragraph{Differentiation formulas} -Each differentiation formula is given in two equivalent forms. -\begin{equation} -\begin{split} -\frac d{dx}\left((1-x)^\al P_n^{(\al,\be)}(x)\right)&= --(n+\al)\,(1-x)^{\al-1} P_n^{(\al-1,\be+1)}(x),\\ -\left((1-x)\frac d{dx}-\al\right)P_n^{(\al,\be)}(x)&= --(n+\al)\,P_n^{(\al-1,\be+1)}(x). -\end{split} -\label{68} -\end{equation} -% -\begin{equation} -\begin{split} -\frac d{dx}\left((1+x)^\be P_n^{(\al,\be)}(x)\right)&= -(n+\be)\,(1+x)^{\be-1} P_n^{(\al+1,\be-1)}(x),\\ -\left((1+x)\frac d{dx}+\be\right)P_n^{(\al,\be)}(x)&= -(n+\be)\,P_n^{(\al+1,\be-1)}(x). -\end{split} -\label{69} -\end{equation} -Formulas \eqref{68} and \eqref{69} follow from -\mycite{DLMF}{(15.5.4), (15.5.6)} -together with (9.8.1). They also follow from each other by \eqref{48}. -% -\paragraph{Generalized Gegenbauer polynomials} -These are defined by -\begin{equation} -S_{2m}^{(\al,\be)}(x):=\const P_m^{(\al,\be)}(2x^2-1),\qquad -S_{2m+1}^{(\al,\be)}(x):=\const x\,P_m^{(\al,\be+1)}(2x^2-1) -\label{70} -\end{equation} -in the notation of \myciteKLS{146}{p.156} -(see also \cite{K27}), while \cite[Section 1.5.2]{K26} -has $C_n^{(\la,\mu)}(x)=\const\allowbreak\times S_n^{(\la-\half,\mu-\half)}(x)$. -For $\al,\be>-1$ we have the orthogonality relation -\begin{equation} -\int_{-1}^1 S_m^{(\al,\be)}(x)\,S_n^{(\al,\be)}(x)\,|x|^{2\be+1}(1-x^2)^\al\,dx -=0\qquad(m\ne n). -\label{71} -\end{equation} -For $\be=\al-1$ generalized Gegenbauer polynomials are limit cases of -continuous $q$-ultraspherical polynomials, see \eqref{176}. - -If we define the {\em Dunkl operator} $T_\mu$ by -\begin{equation} -(T_\mu f)(x):=f'(x)+\mu\,\frac{f(x)-f(-x)}x -\label{72} -\end{equation} -and if we choose the constants in \eqref{70} as -\begin{equation} -S_{2m}^{(\al,\be)}(x)=\frac{(\al+\be+1)_m}{(\be+1)_m}\, P_m^{(\al,\be)}(2x^2-1),\quad -S_{2m+1}^{(\al,\be)}(x)=\frac{(\al+\be+1)_{m+1}}{(\be+1)_{m+1}}\, -x\,P_m^{(\al,\be+1)}(2x^2-1) -\label{73} -\end{equation} -then (see \cite[(1.6)]{K5}) -\begin{equation} -T_{\be+\half}S_n^{(\al,\be)}=2(\al+\be+1)\,S_{n-1}^{(\al+1,\be)}. -\label{74} -\end{equation} -Formula \eqref{74} with \eqref{73} substituted gives rise to two -differentiation formulas involving Jacobi polynomials which are equivalent to -(9.8.7) and \eqref{69}. - -Composition of \eqref{74} with itself gives -\[ -T_{\be+\half}^2S_n^{(\al,\be)}=4(\al+\be+1)(\al+\be+2)\,S_{n-2}^{(\al+2,\be)}, -\] -which is equivalent to the composition of (9.8.7) and \eqref{69}: -\begin{equation} -\left(\frac{d^2}{dx^2}+\frac{2\be+1}x\,\frac d{dx}\right)P_n^{(\al,\be)}(2x^2-1) -=4(n+\al+\be+1)(n+\be)\,P_{n-1}^{(\al+2,\be)}(2x^2-1). -\label{75} -\end{equation} -Formula \eqref{75} was also given in \myciteKLS{322}{(2.4)}. -% -\cite{Abram}, \cite{Ahmed+82}, \cite{Allaway89}, \cite{NAlSalam66}, \cite{AlSalam64}, -\cite{AlSalamChihara72}, \cite{AndrewsAskey85}, \cite{AndrewsAskeyRoy}, \cite{Askey68}, -\cite{Askey70}, \cite{Askey72}, \cite{Askey73}, \cite{Askey74}, \cite{Askey75}, \cite{Askey78}, -\cite{Askey89I}, \cite{Askey2005}, \cite{AskeyFitch}, \cite{AskeyGasper71I}, \cite{AskeyGasper71II}, -\cite{AskeyGasper76}, \cite{AskeyWainger}, \cite{AskeyWilson85}, \cite{Bailey38}, \cite{Brafman51}, -\cite{Brown}, \cite{Carlitz61II}, \cite{Carlitz67}, \cite{ChenIsmail91}, \cite{Chihara78}, -\cite{Chow+}, \cite{Ciesielski}, \cite{Cooper+}, \cite{DetteStudden92}, \cite{DetteStudden95}, -\cite{DijksmaKoorn}, \cite{DimitrovRafaeli}, \cite{DimitrovRodrigues}, \cite{Doha2002I}, -\cite{Doha2003I}, \cite{Doha2004II}, \cite{Dunkl84}, \cite{ElbertLaforgia87II}, -\cite{ElbertLaforgiaRodono}, \cite{Erdelyi+}, \cite{Faldey}, \cite{FoataLeroux}, -\cite{Gasper69}, \cite{Gasper70I}, \cite{Gasper70II}, \cite{Gasper71I}, \cite{Gasper71II}, -\cite{Gasper72I}, \cite{Gasper72II}, \cite{Gasper73I}, \cite{Gasper73II}, \cite{Gasper74}, -\cite{Gasper77}, \cite{Gatteschi87}, \cite{Gautschi2008}, \cite{Gautschi2009I}, -\cite{Gautschi2009II}, \cite{GautschiLeopardi}, \cite{GawronskiShawyer}, \cite{Godoy+}, -\cite{Grad}, \cite{Hajmirzaahmad94}, \cite{HartmannStephan}, \cite{Horton}, \cite{Ismail74}, -\cite{Ismail77}, \cite{Ismail96}, \cite{Ismail2005II}, \cite{IsmailLi}, -\cite{IsmailMassonRahman}, \cite{Kochneff97I}, \cite{Koekoek99}, \cite{Koekoek2000}, -\cite{Koelink96II}, \cite{Koorn73}, \cite{Koorn74}, \cite{Koorn75}, \cite{Koorn77II}, -\cite{Koorn78}, \cite{Koorn72}, \cite{Koorn85}, \cite{Koorn88}, \cite{KuijlaarsMartinez}, -\cite{Kuijlaars+}, \cite{LabelleYehI}, \cite{LabelleYehII}, \cite{Laine}, \cite{Lesky95II}, -\cite{Lesky96}, \cite{Li96}, \cite{Li97}, \cite{LopezTemme2004}, \cite{Luke}, \cite{Martinez}, -\cite{Mathai}, \cite{Meijer}, \cite{Miller89}, \cite{MoakSaffVarga}, \cite{Nikiforov+}, -\cite{NikiforovUvarov}, \cite{Olver}, \cite{Prasad}, \cite{Rahman76I}, \cite{Rahman76II}, -\cite{Rahman77}, \cite{Rahman81I}, \cite{RahmanShah}, \cite{Rainville}, \cite{Rusev}, -\cite{Shi}, \cite{Srivastava69II}, \cite{Srivastava71}, \cite{Srivastava82}, -\cite{SrivastavaSinghal}, \cite{Stanton80I}, \cite{Stanton90}, \cite{Szego75}, \cite{Temme}, -\cite{Vertesi}, \cite{Wimp87}, \cite{WongZhang94}, \cite{WongZhang2006}, \cite{Zarzo+}, -\cite{Zayed}. - -\section*{Special cases} - -\subsection{Gegenbauer / Ultraspherical}\index{Gegenbauer polynomials} -\index{Ultraspherical polynomials} - -\par - -\subsection*{Hypergeometric representation} -The Gegenbauer (or ultraspherical) polynomials are Jacobi polynomials with -$\alpha=\beta=\lambda-\frac{1}{2}$ and another normalization: -\begin{eqnarray} -\label{DefGegenbauer} -C_n^{(\lambda)}(x)&=&\frac{(2\lambda)_n}{(\lambda+\frac{1}{2})_n} -P_n^{(\lambda-\frac{1}{2},\lambda-\frac{1}{2})}(x)\nonumber\\ -&=&\frac{(2\lambda)_n}{n!}\,\hyp{2}{1}{-n,n+2\lambda} -{\lambda+\frac{1}{2}}{\frac{1-x}{2}},\quad\lambda\neq 0. -\end{eqnarray} - -\subsection*{Orthogonality relation} -\begin{eqnarray} -\label{OrtGegenbauer} -& &\int_{-1}^1(1-x^2)^{\lambda-\frac{1}{2}}C_m^{(\lambda)}(x)C_n^{(\lambda)}(x)\,dx\nonumber\\ -& &{}=\frac{\pi\Gamma(n+2\lambda)2^{1-2\lambda}}{\left\{\Gamma(\lambda)\right\}^2(n+\lambda)n!}\,\delta_{mn}, -\quad\lambda>-\frac{1}{2}\quad\lambda\neq 0. -\end{eqnarray} - -\subsection*{Recurrence relation} -\begin{equation} -\label{RecGegenbauer} -2(n+\lambda)xC_n^{(\lambda)}(x)=(n+1)C_{n+1}^{(\lambda)}(x)+(n+2\lambda-1)C_{n-1}^{(\lambda)}(x). -\end{equation} - -\subsection*{Normalized recurrence relation} -\begin{equation} -\label{NormRecGegenbauer} -xp_n(x)=p_{n+1}(x)+\frac{n(n+2\lambda-1)}{4(n+\lambda-1)(n+\lambda)}p_{n-1}(x), -\end{equation} -where -$$C_n^{(\lambda)}(x)=\frac{2^n(\lambda)_n}{n!}p_n(x).$$ - -\subsection*{Differential equation} -\begin{equation} -\label{dvGegenbauer} -(1-x^2)y''(x)-(2\lambda+1)xy'(x)+n(n+2\lambda)y(x)=0,\quad y(x)=C_n^{(\lambda)}(x). -\end{equation} - -\subsection*{Forward shift operator} -\begin{equation} -\label{shift1Gegenbauer} -\frac{d}{dx}C_n^{(\lambda)}(x)=2\lambda C_{n-1}^{(\lambda+1)}(x). -\end{equation} - -\subsection*{Backward shift operator} -\begin{equation} -\label{shift2GegenbauerI} -(1-x^2)\frac{d}{dx}C_n^{(\lambda)}(x)+(1-2\lambda)xC_n^{(\lambda)}(x)= --\frac{(n+1)(2\lambda+n-1)}{2(\lambda-1)} C_{n+1}^{(\lambda-1)}(x) -\end{equation} -or equivalently -\begin{eqnarray} -\label{shift2GegenbauerII} -& &\frac{d}{dx}\left[(1-x^2)^{\lambda-\textstyle\frac{1}{2}}C_n^{(\lambda)}(x)\right]\nonumber\\ -& &{}=-\frac{(n+1)(2\lambda+n-1)}{2(\lambda-1)}(1-x^2)^{\lambda-\textstyle\frac{3}{2}}C_{n+1}^{(\lambda-1)}(x). -\end{eqnarray} - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodGegenbauer} -(1-x^2)^{\lambda-\frac{1}{2}}C_n^{(\lambda)}(x)= -\frac{(2\lambda)_n(-1)^n}{(\lambda+\frac{1}{2})_n2^nn!}\left(\frac{d}{dx}\right)^n -\left[(1-x^2)^{\lambda+n-\frac{1}{2}}\right]. -\end{equation} - -\subsection*{Generating functions} -\begin{equation} -\label{GenGegenbauer1} -(1-2xt+t^2)^{-\lambda}=\sum_{n=0}^{\infty}C_n^{(\lambda)}(x)t^n. -\end{equation} - -\begin{equation} -\label{GenGegenbauer2} -R^{-1}\left(\frac{1+R-xt}{2}\right)^{\frac{1}{2}-\lambda}=\sum_{n=0}^{\infty} -\frac{(\lambda+\frac{1}{2})_n}{(2\lambda)_n}C_n^{(\lambda)}(x)t^n, -\quad R=\sqrt{1-2xt+t^2}. -\end{equation} - -\begin{equation} -\label{GenGegenbauer3} -\hyp{0}{1}{-}{\lambda+\frac{1}{2}}{\frac{(x-1)t}{2}}\, -\hyp{0}{1}{-}{\lambda+\frac{1}{2}}{\frac{(x+1)t}{2}} -=\sum_{n=0}^{\infty}\frac{C_n^{(\lambda)}(x)} -{(2\lambda)_n(\lambda+\frac{1}{2})_n}t^n. -\end{equation} - -\begin{equation} -\label{GenGegenbauer4} -\e^{xt}\,\hyp{0}{1}{-}{\lambda+\frac{1}{2}}{\frac{(x^2-1)t^2}{4}}= -\sum_{n=0}^{\infty}\frac{C_n^{(\lambda)}(x)}{(2\lambda)_n}t^n. -\end{equation} - -\begin{eqnarray} -\label{GenGegenbauer5} -& &\hyp{2}{1}{\gamma,2\lambda-\gamma}{\lambda+\frac{1}{2}}{\frac{1-R-t}{2}}\, -\hyp{2}{1}{\gamma,2\lambda-\gamma}{\lambda+\frac{1}{2}}{\frac{1-R+t}{2}}\nonumber\\ -& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n(2\lambda-\gamma)_n} -{(2\lambda)_n(\lambda+\frac{1}{2})_n}C_n^{(\lambda)}(x)t^n, -\quad R=\sqrt{1-2xt+t^2},\quad\textrm{$\gamma$ arbitrary}. -\end{eqnarray} - -\begin{eqnarray} -\label{GenGegenbauer6} -& &(1-xt)^{-\gamma}\,\hyp{2}{1}{\frac{1}{2}\gamma,\frac{1}{2}\gamma+\frac{1}{2}} -{\lambda+\frac{1}{2}}{\frac{(x^2-1)t^2}{(1-xt)^2}}\nonumber\\ -& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n}{(2\lambda)_n}C_n^{(\lambda)}(x)t^n, -\quad\textrm{$\gamma$ arbitrary}. -\end{eqnarray} - -\subsection*{Limit relation} - -\subsubsection*{Gegenbauer / Ultraspherical $\rightarrow$ Hermite} -The Hermite polynomials given by (\ref{DefHermite}) follow from the Gegenbauer (or -ultraspherical) polynomials by taking $\lambda=\alpha+\frac{1}{2}$ and letting -$\alpha\rightarrow\infty$ in the following way: -\begin{equation} -\lim_{\alpha\rightarrow\infty} -\alpha^{-\frac{1}{2}n}C_n^{(\alpha+\frac{1}{2})}(\alpha^{-\frac{1}{2}}x)=\frac{H_n(x)}{n!}. -\end{equation} - -\subsection*{Remarks} -The case $\lambda=0$ needs another normalization. In that case we have the -Chebyshev polynomials of the first kind described in the next subsection. - -\noindent -The Gegenbauer (or ultraspherical) polynomials are related to the Jacobi polynomials -given by (\ref{DefJacobi}) by the quadratic transformations: -$$C_{2n}^{(\lambda)}(x)=\frac{(\lambda)_n}{(\frac{1}{2})_n} -P_n^{(\lambda-\frac{1}{2},-\frac{1}{2})}(2x^2-1)$$ -and -$$C_{2n+1}^{(\lambda)}(x)=\frac{(\lambda)_{n+1}}{(\frac{1}{2})_{n+1}} -xP_n^{(\lambda-\frac{1}{2},\frac{1}{2})}(2x^2-1).$$ - -\subsection*{References} -\label{sec9.8.1} -% -\paragraph{Notation} -Here the Gegenbauer polynomial is denoted by $C_n^\la$ instead of $C_n^{(\la)}$. -% -\paragraph{Orthogonality relation} -Write the \RHS\ of (9.8.20) as $h_n\,\de_{m,n}$. Then -\begin{equation} -\frac{h_n}{h_0}= -\frac\la{\la+n}\,\frac{(2\la)_n}{n!}\,,\quad -h_0=\frac{\pi^\half\,\Ga(\la+\thalf)}{\Ga(\la+1)},\quad -\frac{h_n}{h_0\,(C_n^\la(1))^2}= -\frac\la{\la+n}\,\frac{n!}{(2\la)_n}\,. -\label{61} -\end{equation} -% -\paragraph{Hypergeometric representation} -Beside (9.8.19) we have also -\begin{equation} -C_n^\lambda(x)=\sum_{\ell=0}^{\lfloor n/2\rfloor}\frac{(-1)^{\ell}(\lambda)_{n-\ell}} -{\ell!\;(n-2\ell)!}\,(2x)^{n-2\ell} -=(2x)^{n}\,\frac{(\lambda)_{n}}{n!}\, -\hyp21{-\thalf n,-\thalf n+\thalf}{1-\la-n}{\frac1{x^2}}. -\label{57} -\end{equation} -See \mycite{DLMF}{(18.5.10)}. -% -\paragraph{Special value} -\begin{equation} -C_n^{\la}(1)=\frac{(2\la)_n}{n!}\,. -\label{49} -\end{equation} -Use (9.8.19) or see \mycite{DLMF}{Table 18.6.1}. -% -\paragraph{Expression in terms of Jacobi} -% -\begin{equation} -\frac{C_n^\la(x)}{C_n^\la(1)}= -\frac{P_n^{(\la-\half,\la-\half)}(x)}{P_n^{(\la-\half,\la-\half)}(1)}\,,\qquad -C_n^\la(x)=\frac{(2\la)_n}{(\la+\thalf)_n}\,P_n^{(\la-\half,\la-\half)}(x). -\label{65} -\end{equation} -% -\paragraph{Re: (9.8.21)} -By iteration of recurrence relation (9.8.21): -\begin{multline} -x^2 C_n^\la(x)= -\frac{(n+1)(n+2)}{4(n+\la)(n+\la+1)}\,C_{n+2}^\la(x)+ -\frac{n^2+2n\la+\la-1}{2(n+\la-1)(n+\la+1)}\,C_n^\la(x)\\ -+\frac{(n+2\la-1)(n+2\la-2)}{4(n+\la)(n+\la-1)}\,C_{n-2}^\la(x). -\label{6} -\end{multline} -% -\paragraph{Bilateral generating functions} -\begin{multline} -\sum_{n=0}^\iy\frac{n!}{(2\la)_n}\,r^n\,C_n^\la(x)\,C_n^\la(y) -=\frac1{(1-2rxy+r^2)^\la}\,\hyp21{\thalf\la,\thalf(\la+1)}{\la+\thalf} -{\frac{4r^2(1-x^2)(1-y^2)}{(1-2rxy+r^2)^2}}\\ -(r\in(-1,1),\;x,y\in[-1,1]). -\label{66} -\end{multline} -For the proof put $\be:=\al$ in \eqref{58}, then use \eqref{63} and \eqref{65}. -The Poisson kernel for Gegenbauer polynomials can be derived in a similar way -from \eqref{59}, or alternatively by applying the operator -$r^{-\la+1}\frac d{dr}\circ r^\la$ to both sides of \eqref{66}: -\begin{multline} -\sum_{n=0}^\iy\frac{\la+n}\la\,\frac{n!}{(2\la)_n}\,r^n\,C_n^\la(x)\,C_n^\la(y) -=\frac{1-r^2}{(1-2rxy+r^2)^{\la+1}}\\ -\times\hyp21{\thalf(\la+1),\thalf(\la+2)}{\la+\thalf} -{\frac{4r^2(1-x^2)(1-y^2)}{(1-2rxy+r^2)^2}}\qquad -(r\in(-1,1),\;x,y\in[-1,1]). -\label{67} -\end{multline} -Formula \eqref{67} was obtained by Gasper \& Rahman \myciteKLS{234}{(4.4)} -as a limit case of their formula for the Poisson kernel for continuous -$q$-ultraspherical polynomials. -% -\paragraph{Trigonometric expansions} -By \mycite{DLMF}{(18.5.11), (15.8.1)}: -\begin{align} -C_n^{\la}(\cos\tha) -&=\sum_{k=0}^n\frac{(\la)_k(\la)_{n-k}}{k!\,(n-k)!}\,e^{i(n-2k)\tha} -=e^{in\tha}\frac{(\la)_n}{n!}\, -\hyp21{-n,\la}{1-\la-n}{e^{-2i\tha}}\label{103}\\ -&=\frac{(\la)_n}{2^\la n!}\, -e^{-\half i\la\pi}e^{i(n+\la)\tha}\,(\sin\tha)^{-\la}\, -\hyp21{\la,1-\la}{1-\la-n}{\frac{i e^{-i\tha}}{2\sin\tha}}\label{104}\\ -&=\frac{(\la)_n}{n!}\,\sum_{k=0}^\iy\frac{(\la)_k(1-\la)_k}{(1-\la-n)_k k!}\, -\frac{\cos((n-k+\la)\tha+\thalf(k-\la)\pi)}{(2\sin\tha)^{k+\la}}\,.\label{105} -\end{align} -In \eqref{104} and \eqref{105} we require that -$\tfrac16\pi<\tha<\tfrac56\pi$. Then the convergence is absolute for $\la>\thalf$ -and conditional for $0<\la\le\thalf$. - -By \mycite{DLMF}{(14.13.1), (14.3.21), (15.8.1)]}: -\begin{align} -C_n^\la(\cos\tha)&=\frac{2\Ga(\la+\thalf)}{\pi^\half\Ga(\la+1)}\, -\frac{(2\la)_n}{(\la+1)_n}\,(\sin\tha)^{1-2\la}\, -\sum_{k=0}^\iy\frac{(1-\la)_k(n+1)_k}{(n+\la+1)_k k!}\, -\sin\big((2k+n+1)\tha\big) -\label{7}\\ -&=\frac{2\Ga(\la+\thalf)}{\pi^\half\Ga(\la+1)}\, -\frac{(2\la)_n}{(\la+1)_n}\,(\sin\tha)^{1-2\la}\, -\Im\!\!\left(e^{i(n+1)\tha}\,\hyp21{1-\la,n+1}{n+\la+1}{e^{2i\tha}}\right)\nonumber\\ -&=\frac{2^\la\Ga(\la+\thalf)}{\pi^\half\Ga(\la+1)}\, -\frac{(2\la)_n}{(\la+1)_n}\,(\sin\tha)^{-\la}\, -\Re\!\!\left(e^{-\thalf i\la\pi}e^{i(n+\la)\tha}\, -\hyp21{\la,1-\la}{1+\la+n}{\frac{e^{i\tha}}{2i\sin\tha}}\right)\nonumber\\ -&=\frac{2^{2\la}\Ga(\la+\thalf)}{\pi^\half\Ga(\la+1)}\,\frac{(2\la)_n}{(\la+1)_n}\, -\sum_{k=0}^\iy\frac{(\la)_k(1-\la)_k}{(1+\la+n)_k k!}\, -\frac{\cos((n+k+\la)\tha-\thalf(k+\la)\pi)}{(2\sin\tha)^{k+\la}}\,. -\label{106} -\end{align} -We require that $0<\tha<\pi$ in \eqref{7} and $\tfrac16\pi<\tha<\tfrac56\pi$ in -\eqref{106} The convergence is absolute for $\la>\thalf$ and conditional for -$0<\la\le\thalf$. -For $\la\in\Zpos$ the above series terminate after the term with -$k=\la-1$. -Formulas \eqref{7} and \eqref{106} are also given in -\mycite{Sz}{(4.9.22), (4.9.25)}. -% -\paragraph{Fourier transform} -\begin{equation} -\frac{\Ga(\la+1)}{\Ga(\la+\thalf)\,\Ga(\thalf)}\, -\int_{-1}^1 \frac{C_n^\la(y)}{C_n^\la(1)}\,(1-y^2)^{\la-\half}\, -e^{ixy}\,dy -=i^n\,2^\la\,\Ga(\la+1)\,x^{-\la}\,J_{\la+n}(x). -\label{8} -\end{equation} -See \mycite{DLMF}{(18.17.17) and (18.17.18)}. -% -\paragraph{Laplace transforms} -\begin{equation} -\frac2{n!\,\Ga(\la)}\, -\int_0^\iy H_n(tx)\,t^{n+2\la-1}\,e^{-t^2}\,dt=C_n^\la(x). -\label{56} -\end{equation} -See Nielsen \cite[p.48, (4) with p.47, (1) and p.28, (10)]{K4} (1918) -or Feldheim \cite[(28)]{K3} (1942). -\begin{equation} -\frac2{\Ga(\la+\thalf)}\,\int_0^1 \frac{C_n^\la(t)}{C_n^\la(1)}\, -(1-t^2)^{\la-\half}\,t^{-1}\,(x/t)^{n+2\la+1}\,e^{-x^2/t^2}\,dt -=2^{-n}\,H_n(x)\,e^{-x^2}\quad(\la>-\thalf). -\label{46} -\end{equation} -Use Askey \& Fitch \cite[(3.29)]{K2} for $\al=\pm\thalf$ together with -\eqref{48}, \eqref{51}, \eqref{52}, \eqref{54} and \eqref{55}. -\paragraph{Addition formula} (see \mycite{AAR}{(9.8.5$'$)]}) -\begin{multline} -R_n^{(\al,\al)}\big(xy+(1-x^2)^\half(1-y^2)^\half t\big) -=\sum_{k=0}^n \frac{(-1)^k(-n)_k\,(n+2\al+1)_k}{2^{2k}((\al+1)_k)^2}\\ -\times(1-x^2)^{k/2} R_{n-k}^{(\al+k,\al+k)}(x)\,(1-y^2)^{k/2} R_{n-k}^{(\al+k,\al+k)}(y)\, -\om_k^{(\al-\half,\al-\half)}\,R_k^{(\al-\half,\al-\half)}(t), -\label{108} -\end{multline} -where -\[ -R_n^{(\al,\be)}(x):=P_n^{(\al,\be)}(x)/P_n^{(\al,\be)}(1),\quad -\om_n^{(\al,\be)}:=\frac{\int_{-1}^1 (1-x)^\al(1+x)^\be\,dx} -{\int_{-1}^1 (R_n^{(\al,\be)}(x))^2\,(1-x)^\al(1+x)^\be\,dx}\,. -\] -% -\cite{Abram}, \cite{Ahmed+86}, \cite{AndrewsAskeyRoy}, \cite{Area+I}, \cite{Askey67}, \cite{Askey74}, \cite{Askey75}, -\cite{Askey89I}, \cite{AskeyFitch}, \cite{AskeyKoornRahman}, \cite{Berg}, \cite{BilodeauI}, -\cite{BojanovNikolov}, \cite{Brafman51}, \cite{Brafman57I}, \cite{Brown}, -\cite{BustozIsmail82}, \cite{BustozIsmail83}, \cite{BustozIsmail89}, \cite{BustozSavage79}, -\cite{BustozSavage80}, \cite{Carlitz61II}, \cite{Chihara78}, \cite{Common}, \cite{Danese}, -\cite{Dette94}, \cite{DijksmaKoorn}, \cite{DilcherStolarsky}, \cite{Dimitrov96}, -\cite{Dimitrov2003}, \cite{Doha2002II}, \cite{Driver}, \cite{ElbertLaforgia86I}, -\cite{ElbertLaforgia86II}, \cite{ElbertLaforgia90}, \cite{Erdelyi+}, \cite{Exton96}, -\cite{Gasper69}, \cite{Gasper72II}, \cite{Gasper85}, \cite{Grad}, \cite{Ismail74}, -\cite{Ismail2005II}, \cite{Koekoek2000}, \cite{Koorn88}, \cite{Laforgia}, \cite{LewanowiczI}, -\cite{Lorch}, \cite{Mathai}, \cite{Nagel}, \cite{Nikiforov+}, \cite{NikiforovUvarov}, -\cite{RahmanShah}, \cite{Rainville}, \cite{Reimer}, \cite{Sartoretto}, \cite{Srivastava71}, -\cite{Szego75}, \cite{Temme}, \cite{Viswanathan}, \cite{Zayed}. - -\subsection{Chebyshev}\index{Chebyshev polynomials} - -\par - -\subsection*{Hypergeometric representation} -The Chebyshev polynomials of the first kind can be obtained from the Jacobi -polynomials by taking $\alpha=\beta=-\frac{1}{2}$: -\begin{equation} -\label{DefChebyshevI} -T_n(x)=\frac{P_n^{(-\frac{1}{2},-\frac{1}{2})}(x)}{P_n^{(-\frac{1}{2},-\frac{1}{2})}(1)} -=\hyp{2}{1}{-n,n}{\frac{1}{2}}{\frac{1-x}{2}} -\end{equation} -and the Chebyshev polynomials of the second kind can be obtained from the -Jacobi polynomials by taking $\alpha=\beta=\frac{1}{2}$: -\begin{equation} -\label{DefChebyshevII} -U_n(x)=(n+1)\frac{P_n^{(\frac{1}{2},\frac{1}{2})}(x)}{P_n^{(\frac{1}{2},\frac{1}{2})}(1)} -=(n+1)\,\hyp{2}{1}{-n,n+2}{\frac{3}{2}}{\frac{1-x}{2}}. -\end{equation} - -\subsection*{Orthogonality relation} -\begin{equation} -\label{OrtChebyshevI} -\int_{-1}^1(1-x^2)^{-\frac{1}{2}}T_m(x)T_n(x)\,dx= -\left\{\begin{array}{ll} -\displaystyle\frac{\cpi}{2}\,\delta_{mn}, & n\neq 0\\[5mm] -\cpi\,\delta_{mn}, & n=0. -\end{array}\right. -\end{equation} - -\begin{equation} -\label{OrtChebyshevII} -\int_{-1}^1(1-x^2)^{\frac{1}{2}}U_m(x)U_n(x)\,dx=\frac{\cpi}{2}\,\delta_{mn}. -\end{equation} - -\subsection*{Recurrence relations} -\begin{equation} -\label{RecChebyshevI} -2xT_n(x)=T_{n+1}(x)+T_{n-1}(x),\quad T_{0}(x)=1\quad\textrm{and}\quad T_1(x)=x. -\end{equation} - -\begin{equation} -\label{RecChebyshevII} -2xU_n(x)=U_{n+1}(x)+U_{n-1}(x),\quad U_{0}(x)=1\quad\textrm{and}\quad U_1(x)=2x. -\end{equation} - -\subsection*{Normalized recurrence relations} -\begin{equation} -\label{NormRecChebyshevI} -xp_n(x)=p_{n+1}(x)+\frac{1}{4}p_{n-1}(x), -\end{equation} -where -$$T_1(x)=p_1(x)=x\quad\textrm{and}\quad T_n(x)=2^np_n(x),\quad n\neq 1.$$ - -\begin{equation} -\label{NormRecChebyshevII} -xp_n(x)=p_{n+1}(x)+\frac{1}{4}p_{n-1}(x), -\end{equation} -where -$$U_n(x)=2^np_n(x).$$ - -\subsection*{Differential equations} -\begin{equation} -\label{dvChebyshevI} -(1-x^2)y''(x)-xy'(x)+n^2y(x)=0,\quad y(x)=T_n(x). -\end{equation} - -\begin{equation} -\label{dvChebyshevII} -(1-x^2)y''(x)-3xy'(x)+n(n+2)y(x)=0,\quad y(x)=U_n(x). -\end{equation} - -\subsection*{Forward shift operator} -\begin{equation} -\label{shift1Chebyshev} -\frac{d}{dx}T_n(x)=nU_{n-1}(x). -\end{equation} - -\subsection*{Backward shift operator} -\begin{equation} -\label{shift2ChebyshevI} -(1-x^2)\frac{d}{dx}U_n(x)-xU_n(x)=-(n+1)T_{n+1}(x) -\end{equation} -or equivalently -\begin{equation} -\label{shift2ChebyshevII} -\frac{d}{dx}\left[\left(1-x^2\right)^{\frac{1}{2}}U_n(x)\right] -=-(n+1)\left(1-x^2\right)^{-\frac{1}{2}}T_{n+1}(x). -\end{equation} - -\subsection*{Rodrigues-type formulas} -\begin{equation} -\label{RodChebyshevI} -(1-x^2)^{-\frac{1}{2}}T_n(x)=\frac{(-1)^n}{(\frac{1}{2})_n2^n} -\left(\frac{d}{dx}\right)^n\left[(1-x^2)^{n-\frac{1}{2}}\right]. -\end{equation} - -\begin{equation} -\label{RodChebyshevII} -(1-x^2)^{\frac{1}{2}}U_n(x)=\frac{(n+1)(-1)^n}{(\frac{3}{2})_n2^n} -\left(\frac{d}{dx}\right)^n\left[(1-x^2)^{n+\frac{1}{2}}\right]. -\end{equation} - -\subsection*{Generating functions} -\begin{equation} -\label{GenChebyshevI1} -\frac{1-xt}{1-2xt+t^2}=\sum_{n=0}^{\infty}T_n(x)t^n. -\end{equation} - -\begin{equation} -\label{GenChebyshevI2} -R^{-1}\sqrt{\frac{1}{2}(1+R-xt)}=\sum_{n=0}^{\infty} -\frac{\left(\frac{1}{2}\right)_n}{n!}T_n(x)t^n,\quad R=\sqrt{1-2xt+t^2}. -\end{equation} - -\begin{equation} -\label{GenChebyshevI3} -\hyp{0}{1}{-}{\frac{1}{2}}{\frac{(x-1)t}{2}}\, -\hyp{0}{1}{-}{\frac{1}{2}}{\frac{(x+1)t}{2}}= -\sum_{n=0}^{\infty}\frac{T_n(x)}{\left(\frac{1}{2}\right)_nn!}t^n. -\end{equation} - -\begin{equation} -\label{GenChebyshevI4} -\e^{xt}\,\hyp{0}{1}{-}{\frac{1}{2}}{\frac{(x^2-1)t^2}{4}}= -\sum_{n=0}^{\infty}\frac{T_n(x)}{n!}t^n. -\end{equation} - -\begin{eqnarray} -\label{GenChebyshevI5} -& &\hyp{2}{1}{\gamma,-\gamma}{\frac{1}{2}}{\frac{1-R-t}{2}}\, -\hyp{2}{1}{\gamma,-\gamma}{\frac{1}{2}}{\frac{1-R+t}{2}}\nonumber\\ -& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n(-\gamma)_n}{\left(\frac{1}{2}\right)_nn!}T_n(x)t^n, -\quad R=\sqrt{1-2xt+t^2},\quad\textrm{$\gamma$ arbitrary}. -\end{eqnarray} - -\begin{eqnarray} -\label{GenChebyshevI6} -& &(1-xt)^{-\gamma}\,\hyp{2}{1}{\frac{1}{2}\gamma,\frac{1}{2}\gamma+\frac{1}{2}}{\frac{1}{2}} -{\frac{(x^2-1)t^2}{(1-xt)^2}}\nonumber\\ -& &=\sum_{n=0}^{\infty}\frac{(\gamma)_n}{n!}T_n(x)t^n,\quad\textrm{$\gamma$ arbitrary}. -\end{eqnarray} - -\begin{equation} -\label{GenChebyshevII1} -\frac{1}{1-2xt+t^2}=\sum_{n=0}^{\infty}U_n(x)t^n. -\end{equation} - -\begin{equation} -\label{GenChebyshevII2} -\frac{1}{R\sqrt{\frac{1}{2}(1+R-xt)}}=\sum_{n=0}^{\infty} -\frac{\left(\frac{3}{2}\right)_n}{(n+1)!}U_n(x)t^n,\quad R=\sqrt{1-2xt+t^2}. -\end{equation} - -\begin{equation} -\label{GenChebyshevII3} -\hyp{0}{1}{-}{\frac{3}{2}}{\frac{(x-1)t}{2}}\, -\hyp{0}{1}{-}{\frac{3}{2}}{\frac{(x+1)t}{2}}= -\sum_{n=0}^{\infty}\frac{U_n(x)}{\left(\frac{3}{2}\right)_n(n+1)!}t^n. -\end{equation} - -\begin{equation} -\label{GenChebyshevII4} -\e^{xt}\,\hyp{0}{1}{-}{\frac{3}{2}}{\frac{(x^2-1)t^2}{4}}= -\sum_{n=0}^{\infty}\frac{U_n(x)}{(n+1)!}t^n. -\end{equation} - -\begin{eqnarray} -\label{GenChebyshevII5} -& &\hyp{2}{1}{\gamma,2-\gamma}{\frac{3}{2}}{\frac{1-R-t}{2}}\, -\hyp{2}{1}{\gamma,2-\gamma}{\frac{3}{2}}{\frac{1-R+t}{2}}\nonumber\\ -& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n(2-\gamma)_n}{\left(\frac{3}{2}\right)_n(n+1)!}U_n(x)t^n, -\quad R=\sqrt{1-2xt+t^2},\quad\textrm{$\gamma$ arbitrary}. -\end{eqnarray} - -\begin{eqnarray} -\label{GenChebyshevII6} -& &(1-xt)^{-\gamma}\,\hyp{2}{1}{\frac{1}{2}\gamma,\frac{1}{2}\gamma+\frac{1}{2}}{\frac{3}{2}} -{\frac{(x^2-1)t^2}{(1-xt)^2}}\nonumber\\ -& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n}{(n+1)!}U_n(x)t^n,\quad\textrm{$\gamma$ arbitrary}. -\end{eqnarray} - -\subsection*{Remarks} -The Chebyshev polynomials can also be written as: -$$T_n(x)=\cos(n\theta),\quad x=\cos\theta$$ -and -$$U_n(x)=\frac{\sin (n+1)\theta}{\sin\theta},\quad x=\cos\theta.$$ -Further we have -$$U_n(x)=C_n^{(1)}(x)$$ -where $C_n^{(\lambda)}(x)$ denotes the Gegenbauer (or ultraspherical) -polynomial given by (\ref{DefGegenbauer}) in the preceding subsection. - -\subsection*{References} -\label{sec9.8.2} -In addition to the Chebyshev polynomials $T_n$ of the first kind (9.8.35) -and $U_n$ of the second kind (9.8.36), -\begin{align} -T_n(x)&:=\frac{P_n^{(-\half,-\half)}(x)}{P_n^{(-\half,-\half)}(1)} -=\cos(n\tha),\quad x=\cos\tha,\\ -U_n(x)&:=(n+1)\,\frac{P_n^{(\half,\half)}(x)}{P_n^{(\half,\half)}(1)} -=\frac{\sin((n+1)\tha)}{\sin\tha}\,,\quad x=\cos\tha, -\end{align} -we have Chebyshev polynomials $V_n$ {\em of the third kind} -and $W_n$ {\em of the fourth kind}, -\begin{align} -V_n(x)&:=\frac{P_n^{(-\half,\half)}(x)}{P_n^{(-\half,\half)}(1)} -=\frac{\cos((n+\thalf)\tha)}{\cos(\thalf\tha)}\,,\quad x=\cos\tha,\\ -W_n(x)&:=(2n+1)\,\frac{P_n^{(\half,-\half)}(x)}{P_n^{(\half,-\half)}(1)} -=\frac{\sin((n+\thalf)\tha)}{\sin(\thalf\tha)}\,,\quad x=\cos\tha, -\end{align} -see \cite[Section 1.2.3]{K20}. Then there is the symmetry -\begin{equation} -V_n(-x)=(-1)^n W_n(x). -\label{140} -\end{equation} - -The names of Chebyshev polynomials of the third and fourth kind -and the notation $V_n(x)$ are due to Gautschi \cite{K21}. -The notation $W_n(x)$ was first used by Mason \cite{K22}. -Names and notations for Chebyshev polynomials of the third and fourth -kind are interchanged in \mycite{AAR}{Remark 2.5.3} and -\mycite{DLMF}{Table 18.3.1}. -% -\cite{Abram}, \cite{AskeyFitch}, \cite{AskeyGasperHarris}, \cite{AskeyIsmail76}, -\cite{Bavinck95}, \cite{Chihara78}, \cite{Danese}, \cite{DilcherStolarsky}, -\cite{Erdelyi+}, \cite{Grad}, \cite{HartmannStephan}, \cite{Ismail2005II}, \cite{Koekoek2000}, -\cite{Luke}, \cite{Mathai}, \cite{Nikiforov+}, \cite{NikiforovUvarov}, \cite{Rainville}, -\cite{Rayes+}, \cite{Rivlin}, \cite{SainteViennot}, \cite{Szego75}, \cite{Temme}, -\cite{Wilson70I}, \cite{Zayed}, \cite{Zhang}, \cite{ZhangWang}. - -\subsection{Legendre / Spherical}\index{Legendre polynomials} -\index{Spherical polynomials} - -\par - -\subsection*{Hypergeometric representation} -The Legendre (or spherical) polynomials are Jacobi polynomials with $\alpha=\beta=0$: -\begin{equation} -\label{DefLegendre} -P_n(x)=P_n^{(0,0)}(x)=\hyp{2}{1}{-n,n+1}{1}{\frac{1-x}{2}}. -\end{equation} - -\subsection*{Orthogonality relation} -\begin{equation} -\label{OrtLegendre} -\int_{-1}^1P_m(x)P_n(x)\,dx=\frac{2}{2n+1}\,\delta_{mn}. -\end{equation} - -\subsection*{Recurrence relation} -\begin{equation} -\label{RecLegendre} -(2n+1)xP_n(x)=(n+1)P_{n+1}(x)+nP_{n-1}(x). -\end{equation} - -\subsection*{Normalized recurrence relation} -\begin{equation} -\label{NormRecLegendre} -xp_n(x)=p_{n+1}(x)+\frac{n^2}{(2n-1)(2n+1)}p_{n-1}(x), -\end{equation} -where -$$P_n(x)=\binom{2n}{n}\frac{1}{2^n}p_n(x).$$ - -\subsection*{Differential equation} -\begin{equation} -\label{dvLegendre} -(1-x^2)y''(x)-2xy'(x)+n(n+1)y(x)=0,\quad y(x)=P_n(x). -\end{equation} - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodLegendre} -P_n(x)=\frac{(-1)^n}{2^nn!}\left(\frac{d}{dx}\right)^n\left[(1-x^2)^n\right]. -\end{equation} - -\subsection*{Generating functions} -\begin{equation} -\label{GenLegendre1} -\frac{1}{\sqrt{1-2xt+t^2}}=\sum_{n=0}^{\infty}P_n(x)t^n. -\end{equation} - -\begin{equation} -\label{GenLegendre2} -\hyp{0}{1}{-}{1}{\frac{(x-1)t}{2}}\,\hyp{0}{1}{-}{1}{\frac{(x+1)t}{2}}= -\sum_{n=0}^{\infty}\frac{P_n(x)}{(n!)^2}t^n. -\end{equation} - -\begin{equation} -\label{GenLegendre3} -\e^{xt}\,\hyp{0}{1}{-}{1}{\frac{(x^2-1)t^2}{4}}= -\sum_{n=0}^{\infty}\frac{P_n(x)}{n!}t^n. -\end{equation} - -\begin{eqnarray} -\label{GenLegendre4} -& &\hyp{2}{1}{\gamma,1-\gamma}{1}{\frac{1-R-t}{2}}\, -\hyp{2}{1}{\gamma,1-\gamma}{1}{\frac{1-R+t}{2}}\nonumber\\ -& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n(1-\gamma)_n}{(n!)^2}P_n(x)t^n, -\quad R=\sqrt{1-2xt+t^2},\quad\textrm{$\gamma$ arbitrary}. -\end{eqnarray} - -\begin{eqnarray} -\label{GenLegendre5} -& &(1-xt)^{-\gamma}\,\hyp{2}{1}{\frac{1}{2}\gamma,\frac{1}{2}\gamma+\frac{1}{2}}{1} -{\frac{(x^2-1)t^2}{(1-xt)^2}}\nonumber\\ -& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n}{n!}P_n(x)t^n,\quad\textrm{$\gamma$ arbitrary}. -\end{eqnarray} - -\subsection*{References} -\cite{Abram}, \cite{Alladi}, \cite{AlSalam90}, \cite{Bhonsle}, \cite{Brafman51}, -\cite{Carlitz57II}, \cite{Chihara78}, \cite{Danese}, \cite{Dattoli2001}, -\cite{DilcherStolarsky}, \cite{ElbertLaforgia94}, \cite{Erdelyi+}, \cite{Grad}, -\cite{Mathai}, \cite{Nikiforov+}, \cite{NikiforovUvarov}, \cite{Olver}, \cite{Rainville}, -\cite{Szego75}, \cite{Temme}, \cite{Zayed}. - -\newpage - -\section{Pseudo Jacobi}\index{Pseudo Jacobi polynomials}\index{Jacobi polynomials!Pseudo} - -\par\setcounter{equation}{0} - -\subsection*{Hypergeometric representation} -\begin{eqnarray} -\label{DefPseudoJacobi} -P_n(x;\nu,N)&=&\frac{(-2i)^n(-N+i\nu)_n}{(n-2N-1)_n}\,\hyp{2}{1}{-n,n-2N-1}{-N+i\nu}{\frac{1-ix}{2}}\\ -&=&(x+i)^n\,\hyp{2}{1}{-n,N+1-n-i\nu}{2N+2-2n}{\frac{2}{1-ix}},\quad n=0,1,2,\ldots,N.\nonumber -\end{eqnarray} - -\subsection*{Orthogonality relation} -\begin{eqnarray} -\label{OrtPseudoJacobi} -& &\frac{1}{2\pi}\int_{-\infty}^{\infty}(1+x^2)^{-N-1}\e^{2\nu\arctan x}P_m(x;\nu,N)P_n(x;\nu,N)\,dx\nonumber\\ -& &{}=\frac{\Gamma(2N+1-2n)\Gamma(2N+2-2n)2^{2n-2N-1}n!}{\Gamma(2N+2-n)\left|\Gamma(N+1-n+i\nu)\right|^2}\,\delta_{mn}. -\end{eqnarray} - -\subsection*{Recurrence relation} -\begin{eqnarray} -\label{RecPseudoJacobi} -xP_n(x;\nu,N)&=&P_{n+1}(x;\nu,N)+\frac{(N+1)\nu}{(n-N-1)(n-N)}P_n(x;\nu,N)\nonumber\\ -& &{}\mathindent{}-\frac{n(n-2N-2)}{(2n-2N-3)(n-N-1)^2(2n-2N-1)}\nonumber\\ -& &{}\mathindent\mathindent\times(n-N-1-i\nu)(n-N-1+i\nu)P_{n-1}(x;\nu,N). -\end{eqnarray} - -\subsection*{Normalized recurrence relation} -\begin{eqnarray} -\label{NormRecPseudoJacobi} -xp_n(x)&=&p_{n+1}(x)+\frac{(N+1)\nu}{(n-N-1)(n-N)}p_n(x)\nonumber\\ -& &{}\mathindent{}-\frac{n(n-2N-2)(n-N-1-i\nu)(n-N-1+i\nu)}{(2n-2N-3)(n-N-1)^2(2n-2N-1)}p_{n-1}(x), -\end{eqnarray} -where -$$P_n(x;\nu,N)=p_n(x).$$ - -\subsection*{Differential equation} -\begin{equation} -\label{dvPseudoJacobi} -(1+x^2)y''(x)+2\left(\nu-Nx\right)y'(x)-n(n-2N-1)y(x)=0, -\end{equation} -where -$$y(x)=P_n(x;\nu,N).$$ - -\subsection*{Forward shift operator} -\begin{equation} -\label{shift1PseudoJacobi} -\frac{d}{dx}P_n(x;\nu,N)=nP_{n-1}(x;\nu,N-1). -\end{equation} - -\subsection*{Backward shift operator} -\begin{eqnarray} -\label{shift2PseudoJacobiI} -& &(1+x^2)\frac{d}{dx}P_n(x;\nu,N)+2\left[\nu-(N+1)x\right]P_n(x;\nu,N)\nonumber\\ -& &{}=(n-2N-2)P_{n+1}(x;\nu,N+1) -\end{eqnarray} -or equivalently -\begin{eqnarray} -\label{shift2PseudoJacobiII} -& &\frac{d}{dx}\left[(1+x^2)^{-N-1}\e^{2\nu\arctan x}P_n(x;\nu,N)\right]\nonumber\\ -& &{}=(n-2N-2)(1+x^2)^{-N-2}\e^{2\nu\arctan x}P_{n+1}(x;\nu,N+1). -\end{eqnarray} - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodPseudoJacobi} -P_n(x;\nu,N)=\frac{(1+x^2)^{N+1}\expe^{-2\nu\arctan x}}{(n-2N-1)_n} -\left(\frac{d}{dx}\right)^n\left[(1+x^2)^{n-N-1}\expe^{2\nu\arctan x}\right]. -\end{equation} - -\subsection*{Generating function} -\begin{eqnarray} -\label{GenPseudoJacobi} -& &\left[\hyp{0}{1}{-}{-N+i\nu}{(x+i)t}\,\hyp{0}{1}{-}{-N-i\nu}{(x-i)t}\right]_N\nonumber\\ -& &{}=\sum_{n=0}^N\frac{(n-2N-1)_n}{(-N+i\nu)_n(-N-i\nu)_nn!}P_n(x;\nu,N)t^n. -\end{eqnarray} - -\subsection*{Limit relation} - -\subsubsection*{Continuous Hahn $\rightarrow$ Pseudo Jacobi} -The pseudo Jacobi polynomials follow from the continuous Hahn polynomials given by -(\ref{DefContHahn}) by the substitutions $x\rightarrow xt$, $a=\frac{1}{2}(-N+i\nu-2t)$, -$b=\frac{1}{2}(-N-i\nu+2t)$, $c=\frac{1}{2}(-N-i\nu-2t)$ and $d=\frac{1}{2}(-N+i\nu+2t)$, -division by $t^n$ and the limit $t\rightarrow\infty$: -\begin{eqnarray*} -& &\lim_{t\rightarrow\infty}\frac{p_n(xt;\frac{1}{2}(-N+i\nu-2t),\frac{1}{2}(-N-i\nu+2t), -\frac{1}{2}(-N+i\nu-2t),\frac{1}{2}(-N-i\nu+2t))}{t^n}\\ -& &{}=\frac{(n-2N-1)_n}{n!}P_n(x;\nu,N). -\end{eqnarray*} - -\subsection*{Remarks} -Since we have for $k 0$ & -% $0 < c < 1$ } -\end{equation} - -\subsection*{Recurrence relation} -\begin{eqnarray} -\label{RecMeixner} -(c-1)xM_n(x;\beta,c)&=&c(n+\beta)M_{n+1}(x;\beta,c)\nonumber\\ -& &{}\mathindent{}-\left[n+(n+\beta)c\right]M_n(x;\beta,c)+nM_{n-1}(x;\beta,c). -\end{eqnarray} - -\subsection*{Normalized recurrence relation} -\begin{equation} -\label{NormRecMeixner} -xp_n(x)=p_{n+1}(x)+\frac{n+(n+\beta)c}{1-c}p_n(x)+ -\frac{n(n+\beta-1)c}{(1-c)^2}p_{n-1}(x), -\end{equation} -where -$$M_n(x;\beta,c)=\frac{1}{(\beta)_n}\left(\frac{c-1}{c}\right)^np_n(x).$$ - -\subsection*{Difference equation} -\begin{equation} -\label{dvMeixner} -n(c-1)y(x)=c(x+\beta)y(x+1)-\left[x+(x+\beta)c\right]y(x)+xy(x-1), -\end{equation} -where -$$y(x)=M_n(x;\beta,c).$$ - -\subsection*{Forward shift operator} -\begin{equation} -\label{shift1MeixnerI} -M_n(x+1;\beta,c)-M_n(x;\beta,c)= -\frac{n}{\beta}\left(\frac{c-1}{c}\right)M_{n-1}(x;\beta+1,c) -\end{equation} -or equivalently -\begin{equation} -\label{shift1MeixnerII} -\Delta M_n(x;\beta,c)=\frac{n}{\beta}\left(\frac{c-1}{c}\right)M_{n-1}(x;\beta+1,c). -\end{equation} - -\subsection*{Backward shift operator} -\begin{equation} -\label{shift2MeixnerI} -c(\beta+x-1)M_n(x;\beta,c)-xM_n(x-1;\beta,c)=c(\beta-1)M_{n+1}(x;\beta-1,c) -\end{equation} -or equivalently -\begin{equation} -\label{shift2MeixnerII} -\nabla\left[\frac{(\beta)_xc^x}{x!}M_n(x;\beta,c)\right]= -\frac{(\beta-1)_xc^x}{x!}M_{n+1}(x;\beta-1,c). -\end{equation} - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodMeixner} -\frac{(\beta)_xc^x}{x!}M_n(x;\beta,c)=\nabla^n\left[\frac{(\beta+n)_xc^x}{x!}\right]. -\end{equation} - -\subsection*{Generating functions} -\begin{equation} -\label{GenMeixner1} -\left(1-\frac{t}{c}\right)^x(1-t)^{-x-\beta}= -\sum_{n=0}^{\infty}\frac{(\beta)_n}{n!}M_n(x;\beta,c)t^n. -\end{equation} - -\begin{equation} -\label{GenMeixner2} -\e^t\,\hyp{1}{1}{-x}{\beta}{\left(\frac{1-c}{c}\right)t}= -\sum_{n=0}^{\infty}\frac{M_n(x;\beta,c)}{n!}t^n. -\end{equation} - -\begin{equation} -\label{GenMeixner3} -(1-t)^{-\gamma}\,\hyp{2}{1}{\gamma,-x}{\beta}{\frac{(1-c)t}{c(1-t)}} -=\sum_{n=0}^{\infty}\frac{(\gamma)_n}{n!}M_n(x;\beta,c)t^n, -\quad\textrm{$\gamma$ arbitrary}. -\end{equation} - -\subsection*{Limit relations} - -\subsubsection*{Hahn $\rightarrow$ Meixner} -If we take $\alpha=b-1$, $\beta=N(1-c)c^{-1}$ in the definition (\ref{DefHahn}) -of the Hahn polynomials and let $N\rightarrow\infty$ we find the Meixner polynomials: -$$\lim_{N\rightarrow\infty} -Q_n(x;b-1,N(1-c)c^{-1},N)=M_n(x;b,c).$$ - -\subsubsection*{Dual Hahn $\rightarrow$ Meixner} -To obtain the Meixner polynomials from the dual Hahn polynomials we have to take -$\gamma=\beta-1$ and $\delta=N(1-c)c^{-1}$ in the definition (\ref{DefDualHahn}) of -the dual Hahn polynomials and let $N\rightarrow\infty$: -$$\lim_{N\rightarrow\infty} -R_n(\lambda(x);\beta-1,N(1-c)c^{-1},N)=M_n(x;\beta,c).$$ - -\subsubsection*{Meixner $\rightarrow$ Laguerre} -The Laguerre polynomials given by (\ref{DefLaguerre}) are obtained from the Meixner polynomials -if we take $\beta=\alpha+1$ and $x\rightarrow (1-c)^{-1}x$ and let $c\rightarrow 1$: -\begin{equation} -\lim_{c\rightarrow 1} -M_n((1-c)^{-1}x;\alpha+1,c)=\frac{L_n^{(\alpha)}(x)}{L_n^{(\alpha)}(0)}. -\end{equation} - -\subsubsection*{Meixner $\rightarrow$ Charlier} -The Charlier polynomials given by (\ref{DefCharlier}) are obtained from the Meixner polynomials -if we take $c=(a+\beta)^{-1}a$ and let $\beta\rightarrow\infty$: -\begin{equation} -\lim_{\beta\rightarrow\infty} -M_n(x;\beta,(a+\beta)^{-1}a)=C_n(x;a). -\end{equation} - -\subsection*{Remarks} -The Meixner polynomials are related to the Jacobi polynomials given by (\ref{DefJacobi}) -in the following way: -$$\frac{(\beta)_n}{n!}M_n(x;\beta,c)=P_n^{(\beta-1,-n-\beta-x)}((2-c)c^{-1}).$$ - -\noindent -The Meixner polynomials are also related to the Krawtchouk polynomials given by -(\ref{DefKrawtchouk}) in the following way: -$$K_n(x;p,N)=M_n(x;-N,(p-1)^{-1}p).$$ - -\subsection*{References} -\label{sec9.9} -In this section in \mycite{KLS} the pseudo Jacobi polynomial $P_n(x;\nu,N)$ in (9.9.1) -is considered -for $N\in\ZZ_{\ge0}$ and $n=0,1,\ldots,n$. However, we can more generally take -$-\thalf\be>0\;\;{\rm or}\;\;\al<\be<0). -\] -Then $P_n$ can be expressed as a Meixner polynomial: -\[ -P_n(x)=(-k_2(\al\be)^{-1})_n\,\be^n\, -M_n\left(-\,\frac{x+k_2\al^{-1}}{\al-\be},-k_2(\al\be)^{-1},\be\al^{-1}\right). -\] - -In 1938 Gottlieb \cite[\S2]{K1} introduces polynomials $l_n$ ``of Laguerre type'' -which turn out to be special Meixner polynomials: -$l_n(x)=e^{-n\la} M_n(x;1,e^{-\la})$. -% -\paragraph{Uniqueness of orthogonality measure} -The coefficient of $p_{n-1}(x)$ in (9.10.4) behaves as $O(n^2)$ as $n\to\iy$. -Hence \eqref{93} holds, by which the orthogonality measure is unique. -% -\cite{AlSalam90}, \cite{AndrewsAskey85}, \cite{Area+II}, \cite{Askey75}, \cite{Askey89I}, -\cite{AskeyGasper77}, \cite{AskeyWilson85}, \cite{AtakRahmanSuslov}, \cite{Campigotto+}, -\cite{LChiharaStanton}, \cite{Chihara78}, \cite{Dette95}, \cite{Dominici}, \cite{Dunkl76}, -\cite{Dunkl84}, \cite{DunklRamirez}, \cite{Erdelyi+}, \cite{FeinsilverSchott}, -\cite{Gasper73I}, \cite{Gasper74}, \cite{HoareRahman}, \cite{Ismail2005II}, \cite{Karlin58}, -\cite{Koorn82}, \cite{Koorn88}, \cite{LabelleYehI}, \cite{LabelleYehII}, \cite{Lesky62}, -\cite{Lesky89}, \cite{Lesky94I}, \cite{Lesky95II}, \cite{LewanowiczII}, \cite{Nikiforov+}, -\cite{NikiforovUvarov}, \cite{Qiu}, \cite{Rahman78I}, \cite{Rahman79}, \cite{Stanton84}, -\cite{Stanton90}, \cite{Szego75}, \cite{Zarzo+}, \cite{Zeng90}. - - -\section{Laguerre}\index{Laguerre polynomials} - -\par\setcounter{equation}{0} - -\subsection*{Hypergeometric representation} -\begin{equation} -\label{DefLaguerre} -L_n^{(\alpha)}(x)=\frac{(\alpha+1)_n}{n!}\,\hyp{1}{1}{-n}{\alpha+1}{x}. -\end{equation} - -\subsection*{Orthogonality relation} -\begin{equation} -\label{OrtLaguerre} -\int_0^{\infty}\expe^{-x}x^{\alpha}L_m^{(\alpha)}(x)L_n^{(\alpha)}(x)\,dx= -\frac{\Gamma(n+\alpha+1)}{n!}\,\delta_{mn},\quad\alpha > -1. -\end{equation} - -\subsection*{Recurrence relation} -\begin{equation} -\label{RecLaguerre} -(n+1)L_{n+1}^{(\alpha)}(x)-(2n+\alpha+1-x)L_n^{(\alpha)}(x)+(n+\alpha)L_{n-1}^{(\alpha)}(x)=0. -\end{equation} - -\subsection*{Normalized recurrence relation} -\begin{equation} -\label{NormRecLaguerre} -xp_n(x)=p_{n+1}(x)+(2n+\alpha+1)p_n(x)+n(n+\alpha)p_{n-1}(x), -\end{equation} -where -$$L_n^{(\alpha)}(x)=\frac{(-1)^n}{n!}p_n(x).$$ - -\subsection*{Differential equation} -\begin{equation} -\label{dvLaguerre} -xy''(x)+(\alpha+1-x)y'(x)+ny(x)=0,\quad y(x)=L_n^{(\alpha)}(x). -\end{equation} - -\newpage - -\subsection*{Forward shift operator} -\begin{equation} -\label{shift1Laguerre} -\frac{d}{dx}L_n^{(\alpha)}(x)=-L_{n-1}^{(\alpha+1)}(x). -\end{equation} - -\subsection*{Backward shift operator} -\begin{equation} -\label{shift2LaguerreI} -x\frac{d}{dx}L_n^{(\alpha)}(x)+(\alpha-x)L_n^{(\alpha)}(x)=(n+1)L_{n+1}^{(\alpha-1)}(x) -\end{equation} -or equivalently -\begin{equation} -\label{shift2LaguerreII} -\frac{d}{dx}\left[\expe^{-x}x^{\alpha}L_n^{(\alpha)}(x)\right]=(n+1)\expe^{-x}x^{\alpha-1}L^{(\alpha-1)}_{n+1}(x). -\end{equation} - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodLaguerre} -\e^{-x}x^{\alpha}L_n^{(\alpha)}(x)=\frac{1}{n!}\left(\frac{d}{dx}\right)^n\left[\expe^{-x}x^{n+\alpha}\right]. -\end{equation} - -\subsection*{Generating functions} -\begin{equation} -\label{GenLaguerre1} -(1-t)^{-\alpha-1}\exp\left(\frac{xt}{t-1}\right)= -\sum_{n=0}^{\infty}L_n^{(\alpha)}(x)t^n. -\end{equation} - -\begin{equation} -\label{GenLaguerre2} -\e^t\,\hyp{0}{1}{-}{\alpha+1}{-xt} -=\sum_{n=0}^{\infty}\frac{L_n^{(\alpha)}(x)}{(\alpha+1)_n}t^n. -\end{equation} - -\begin{equation} -\label{GenLaguerre3} -(1-t)^{-\gamma}\,\hyp{1}{1}{\gamma}{\alpha+1}{\frac{xt}{t-1}} -=\sum_{n=0}^{\infty}\frac{(\gamma)_n}{(\alpha+1)_n}L_n^{(\alpha)}(x)t^n, -\quad\textrm{$\gamma$ arbitrary}. -\end{equation} - -\subsection*{Limit relations} - -\subsubsection*{Meixner-Pollaczek $\rightarrow$ Laguerre} -The Laguerre polynomials can be obtained from the Meixner-Pollaczek polynomials given by -(\ref{DefMP}) by the substitution $\lambda=\frac{1}{2}(\alpha+1)$, -$x\rightarrow -\frac{1}{2}\phi^{-1}x$ and the limit $\phi\rightarrow 0$: -$$\lim_{\phi\rightarrow 0} -P_n^{(\frac{1}{2}\alpha+\frac{1}{2})}(-\textstyle\frac{1}{2}\phi^{-1}x;\phi)=L_n^{(\alpha)}(x).$$ - -\subsubsection*{Jacobi $\rightarrow$ Laguerre} -The Laguerre polynomials are obtained from the Jacobi polynomials given by (\ref{DefJacobi}) -if we set $x\rightarrow 1-2\beta^{-1}x$ and then take the limit $\beta\rightarrow\infty$: -$$\lim_{\beta\rightarrow\infty} -P_n^{(\alpha,\beta)}(1-2\beta^{-1}x)=L_n^{(\alpha)}(x).$$ - -\subsubsection*{Meixner $\rightarrow$ Laguerre} -If we take $\beta=\alpha+1$ and $x\rightarrow (1-c)^{-1}x$ in the definition -(\ref{DefMeixner}) of the Meixner polynomials and let $c\rightarrow 1$ we obtain -the Laguerre polynomials: -$$\lim_{c\rightarrow 1} -M_n((1-c)^{-1}x;\alpha+1,c)=\frac{L_n^{(\alpha)}(x)}{L_n^{(\alpha)}(0)}.$$ - -\subsubsection*{Laguerre $\rightarrow$ Hermite} -The Hermite polynomials given by (\ref{DefHermite}) can be obtained from the Laguerre -polynomials by taking the limit $\alpha\rightarrow\infty$ in the following way: -\begin{equation} -\lim_{\alpha\rightarrow\infty} -\left(\frac{2}{\alpha}\right)^{\frac{1}{2}n} -L_n^{(\alpha)}((2\alpha)^{\frac{1}{2}}x+\alpha)=\frac{(-1)^n}{n!}H_n(x). -\end{equation} - -\subsection*{Remarks} -The definition (\ref{DefLaguerre}) of the Laguerre polynomials can also be -written as: -$$L_n^{(\alpha)}(x)=\frac{1}{n!}\sum_{k=0}^n\frac{(-n)_k}{k!}(\alpha+k+1)_{n-k}x^k.$$ -In this way the Laguerre polynomials can also be seen as polynomials in the parameter $\alpha$. -Therefore they can be defined for all $\alpha$. - -\noindent -The Laguerre polynomials are related to the Bessel polynomials given by (\ref{DefBessel}) -in the following way: -$$L_n^{(\alpha)}(x)=\frac{(-x)^n}{n!}y_n(2x^{-1};-2n-\alpha-1).$$ - -\noindent -The Laguerre polynomials are related to the Charlier polynomials given by (\ref{DefCharlier}) -in the following way: -$$\frac{(-a)^n}{n!}C_n(x;a)=L_n^{(x-n)}(a).$$ - -\noindent -The Laguerre polynomials and the Hermite polynomials given by (\ref{DefHermite}) are also -connected by the following quadratic transformations: -$$H_{2n}(x)=(-1)^nn!\,2^{2n}L_n^{(-\frac{1}{2})}(x^2)$$ -and -$$H_{2n+1}(x)=(-1)^nn!\,2^{2n+1}xL_n^{(\frac{1}{2})}(x^2).$$ - -\noindent -In combinatorics the Laguerre polynomials with $\alpha=0$ are often called Rook -polynomials. - -\subsection*{References} -\label{sec9.11} -% -\paragraph{Special values} -By (9.11.1) and the binomial formula: -\begin{equation} -K_n(0;p,N)=1,\qquad -K_n(N;p,N)=(1-p^{-1})^n. -\label{9} -\end{equation} -The self-duality (p.240, Remarks, first formula) -\begin{equation} -K_n(x;p,N)=K_x(n;p,N)\qquad (n,x\in \{0,1,\ldots,N\}) -\label{147} -\end{equation} -combined with \eqref{9} yields: -\begin{equation} -K_N(x;p,N)=(1-p^{-1})^x\qquad(x\in\{0,1,\ldots,N\}). -\label{148} -\end{equation} -% -\paragraph{Symmetry} -By the orthogonality relation (9.11.2): -\begin{equation} -\frac{K_n(N-x;p,N)}{K_n(N;p,N)}=K_n(x;1-p,N). -\label{10} -\end{equation} -By \eqref{10} and \eqref{147} we have also -\begin{equation} -\frac{K_{N-n}(x;p,N)}{K_N(x;p,N)}=K_n(x;1-p,N) -\qquad(n,x\in\{0,1,\ldots,N\}), -\label{149} -\end{equation} -and, by \eqref{149}, \eqref{10} and \eqref{9}, -\begin{equation} -K_{N-n}(N-x;p,N)=\left(\frac p{p-1}\right)^{n+x-N}K_n(x;p,N) -\qquad(n,x\in\{0,1,\ldots,N\}). -\label{150} -\end{equation} -A particular case of \eqref{10} is: -\begin{equation} -K_n(N-x;\thalf,N)=(-1)^n K_n(x;\thalf,N). -\label{11} -\end{equation} -Hence -\begin{equation} -K_{2m+1}(N;\thalf,2N)=0. -\label{12} -\end{equation} -From (9.11.11): -\begin{equation} -K_{2m}(N;\thalf,2N)=\frac{(\thalf)_m}{(-N+\thalf)_m}\,. -\label{13} -\end{equation} -% -\paragraph{Quadratic transformations} -\begin{align} -K_{2m}(x+N;\thalf,2N)&=\frac{(\thalf)_m}{(-N+\thalf)_m}\, -R_m(x^2;-\thalf,-\thalf,N), -\label{31}\\ -K_{2m+1}(x+N;\thalf,2N)&=-\,\frac{(\tfrac32)_m}{N\,(-N+\thalf)_m}\, -x\,R_m(x^2-1;\thalf,\thalf,N-1), -\label{33}\\ -K_{2m}(x+N+1;\thalf,2N+1)&=\frac{(\tfrac12)_m}{(-N-\thalf)_m}\, -R_m(x(x+1);-\thalf,\thalf,N), -\label{32}\\ -K_{2m+1}(x+N+1;\thalf,2N+1)&=\frac{(\tfrac32)_m}{(-N-\thalf)_{m+1}}\, -(x+\thalf)\,R_m(x(x+1);\thalf,-\thalf,N), -\label{34} -\end{align} -where $R_m$ is a dual Hahn polynomial (9.6.1). For the proofs use -(9.6.2), (9.11.2), (9.6.4) and (9.11.4). -% -\paragraph{Generating functions} -\begin{multline} -\sum_{x=0}^N\binom Nx K_m(x;p,N)K_n(x;q,N)z^x\\ -=\left(\frac{p-z+pz}p\right)^m -\left(\frac{q-z+qz}q\right)^n -(1+z)^{N-m-n} -K_m\left(n;-\,\frac{(p-z+pz)(q-z+qz)}z,N\right). -\label{107} -\end{multline} -This follows immediately from Rosengren \cite[(3.5)]{K8}, which goes back -to Meixner \cite{K9}. -% -\cite{Abdul}, \cite{Abram}, \cite{Ahmed+82}, \cite{Allaway76}, \cite{Allaway91}, -\cite{NAlSalam66}, \cite{AlSalam64}, \cite{AlSalam90}, \cite{AlSalamChihara72}, -\cite{AlSalamChihara76}, \cite{AndrewsAskey85}, \cite{AndrewsAskeyRoy}, \cite{Askey68}, -\cite{Askey75}, \cite{Askey89I}, \cite{Askey2005}, \cite{AskeyGasper76}, \cite{AskeyGasper77}, -\cite{AskeyIsmail76}, \cite{AskeyIsmailKoorn}, \cite{AskeyWilson85}, \cite{Barrett}, -\cite{Bavinck96}, \cite{Brafman51}, \cite{Brafman57II}, \cite{Brenke}, \cite{Brown}, -\cite{BustozSavage79}, \cite{BustozSavage80}, \cite{Carlitz60}, \cite{Carlitz61I}, -\cite{Carlitz61II}, \cite{Carlitz62}, \cite{Carlitz68}, \cite{ChenSrivastava}, -\cite{ChenIsmail91}, \cite{Chihara68II}, \cite{Chihara78}, \cite{Cohen}, \cite{Cooper+}, -\cite{Dattoli2001}, \cite{DetteStudden92}, \cite{DetteStudden95}, \cite{Dimitrov2003}, -\cite{Doha2003II}, \cite{ElbertLaforgia87I}, \cite{Erdelyi}, \cite{Erdelyi+}, \cite{Exton98}, -\cite{Faldey}, \cite{Gasper73II}, \cite{Gasper77}, \cite{Gatteschi2002}, \cite{Gawronski87}, -\cite{Gawronski93}, \cite{Gillis}, \cite{GillisIsmailOffer}, \cite{Godoy+}, \cite{Grad}, -\cite{Hajmirzaahmad95}, \cite{Ismail74}, \cite{Ismail77}, \cite{Ismail2005II}, -\cite{IsmailLetVal88}, \cite{IsmailLi}, \cite{IsmailMassonRahman}, \cite{IsmailMuldoon}, -\cite{IsmailStanton97}, \cite{IsmailTamhankar}, \cite{Jackson}, \cite{Karlin58}, -\cite{Kochneff95}, \cite{Kochneff97II}, \cite{Koekoek2000}, \cite{Koorn77I}, \cite{Koorn78}, -\cite{Koorn85}, \cite{Koorn88}, \cite{Krasikov2003}, \cite{Krasikov2005}, -\cite{KuijlaarsMcLaughlin}, \cite{KwonLittle}, \cite{LabelleYehI}, \cite{LabelleYehII}, -\cite{LabelleYehIII}, \cite{Lee97I}, \cite{Lee97II}, \cite{Lee97III}, \cite{Lesky95II}, -\cite{Lesky96}, \cite{LewanowiczI}, \cite{LopezTemme2004}, \cite{Mathai}, \cite{Meixner}, -\cite{Nikiforov+}, \cite{NikiforovUvarov}, \cite{Olver}, \cite{PerezPinar}, \cite{Pittaluga}, -\cite{Rainville}, \cite{SainteViennot}, \cite{Srivastava66}, \cite{Srivastava69I}, -\cite{Srivastava69II}, \cite{Srivastava70}, \cite{Srivastava71}, \cite{Szego75}, \cite{Temme}, -\cite{Trickovic}, \cite{Trivedi}, \cite{Viennot}. - - -\section{Bessel}\index{Bessel polynomials} - -\par\setcounter{equation}{0} - -\subsection*{Hypergeometric representation} -\begin{eqnarray} -\label{DefBessel} -y_n(x;a)&=&\hyp{2}{0}{-n,n+a+1}{-}{-\frac{x}{2}}\\ -&=&(n+a+1)_n\left(\frac{x}{2}\right)^n\,\hyp{1}{1}{-n}{-2n-a}{\frac{2}{x}}, -\quad n=0,1,2,\ldots,N.\nonumber -\end{eqnarray} - -\subsection*{Orthogonality relation} -\begin{eqnarray} -\label{OrtBessel} -& &\int_0^{\infty}x^a\e^{-\frac{2}{x}}y_m(x;a)y_n(x;a)\,dx\nonumber\\ -& &=-\frac{2^{a+1}}{2n+a+1}\Gamma(-n-a)n!\,\delta_{mn},\quad a<-2N-1. -\end{eqnarray} - -\subsection*{Recurrence relation} -\begin{eqnarray} -\label{RecBessel} -& &2(n+a+1)(2n+a)y_{n+1}(x;a)\nonumber\\ -& &{}=(2n+a+1)\left[2a+(2n+a)(2n+a+2)x\right]y_n(x;a)\nonumber\\ -& &{}\mathindent{}+2n(2n+a+2)y_{n-1}(x;a). -\end{eqnarray} - -\subsection*{Normalized recurrence relation} -\begin{eqnarray} -\label{NormRecBessel} -xp_n(x)&=&p_{n+1}(x)-\frac{2a}{(2n+a)(2n+a+2)}p_n(x)\nonumber\\ -& &{}\mathindent{}-\frac{4n(n+a)}{(2n+a-1)(2n+a)^2(2n+a+1)}p_{n-1}(x), -\end{eqnarray} -where -$$y_n(x;a)=\frac{(n+a+1)_n}{2^n}p_n(x).$$ - -\subsection*{Differential equation} -\begin{equation} -\label{dvBessel} -x^2y''(x)+\left[(a+2)x+2\right]y'(x)-n(n+a+1)y(x)=0,\quad y(x)=y_n(x;a). -\end{equation} - -\subsection*{Forward shift operator} -\begin{equation} -\label{shift1Bessel} -\frac{d}{dx}y_n(x;a)=\frac{n(n+a+1)}{2}y_{n-1}(x;a+2). -\end{equation} - -\subsection*{Backward shift operator} -\begin{equation} -\label{shift2BesselI} -x^2\frac{d}{dx}y_n(x;a)+(ax+2)y_n(x;a)=2y_{n+1}(x;a-2) -\end{equation} -or equivalently -\begin{equation} -\label{shift2BesselII} -\frac{d}{dx}\left[x^a\expe^{-\frac{2}{x}}y_n(x;a)\right]=2x^{a-2}\expe^{-\frac{2}{x}}y_{n+1}(x;a-2). -\end{equation} - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodBessel} -y_n(x;a)=2^{-n}x^{-a}\expe^{\frac{2}{x}}D^n\left(x^{2n+a}\expe^{-\frac{2}{x}}\right). -\end{equation} - -\subsection*{Generating function} -\begin{equation} -\label{GenBessel} -\left(1-2xt\right)^{-\frac{1}{2}}\left(\frac{2}{1+\sqrt{1-2xt}}\right)^a -\exp\left(\frac{2t}{1+\sqrt{1-2xt}}\right)=\sum_{n=0}^{\infty}y_n(x;a)\frac{t^n}{n!}. -\end{equation} - -\subsection*{Limit relation} - -\subsubsection*{Jacobi $\rightarrow$ Bessel} -If we take $\beta=a-\alpha$ in the definition (\ref{DefJacobi}) of the Jacobi polynomials -and let $\alpha\rightarrow -\infty$ we find the Bessel polynomials: -$$\lim_{\alpha\rightarrow -\infty} -\frac{P_n^{(\alpha,a-\alpha)}(1+\alpha x)}{P_n^{(\alpha,a-\alpha)}(1)}=y_n(x;a).$$ - -\subsection*{Remarks} -The following notations are also used for the Bessel polynomials: -$$y_n(x;a,b)=y_n(2b^{-1}x;a)\quad\textrm{and}\quad\theta_n(x;a,b)=x^ny_n(x^{-1};a,b).$$ -However, the Bessel polynomials essentially depend on only one parameter. - -\noindent -The Bessel polynomials are related to the Laguerre polynomials given by (\ref{DefLaguerre}) -in the following way: -$$L_n^{(\alpha)}(x)=\frac{(-x)^n}{n!}y_n(2x^{-1};-2n-\alpha-1).$$ - -\noindent -The special case $a=-2N-2$ of the Bessel polynomials can be obtained from the pseudo Jacobi -polynomials by setting $x\rightarrow\nu x$ in the definition (\ref{DefPseudoJacobi}) of the -pseudo Jacobi polynomials and taking the limit $\nu\rightarrow\infty$ in the following way: -$$\lim\limits_{\nu\rightarrow\infty}\frac{P_n(\nu x;\nu,N)}{\nu^n} -=\frac{2^n}{(n-2N-1)_n}y_n(x;-2N-2).$$ - -\subsection*{References} -\cite{Andrade}, \cite{BergVignat}, \cite{Carlitz57I}, \cite{Dattoli2003}, -\cite{DohaAhmed2004}, \cite{DohaAhmed2006}, \cite{Grosswald}, \cite{Ismail2005II}, -\cite{KrallFrink}, \cite{Lesky98}, \cite{NikiforovUvarov}. - - -\section{Charlier}\index{Charlier polynomials} - -\par\setcounter{equation}{0} - -\subsection*{Hypergeometric representation} -\begin{equation} -\label{DefCharlier} -C_n(x;a)=\hyp{2}{0}{-n,-x}{-}{-\frac{1}{a}}. -\end{equation} - -\subsection*{Orthogonality relation} -\begin{equation} -\label{OrtCharlier} -\sum_{x=0}^{\infty}\frac{a^x}{x!}C_m(x;a)C_n(x;a)= -a^{-n}\expe^an!\,\delta_{mn},\quad a > 0. -\end{equation} - -\subsection*{Recurrence relation} -\begin{equation} -\label{RecCharlier} --xC_n(x;a)=aC_{n+1}(x;a)-(n+a)C_n(x;a)+nC_{n-1}(x;a). -\end{equation} - -\subsection*{Normalized recurrence relation} -\begin{equation} -\label{NormRecCharlier} -xp_n(x)=p_{n+1}(x)+(n+a)p_n(x)+nap_{n-1}(x), -\end{equation} -where -$$C_n(x;a)=\left(-\frac{1}{a}\right)^np_n(x).$$ - -\subsection*{Difference equation} -\begin{equation} -\label{dvCharlier} --ny(x)=ay(x+1)-(x+a)y(x)+xy(x-1),\quad y(x)=C_n(x;a). -\end{equation} - -\subsection*{Forward shift operator} -\begin{equation} -\label{shift1CharlierI} -C_n(x+1;a)-C_n(x;a)=-\frac{n}{a}C_{n-1}(x;a) -\end{equation} -or equivalently -\begin{equation} -\label{shift1CharlierII} -\Delta C_n(x;a)=-\frac{n}{a}C_{n-1}(x;a). -\end{equation} - -\subsection*{Backward shift operator} -\begin{equation} -\label{shift2CharlierI} -C_n(x;a)-\frac{x}{a}C_n(x-1;a)=C_{n+1}(x;a) -\end{equation} -or equivalently -\begin{equation} -\label{shift2CharlierII} -\nabla\left[\frac{a^x}{x!}C_n(x;a)\right]=\frac{a^x}{x!}C_{n+1}(x;a). -\end{equation} - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodCharlier} -\frac{a^x}{x!}C_n(x;a)=\nabla^n\left[\frac{a^x}{x!}\right]. -\end{equation} - -\subsection*{Generating function} -\begin{equation} -\label{GenCharlier} -\e^t\left(1-\frac{t}{a}\right)^x=\sum_{n=0}^{\infty}\frac{C_n(x;a)}{n!}t^n. -\end{equation} - -\subsection*{Limit relations} - -\subsubsection*{Meixner $\rightarrow$ Charlier} -If we take $c=(a+\beta)^{-1}a$ in the definition (\ref{DefMeixner}) of the Meixner polynomials -and let $\beta\rightarrow\infty$ we find the Charlier polynomials: -$$\lim_{\beta\rightarrow\infty}M_n(x;\beta,(a+\beta)^{-1}a)=C_n(x;a).$$ - -\subsubsection*{Krawtchouk $\rightarrow$ Charlier} -The Charlier polynomials can be found from the Krawtchouk polynomials given by -(\ref{DefKrawtchouk}) by taking $p=N^{-1}a$ and letting $N\rightarrow\infty$: -$$\lim_{N\rightarrow\infty}K_n(x;N^{-1}a,N)=C_n(x;a).$$ - -\subsubsection*{Charlier $\rightarrow$ Hermite} -The Hermite polynomials given by (\ref{DefHermite}) are obtained from the Charlier polynomials -if we set $x\rightarrow (2a)^{1/2}x+a$ and let $a\rightarrow\infty$. In fact we have -\begin{equation} -\lim_{a\rightarrow\infty} -(2a)^{\frac{1}{2}n}C_n((2a)^{\frac{1}{2}}x+a;a)=(-1)^nH_n(x). -\end{equation} - -\subsection*{Remark} -The Charlier polynomials are related to the Laguerre polynomials given by (\ref{DefLaguerre}) -in the following way: -$$\frac{(-a)^n}{n!}C_n(x;a)=L_n^{(x-n)}(a).$$ - -\subsection*{References} -\label{sec9.12} -\paragraph{Notation} -Here the Laguerre polynomial is denoted by $L_n^\al$ instead of -$L_n^{(\al)}$. -% -\paragraph{Hypergeometric representation} -\begin{align} -L_n^\al(x)&= -\frac{(\al+1)_n}{n!}\,\hyp11{-n}{\al+1}x -\label{182}\\ -&=\frac{(-x)^n}{n!} \hyp20{-n,-n-\al}-{-\,\frac1x} -\label{183}\\ -&=\frac{(-x)^n}{n!}\,C_n(n+\al;x), -\label{184} -\end{align} -where $C_n$ in \eqref{184} is a -\hyperref[sec9.14]{Charlier polynomial}. -Formula \eqref{182} is (9.12.1). Then \eqref{183} follows by reversal -of summation. Finally \eqref{184} follows by \eqref{183} and \eqref{179}. -It is also the remark on top of p.244 in \mycite{KLS}, and it is essentially -\myciteKLS{416}{(2.7.10)}. -% -\paragraph{Uniqueness of orthogonality measure} -The coefficient of $p_{n-1}(x)$ in (9.12.4) behaves as $O(n^2)$ as $n\to\iy$. -Hence \eqref{93} holds, by which the orthogonality measure is unique. -% -\paragraph{Special value} -\begin{equation} -L_n^{\al}(0)=\frac{(\al+1)_n}{n!}\,. -\label{53} -\end{equation} -Use (9.12.1) or see \mycite{DLMF}{18.6.1)}. -% -\paragraph{Quadratic transformations} -\begin{align} -H_{2n}(x)&=(-1)^n\,2^{2n}\,n!\,L_n^{-1/2}(x^2), -\label{54}\\ -H_{2n+1}(x)&=(-1)^n\,2^{2n+1}\,n!\,x\,L_n^{1/2}(x^2). -\label{55} -\end{align} -See p.244, Remarks, last two formulas. -Or see \mycite{DLMF}{(18.7.19), (18.7.20)}. -% -\paragraph{Fourier transform} -\begin{equation} -\frac1{\Ga(\al+1)}\,\int_0^\iy \frac{L_n^\al(y)}{L_n^\al(0)}\, -e^{-y}\,y^\al\,e^{ixy}\,dy= -i^n\,\frac{y^n}{(iy+1)^{n+\al+1}}\,, -\label{14} -\end{equation} -see \mycite{DLMF}{(18.17.34)}. -% -\paragraph{Differentiation formulas} -Each differentiation formula is given in two equivalent forms. -\begin{equation} -\frac d{dx}\left(x^\al L_n^\al(x)\right)= -(n+\al)\,x^{\al-1} L_n^{\al-1}(x),\qquad -\left(x\frac d{dx}+\al\right)L_n^\al(x)= -(n+\al)\,L_n^{\al-1}(x). -\label{76} -\end{equation} -% -\begin{equation} -\frac d{dx}\left(e^{-x} L_n^\al(x)\right)= --e^{-x} L_n^{\al+1}(x),\qquad -\left(\frac d{dx}-1\right)L_n^\al(x)= --L_n^{\al+1}(x). -\label{77} -\end{equation} -% -Formulas \eqref{76} and \eqref{77} follow from -\mycite{DLMF}{(13.3.18), (13.3.20)} -together with (9.12.1). -% -\paragraph{Generalized Hermite polynomials} -See \myciteKLS{146}{p.156}, \cite[Section 1.5.1]{K26}. -These are defined by -\begin{equation} -H_{2m}^\mu(x):=\const L_m^{\mu-\half}(x^2),\qquad -H_{2m+1}^\mu(x):=\const x\,L_m^{\mu+\half}(x^2). -\label{78} -\end{equation} -Then for $\mu>-\thalf$ we have orthogonality relation -\begin{equation} -\int_{-\iy}^{\iy} H_m^\mu(x)\,H_n^\mu(x)\,|x|^{2\mu}e^{-x^2}\,dx -=0\qquad(m\ne n). -\label{79} -\end{equation} -Let the Dunkl operator $T_\mu$ be defined by \eqref{72}. -If we choose the constants in \eqref{78} as -\begin{equation} -H_{2m}^\mu(x)=\frac{(-1)^m(2m)!}{(\mu+\thalf)_m}\,L_m^{\mu-\half}(x^2),\qquad -H_{2m+1}^\mu(x)=\frac{(-1)^m(2m+1)!}{(\mu+\thalf)_{m+1}}\, - x\,L_m^{\mu+\half}(x^2) - \label{80} -\end{equation} -then (see \cite[(1.6)]{K5}) -\begin{equation} -T_\mu H_n^\mu=2n\,H_{n-1}^\mu. -\label{81} -\end{equation} -Formula \eqref{81} with \eqref{80} substituted gives rise to two -differentiation formulas involving Laguerre polynomials which are equivalent to -(9.12.6) and \eqref{76}. - -Composition of \eqref{81} with itself gives -\[ -T_\mu^2 H_n^\mu=4n(n-1)\,H_{n-2}^\mu, -\] -which is equivalent to the composition of (9.12.6) and \eqref{76}: -\begin{equation} -\left(\frac{d^2}{dx^2}+\frac{2\al+1}x\,\frac d{dx}\right)L_n^\al(x^2) -=-4(n+\al)\,L_{n-1}^\al(x^2). -\label{82} -\end{equation} -% -\cite{Allaway76}, \cite{NAlSalam66}, \cite{AlSalam90}, \cite{AlSalamChihara76}, -\cite{AlSalamIsmail76}, \cite{AndrewsAskey85}, \cite{Area+II}, \cite{Askey75}, -\cite{AskeyGasper77}, \cite{AskeyWilson85}, \cite{AtakRahmanSuslov}, \cite{Bavinck98}, -\cite{BavinckKoekoek}, \cite{Chihara78}, \cite{Chihara79}, \cite{Dunkl76}, \cite{Erdelyi+}, -\cite{Gasper73I}, \cite{Gasper74}, \cite{Goh}, \cite{HoareRahman}, \cite{IsmailLetVal88}, -\cite{Koekoek2000}, \cite{Koorn88}, \cite{Krasikov2002}, \cite{LabelleYehI}, -\cite{LabelleYehII}, \cite{LabelleYehIII}, \cite{Lesky62}, \cite{Lesky89}, \cite{Lesky94I}, -\cite{Lesky95II}, \cite{LewanowiczII}, \cite{LopezTemme2004}, \cite{Meixner}, \cite{Nikiforov+}, -\cite{NikiforovUvarov}, \cite{Szafraniec}, \cite{Szego75}, \cite{Viennot}, \cite{Zarzo+}, -\cite{Zeng90}. - - -\section{Hermite}\index{Hermite polynomials} - -\par\setcounter{equation}{0} - -\subsection*{Hypergeometric representation} -\begin{equation} -\label{DefHermite} -H_n(x)=(2x)^n\,\hyp{2}{0}{-n/2,-(n-1)/2}{-}{-\frac{1}{x^2}}. -\end{equation} - -\subsection*{Orthogonality relation} -\begin{equation} -\label{OrtHermite} -\frac{1}{\sqrt{\cpi}}\int_{-\infty}^{\infty}\expe^{-x^2}H_m(x)H_n(x)\,dx -=2^nn!\,\delta_{mn}. -\end{equation} - -\subsection*{Recurrence relation} -\begin{equation} -\label{RecHermite} -H_{n+1}(x)-2xH_n(x)+2nH_{n-1}(x)=0. -\end{equation} - -\subsection*{Normalized recurrence relation} -\begin{equation} -\label{NormRecHermite} -xp_n(x)=p_{n+1}(x)+\frac{n}{2}p_{n-1}(x), -\end{equation} -where -$$H_n(x)=2^np_n(x).$$ - -\subsection*{Differential equation} -\begin{equation} -\label{dvHermite} -y''(x)-2xy'(x)+2ny(x)=0,\quad y(x)=H_n(x). -\end{equation} - -\subsection*{Forward shift operator} -\begin{equation} -\label{shift1Hermite} -\frac{d}{dx}H_n(x)=2nH_{n-1}(x). -\end{equation} - -\subsection*{Backward shift operator} -\begin{equation} -\label{shift2HermiteI} -\frac{d}{dx}H_n(x)-2xH_n(x)=-H_{n+1}(x) -\end{equation} -or equivalently -\begin{equation} -\label{shift2HermiteII} -\frac{d}{dx}\left[\expe^{-x^2}H_n(x)\right]=-\expe^{-x^2}H_{n+1}(x). -\end{equation} - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodHermite} -\e^{-x^2}H_n(x)=(-1)^n\left(\frac{d}{dx}\right)^n\left[\expe^{-x^2}\right]. -\end{equation} - -\subsection*{Generating functions} -\begin{equation} -\label{GenHermite1} -\exp\left(2xt-t^2\right)=\sum_{n=0}^{\infty}\frac{H_n(x)}{n!}t^n. -\end{equation} - -\begin{equation} -\label{GenHermite2} -\left\{\begin{array}{l} -\displaystyle\expe^t\cos(2x\sqrt{t})=\sum_{n=0}^{\infty} -\frac{(-1)^n}{(2n)!}H_{2n}(x)t^n\\[5mm] -\displaystyle\frac{\expe^t}{\sqrt{t}}\sin(2x\sqrt{t})=\sum_{n=0}^{\infty} -\frac{(-1)^n}{(2n+1)!}H_{2n+1}(x)t^n. -\end{array}\right. -\end{equation} - -\begin{equation} -\label{GenHermite3} -\left\{\begin{array}{l} -\displaystyle \expe^{-t^2}\cosh(2xt)=\sum_{n=0}^{\infty} -\frac{H_{2n}(x)}{(2n)!}t^{2n}\\[5mm] -\displaystyle\expe^{-t^2}\sinh(2xt)=\sum_{n=0}^{\infty} -\frac{H_{2n+1}(x)}{(2n+1)!}t^{2n+1}. -\end{array}\right. -\end{equation} - -\begin{equation} -\label{GenHermite4} -\left\{\begin{array}{l} -\displaystyle (1+t^2)^{-\gamma}\,\hyp{1}{1}{\gamma}{\frac{1}{2}}{\frac{x^2t^2}{1+t^2}}= -\sum_{n=0}^{\infty}\frac{(\gamma)_n}{(2n)!}H_{2n}(x)t^{2n}\\[5mm] -\displaystyle\frac{xt}{\sqrt{1+t^2}}\, -\hyp{1}{1}{\gamma+\frac{1}{2}}{\frac{3}{2}}{\frac{x^2t^2}{1+t^2}} -=\sum_{n=0}^{\infty}\frac{(\gamma+\frac{1}{2})_n}{(2n+1)!}H_{2n+1}(x)t^{2n+1} -\end{array}\right. -\end{equation} -with $\gamma$ arbitrary. - -\begin{equation} -\label{GenHermite5} -\frac{1+2xt+4t^2}{(1+4t^2)^{\frac{3}{2}}}\exp\left(\frac{4x^2t^2}{1+4t^2}\right) -=\sum_{n=0}^{\infty}\frac{H_n(x)}{\lfloor n/2\rfloor\,!}t^n, -\end{equation} -where $\lfloor n/2\rfloor$ denotes the largest integer smaller than or equal to $n/2$. - -\subsection*{Limit relations} - -\subsubsection*{Meixner-Pollaczek $\rightarrow$ Hermite} -If we take $x\rightarrow (\sin\phi)^{-1}(x\sqrt{\lambda}-\lambda\cos\phi)$ -in the definition (\ref{DefMP}) of the Meixner-Pollaczek polynomials and -then let $\lambda\rightarrow\infty$ we obtain the Hermite polynomials: -$$\lim_{\lambda\rightarrow\infty} -\lambda^{-\frac{1}{2}n}P_n^{(\lambda)} -((\sin\phi)^{-1}(x\sqrt{\lambda}-\lambda\cos\phi);\phi)=\frac{H_n(x)}{n!}.$$ - -\subsubsection*{Jacobi $\rightarrow$ Hermite} -The Hermite polynomials follow from the Jacobi polynomials given by (\ref{DefJacobi}) by -taking $\beta=\alpha$ and letting $\alpha\rightarrow\infty$ in the following way: -$$\lim_{\alpha\rightarrow\infty} -\alpha^{-\frac{1}{2}n}P_n^{(\alpha,\alpha)}(\alpha^{-\frac{1}{2}}x)=\frac{H_n(x)}{2^nn!}.$$ - -\subsubsection*{Gegenbauer / Ultraspherical $\rightarrow$ Hermite} -The Hermite polynomials follow from the Gegenbauer (or ultraspherical) polynomials given by -(\ref{DefGegenbauer}) by taking $\lambda=\alpha+\frac{1}{2}$ and letting -$\alpha\rightarrow\infty$ in the following way: -$$\lim_{\alpha\rightarrow\infty} -\alpha^{-\frac{1}{2}n}C_n^{(\alpha+\frac{1}{2})}(\alpha^{-\frac{1}{2}}x)=\frac{H_n(x)}{n!}.$$ - -\subsubsection*{Krawtchouk $\rightarrow$ Hermite} -The Hermite polynomials follow from the Krawtchouk polynomials given by (\ref{DefKrawtchouk}) -by setting $x\rightarrow pN+x\sqrt{2p(1-p)N}$ and then letting $N\rightarrow\infty$: -$$\lim_{N\rightarrow\infty} -\sqrt{\binom{N}{n}}K_n(pN+x\sqrt{2p(1-p)N};p,N) -=\frac{\displaystyle (-1)^nH_n(x)}{\displaystyle\sqrt{2^nn!\left(\frac{p}{1-p}\right)^n}}.$$ - -\subsubsection*{Laguerre $\rightarrow$ Hermite} -The Hermite polynomials can be obtained from the Laguerre polynomials given by -(\ref{DefLaguerre}) by taking the limit $\alpha\rightarrow\infty$ in the following way: -$$\lim_{\alpha\rightarrow\infty} -\left(\frac{2}{\alpha}\right)^{\frac{1}{2}n} -L_n^{(\alpha)}((2\alpha)^{\frac{1}{2}}x+\alpha)=\frac{(-1)^n}{n!}H_n(x).$$ - -\subsubsection*{Charlier $\rightarrow$ Hermite} -If we set $x\rightarrow (2a)^{1/2}x+a$ in the definition (\ref{DefCharlier}) -of the Charlier polynomials and let $a\rightarrow\infty$ we find the Hermite -polynomials. In fact we have -$$\lim_{a\rightarrow\infty} -(2a)^{\frac{1}{2}n}C_n((2a)^{\frac{1}{2}}x+a;a)=(-1)^nH_n(x).$$ - -\subsection*{Remarks} -The Hermite polynomials can also be written as: -$$\frac{H_n(x)}{n!}=\sum_{k=0}^{\lfloor n/2\rfloor} -\frac{(-1)^k(2x)^{n-2k}}{k!\,(n-2k)!},$$ -where $\lfloor n/2\rfloor$ denotes the largest integer smaller than or equal to $n/2$. - -\noindent -The Hermite polynomials and the Laguerre polynomials given by (\ref{DefLaguerre}) are also -connected by the following quadratic transformations: -$$H_{2n}(x)=(-1)^nn!\,2^{2n}L_n^{(-\frac{1}{2})}(x^2)$$ -and -$$H_{2n+1}(x)=(-1)^nn!\,2^{2n+1}xL_n^{(\frac{1}{2})}(x^2).$$ - -\subsection*{References} -\label{sec9.14} -% -\paragraph{Hypergeometric representation} -\begin{align} -C_n(x;a)&=\hyp20{-n,-x}-{-\,\frac1a} -\label{179}\\ -&=\frac{(-x)_n}{a^n} \hyp11{-n}{x-n+1}a -\label{180}\\ -&=\frac{n!}{(-a)^n}\,L_n^{x-n}(a), -\label{181} -\end{align} -where $L_n^\al(x)$ is a -\hyperref[sec9.12]{Laguerre polynomial}. -Formula \eqref{179} is (9.14.1). Then \eqref{180} follows by reversal -of the summation. Finally \eqref{181} follows by \eqref{180} and -(9.12.1). It is also the Remark on p.249 of \mycite{KLS}, and it -was earlier given in \myciteKLS{416}{(2.7.10)}. -% -\paragraph{Uniqueness of orthogonality measure} -The coefficient of $p_{n-1}(x)$ in (9.14.4) behaves as $O(n)$ as $n\to\iy$. -Hence \eqref{93} holds, by which the orthogonality measure is unique. -% -\cite{Abram}, \cite{NAlSalam66}, \cite{AlSalam90}, \cite{AlSalamChihara72}, -\cite{AlSalamChihara76}, \cite{AndrewsAskey85}, \cite{AndrewsAskeyRoy}, \cite{Area+I}, -\cite{Askey68}, \cite{Askey75}, \cite{Askey89I}, \cite{AskeyGasper76}, \cite{AskeyWilson85}, -\cite{Azor}, \cite{Berg}, \cite{BilodeauII}, \cite{Brafman51}, \cite{Brafman57II}, -\cite{Brenke}, \cite{CarlitzSrivastava}, \cite{ChenSrivastava}, \cite{Chihara78}, -\cite{Cohen}, \cite{Danese}, \cite{DetteStudden92}, \cite{DetteStudden95}, \cite{Doha2004I}, -\cite{Erdelyi+}, \cite{Faldey}, \cite{Gawronski87}, \cite{Gawronski93}, \cite{Gillis+}, -\cite{Godoy+}, \cite{Grad}, \cite{Ismail74}, \cite{Ismail2005II}, \cite{IsmailStanton97}, -\cite{Koekoek2000}, \cite{Koorn88}, \cite{Krasikov2004}, \cite{Kwon+}, \cite{LabelleYehIII}, -\cite{Lesky95II}, \cite{Lesky96}, \cite{LewanowiczI}, \cite{LopezTemme99}, \cite{Mathai}, -\cite{Meixner}, \cite{Nikiforov+}, \cite{NikiforovUvarov}, \cite{Olver}, \cite{Pittaluga}, -\cite{Rainville}, \cite{SainteViennot}, \cite{Srivastava71}, \cite{SrivastavaMathur}, -\cite{Szego75}, \cite{Temme}, \cite{TemmeLopez2000}, \cite{Trickovic}, \cite{Viennot}, -\cite{Weisner59}, \cite{Wyman}. - -\end{document} diff --git a/KLSadd_insertion/tempchap14.tex b/KLSadd_insertion/tempchap14.tex deleted file mode 100644 index 0b5f396..0000000 --- a/KLSadd_insertion/tempchap14.tex +++ /dev/null @@ -1,5734 +0,0 @@ -\documentclass[envcountchap,graybox]{svmono} - -\addtolength{\textwidth}{1mm} - -\usepackage{amsmath,amssymb} - -\usepackage{amsfonts} -%\usepackage{breqn} -\usepackage{DLMFmath} -\usepackage{DRMFfcns} - -\usepackage{mathptmx} -\usepackage{helvet} -\usepackage{courier} - -\usepackage{makeidx} -\usepackage{graphicx} - -\usepackage{multicol} -\usepackage[bottom]{footmisc} - -\makeindex - -\def\bibname{Bibliography} -\def\refname{Bibliography} - -\def\theequation{\thesection.\arabic{equation}} - -\smartqed - -\let\corollary=\undefined -\spnewtheorem{corollary}[theorem]{Corollary}{\bfseries}{\itshape} - -\newcounter{rom} - -\newcommand{\hyp}[5]{\mbox{}_{#1}{F}_{#2} -\left(\genfrac{}{}{0pt}{}{#3}{#4}\,;\,#5\right)} - -\newcommand{\qhyp}[5]{\mbox{}_{#1}{\phi}_{#2} -\left(\genfrac{}{}{0pt}{}{#3}{#4}\,;\,q,\,#5\right)} - -\newcommand{\qhypK}[5]{\,\mbox{}_{#1}\phi_{#2}\!\left( - \genfrac{}{}{0pt}{}{#3}{#4};#5\right)} - -\newcommand{\mathindent}{\hspace{7.5mm}} - -\newcommand{\e}{\textrm{e}} - -\renewcommand{\E}{\textrm{E}} - -\renewcommand{\textfraction}{-1} - -\renewcommand{\Gamma}{\varGamma} - -\renewcommand{\leftlegendglue}{\hfil} - -\settowidth{\tocchpnum}{14\enspace} -\settowidth{\tocsecnum}{14.30\enspace} -\settowidth{\tocsubsecnum}{14.12.1\enspace} - -\makeatletter -\def\cleartoversopage{\clearpage\if@twoside\ifodd\c@page - \hbox{}\newpage\if@twocolumn\hbox{}\newpage\fi - \else\fi\fi} - -\newcommand{\clearemptyversopage}{ - \clearpage{\pagestyle{empty}\cleartoversopage}} -\makeatother - -\oddsidemargin -1.5cm -\topmargin -2.0cm -\textwidth 16.3cm -\textheight 25cm - -\begin{document} - -\author{Roelof Koekoek\\[2.5mm]Peter A. Lesky\\[2.5mm]Ren\'e F. Swarttouw} -\title{Hypergeometric orthogonal polynomials and their\\$q$-analogues} -\subtitle{-- Monograph --} -\maketitle - -\frontmatter - -\large - -\pagenumbering{roman} -\addtocounter{chapter}{13} - -\chapter{Basic hypergeometric orthogonal polynomials} -\label{BasicHyperOrtPol} - - -In this chapter we deal with all families of basic hypergeometric orthogonal polynomials -appearing in the $q$-analogue of the Askey scheme on the pages~\pageref{qscheme1} and -\pageref{qscheme2}. For each family of orthogonal polynomials we state the most important -properties such as a representation as a basic hypergeometric function, orthogonality -relation(s), the three-term recurrence relation, the second-order $q$-difference equation, -the forward shift (or degree lowering) and backward shift (or degree raising) operator, a -Rodrigues-type formula and some generating functions. Throughout this chapter we assume -that $01$ and $b,c,d$ are real or one is real and the other two are complex conjugates,\\ -$\max(|b|,|c|,|d|)<1$ and the pairwise products of $a,b,c$ and $d$ have -absolute value less than $1$, then we have another orthogonality relation -given by: -\begin{eqnarray} -\label{OrtAskeyWilsonII} -& &\frac{1}{2\pi}\int_{-1}^1\frac{w(x)}{\sqrt{1-x^2}}p_m(x;a,b,c,d|q)p_n(x;a,b,c,d|q)\,dx\nonumber\\ -& &{}\mathindent{}+\sum_{\begin{array}{c}\scriptstyle k\\ \scriptstyle 11$ and $b$ and $c$ are real or complex conjugates, -$\max(|b|,|c|)<1$ and the pairwise products of $a,b$ and $c$ have -absolute value less than $1$, then we have another orthogonality relation -given by: -\begin{eqnarray} -\label{OrtContDualqHahnII} -& &\frac{1}{2\pi}\int_{-1}^1\frac{w(x)}{\sqrt{1-x^2}}p_m(x;a,b,c|q)p_n(x;a,b,c|q)\,dx\nonumber\\ -& &{}\mathindent{}+\sum_{\begin{array}{c}\scriptstyle k\\ \scriptstyle 10$) in the definition (\ref{DefBigqJacobi}) -of the big $q$-Jacobi polynomials and take the limit $c\rightarrow -\infty$ we -obtain the $q$-Meixner polynomials given by (\ref{DefqMeixner}): -\begin{equation} -\lim_{c\rightarrow -\infty}P_n(q^{-x};a,-a^{-1}cd^{-1},c;q)=M_n(q^{-x};a,d;q). -\end{equation} - -\subsubsection*{Big $q$-Jacobi $\rightarrow$ Jacobi} -If we set $c=0$, $a=q^{\alpha}$ and $b=q^{\beta}$ in the definition (\ref{DefBigqJacobi}) -of the big $q$-Jacobi polynomials and let $q\rightarrow 1$ we find the Jacobi -polynomials given by (\ref{DefJacobi}): -\begin{equation} -\lim_{q\rightarrow 1}P_n(x;q^{\alpha},q^{\beta},0;q)=\frac{P_n^{(\alpha,\beta)}(2x-1)}{P_n^{(\alpha,\beta)}(1)}. -\end{equation} -If we take $c=-q^{\gamma}$ for arbitrary real $\gamma$ instead of $c=0$ we -find -\begin{equation} -\lim_{q\rightarrow 1}P_n(x;q^{\alpha},q^{\beta},-q^{\gamma};q)=\frac{P_n^{(\alpha,\beta)}(x)}{P_n^{(\alpha,\beta)}(1)}. -\end{equation} - -\subsection*{Remarks} -The big $q$-Jacobi polynomials with $c=0$ and the little -$q$-Jacobi polynomials given by (\ref{DefLittleqJacobi}) are related -in the following way: -$$P_n(x;a,b,0;q)=\frac{(bq;q)_n}{(aq;q)_n}(-1)^na^nq^{n+\binom{n}{2}}p_n(a^{-1}q^{-1}x;b,a|q).$$ - -\noindent -Sometimes the big $q$-Jacobi polynomials are defined in terms of four -parameters instead of three. In fact the polynomials given by the definition -$$P_n(x;a,b,c,d;q)=\qhyp{3}{2}{q^{-n},abq^{n+1},ac^{-1}qx}{aq,-ac^{-1}dq}{q}$$ -are orthogonal on the interval $[-d,c]$ with respect to the weight function -$$\frac{(c^{-1}qx,-d^{-1}qx;q)_{\infty}}{(ac^{-1}qx,-bd^{-1}qx;q)_{\infty}}d_qx.$$ -These polynomials are not really different from those given by -(\ref{DefBigqJacobi}) since we have -$$P_n(x;a,b,c,d;q)=P_n(ac^{-1}qx;a,b,-ac^{-1}d;q)$$ -and -$$P_n(x;a,b,c;q)=P_n(x;a,b,aq,-cq;q).$$ - -\subsection*{References} -\cite{NAlSalam89}, \cite{AlSalam90}, \cite{AndrewsAskey85}, \cite{AtakKlimyk2004}, -\cite{AtakRahmanSuslov}, \cite{DattaGriffin}, \cite{FloreaniniVinetII}, -\cite{GasperRahman90}, \cite{GrunbaumHaine96}, \cite{Gupta92}, \cite{Hahn}, -\cite{Ismail86I}, \cite{IsmailStanton97}, \cite{IsmailWilson}, \cite{KalninsMiller88}, -\cite{KoelinkE}, \cite{Koorn90II}, \cite{Koorn93}, \cite{Koorn2007}, \cite{Miller89}, -\cite{Nikiforov+}, \cite{NoumiMimachi90II}, \cite{NoumiMimachi90III}, -\cite{NoumiMimachi91}, \cite{Spiridonov97}, \cite{SrivastavaJain90}. - - -\section*{Special case} - -\subsection{Big $q$-Legendre} -\index{Big q-Legendre polynomials@Big $q$-Legendre polynomials} -\index{q-Legendre polynomials@$q$-Legendre polynomials!Big} -\par - -\subsection*{Basic hypergeometric representation} The big $q$-Legendre polynomials are big $q$-Jacobi polynomials -with $a=b=1$: -\begin{equation} -\label{DefBigqLegendre} -P_n(x;c;q)=\qhyp{3}{2}{q^{-n},q^{n+1},x}{q,cq}{q}. -\end{equation} - -\subsection*{Orthogonality relation} -\begin{eqnarray} -\label{OrtBigqLegendre} -& &\int_{cq}^{q}P_m(x;c;q)P_n(x;c;q)\,d_qx\nonumber\\ -& &{}=q(1-c)\frac{(1-q)}{(1-q^{2n+1})} -\frac{(c^{-1}q;q)_n}{(cq;q)_n}(-cq^2)^nq^{\binom{n}{2}}\,\delta_{mn},\quad c<0. -\end{eqnarray} - -\subsection*{Recurrence relation} -\begin{eqnarray} -\label{RecBigqLegendre} -(x-1)P_n(x;c;q)&=&A_nP_{n+1}(x;c;q)-\left(A_n+C_n\right)P_n(x;c;q)\nonumber\\ -& &{}\mathindent{}+C_nP_{n-1}(x;c;q), -\end{eqnarray} -where -$$\left\{\begin{array}{l} -\displaystyle A_n=\frac{(1-q^{n+1})(1-cq^{n+1})}{(1+q^{n+1})(1-q^{2n+1})}\\ -\\ -\displaystyle C_n=-cq^{n+1}\frac{(1-q^n)(1-c^{-1}q^n)}{(1+q^n)(1-q^{2n+1})}. -\end{array}\right.$$ - -\subsection*{Normalized recurrence relation} -\begin{equation} -\label{NormRecBigqLegendre} -xp_n(x)=p_{n+1}(x)+\left[1-(A_n+C_n)\right]p_n(x)+A_{n-1}C_np_{n-1}(x), -\end{equation} -where -$$P_n(x;c;q)=\frac{(q^{n+1};q)_n}{(q,cq;q)_n}p_n(x).$$ - -\subsection*{$q$-Difference equation} -\begin{eqnarray} -\label{dvBigqLegendre} -& &q^{-n}(1-q^n)(1-q^{n+1})x^2y(x)\nonumber\\ -& &{}=B(x)y(qx)-\left[B(x)+D(x)\right]y(x)+D(x)y(q^{-1}x), -\end{eqnarray} -where -$$y(x)=P_n(x;c;q)$$ -and -$$\left\{\begin{array}{l}\displaystyle B(x)=q(x-1)(x-c)\\ -\\ -\displaystyle D(x)=(x-q)(x-cq).\end{array}\right.$$ - -\subsection*{Rodrigues-type formula} -\begin{eqnarray} -\label{RodBigqLegendre} -P_n(x;c;q)&=&\frac{c^nq^{n(n+1)}(1-q)^n}{(q,cq;q)_n} -\left(\mathcal{D}_q\right)^n\left[(q^{-n}x,c^{-1}q^{-n}x;q)_n\right]\\ -&=&\frac{(1-q)^n}{(q,cq;q)_n}\left(\mathcal{D}_q\right)^n -\left[(qx^{-1},cqx^{-1};q)_nx^{2n}\right].\nonumber -\end{eqnarray} - -\subsection*{Generating functions} -\begin{equation} -\label{GenBigqLegendre1} -\qhyp{2}{1}{qx^{-1},0}{q}{xt}\,\qhyp{1}{1}{c^{-1}x}{q}{cqt} -=\sum_{n=0}^{\infty}\frac{(cq;q)_n}{(q,q;q)_n}P_n(x;c;q)t^n. -\end{equation} - -\begin{equation} -\label{GenBigqLegendre2} -\qhyp{2}{1}{cqx^{-1},0}{cq}{xt}\,\qhyp{1}{1}{c^{-1}x}{c^{-1}q}{qt} -=\sum_{n=0}^{\infty}\frac{P_n(x;c;q)}{(c^{-1}q;q)_n}t^n. -\end{equation} - -\subsection*{Limit relations} - -\subsubsection*{Big $q$-Legendre $\rightarrow$ Legendre / Spherical} -If we set $c=0$ in the definition (\ref{DefBigqLegendre}) of the big -$q$-Legendre polynomials and let $q\rightarrow 1$ we simply obtain the Legendre -(or spherical) polynomials given by (\ref{DefLegendre}): -\begin{equation} -\lim_{q\rightarrow 1}P_n(x;0;q)=P_n(2x-1). -\end{equation} -If we take $c=-q^{\gamma}$ for arbitrary real $\gamma$ instead of $c=0$ we -find -\begin{equation} -\lim_{q\rightarrow 1}P_n(x;-q^{\gamma};q)=P_n(x). -\end{equation} - -\subsection*{References} -\cite{Koelink95I}, \cite{Koorn90II}. - - -\section{$q$-Hahn}\index{q-Hahn polynomials@$q$-Hahn polynomials} -\par\setcounter{equation}{0} - -\subsection*{Basic hypergeometric representation} -\begin{equation} -\label{DefqHahn} -Q_n(q^{-x};\alpha,\beta,N|q)=\qhyp{3}{2}{q^{-n},\alpha\beta q^{n+1},q^{-x}} -{\alpha q,q^{-N}}{q},\quad n=0,1,2,\ldots,N. -\end{equation} - -\subsection*{Orthogonality relation} -For $0<\alpha q<1$ and $0<\beta q<1$, or for $\alpha>q^{-N}$ and $\beta>q^{-N}$, we have -\begin{eqnarray} -\label{OrtqHahn} -& &\sum_{x=0}^N\frac{(\alpha q,q^{-N};q)_x}{(q,\beta^{-1}q^{-N};q)_x}(\alpha\beta q)^{-x} -Q_m(q^{-x};\alpha,\beta,N|q)Q_n(q^{-x};\alpha,\beta,N|q)\nonumber\\ -& &{}=\frac{(\alpha\beta q^2;q)_N}{(\beta q;q)_N(\alpha q)^N} -\frac{(q,\alpha\beta q^{N+2},\beta q;q)_n}{(\alpha q,\alpha\beta q,q^{-N};q)_n}\, -\frac{(1-\alpha\beta q)(-\alpha q)^n}{(1-\alpha\beta q^{2n+1})} -q^{\binom{n}{2}-Nn}\,\delta_{mn}. -\end{eqnarray} - -\newpage - -\subsection*{Recurrence relation} -\begin{eqnarray} -\label{RecqHahn} --\left(1-q^{-x}\right)Q_n(q^{-x})&=&A_nQ_{n+1}(q^{-x})-\left(A_n+C_n\right)Q_n(q^{-x})\nonumber\\ -& &{}\mathindent{}+C_nQ_{n-1}(q^{-x}), -\end{eqnarray} -where -$$Q_n(q^{-x}):=Q_n(q^{-x};\alpha,\beta,N|q)$$ -and -$$\left\{\begin{array}{l} -\displaystyle A_n=\frac{(1-q^{n-N})(1-\alpha q^{n+1})(1-\alpha\beta q^{n+1})}{(1-\alpha\beta q^{2n+1})(1-\alpha\beta q^{2n+2})}\\ -\\ -\displaystyle C_n=-\frac{\alpha q^{n-N}(1-q^n)(1-\alpha\beta q^{n+N+1})(1-\beta q^n)}{(1-\alpha\beta q^{2n})(1-\alpha\beta q^{2n+1})}. -\end{array}\right.$$ - -\subsection*{Normalized recurrence relation} -\begin{equation} -\label{NormRecqHahn} -xp_n(x)=p_{n+1}(x)+\left[1-(A_n+C_n)\right]p_n(x)+A_{n-1}C_np_{n-1}(x), -\end{equation} -where -$$Q_n(q^{-x};\alpha,\beta,N|q)= -\frac{(\alpha\beta q^{n+1};q)_n}{(\alpha q,q^{-N};q)_n}p_n(q^{-x}).$$ - -\subsection*{$q$-Difference equation} -\begin{eqnarray} -\label{dvqHahn} -& &q^{-n}(1-q^n)(1-\alpha\beta q^{n+1})y(x)\nonumber\\ -& &{}=B(x)y(x+1)-\left[B(x)+D(x)\right]y(x)+D(x)y(x-1), -\end{eqnarray} -where -$$y(x)=Q_n(q^{-x};\alpha,\beta,N|q)$$ -and -$$\left\{\begin{array}{l}\displaystyle B(x)=(1-q^{x-N})(1-\alpha q^{x+1})\\ -\\ -\displaystyle D(x)=\alpha q(1-q^x)(\beta-q^{x-N-1}).\end{array}\right.$$ - -\newpage - -\subsection*{Forward shift operator} -\begin{eqnarray} -\label{shift1qHahnI} -& &Q_n(q^{-x-1};\alpha,\beta,N|q)-Q_n(q^{-x};\alpha,\beta,N|q)\nonumber\\ -& &{}=\frac{q^{-n-x}(1-q^n)(1-\alpha\beta q^{n+1})}{(1-\alpha q)(1-q^{-N})} -Q_{n-1}(q^{-x};\alpha q,\beta q,N-1|q) -\end{eqnarray} -or equivalently -\begin{eqnarray} -\label{shift1qHahnII} -\frac{\Delta Q_n(q^{-x};\alpha,\beta,N|q)}{\Delta q^{-x}} -&=&\frac{q^{-n+1}(1-q^n)(1-\alpha\beta q^{n+1})}{(1-q)(1-\alpha q)(1-q^{-N})}\nonumber\\ -& &{}\mathindent{}\times Q_{n-1}(q^{-x};\alpha q,\beta q,N-1|q). -\end{eqnarray} - -\subsection*{Backward shift operator} -\begin{eqnarray} -\label{shift2qHahnI} -& &(1-\alpha q^x)(1-q^{x-N-1})Q_n(q^{-x};\alpha,\beta,N|q)\nonumber\\ -& &{}\mathindent{}-\alpha(1-q^x)(\beta-q^{x-N-1})Q_n(q^{-x+1};\alpha,\beta,N|q)\nonumber\\ -& &{}=q^x(1-\alpha)(1-q^{-N-1})Q_{n+1}(q^{-x};\alpha q^{-1},\beta q^{-1},N+1|q) -\end{eqnarray} -or equivalently -\begin{eqnarray} -\label{shift2qHahnII} -& &\frac{\nabla\left[w(x;\alpha,\beta,N|q)Q_n(q^{-x};\alpha,\beta,N|q)\right]}{\nabla q^{-x}}\nonumber\\ -& &{}=\frac{1}{1-q}w(x;\alpha q^{-1},\beta q^{-1},N+1|q)Q_{n+1}(q^{-x};\alpha q^{-1},\beta q^{-1},N+1|q), -\end{eqnarray} -where -$$w(x;\alpha,\beta,N|q)=\frac{(\alpha q,q^{-N};q)_x}{(q,\beta^{-1}q^{-N};q)_x}(\alpha\beta)^{-x}.$$ - -\subsection*{Rodrigues-type formula} -\begin{eqnarray} -\label{RodqHahn} -& &w(x;\alpha,\beta,N|q)Q_n(q^{-x};\alpha,\beta,N|q)\nonumber\\ -& &{}=(1-q)^n\left(\nabla_q\right)^n\left[w(x;\alpha q^n,\beta q^n,N-n|q)\right], -\end{eqnarray} -where -$$\nabla_q:=\frac{\nabla}{\nabla q^{-x}}.$$ - -\subsection*{Generating functions} For $x=0,1,2,\ldots,N$ we have -\begin{eqnarray} -\label{GenqHahn1} -& &\qhyp{1}{1}{q^{-x}}{\alpha q}{\alpha qt}\,\qhyp{2}{1}{q^{x-N},0}{\beta q}{q^{-x}t}\nonumber\\ -& &{}=\sum_{n=0}^N\frac{(q^{-N};q)_n}{(\beta q,q;q)_n}Q_n(q^{-x};\alpha,\beta,N|q)t^n. -\end{eqnarray} - -\begin{eqnarray} -\label{GenqHahn2} -& &\qhyp{2}{1}{q^{-x},\beta q^{N+1-x}}{0}{-\alpha q^{x-N+1}t}\, -\qhyp{2}{0}{q^{x-N},\alpha q^{x+1}}{-}{-q^{-x}t}\nonumber\\ -& &{}=\sum_{n=0}^N\frac{(\alpha q,q^{-N};q)_n}{(q;q)_n}q^{-\binom{n}{2}}Q_n(q^{-x};\alpha,\beta,N|q)t^n. -\end{eqnarray} - -\subsection*{Limit relations} - -\subsubsection*{$q$-Racah $\rightarrow$ $q$-Hahn} -The $q$-Hahn polynomials follow from the $q$-Racah polynomials by the substitution -$\delta=0$ and $\gamma q=q^{-N}$ in the definition (\ref{DefqRacah}) of the -$q$-Racah polynomials: -$$R_n(\mu(x);\alpha,\beta,q^{-N-1},0|q)=Q_n(q^{-x};\alpha,\beta,N|q).$$ -Another way to obtain the $q$-Hahn polynomials from the $q$-Racah -polynomials is by setting $\gamma=0$ and $\delta=\beta^{-1}q^{-N-1}$ in the definition -(\ref{DefqRacah}): -$$R_n(\mu(x);\alpha,\beta,0,\beta^{-1}q^{-N-1}|q)=Q_n(q^{-x};\alpha,\beta,N|q).$$ -And if we take $\alpha q=q^{-N}$, $\beta\rightarrow\beta\gamma q^{N+1}$ and $\delta=0$ in the -definition (\ref{DefqRacah}) of the $q$-Racah polynomials we find the -$q$-Hahn polynomials given by (\ref{DefqHahn}) in the following way: -$$R_n(\mu(x);q^{-N-1},\beta\gamma q^{N+1},\gamma,0|q)=Q_n(q^{-x};\gamma,\beta,N|q).$$ -Note that $\mu(x)=q^{-x}$ in each case. - -\subsubsection*{$q$-Hahn $\rightarrow$ Little $q$-Jacobi} -If we set $x\rightarrow N-x$ in the definition (\ref{DefqHahn}) of the $q$-Hahn -polynomials and take the limit $N\rightarrow\infty$ we find the little $q$-Jacobi polynomials: -\begin{equation} -\lim _{N\rightarrow\infty} Q_n(q^{x-N};\alpha,\beta,N|q)=p_n(q^x;\alpha,\beta|q), -\end{equation} -where $p_n(q^x;\alpha,\beta|q)$ is given by (\ref{DefLittleqJacobi}). - -\subsubsection*{$q$-Hahn $\rightarrow$ $q$-Meixner} -The $q$-Meixner polynomials given by (\ref{DefqMeixner}) can be obtained from the $q$-Hahn polynomials -by setting $\alpha=b$ and $\beta=-b^{-1}c^{-1}q^{-N-1}$ in the definition (\ref{DefqHahn}) of the -$q$-Hahn polynomials and letting $N\rightarrow\infty$: -\begin{equation} -\lim_{N\rightarrow\infty} -Q_n(q^{-x};b,-b^{-1}c^{-1}q^{-N-1},N|q)=M_n(q^{-x};b,c;q). -\end{equation} - -\subsubsection*{$q$-Hahn $\rightarrow$ Quantum $q$-Krawtchouk} -The quantum $q$-Krawtchouk polynomials given by (\ref{DefQuantumqKrawtchouk}) -simply follow from the $q$-Hahn polynomials by setting $\beta=p$ in the definition (\ref{DefqHahn}) of -the $q$-Hahn polynomials and taking the limit $\alpha\rightarrow\infty$: -\begin{equation} -\lim_{\alpha\rightarrow\infty}Q_n(q^{-x};\alpha,p,N|q)=K_n^{qtm}(q^{-x};p,N;q). -\end{equation} - -\subsubsection*{$q$-Hahn $\rightarrow$ $q$-Krawtchouk} -If we set $\beta=-\alpha^{-1}q^{-1}p$ in the definition (\ref{DefqHahn}) of the $q$-Hahn polynomials -and then let $\alpha\rightarrow 0$ we obtain the $q$-Krawtchouk polynomials given by -(\ref{DefqKrawtchouk}): -\begin{equation} -\lim_{\alpha\rightarrow 0} -Q_n(q^{-x};\alpha,-\alpha^{-1}q^{-1}p,N|q)=K_n(q^{-x};p,N;q). -\end{equation} - -\subsubsection*{$q$-Hahn $\rightarrow$ Affine $q$-Krawtchouk} -The affine $q$-Krawtchouk polynomials given by (\ref{DefAffqKrawtchouk}) -can be obtained from the $q$-Hahn polynomials by the substitution $\alpha=p$ and -$\beta=0$ in (\ref{DefqHahn}): -\begin{equation} -Q_n(q^{-x};p,0,N|q)=K_n^{Aff}(q^{-x};p,N;q). -\end{equation} - -\subsubsection*{$q$-Hahn $\rightarrow$ Hahn} -The Hahn polynomials given by (\ref{DefHahn}) simply follow from the $q$-Hahn -polynomials given by (\ref{DefqHahn}), after setting $\alpha\rightarrow q^{\alpha}$ -and $\beta\rightarrow q^{\beta}$, in the following way: -\begin{equation} -\lim_{q\rightarrow 1}Q_n(q^{-x};q^{\alpha},q^{\beta},N|q)=Q_n(x;\alpha,\beta,N). -\end{equation} - -\subsection*{Remark} -The $q$-Hahn polynomials given by (\ref{DefqHahn}) and the dual $q$-Hahn -polynomials given by (\ref{DefDualqHahn}) are related in the following way: -$$Q_n(q^{-x};\alpha,\beta,N|q)=R_x(\mu(n);\alpha,\beta,N|q),$$ -with -$$\mu(n)=q^{-n}+\alpha\beta q^{n+1}$$ -or -$$R_n(\mu(x);\gamma,\delta,N|q)=Q_x(q^{-n};\gamma,\delta,N|q),$$ -where -$$\mu(x)=q^{-x}+\gamma\delta q^{x+1}.$$ - -\subsection*{References} -\cite{AlSalam90}, \cite{AlvarezRonveaux}, \cite{AndrewsAskey85}, -\cite{AskeyWilson79}, \cite{AskeyWilson85}, \cite{AtakRahmanSuslov}, -\cite{Dunkl78II}, \cite{Fischer95}, \cite{GasperRahman84}, -\cite{GasperRahman90}, \cite{Hahn}, \cite{KalninsMiller88}, -\cite{Koelink96I}, \cite{KoelinkKoorn}, \cite{Koorn89III}, \cite{Koorn90II}, -\cite{Nikiforov+}, \cite{NoumiMimachi90II}, \cite{Rahman82}, -\cite{Stanton80II}, \cite{Stanton84}, \cite{Stanton90}. - - -\section{Dual $q$-Hahn}\index{Dual q-Hahn polynomials@Dual $q$-Hahn polynomials} -\index{q-Hahn polynomials@$q$-Hahn polynomials!Dual} -\par\setcounter{equation}{0} - -\subsection*{Basic hypergeometric representation} -\begin{equation} -\label{DefDualqHahn} -R_n(\mu(x);\gamma,\delta,N|q)= -\qhyp{3}{2}{q^{-n},q^{-x},\gamma\delta q^{x+1}}{\gamma q,q^{-N}}{q},\quad n=0,1,2,\ldots,N, -\end{equation} -where -$$\mu(x):=q^{-x}+\gamma\delta q^{x+1}.$$ - -\newpage - -\subsection*{Orthogonality relation} -For $0<\gamma q<1$ and $0<\delta q<1$, or for $\gamma>q^{-N}$ and $\delta>q^{-N}$, we have -\begin{eqnarray} -\label{OrtDualqHahn} -& &\sum_{x=0}^N\frac{(\gamma q,\gamma\delta q,q^{-N};q)_x}{(q,\gamma\delta q^{N+2},\delta q;q)_x} -\frac{(1-\gamma\delta q^{2x+1})}{(1-\gamma\delta q)(-\gamma q)^x}q^{Nx-\binom{x}{2}}\nonumber\\ -& &{}\mathindent{}\times R_m(\mu(x);\gamma,\delta,N|q)R_n(\mu(x);\gamma,\delta,N|q)\nonumber\\ -& &{}=\frac{(\gamma\delta q^2;q)_N}{(\delta q;q)_N}(\gamma q)^{-N} -\frac{(q,\delta^{-1}q^{-N};q)_n}{(\gamma q,q^{-N};q)_n}(\gamma\delta q)^n\,\delta_{mn}. -\end{eqnarray} - -\subsection*{Recurrence relation} -\begin{eqnarray} -\label{RecDualqHahn} -& &-\left(1-q^{-x}\right)\left(1-\gamma\delta q^{x+1}\right)R_n(\mu(x))\nonumber\\ -& &{}=A_nR_{n+1}(\mu(x))-\left(A_n+C_n\right)R_n(\mu(x))+C_nR_{n-1}(\mu(x)), -\end{eqnarray} -where -$$R_n(\mu(x)):=R_n(\mu(x);\gamma,\delta,N|q)$$ -and -$$\left\{\begin{array}{l} -\displaystyle A_n=\left(1-q^{n-N}\right)\left(1-\gamma q^{n+1}\right)\\ -\\ -\displaystyle C_n=\gamma q\left(1-q^n\right)\left(\delta -q^{n-N-1}\right). -\end{array}\right.$$ - -\subsection*{Normalized recurrence relation} -\begin{eqnarray} -\label{NormRecDualqHahn} -xp_n(x)&=&p_{n+1}(x)+\left[1+\gamma\delta q-(A_n+C_n)\right]p_n(x)\nonumber\\ -& &{}\mathindent{}+\gamma q(1-q^n)(1-\gamma q^n)\nonumber\\ -& &{}\mathindent\mathindent{}\times(1-q^{n-N-1})(\delta-q^{n-N-1})p_{n-1}(x), -\end{eqnarray} -where -$$R_n(\mu(x);\gamma,\delta,N|q)=\frac{1}{(\gamma q,q^{-N};q)_n}p_n(\mu(x)).$$ - -\subsection*{$q$-Difference equation} -\begin{equation} -\label{dvDualqHahn} -q^{-n}(1-q^n)y(x)=B(x)y(x+1)-\left[B(x)+D(x)\right]y(x) -+D(x)y(x-1), -\end{equation} -where -$$y(x)=R_n(\mu(x);\gamma,\delta,N|q)$$ -and -$$\left\{\begin{array}{l}\displaystyle B(x)=\frac{(1-q^{x-N})(1-\gamma q^{x+1})(1-\gamma\delta q^{x+1})} -{(1-\gamma\delta q^{2x+1})(1-\gamma\delta q^{2x+2})}\\ -\\ -\displaystyle D(x)=-\frac{\gamma q^{x-N}(1-q^x)(1-\gamma\delta q^{x+N+1})(1-\delta q^x)} -{(1-\gamma\delta q^{2x})(1-\gamma\delta q^{2x+1})}.\end{array}\right.$$ - -\subsection*{Forward shift operator} -\begin{eqnarray} -\label{shift1DualqHahnI} -& &R_n(\mu(x+1);\gamma,\delta,N|q)-R_n(\mu(x);\gamma,\delta,N|q)\nonumber\\ -& &{}=\frac{q^{-n-x}(1-q^n)(1-\gamma\delta q^{2x+2})}{(1-\gamma q)(1-q^{-N})} -R_{n-1}(\mu(x);\gamma q,\delta,N-1|q) -\end{eqnarray} -or equivalently -\begin{eqnarray} -\label{shift1DualqHahnII} -& &\frac{\Delta R_n(\mu(x);\gamma,\delta,N|q)}{\Delta\mu(x)}\nonumber\\ -& &{}=\frac{q^{-n+1}(1-q^n)}{(1-q)(1-\gamma q)(1-q^{-N})}R_{n-1}(\mu(x);\gamma q,\delta,N-1|q). -\end{eqnarray} - -\subsection*{Backward shift operator} -\begin{eqnarray} -\label{shift2DualqHahnI} -& &(1-\gamma q^x)(1-\gamma\delta q^x)(1-q^{x-N-1})R_n(\mu(x);\gamma,\delta,N|q)\nonumber\\ -& &{}\mathindent{}+\gamma q^{x-N-1}(1-q^x)(1-\gamma\delta q^{x+N+1})(1-\delta q^x)R_n(\mu(x-1);\gamma,\delta,N|q)\nonumber\\ -& &{}=q^x(1-\gamma)(1-q^{-N-1})(1-\gamma\delta q^{2x})R_{n+1}(\mu(x);\gamma q^{-1},\delta,N+1|q) -\end{eqnarray} -or equivalently -\begin{eqnarray} -\label{shift2DualqHahnII} -& &\frac{\nabla\left[w(x;\gamma,\delta,N|q)R_n(\mu(x);\gamma,\delta,N|q)\right]}{\nabla\mu(x)}\nonumber\\ -& &{}=\frac{1}{(1-q)(1-\gamma\delta)}w(x;\gamma q^{-1},\delta,N+1|q)\nonumber\\ -& &{}\mathindent{}\times R_{n+1}(\mu(x);\gamma q^{-1},\delta,N+1|q), -\end{eqnarray} -where -$$w(x;\gamma,\delta,N|q)=\frac{(\gamma q,\gamma\delta q,q^{-N};q)_x} -{(q,\gamma\delta q^{N+2},\delta q;q)_x}(-\gamma^{-1})^x q^{Nx-\binom{x}{2}}.$$ - -\subsection*{Rodrigues-type formula} -\begin{eqnarray} -\label{RodDualqHahn} -& &w(x;\gamma,\delta,N|q)R_n(\mu(x);\gamma,\delta,N|q)\nonumber\\ -& &{}=(1-q)^n(\gamma\delta q;q)_n -\left(\nabla_{\mu}\right)^n\left[w(x;\gamma q^n,\delta,N-n|q)\right], -\end{eqnarray} -where -$$\nabla_{\mu}:=\frac{\nabla}{\nabla\mu(x)}.$$ - -\subsection*{Generating functions} For $x=0,1,2,\ldots,N$ we have -\begin{eqnarray} -\label{GenDualqHahn1} -& &(q^{-N}t;q)_{N-x}\cdot\qhyp{2}{1}{q^{-x},\delta^{-1}q^{-x}}{\gamma q}{\gamma\delta q^{x+1}t}\nonumber\\ -& &{}=\sum_{n=0}^N\frac{(q^{-N};q)_n}{(q;q)_n}R_n(\mu(x);\gamma,\delta,N|q)t^n. -\end{eqnarray} - -\begin{eqnarray} -\label{GenDualqHahn2} -& &(\gamma\delta qt;q)_x\cdot\qhyp{2}{1}{q^{x-N},\gamma q^{x+1}}{\delta^{-1}q^{-N}}{q^{-x}t}\nonumber\\ -& &{}=\sum_{n=0}^N -\frac{(q^{-N},\gamma q;q)_n}{(\delta^{-1}q^{-N},q;q)_n}R_n(\mu(x);\gamma,\delta,N|q)t^n. -\end{eqnarray} - -\subsection*{Limit relations} - -\subsubsection*{$q$-Racah $\rightarrow$ Dual $q$-Hahn} -To obtain the dual $q$-Hahn polynomials from the $q$-Racah polynomials we have to -take $\beta=0$ and $\alpha q=q^{-N}$ in (\ref{DefqRacah}): -$$R_n(\mu(x);q^{-N-1},0,\gamma,\delta|q)=R_n(\mu(x);\gamma,\delta,N|q),$$ -with -$$\mu(x)=q^{-x}+\gamma\delta q^{x+1}.$$ -We may also take $\alpha=0$ and $\beta=\delta^{-1}q^{-N-1}$ in (\ref{DefqRacah}) to obtain -the dual $q$-Hahn polynomials from the $q$-Racah polynomials: -$$R_n(\mu(x);0,\delta^{-1}q^{-N-1},\gamma,\delta|q)=R_n(\mu(x);\gamma,\delta,N|q).$$ -And if we take $\gamma q=q^{-N}$, $\delta\rightarrow\alpha\delta q^{N+1}$ and $\beta=0$ in the -definition (\ref{DefqRacah}) of the $q$-Racah polynomials we find the dual -$q$-Hahn polynomials given by (\ref{DefDualqHahn}) in the following way: -$$R_n(\mu(x);\alpha,0,q^{-N-1},\alpha\delta q^{N+1}|q)=R_n({\tilde \mu}(x);\alpha,\delta,N|q),$$ -with -$${\tilde \mu}(x)=q^{-x}+\alpha\delta q^{x+1}.$$ - -\subsubsection*{Dual $q$-Hahn $\rightarrow$ Affine $q$-Krawtchouk} -The affine $q$-Krawtchouk polynomials given by (\ref{DefAffqKrawtchouk}) -can be obtained from the dual $q$-Hahn polynomials by the substitution $\gamma=p$ -and $\delta=0$ in (\ref{DefDualqHahn}): -\begin{equation} -R_n(\mu(x);p,0,N|q)=K_n^{Aff}(q^{-x};p,N;q). -\end{equation} -Note that $\mu(x)=q^{-x}$ in this case. - -\subsubsection*{Dual $q$-Hahn $\rightarrow$ Dual $q$-Krawtchouk} -The dual $q$-Krawtchouk polynomials given by (\ref{DefDualqKrawtchouk}) can -be obtained from the dual $q$-Hahn polynomials by setting $\delta=c\gamma^{-1}q^{-N-1}$ -in (\ref{DefDualqHahn}) and letting $\gamma\rightarrow 0$: -\begin{equation} -\lim_{\gamma\rightarrow 0} -R_n(\mu(x);\gamma,c\gamma^{-1}q^{-N-1},N|q)=K_n(\lambda(x);c,N|q). -\end{equation} - -\subsubsection*{Dual $q$-Hahn $\rightarrow$ Dual Hahn} -The dual Hahn polynomials given by (\ref{DefDualHahn}) follow from the dual $q$-Hahn -polynomials by simply taking the limit $q\rightarrow 1$ in the definition (\ref{DefDualqHahn}) -of the dual $q$-Hahn polynomials after applying the substitution -$\gamma\rightarrow q^{\gamma}$ and $\delta\rightarrow q^{\delta}$: -\begin{equation} -\lim_{q\rightarrow 1}R_n(\mu(x);q^{\gamma},q^{\delta},N|q)=R_n(\lambda(x);\gamma,\delta,N), -\end{equation} -where -$$\left\{\begin{array}{l} -\displaystyle\mu(x)=q^{-x}+q^{x+\gamma+\delta+1}\\[5mm] -\displaystyle\lambda(x)=x(x+\gamma+\delta+1). -\end{array}\right.$$ - -\subsection*{Remark} -The dual $q$-Hahn polynomials given by (\ref{DefDualqHahn}) and the -$q$-Hahn polynomials given by (\ref{DefqHahn}) are related in the following way: -$$Q_n(q^{-x};\alpha,\beta,N|q)=R_x(\mu(n);\alpha,\beta,N|q),$$ -with -$$\mu(n)=q^{-n}+\alpha\beta q^{n+1}$$ -or -$$R_n(\mu(x);\gamma,\delta,N|q)=Q_x(q^{-n};\gamma,\delta,N|q),$$ -where -$$\mu(x)=q^{-x}+\gamma\delta q^{x+1}.$$ - -\subsection*{References} -\cite{AlvarezSmirnov}, \cite{AndrewsAskey85}, \cite{AskeyWilson79}, -\cite{AskeyWilson85}, \cite{AtakRahmanSuslov}, \cite{GasperRahman90}, -\cite{KoelinkKoorn}, \cite{Nikiforov+}, \cite{Stanton84}. - - -\section{Al-Salam-Chihara}\index{Al-Salam-Chihara polynomials} -\par\setcounter{equation}{0} - -\subsection*{Basic hypergeometric representation} -\begin{eqnarray} -\label{DefAlSalamChihara} -Q_n(x;a,b|q)&=&\frac{(ab;q)_n}{a^n}\, -\qhyp{3}{2}{q^{-n},a\e^{i\theta},a\e^{-i\theta}}{ab,0}{q}\\ -&=&(a\e^{i\theta};q)_n\e^{-in\theta}\,\qhyp{2}{1}{q^{-n},b\e^{-i\theta}} -{a^{-1}q^{-n+1}\e^{-i\theta}}{a^{-1}q\e^{i\theta}}\nonumber\\ -&=&(b\e^{-i\theta};q)_n\e^{in\theta}\,\qhyp{2}{1}{q^{-n},a\e^{i\theta}} -{b^{-1}q^{-n+1}\e^{i\theta}}{b^{-1}q\e^{-i\theta}},\quad x=\cos\theta.\nonumber -\end{eqnarray} - -\subsection*{Orthogonality relation} -If $a$ and $b$ are real or complex conjugates and $\max(|a|,|b|)<1$, then we have the following orthogonality relation -\begin{equation} -\label{OrtAlSalamChiharaI} -\frac{1}{2\pi}\int_{-1}^1\frac{w(x)}{\sqrt{1-x^2}}Q_m(x;a,b|q)Q_n(x;a,b|q)\,dx -=\frac{\,\delta_{mn}}{(q^{n+1},abq^n;q)_{\infty}}, -\end{equation} -where -$$w(x):=w(x;a,b|q)=\left|\frac{(\e^{2i\theta};q)_{\infty}} -{(a\e^{i\theta},b\e^{i\theta};q)_{\infty}}\right|^2 -=\frac{h(x,1)h(x,-1)h(x,q^{\frac{1}{2}})h(x,-q^{\frac{1}{2}})}{h(x,a)h(x,b)},$$ -with -$$h(x,\alpha):=\prod_{k=0}^{\infty}\left(1-2\alpha xq^k+\alpha^2q^{2k}\right) -=\left(\alpha\e^{i\theta},\alpha\e^{-i\theta};q\right)_{\infty},\quad x=\cos\theta.$$ -If $a>1$ and $|ab|<1$, then we have another orthogonality relation given by: -\begin{eqnarray} -\label{OrtAlSalamChiharaII} -& &\frac{1}{2\pi}\int_{-1}^1\frac{w(x)}{\sqrt{1-x^2}}Q_m(x;a,b|q)Q_n(x;a,b|q)\,dx\nonumber\\ -& &{}\mathindent{}+\sum_{\begin{array}{c}\scriptstyle k\\ \scriptstyle 10. -\end{eqnarray} - -\subsection*{Recurrence relation} -\begin{eqnarray} -\label{RecqMeixner} -& &q^{2n+1}(1-q^{-x})M_n(q^{-x})\nonumber\\ -& &{}=c(1-bq^{n+1})M_{n+1}(q^{-x})\nonumber\\ -& &{}\mathindent{}-\left[c(1-bq^{n+1})+q(1-q^n)(c+q^n)\right]M_n(q^{-x})\nonumber\\ -& &{}\mathindent\mathindent{}+q(1-q^n)(c+q^n)M_{n-1}(q^{-x}), -\end{eqnarray} -where -$$M_n(q^{-x}):=M_n(q^{-x};b,c;q).$$ - -\subsection*{Normalized recurrence relation} -\begin{eqnarray} -\label{NormRecqMeixner} -xp_n(x)&=&p_{n+1}(x)+ -\left[1+q^{-2n-1}\left\{c(1-bq^{n+1})+q(1-q^n)(c+q^n)\right\}\right]p_n(x)\nonumber\\ -& &{}\mathindent{}+cq^{-4n+1}(1-q^n)(1-bq^n)(c+q^n)p_{n-1}(x), -\end{eqnarray} -where -$$M_n(q^{-x};b,c;q)=\frac{(-1)^nq^{n^2}}{(bq;q)_nc^n}p_n(q^{-x}).$$ - -\subsection*{$q$-Difference equation} -\begin{equation} -\label{dvqMeixner} --(1-q^n)y(x)=B(x)y(x+1)-\left[B(x)+D(x)\right]y(x)+D(x)y(x-1), -\end{equation} -where -$$y(x)=M_n(q^{-x};b,c;q)$$ -and -$$\left\{\begin{array}{l}\displaystyle B(x)=cq^x(1-bq^{x+1})\\ -\\ -\displaystyle D(x)=(1-q^x)(1+bcq^x).\end{array}\right.$$ - -\subsection*{Forward shift operator} -\begin{eqnarray} -\label{shift1qMeixnerI} -& &M_n(q^{-x-1};b,c;q)-M_n(q^{-x};b,c;q)\nonumber\\ -& &{}=-\frac{q^{-x}(1-q^n)}{c(1-bq)}M_{n-1}(q^{-x};bq,cq^{-1};q) -\end{eqnarray} -or equivalently -\begin{equation} -\label{shift1qMeixnerII} -\frac{\Delta M_n(q^{-x};b,c;q)}{\Delta q^{-x}} -=-\frac{q(1-q^n)}{c(1-q)(1-bq)}M_{n-1}(q^{-x};bq,cq^{-1};q). -\end{equation} - -\subsection*{Backward shift operator} -\begin{eqnarray} -\label{shift2qMeixnerI} -& &cq^x(1-bq^x)M_n(q^{-x};b,c;q)-(1-q^x)(1+bcq^x)M_n(q^{-x+1};b,c;q)\nonumber\\ -& &{}=cq^x(1-b)M_{n+1}(q^{-x};bq^{-1},cq;q) -\end{eqnarray} -or equivalently -\begin{eqnarray} -\label{shift2qMeixnerII} -& &\frac{\nabla\left[w(x;b,c;q)M_n(q^{-x};b,c;q)\right]}{\nabla q^{-x}}\nonumber\\ -& &{}=\frac{1}{1-q}w(x;bq^{-1},cq;q)M_{n+1}(q^{-x};bq^{-1},cq;q), -\end{eqnarray} -where -$$w(x;b,c;q)=\frac{(bq;q)_x}{(q,-bcq;q)_x}c^xq^{\binom{x+1}{2}}.$$ - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodqMeixner} -w(x;b,c;q)M_n(q^{-x};b,c;q)=(1-q)^n\left(\nabla_q\right)^n\left[w(x;bq^n,cq^{-n};q)\right], -\end{equation} -where -$$\nabla_q:=\frac{\nabla}{\nabla q^{-x}}.$$ - -\subsection*{Generating functions} -\begin{equation} -\label{GenqMeixner1} -\frac{1}{(t;q)_{\infty}}\,\qhyp{1}{1}{q^{-x}}{bq}{-c^{-1}qt} -=\sum_{n=0}^{\infty}\frac{M_n(q^{-x};b,c;q)}{(q;q)_n}t^n. -\end{equation} - -\begin{eqnarray} -\label{GenqMeixner2} -& &\frac{1}{(t;q)_{\infty}}\,\qhyp{1}{1}{-b^{-1}c^{-1}q^{-x}}{-c^{-1}q}{bqt}\nonumber\\ -& &{}=\sum_{n=0}^{\infty}\frac{(bq;q)_n}{(-c^{-1}q,q;q)_n}M_n(q^{-x};b,c;q)t^n. -\end{eqnarray} - -\subsection*{Limit relations} - -\subsubsection*{Big $q$-Jacobi $\rightarrow$ $q$-Meixner} -If we set $b=-a^{-1}cd^{-1}$ (with $d>0$) in the definition (\ref{DefBigqJacobi}) -of the big $q$-Jacobi polynomials and take the limit $c\rightarrow -\infty$ we -obtain the $q$-Meixner polynomials given by (\ref{DefqMeixner}): -\begin{equation} -\lim_{c\rightarrow -\infty}P_n(q^{-x};a,-a^{-1}cd^{-1},c;q)=M_n(q^{-x};a,d;q). -\end{equation} - -\subsubsection*{$q$-Hahn $\rightarrow$ $q$-Meixner} -The $q$-Meixner polynomials given by (\ref{DefqMeixner}) can be obtained from the $q$-Hahn polynomials -by setting $\alpha=b$ and $\beta=-b^{-1}c^{-1}q^{-N-1}$ in the definition (\ref{DefqHahn}) of the -$q$-Hahn polynomials and letting $N\rightarrow\infty$: -$$\lim_{N\rightarrow\infty}Q_n(q^{-x};b,-b^{-1}c^{-1}q^{-N-1},N|q)=M_n(q^{-x};b,c;q).$$ - -\subsubsection*{$q$-Meixner $\rightarrow$ $q$-Laguerre} -The $q$-Laguerre polynomials given by (\ref{DefqLaguerre}) can be obtained -from the $q$-Meixner polynomials given by (\ref{DefqMeixner}) by setting -$b=q^{\alpha}$ and $q^{-x}\rightarrow cq^{\alpha}x$ in the definition -(\ref{DefqMeixner}) of the $q$-Meixner polynomials and then taking the limit -$c\rightarrow\infty$: -\begin{equation} -\lim_{c\rightarrow\infty}M_n(cq^{\alpha}x;q^{\alpha},c;q)= -\frac{(q;q)_n}{(q^{\alpha+1};q)_n}L_n^{(\alpha)}(x;q). -\end{equation} - -\subsubsection*{$q$-Meixner $\rightarrow$ $q$-Charlier} -The $q$-Charlier polynomials given by (\ref{DefqCharlier}) can easily be -obtained from the $q$-Meixner given by (\ref{DefqMeixner}) by setting $b=0$ -in the definition (\ref{DefqMeixner}) of the $q$-Meixner polynomials: -\begin{equation} -M_n(x;0,c;q)=C_n(x;c;q). -\end{equation} - -\subsubsection*{$q$-Meixner $\rightarrow$ Al-Salam-Carlitz~II} -The Al-Salam-Carlitz~II polynomials given by (\ref{DefAlSalamCarlitzII}) -can be obtained from the $q$-Meixner polynomials given by (\ref{DefqMeixner}) -by setting $b=-ac^{-1}$ in the definition (\ref{DefqMeixner}) of the -$q$-Meixner polynomials and then taking the limit $c\rightarrow 0$: -\begin{equation} -\lim_{c\rightarrow 0}M_n(x;-ac^{-1},c;q)= -\left(-\frac{1}{a}\right)^nq^{\binom{n}{2}}V_n^{(a)}(x;q). -\end{equation} - -\subsubsection*{$q$-Meixner $\rightarrow$ Meixner} -To find the Meixner polynomials given by (\ref{DefMeixner}) from the $q$-Meixner polynomials given by -(\ref{DefqMeixner}) we set $b=q^{\beta-1}$ and $c\rightarrow (1-c)^{-1}c$ and let $q\rightarrow 1$: -\begin{equation} -\lim_{q\rightarrow 1}M_n(q^{-x};q^{\beta-1},(1-c)^{-1}c;q)=M_n(x;\beta,c). -\end{equation} - -\subsection*{Remarks} -The $q$-Meixner polynomials given by (\ref{DefqMeixner}) and the little -$q$-Jacobi polynomials given by (\ref{DefLittleqJacobi}) are related in the -following way: -$$M_n(q^{-x};b,c;q)=p_n(-c^{-1}q^n;b,b^{-1}q^{-n-x-1}|q).$$ - -\noindent -The $q$-Meixner polynomials and the quantum $q$-Krawtchouk polynomials -given by (\ref{DefQuantumqKrawtchouk}) are related in the following way: -$$K_n^{qtm}(q^{-x};p,N;q)=M_n(q^{-x};q^{-N-1},-p^{-1};q).$$ - -\subsection*{References} -\cite{AlSalam90}, \cite{AlSalamVerma82II}, \cite{AlSalamVerma88}, \cite{AlvarezRonveaux}, -\cite{AtakAtakKlimyk}, \cite{AtakRahmanSuslov}, \cite{Campigotto+}, \cite{GasperRahman90}, -\cite{Hahn}, \cite{Ismail2005I}, \cite{Nikiforov+}, \cite{Smirnov}. - - -\section{Quantum $q$-Krawtchouk} -\index{Quantum q-Krawtchouk polynomials@Quantum $q$-Krawtchouk polynomials} -\index{q-Krawtchouk polynomials@$q$-Krawtchouk polynomials!Quantum} -\par\setcounter{equation}{0} - -\subsection*{Basic hypergeometric representation} -\begin{equation} -\label{DefQuantumqKrawtchouk} -K_n^{qtm}(q^{-x};p,N;q)= -\qhyp{2}{1}{q^{-n},q^{-x}}{q^{-N}}{pq^{n+1}},\quad n=0,1,2,\ldots,N. -\end{equation} - -\subsection*{Orthogonality relation} -\begin{eqnarray} -\label{OrtQuantumqKrawtchouk} -& &\sum_{x=0}^N\frac{(pq;q)_{N-x}}{(q;q)_x(q;q)_{N-x}}(-1)^{N-x}q^{\binom{x}{2}} -K_m^{qtm}(q^{-x};p,N;q)K_n^{qtm}(q^{-x};p,N;q)\nonumber\\ -& &{}=\frac{(-1)^np^N(q;q)_{N-n}(q,pq;q)_n}{(q,q;q)_N} -q^{\binom{N+1}{2}-\binom{n+1}{2}+Nn}\,\delta_{mn},\quad p>q^{-N}. -\end{eqnarray} - -\subsection*{Recurrence relation} -\begin{eqnarray} -\label{RecQuantumqKrawtchouk} -& &-pq^{2n+1}(1-q^{-x})K_n^{qtm}(q^{-x})\nonumber\\ -& &{}=(1-q^{n-N})K_{n+1}^{qtm}(q^{-x})\nonumber\\ -& &{}\mathindent{}-\left[(1-q^{n-N})+q(1-q^n)(1-pq^n)\right]K_n^{qtm}(q^{-x})\nonumber\\ -& &{}\mathindent\mathindent{}+q(1-q^n)(1-pq^n)K_{n-1}^{qtm}(q^{-x}), -\end{eqnarray} -where -$$K_n^{qtm}(q^{-x}):=K_n^{qtm}(q^{-x};p,N;q).$$ - -\subsection*{Normalized recurrence relation} -\begin{eqnarray} -\label{NormRecQuantumqKrawtchouk} -xp_n(x)&=&p_{n+1}(x)+ -\left[1-p^{-1}q^{-2n-1}\left\{(1-q^{n-N})+q(1-q^n)(1-pq^n)\right\}\right]p_n(x)\nonumber\\ -& &{}\mathindent{}+p^{-2}q^{-4n+1}(1-q^n)(1-pq^n)(1-q^{n-N-1})p_{n-1}(x), -\end{eqnarray} -where -$$K_n^{qtm}(q^{-x};p,N;q)=\frac{p^nq^{n^2}}{(q^{-N};q)_n}p_n(q^{-x}).$$ - -\subsection*{$q$-Difference equation} -\begin{equation} -\label{dvQuantumqKrawtchouk} --p(1-q^n)y(x)=B(x)y(x+1)-\left[B(x)+D(x)\right]y(x)+D(x)y(x-1), -\end{equation} -where -$$y(x)=K_n^{qtm}(q^{-x};p,N;q)$$ -and -$$\left\{\begin{array}{l}\displaystyle B(x)=-q^x(1-q^{x-N})\\ -\\ -\displaystyle D(x)=(1-q^x)(p-q^{x-N-1}).\end{array}\right.$$ - -\subsection*{Forward shift operator} -\begin{eqnarray} -\label{shift1QuantumqKrawtchoukI} -& &K_n^{qtm}(q^{-x-1};p,N;q)-K_n^{qtm}(q^{-x};p,N;q)\nonumber\\ -& &{}=\frac{pq^{-x}(1-q^n)}{1-q^{-N}}K_{n-1}^{qtm}(q^{-x};pq,N-1;q) -\end{eqnarray} -or equivalently -\begin{equation} -\label{shift1QuantumqKrawtchoukII} -\frac{\Delta K_n^{qtm}(q^{-x};p,N;q)}{\Delta q^{-x}}= -\frac{pq(1-q^n)}{(1-q)(1-q^{-N})}K_{n-1}^{qtm}(q^{-x};pq,N-1;q). -\end{equation} - -\subsection*{Backward shift operator} -\begin{eqnarray} -\label{shift2QuantumqKrawtchoukI} -& &(1-q^{x-N-1})K_n^{qtm}(q^{-x};p,N;q)\nonumber\\ -& &{}\mathindent{}+q^{-x}(1-q^x)(p-q^{x-N-1})K_n^{qtm}(q^{-x+1};p,N;q)\nonumber\\ -& &{}=(1-q^{-N-1})K_{n+1}^{qtm}(q^{-x};pq^{-1},N+1;q) -\end{eqnarray} -or equivalently -\begin{eqnarray} -\label{shift2QuantumqKrawtchoukII} -& &\frac{\nabla\left[w(x;p,N;q)K_n^{qtm}(q^{-x};p,N;q)\right]}{\nabla q^{-x}}\nonumber\\ -& &{}=\frac{1}{1-q}w(x;pq^{-1},N+1;q)K_{n+1}^{qtm}(q^{-x};pq^{-1},N+1;q), -\end{eqnarray} -where -$$w(x;p,N;q)=\frac{(q^{-N};q)_x}{(q,p^{-1}q^{-N};q)_x}(-p)^{-x}q^{\binom{x+1}{2}}.$$ - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodQuantumqKrawtchouk} -w(x;p,N;q)K_n^{qtm}(q^{-x};p,N;q)=(1-q)^n\left(\nabla_q\right)^n\left[w(x;pq^n,N-n;q)\right], -\end{equation} -where -$$\nabla_q:=\frac{\nabla}{\nabla q^{-x}}.$$ - -\subsection*{Generating functions} For $x=0,1,2,\ldots,N$ we have -\begin{eqnarray} -\label{GenQuantumqKrawtchouk1} -& &(q^{x-N}t;q)_{N-x}\cdot\qhyp{2}{1}{q^{-x},pq^{N+1-x}}{0}{q^{x-N}t}\nonumber\\ -& &{}=\sum_{n=0}^N\frac{(q^{-N};q)_n}{(q;q)_n}K_n^{qtm}(q^{-x};p,N;q)t^n. -\end{eqnarray} - -\begin{eqnarray} -\label{GenQuantumqKrawtchouk2} -& &(q^{-x}t;q)_x\cdot\qhyp{2}{1}{q^{x-N},0}{pq}{q^{-x}t}\nonumber\\ -& &{}=\sum_{n=0}^N\frac{(q^{-N};q)_n}{(pq,q;q)_n}K_n^{qtm}(q^{-x};p,N;q)t^n. -\end{eqnarray} - -\subsection*{Limit relations} - -\subsubsection*{$q$-Hahn $\rightarrow$ Quantum $q$-Krawtchouk} -The quantum $q$-Krawtchouk polynomials given by (\ref{DefQuantumqKrawtchouk}) -simply follow from the $q$-Hahn polynomials by setting $\beta=p$ in the definition (\ref{DefqHahn}) of -the $q$-Hahn polynomials and taking the limit $\alpha\rightarrow\infty$: -$$\lim_{\alpha\rightarrow\infty}Q_n(q^{-x};\alpha,p,N|q)=K_n^{qtm}(q^{-x};p,N;q).$$ - -\subsubsection*{Quantum $q$-Krawtchouk $\rightarrow$ Al-Salam-Carlitz~II} -If we set $p=a^{-1}q^{-N-1}$ in the definition (\ref{DefQuantumqKrawtchouk}) -of the quantum $q$-Krawtchouk polynomials and let $N\rightarrow\infty$ we -obtain the Al-Salam-Carlitz~II polynomials given by -(\ref{DefAlSalamCarlitzII}). In fact we have -\begin{equation} -\lim_{N\rightarrow\infty}K_n^{qtm}(x;a^{-1}q^{-N-1},N;q)= -\left(-\frac{1}{a}\right)^nq^{\binom{n}{2}}V_n^{(a)}(x;q). -\end{equation} - -\subsubsection*{Quantum $q$-Krawtchouk $\rightarrow$ Krawtchouk} -The Krawtchouk polynomials given by (\ref{DefKrawtchouk}) easily follow from -the quantum $q$-Krawtchouk polynomials given by (\ref{DefQuantumqKrawtchouk}) -in the following way: -\begin{equation} -\lim_{q\rightarrow 1}K_n^{qtm}(q^{-x};p,N;q)=K_n(x;p^{-1},N). -\end{equation} - -\subsection*{Remarks} -The quantum $q$-Krawtchouk polynomials given by -(\ref{DefQuantumqKrawtchouk}) and the $q$-Meixner polynomials given by -(\ref{DefqMeixner}) are related in the following way: -$$K_n^{qtm}(q^{-x};p,N;q)=M_n(q^{-x};q^{-N-1},-p^{-1};q).$$ - -\noindent -The quantum $q$-Krawtchouk polynomials are related to the affine -$q$-Krawtchouk polynomials given by (\ref{DefAffqKrawtchouk}) -by the transformation $q\leftrightarrow q^{-1}$ in the following way: -$$K_n^{qtm}(q^x;p,N;q^{-1})=(p^{-1}q;q)_n\left(-\frac{p}{q}\right)^nq^{-\binom{n}{2}} -K_n^{Aff}(q^{x-N};p^{-1},N;q).$$ - -\subsection*{References} -\cite{GasperRahman90}, \cite{Koorn89III}, \cite{Koorn90II}, \cite{Smirnov}. - - -\section{$q$-Krawtchouk}\index{q-Krawtchouk polynomials@$q$-Krawtchouk polynomials} -\par\setcounter{equation}{0} - -\subsection*{Basic hypergeometric representation} For $n=0,1,2,\ldots,N$ we have -\begin{eqnarray} -\label{DefqKrawtchouk} -K_n(q^{-x};p,N;q)&=&\qhyp{3}{2}{q^{-n},q^{-x},-pq^n}{q^{-N},0}{q}\\ -&=&\frac{(q^{x-N};q)_n}{(q^{-N};q)_nq^{nx}}\, -\qhyp{2}{1}{q^{-n},q^{-x}}{q^{N-x-n+1}}{-pq^{n+N+1}}.\nonumber -\end{eqnarray} - -\subsection*{Orthogonality relation} -\begin{eqnarray} -\label{OrtqKrawtchouk} -& &\sum_{x=0}^N\frac{(q^{-N};q)_x}{(q;q)_x}(-p)^{-x}K_m(q^{-x};p,N;q)K_n(q^{-x};p,N;q)\nonumber\\ -& &{}=\frac{(q,-pq^{N+1};q)_n}{(-p,q^{-N};q)_n}\frac{(1+p)}{(1+pq^{2n})}\nonumber\\ -& &{}\mathindent{}\times (-pq;q)_Np^{-N}q^{-\binom{N+1}{2}} -\left(-pq^{-N}\right)^nq^{n^2}\,\delta_{mn},\quad p>0. -\end{eqnarray} - -\subsection*{Recurrence relation} -\begin{eqnarray} -\label{RecqKrawtchouk} --\left(1-q^{-x}\right)K_n(q^{-x})&=&A_nK_{n+1}(q^{-x})-\left(A_n+C_n\right)K_n(q^{-x})\nonumber\\ -& &{}\mathindent{}+C_nK_{n-1}(q^{-x}), -\end{eqnarray} -where -$$K_n(q^{-x}):=K_n(q^{-x};p,N;q)$$ -and -$$\left\{\begin{array}{l} -\displaystyle A_n=\frac{(1-q^{n-N})(1+pq^n)}{(1+pq^{2n})(1+pq^{2n+1})}\\ -\\ -\displaystyle C_n=-pq^{2n-N-1}\frac{(1+pq^{n+N})(1-q^n)}{(1+pq^{2n-1})(1+pq^{2n})}. -\end{array}\right.$$ - -\subsection*{Normalized recurrence relation} -\begin{equation} -\label{NormRecqKrawtchouk} -xp_n(x)=p_{n+1}(x)+\left[1-(A_n+C_n)\right]p_n(x)+A_{n-1}C_np_{n-1}(x), -\end{equation} -where -$$K_n(q^{-x};p,N;q)=\frac{(-pq^n;q)_n}{(q^{-N};q)_n}p_n(q^{-x}).$$ - -\subsection*{$q$-Difference equation} -\begin{eqnarray} -\label{dvqKrawtchouk} -& &q^{-n}(1-q^n)(1+pq^n)y(x)\nonumber\\ -& &{}=(1-q^{x-N})y(x+1)-\left[(1-q^{x-N})-p(1-q^x)\right]y(x)\nonumber\\ -& &{}\mathindent{}-p(1-q^x)y(x-1), -\end{eqnarray} -where -$$y(x)=K_n(q^{-x};p,N;q).$$ - -\subsection*{Forward shift operator} -\begin{eqnarray} -\label{shift1qKrawtchoukI} -& &K_n(q^{-x-1};p,N;q)-K_n(q^{-x};p,N;q)\nonumber\\ -& &{}=\frac{q^{-n-x}(1-q^n)(1+pq^n)}{1-q^{-N}}K_{n-1}(q^{-x};pq^2,N-1;q) -\end{eqnarray} -or equivalently -\begin{equation} -\label{shift1qKrawtchoukII} -\frac{\Delta K_n(q^{-x};p,N;q)}{\Delta q^{-x}}= -\frac{q^{-n+1}(1-q^n)(1+pq^n)}{(1-q)(1-q^{-N})}K_{n-1}(q^{-x};pq^2,N-1;q). -\end{equation} - -\subsection*{Backward shift operator} -\begin{eqnarray} -\label{shift2qKrawtchoukI} -& &(1-q^{x-N-1})K_n(q^{-x};p,N;q)+pq^{-1}(1-q^x)K_n(q^{-x+1};p,N;q)\nonumber\\ -& &{}=q^x(1-q^{-N-1})K_{n+1}(q^{-x};pq^{-2},N+1;q) -\end{eqnarray} -or equivalently -\begin{eqnarray} -\label{shift2qKrawtchoukII} -& &\frac{\nabla\left[w(x;p,N;q)K_n(q^{-x};p,N;q)\right]}{\nabla q^{-x}}\nonumber\\ -& &{}=\frac{1}{1-q}w(x;pq^{-2},N+1;q)K_{n+1}(q^{-x};pq^{-2},N+1;q), -\end{eqnarray} -where -$$w(x;p,N;q)=\frac{(q^{-N};q)_x}{(q;q)_x}\left(-\frac{q}{p}\right)^x.$$ - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodqKrawtchouk} -w(x;p,N;q)K_n(q^{-x};p,N;q)=(1-q)^n\left(\nabla_q\right)^n\left[w(x;pq^{2n},N-n;q)\right], -\end{equation} -where -$$\nabla_q:=\frac{\nabla}{\nabla q^{-x}}.$$ - -\subsection*{Generating function} For $x=0,1,2,\ldots,N$ we have -\begin{eqnarray} -\label{GenqKrawtchouk} -& &\qhyp{1}{1}{q^{-x}}{0}{pqt}\,\qhyp{2}{0}{q^{x-N},0}{-}{-q^{-x}t}\nonumber\\ -& &{}=\sum_{n=0}^N\frac{(q^{-N};q)_n}{(q;q)_n}q^{-\binom{n}{2}}K_n(q^{-x};p,N;q)t^n. -\end{eqnarray} - -\subsection*{Limit relations} - -\subsubsection*{$q$-Racah $\rightarrow$ $q$-Krawtchouk} -The $q$-Krawtchouk polynomials given by (\ref{DefqKrawtchouk}) can be obtained from -the $q$-Racah polynomials by setting $\alpha q=q^{-N}$, $\beta=-pq^N$ and -$\gamma=\delta=0$ in the definition (\ref{DefqRacah}) of the $q$-Racah polynomials: -$$R_n(q^{-x};q^{-N-1},-pq^N,0,0|q)=K_n(q^{-x};p,N;q).$$ -Note that $\mu(x)=q^{-x}$ in this case. - -\subsubsection*{$q$-Hahn $\rightarrow$ $q$-Krawtchouk} -If we set $\beta=-\alpha^{-1}q^{-1}p$ in the definition (\ref{DefqHahn}) of the $q$-Hahn polynomials -and then let $\alpha\rightarrow 0$ we obtain the $q$-Krawtchouk polynomials given by -(\ref{DefqKrawtchouk}): -$$\lim_{\alpha\rightarrow 0} -Q_n(q^{-x};\alpha,-\alpha^{-1}q^{-1}p,N|q)=K_n(q^{-x};p,N;q).$$ - -\subsubsection*{$q$-Krawtchouk $\rightarrow$ $q$-Bessel} -If we set $x\rightarrow N-x$ in the definition (\ref{DefqKrawtchouk}) of the -$q$-Krawtchouk polynomials and then take the limit $N\rightarrow\infty$ we -obtain the $q$-Bessel polynomials given by (\ref{DefqBessel}): -\begin{equation} -\lim_{N\rightarrow\infty}K_n(q^{x-N};p,N;q)=y_n(q^x;p;q). -\end{equation} - -\subsubsection*{$q$-Krawtchouk $\rightarrow$ $q$-Charlier} -By setting $p=a^{-1}q^{-N}$ in the definition (\ref{DefqKrawtchouk}) of the -$q$-Krawtchouk polynomials and then taking the limit $N\rightarrow\infty$ we -obtain the $q$-Charlier polynomials given by (\ref{DefqCharlier}): -\begin{equation} -\lim_{N\rightarrow\infty}K_n(q^{-x};a^{-1}q^{-N},N;q)=C_n(q^{-x};a;q). -\end{equation} - -\subsubsection*{$q$-Krawtchouk $\rightarrow$ Krawtchouk} -If we take the limit $q\rightarrow 1$ in the definition (\ref{DefqKrawtchouk}) of the $q$-Krawtchouk -polynomials we simply find the Krawtchouk polynomials given by (\ref{DefKrawtchouk}) in the -following way: -\begin{equation} -\lim_{q\rightarrow 1}K_n(q^{-x};p,N;q)=K_n(x;(p+1)^{-1},N). -\end{equation} - -\subsection*{Remark} -The $q$-Krawtchouk polynomials given by (\ref{DefqKrawtchouk}) and the -dual $q$-Krawtchouk polynomials given by (\ref{DefDualqKrawtchouk}) are -related in the following way: -$$K_n(q^{-x};p,N;q)=K_x(\lambda(n);-pq^N,N|q)$$ -with -$$\lambda(n)=q^{-n}-pq^n$$ -or -$$K_n(\lambda(x);c,N|q)=K_x(q^{-n};-cq^{-N},N;q)$$ -with -$$\lambda(x)=q^{-x}+cq^{x-N}.$$ - -\subsection*{References} -\cite{AlvarezRonveaux}, \cite{AskeyWilson79}, \cite{AtakRahmanSuslov}, -\cite{Campigotto+}, \cite{GasperRahman90}, \cite{Nikiforov+}, -\cite{NoumiMimachi91}, \cite{Stanton80III}, \cite{Stanton84}. - - -\newpage - -\section{Affine $q$-Krawtchouk} -\index{Affine q-Krawtchouk polynomials@Affine $q$-Krawtchouk polynomials} -\index{q-Krawtchouk polynomials@$q$-Krawtchouk polynomials!Affine} -\par\setcounter{equation}{0} - -\subsection*{Basic hypergeometric representation} -\begin{eqnarray} -\label{DefAffqKrawtchouk} -K_n^{Aff}(q^{-x};p,N;q)&=&\qhyp{3}{2}{q^{-n},0,q^{-x}}{pq,q^{-N}}{q}\\ -&=&\frac{(-pq)^nq^{\binom{n}{2}}}{(pq;q)_n}\, -\qhyp{2}{1}{q^{-n},q^{x-N}}{q^{-N}}{\frac{q^{-x}}{p}},\quad n=0,1,2,\ldots,N.\nonumber -\end{eqnarray} - -\subsection*{Orthogonality relation} -\begin{eqnarray} -\label{OrtAffqKrawtchouk} -& &\sum_{x=0}^N\frac{(pq;q)_x(q;q)_N}{(q;q)_x(q;q)_{N-x}}(pq)^{-x}K_m^{Aff}(q^{-x};p,N;q)K_n^{Aff}(q^{-x};p,N;q)\nonumber\\ -& &{}=(pq)^{n-N}\frac{(q;q)_n(q;q)_{N-n}}{(pq;q)_n(q;q)_N}\,\delta_{mn},\quad 01$, then we have another orthogonality relation given by: -\begin{eqnarray} -\label{OrtContBigqHermite2} -& &\frac{1}{2\pi}\int_{-1}^1\frac{w(x)}{\sqrt{1-x^2}}H_m(x;a|q)H_n(x;a|q)\,dx\nonumber\\ -& &{}\mathindent{}+\sum_{\begin{array}{c}\scriptstyle k\\ \scriptstyle 1-1 -\end{eqnarray} -and -\begin{eqnarray} -\label{OrtqLaguerre2} -& &\sum_{k=-\infty}^{\infty}\frac{q^{k\alpha+k}}{(-cq^k;q)_{\infty}}L_m^{(\alpha)}(cq^k;q)L_n^{(\alpha)}(cq^k;q)\nonumber\\ -& &{}=\frac{(q,-cq^{\alpha+1},-c^{-1}q^{-\alpha};q)_{\infty}} -{(q^{\alpha+1},-c,-c^{-1}q;q)_{\infty}}\frac{(q^{\alpha+1};q)_n}{(q;q)_nq^n}\,\delta_{mn}, -\quad\alpha>-1,\quad c>0. -\end{eqnarray} -For $c=1$ the latter orthogonality relation can also be written as -\begin{eqnarray} -\label{OrtqLaguerre3} -& &\int_0^{\infty}\frac{x^{\alpha}}{(-x;q)_{\infty}}L_m^{(\alpha)}(x;q)L_n^{(\alpha)}(x;q)\,d_qx\nonumber\\ -& &{}=\frac{1-q}{2}\,\frac{(q,-q^{\alpha+1},-q^{-\alpha};q)_{\infty}} -{(q^{\alpha+1},-q,-q;q)_{\infty}}\frac{(q^{\alpha+1};q)_n}{(q;q)_nq^n}\,\delta_{mn},\quad\alpha>-1. -\end{eqnarray} - -\newpage - -\subsection*{Recurrence relation} -\begin{eqnarray} -\label{RecqLaguerre} --q^{2n+\alpha+1}xL_n^{(\alpha)}(x;q)&=&(1-q^{n+1})L_{n+1}^{(\alpha)}(x;q)\nonumber\\ -& &{}\mathindent{}-\left[(1-q^{n+1})+q(1-q^{n+\alpha})\right]L_n^{(\alpha)}(x;q)\nonumber\\ -& &{}\mathindent\mathindent{}+q(1-q^{n+\alpha})L_{n-1}^{(\alpha)}(x;q). -\end{eqnarray} - -\subsection*{Normalized recurrence relation} -\begin{eqnarray} -\label{NormRecqLaguerre} -xp_n(x)&=&p_{n+1}(x)+q^{-2n-\alpha-1}\left[(1-q^{n+1})+q(1-q^{n+\alpha})\right]p_n(x)\nonumber\\ -& &{}\mathindent{}+q^{-4n-2\alpha+1}(1-q^n)(1-q^{n+\alpha})p_{n-1}(x), -\end{eqnarray} -where -$$L_n^{(\alpha)}(x;q)=\frac{(-1)^nq^{n(n+\alpha)}}{(q;q)_n}p_n(x).$$ - -\subsection*{$q$-Difference equation} -\begin{equation} -\label{dvqLaguerre} --q^{\alpha}(1-q^n)xy(x)=q^{\alpha}(1+x)y(qx)-\left[1+q^{\alpha}(1+x)\right]y(x)+y(q^{-1}x), -\end{equation} -where -$$y(x)=L_n^{(\alpha)}(x;q).$$ - -\subsection*{Forward shift operator} -\begin{equation} -\label{shift1qLaguerreI} -L_n^{(\alpha)}(x;q)-L_n^{(\alpha)}(qx;q)=-q^{\alpha+1}xL_{n-1}^{(\alpha+1)}(qx;q) -\end{equation} -or equivalently -\begin{equation} -\label{shift1qLaguerreII} -\mathcal{D}_qL_n^{(\alpha)}(x;q)=-\frac{q^{\alpha+1}}{1-q}L_{n-1}^{(\alpha+1)}(qx;q). -\end{equation} - -\subsection*{Backward shift operator} -\begin{equation} -\label{shift2qLaguerreI} -L_n^{(\alpha)}(x;q)-q^{\alpha}(1+x)L_n^{(\alpha)}(qx;q)=(1-q^{n+1})L_{n+1}^{(\alpha-1)}(x;q) -\end{equation} -or equivalently -\begin{equation} -\label{shift2qLaguerreII} -\mathcal{D}_q\left[w(x;\alpha;q)L_n^{(\alpha)}(x;q)\right]= -\frac{1-q^{n+1}}{1-q}w(x;\alpha-1;q)L_{n+1}^{(\alpha-1)}(x;q), -\end{equation} -where -$$w(x;\alpha;q)=\frac{x^{\alpha}}{(-x;q)_{\infty}}.$$ - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodqLaguerre} -w(x;\alpha;q)L_n^{(\alpha)}(x;q)= -\frac{(1-q)^n}{(q;q)_n}\left(\mathcal{D}_q\right)^n\left[w(x;\alpha+n;q)\right]. -\end{equation} - -\subsection*{Generating functions} -\begin{equation} -\label{GenqLaguerre1} -\frac{1}{(t;q)_{\infty}}\,\qhyp{1}{1}{-x}{0}{q^{\alpha+1}t} -=\sum_{n=0}^{\infty}L_n^{(\alpha)}(x;q)t^n. -\end{equation} - -\begin{equation} -\label{GenqLaguerre2} -\frac{1}{(t;q)_{\infty}}\,\qhyp{0}{1}{-}{q^{\alpha+1}}{-q^{\alpha+1}xt} -=\sum_{n=0}^{\infty}\frac{L_n^{(\alpha)}(x;q)}{(q^{\alpha+1};q)_n}t^n. -\end{equation} - -\begin{equation} -\label{GenqLaguerre3} -(t;q)_{\infty}\cdot\qhyp{0}{2}{-}{q^{\alpha+1},t}{-q^{\alpha+1}xt} -=\sum_{n=0}^{\infty}\frac{(-1)^nq^{\binom{n}{2}}}{(q^{\alpha+1};q)_n}L_n^{(\alpha)}(x;q)t^n. -\end{equation} - -\begin{eqnarray} -\label{GenqLaguerre4} -& &\frac{(\gamma t;q)_{\infty}}{(t;q)_{\infty}}\,\qhyp{1}{2}{\gamma}{q^{\alpha+1},\gamma t}{-q^{\alpha+1}xt}\nonumber\\ -& &{}=\sum_{n=0}^{\infty}\frac{(\gamma;q)_n}{(q^{\alpha+1};q)_n}L_n^{(\alpha)}(x;q)t^n, -\quad\textrm{$\gamma$ arbitrary}. -\end{eqnarray} - -\subsection*{Limit relations} - -\subsubsection*{Little $q$-Jacobi $\rightarrow$ $q$-Laguerre} -If we substitute $a=q^{\alpha}$ and $x\rightarrow -b^{-1}q^{-1}x$ in the definition -(\ref{DefLittleqJacobi}) of the little $q$-Jacobi polynomials and then take the limit -$b\rightarrow -\infty$ we find the $q$-Laguerre polynomials given by (\ref{DefqLaguerre}): -$$\lim_{b\rightarrow -\infty}p_n(-b^{-1}q^{-1}x;q^{\alpha},b|q)= -\frac{(q;q)_n}{(q^{\alpha+1};q)_n}L_n^{(\alpha)}(x;q).$$ - -\subsubsection*{$q$-Meixner $\rightarrow$ $q$-Laguerre} -The $q$-Laguerre polynomials given by (\ref{DefqLaguerre}) can be obtained -from the $q$-Meixner polynomials given by (\ref{DefqMeixner}) by setting -$b=q^{\alpha}$ and $q^{-x}\rightarrow cq^{\alpha}x$ in the definition -(\ref{DefqMeixner}) of the $q$-Meixner polynomials and then taking the limit -$c\rightarrow\infty$: -$$\lim_{c\rightarrow\infty}M_n(cq^{\alpha}x;q^{\alpha},c;q)= -\frac{(q;q)_n}{(q^{\alpha+1};q)_n}L_n^{(\alpha)}(x;q).$$ - -\subsubsection*{$q$-Laguerre $\rightarrow$ Stieltjes-Wigert} -If we set $x\rightarrow xq^{-\alpha}$ in the definition -(\ref{DefqLaguerre}) of the $q$-Laguerre polynomials and take the limit -$\alpha\rightarrow\infty$ we simply obtain the Stieltjes-Wigert polynomials given -by (\ref{DefStieltjesWigert}): -\begin{equation} -\lim_{\alpha\rightarrow\infty}L_n^{(\alpha)}\left(xq^{-\alpha};q\right)=S_n(x;q). -\end{equation} - -\subsubsection*{$q$-Laguerre $\rightarrow$ Laguerre / Charlier} -If we set $x\rightarrow (1-q)x$ in the definition (\ref{DefqLaguerre}) -of the $q$-Laguerre polynomials and take the limit $q\rightarrow 1$ we obtain -the Laguerre polynomials given by (\ref{DefLaguerre}): -\begin{equation} -\lim_{q\rightarrow 1}L_n^{(\alpha)}((1-q)x;q)=L_n^{(\alpha)}(x). -\end{equation} - -If we set $x\rightarrow -q^{-x}$ and $q^{\alpha}=a^{-1}(q-1)^{-1}$ (or -$\alpha=-(\ln q)^{-1}\ln (q-1)a$) in the definition (\ref{DefqLaguerre}) of the -$q$-Laguerre polynomials, multiply by $(q;q)_n$, and take the limit -$q\rightarrow 1$ we obtain the Charlier polynomials given by (\ref{DefCharlier}): -\begin{equation} -\lim_{q\rightarrow 1}\,(q;q)_nL_n^{(\alpha)}(-q^{-x};q)=C_n(x;a), -\end{equation} -where -$$q^{\alpha}=\frac{1}{a(q-1)}\quad\textrm{or}\quad\alpha=-\frac{\ln (q-1)a}{\ln q}.$$ - -\subsection*{Remarks} -The $q$-Laguerre polynomials are sometimes called the -generalized Stieltjes-Wigert polynomials. - -If we replace $q$ by $q^{-1}$ we obtain the little $q$-Laguerre -(or Wall) polynomials given by (\ref{DefLittleqLaguerre}) in the following -way: -$$L_n^{(\alpha)}(x;q^{-1})=\frac{(q^{\alpha+1};q)_n}{(q;q)_nq^{n\alpha}}p_n(-x;q^{\alpha}|q).$$ - -\noindent -The $q$-Laguerre polynomials given by (\ref{DefqLaguerre}) and the $q$-Bessel polynomials -given by (\ref{DefqBessel}) are related in the following way: -$$\frac{y_n(q^x;a;q)}{(q;q)_n}=L_n^{(x-n)}(aq^n;q).$$ - -\noindent -The $q$-Laguerre polynomials given by (\ref{DefqLaguerre}) and the -$q$-Charlier polynomials given by (\ref{DefqCharlier}) are related in the -following way: -$$\frac{C_n(-x;-q^{-\alpha};q)}{(q;q)_n}=L_n^{(\alpha)}(x;q).$$ - -\noindent -Since the Stieltjes and Hamburger moment problems corresponding to the -$q$-Laguerre polynomials are indeterminate there exist many different weight -functions. - -\subsection*{References} -\cite{NAlSalam89}, \cite{AlSalam90}, \cite{Askey86}, \cite{Askey89I}, \cite{AskeyWilson85}, -\cite{AtakAtakI}, \cite{ChenIsmailMuttalib}, \cite{Chihara68II}, \cite{Chihara78}, -\cite{Chihara79}, \cite{Chris}, \cite{Exton77}, \cite{GasperRahman90}, \cite{GrunbaumHaine96}, -\cite{Ismail2005I}, \cite{IsmailRahman98}, \cite{Jain95}, \cite{Moak}. - - -\section{$q$-Bessel} -\index{q-Bessel polynomials@$q$-Bessel polynomials} -\par\setcounter{equation}{0} - -\subsection*{Basic hypergeometric representation} -\begin{eqnarray} -\label{DefqBessel} -y_n(x;a;q)&=&\qhyp{2}{1}{q^{-n},-aq^n}{0}{qx}\\ -&=&(q^{-n+1}x;q)_n\cdot\qhyp{1}{1}{q^{-n}}{q^{-n+1}x}{-aq^{n+1}x}\nonumber\\ -&=&\left(-aq^nx\right)^n\cdot\qhyp{2}{1}{q^{-n},x^{-1}}{0}{-\frac{q^{-n+1}}{a}}.\nonumber -\end{eqnarray} - -\newpage - -\subsection*{Orthogonality relation} -\begin{eqnarray} -\label{OrtqBessel} -& &\sum_{k=0}^{\infty}\frac{a^k}{(q;q)_k}q^{\binom{k+1}{2}}y_m(q^k;a;q)y_n(q^k;a;q)\nonumber\\ -& &{}=(q;q)_n(-aq^n;q)_{\infty}\frac{a^nq^{\binom{n+1}{2}}}{(1+aq^{2n})}\,\delta_{mn},\quad a>0. -\end{eqnarray} - -\subsection*{Recurrence relation} -\begin{equation} -\label{RecqBessel} --xy_n(x;a;q)=A_ny_{n+1}(x;a;q)-(A_n+C_n)y_n(x;a;q)+C_ny_{n-1}(x;a;q), -\end{equation} -where -$$\left\{\begin{array}{l}\displaystyle A_n=q^n\frac{(1+aq^n)}{(1+aq^{2n})(1+aq^{2n+1})}\\ -\\ -\displaystyle C_n=aq^{2n-1}\frac{(1-q^n)}{(1+aq^{2n-1})(1+aq^{2n})}.\end{array}\right.$$ - -\subsection*{Normalized recurrence relation} -\begin{equation} -\label{NormRecqBessel} -xp_n(x)=p_{n+1}(x)+(A_n+C_n)p_n(x)+A_{n-1}C_np_{n-1}(x), -\end{equation} -where -$$y_n(x;a;q)=(-1)^nq^{-\binom{n}{2}}(-aq^n;q)_np_n(x).$$ - -\subsection*{$q$-Difference equation} -\begin{eqnarray} -\label{dvqBessel} -& &-q^{-n}(1-q^n)(1+aq^n)xy(x)\nonumber\\ -& &{}=axy(qx)-(ax+1-x)y(x)+(1-x)y(q^{-1}x), -\end{eqnarray} -where -$$y(x)=y_n(x;a;q).$$ - -\newpage - -\subsection*{Forward shift operator} -\begin{equation} -\label{shift1qBesselI} -y_n(x;a;q)-y_n(qx;a;q)=-q^{-n+1}(1-q^n)(1+aq^n)xy_{n-1}(x;aq^2;q) -\end{equation} -or equivalently -\begin{equation} -\label{shift1qBesselII} -\mathcal{D}_qy_n(x;a;q)=-\frac{q^{-n+1}(1-q^n)(1+aq^n)}{1-q}y_{n-1}(x;aq^2;q). -\end{equation} - -\subsection*{Backward shift operator} -\begin{equation} -\label{shift2qBesselI} -aq^{x-1}y_n(q^x;a;q)-(1-q^x)y_n(q^{x-1}x;a;q)=-y_{n+1}(q^x;aq^{-2};q) -\end{equation} -or equivalently -\begin{equation} -\label{shift2qBesselII} -\frac{\nabla\left[w(x;a;q)y_n(q^x;a;q)\right]}{\nabla q^x}= -\frac{q^2}{a(1-q)}w(x;aq^{-2};q)y_{n+1}(q^x;aq^{-2};q), -\end{equation} -where -$$w(x;a;q)=\frac{a^xq^{\binom{x}{2}}}{(q;q)_x}.$$ - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodqBessel} -w(x;a;q)y_n(q^x;a;q)=a^n(1-q)^nq^{n(n-1)}\left(\nabla_q\right)^n\left[w(x;aq^{2n};q)\right], -\end{equation} -where -$$\nabla_q:=\frac{\nabla}{\nabla q^x}.$$ - -\subsection*{Generating functions} -\begin{eqnarray} -\label{GenqBessel1} -& &\qhyp{0}{1}{-}{0}{-aq^{x+1}t}\,\qhyp{2}{0}{q^{-x},0}{-}{q^xt}\nonumber\\ -& &{}=\sum_{n=0}^{\infty}\frac{y_n(q^x;a;q)}{(q;q)_n}t^n,\quad x=0,1,2,\ldots. -\end{eqnarray} - -\begin{equation} -\label{GenqBessel2} -\frac{(t;q)_{\infty}}{(xt;q)_{\infty}}\,\qhyp{1}{3}{xt}{0,0,t}{-aqxt}= -\sum_{n=0}^{\infty}\frac{(-1)^nq^{\binom{n}{2}}}{(q;q)_n}y_n(x;a;q)t^n. -\end{equation} - -\subsection*{Limit relations} - -\subsubsection*{Little $q$-Jacobi $\rightarrow$ $q$-Bessel} -If we set $b\rightarrow -a^{-1}q^{-1}b$ in the definition (\ref{DefLittleqJacobi}) of -the little $q$-Jacobi polynomials and then take the limit $a\rightarrow 0$ we obtain -the $q$-Bessel polynomials given by (\ref{DefqBessel}): -$$\lim_{a\rightarrow 0}p_n(x;a,-a^{-1}q^{-1}b|q)=y_n(x;b;q).$$ - -\subsubsection*{$q$-Krawtchouk $\rightarrow$ $q$-Bessel} -If we set $x\rightarrow N-x$ in the definition (\ref{DefqKrawtchouk}) of the -$q$-Krawtchouk polynomials and then take the limit $N\rightarrow\infty$ we -obtain the $q$-Bessel polynomials given by (\ref{DefqBessel}): -$$\lim_{N\rightarrow\infty}K_n(q^{x-N};p,N;q)=y_n(q^x;p;q).$$ - -\subsubsection*{$q$-Bessel $\rightarrow$ Stieltjes-Wigert} -The Stieltjes-Wigert polynomials given by (\ref{DefStieltjesWigert}) can be obtained -from the $q$-Bessel polynomials by setting $x\rightarrow a^{-1}x$ in the definition -(\ref{DefqBessel}) of the $q$-Bessel polynomials and then taking the limit -$a\rightarrow\infty$. In fact we have -\begin{equation} -\lim_{a\rightarrow\infty}y_n(a^{-1}x;a;q)=(q;q)_nS_n(x;q). -\end{equation} - -\subsubsection*{$q$-Bessel $\rightarrow$ Bessel} -If we set $x\rightarrow -\frac{1}{2}(1-q)^{-1}x$ and $a\rightarrow -q^{a+1}$ in the definition -(\ref{DefqBessel}) of the $q$-Bessel polynomials and take the limit $q\rightarrow 1$ -we find the Bessel polynomials given by (\ref{DefBessel}): -\begin{equation} -\lim_{q\rightarrow 1}y_n(-\textstyle\frac{1}{2}(1-q)^{-1}x;-q^{a+1};q)=y_n(x;a). -\end{equation} - -\subsubsection*{$q$-Bessel $\rightarrow$ Charlier} -If we set $x\rightarrow q^x$ and $a\rightarrow a(1-q)$ in the definition (\ref{DefqBessel}) -of the $q$-Bessel polynomials and take the limit $q\rightarrow 1$ we find the Charlier -polynomials given by (\ref{DefCharlier}): -\begin{equation} -\lim_{q\rightarrow 1}\frac{y_n(q^x;a(1-q);q)}{(q-1)^n}=a^nC_n(x;a). -\end{equation} - -\subsection*{Remark} -In \cite{Koekoek94} and \cite{Koekoek98} these $q$-Bessel polynomials were called -\emph{alternative $q$-Charlier polynomials}. - -\noindent -The $q$-Bessel polynomials given by (\ref{DefqBessel}) and the $q$-Laguerre -polynomials given by (\ref{DefqLaguerre}) are related in the following way: -$$\frac{y_n(q^x;a;q)}{(q;q)_n}=L_n^{(x-n)}(aq^n;q).$$ - -\subsection*{Reference} -\cite{DattaGriffin}. - - -\section{$q$-Charlier}\index{q-Charlier polynomials@$q$-Charlier polynomials} -\par\setcounter{equation}{0} - -\subsection*{Basic hypergeometric representation} -\begin{eqnarray} -\label{DefqCharlier} -C_n(q^{-x};a;q)&=&\qhyp{2}{1}{q^{-n},q^{-x}}{0}{-\frac{q^{n+1}}{a}}\\ -&=&(-a^{-1}q;q)_n\cdot\qhyp{1}{1}{q^{-n}}{-a^{-1}q}{-\frac{q^{n+1-x}}{a}}.\nonumber -\end{eqnarray} - -\subsection*{Orthogonality relation} -\begin{eqnarray} -\label{OrtqCharlier} -& &\sum_{x=0}^{\infty}\frac{a^x}{(q;q)_x}q^{\binom{x}{2}}C_m(q^{-x};a;q)C_n(q^{-x};a;q)\nonumber\\ -& &{}=q^{-n}(-a;q)_{\infty}(-a^{-1}q,q;q)_n\,\delta_{mn},\quad a>0. -\end{eqnarray} - -\newpage - -\subsection*{Recurrence relation} -\begin{eqnarray} -\label{RecqCharlier} -& &q^{2n+1}(1-q^{-x})C_n(q^{-x})\nonumber\\ -& &{}=aC_{n+1}(q^{-x})-\left[a+q(1-q^n)(a+q^n)\right]C_n(q^{-x})\nonumber\\ -& &{}\mathindent{}+q(1-q^n)(a+q^n)C_{n-1}(q^{-x}), -\end{eqnarray} -where -$$C_n(q^{-x}):=C_n(q^{-x};a;q).$$ - -\subsection*{Normalized recurrence relation} -\begin{eqnarray} -\label{NormRecqCharlier} -xp_n(x)&=&p_{n+1}(x)+\left[1+q^{-2n-1}\left\{a+q(1-q^n)(a+q^n)\right\}\right]p_n(x)\nonumber\\ -& &{}\mathindent{}+aq^{-4n+1}(1-q^n)(a+q^n)p_{n-1}(x), -\end{eqnarray} -where -$$C_n(q^{-x};a;q)=\frac{(-1)^nq^{n^2}}{a^n}p_n(q^{-x}).$$ - -\subsection*{$q$-Difference equation} -\begin{equation} -\label{dvqCharlier} -q^ny(x)=aq^xy(x+1)-q^x(a-1)y(x)+(1-q^x)y(x-1), -\end{equation} -where -$$y(x)=C_n(q^{-x};a;q).$$ - -\subsection*{Forward shift operator} -\begin{equation} -\label{shift1qCharlierI} -C_n(q^{-x-1};a;q)-C_n(q^{-x};a;q)=-a^{-1}q^{-x}(1-q^n)C_{n-1}(q^{-x};aq^{-1};q) -\end{equation} -or equivalently -\begin{equation} -\label{shift1qCharlierII} -\frac{\Delta C_n(q^{-x};a;q)}{\Delta q^{-x}}=-\frac{q(1-q^n)}{a(1-q)}C_{n-1}(q^{-x};aq^{-1};q). -\end{equation} - -\newpage - -\subsection*{Backward shift operator} -\begin{equation} -\label{shift2qCharlierI} -C_n(q^{-x};a;q)-a^{-1}q^{-x}(1-q^x)C_n(q^{-x+1};a;q)=C_{n+1}(q^{-x};aq;q) -\end{equation} -or equivalently -\begin{equation} -\label{shift2qCharlierII} -\frac{\nabla\left[w(x;a;q)C_n(q^{-x};a;q)\right]}{\nabla q^{-x}} -=\frac{1}{1-q}w(x;aq;q)C_{n+1}(q^{-x};aq;q), -\end{equation} -where -$$w(x;a;q)=\frac{a^xq^{\binom{x+1}{2}}}{(q;q)_x}.$$ - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodqCharlier} -w(x;a;q)C_n(q^{-x};a;q)=(1-q)^n\left(\nabla_q\right)^n\left[w(x;aq^{-n};q)\right], -\end{equation} -where -$$\nabla_q:=\frac{\nabla}{\nabla q^{-x}}.$$ - -\subsection*{Generating functions} -\begin{equation} -\label{GenqCharlier1} -\frac{1}{(t;q)_{\infty}}\,\qhyp{1}{1}{q^{-x}}{0}{-a^{-1}qt} -=\sum_{n=0}^{\infty}\frac{C_n(q^{-x};a;q)}{(q;q)_n}t^n. -\end{equation} - -\begin{equation} -\label{GenqCharlier2} -\frac{1}{(t;q)_{\infty}}\,\qhyp{0}{1}{-}{-a^{-1}q}{-a^{-1}q^{-x+1}t} -=\sum_{n=0}^{\infty}\frac{C_n(q^{-x};a;q)}{(-a^{-1}q,q;q)_n}t^n. -\end{equation} - -\subsection*{Limit relations} - -\subsubsection*{$q$-Meixner $\rightarrow$ $q$-Charlier} -The $q$-Charlier polynomials given by (\ref{DefqCharlier}) can easily be -obtained from the $q$-Meixner given by (\ref{DefqMeixner}) by setting $b=0$ -in the definition (\ref{DefqMeixner}) of the $q$-Meixner polynomials: -\begin{equation} -M_n(x;0,c;q)=C_n(x;c;q). -\end{equation} - -\subsubsection*{$q$-Krawtchouk $\rightarrow$ $q$-Charlier} -By setting $p=a^{-1}q^{-N}$ in the definition (\ref{DefqKrawtchouk}) of the -$q$-Krawtchouk polynomials and then taking the limit $N\rightarrow\infty$ we -obtain the $q$-Charlier polynomials given by (\ref{DefqCharlier}): -$$\lim_{N\rightarrow\infty}K_n(q^{-x};a^{-1}q^{-N},N;q)=C_n(q^{-x};a;q).$$ - -\subsubsection*{$q$-Charlier $\rightarrow$ Stieltjes-Wigert} -If we set $q^{-x}\rightarrow ax$ in the definition (\ref{DefqCharlier}) of the -$q$-Charlier polynomials and take the limit $a\rightarrow\infty$ we obtain -the Stieltjes-Wigert polynomials given by (\ref{DefStieltjesWigert}) in the -following way: -\begin{equation} -\lim_{a\rightarrow\infty}C_n(ax;a;q)=(q;q)_nS_n(x;q). -\end{equation} - -\subsubsection*{$q$-Charlier $\rightarrow$ Charlier} -If we set $a\rightarrow a(1-q)$ in the definition (\ref{DefqCharlier}) -of the $q$-Charlier polynomials and take the limit $q\rightarrow 1$ we obtain the -Charlier polynomials given by (\ref{DefCharlier}): -\begin{equation} -\lim_{q\rightarrow 1}C_n(q^{-x};a(1-q);q)=C_n(x;a). -\end{equation} - -\subsection*{Remark} -The $q$-Charlier polynomials given by (\ref{DefqCharlier}) and the -$q$-Laguerre polynomials given by (\ref{DefqLaguerre}) are related in the -following way: -$$\frac{C_n(-x;-q^{-\alpha};q)}{(q;q)_n}=L_n^{(\alpha)}(x;q).$$ - -\subsection*{References} -\cite{AlvarezRonveaux}, \cite{AtakRahmanSuslov}, \cite{GasperRahman90}, -\cite{Hahn}, \cite{Koelink96III}, \cite{Nikiforov+}, \cite{Zeng95}. - - -\section{Al-Salam-Carlitz~I}\index{Al-Salam-Carlitz~I polynomials} -\par\setcounter{equation}{0} - -\subsection*{Basic hypergeometric representation} -\begin{equation} -\label{DefAlSalamCarlitzI} -U_n^{(a)}(x;q)=(-a)^nq^{\binom{n}{2}}\,\qhyp{2}{1}{q^{-n},x^{-1}}{0}{\frac{qx}{a}}. -\end{equation} - -\subsection*{Orthogonality relation} -\begin{eqnarray} -\label{OrtAlSalamCarlitzI} -& &\int_a^1(qx,a^{-1}qx;q)_{\infty}U_m^{(a)}(x;q)U_n^{(a)}(x;q)\,d_qx\nonumber\\ -& &{}=(-a)^n(1-q)(q;q)_n(q,a,a^{-1}q;q)_{\infty}q^{\binom{n}{2}}\,\delta_{mn},\quad a<0. -\end{eqnarray} - -\subsection*{Recurrence relation} -\begin{equation} -\label{RecAlSalamCarlitzI} -xU_n^{(a)}(x;q)=U_{n+1}^{(a)}(x;q)+(a+1)q^nU_n^{(a)}(x;q) --aq^{n-1}(1-q^n)U_{n-1}^{(a)}(x;q). -\end{equation} - -\subsection*{Normalized recurrence relation} -\begin{equation} -\label{NormRecAlSalamCarlitzI} -xp_n(x)=p_{n+1}(x)+(a+1)q^np_n(x)-aq^{n-1}(1-q^n)p_{n-1}(x), -\end{equation} -where -$$U_n^{(a)}(x;q)=p_n(x).$$ - -\subsection*{$q$-Difference equation} -\begin{eqnarray} -\label{dvAlSalamCarlitzI} -(1-q^n)x^2y(x)&=&aq^{n-1}y(qx)-\left[aq^{n-1}+q^n(1-x)(a-x)\right]y(x)\nonumber\\ -& &{}\mathindent{}+q^n(1-x)(a-x)y(q^{-1}x), -\end{eqnarray} -where -$$y(x)=U_n^{(a)}(x;q).$$ - -\subsection*{Forward shift operator} -\begin{equation} -\label{shift1AlSalamCarlitzI-I} -U_n^{(a)}(x;q)-U_n^{(a)}(qx;q)=(1-q^n)xU_{n-1}^{(a)}(x;q) -\end{equation} -or equivalently -\begin{equation} -\label{shift1AlSalamCarlitzI-II} -\mathcal{D}_qU_n^{(a)}(x;q)=\frac{1-q^n}{1-q}U_{n-1}^{(a)}(x;q). -\end{equation} - -\subsection*{Backward shift operator} -\begin{equation} -\label{shift2AlSalamCarlitzI-I} -aU_n^{(a)}(x;q)-(1-x)(a-x)U_n^{(a)}(q^{-1}x;q)=-q^{-n}xU_{n+1}^{(a)}(x;q) -\end{equation} -or equivalently -\begin{equation} -\label{shift2AlSalamCarlitzI-II} -\mathcal{D}_{q^{-1}}\left[w(x;a;q)U_n^{(a)}(x;q)\right]= -\frac{q^{-n+1}}{a(1-q)}w(x;a;q)U_{n+1}^{(a)}(x;q), -\end{equation} -where -$$w(x;a;q)=(qx,a^{-1}qx;q)_{\infty}.$$ - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodAlSalamCarlitzI} -w(x;a;q)U_n^{(a)}(x;q)=a^nq^{\frac{1}{2}n(n-3)}(1-q)^n -\left(\mathcal{D}_{q^{-1}}\right)^n\left[w(x;a;q)\right]. -\end{equation} - -\subsection*{Generating function} -\begin{equation} -\label{GenAlSalamCarlitzI} -\frac{(t,at;q)_{\infty}}{(xt;q)_{\infty}}= -\sum_{n=0}^{\infty}\frac{U_n^{(a)}(x;q)}{(q;q)_n}t^n. -\end{equation} - -\subsection*{Limit relations} - -\subsubsection*{Big $q$-Laguerre $\rightarrow$ Al-Salam-Carlitz~I} -If we set $x\rightarrow aqx$ and $b\rightarrow ab$ in the definition -(\ref{DefBigqLaguerre}) of the big $q$-Laguerre polynomials and take the -limit $a\rightarrow 0$ we obtain the Al-Salam-Carlitz~I polynomials given by -(\ref{DefAlSalamCarlitzI}): -$$\lim_{a\rightarrow 0}\frac{P_n(aqx;a,ab;q)}{a^n}=q^nU_n^{(b)}(x;q).$$ - -\subsubsection*{Dual $q$-Krawtchouk $\rightarrow$ Al-Salam-Carlitz~I} -If we set $c=a^{-1}$ in the definition (\ref{DefDualqKrawtchouk}) -of the dual $q$-Krawtchouk polynomials and take the limit -$N\rightarrow\infty$ we simply obtain the Al-Salam-Carlitz~I polynomials -given by (\ref{DefAlSalamCarlitzI}): -$$\lim_{N\rightarrow\infty}K_n(\lambda(x);a^{-1},N|q)= -\left(-\frac{1}{a}\right)^nq^{-\binom{n}{2}}U_n^{(a)}(q^x;q).$$ -Note that $\lambda(x)=q^{-x}+a^{-1}q^{x-N}$. - -\subsubsection*{Al-Salam-Carlitz~I $\rightarrow$ Discrete $q$-Hermite~I} -The discrete $q$-Hermite~I polynomials given by -(\ref{DefDiscreteqHermiteI}) can easily be obtained from the -Al-Salam-Carlitz~I polynomials given by (\ref{DefAlSalamCarlitzI}) by the -substitution $a=-1$: -\begin{equation} -U_n^{(-1)}(x;q)=h_n(x;q). -\end{equation} - -\subsubsection*{Al-Salam-Carlitz~I $\rightarrow$ Charlier / Hermite} -If we set $a\rightarrow a(q-1)$ and $x\rightarrow q^x$ in the definition -(\ref{DefAlSalamCarlitzI}) of the Al-Salam-Carlitz~I polynomials and take the -limit $q\rightarrow 1$ after dividing by $a^n(1-q)^n$ we obtain the Charlier -polynomials given by (\ref{DefCharlier}): -\begin{equation} -\lim_{q\rightarrow 1}\frac{U_n^{(a(q-1))}(q^x;q)}{(1-q)^n}=a^nC_n(x;a). -\end{equation} - -If we set $x\rightarrow x\sqrt{1-q^2}$ and $a\rightarrow a\sqrt{1-q^2}-1$ -in the definition (\ref{DefAlSalamCarlitzI}) of the Al-Salam-Carlitz~I -polynomials, divide by $(1-q^2)^{\frac{n}{2}}$, and let $q$ tend to $1$ we -obtain the Hermite polynomials given by (\ref{DefHermite}) with shifted -argument. In fact we have -\begin{equation} -\lim_{q\rightarrow 1}\frac{U_n^{(a\sqrt{1-q^2}-1)}(x\sqrt{1-q^2};q)} -{(1-q^2)^{\frac{n}{2}}}=\frac{H_n(x-a)}{2^n}. -\end{equation} - -\subsection*{Remark} -The Al-Salam-Carlitz~I polynomials are related to the Al-Salam-Carlitz~II -polynomials given by (\ref{DefAlSalamCarlitzII}) in the following way: -$$U_n^{(a)}(x;q^{-1})=V_n^{(a)}(x;q).$$ - -\subsection*{References} -\cite{AlSalam90}, \cite{AlSalamCarlitz}, \cite{AlSalamChihara76}, \cite{AskeySuslovII}, -\cite{AtakRahmanSuslov}, \cite{Chihara68II}, \cite{Chihara78}, \cite{DattaGriffin}, -\cite{Dehesa}, \cite{DohaAhmed2005}, \cite{GasperRahman90}, \cite{Ismail85I}, -\cite{IsmailMuldoon}, \cite{Kim}, \cite{Zeng95}. - - -\section{Al-Salam-Carlitz~II}\index{Al-Salam-Carlitz~II polynomials} -\par\setcounter{equation}{0} - -\subsection*{Basic hypergeometric representation} -\begin{equation} -\label{DefAlSalamCarlitzII} -V_n^{(a)}(x;q)= -(-a)^nq^{-\binom{n}{2}}\,\qhyp{2}{0}{q^{-n},x}{-}{\frac{q^n}{a}}. -\end{equation} - -\subsection*{Orthogonality relation} -\begin{eqnarray} -\label{OrtAlSalamCarlitzII} -& &\sum_{k=0}^{\infty}\frac{q^{k^2}a^k}{(q;q)_k(aq;q)_k} -V_m^{(a)}(q^{-k};q)V_n^{(a)}(q^{-k};q)\nonumber\\ -& &{}=\frac{(q;q)_na^n}{(aq;q)_{\infty}q^{n^2}}\,\delta_{mn},\quad 00,\quad\textrm{with}\quad -\gamma^2=-\frac{1}{2\ln q}.$$ - -\subsection*{References} -\cite{Askey86}, \cite{Askey89I}, \cite{AtakAtakIII}, \cite{Chihara70}, \cite{Chihara78}, -\cite{Dehesa}, \cite{Ismail2005I}, \cite{Nikiforov+}, \cite{Stieltjes}, \cite{Szego75}, -\cite{ValentAssche}, \cite{Wigert}. - - -\section{Discrete $q$-Hermite~I} -\index{Discrete q-Hermite~I polynomials@Discrete $q$-Hermite~I polynomials} -\index{q-Hermite~I polynomials@$q$-Hermite~I polynomials!Discrete} -\par\setcounter{equation}{0} - -\subsection*{Basic hypergeometric representation} -The discrete $q$-Hermite~I polynomials are Al-Salam-Carlitz~I polynomials -with $a=-1$: -\begin{eqnarray} -\label{DefDiscreteqHermiteI} -h_n(x;q)=U_n^{(-1)}(x;q)&=&q^{\binom{n}{2}}\,\qhyp{2}{1}{q^{-n},x^{-1}}{0}{-qx}\\ -&=&x^n\qhypK{2}{0}{q^{-n},q^{-n+1}}{0}{q^2,\frac{q^{2n-1}}{x^2}}\nonumber -\end{eqnarray} - -\subsection*{Orthogonality relation} -\begin{eqnarray} -\label{OrtDiscreteqHermiteI} -& &\int_{-1}^1(qx,-qx;q)_{\infty}h_m(x;q)h_n(x;q)\,d_qx\nonumber\\ -& &{}=(1-q)(q;q)_n(q,-1,-q;q)_{\infty}q^{\binom{n}{2}}\,\delta_{mn}. -\end{eqnarray} - -\subsection*{Recurrence relation} -\begin{equation} -\label{RecDiscreteqHermiteI} -xh_n(x;q)=h_{n+1}(x;q)+q^{n-1}(1-q^n)h_{n-1}(x;q). -\end{equation} - -\subsection*{Normalized recurrence relation} -\begin{equation} -\label{NormRecDiscreteqHermiteI} -xp_n(x)=p_{n+1}(x)+q^{n-1}(1-q^n)p_{n-1}(x), -\end{equation} -where -$$h_n(x;q)=p_n(x).$$ - -\subsection*{$q$-Difference equation} -\begin{equation} -\label{dvDiscreteqHermiteI} --q^{-n+1}x^2y(x)=y(qx)-(1+q)y(x)+q(1-x^2)y(q^{-1}x), -\end{equation} -where -$$y(x)=h_n(x;q).$$ - -\subsection*{Forward shift operator} -\begin{equation} -\label{shift1DiscreteqHermiteI-I} -h_n(x;q)-h_n(qx;q)=(1-q^n)xh_{n-1}(x;q) -\end{equation} -or equivalently -\begin{equation} -\label{shift1DiscreteqHermiteI-II} -\mathcal{D}_qh_n(x;q)=\frac{1-q^n}{1-q}h_{n-1}(x;q). -\end{equation} - -\subsection*{Backward shift operator} -\begin{equation} -\label{shift2DiscreteqHermiteI-I} -h_n(x;q)-(1-x^2)h_n(q^{-1}x;q)=q^{-n}xh_{n+1}(x;q) -\end{equation} -or equivalently -\begin{equation} -\label{shift2DiscreteqHermiteI-II} -\mathcal{D}_{q^{-1}}\left[w(x;q)h_n(x;q)\right] -=-\frac{q^{-n+1}}{1-q}w(x;q)h_{n+1}(x;q), -\end{equation} -where -$$w(x;q)=(qx,-qx;q)_{\infty}.$$ - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodDiscreteqHermiteI} -w(x;q)h_n(x;q)=(q-1)^nq^{\frac{1}{2}n(n-3)} -\left(\mathcal{D}_{q^{-1}}\right)^n\left[w(x;q)\right]. -\end{equation} - -\subsection*{Generating function} -\begin{equation} -\label{GenDiscreteqHermiteI} -\frac{(t^2;q^2)_{\infty}}{(xt;q)_{\infty}}= -\sum_{n=0}^{\infty}\frac{h_n(x;q)}{(q;q)_n}t^n. -\end{equation} - -\subsection*{Limit relations} - -\subsubsection*{Al-Salam-Carlitz~I $\rightarrow$ Discrete $q$-Hermite~I} -The discrete $q$-Hermite~I polynomials given by -(\ref{DefDiscreteqHermiteI}) can easily be obtained from the -Al-Salam-Carlitz~I polynomials given by (\ref{DefAlSalamCarlitzI}) by the -substitution $a=-1$: -$$U_n^{(-1)}(x;q)=h_n(x;q).$$ - -\subsubsection*{Discrete $q$-Hermite~I $\rightarrow$ Hermite} -The Hermite polynomials given by (\ref{DefHermite}) can be found -from the discrete $q$-Hermite~I polynomials given by -(\ref{DefDiscreteqHermiteI}) in the following way: -\begin{equation} -\lim_{q\rightarrow 1}\frac{h_n(x\sqrt{1-q^2};q)} -{(1-q^2)^{\frac{n}{2}}}=\frac{H_n(x)}{2^n}. -\end{equation} - -\subsection*{Remark} The discrete $q$-Hermite~I polynomials are related to the -discrete $q$-Hermite~II polynomials given by (\ref{DefDiscreteqHermiteII}) -in the following way: -$$h_n(ix;q^{-1})=i^n{\tilde h}_n(x;q).$$ - -\subsection*{References} -\cite{AlSalam90}, \cite{AlSalamCarlitz}, \cite{AtakRahmanSuslov}, -\cite{BergIsmail}, \cite{BustozIsmail82}, \cite{GasperRahman90}, \cite{Hahn}, -\cite{Koorn97}. - - -\newpage - -\section{Discrete $q$-Hermite~II} -\index{Discrete q-Hermite~II polynomials@Discrete $q$-Hermite~II polynomials} -\index{q-Hermite~II polynomials@$q$-Hermite~II polynomials!Discrete} -\par\setcounter{equation}{0} - -\subsection*{Basic hypergeometric representation} -The discrete $q$-Hermite~II polynomials are Al-Salam-Carlitz~II polynomials -with $a=-1$: -\begin{eqnarray} -\label{DefDiscreteqHermiteII} -{\tilde h}_n(x;q)=i^{-n}V_n^{(-1)}(ix;q) -&=&i^{-n}q^{-\binom{n}{2}}\,\qhyp{2}{0}{q^{-n},ix}{-}{-q^n}\\ -&=&x^n -\qhypK{2}{1}{q^{-n},q^{-n+1}}{0}{q^2,-\frac{q^2}{x^2}}\nonumber -\end{eqnarray} - -\subsection*{Orthogonality relation} -\begin{eqnarray} -\label{OrtDiscreteqHermiteIIa} -& &\sum_{k=-\infty}^{\infty}\left[{\tilde h}_m(cq^k;q) -{\tilde h}_n(cq^k;q)+{\tilde h}_m(-cq^k;q){\tilde h}_n(-cq^k;q)\right] -w(cq^k;q)q^k\nonumber\\ -& &{}=2\frac{(q^2,-c^2q,-c^{-2}q;q^2)_{\infty}}{(q,-c^2,-c^{-2}q^2;q^2)_{\infty}} -\frac{(q;q)_n}{q^{n^2}}\,\delta_{mn},\quad c>0, -\end{eqnarray} -where -$$w(x;q)=\frac{1}{(ix,-ix;q)_{\infty}}=\frac{1}{(-x^2;q^2)_{\infty}}.$$ -For $c=1$ this orthogonality relation can also be written as -\begin{equation} -\label{OrtDiscreteqHermiteIIb} -\int_{-\infty}^{\infty}\frac{{\tilde h}_m(x;q){\tilde h}_n(x;q)}{(-x^2;q^2)_{\infty}}\,d_qx -=\frac{\left(q^2,-q,-q;q^2\right)_{\infty}}{\left(q^3,-q^2,-q^2;q^2\right)_{\infty}} -\frac{(q;q)_n}{q^{n^2}}\,\delta_{mn}. -\end{equation} - -\subsection*{Recurrence relation} -\begin{equation} -\label{RecDiscreteqHermiteII} -x{\tilde h}_n(x;q)={\tilde h}_{n+1}(x;q)+q^{-2n+1}(1-q^n){\tilde h}_{n-1}(x;q). -\end{equation} - -\subsection*{Normalized recurrence relation} -\begin{equation} -\label{NormRecDiscreteqHermiteII} -xp_n(x)=p_{n+1}(x)+q^{-2n+1}(1-q^n)p_{n-1}(x), -\end{equation} -where -$${\tilde h}_n(x;q)=p_n(x).$$ - -\subsection*{$q$-Difference equation} -\begin{eqnarray} -\label{dvDiscreteqHermiteII} -& &-(1-q^n)x^2{\tilde h}_n(x;q)\nonumber\\ -& &{}=(1+x^2){\tilde h}_n(qx;q)-(1+x^2+q){\tilde h}_n(x;q)+q{\tilde h}_n(q^{-1}x;q). -\end{eqnarray} - -\subsection*{Forward shift operator} -\begin{equation} -\label{shift1DiscreteqHermiteII-I} -\tilde{h}_n(x;q)-\tilde{h}_n(qx;q)=q^{-n+1}(1-q^n)x\tilde{h}_{n-1}(qx;q) -\end{equation} -or equivalently -\begin{equation} -\label{shift1DiscreteqHermiteII-II} -\mathcal{D}_q\tilde{h}_n(x;q)=\frac{q^{-n+1}(1-q^n)}{1-q}\tilde{h}_{n-1}(qx;q). -\end{equation} - -\subsection*{Backward shift operator} -\begin{equation} -\label{shift2DiscreteqHermiteII-I} -\tilde{h}_n(x;q)-(1+x^2)\tilde{h}_n(qx;q)=-q^nx\tilde{h}_{n+1}(x;q) -\end{equation} -or equivalently -\begin{equation} -\label{shift2DiscreteqHermiteII-II} -\mathcal{D}_q\left[w(x;q)\tilde{h}_n(x;q)\right] -=-\frac{q^n}{1-q}w(x;q)\tilde{h}_{n+1}(x;q). -\end{equation} - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodDiscreteqHermiteII} -w(x;q)\tilde{h}_n(x;q)=(q-1)^nq^{-\binom{n}{2}} -\left(\mathcal{D}_q\right)^n\left[w(x;q)\right]. -\end{equation} - -\subsection*{Generating functions} -\begin{equation} -\label{GenDiscreteqHermiteII1} -\frac{(-xt;q)_{\infty}}{(-t^2;q^2)_{\infty}}=\sum_{n=0}^{\infty} -\frac{q^{\binom{n}{2}}}{(q;q)_n}{\tilde h}_n(x;q)t^n. -\end{equation} - -\begin{equation} -\label{GenDiscreteqHermiteII2} -(-it;q)_{\infty}\cdot\qhyp{1}{1}{ix}{-it}{it}=\sum_{n=0}^{\infty} -\frac{(-1)^nq^{n(n-1)}}{(q;q)_n}{\tilde h}_n(x;q)t^n. -\end{equation} - -\subsection*{Limit relations} - -\subsubsection*{Al-Salam-Carlitz~II $\rightarrow$ Discrete $q$-Hermite~II} -The discrete $q$-Hermite~II polynomials given by -(\ref{DefDiscreteqHermiteII}) follow from the Al-Salam-Carlitz~II -polynomials given by (\ref{DefAlSalamCarlitzII}) by the substitution $a=-1$ -in the following way: -$$i^{-n}V_n^{(-1)}(ix;q)={\tilde h}_n(x;q).$$ - -\subsubsection*{Discrete $q$-Hermite~II $\rightarrow$ Hermite} -The Hermite polynomials given by (\ref{DefHermite}) can be found -from the discrete $q$-Hermite~II polynomials given by -(\ref{DefDiscreteqHermiteII}) in the following way: -\begin{equation} -\lim_{q\rightarrow 1}\frac{{\tilde h}_n(x\sqrt{1-q^2};q)} -{(1-q^2)^{\frac{n}{2}}}=\frac{H_n(x)}{2^n}. -\end{equation} - -\subsection*{Remark} The discrete $q$-Hermite~II polynomials are related to the -discrete $q$-Hermite~I polynomials given by (\ref{DefDiscreteqHermiteI}) -in the following way: -$${\tilde h}_n(x;q^{-1})=i^{-n}h_n(ix;q).$$ - -\subsection*{References} -\cite{BergIsmail}, \cite{Koorn97}. - -\end{document} From 51ed451f7dba62ed3a831ceb08f2fff3aacf5ee2 Mon Sep 17 00:00:00 2001 From: Rahul Shah Date: Wed, 17 Feb 2016 16:17:01 -0500 Subject: [PATCH 005/402] Updating README test --- README.md | 4 +++- 1 file changed, 3 insertions(+), 1 deletion(-) diff --git a/README.md b/README.md index 75cc520..1da778f 100644 --- a/README.md +++ b/README.md @@ -29,5 +29,7 @@ To achieve the desired result, one should execute the progams in the following o ## KLSadd Insertion Project -Edits the DRMF chapter files to include rel +Edits the DRMF chapter files to include the relevant KLS addendum additions. Additions are (currently)only being added right before the "References" paragraphs in each section. The linetest.py file is the first working model of the code, but it is very messy and esoteric. Comments have been made to aid interpretation. The new project file, updateChapters.py is a more readable version, but not quite up to date. +Left to do in updateChapters.py: +import over the necessary files to make the pdf work, change all instances of "section" to "paragraph" and add "\large\bf" to make font large and bold like the other chapter headings. Also necessary to add initials of programmer wherever the new additions were added (ex. "%RS added, %RS end") From f6b7b72f83c56cb2c911a6960f534408688675f6 Mon Sep 17 00:00:00 2001 From: Rahul Shah Date: Wed, 17 Feb 2016 16:32:30 -0500 Subject: [PATCH 006/402] updated README --- README.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/README.md b/README.md index 1da778f..b9ec267 100644 --- a/README.md +++ b/README.md @@ -28,7 +28,7 @@ To achieve the desired result, one should execute the progams in the following o ## BMP Seeding Project ## KLSadd Insertion Project - +linetest.py and updateChapters (mostly) written by Rahul Shah (ILIKEFUUD) Edits the DRMF chapter files to include the relevant KLS addendum additions. Additions are (currently)only being added right before the "References" paragraphs in each section. The linetest.py file is the first working model of the code, but it is very messy and esoteric. Comments have been made to aid interpretation. The new project file, updateChapters.py is a more readable version, but not quite up to date. Left to do in updateChapters.py: From 2b579f2462997108415bca53f57abf8948043095 Mon Sep 17 00:00:00 2001 From: Rahul Shah Date: Wed, 17 Feb 2016 16:48:44 -0500 Subject: [PATCH 007/402] Removed chapter files --- KLSadd_insertion/newtempchap09.pdf | Bin 319561 -> 0 bytes KLSadd_insertion/updated14.tex | 4162 ---------------------------- KLSadd_insertion/updated9.tex | 4162 ---------------------------- README.md | 5 +- 4 files changed, 4 insertions(+), 8325 deletions(-) delete mode 100644 KLSadd_insertion/newtempchap09.pdf delete mode 100644 KLSadd_insertion/updated14.tex delete mode 100644 KLSadd_insertion/updated9.tex diff --git a/KLSadd_insertion/newtempchap09.pdf b/KLSadd_insertion/newtempchap09.pdf deleted file mode 100644 index bec3d85a50c47d3eae72c71fd38c175ac81a2ab1..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 319561 zcma&NW2|UFvn{%8+qSK}Y}>YN+qP}nwry)K+s57Jy_1)hd?&f5Gk?rqo%F1pHEN7e zMJg{WO3O&c3Pn1EUmX^xu|H}<8%VvcZQ+E88qt*gd~X}%{C>pD7-yGQ&hJ@jZNB8QXjuyFp57gbKmV%2%v2=4qg9}Pi!qsI` zjtQE1uOFJX4+qB7_{#3hRGGoXSnU2H1Z7Fq-qXpJA?S&*UjyzBo*c5SYR+f1`AX00 z+)v&39r*rQ3DTKwO~B|73?JV(aJ+qEdfh7&2SymO&|lb?2oAB`7=;TtcF`6~9aIlK z@}`t4Nr-BZ`TJcC4wN9<2fxX~N9)Ct+LZVbw`nV^y7c>QmoUQcewQJ(HCq9CTraVN zlFiNDZN_2qeV6wExaoNlqwul?pB_~ljM$uCvF$F(syfvgjpNkHw3wDdxiMNBx{sA- z_2HANY4M>bb+X(LUPomSaj>FYTqyN@0GpPKA3~Yf8vl31{=NNIv^hBcGwdAy9drVM z|9jYh0sbw(|J;|8`9Jq%WMN|a-(IH?H3_>dQIy_GwL9^G)!?T1YzqYtTKOW{c_D0I z&UViEWpN@RN0h|KL%+RO(*p4TDjYEpgZ}K}nb#MQJTw-`{Wwu+M8cwMD8Y>ANzeN? z!iZJ-XRl<0T4U$NU>6c5QKW<+Py8OOua38OgZmTRMhIQU_9#3y;8c=GQTWRE$B{)6Hk&?XttAw%Hoo+4C!>qXbvKB|6TEgM zN##sUTYt$4B8v-;@74aM?9uPqi|xv7)G$2t0kR0WP2Ad+k+7AD3pZPptM3J{5vmJI zz4KWfW^A`6t8(DCr#qwHxQ_I+?${VTM&XCiC!SQ}VKqQh+;!y*_`;c-mlCh5>hpg_b&F=q7Wt zU{&uM)wA~9s@+rdDBwWI>Fw9hd-UMAR!c8;Dl64iv8bsy3?#T%Cjr}0VB-}*vD(1A z=e2J4d;b(hw^Ln3L_)XOJQu9w*!nw!oNy(L^E-|dxwmk#k@j{uZ8hbp{NmibqP^de z$Z)-h-PUp?GlNE{UZM^gUcc1P#o1J4A>Dc~_3wZ>{?;y)SgcYn)iOrk8|&Dl(!{W1 z>As?B;>j(Tg){fk@`V04XOHam6Q>SWMeKJVe;M=a)(!}8$93tnobMn#w*UfNw#3ZQ znxDLQ1$E1H)@0IV4(Fi7h8>q&n^v;RGb$DZw5>Ph>r=B8mx|jR)iUFrx4=iaUNTAuN zgNo;-xt2rDwk!M7AQO)XE2*>=J{&U41+=%X%;66qavxVic5?#1WEu4G75(d6zwz!M z?o8rZxxXRNm4sD9WB;h_Kg$wBH-|3Q$!BrRp2rB4znx*kj;@JLispEJzTwj#$Cm4`*dW zjB=3GZ(5Z{z&({vPDhtqm#CUFi3#M2DN#@&yX~QI`Ch*2gemwGNkm1;B-ba4h=+2p zSOVJ)FW$0A z4c>6e#V;5k{Vvk~$;Mm6mlk~Tb;-2@GfgFP_11LAP+FW@LnLntGV}L!oscTL+|<1) zd{X_S3}H0P<8;&9C<~^Us7L-I+k;Pj&^*Oturv2AI

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zw`Gpt+$a3P3C|=og_bhoNtsmc^@E_0liN0Po@G0A@SZWs&q}q^4>#RhaXdSd*F8fl zHItrG<3vgE=94K0+?T1yUvnSZUke;SH?f~J@wt;%$YU(rn>FZ(6&~^JRjP5JJ8AA? zW01#;jxlt<=I^UR8~HQf0C06@PtnKS)a!&M>EQRSQ{RN|=T9gfjN8`zuL%T=bYKnP zeV5@9>}(C`4!bUq3siiFXZGJ)m@aYX9JF~kldNt^JuRK`0IW7b z3Kl4$np-F=6U+t!_LTOY$!jp_&XfSUH`V8&Te-3{Ri+|c*@hHbx_Q;|B6T_MlpQ)L z$=6k(fpyHn-%v#Kyy`Bcd&}z>)dJlbPq@>{1n;Bp<@Rn#2cdeX)5} zl0T#IxHCc8iM@jw3t2`(MEyk$#69Q+#1WtNw-k77yaO7kXP2t|_#1Hwe@%V}HsTcj z>U?Wk(oGo%P#;{_;ke&-99&4^xSzNB)xZ(w*ny!pI`hcw*uUViga~BHitu_jj9_rW zHVr=HTMNo^LUa&gTwDvkdoV4T^(Bq;R7ZY9shZ8zq z-_!Kb6C#>djew7BSb;Fg86KWxiu@D$)Hb~6n<=C=8U0T*>?{*R`EJc6m{k+lKTwEr z{UE)UL`SJ6U7R4l-L0jK#`rTEJF=Tx)%hR2w-6Jf=0q^jrNaYz*iE48jLsaqh1iW7 z!UAOk*Ma6o_9%K8{{NE^S+EsI5EzxW6&XT`5J3C*{~aNmc~t5Ewsx|7G(3E0{e(&W fzyD9Vx|z7Rd4Vi)2<$*k9&Qc0$ and non-real parameters occur in conjugate pairs, then -\begin{eqnarray} -\label{OrthIWilson} -& &\frac{1}{2\pi}\int_0^{\infty} -\left|\frac{\Gamma(a+ix)\Gamma(b+ix)\Gamma(c+ix)\Gamma(d+ix)}{\Gamma(2ix)}\right|^2\nonumber\\ -& &{}\mathindent\times W_m(x^2;a,b,c,d)W_n(x^2;a,b,c,d)\,dx\nonumber\\ -& &{}=\frac{\Gamma(n+a+b)\cdots\Gamma(n+c+d)}{\Gamma(2n+a+b+c+d)}(n+a+b+c+d-1)_nn!\,\delta_{mn}, -\end{eqnarray} -where -\begin{eqnarray*} -& &\Gamma(n+a+b)\cdots\Gamma(n+c+d)\\ -& &{}=\Gamma(n+a+b)\Gamma(n+a+c)\Gamma(n+a+d)\Gamma(n+b+c)\Gamma(n+b+d)\Gamma(n+c+d). -\end{eqnarray*} -If $a<0$ and $a+b$, $a+c$, $a+d$ are positive or a pair of complex conjugates -occur with positive real parts, then -\begin{eqnarray} -\label{OrtIIWilson} -& &\frac{1}{2\pi}\int_0^{\infty}\left|\frac{\Gamma(a+ix)\Gamma(b+ix)\Gamma(c+ix)\Gamma(d+ix)}{\Gamma(2ix)}\right|^2\nonumber\\ -& &{}\mathindent\mathindent\times W_m(x^2;a,b,c,d)W_n(x^2;a,b,c,d)\,dx\nonumber\\ -& &{}\mathindent{}+\frac{\Gamma(a+b)\Gamma(a+c)\Gamma(a+d)\Gamma(b-a)\Gamma(c-a)\Gamma(d-a)}{\Gamma(-2a)}\nonumber\\ -& &{}\mathindent\mathindent{}\times\sum_{\begin{array}{c} -{\scriptstyle k=0,1,2\ldots}\\{\scriptstyle a+k<0}\end{array}} -\frac{(2a)_k(a+1)_k(a+b)_k(a+c)_k(a+d)_k}{(a)_k(a-b+1)_k(a-c+1)_k(a-d+1)_kk!}\nonumber\\ -& &{}\mathindent\mathindent\mathindent\mathindent{}\times W_m(-(a+k)^2;a,b,c,d)W_n(-(a+k)^2;a,b,c,d)\nonumber\\ -& &{}=\frac{\Gamma(n+a+b)\cdots\Gamma(n+c+d)}{\Gamma(2n+a+b+c+d)}(n+a+b+c+d-1)_nn!\,\delta_{mn}. -\end{eqnarray} - -\subsection*{Recurrence relation} -\begin{equation} -\label{RecWilson} --\left(a^2+x^2\right){\tilde{W}}_n(x^2) -=A_n{\tilde{W}}_{n+1}(x^2)-\left(A_n+C_n\right){\tilde{W}}_n(x^2)+C_n{\tilde{W}}_{n-1}(x^2), -\end{equation} -where -$${\tilde{W}}_n(x^2):={\tilde{W}}_n(x^2;a,b,c,d)=\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}$$ -and -$$\left\{\begin{array}{l} -\displaystyle A_n=\frac{(n+a+b+c+d-1)(n+a+b)(n+a+c)(n+a+d)}{(2n+a+b+c+d-1)(2n+a+b+c+d)}\\[5mm] -\displaystyle C_n=\frac{n(n+b+c-1)(n+b+d-1)(n+c+d-1)}{(2n+a+b+c+d-2)(2n+a+b+c+d-1)}. -\end{array}\right.$$ - -\subsection*{Normalized recurrence relation} -\begin{equation} -\label{NormRecWilson} -xp_n(x)=p_{n+1}(x)+(A_n+C_n-a^2)p_n(x)+A_{n-1}C_np_{n-1}(x), -\end{equation} -where -$$W_n(x^2;a,b,c,d)=(-1)^n(n+a+b+c+d-1)_np_n(x^2).$$ - -\subsection*{Difference equation} -\begin{eqnarray} -\label{dvWilson} -& &n(n+a+b+c+d-1)y(x)\nonumber\\ -& &{}=B(x)y(x+i)-\left[B(x)+D(x)\right]y(x)+D(x)y(x-i), -\end{eqnarray} -where -$$y(x)=W_n(x^2;a,b,c,d)$$ -and -$$\left\{\begin{array}{l} -\displaystyle B(x)=\frac{(a-ix)(b-ix)(c-ix)(d-ix)}{2ix(2ix-1)}\\[5mm] -\displaystyle D(x)=\frac{(a+ix)(b+ix)(c+ix)(d+ix)}{2ix(2ix+1)}. -\end{array}\right.$$ - -\subsection*{Forward shift operator} -\begin{eqnarray} -\label{shift1WilsonI} -& &W_n((x+\textstyle\frac{1}{2}i)^2;a,b,c,d) --W_n((x-\textstyle\frac{1}{2}i)^2;a,b,c,d)\nonumber\\ -& &{}=-2inx(n+a+b+c+d-1)W_{n-1}(x^2;a+\textstyle\frac{1}{2},b+\textstyle\frac{1}{2}, -c+\textstyle\frac{1}{2},d+\textstyle\frac{1}{2}) -\end{eqnarray} -or equivalently -\begin{eqnarray} -\label{shift1WilsonII} -& &\frac{\delta W_n(x^2;a,b,c,d)}{\delta x^2}\nonumber\\ -& &{}=-n(n+a+b+c+d-1)W_{n-1}(x^2;a+\textstyle\frac{1}{2},b+\textstyle\frac{1}{2}, -c+\textstyle\frac{1}{2},d+\textstyle\frac{1}{2}). -\end{eqnarray} - -\newpage - -\subsection*{Backward shift operator} -\begin{eqnarray} -\label{shift2WilsonI} -& &(a-\textstyle\frac{1}{2}-ix)(b-\textstyle\frac{1}{2}-ix) -(c-\textstyle\frac{1}{2}-ix)(d-\textstyle\frac{1}{2}-ix) -W_n((x+\textstyle\frac{1}{2}i)^2;a,b,c,d)\nonumber\\ -& &{}\mathindent{}-(a-\textstyle\frac{1}{2}+ix)(b-\textstyle\frac{1}{2}+ix) -(c-\textstyle\frac{1}{2}+ix)(d-\textstyle\frac{1}{2}+ix) -W_n((x-\textstyle\frac{1}{2}i)^2;a,b,c,d)\nonumber\\ -& &{}=-2ix W_{n+1}(x^2;a-\textstyle\frac{1}{2},b-\textstyle\frac{1}{2}, -c-\textstyle\frac{1}{2},d-\textstyle\frac{1}{2}) -\end{eqnarray} -or equivalently -\begin{eqnarray} -\label{shift2WilsonII} -& &\frac{\delta\left[\omega(x;a,b,c,d)W_n(x^2;a,b,c,d)\right]}{\delta x^2}\nonumber\\ -& &{}=\omega(x;a-\textstyle\frac{1}{2},b-\textstyle\frac{1}{2}, -c-\textstyle\frac{1}{2},d-\textstyle\frac{1}{2}) -W_{n+1}(x^2;a-\textstyle\frac{1}{2},b-\textstyle\frac{1}{2},c-\textstyle\frac{1}{2},d-\textstyle\frac{1}{2}), -\end{eqnarray} -where -$$\omega(x;a,b,c,d):=\frac{1}{2ix}\left|\frac{\Gamma(a+ix)\Gamma(b+ix)\Gamma(c+ix)\Gamma(d+ix)}{\Gamma(2ix)}\right|^2.$$ - -\subsection*{Rodrigues-type formula} -\begin{eqnarray} -\label{RodWilson} -& &\omega(x;a,b,c,d)W_n(x^2;a,b,c,d)\nonumber\\ -& &{}=\left(\frac{\delta}{\delta x^2}\right)^n -\left[\omega(x;a+\textstyle\frac{1}{2}n,b+\textstyle\frac{1}{2}n, -c+\textstyle\frac{1}{2}n,d+\textstyle\frac{1}{2}n)\right]. -\end{eqnarray} - -\subsection*{Generating functions} -\begin{equation} -\label{GenWilson1} -\hyp{2}{1}{a+ix,b+ix}{a+b}{t}\,\hyp{2}{1}{c-ix,d-ix}{c+d}{t}= -\sum_{n=0}^{\infty}\frac{W_n(x^2;a,b,c,d)t^n}{(a+b)_n(c+d)_nn!}. -\end{equation} - -\begin{equation} -\label{GenWilson2} -\hyp{2}{1}{a+ix,c+ix}{a+c}{t}\,\hyp{2}{1}{b-ix,d-ix}{b+d}{t}= -\sum_{n=0}^{\infty}\frac{W_n(x^2;a,b,c,d)t^n}{(a+c)_n(b+d)_nn!}. -\end{equation} - -\begin{equation} -\label{GenWilson3} -\hyp{2}{1}{a+ix,d+ix}{a+d}{t}\,\hyp{2}{1}{b-ix,c-ix}{b+c}{t}= -\sum_{n=0}^{\infty}\frac{W_n(x^2;a,b,c,d)t^n}{(a+d)_n(b+c)_nn!}. -\end{equation} - -\begin{eqnarray} -\label{GenWilson4} -& &(1-t)^{1-a-b-c-d}\nonumber\\ -& &{}\mathindent\times\hyp{4}{3}{\frac{1}{2}(a+b+c+d-1),\frac{1}{2}(a+b+c+d),a+ix,a-ix} -{a+b,a+c,a+d}{-\frac{4t}{(1-t)^2}}\nonumber\\ -& &{}=\sum_{n=0}^{\infty}\frac{(a+b+c+d-1)_n}{(a+b)_n(a+c)_n(a+d)_nn!}W_n(x^2;a,b,c,d)t^n. -\end{eqnarray} - -\subsection*{Limit relations} - -\subsubsection*{Wilson $\rightarrow$ Continuous dual Hahn} -The continuous dual Hahn polynomials given by (\ref{DefContDualHahn}) can be found from the -Wilson polynomials by dividing by $(a+d)_n$ and letting $d\rightarrow\infty$: -\begin{equation} -\lim_{d\rightarrow\infty}\frac{W_n(x^2;a,b,c,d)}{(a+d)_n}=S_n(x^2;a,b,c). -\end{equation} - -\subsubsection*{Wilson $\rightarrow$ Continuous Hahn} -The continuous Hahn polynomials given by (\ref{DefContHahn}) are obtained from the Wilson -polynomials by the substitutions $a\rightarrow a-it$, $b\rightarrow b-it$, $c\rightarrow c+it$, -$d\rightarrow d+it$ and $x\rightarrow x+t$ and the limit $t\rightarrow\infty$ in the following -way: -\begin{equation} -\lim_{t\rightarrow\infty} -\frac{W_n((x+t)^2;a-it,b-it,c+it,d+it)}{(-2t)^nn!}=p_n(x;a,b,c,d). -\end{equation} - -\subsubsection*{Wilson $\rightarrow$ Jacobi} -The Jacobi polynomials given by (\ref{DefJacobi}) can be found from the Wilson -polynomials by substituting $a=b=\frac{1}{2}(\alpha+1)$, $c=\frac{1}{2}(\beta+1)+it$, -$d=\frac{1}{2}(\beta+1)-it$ and $x\rightarrow t\sqrt{\frac{1}{2}(1-x)}$ in the -definition (\ref{DefWilson}) of the Wilson polynomials and taking the limit -$t\rightarrow\infty$. In fact we have -\begin{eqnarray} -& &\lim_{t\rightarrow\infty}\frac{W_n(\frac{1}{2}(1-x)t^2;\frac{1}{2}(\alpha+1), -\frac{1}{2}(\alpha+1),\frac{1}{2}(\beta+1)+it,\frac{1}{2}(\beta+1)-it)}{t^{2n}n!}\nonumber\\ -& &{}=P_n^{(\alpha,\beta)}(x). -\end{eqnarray} - -\subsection*{Remarks} -Note that for $k0$, $c=\bar{a}$ and $d=\bar{b}$, then -\begin{eqnarray} -\label{OrtContHahn} -& &\frac{1}{2\pi}\int_{-\infty}^{\infty}\Gamma(a+ix)\Gamma(b+ix)\Gamma(c-ix)\Gamma(d-ix) -p_m(x;a,b,c,d)p_n(x;a,b,c,d)\,dx\nonumber\\ -& &{}=\frac{\Gamma(n+a+c)\Gamma(n+a+d)\Gamma(n+b+c)\Gamma(n+b+d)} -{(2n+a+b+c+d-1)\Gamma(n+a+b+c+d-1)n!}\,\delta_{mn}. -\end{eqnarray} - -\subsection*{Recurrence relation} -\begin{equation} -\label{RecContHahn} -(a+ix)\tilde{p}_n(x)=A_n\tilde{p}_{n+1}(x)-\left(A_n+C_n\right)\tilde{p}_n(x)+C_n\tilde{p}_{n-1}(x), -\end{equation} -where -$$\tilde{p}_n(x):=\tilde{p}_n(x;a,b,c,d)=\frac{n!}{i^n(a+c)_n(a+d)_n}p_n(x;a,b,c,d)$$ -and -$$\left\{\begin{array}{l} -\displaystyle A_n=-\frac{(n+a+b+c+d-1)(n+a+c)(n+a+d)}{(2n+a+b+c+d-1)(2n+a+b+c+d)}\\[5mm] -\displaystyle C_n=\frac{n(n+b+c-1)(n+b+d-1)}{(2n+a+b+c+d-2)(2n+a+b+c+d-1)}. -\end{array}\right.$$ - -\subsection*{Normalized recurrence relation} -\begin{equation} -\label{NormRecContHahn} -xp_n(x)=p_{n+1}(x)+i(A_n+C_n+a)p_n(x)-A_{n-1}C_np_{n-1}(x), -\end{equation} -where -$$p_n(x;a,b,c,d)=\frac{(n+a+b+c+d-1)_n}{n!}p_n(x).$$ - -\subsection*{Difference equation} -\begin{eqnarray} -\label{dvContHahn} -& &n(n+a+b+c+d-1)y(x)\nonumber\\ -& &{}=B(x)y(x+i)-\left[B(x)+D(x)\right]y(x)+D(x)y(x-i), -\end{eqnarray} -where -$$y(x)=p_n(x;a,b,c,d)$$ -and -$$\left\{\begin{array}{l} -\displaystyle B(x)=(c-ix)(d-ix)\\[5mm] -\displaystyle D(x)=(a+ix)(b+ix). -\end{array}\right.$$ - -\subsection*{Forward shift operator} -\begin{eqnarray} -\label{shift1ContHahnI} -& &p_n(x+\textstyle\frac{1}{2}i;a,b,c,d)-p_n(x-\textstyle\frac{1}{2}i;a,b,c,d)\nonumber\\ -& &{}=i(n+a+b+c+d-1)p_{n-1}(x;a+\textstyle\frac{1}{2}, -b+\textstyle\frac{1}{2},c+\textstyle\frac{1}{2},d+\textstyle\frac{1}{2}) -\end{eqnarray} -or equivalently -\begin{equation} -\label{shift1ContHahnII} -\frac{\delta p_n(x;a,b,c,d)}{\delta x}=(n+a+b+c+d-1) -p_{n-1}(x;a+\textstyle\frac{1}{2},b+\textstyle\frac{1}{2}, -c+\textstyle\frac{1}{2},d+\textstyle\frac{1}{2}). -\end{equation} - -\subsection*{Backward shift operator} -\begin{eqnarray} -\label{shift2ContHahnI} -& &(c-\textstyle\frac{1}{2}-ix)(d-\textstyle\frac{1}{2}-ix) -p_n(x+\textstyle\frac{1}{2}i;a,b,c,d)\nonumber\\ -& &{}\mathindent{}-(a-\textstyle\frac{1}{2}+ix)(b-\textstyle\frac{1}{2}+ix) -p_n(x-\textstyle\frac{1}{2}i;a,b,c,d)\nonumber\\ -& &{}=\frac{n+1}{i}p_{n+1}(x;a-\textstyle\frac{1}{2}, -b-\textstyle\frac{1}{2},c-\textstyle\frac{1}{2},d-\textstyle\frac{1}{2}) -\end{eqnarray} -or equivalently -\begin{eqnarray} -\label{shift2ContHahnII} -& &\frac{\delta\left[\omega(x;a,b,c,d)p_n(x;a,b,c,d)\right]}{\delta x}\nonumber\\ -& &{}=-(n+1)\omega(x;a-\textstyle\frac{1}{2},b-\textstyle\frac{1}{2}, -c-\textstyle\frac{1}{2},d-\textstyle\frac{1}{2})\nonumber\\ -& &{}\mathindent\times p_{n+1}(x;a-\textstyle\frac{1}{2},b-\textstyle\frac{1}{2},c-\textstyle\frac{1}{2},d-\textstyle\frac{1}{2}), -\end{eqnarray} -where -$$\omega(x;a,b,c,d)=\Gamma(a+ix)\Gamma(b+ix)\Gamma(c-ix)\Gamma(d-ix).$$ - -\subsection*{Rodrigues-type formula} -\begin{eqnarray} -\label{RodContHahn} -& &\omega(x;a,b,c,d)p_n(x;a,b,c,d)\nonumber\\ -& &{}=\frac{(-1)^n}{n!}\left(\frac{\delta}{\delta x}\right)^n -\left[\omega(x;a+\textstyle\frac{1}{2}n,b+\textstyle\frac{1}{2}n, -c+\textstyle\frac{1}{2}n,d+\textstyle\frac{1}{2}n)\right]. -\end{eqnarray} - -\subsection*{Generating functions} -\begin{equation} -\label{GenContHahn1} -\hyp{1}{1}{a+ix}{a+c}{-it}\,\hyp{1}{1}{d-ix}{b+d}{it}= -\sum_{n=0}^{\infty}\frac{p_n(x;a,b,c,d)}{(a+c)_n(b+d)_n}t^n. -\end{equation} - -\begin{equation} -\label{GenContHahn2} -\hyp{1}{1}{a+ix}{a+d}{-it}\,\hyp{1}{1}{c-ix}{b+c}{it}= -\sum_{n=0}^{\infty}\frac{p_n(x;a,b,c,d)}{(a+d)_n(b+c)_n}t^n. -\end{equation} - -\begin{eqnarray} -\label{GenContHahn3} -& &(1-t)^{1-a-b-c-d}\,\hyp{3}{2}{\frac{1}{2}(a+b+c+d-1),\frac{1}{2}(a+b+c+d),a+ix} -{a+c,a+d}{-\frac{4t}{(1-t)^2}}\nonumber\\ -& &{}=\sum_{n=0}^{\infty} -\frac{(a+b+c+d-1)_n}{(a+c)_n(a+d)_ni^n}p_n(x;a,b,c,d)t^n. -\end{eqnarray} - -\subsection*{Limit relations} - -\subsubsection*{Wilson $\rightarrow$ Continuous Hahn} -The continuous Hahn polynomials are obtained from the Wilson polynomials given by -(\ref{DefWilson}) by the substitution $a\rightarrow a-it$, $b\rightarrow b-it$, -$c\rightarrow c+it$, $d\rightarrow d+it$ and $x\rightarrow x+t$ and the limit -$t\rightarrow\infty$ in the following way: -$$\lim_{t\rightarrow\infty} -\frac{W_n((x+t)^2;a-it,b-it,c+it,d+it)}{(-2t)^nn!}=p_n(x;a,b,c,d).$$ - -\subsubsection*{Continuous Hahn $\rightarrow$ Meixner-Pollaczek} -The Meixner-Pollaczek polynomials given by (\ref{DefMP}) can be obtained from the -continuous Hahn polynomials by setting $x\rightarrow x+t$, $a=\lambda-it$, $c=\lambda+it$ -and $b=d=t\tan\phi$ and taking the limit $t\rightarrow\infty$: -\begin{equation} -\lim_{t\rightarrow\infty}\frac{p_n(x+t;\lambda-it,t\tan\phi,\lambda+it,t\tan\phi)}{t^n} -=\frac{P_n^{(\lambda)}(x;\phi)}{(\cos\phi)^n}. -\end{equation} - -\subsubsection*{Continuous Hahn $\rightarrow$ Jacobi} -The Jacobi polynomials given by (\ref{DefJacobi}) follow from the continuous Hahn polynomials -by the substitution $x\rightarrow \frac{1}{2}xt$, $a=\frac{1}{2}(\alpha+1-it)$, -$b=\frac{1}{2}(\beta+1+it)$, $c=\frac{1}{2}(\alpha+1+it)$ and $d=\frac{1}{2}(\beta+1-it)$, -division by $t^n$ and the limit $t\rightarrow\infty$: -\begin{eqnarray} -& &\lim_{t\rightarrow\infty} -\frac{p_n(\frac{1}{2}xt;\frac{1}{2}(\alpha+1-it),\frac{1}{2}(\beta+1+it), -\frac{1}{2}(\alpha+1+it),\frac{1}{2}(\beta+1-it))}{t^n}\nonumber\\ -& &{}=P_n^{(\alpha,\beta)}(x). -\end{eqnarray} - -\subsubsection*{Continuous Hahn $\rightarrow$ Pseudo Jacobi} -The pseudo Jacobi polynomials given by (\ref{DefPseudoJacobi}) follow from the continuous -Hahn polynomials by the substitution $x\rightarrow xt$, $a=\frac{1}{2}(-N+i\nu-2t)$, -$b=\frac{1}{2}(-N-i\nu+2t)$, $c=\frac{1}{2}(-N-i\nu-2t)$ and $d=\frac{1}{2}(-N+i\nu+2t)$, -division by $t^n$ and the limit $t\rightarrow\infty$: -\begin{eqnarray} -& &\lim_{t\rightarrow\infty}\frac{p_n(xt;\frac{1}{2}(-N+i\nu-2t),\frac{1}{2}(-N-i\nu+2t), -\frac{1}{2}(-N+i\nu-2t),\frac{1}{2}(-N-i\nu+2t))}{t^n}\nonumber\\ -& &{}=\frac{(n-2N-1)_n}{n!}P_n(x;\nu,N). -\end{eqnarray} - -\subsection*{Remark} -Since we have for $k0$ and $(c,d)=(\overline a,\overline b)$ or $(\overline b,\overline a)$.\\ -Thus, under these assumptions, the continuous Hahn polynomial -$p_n(x;a,b,c,d)$ -is symmetric in $a,b$ and in $c,d$. -This follows from the orthogonality relation (9.4.2) -together with the value of its coefficient of $x^n$ given in (9.4.4b).\\ -As a consequence, it is sufficient to give generating function (9.4.11). Then the generating -function (9.4.12) will follow by symmetry in the parameters. -% -\paragraph{Uniqueness of orthogonality measure} -The coefficient of $p_{n-1}(x)$ in (9.4.4) behaves as $O(n^2)$ as $n\to\iy$. -Hence \eqref{93} holds, by which the orthogonality measure is unique. -% -\paragraph{Special cases} -In the following special case there is a reduction to -Meixner-Pollaczek: -\begin{equation} -p_n(x;a,a+\thalf,a,a+\thalf)= -\frac{(2a)_n (2a+\thalf)_n}{(4a)_n}\,P_n^{(2a)}(2x;\thalf\pi). -\end{equation} -See \myciteKLS{342}{(2.6)} (note that in \myciteKLS{342}{(2.3)} the -Meixner-Pollaczek polynonmials are defined different from (9.7.1), -without a constant factor in front). - -For $0-1$ and $\beta>-1$, or for $\alpha<-N$ and $\beta<-N$, we have -\begin{eqnarray} -\label{OrtHahn} -& &\sum_{x=0}^N\binom{\alpha +x}{x}\binom{\beta+N-x}{N-x}Q_m(x;\alpha,\beta,N)Q_n(x;\alpha,\beta,N)\nonumber\\ -& &{}=\frac{(-1)^n(n+\alpha+\beta+1)_{N+1}(\beta+1)_nn!}{(2n+\alpha+\beta+1)(\alpha+1)_n(-N)_nN!}\,\delta_{mn}. -\end{eqnarray} - -\subsection*{Recurrence relation} -\begin{equation} -\label{RecHahn} --xQ_n(x)=A_nQ_{n+1}(x)-\left(A_n+C_n\right)Q_n(x)+C_nQ_{n-1}(x), -\end{equation} -where -$$Q_n(x):=Q_n(x;\alpha,\beta,N)$$ -and -$$\left\{\begin{array}{l} -\displaystyle A_n=\frac{(n+\alpha+\beta+1)(n+\alpha+1)(N-n)}{(2n+\alpha+\beta+1)(2n+\alpha+\beta+2)}\\[5mm] -\displaystyle C_n=\frac{n(n+\alpha+\beta+N+1)(n+\beta)}{(2n+\alpha+\beta)(2n+\alpha+\beta+1)}. -\end{array}\right.$$ - -\subsection*{Normalized recurrence relation} -\begin{equation} -\label{NormRecHahn} -xp_n(x)=p_{n+1}(x)+\left(A_n+C_n\right)p_n(x)+A_{n-1}C_np_{n-1}(x), -\end{equation} -where -$$Q_n(x;\alpha,\beta,N)=\frac{(n+\alpha+\beta+1)_n}{(\alpha+1)_n(-N)_n}p_n(x).$$ - -\subsection*{Difference equation} -\begin{equation} -\label{dvHahn} -n(n+\alpha+\beta+1)y(x)=B(x)y(x+1)-\left[B(x)+D(x)\right]y(x)+D(x)y(x-1), -\end{equation} -where -$$y(x)=Q_n(x;\alpha,\beta,N)$$ -and -$$\left\{\begin{array}{l} -\displaystyle B(x)=(x+\alpha+1)(x-N)\\[5mm] -\displaystyle D(x)=x(x-\beta-N-1). -\end{array}\right.$$ - -\subsection*{Forward shift operator} -\begin{eqnarray} -\label{shift1HahnI} -& &Q_n(x+1;\alpha,\beta,N)-Q_n(x;\alpha,\beta,N)\nonumber\\ -& &{}=-\frac{n(n+\alpha+\beta+1)}{(\alpha+1)N}Q_{n-1}(x;\alpha+1,\beta+1,N-1) -\end{eqnarray} -or equivalently -\begin{equation} -\label{shift1HahnII} -\Delta Q_n(x;\alpha,\beta,N)=-\frac{n(n+\alpha+\beta+1)}{(\alpha+1)N}Q_{n-1}(x;\alpha+1,\beta+1,N-1). -\end{equation} - -\subsection*{Backward shift operator} -\begin{eqnarray} -\label{shift2HahnI} -& &(x+\alpha)(N+1-x)Q_n(x;\alpha,\beta,N)-x(\beta+N+1-x)Q_n(x-1;\alpha,\beta,N)\nonumber\\ -& &{}=\alpha(N+1)Q_{n+1}(x;\alpha-1,\beta-1,N+1) -\end{eqnarray} -or equivalently -\begin{eqnarray} -\label{shift2HahnII} -& &\nabla\left[\omega(x;\alpha,\beta,N)Q_n(x;\alpha,\beta,N)\right]\nonumber\\ -& &{}=\frac{N+1}{\beta}\omega(x;\alpha-1,\beta-1,N+1)Q_{n+1}(x;\alpha-1,\beta-1,N+1), -\end{eqnarray} -where -$$\omega(x;\alpha,\beta,N)=\binom{\alpha+x}{x}\binom{\beta+N-x}{N-x}.$$ - -\subsection*{Rodrigues-type formula} -\begin{eqnarray} -\label{RodHahn} -& &\omega(x;\alpha,\beta,N)Q_n(x;\alpha,\beta,N)\nonumber\\ -& &{}=\frac{(-1)^n(\beta+1)_n}{(-N)_n}\nabla^n\left[\omega(x;\alpha+n,\beta+n,N-n)\right]. -\end{eqnarray} - -\subsection*{Generating functions} -For $x=0,1,2,\ldots,N$ we have -\begin{equation} -\label{GenHahn1} -\hyp{1}{1}{-x}{\alpha+1}{-t}\,\hyp{1}{1}{x-N}{\beta+1}{t} -=\sum_{n=0}^N\frac{(-N)_n}{(\beta+1)_nn!}Q_n(x;\alpha,\beta,N)t^n. -\end{equation} - -\begin{eqnarray} -\label{GenHahn2} -& &\hyp{2}{0}{-x,-x+\beta+N+1}{-}{-t}\,\hyp{2}{0}{x-N,x+\alpha+1}{-}{t}\nonumber\\ -& &{}=\sum_{n=0}^N\frac{(-N)_n(\alpha+1)_n}{n!}Q_n(x;\alpha,\beta,N)t^n. -\end{eqnarray} - -\begin{eqnarray} -\label{GenHahn3} -& &\left[(1-t)^{-\alpha-\beta-1}\,\hyp{3}{2}{\frac{1}{2}(\alpha+\beta+1),\frac{1}{2}(\alpha+\beta+2),-x} -{\alpha+1,-N}{-\frac{4t}{(1-t)^2}}\right]_N\nonumber\\ -& &{}=\sum_{n=0}^N\frac{(\alpha+\beta+1)_n}{n!}Q_n(x;\alpha,\beta,N)t^n. -\end{eqnarray} - -\subsection*{Limit relations} - -\subsubsection*{Racah $\rightarrow$ Hahn} -If we take $\gamma+1=-N$ and let $\delta\rightarrow\infty$ in the definition (\ref{DefRacah}) -of the Racah polynomials, we obtain the Hahn polynomials. Hence -$$\lim_{\delta\rightarrow\infty} -R_n(\lambda(x);\alpha,\beta,-N-1,\delta)=Q_n(x;\alpha,\beta,N).$$ -And if we take $\delta=-\beta-N-1$ and let $\gamma\rightarrow\infty$ in the definition (\ref{DefRacah}) -of the Racah polynomials, we also obtain the Hahn polynomials: -$$\lim_{\gamma\rightarrow\infty} -R_n(\lambda(x);\alpha,\beta,\gamma,-\beta-N-1)=Q_n(x;\alpha,\beta,N).$$ -Another way to do this is to take $\alpha+1=-N$ and $\beta\rightarrow\beta+\gamma+N+1$ in -the definition (\ref{DefRacah}) of the Racah polynomials and then take the limit -$\delta\rightarrow\infty$. In that case we obtain the Hahn polynomials in the following way: -$$\lim_{\delta\rightarrow\infty} -R_n(\lambda(x);-N-1,\beta+\gamma+N+1,\gamma,\delta)=Q_n(x;\gamma,\beta,N).$$ - -\subsubsection*{Hahn $\rightarrow$ Jacobi} -To find the Jacobi polynomials given by (\ref{DefJacobi}) from the Hahn polynomials we take -$x\rightarrow Nx$ and let $N\rightarrow\infty$. In fact we have -\begin{equation} -\lim_{N\rightarrow\infty} -Q_n(Nx;\alpha,\beta,N)=\frac{P_n^{(\alpha,\beta)}(1-2x)}{P_n^{(\alpha,\beta)}(1)}. -\end{equation} - -\subsubsection*{Hahn $\rightarrow$ Meixner} -The Meixner polynomials given by (\ref{DefMeixner}) can be obtained from the Hahn polynomials -by taking $\alpha=b-1$, $\beta=N(1-c)c^{-1}$ and letting $N\rightarrow\infty$: -\begin{equation} -\lim_{N\rightarrow\infty} -Q_n(x;b-1,N(1-c)c^{-1},N)=M_n(x;b,c). -\end{equation} - -\subsubsection*{Hahn $\rightarrow$ Krawtchouk} -The Krawtchouk polynomials given by (\ref{DefKrawtchouk}) are obtained from the Hahn polynomials -if we take $\alpha=pt$ and $\beta=(1-p)t$ and let $t\rightarrow\infty$: -\begin{equation} -\lim_{t\rightarrow\infty}Q_n(x;pt,(1-p)t,N)=K_n(x;p,N). -\end{equation} - -\subsection*{Remark} -If we interchange the role of $x$ and $n$ in (\ref{DefHahn}) we obtain the -dual Hahn polynomials given by (\ref{DefDualHahn}). - -\noindent -Since -$$Q_n(x;\alpha,\beta,N)=R_x(\lambda(n);\alpha,\beta,N)$$ -we obtain the dual orthogonality relation for the Hahn polynomials from the -orthogonality relation (\ref{OrtDualHahn}) of the dual Hahn polynomials: -\begin{eqnarray*} -& &\sum_{n=0}^N\frac{(2n+\alpha+\beta+1)(\alpha+1)_n(-N)_nN!}{(-1)^n(n+\alpha+\beta+1)_{N+1}(\beta+1)_nn!} -Q_n(x;\alpha,\beta,N)Q_n(y;\alpha,\beta,N)\\ -& &{}=\frac{\delta_{xy}}{\dbinom{\alpha+x}{x}\dbinom{\beta+N-x}{N-x}},\quad x,y \in \{0,1,2,\ldots,N\}. -\subsection*{9.5 Hahn} -\label{sec9.5} -% -\paragraph{Special values} -\begin{equation} -Q_n(0;\al,\be,N)=1,\quad -Q_n(N;\al,\be,N)=\frac{(-1)^n(\be+1)_n}{(\al+1)_n}\,. -\label{95} -\end{equation} -Use (9.5.1) and compare with (9.8.1) and \eqref{50}. - -From (9.5.3) and \eqref{1} it follows that -\begin{equation} -Q_{2n}(N;\al,\al,2N)=\frac{(\thalf)_n(N+\al+1)_n}{(-N+\thalf)_n(\al+1)_n}\,. -\label{30} -\end{equation} -From (9.5.1) and \mycite{DLMF}{(15.4.24)} it follows that -\begin{equation} -Q_N(x;\al,\be,N)=\frac{(-N-\be)_x}{(\al+1)_x}\qquad(x=0,1,\ldots,N). -\label{44} -\end{equation} -% -\paragraph{Symmetries} -By the orthogonality relation (9.5.2): -\begin{equation} -\frac{Q_n(N-x;\al,\be,N)}{Q_n(N;\al,\be,N)}=Q_n(x;\be,\al,N), -\label{96} -\end{equation} -It follows from \eqref{97} and \eqref{45} that -\begin{equation} -\frac{Q_{N-n}(x;\al,\be,N)}{Q_N(x;\al,\be,N)} -=Q_n(x;-N-\be-1,-N-\al-1,N) -\qquad(x=0,1,\ldots,N). -\label{100} -\end{equation} -% -\paragraph{Duality} -The Remark on p.208 gives the duality between Hahn and dual Hahn polynomials: -% -\begin{equation} -Q_n(x;\al,\be,N)=R_x(n(n+\al+\be+1);\al,\be,N)\quad(n,x\in\{0,1,\ldots N\}). -\label{45} -\end{equation} -\end{eqnarray*} - -\subsection*{References} -\cite{AlSalam90}, \cite{AndrewsAskey85}, \cite{Area+II}, \cite{Askey75}, -\cite{Askey89I}, \cite{Askey2005}, \cite{AskeyGasper77}, \cite{AskeyWilson85}, -\cite{AtakRahmanSuslov}, \cite{AtakSuslov88}, \cite{Chihara78}, -\cite{Ciesielski}, \cite{Cooper+}, \cite{Dette95}, \cite{Dunkl76}, -\cite{Dunkl78I}, \cite{Gasper73I}, \cite{Gasper74}, \cite{HoareRahman}, -\cite{Ismail77}, \cite{Ismail2005II}, \cite{Karlin61}, \cite{Koorn81}, \cite{Koorn88}, -\cite{LabelleYehI}, \cite{LabelleYehII}, \cite{Laine}, \cite{Lesky62}, -\cite{Lesky88}, \cite{Lesky89}, \cite{Lesky94I}, \cite{Lesky95II}, -\cite{LewanowiczII}, \cite{Neuman}, \cite{Nikiforov+}, \cite{NikiforovUvarov}, -\cite{Rahman76III}, \cite{Rahman78I}, \cite{Rahman78II}, \cite{Rahman81III}, -\cite{Sablonniere}, \cite{Stanton84}, \cite{Stanton90}, \cite{Wilson80}, \cite{Wilson70II}, -\cite{Zarzo+}. - - -\section{Dual Hahn}\index{Dual Hahn polynomials}\index{Hahn polynomials!Dual} - -\par\setcounter{equation}{0} - -\subsection*{Hypergeometric representation} -\begin{equation} -\label{DefDualHahn} -R_n(\lambda(x);\gamma,\delta,N)= -\hyp{3}{2}{-n,-x,x+\gamma+\delta+1}{\gamma+1,-N}{1},\quad n=0,1,2,\ldots,N, -\end{equation} -where -$$\lambda(x)=x(x+\gamma+\delta+1).$$ - -\subsection*{Orthogonality relation} -For $\gamma>-1$ and $\delta>-1$, or for $\gamma<-N$ and $\delta<-N$, we have -\begin{eqnarray} -\label{OrtDualHahn} -& &\sum_{x=0}^N\frac{(2x+\gamma+\delta+1)(\gamma+1)_x(-N)_xN!}{(-1)^x(x+\gamma+\delta+1)_{N+1}(\delta+1)_xx!} -R_m(\lambda(x);\gamma,\delta,N)R_n(\lambda(x);\gamma,\delta,N)\nonumber\\ -& &{}=\frac{\,\delta_{mn}}{\dbinom{\gamma+n}{n}\dbinom{\delta+N-n}{N-n}}. -\end{eqnarray} - -\subsection*{Recurrence relation} -\begin{equation} -\label{RecDualHahn} -\lambda(x)R_n(\lambda(x)) -=A_nR_{n+1}(\lambda(x))-\left(A_n+C_n\right)R_n(\lambda(x))+C_nR_{n-1}(\lambda(x)), -\end{equation} -where -$$R_n(\lambda(x)):=R_n(\lambda(x);\gamma,\delta,N)$$ -and -$$\left\{\begin{array}{l} -\displaystyle A_n=(n+\gamma+1)(n-N)\\[5mm] -\displaystyle C_n=n(n-\delta-N-1). -\end{array}\right.$$ - -\subsection*{Normalized recurrence relation} -\begin{equation} -\label{NormRecDualHahn} -xp_n(x)=p_{n+1}(x)-(A_n+C_n)p_n(x)+A_{n-1}C_np_{n-1}(x), -\end{equation} -where -$$R_n(\lambda(x);\gamma,\delta,N)=\frac{1}{(\gamma+1)_n(-N)_n}p_n(\lambda(x)).$$ - -\subsection*{Difference equation} -\begin{equation} -\label{dvDualHahn} --ny(x)=B(x)y(x+1)-\left[B(x)+D(x)\right]y(x)+D(x)y(x-1), -\end{equation} -where -$$y(x)=R_n(\lambda(x);\gamma,\delta,N)$$ -and -$$\left\{\begin{array}{l} -\displaystyle B(x)=\frac{(x+\gamma+1)(x+\gamma+\delta+1)(N-x)}{(2x+\gamma+\delta+1)(2x+\gamma+\delta+2)}\\[5mm] -\displaystyle D(x)=\frac{x(x+\gamma+\delta+N+1)(x+\delta)}{(2x+\gamma+\delta)(2x+\gamma+\delta+1)}. -\end{array}\right.$$ - -\subsection*{Forward shift operator} -\begin{eqnarray} -\label{shift1DualHahnI} -& &R_n(\lambda(x+1);\gamma,\delta,N)-R_n(\lambda(x);\gamma,\delta,N)\nonumber\\ -& &{}=-\frac{n(2x+\gamma+\delta+2)}{(\gamma+1)N}R_{n-1}(\lambda(x);\gamma+1,\delta,N-1) -\end{eqnarray} -or equivalently -\begin{equation} -\label{shift1DualHahnII} -\frac{\Delta R_n(\lambda(x);\gamma,\delta,N)}{\Delta\lambda(x)}= --\frac{n}{(\gamma+1)N}R_{n-1}(\lambda(x);\gamma+1,\delta,N-1). -\end{equation} - -\subsection*{Backward shift operator} -\begin{eqnarray} -\label{shift2DualHahnI} -& &(x+\gamma)(x+\gamma+\delta)(N+1-x)R_n(\lambda(x);\gamma,\delta,N)\nonumber\\ -& &{}\mathindent{}-x(x+\gamma+\delta+N+1)(x+\delta)R_n(\lambda(x-1);\gamma,\delta,N)\nonumber\\ -& &{}=\gamma(N+1)(2x+\gamma+\delta)R_{n+1}(\lambda(x);\gamma-1,\delta,N+1) -\end{eqnarray} -or equivalently -\begin{eqnarray} -\label{shift2DualHahnII} -& &\frac{\nabla\left[\omega(x;\gamma,\delta,N)R_n(\lambda(x);\gamma,\delta,N)\right]}{\nabla\lambda(x)}\nonumber\\ -& &{}=\frac{1}{\gamma+\delta}\omega(x;\gamma-1,\delta,N+1)R_{n+1}(\lambda(x);\gamma-1,\delta,N+1), -\end{eqnarray} -where -$$\omega(x;\gamma,\delta,N)=\frac{(-1)^x(\gamma+1)_x(\gamma+\delta+1)_x(-N)_x} -{(\gamma+\delta+N+2)_x(\delta+1)_xx!}.$$ - -\newpage - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodDualHahn} -\omega(x;\gamma,\delta,N)R_n(\lambda(x);\gamma,\delta,N) -=(\gamma+\delta+1)_n\left(\nabla_{\lambda}\right)^n\left[\omega(x;\gamma+n,\delta,N-n)\right], -\end{equation} -where -$$\nabla_{\lambda}:=\frac{\nabla}{\nabla\lambda(x)}.$$ - -\subsection*{Generating functions} -For $x=0,1,2,\ldots,N$ we have -\begin{equation} -\label{GenDualHahn1} -(1-t)^{N-x}\,\hyp{2}{1}{-x,-x-\delta}{\gamma+1}{t} -=\sum_{n=0}^N\frac{(-N)_n}{n!}R_n(\lambda(x);\gamma,\delta,N)t^n. -\end{equation} - -\begin{eqnarray} -\label{GenDualHahn2} -& &(1-t)^x\,\hyp{2}{1}{x-N,x+\gamma+1}{-\delta-N}{t}\nonumber\\ -& &=\sum_{n=0}^N\frac{(\gamma+1)_n(-N)_n}{(-\delta-N)_nn!}R_n(\lambda(x);\gamma,\delta,N)t^n. -\end{eqnarray} - -\begin{equation} -\label{GenDualHahn3} -\left[\expe^t\,\hyp{2}{2}{-x,x+\gamma+\delta+1}{\gamma+1,-N}{-t}\right]_N=\sum_{n=0}^N -\frac{R_n(\lambda(x);\gamma,\delta,N)}{n!}t^n. -\end{equation} - -\begin{eqnarray} -\label{GenDualHahn4} -& &\left[(1-t)^{-\epsilon}\,\hyp{3}{2}{\epsilon,-x,x+\gamma+\delta+1}{\gamma+1,-N}{\frac{t}{t-1}}\right]_N\nonumber\\ -& &{}=\sum_{n=0}^N\frac{(\epsilon)_n}{n!}R_n(\lambda(x);\gamma,\delta,N)t^n, -\quad\textrm{$\epsilon$ arbitrary}. -\end{eqnarray} - -\subsection*{Limit relations} - -\subsubsection*{Racah $\rightarrow$ Dual Hahn} -If we take $\alpha+1=-N$ and let $\beta\rightarrow\infty$ in the definition (\ref{DefRacah}) -of the Racah polynomials, then we obtain the dual Hahn polynomials: -$$\lim_{\beta\rightarrow\infty} -R_n(\lambda(x);-N-1,\beta,\gamma,\delta)=R_n(\lambda(x);\gamma,\delta,N).$$ -And if we take $\beta=-\delta-N-1$ and let $\alpha\rightarrow\infty$ in the definition (\ref{DefRacah}) -of the Racah polynomials, then we also obtain the dual Hahn polynomials: -$$\lim_{\alpha\rightarrow\infty} -R_n(\lambda(x);\alpha,-\delta-N-1,\gamma,\delta)=R_n(\lambda(x);\gamma,\delta,N).$$ -Finally, if we take $\gamma+1=-N$ and $\delta\rightarrow\alpha+\delta+N+1$ in the definition -(\ref{DefRacah}) of the Racah polynomials and take the limit -$\beta\rightarrow\infty$ we find the dual Hahn polynomials in the following way: -$$\lim_{\beta\rightarrow\infty} -R_n(\lambda(x);\alpha,\beta,-N-1,\alpha+\delta+N+1)=R_n(\lambda(x);\alpha,\delta,N).$$ - -\subsubsection*{Dual Hahn $\rightarrow$ Meixner} -The Meixner polynomials given by (\ref{DefMeixner}) are obtained from the dual Hahn polynomials -if we take $\gamma=\beta-1$ and $\delta=N(1-c)c^{-1}$ and let $N\rightarrow\infty$: -\begin{equation} -\lim_{N\rightarrow\infty} -R_n(\lambda(x);\beta-1,N(1-c)c^{-1},N)=M_n(x;\beta,c). -\end{equation} - -\subsubsection*{Dual Hahn $\rightarrow$ Krawtchouk} -The Krawtchouk polynomials given by (\ref{DefKrawtchouk}) can be obtained from the dual -Hahn polynomials by setting $\gamma=pt$, $\delta=(1-p)t$ and letting $t\rightarrow\infty$: -\begin{equation} -\lim_{t\rightarrow\infty}R_n(\lambda(x);pt,(1-p)t,N)=K_n(x;p,N). -\end{equation} - -\subsection*{Remark} -If we interchange the role of $x$ and $n$ in the definition (\ref{DefDualHahn}) -of the dual Hahn polynomials we obtain the Hahn polynomials given by (\ref{DefHahn}). - -\noindent -Since -$$R_n(\lambda(x);\gamma,\delta,N)=Q_x(n;\gamma,\delta,N)$$ -we obtain the dual orthogonality relation for the dual Hahn polynomials -from the orthogonality relation (\ref{OrtHahn}) for the Hahn polynomials: -\begin{eqnarray*} -& &\sum_{n=0}^N\binom{\gamma+n}{n}\binom{\delta+N-n}{N-n} R_n(\lambda(x);\gamma,\delta,N)R_n(\lambda(y);\gamma,\delta,N)\\ -& &{}=\frac{(-1)^x(x+\gamma+\delta+1)_{N+1}(\delta+1)_xx!} -{(2x+\gamma+\delta+1)(\gamma+1)_x(-N)_xN!}\delta_{xy},\quad x,y \in \{0,1,2,\ldots,N\}. -\subsection*{9.6 Dual Hahn} -\label{sec9.6} -% -\paragraph{Special values} -By \eqref{44} and \eqref{45} we have -\begin{equation} -R_n(N(N+\ga+\de+1);\ga,\de,N)=\frac{(-N-\de)_n}{(\ga+1)_n}\,. -\label{47} -\end{equation} -It follows from \eqref{95} and \eqref{45} that -\begin{equation} -R_N(x(x+\ga+\de+1);\ga,\de,N) -=\frac{(-1)^x(\de+1)_x}{(\ga+1)_x}\qquad(x=0,1,\ldots,N). -\label{101} -\end{equation} -% -\paragraph{Symmetries} -Write the weight in (9.6.2) as -\begin{equation} -w_x(\al,\be,N):=N!\,\frac{2x+\ga+\de+1}{(x+\ga+\de+1)_{N+1}}\, -\frac{(\ga+1)_x}{(\de+1)_x}\,\binom Nx. -\label{98} -\end{equation} -Then -\begin{equation} -(\de+1)_N\,w_{N-x}(\ga,\de,N)= -(-\ga-N)_N\,w_x(-\de-N-1,-\ga-N-1,N). -\label{99} -\end{equation} -Hence, by (9.6.2), -\begin{equation} -\frac{R_n((N-x)(N-x+\ga+\de+1);\ga,\de,N)}{R_n(N(N+\ga+\de+1);\ga,\de,N)} -=R_n(x(x-2N-\ga-\de-1);-N-\de-1,-N-\ga-1,N). -\label{97} -\end{equation} -Alternatively, \eqref{97} follows from (9.6.1) and -\mycite{DLMF}{(16.4.11)}. - -It follows from \eqref{96} and \eqref{45} that -\begin{equation} -\frac{R_{N-n}(x(x+\ga+\de+1);\ga,\de,N)} -{R_N(x(x+\ga+\de+1);\ga,\de,N)} -=R_n(x(x+\ga+\de+1);\de,\ga,N)\qquad(x=0,1,\ldots,N). -\label{102} -\end{equation} -% -\paragraph{Re: (9.6.11).} -The generating function (9.6.11) can be written in a more conceptual way as -\begin{equation} -(1-t)^x\,\hyp21{x-N,x+\ga+1}{-\de-N}t=\frac{N!}{(\de+1)_N}\, -\sum_{n=0}^N \om_n\,R_n(\la(x);\ga,\de,N)\,t^n, -\label{2} -\end{equation} -where -\begin{equation} -\om_n:=\binom{\ga+n}n \binom{\de+N-n}{N-n}, -\label{3} -\end{equation} -i.e., the denominator on the \RHS\ of (9.6.2). -By the duality between Hahn polynomials and dual Hahn polynomials (see \eqref{45}) the above generating function can be rewritten in -terms of Hahn polynomials: -\begin{equation} -(1-t)^n\,\hyp21{n-N,n+\al+1}{-\be-N}t=\frac{N!}{(\be+1)_N}\, -\sum_{x=0}^N w_x\,Q_n(x;\al,\be,N)\,t^x, -\label{4} -\end{equation} -where -\begin{equation} -w_x:=\binom{\al+x}x \binom{\be+N-x}{N-x}, -\label{5} -\end{equation} -i.e., the weight occurring in the orthogonality relation (9.5.2) -for Hahn polynomials. -\paragraph{Re: (9.6.15).} -There should be a closing bracket before the equality sign. -\end{eqnarray*} - -\subsection*{References} -\cite{Askey2005}, \cite{AskeyWilson85}, \cite{AtakRahmanSuslov}, \cite{AtakSuslov88}, -\cite{Ismail2005II}, \cite{Karlin61}, \cite{Koorn81}, \cite{Koorn88}, \cite{Lesky93}, -\cite{Lesky94I}, \cite{Lesky95I}, \cite{Lesky95II}, \cite{Nikiforov+}, \cite{NikiforovUvarov}, -\cite{Rahman81II}, \cite{Stanton84}, \cite{Wilson80}. - - -\section{Meixner-Pollaczek}\index{Meixner-Pollaczek polynomials} - -\par\setcounter{equation}{0} - -\subsection*{Hypergeometric representation} -\begin{equation} -\label{DefMP} -P_n^{(\lambda)}(x;\phi)=\frac{(2\lambda)_n}{n!}\expe^{in\phi}\, -\hyp{2}{1}{-n,\lambda+ix}{2\lambda}{1-\expe^{-2i\phi}}. -\end{equation} - -\subsection*{Orthogonality relation} -\begin{equation} -\label{OrtMP} -\frac{1}{2\pi}\int_{-\infty}^{\infty} -\e^{(2\phi-\pi)x}\left|\Gamma(\lambda+ix)\right|^2 -P_m^{(\lambda)}(x;\phi)P_n^{(\lambda)}(x;\phi)\,dx -{}=\frac{\Gamma(n+2\lambda)}{(2\sin\phi)^{2\lambda}n!}\,\delta_{mn}, -% \constraint{ -% $\lambda > 0$ & -% $0 < \phi < \pi$ } -\end{equation} - -\subsection*{Recurrence relation} -\begin{eqnarray} -\label{RecMP} -& &(n+1)P_{n+1}^{(\lambda)}(x;\phi)-2\left[x\sin\phi+(n+\lambda)\cos\phi\right] -P_n^{(\lambda)}(x;\phi)\nonumber\\ -& &{}\mathindent{}+(n+2\lambda-1)P_{n-1}^{(\lambda)}(x;\phi)=0. -\end{eqnarray} - -\subsection*{Normalized recurrence relation} -\begin{equation} -\label{NormRecMP} -xp_n(x)=p_{n+1}(x)-\left(\frac{n+\lambda}{\tan\phi}\right)p_n(x)+ -\frac{n(n+2\lambda-1)}{4\sin^2\phi}p_{n-1}(x), -\end{equation} -where -$$P_n^{(\lambda)}(x;\phi)=\frac{(2\sin\phi)^n}{n!}p_n(x).$$ - -\subsection*{Difference equation} -\begin{eqnarray} -\label{dvMP} -& &\e^{i\phi}(\lambda-ix)y(x+i)+2i\left[x\cos\phi-(n+\lambda)\sin\phi\right]y(x)\nonumber\\ -& &{}\mathindent{}-\e^{-i\phi}(\lambda+ix)y(x-i)=0,\quad y(x)=P_n^{(\lambda)}(x;\phi). -\end{eqnarray} - -\subsection*{Forward shift operator} -\begin{equation} -\label{shift1MPI} -P_n^{(\lambda)}(x+\textstyle\frac{1}{2}i;\phi)- -P_n^{(\lambda)}(x-\textstyle\frac{1}{2}i;\phi)=(\expe^{i\phi}-\expe^{-i\phi}) -P_{n-1}^{(\lambda+\frac{1}{2})}(x;\phi) -\end{equation} -or equivalently -\begin{equation} -\label{shift1MPII} -\frac{\delta P_n^{(\lambda)}(x;\phi)}{\delta x} -=2\sin\phi\,P_{n-1}^{(\lambda+\frac{1}{2})}(x;\phi). -\end{equation} - -\subsection*{Backward shift operator} -\begin{eqnarray} -\label{shift2MPI} -& &\e^{i\phi}(\lambda-\textstyle\frac{1}{2}-ix) -P_n^{(\lambda)}(x+\textstyle\frac{1}{2}i;\phi)+ -\e^{-i\phi}(\lambda-\textstyle\frac{1}{2}+ix) -P_n^{(\lambda)}(x-\textstyle\frac{1}{2}i;\phi)\nonumber\\ -& &{}=(n+1)P_{n+1}^{(\lambda-\frac{1}{2})}(x;\phi) -\end{eqnarray} -or equivalently -\begin{equation} -\label{shift2MPII} -\frac{\delta\left[\omega(x;\lambda,\phi)P_n^{(\lambda)}(x;\phi)\right]}{\delta x}= --(n+1)\omega(x;\lambda-\textstyle\frac{1}{2},\phi) -P_{n+1}^{(\lambda-\frac{1}{2})}(x;\phi), -\end{equation} -where -$$\omega(x;\lambda,\phi)=\Gamma(\lambda+ix)\Gamma(\lambda-ix)\e^{(2\phi-\pi)x}.$$ - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodMP} -\omega(x;\lambda,\phi)P_n^{(\lambda)}(x;\phi)=\frac{(-1)^n}{n!} -\left(\frac{\delta}{\delta x}\right)^n\left[\omega(x;\lambda+\textstyle\frac{1}{2}n,\phi)\right]. -\end{equation} - -\newpage - -\subsection*{Generating functions} -\begin{equation} -\label{GenMP1} -(1-\expe^{i\phi}t)^{-\lambda+ix}(1-\expe^{-i\phi}t)^{-\lambda-ix}= -\sum_{n=0}^{\infty}P_n^{(\lambda)}(x;\phi)t^n. -\end{equation} - -\begin{equation} -\label{GenMP2} -\e^t\,\hyp{1}{1}{\lambda+ix}{2\lambda}{(\expe^{-2i\phi}-1)t}= -\sum_{n=0}^{\infty}\frac{P_n^{(\lambda)}(x;\phi)}{(2\lambda)_n\expe^{in\phi}}t^n. -\end{equation} - -\begin{eqnarray} -\label{GenMP3} -& &(1-t)^{-\gamma}\,\hyp{2}{1}{\gamma,\lambda+ix}{2\lambda}{\frac{(1-\e^{-2i\phi})t}{t-1}}\nonumber\\ -& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n}{(2\lambda)_n}\frac{P_n^{(\lambda)}(x;\phi)}{\e^{in\phi}}t^n, -\quad\textrm{$\gamma$ arbitrary}. -\end{eqnarray} - -\subsection*{Limit relations} - -\subsubsection*{Continuous dual Hahn $\rightarrow$ Meixner-Pollaczek} -The Meixner-Pollaczek polynomials can be obtained from the continuous dual Hahn polynomials -given by (\ref{DefContDualHahn}) by the substitutions $x\rightarrow x-t$, $a=\lambda+it$, -$b=\lambda-it$ and $c=t\cot\phi$ and the limit $t\rightarrow\infty$: -$$\lim_{t\rightarrow\infty}\frac{S_n((x-t)^2;\lambda+it,\lambda-it,t\cot\phi)}{t^nn!} -=\frac{P_n^{(\lambda)}(x;\phi)}{(\sin\phi)^n}.$$ - -\subsubsection*{Continuous Hahn $\rightarrow$ Meixner-Pollaczek} -By setting $x\rightarrow x+t$, $a=\lambda-it$, $c=\lambda+it$ and $b=d=t\tan\phi$ in the -definition (\ref{DefContHahn}) of the continuous Hahn polynomials and taking the limit -$t\rightarrow\infty$ we obtain the Meixner-Pollaczek polynomials: -$$\lim_{t\rightarrow\infty}\frac{p_n(x+t;\lambda-it,t\tan\phi,\lambda+it,t\tan\phi)}{t^nn!} -=\frac{P_n^{(\lambda)}(x;\phi)}{(\cos\phi)^n}.$$ - -\newpage - -\subsubsection*{Meixner-Pollaczek $\rightarrow$ Laguerre} -The Laguerre polynomials given by (\ref{DefLaguerre}) can be obtained from the -Meixner-Pollaczek polynomials by the substitution $\lambda=\frac{1}{2}(\alpha+1)$, -$x\rightarrow -\frac{1}{2}\phi^{-1}x$ and the limit $\phi\rightarrow 0$: -\begin{equation} -\lim_{\phi\rightarrow 0} -P_n^{(\frac{1}{2}\alpha+\frac{1}{2})}(-\textstyle\frac{1}{2}\phi^{-1}x;\phi)=L_n^{(\alpha)}(x). -\end{equation} - -\subsubsection*{Meixner-Pollaczek $\rightarrow$ Hermite} -The Hermite polynomials given by (\ref{DefHermite}) are obtained from the Meixner-Pollaczek -polynomials if we substitute $x\rightarrow (\sin\phi)^{-1}(x\sqrt{\lambda}-\lambda\cos\phi)$ -and then let $\lambda\rightarrow\infty$: -\begin{equation} -\lim_{\lambda\rightarrow\infty} -\lambda^{-\frac{1}{2}n}P_n^{(\lambda)} -((\sin\phi)^{-1}(x\sqrt{\lambda}-\lambda\cos\phi);\phi)=\frac{H_n(x)}{n!}. -\end{equation} - -\subsection*{Remark} -Since we have for $k-1$ and $\beta>-1$ we have -\begin{eqnarray} -\label{OrtJacobi1} -& &\int_{-1}^1(1-x)^{\alpha}(1+x)^{\beta}P_m^{(\alpha,\beta)}(x)P_n^{(\alpha,\beta)}(x)\,dx\nonumber\\ -& &{}=\frac{2^{\alpha+\beta+1}}{2n+\alpha+\beta+1}\frac{\Gamma(n+\alpha+1)\Gamma(n+\beta+1)}{\Gamma(n+\alpha+\beta+1)n!}\,\delta_{mn}. -\end{eqnarray} -For $\alpha+\beta<-2N-1$, $\beta>-1$ and $m,n\in\{0,1,2,\ldots,N\}$ we also have -\begin{eqnarray} -\label{OrtJacobi2} -& &\int_1^{\infty}(x+1)^{\alpha}(x-1)^{\beta}P_m^{(\alpha,\beta)}(-x)P_n^{(\alpha,\beta)}(-x)\,dx\nonumber\\ -& &{}=-\frac{2^{\alpha+\beta+1}}{2n+\alpha+\beta+1}\frac{\Gamma(-n-\alpha-\beta)\Gamma(n+\alpha+\beta+1)}{\Gamma(-n-\alpha)n!}\,\delta_{mn}. -\end{eqnarray} - -\subsection*{Recurrence relation} -\begin{eqnarray} -\label{RecJacobi} -xP_n^{(\alpha,\beta)}(x)&=&\frac{2(n+1)(n+\alpha+\beta+1)}{(2n+\alpha+\beta+1)(2n+\alpha+\beta+2)}P_{n+1}^{(\alpha,\beta)}(x)\nonumber\\ -& &{}\mathindent{}+\frac{\beta^2-\alpha^2}{(2n+\alpha+\beta)(2n+\alpha+\beta+2)}P_n^{(\alpha,\beta)}(x)\nonumber\\ -& &{}\mathindent\mathindent{}+\frac{2(n+\alpha)(n+\beta)}{(2n+\alpha+\beta)(2n+\alpha+\beta+1)}P_{n-1}^{(\alpha,\beta)}(x). -\end{eqnarray} - -\subsection*{Normalized recurrence relation} -\begin{eqnarray} -\label{NormRecJacobi} -xp_n(x)&=&p_{n+1}(x)+\frac{\beta^2-\alpha^2}{(2n+\alpha+\beta)(2n+\alpha+\beta+2)}p_n(x)\nonumber\\ -& &{}\mathindent{}+\frac{4n(n+\alpha)(n+\beta)(n+\alpha+\beta)} -{(2n+\alpha+\beta-1)(2n+\alpha+\beta)^2(2n+\alpha+\beta+1)}p_{n-1}(x) -\end{eqnarray} -where -$$P_n^{(\alpha,\beta)}(x)=\frac{(n+\alpha+\beta+1)_n}{2^nn!}p_n(x).$$ - -\subsection*{Differential equation} -\begin{eqnarray} -\label{dvJacobi} -& &(1-x^2)y''(x)+\left[\beta-\alpha-(\alpha+\beta+2)x\right]y'(x)\nonumber\\ -& &{}\mathindent{}+n(n+\alpha+\beta+1)y(x)=0,\quad y(x)=P_n^{(\alpha,\beta)}(x). -\end{eqnarray} - -\subsection*{Forward shift operator} -\begin{equation} -\label{shift1Jacobi} -\frac{d}{dx}P_n^{(\alpha,\beta)}(x)=\frac{n+\alpha+\beta+1}{2}P_{n-1}^{(\alpha+1,\beta+1)}(x). -\end{equation} - -\subsection*{Backward shift operator} -\begin{eqnarray} -\label{shift2JacobiI} -& &(1-x^2)\frac{d}{dx}P_n^{(\alpha,\beta)}(x)+ -\left[(\beta-\alpha)-(\alpha+\beta)x\right]P_n^{(\alpha,\beta)}(x)\nonumber\\ -& &{}=-2(n+1)P_{n+1}^{(\alpha-1,\beta-1)}(x) -\end{eqnarray} -or equivalently -\begin{eqnarray} -\label{shift2JacobiII} -& &\frac{d}{dx}\left[(1-x)^\alpha(1+x)^\beta P_n^{(\alpha,\beta)}(x)\right]\nonumber\\ -& &{}=-2(n+1)(1-x)^{\alpha-1}(1+x)^{\beta-1}P_{n+1}^{(\alpha-1,\beta-1)}(x). -\end{eqnarray} - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodJacobi} -(1-x)^{\alpha}(1+x)^{\beta}P_n^{(\alpha,\beta)}(x)= -\frac{(-1)^n}{2^nn!}\left(\frac{d}{dx}\right)^n -\left[(1-x)^{n+\alpha}(1+x)^{n+\beta}\right]. -\end{equation} - -\subsection*{Generating functions} -\begin{equation} -\label{GenJacobi1} -\frac{2^{\alpha+\beta}}{R(1+R-t)^{\alpha}(1+R+t)^{\beta}}= -\sum_{n=0}^{\infty}P_n^{(\alpha,\beta)}(x)t^n,\quad R=\sqrt{1-2xt+t^2}. -\end{equation} - -\begin{eqnarray} -\label{GenJacobi2} -& &\hyp{0}{1}{-}{\alpha+1}{\frac{(x-1)t}{2}}\,\hyp{0}{1}{-}{\beta+1}{\frac{(x+1)t}{2}}\nonumber\\ -& &{}=\sum_{n=0}^{\infty}\frac{P_n^{(\alpha,\beta)}(x)}{(\alpha+1)_n(\beta+1)_n}t^n. -\end{eqnarray} - -\begin{eqnarray} -\label{GenJacobi3} -& &(1-t)^{-\alpha-\beta-1}\,\hyp{2}{1}{\frac{1}{2}(\alpha+\beta+1),\frac{1}{2}(\alpha+\beta+2)} -{\alpha+1}{\frac{2(x-1)t}{(1-t)^2}}\nonumber\\ -& &{}=\sum_{n=0}^{\infty}\frac{(\alpha+\beta+1)_n}{(\alpha+1)_n}P_n^{(\alpha,\beta)}(x)t^n. -\end{eqnarray} - -\begin{eqnarray} -\label{GenJacobi4} -& &(1+t)^{-\alpha-\beta-1}\,\hyp{2}{1}{\frac{1}{2}(\alpha+\beta+1),\frac{1}{2}(\alpha+\beta+2)} -{\beta+1}{\frac{2(x+1)t}{(1+t)^2}}\nonumber\\ -& &{}=\sum_{n=0}^{\infty}\frac{(\alpha+\beta+1)_n}{(\beta+1)_n}P_n^{(\alpha,\beta)}(x)t^n. -\end{eqnarray} - -\begin{eqnarray} -\label{GenJacobi5} -& &\hyp{2}{1}{\gamma,\alpha+\beta+1-\gamma}{\alpha+1}{\frac{1-R-t}{2}}\, -\hyp{2}{1}{\gamma,\alpha+\beta+1-\gamma}{\beta+1}{\frac{1-R+t}{2}}\nonumber\\ -& &{}=\sum_{n=0}^{\infty} -\frac{(\gamma)_n(\alpha+\beta+1-\gamma)_n}{(\alpha+1)_n(\beta+1)_n}P_n^{(\alpha,\beta)}(x)t^n, -\quad R=\sqrt{1-2xt+t^2} -\end{eqnarray} -with $\gamma$ arbitrary. - -\subsection*{Limit relations} - -\subsubsection*{Wilson $\rightarrow$ Jacobi} -The Jacobi polynomials can be found from the Wilson polynomials given by (\ref{DefWilson}) by -substituting $a=b=\frac{1}{2}(\alpha+1)$, $c=\frac{1}{2}(\beta+1)+it$, -$d=\frac{1}{2}(\beta+1)-it$ and $x\rightarrow t\sqrt{\frac{1}{2}(1-x)}$ in the -definition (\ref{DefWilson}) of the Wilson polynomials and taking the limit -$t\rightarrow\infty$. In fact we have -$$\lim_{t\rightarrow\infty} -\frac{W_n(\frac{1}{2}(1-x)t^2;\frac{1}{2}(\alpha+1), -\frac{1}{2}(\alpha+1),\frac{1}{2}(\beta+1)+it,\frac{1}{2}(\beta+1)-it)} -{t^{2n}n!}=P_n^{(\alpha,\beta)}(x).$$ - -\subsubsection*{Continuous Hahn $\rightarrow$ Jacobi} -The Jacobi polynomials follow from the continuous Hahn polynomials given by (\ref{DefContHahn}) -by using the substitution $x\rightarrow \frac{1}{2}xt$, $a=\frac{1}{2}(\alpha+1-it)$, -$b=\frac{1}{2}(\beta+1+it)$, $c=\frac{1}{2}(\alpha+1+it)$ and $d=\frac{1}{2}(\beta+1-it)$ -in (\ref{DefContHahn}), division by $t^n$ and the limit $t\rightarrow\infty$: -$$\lim_{t\rightarrow\infty} -\frac{p_n(\frac{1}{2}xt;\frac{1}{2}(\alpha+1-it),\frac{1}{2}(\beta+1+it), -\frac{1}{2}(\alpha+1+it),\frac{1}{2}(\beta+1-it))}{t^n}=P_n^{(\alpha,\beta)}(x).$$ - -\subsubsection*{Hahn $\rightarrow$ Jacobi} -To find the Jacobi polynomials from the Hahn polynomials given by (\ref{DefHahn}) we take -$x\rightarrow Nx$ in (\ref{DefHahn}) and let $N\rightarrow\infty$. In fact we have -$$\lim_{N\rightarrow\infty} -Q_n(Nx;\alpha,\beta,N)=\frac{P_n^{(\alpha,\beta)}(1-2x)}{P_n^{(\alpha,\beta)}(1)}.$$ - -\subsubsection*{Jacobi $\rightarrow$ Laguerre} -The Laguerre polynomials given by (\ref{DefLaguerre}) can be obtained from the Jacobi -polynomials by setting $x\rightarrow 1-2\beta^{-1}x$ and then the limit $\beta\rightarrow\infty$: -\begin{equation} -\lim_{\beta\rightarrow\infty} -P_n^{(\alpha,\beta)}(1-2\beta^{-1}x)=L_n^{(\alpha)}(x). -\end{equation} - -\subsubsection*{Jacobi $\rightarrow$ Bessel} -The Bessel polynomials given by (\ref{DefBessel}) are obtained from the Jacobi polynomials -if we take $\beta=a-\alpha$ and let $\alpha\rightarrow-\infty$: -\begin{equation} -\lim_{\alpha\rightarrow-\infty} -\frac{P_n^{(\alpha,a-\alpha)}(1+\alpha x)}{P_n^{(\alpha,a-\alpha)}(1)}=y_n(x;a). -\end{equation} - -\subsubsection*{Jacobi $\rightarrow$ Hermite} -The Hermite polynomials given by (\ref{DefHermite}) follow from the Jacobi polynomials -by taking $\beta=\alpha$ and letting $\alpha\rightarrow\infty$ in the following way: -\begin{equation} -\lim_{\alpha\rightarrow\infty} -\alpha^{-\frac{1}{2}n}P_n^{(\alpha,\alpha)}(\alpha^{-\frac{1}{2}}x)=\frac{H_n(x)}{2^nn!}. -\end{equation} - -\subsection*{Remarks} -The definition (\ref{DefJacobi}) of the Jacobi polynomials can also be written as: -$$P_n^{(\alpha,\beta)}(x)=\frac{1}{n!}\sum_{k=0}^n\frac{(-n)_k}{k!} -(n+\alpha+\beta+1)_k(\alpha+k+1)_{n-k}\left(\frac{1-x}{2}\right)^k.$$ -In this way the Jacobi polynomials can also be seen as polynomials in the parameters $\alpha$ -and $\beta$. Therefore they can be defined for all $\alpha$ and $\beta$. Then we have the -following connection with the Meixner polynomials given by (\ref{DefMeixner}): -$$\frac{(\beta)_n}{n!}M_n(x;\beta,c)=P_n^{(\beta-1,-n-\beta-x)}((2-c)c^{-1}).$$ - -\noindent -The Jacobi polynomials are related to the pseudo Jacobi polynomials defined by -(\ref{DefPseudoJacobi}) in the following way: -$$P_n(x;\nu,N)=\frac{(-2i)^nn!}{(n-2N-1)_n}P_n^{(-N-1+i\nu,-N-1-i\nu)}(ix).$$ - -\noindent -The Jacobi polynomials are also related to the Gegenbauer (or ultraspherical) polynomials -given by (\ref{DefGegenbauer}) by the quadratic transformations: -$$C_{2n}^{(\lambda)}(x)=\frac{(\lambda)_n}{(\frac{1}{2})_n} -P_n^{(\lambda-\frac{1}{2},-\frac{1}{2})}(2x^2-1)$$ -and -$$C_{2n+1}^{(\lambda)}(x)=\frac{(\lambda)_{n+1}}{(\frac{1}{2})_{n+1}} -\subsection*{9.8 Jacobi} -\label{sec9.8} -% -\paragraph{Orthogonality relation} -Write the \RHS\ of (9.8.2) as $h_n\,\de_{m,n}$. Then -\begin{equation} -\begin{split} -&\frac{h_n}{h_0}= -\frac{n+\al+\be+1}{2n+\al+\be+1}\, -\frac{(\al+1)_n(\be+1)_n}{(\al+\be+2)_n\,n!}\,,\quad -h_0=\frac{2^{\al+\be+1}\Ga(\al+1)\Ga(\be+1)}{\Ga(\al+\be+2)}\,,\sLP -&\frac{h_n}{h_0\,(P_n^{(\al,\be)}(1))^2}= -\frac{n+\al+\be+1}{2n+\al+\be+1}\, -\frac{(\be+1)_n\,n!}{(\al+1)_n\,(\al+\be+2)_n}\,. -\end{split} -\label{60} -\end{equation} - -In (9.8.3) the numerator factor $\Ga(n+\al+\be+1)$ in the last line should be -$\Ga(\be+1)$. When thus corrected, (9.8.3) can be rewritten as: -\begin{equation} -\begin{split} -&\int_1^\iy P_m^{(\al,\be)}(x)\,P_n^{(\al,\be)}(x)\,(x-1)^\al (x+1)^\be\,dx=h_n\,\de_{m,n}\,,\\ -&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad-1-\be>\al>-1,\quad m,n<-\thalf(\al+\be+1),\\ -&\frac{h_n}{h_0}= -\frac{n+\al+\be+1}{2n+\al+\be+1}\, -\frac{(\al+1)_n(\be+1)_n}{(\al+\be+2)_n\,n!}\,,\quad -h_0=\frac{2^{\al+\be+1}\Ga(\al+1)\Ga(-\al-\be-1)}{\Ga(-\be)}\,. -\end{split} -\label{122} -\end{equation} - -% -\paragraph{Symmetry} -\begin{equation} -P_n^{(\al,\be)}(-x)=(-1)^n\,P_n^{(\be,\al)}(x). -\label{48} -\end{equation} -Use (9.8.2) and (9.8.5b) or see \mycite{DLMF}{Table 18.6.1}. -% -\paragraph{Special values} -\begin{equation} -P_n^{(\al,\be)}(1)=\frac{(\al+1)_n}{n!}\,,\quad -P_n^{(\al,\be)}(-1)=\frac{(-1)^n(\be+1)_n}{n!}\,,\quad -\frac{P_n^{(\al,\be)}(-1)}{P_n^{(\al,\be)}(1)}=\frac{(-1)^n(\be+1)_n}{(\al+1)_n}\,. -\label{50} -\end{equation} -Use (9.8.1) and \eqref{48} or see \mycite{DLMF}{Table 18.6.1}. -% -\paragraph{Generating functions} -Formula (9.8.15) was first obtained by Brafman \myciteKLS{109}. -% -\paragraph{Bilateral generating functions} -For $0\le r<1$ and $x,y\in[-1,1]$ we have in terms of $F_4$ (see~\eqref{62}): -\begin{align} -&\sum_{n=0}^\iy\frac{(\al+\be+1)_n\,n!}{(\al+1)_n(\be+1)_n}\,r^n\, -P_n^{(\al,\be)}(x)\,P_n^{(\al,\be)}(y) -=\frac1{(1+r)^{\al+\be+1}} -\nonumber\\ -&\qquad\quad\times F_4\Big(\thalf(\al+\be+1),\thalf(\al+\be+2);\al+1,\be+1; -\frac{r(1-x)(1-y)}{(1+r)^2},\frac{r(1+x)(1+y)}{(1+r)^2}\Big), -\label{58}\sLP -&\sum_{n=0}^\iy\frac{2n+\al+\be+1}{n+\al+\be+1} -\frac{(\al+\be+2)_n\,n!}{(\al+1)_n(\be+1)_n}\,r^n\, -P_n^{(\al,\be)}(x)\,P_n^{(\al,\be)}(y) -=\frac{1-r}{(1+r)^{\al+\be+2}}\nonumber\\ -&\qquad\quad\times F_4\Big(\thalf(\al+\be+2),\thalf(\al+\be+3);\al+1,\be+1; -\frac{r(1-x)(1-y)}{(1+r)^2},\frac{r(1+x)(1+y)}{(1+r)^2}\Big). -\label{59} -\end{align} -Formulas \eqref{58} and \eqref{59} were first -given by Bailey \myciteKLS{91}{(2.1), (2.3)}. -See Stanton \myciteKLS{485} for a shorter proof. -(However, in the second line of -\myciteKLS{485}{(1)} $z$ and $Z$ should be interchanged.)$\;$ -As observed in Bailey \myciteKLS{91}{p.10}, \eqref{59} follows -from \eqref{58} -by applying the operator $r^{-\half(\al+\be-1)}\,\frac d{dr}\circ r^{\half(\al+\be+1)}$ -to both sides of \eqref{58}. -In view of \eqref{60}, formula \eqref{59} is the Poisson kernel for Jacobi -polynomials. The \RHS\ of \eqref{59} makes clear that this kernel is positive. -See also the discussion in Askey \myciteKLS{46}{following (2.32)}. -% -\paragraph{Quadratic transformations} -\begin{align} -\frac{C_{2n}^{(\al+\half)}(x)}{C_{2n}^{(\al+\half)}(1)} -=\frac{P_{2n}^{(\al,\al)}(x)}{P_{2n}^{(\al,\al)}(1)} -&=\frac{P_n^{(\al,-\half)}(2x^2-1)}{P_n^{(\al,-\half)}(1)}\,, -\label{51}\\ -\frac{C_{2n+1}^{(\al+\half)}(x)}{C_{2n+1}^{(\al+\half)}(1)} -=\frac{P_{2n+1}^{(\al,\al)}(x)}{P_{2n+1}^{(\al,\al)}(1)} -&=\frac{x\,P_n^{(\al,\half)}(2x^2-1)}{P_n^{(\al,\half)}(1)}\,. -\label{52} -\end{align} -See p.221, Remarks, last two formulas together with \eqref{50} and \eqref{49}. -Or see \mycite{DLMF}{(18.7.13), (18.7.14)}. -% -\paragraph{Differentiation formulas} -Each differentiation formula is given in two equivalent forms. -\begin{equation} -\begin{split} -\frac d{dx}\left((1-x)^\al P_n^{(\al,\be)}(x)\right)&= --(n+\al)\,(1-x)^{\al-1} P_n^{(\al-1,\be+1)}(x),\\ -\left((1-x)\frac d{dx}-\al\right)P_n^{(\al,\be)}(x)&= --(n+\al)\,P_n^{(\al-1,\be+1)}(x). -\end{split} -\label{68} -\end{equation} -% -\begin{equation} -\begin{split} -\frac d{dx}\left((1+x)^\be P_n^{(\al,\be)}(x)\right)&= -(n+\be)\,(1+x)^{\be-1} P_n^{(\al+1,\be-1)}(x),\\ -\left((1+x)\frac d{dx}+\be\right)P_n^{(\al,\be)}(x)&= -(n+\be)\,P_n^{(\al+1,\be-1)}(x). -\end{split} -\label{69} -\end{equation} -Formulas \eqref{68} and \eqref{69} follow from -\mycite{DLMF}{(15.5.4), (15.5.6)} -together with (9.8.1). They also follow from each other by \eqref{48}. -% -\paragraph{Generalized Gegenbauer polynomials} -These are defined by -\begin{equation} -S_{2m}^{(\al,\be)}(x):=\const P_m^{(\al,\be)}(2x^2-1),\qquad -S_{2m+1}^{(\al,\be)}(x):=\const x\,P_m^{(\al,\be+1)}(2x^2-1) -\label{70} -\end{equation} -in the notation of \myciteKLS{146}{p.156} -(see also \cite{K27}), while \cite[Section 1.5.2]{K26} -has $C_n^{(\la,\mu)}(x)=\const\allowbreak\times S_n^{(\la-\half,\mu-\half)}(x)$. -For $\al,\be>-1$ we have the orthogonality relation -\begin{equation} -\int_{-1}^1 S_m^{(\al,\be)}(x)\,S_n^{(\al,\be)}(x)\,|x|^{2\be+1}(1-x^2)^\al\,dx -=0\qquad(m\ne n). -\label{71} -\end{equation} -For $\be=\al-1$ generalized Gegenbauer polynomials are limit cases of -continuous $q$-ultraspherical polynomials, see \eqref{176}. - -If we define the {\em Dunkl operator} $T_\mu$ by -\begin{equation} -(T_\mu f)(x):=f'(x)+\mu\,\frac{f(x)-f(-x)}x -\label{72} -\end{equation} -and if we choose the constants in \eqref{70} as -\begin{equation} -S_{2m}^{(\al,\be)}(x)=\frac{(\al+\be+1)_m}{(\be+1)_m}\, P_m^{(\al,\be)}(2x^2-1),\quad -S_{2m+1}^{(\al,\be)}(x)=\frac{(\al+\be+1)_{m+1}}{(\be+1)_{m+1}}\, -x\,P_m^{(\al,\be+1)}(2x^2-1) -\label{73} -\end{equation} -then (see \cite[(1.6)]{K5}) -\begin{equation} -T_{\be+\half}S_n^{(\al,\be)}=2(\al+\be+1)\,S_{n-1}^{(\al+1,\be)}. -\label{74} -\end{equation} -Formula \eqref{74} with \eqref{73} substituted gives rise to two -differentiation formulas involving Jacobi polynomials which are equivalent to -(9.8.7) and \eqref{69}. - -Composition of \eqref{74} with itself gives -\[ -T_{\be+\half}^2S_n^{(\al,\be)}=4(\al+\be+1)(\al+\be+2)\,S_{n-2}^{(\al+2,\be)}, -\] -which is equivalent to the composition of (9.8.7) and \eqref{69}: -\begin{equation} -\left(\frac{d^2}{dx^2}+\frac{2\be+1}x\,\frac d{dx}\right)P_n^{(\al,\be)}(2x^2-1) -=4(n+\al+\be+1)(n+\be)\,P_{n-1}^{(\al+2,\be)}(2x^2-1). -\label{75} -\end{equation} -Formula \eqref{75} was also given in \myciteKLS{322}{(2.4)}. -xP_n^{(\lambda-\frac{1}{2},\frac{1}{2})}(2x^2-1).$$ - -\subsection*{References} -\cite{Abram}, \cite{Ahmed+82}, \cite{Allaway89}, \cite{NAlSalam66}, \cite{AlSalam64}, -\cite{AlSalamChihara72}, \cite{AndrewsAskey85}, \cite{AndrewsAskeyRoy}, \cite{Askey68}, -\cite{Askey70}, \cite{Askey72}, \cite{Askey73}, \cite{Askey74}, \cite{Askey75}, \cite{Askey78}, -\cite{Askey89I}, \cite{Askey2005}, \cite{AskeyFitch}, \cite{AskeyGasper71I}, \cite{AskeyGasper71II}, -\cite{AskeyGasper76}, \cite{AskeyWainger}, \cite{AskeyWilson85}, \cite{Bailey38}, \cite{Brafman51}, -\cite{Brown}, \cite{Carlitz61II}, \cite{Carlitz67}, \cite{ChenIsmail91}, \cite{Chihara78}, -\cite{Chow+}, \cite{Ciesielski}, \cite{Cooper+}, \cite{DetteStudden92}, \cite{DetteStudden95}, -\cite{DijksmaKoorn}, \cite{DimitrovRafaeli}, \cite{DimitrovRodrigues}, \cite{Doha2002I}, -\cite{Doha2003I}, \cite{Doha2004II}, \cite{Dunkl84}, \cite{ElbertLaforgia87II}, -\cite{ElbertLaforgiaRodono}, \cite{Erdelyi+}, \cite{Faldey}, \cite{FoataLeroux}, -\cite{Gasper69}, \cite{Gasper70I}, \cite{Gasper70II}, \cite{Gasper71I}, \cite{Gasper71II}, -\cite{Gasper72I}, \cite{Gasper72II}, \cite{Gasper73I}, \cite{Gasper73II}, \cite{Gasper74}, -\cite{Gasper77}, \cite{Gatteschi87}, \cite{Gautschi2008}, \cite{Gautschi2009I}, -\cite{Gautschi2009II}, \cite{GautschiLeopardi}, \cite{GawronskiShawyer}, \cite{Godoy+}, -\cite{Grad}, \cite{Hajmirzaahmad94}, \cite{HartmannStephan}, \cite{Horton}, \cite{Ismail74}, -\cite{Ismail77}, \cite{Ismail96}, \cite{Ismail2005II}, \cite{IsmailLi}, -\cite{IsmailMassonRahman}, \cite{Kochneff97I}, \cite{Koekoek99}, \cite{Koekoek2000}, -\cite{Koelink96II}, \cite{Koorn73}, \cite{Koorn74}, \cite{Koorn75}, \cite{Koorn77II}, -\cite{Koorn78}, \cite{Koorn72}, \cite{Koorn85}, \cite{Koorn88}, \cite{KuijlaarsMartinez}, -\cite{Kuijlaars+}, \cite{LabelleYehI}, \cite{LabelleYehII}, \cite{Laine}, \cite{Lesky95II}, -\cite{Lesky96}, \cite{Li96}, \cite{Li97}, \cite{LopezTemme2004}, \cite{Luke}, \cite{Martinez}, -\cite{Mathai}, \cite{Meijer}, \cite{Miller89}, \cite{MoakSaffVarga}, \cite{Nikiforov+}, -\cite{NikiforovUvarov}, \cite{Olver}, \cite{Prasad}, \cite{Rahman76I}, \cite{Rahman76II}, -\cite{Rahman77}, \cite{Rahman81I}, \cite{RahmanShah}, \cite{Rainville}, \cite{Rusev}, -\cite{Shi}, \cite{Srivastava69II}, \cite{Srivastava71}, \cite{Srivastava82}, -\cite{SrivastavaSinghal}, \cite{Stanton80I}, \cite{Stanton90}, \cite{Szego75}, \cite{Temme}, -\cite{Vertesi}, \cite{Wimp87}, \cite{WongZhang94}, \cite{WongZhang2006}, \cite{Zarzo+}, -\cite{Zayed}. - -\section*{Special cases} - -\subsection{Gegenbauer / Ultraspherical}\index{Gegenbauer polynomials} -\index{Ultraspherical polynomials} - -\par - -\subsection*{Hypergeometric representation} -The Gegenbauer (or ultraspherical) polynomials are Jacobi polynomials with -$\alpha=\beta=\lambda-\frac{1}{2}$ and another normalization: -\begin{eqnarray} -\label{DefGegenbauer} -C_n^{(\lambda)}(x)&=&\frac{(2\lambda)_n}{(\lambda+\frac{1}{2})_n} -P_n^{(\lambda-\frac{1}{2},\lambda-\frac{1}{2})}(x)\nonumber\\ -&=&\frac{(2\lambda)_n}{n!}\,\hyp{2}{1}{-n,n+2\lambda} -{\lambda+\frac{1}{2}}{\frac{1-x}{2}},\quad\lambda\neq 0. -\end{eqnarray} - -\subsection*{Orthogonality relation} -\begin{eqnarray} -\label{OrtGegenbauer} -& &\int_{-1}^1(1-x^2)^{\lambda-\frac{1}{2}}C_m^{(\lambda)}(x)C_n^{(\lambda)}(x)\,dx\nonumber\\ -& &{}=\frac{\pi\Gamma(n+2\lambda)2^{1-2\lambda}}{\left\{\Gamma(\lambda)\right\}^2(n+\lambda)n!}\,\delta_{mn}, -\quad\lambda>-\frac{1}{2}\quad\lambda\neq 0. -\end{eqnarray} - -\subsection*{Recurrence relation} -\begin{equation} -\label{RecGegenbauer} -2(n+\lambda)xC_n^{(\lambda)}(x)=(n+1)C_{n+1}^{(\lambda)}(x)+(n+2\lambda-1)C_{n-1}^{(\lambda)}(x). -\end{equation} - -\subsection*{Normalized recurrence relation} -\begin{equation} -\label{NormRecGegenbauer} -xp_n(x)=p_{n+1}(x)+\frac{n(n+2\lambda-1)}{4(n+\lambda-1)(n+\lambda)}p_{n-1}(x), -\end{equation} -where -$$C_n^{(\lambda)}(x)=\frac{2^n(\lambda)_n}{n!}p_n(x).$$ - -\subsection*{Differential equation} -\begin{equation} -\label{dvGegenbauer} -(1-x^2)y''(x)-(2\lambda+1)xy'(x)+n(n+2\lambda)y(x)=0,\quad y(x)=C_n^{(\lambda)}(x). -\end{equation} - -\subsection*{Forward shift operator} -\begin{equation} -\label{shift1Gegenbauer} -\frac{d}{dx}C_n^{(\lambda)}(x)=2\lambda C_{n-1}^{(\lambda+1)}(x). -\end{equation} - -\subsection*{Backward shift operator} -\begin{equation} -\label{shift2GegenbauerI} -(1-x^2)\frac{d}{dx}C_n^{(\lambda)}(x)+(1-2\lambda)xC_n^{(\lambda)}(x)= --\frac{(n+1)(2\lambda+n-1)}{2(\lambda-1)} C_{n+1}^{(\lambda-1)}(x) -\end{equation} -or equivalently -\begin{eqnarray} -\label{shift2GegenbauerII} -& &\frac{d}{dx}\left[(1-x^2)^{\lambda-\textstyle\frac{1}{2}}C_n^{(\lambda)}(x)\right]\nonumber\\ -& &{}=-\frac{(n+1)(2\lambda+n-1)}{2(\lambda-1)}(1-x^2)^{\lambda-\textstyle\frac{3}{2}}C_{n+1}^{(\lambda-1)}(x). -\end{eqnarray} - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodGegenbauer} -(1-x^2)^{\lambda-\frac{1}{2}}C_n^{(\lambda)}(x)= -\frac{(2\lambda)_n(-1)^n}{(\lambda+\frac{1}{2})_n2^nn!}\left(\frac{d}{dx}\right)^n -\left[(1-x^2)^{\lambda+n-\frac{1}{2}}\right]. -\end{equation} - -\subsection*{Generating functions} -\begin{equation} -\label{GenGegenbauer1} -(1-2xt+t^2)^{-\lambda}=\sum_{n=0}^{\infty}C_n^{(\lambda)}(x)t^n. -\end{equation} - -\begin{equation} -\label{GenGegenbauer2} -R^{-1}\left(\frac{1+R-xt}{2}\right)^{\frac{1}{2}-\lambda}=\sum_{n=0}^{\infty} -\frac{(\lambda+\frac{1}{2})_n}{(2\lambda)_n}C_n^{(\lambda)}(x)t^n, -\quad R=\sqrt{1-2xt+t^2}. -\end{equation} - -\begin{equation} -\label{GenGegenbauer3} -\hyp{0}{1}{-}{\lambda+\frac{1}{2}}{\frac{(x-1)t}{2}}\, -\hyp{0}{1}{-}{\lambda+\frac{1}{2}}{\frac{(x+1)t}{2}} -=\sum_{n=0}^{\infty}\frac{C_n^{(\lambda)}(x)} -{(2\lambda)_n(\lambda+\frac{1}{2})_n}t^n. -\end{equation} - -\begin{equation} -\label{GenGegenbauer4} -\e^{xt}\,\hyp{0}{1}{-}{\lambda+\frac{1}{2}}{\frac{(x^2-1)t^2}{4}}= -\sum_{n=0}^{\infty}\frac{C_n^{(\lambda)}(x)}{(2\lambda)_n}t^n. -\end{equation} - -\begin{eqnarray} -\label{GenGegenbauer5} -& &\hyp{2}{1}{\gamma,2\lambda-\gamma}{\lambda+\frac{1}{2}}{\frac{1-R-t}{2}}\, -\hyp{2}{1}{\gamma,2\lambda-\gamma}{\lambda+\frac{1}{2}}{\frac{1-R+t}{2}}\nonumber\\ -& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n(2\lambda-\gamma)_n} -{(2\lambda)_n(\lambda+\frac{1}{2})_n}C_n^{(\lambda)}(x)t^n, -\quad R=\sqrt{1-2xt+t^2},\quad\textrm{$\gamma$ arbitrary}. -\end{eqnarray} - -\begin{eqnarray} -\label{GenGegenbauer6} -& &(1-xt)^{-\gamma}\,\hyp{2}{1}{\frac{1}{2}\gamma,\frac{1}{2}\gamma+\frac{1}{2}} -{\lambda+\frac{1}{2}}{\frac{(x^2-1)t^2}{(1-xt)^2}}\nonumber\\ -& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n}{(2\lambda)_n}C_n^{(\lambda)}(x)t^n, -\quad\textrm{$\gamma$ arbitrary}. -\end{eqnarray} - -\subsection*{Limit relation} - -\subsubsection*{Gegenbauer / Ultraspherical $\rightarrow$ Hermite} -The Hermite polynomials given by (\ref{DefHermite}) follow from the Gegenbauer (or -ultraspherical) polynomials by taking $\lambda=\alpha+\frac{1}{2}$ and letting -$\alpha\rightarrow\infty$ in the following way: -\begin{equation} -\lim_{\alpha\rightarrow\infty} -\alpha^{-\frac{1}{2}n}C_n^{(\alpha+\frac{1}{2})}(\alpha^{-\frac{1}{2}}x)=\frac{H_n(x)}{n!}. -\end{equation} - -\subsection*{Remarks} -The case $\lambda=0$ needs another normalization. In that case we have the -Chebyshev polynomials of the first kind described in the next subsection. - -\noindent -The Gegenbauer (or ultraspherical) polynomials are related to the Jacobi polynomials -given by (\ref{DefJacobi}) by the quadratic transformations: -$$C_{2n}^{(\lambda)}(x)=\frac{(\lambda)_n}{(\frac{1}{2})_n} -P_n^{(\lambda-\frac{1}{2},-\frac{1}{2})}(2x^2-1)$$ -and -$$C_{2n+1}^{(\lambda)}(x)=\frac{(\lambda)_{n+1}}{(\frac{1}{2})_{n+1}} -\subsection*{9.8.1 Gegenbauer / Ultraspherical} -\label{sec9.8.1} -% -\paragraph{Notation} -Here the Gegenbauer polynomial is denoted by $C_n^\la$ instead of $C_n^{(\la)}$. -% -\paragraph{Orthogonality relation} -Write the \RHS\ of (9.8.20) as $h_n\,\de_{m,n}$. Then -\begin{equation} -\frac{h_n}{h_0}= -\frac\la{\la+n}\,\frac{(2\la)_n}{n!}\,,\quad -h_0=\frac{\pi^\half\,\Ga(\la+\thalf)}{\Ga(\la+1)},\quad -\frac{h_n}{h_0\,(C_n^\la(1))^2}= -\frac\la{\la+n}\,\frac{n!}{(2\la)_n}\,. -\label{61} -\end{equation} -% -\paragraph{Hypergeometric representation} -Beside (9.8.19) we have also -\begin{equation} -C_n^\lambda(x)=\sum_{\ell=0}^{\lfloor n/2\rfloor}\frac{(-1)^{\ell}(\lambda)_{n-\ell}} -{\ell!\;(n-2\ell)!}\,(2x)^{n-2\ell} -=(2x)^{n}\,\frac{(\lambda)_{n}}{n!}\, -\hyp21{-\thalf n,-\thalf n+\thalf}{1-\la-n}{\frac1{x^2}}. -\label{57} -\end{equation} -See \mycite{DLMF}{(18.5.10)}. -% -\paragraph{Special value} -\begin{equation} -C_n^{\la}(1)=\frac{(2\la)_n}{n!}\,. -\label{49} -\end{equation} -Use (9.8.19) or see \mycite{DLMF}{Table 18.6.1}. -% -\paragraph{Expression in terms of Jacobi} -% -\begin{equation} -\frac{C_n^\la(x)}{C_n^\la(1)}= -\frac{P_n^{(\la-\half,\la-\half)}(x)}{P_n^{(\la-\half,\la-\half)}(1)}\,,\qquad -C_n^\la(x)=\frac{(2\la)_n}{(\la+\thalf)_n}\,P_n^{(\la-\half,\la-\half)}(x). -\label{65} -\end{equation} -% -\paragraph{Re: (9.8.21)} -By iteration of recurrence relation (9.8.21): -\begin{multline} -x^2 C_n^\la(x)= -\frac{(n+1)(n+2)}{4(n+\la)(n+\la+1)}\,C_{n+2}^\la(x)+ -\frac{n^2+2n\la+\la-1}{2(n+\la-1)(n+\la+1)}\,C_n^\la(x)\\ -+\frac{(n+2\la-1)(n+2\la-2)}{4(n+\la)(n+\la-1)}\,C_{n-2}^\la(x). -\label{6} -\end{multline} -% -\paragraph{Bilateral generating functions} -\begin{multline} -\sum_{n=0}^\iy\frac{n!}{(2\la)_n}\,r^n\,C_n^\la(x)\,C_n^\la(y) -=\frac1{(1-2rxy+r^2)^\la}\,\hyp21{\thalf\la,\thalf(\la+1)}{\la+\thalf} -{\frac{4r^2(1-x^2)(1-y^2)}{(1-2rxy+r^2)^2}}\\ -(r\in(-1,1),\;x,y\in[-1,1]). -\label{66} -\end{multline} -For the proof put $\be:=\al$ in \eqref{58}, then use \eqref{63} and \eqref{65}. -The Poisson kernel for Gegenbauer polynomials can be derived in a similar way -from \eqref{59}, or alternatively by applying the operator -$r^{-\la+1}\frac d{dr}\circ r^\la$ to both sides of \eqref{66}: -\begin{multline} -\sum_{n=0}^\iy\frac{\la+n}\la\,\frac{n!}{(2\la)_n}\,r^n\,C_n^\la(x)\,C_n^\la(y) -=\frac{1-r^2}{(1-2rxy+r^2)^{\la+1}}\\ -\times\hyp21{\thalf(\la+1),\thalf(\la+2)}{\la+\thalf} -{\frac{4r^2(1-x^2)(1-y^2)}{(1-2rxy+r^2)^2}}\qquad -(r\in(-1,1),\;x,y\in[-1,1]). -\label{67} -\end{multline} -Formula \eqref{67} was obtained by Gasper \& Rahman \myciteKLS{234}{(4.4)} -as a limit case of their formula for the Poisson kernel for continuous -$q$-ultraspherical polynomials. -% -\paragraph{Trigonometric expansions} -By \mycite{DLMF}{(18.5.11), (15.8.1)}: -\begin{align} -C_n^{\la}(\cos\tha) -&=\sum_{k=0}^n\frac{(\la)_k(\la)_{n-k}}{k!\,(n-k)!}\,e^{i(n-2k)\tha} -=e^{in\tha}\frac{(\la)_n}{n!}\, -\hyp21{-n,\la}{1-\la-n}{e^{-2i\tha}}\label{103}\\ -&=\frac{(\la)_n}{2^\la n!}\, -e^{-\half i\la\pi}e^{i(n+\la)\tha}\,(\sin\tha)^{-\la}\, -\hyp21{\la,1-\la}{1-\la-n}{\frac{i e^{-i\tha}}{2\sin\tha}}\label{104}\\ -&=\frac{(\la)_n}{n!}\,\sum_{k=0}^\iy\frac{(\la)_k(1-\la)_k}{(1-\la-n)_k k!}\, -\frac{\cos((n-k+\la)\tha+\thalf(k-\la)\pi)}{(2\sin\tha)^{k+\la}}\,.\label{105} -\end{align} -In \eqref{104} and \eqref{105} we require that -$\tfrac16\pi<\tha<\tfrac56\pi$. Then the convergence is absolute for $\la>\thalf$ -and conditional for $0<\la\le\thalf$. - -By \mycite{DLMF}{(14.13.1), (14.3.21), (15.8.1)]}: -\begin{align} -C_n^\la(\cos\tha)&=\frac{2\Ga(\la+\thalf)}{\pi^\half\Ga(\la+1)}\, -\frac{(2\la)_n}{(\la+1)_n}\,(\sin\tha)^{1-2\la}\, -\sum_{k=0}^\iy\frac{(1-\la)_k(n+1)_k}{(n+\la+1)_k k!}\, -\sin\big((2k+n+1)\tha\big) -\label{7}\\ -&=\frac{2\Ga(\la+\thalf)}{\pi^\half\Ga(\la+1)}\, -\frac{(2\la)_n}{(\la+1)_n}\,(\sin\tha)^{1-2\la}\, -\Im\!\!\left(e^{i(n+1)\tha}\,\hyp21{1-\la,n+1}{n+\la+1}{e^{2i\tha}}\right)\nonumber\\ -&=\frac{2^\la\Ga(\la+\thalf)}{\pi^\half\Ga(\la+1)}\, -\frac{(2\la)_n}{(\la+1)_n}\,(\sin\tha)^{-\la}\, -\Re\!\!\left(e^{-\thalf i\la\pi}e^{i(n+\la)\tha}\, -\hyp21{\la,1-\la}{1+\la+n}{\frac{e^{i\tha}}{2i\sin\tha}}\right)\nonumber\\ -&=\frac{2^{2\la}\Ga(\la+\thalf)}{\pi^\half\Ga(\la+1)}\,\frac{(2\la)_n}{(\la+1)_n}\, -\sum_{k=0}^\iy\frac{(\la)_k(1-\la)_k}{(1+\la+n)_k k!}\, -\frac{\cos((n+k+\la)\tha-\thalf(k+\la)\pi)}{(2\sin\tha)^{k+\la}}\,. -\label{106} -\end{align} -We require that $0<\tha<\pi$ in \eqref{7} and $\tfrac16\pi<\tha<\tfrac56\pi$ in -\eqref{106} The convergence is absolute for $\la>\thalf$ and conditional for -$0<\la\le\thalf$. -For $\la\in\Zpos$ the above series terminate after the term with -$k=\la-1$. -Formulas \eqref{7} and \eqref{106} are also given in -\mycite{Sz}{(4.9.22), (4.9.25)}. -% -\paragraph{Fourier transform} -\begin{equation} -\frac{\Ga(\la+1)}{\Ga(\la+\thalf)\,\Ga(\thalf)}\, -\int_{-1}^1 \frac{C_n^\la(y)}{C_n^\la(1)}\,(1-y^2)^{\la-\half}\, -e^{ixy}\,dy -=i^n\,2^\la\,\Ga(\la+1)\,x^{-\la}\,J_{\la+n}(x). -\label{8} -\end{equation} -See \mycite{DLMF}{(18.17.17) and (18.17.18)}. -% -\paragraph{Laplace transforms} -\begin{equation} -\frac2{n!\,\Ga(\la)}\, -\int_0^\iy H_n(tx)\,t^{n+2\la-1}\,e^{-t^2}\,dt=C_n^\la(x). -\label{56} -\end{equation} -See Nielsen \cite[p.48, (4) with p.47, (1) and p.28, (10)]{K4} (1918) -or Feldheim \cite[(28)]{K3} (1942). -\begin{equation} -\frac2{\Ga(\la+\thalf)}\,\int_0^1 \frac{C_n^\la(t)}{C_n^\la(1)}\, -(1-t^2)^{\la-\half}\,t^{-1}\,(x/t)^{n+2\la+1}\,e^{-x^2/t^2}\,dt -=2^{-n}\,H_n(x)\,e^{-x^2}\quad(\la>-\thalf). -\label{46} -\end{equation} -Use Askey \& Fitch \cite[(3.29)]{K2} for $\al=\pm\thalf$ together with -\eqref{48}, \eqref{51}, \eqref{52}, \eqref{54} and \eqref{55}. -\paragraph{Addition formula} (see \mycite{AAR}{(9.8.5$'$)]}) -\begin{multline} -R_n^{(\al,\al)}\big(xy+(1-x^2)^\half(1-y^2)^\half t\big) -=\sum_{k=0}^n \frac{(-1)^k(-n)_k\,(n+2\al+1)_k}{2^{2k}((\al+1)_k)^2}\\ -\times(1-x^2)^{k/2} R_{n-k}^{(\al+k,\al+k)}(x)\,(1-y^2)^{k/2} R_{n-k}^{(\al+k,\al+k)}(y)\, -\om_k^{(\al-\half,\al-\half)}\,R_k^{(\al-\half,\al-\half)}(t), -\label{108} -\end{multline} -where -\[ -R_n^{(\al,\be)}(x):=P_n^{(\al,\be)}(x)/P_n^{(\al,\be)}(1),\quad -\om_n^{(\al,\be)}:=\frac{\int_{-1}^1 (1-x)^\al(1+x)^\be\,dx} -{\int_{-1}^1 (R_n^{(\al,\be)}(x))^2\,(1-x)^\al(1+x)^\be\,dx}\,. -\] -xP_n^{(\lambda-\frac{1}{2},\frac{1}{2})}(2x^2-1).$$ - -\subsection*{References} -\cite{Abram}, \cite{Ahmed+86}, \cite{AndrewsAskeyRoy}, \cite{Area+I}, \cite{Askey67}, \cite{Askey74}, \cite{Askey75}, -\cite{Askey89I}, \cite{AskeyFitch}, \cite{AskeyKoornRahman}, \cite{Berg}, \cite{BilodeauI}, -\cite{BojanovNikolov}, \cite{Brafman51}, \cite{Brafman57I}, \cite{Brown}, -\cite{BustozIsmail82}, \cite{BustozIsmail83}, \cite{BustozIsmail89}, \cite{BustozSavage79}, -\cite{BustozSavage80}, \cite{Carlitz61II}, \cite{Chihara78}, \cite{Common}, \cite{Danese}, -\cite{Dette94}, \cite{DijksmaKoorn}, \cite{DilcherStolarsky}, \cite{Dimitrov96}, -\cite{Dimitrov2003}, \cite{Doha2002II}, \cite{Driver}, \cite{ElbertLaforgia86I}, -\cite{ElbertLaforgia86II}, \cite{ElbertLaforgia90}, \cite{Erdelyi+}, \cite{Exton96}, -\cite{Gasper69}, \cite{Gasper72II}, \cite{Gasper85}, \cite{Grad}, \cite{Ismail74}, -\cite{Ismail2005II}, \cite{Koekoek2000}, \cite{Koorn88}, \cite{Laforgia}, \cite{LewanowiczI}, -\cite{Lorch}, \cite{Mathai}, \cite{Nagel}, \cite{Nikiforov+}, \cite{NikiforovUvarov}, -\cite{RahmanShah}, \cite{Rainville}, \cite{Reimer}, \cite{Sartoretto}, \cite{Srivastava71}, -\cite{Szego75}, \cite{Temme}, \cite{Viswanathan}, \cite{Zayed}. - -\subsection{Chebyshev}\index{Chebyshev polynomials} - -\par - -\subsection*{Hypergeometric representation} -The Chebyshev polynomials of the first kind can be obtained from the Jacobi -polynomials by taking $\alpha=\beta=-\frac{1}{2}$: -\begin{equation} -\label{DefChebyshevI} -T_n(x)=\frac{P_n^{(-\frac{1}{2},-\frac{1}{2})}(x)}{P_n^{(-\frac{1}{2},-\frac{1}{2})}(1)} -=\hyp{2}{1}{-n,n}{\frac{1}{2}}{\frac{1-x}{2}} -\end{equation} -and the Chebyshev polynomials of the second kind can be obtained from the -Jacobi polynomials by taking $\alpha=\beta=\frac{1}{2}$: -\begin{equation} -\label{DefChebyshevII} -U_n(x)=(n+1)\frac{P_n^{(\frac{1}{2},\frac{1}{2})}(x)}{P_n^{(\frac{1}{2},\frac{1}{2})}(1)} -=(n+1)\,\hyp{2}{1}{-n,n+2}{\frac{3}{2}}{\frac{1-x}{2}}. -\end{equation} - -\subsection*{Orthogonality relation} -\begin{equation} -\label{OrtChebyshevI} -\int_{-1}^1(1-x^2)^{-\frac{1}{2}}T_m(x)T_n(x)\,dx= -\left\{\begin{array}{ll} -\displaystyle\frac{\cpi}{2}\,\delta_{mn}, & n\neq 0\\[5mm] -\cpi\,\delta_{mn}, & n=0. -\end{array}\right. -\end{equation} - -\begin{equation} -\label{OrtChebyshevII} -\int_{-1}^1(1-x^2)^{\frac{1}{2}}U_m(x)U_n(x)\,dx=\frac{\cpi}{2}\,\delta_{mn}. -\end{equation} - -\subsection*{Recurrence relations} -\begin{equation} -\label{RecChebyshevI} -2xT_n(x)=T_{n+1}(x)+T_{n-1}(x),\quad T_{0}(x)=1\quad\textrm{and}\quad T_1(x)=x. -\end{equation} - -\begin{equation} -\label{RecChebyshevII} -2xU_n(x)=U_{n+1}(x)+U_{n-1}(x),\quad U_{0}(x)=1\quad\textrm{and}\quad U_1(x)=2x. -\end{equation} - -\subsection*{Normalized recurrence relations} -\begin{equation} -\label{NormRecChebyshevI} -xp_n(x)=p_{n+1}(x)+\frac{1}{4}p_{n-1}(x), -\end{equation} -where -$$T_1(x)=p_1(x)=x\quad\textrm{and}\quad T_n(x)=2^np_n(x),\quad n\neq 1.$$ - -\begin{equation} -\label{NormRecChebyshevII} -xp_n(x)=p_{n+1}(x)+\frac{1}{4}p_{n-1}(x), -\end{equation} -where -$$U_n(x)=2^np_n(x).$$ - -\subsection*{Differential equations} -\begin{equation} -\label{dvChebyshevI} -(1-x^2)y''(x)-xy'(x)+n^2y(x)=0,\quad y(x)=T_n(x). -\end{equation} - -\begin{equation} -\label{dvChebyshevII} -(1-x^2)y''(x)-3xy'(x)+n(n+2)y(x)=0,\quad y(x)=U_n(x). -\end{equation} - -\subsection*{Forward shift operator} -\begin{equation} -\label{shift1Chebyshev} -\frac{d}{dx}T_n(x)=nU_{n-1}(x). -\end{equation} - -\subsection*{Backward shift operator} -\begin{equation} -\label{shift2ChebyshevI} -(1-x^2)\frac{d}{dx}U_n(x)-xU_n(x)=-(n+1)T_{n+1}(x) -\end{equation} -or equivalently -\begin{equation} -\label{shift2ChebyshevII} -\frac{d}{dx}\left[\left(1-x^2\right)^{\frac{1}{2}}U_n(x)\right] -=-(n+1)\left(1-x^2\right)^{-\frac{1}{2}}T_{n+1}(x). -\end{equation} - -\subsection*{Rodrigues-type formulas} -\begin{equation} -\label{RodChebyshevI} -(1-x^2)^{-\frac{1}{2}}T_n(x)=\frac{(-1)^n}{(\frac{1}{2})_n2^n} -\left(\frac{d}{dx}\right)^n\left[(1-x^2)^{n-\frac{1}{2}}\right]. -\end{equation} - -\begin{equation} -\label{RodChebyshevII} -(1-x^2)^{\frac{1}{2}}U_n(x)=\frac{(n+1)(-1)^n}{(\frac{3}{2})_n2^n} -\left(\frac{d}{dx}\right)^n\left[(1-x^2)^{n+\frac{1}{2}}\right]. -\end{equation} - -\subsection*{Generating functions} -\begin{equation} -\label{GenChebyshevI1} -\frac{1-xt}{1-2xt+t^2}=\sum_{n=0}^{\infty}T_n(x)t^n. -\end{equation} - -\begin{equation} -\label{GenChebyshevI2} -R^{-1}\sqrt{\frac{1}{2}(1+R-xt)}=\sum_{n=0}^{\infty} -\frac{\left(\frac{1}{2}\right)_n}{n!}T_n(x)t^n,\quad R=\sqrt{1-2xt+t^2}. -\end{equation} - -\begin{equation} -\label{GenChebyshevI3} -\hyp{0}{1}{-}{\frac{1}{2}}{\frac{(x-1)t}{2}}\, -\hyp{0}{1}{-}{\frac{1}{2}}{\frac{(x+1)t}{2}}= -\sum_{n=0}^{\infty}\frac{T_n(x)}{\left(\frac{1}{2}\right)_nn!}t^n. -\end{equation} - -\begin{equation} -\label{GenChebyshevI4} -\e^{xt}\,\hyp{0}{1}{-}{\frac{1}{2}}{\frac{(x^2-1)t^2}{4}}= -\sum_{n=0}^{\infty}\frac{T_n(x)}{n!}t^n. -\end{equation} - -\begin{eqnarray} -\label{GenChebyshevI5} -& &\hyp{2}{1}{\gamma,-\gamma}{\frac{1}{2}}{\frac{1-R-t}{2}}\, -\hyp{2}{1}{\gamma,-\gamma}{\frac{1}{2}}{\frac{1-R+t}{2}}\nonumber\\ -& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n(-\gamma)_n}{\left(\frac{1}{2}\right)_nn!}T_n(x)t^n, -\quad R=\sqrt{1-2xt+t^2},\quad\textrm{$\gamma$ arbitrary}. -\end{eqnarray} - -\begin{eqnarray} -\label{GenChebyshevI6} -& &(1-xt)^{-\gamma}\,\hyp{2}{1}{\frac{1}{2}\gamma,\frac{1}{2}\gamma+\frac{1}{2}}{\frac{1}{2}} -{\frac{(x^2-1)t^2}{(1-xt)^2}}\nonumber\\ -& &=\sum_{n=0}^{\infty}\frac{(\gamma)_n}{n!}T_n(x)t^n,\quad\textrm{$\gamma$ arbitrary}. -\end{eqnarray} - -\begin{equation} -\label{GenChebyshevII1} -\frac{1}{1-2xt+t^2}=\sum_{n=0}^{\infty}U_n(x)t^n. -\end{equation} - -\begin{equation} -\label{GenChebyshevII2} -\frac{1}{R\sqrt{\frac{1}{2}(1+R-xt)}}=\sum_{n=0}^{\infty} -\frac{\left(\frac{3}{2}\right)_n}{(n+1)!}U_n(x)t^n,\quad R=\sqrt{1-2xt+t^2}. -\end{equation} - -\begin{equation} -\label{GenChebyshevII3} -\hyp{0}{1}{-}{\frac{3}{2}}{\frac{(x-1)t}{2}}\, -\hyp{0}{1}{-}{\frac{3}{2}}{\frac{(x+1)t}{2}}= -\sum_{n=0}^{\infty}\frac{U_n(x)}{\left(\frac{3}{2}\right)_n(n+1)!}t^n. -\end{equation} - -\begin{equation} -\label{GenChebyshevII4} -\e^{xt}\,\hyp{0}{1}{-}{\frac{3}{2}}{\frac{(x^2-1)t^2}{4}}= -\sum_{n=0}^{\infty}\frac{U_n(x)}{(n+1)!}t^n. -\end{equation} - -\begin{eqnarray} -\label{GenChebyshevII5} -& &\hyp{2}{1}{\gamma,2-\gamma}{\frac{3}{2}}{\frac{1-R-t}{2}}\, -\hyp{2}{1}{\gamma,2-\gamma}{\frac{3}{2}}{\frac{1-R+t}{2}}\nonumber\\ -& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n(2-\gamma)_n}{\left(\frac{3}{2}\right)_n(n+1)!}U_n(x)t^n, -\quad R=\sqrt{1-2xt+t^2},\quad\textrm{$\gamma$ arbitrary}. -\end{eqnarray} - -\begin{eqnarray} -\label{GenChebyshevII6} -& &(1-xt)^{-\gamma}\,\hyp{2}{1}{\frac{1}{2}\gamma,\frac{1}{2}\gamma+\frac{1}{2}}{\frac{3}{2}} -{\frac{(x^2-1)t^2}{(1-xt)^2}}\nonumber\\ -& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n}{(n+1)!}U_n(x)t^n,\quad\textrm{$\gamma$ arbitrary}. -\end{eqnarray} - -\subsection*{Remarks} -The Chebyshev polynomials can also be written as: -$$T_n(x)=\cos(n\theta),\quad x=\cos\theta$$ -and -$$U_n(x)=\frac{\sin (n+1)\theta}{\sin\theta},\quad x=\cos\theta.$$ -Further we have -$$U_n(x)=C_n^{(1)}(x)$$ -where $C_n^{(\lambda)}(x)$ denotes the Gegenbauer (or ultraspherical) -\subsection*{9.8.2 Chebyshev} -\label{sec9.8.2} -In addition to the Chebyshev polynomials $T_n$ of the first kind (9.8.35) -and $U_n$ of the second kind (9.8.36), -\begin{align} -T_n(x)&:=\frac{P_n^{(-\half,-\half)}(x)}{P_n^{(-\half,-\half)}(1)} -=\cos(n\tha),\quad x=\cos\tha,\\ -U_n(x)&:=(n+1)\,\frac{P_n^{(\half,\half)}(x)}{P_n^{(\half,\half)}(1)} -=\frac{\sin((n+1)\tha)}{\sin\tha}\,,\quad x=\cos\tha, -\end{align} -we have Chebyshev polynomials $V_n$ {\em of the third kind} -and $W_n$ {\em of the fourth kind}, -\begin{align} -V_n(x)&:=\frac{P_n^{(-\half,\half)}(x)}{P_n^{(-\half,\half)}(1)} -=\frac{\cos((n+\thalf)\tha)}{\cos(\thalf\tha)}\,,\quad x=\cos\tha,\\ -W_n(x)&:=(2n+1)\,\frac{P_n^{(\half,-\half)}(x)}{P_n^{(\half,-\half)}(1)} -=\frac{\sin((n+\thalf)\tha)}{\sin(\thalf\tha)}\,,\quad x=\cos\tha, -\end{align} -see \cite[Section 1.2.3]{K20}. Then there is the symmetry -\begin{equation} -V_n(-x)=(-1)^n W_n(x). -\label{140} -\end{equation} - -The names of Chebyshev polynomials of the third and fourth kind -and the notation $V_n(x)$ are due to Gautschi \cite{K21}. -The notation $W_n(x)$ was first used by Mason \cite{K22}. -Names and notations for Chebyshev polynomials of the third and fourth -kind are interchanged in \mycite{AAR}{Remark 2.5.3} and -\mycite{DLMF}{Table 18.3.1}. -polynomial given by (\ref{DefGegenbauer}) in the preceding subsection. - -\subsection*{References} -\cite{Abram}, \cite{AskeyFitch}, \cite{AskeyGasperHarris}, \cite{AskeyIsmail76}, -\cite{Bavinck95}, \cite{Chihara78}, \cite{Danese}, \cite{DilcherStolarsky}, -\cite{Erdelyi+}, \cite{Grad}, \cite{HartmannStephan}, \cite{Ismail2005II}, \cite{Koekoek2000}, -\cite{Luke}, \cite{Mathai}, \cite{Nikiforov+}, \cite{NikiforovUvarov}, \cite{Rainville}, -\cite{Rayes+}, \cite{Rivlin}, \cite{SainteViennot}, \cite{Szego75}, \cite{Temme}, -\cite{Wilson70I}, \cite{Zayed}, \cite{Zhang}, \cite{ZhangWang}. - -\subsection{Legendre / Spherical}\index{Legendre polynomials} -\index{Spherical polynomials} - -\par - -\subsection*{Hypergeometric representation} -The Legendre (or spherical) polynomials are Jacobi polynomials with $\alpha=\beta=0$: -\begin{equation} -\label{DefLegendre} -P_n(x)=P_n^{(0,0)}(x)=\hyp{2}{1}{-n,n+1}{1}{\frac{1-x}{2}}. -\end{equation} - -\subsection*{Orthogonality relation} -\begin{equation} -\label{OrtLegendre} -\int_{-1}^1P_m(x)P_n(x)\,dx=\frac{2}{2n+1}\,\delta_{mn}. -\end{equation} - -\subsection*{Recurrence relation} -\begin{equation} -\label{RecLegendre} -(2n+1)xP_n(x)=(n+1)P_{n+1}(x)+nP_{n-1}(x). -\end{equation} - -\subsection*{Normalized recurrence relation} -\begin{equation} -\label{NormRecLegendre} -xp_n(x)=p_{n+1}(x)+\frac{n^2}{(2n-1)(2n+1)}p_{n-1}(x), -\end{equation} -where -$$P_n(x)=\binom{2n}{n}\frac{1}{2^n}p_n(x).$$ - -\subsection*{Differential equation} -\begin{equation} -\label{dvLegendre} -(1-x^2)y''(x)-2xy'(x)+n(n+1)y(x)=0,\quad y(x)=P_n(x). -\end{equation} - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodLegendre} -P_n(x)=\frac{(-1)^n}{2^nn!}\left(\frac{d}{dx}\right)^n\left[(1-x^2)^n\right]. -\end{equation} - -\subsection*{Generating functions} -\begin{equation} -\label{GenLegendre1} -\frac{1}{\sqrt{1-2xt+t^2}}=\sum_{n=0}^{\infty}P_n(x)t^n. -\end{equation} - -\begin{equation} -\label{GenLegendre2} -\hyp{0}{1}{-}{1}{\frac{(x-1)t}{2}}\,\hyp{0}{1}{-}{1}{\frac{(x+1)t}{2}}= -\sum_{n=0}^{\infty}\frac{P_n(x)}{(n!)^2}t^n. -\end{equation} - -\begin{equation} -\label{GenLegendre3} -\e^{xt}\,\hyp{0}{1}{-}{1}{\frac{(x^2-1)t^2}{4}}= -\sum_{n=0}^{\infty}\frac{P_n(x)}{n!}t^n. -\end{equation} - -\begin{eqnarray} -\label{GenLegendre4} -& &\hyp{2}{1}{\gamma,1-\gamma}{1}{\frac{1-R-t}{2}}\, -\hyp{2}{1}{\gamma,1-\gamma}{1}{\frac{1-R+t}{2}}\nonumber\\ -& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n(1-\gamma)_n}{(n!)^2}P_n(x)t^n, -\quad R=\sqrt{1-2xt+t^2},\quad\textrm{$\gamma$ arbitrary}. -\end{eqnarray} - -\begin{eqnarray} -\label{GenLegendre5} -& &(1-xt)^{-\gamma}\,\hyp{2}{1}{\frac{1}{2}\gamma,\frac{1}{2}\gamma+\frac{1}{2}}{1} -{\frac{(x^2-1)t^2}{(1-xt)^2}}\nonumber\\ -& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n}{n!}P_n(x)t^n,\quad\textrm{$\gamma$ arbitrary}. -\subsection*{9.9 Pseudo Jacobi (or Routh-Romanovski)} -\label{sec9.9} -In this section in \mycite{KLS} the pseudo Jacobi polynomial $P_n(x;\nu,N)$ in (9.9.1) -is considered -for $N\in\ZZ_{\ge0}$ and $n=0,1,\ldots,n$. However, we can more generally take -$-\thalf\be>0\;\;{\rm or}\;\;\al<\be<0). -\] -Then $P_n$ can be expressed as a Meixner polynomial: -\[ -P_n(x)=(-k_2(\al\be)^{-1})_n\,\be^n\, -M_n\left(-\,\frac{x+k_2\al^{-1}}{\al-\be},-k_2(\al\be)^{-1},\be\al^{-1}\right). -\] - -In 1938 Gottlieb \cite[\S2]{K1} introduces polynomials $l_n$ ``of Laguerre type'' -which turn out to be special Meixner polynomials: -$l_n(x)=e^{-n\la} M_n(x;1,e^{-\la})$. -% -\paragraph{Uniqueness of orthogonality measure} -The coefficient of $p_{n-1}(x)$ in (9.10.4) behaves as $O(n^2)$ as $n\to\iy$. -Hence \eqref{93} holds, by which the orthogonality measure is unique. -=\frac{2^n}{(n-2N-1)_n}y_n(x;-2N-2).$$ - -\subsection*{References} -\cite{Askey87}, \cite{BorodinOlshanski}, \cite{Lesky96}. - - -\section{Meixner}\index{Meixner polynomials} - -\par\setcounter{equation}{0} - -\subsection*{Hypergeometric representation} -\begin{equation} -\label{DefMeixner} -M_n(x;\beta,c)=\hyp{2}{1}{-n,-x}{\beta}{1-\frac{1}{c}}. -\end{equation} - -\subsection*{Orthogonality relation} -\begin{equation} -\label{OrtMeixner} -\sum_{x=0}^{\infty}\frac{(\beta)_x}{x!}c^xM_m(x;\beta,c)M_n(x;\beta,c) -{}=\frac{c^{-n}n!}{(\beta)_n(1-c)^{\beta}}\,\delta_{mn} -% \constraint{ -% $\beta > 0$ & -% $0 < c < 1$ } -\end{equation} - -\subsection*{Recurrence relation} -\begin{eqnarray} -\label{RecMeixner} -(c-1)xM_n(x;\beta,c)&=&c(n+\beta)M_{n+1}(x;\beta,c)\nonumber\\ -& &{}\mathindent{}-\left[n+(n+\beta)c\right]M_n(x;\beta,c)+nM_{n-1}(x;\beta,c). -\end{eqnarray} - -\subsection*{Normalized recurrence relation} -\begin{equation} -\label{NormRecMeixner} -xp_n(x)=p_{n+1}(x)+\frac{n+(n+\beta)c}{1-c}p_n(x)+ -\frac{n(n+\beta-1)c}{(1-c)^2}p_{n-1}(x), -\end{equation} -where -$$M_n(x;\beta,c)=\frac{1}{(\beta)_n}\left(\frac{c-1}{c}\right)^np_n(x).$$ - -\subsection*{Difference equation} -\begin{equation} -\label{dvMeixner} -n(c-1)y(x)=c(x+\beta)y(x+1)-\left[x+(x+\beta)c\right]y(x)+xy(x-1), -\end{equation} -where -$$y(x)=M_n(x;\beta,c).$$ - -\subsection*{Forward shift operator} -\begin{equation} -\label{shift1MeixnerI} -M_n(x+1;\beta,c)-M_n(x;\beta,c)= -\frac{n}{\beta}\left(\frac{c-1}{c}\right)M_{n-1}(x;\beta+1,c) -\end{equation} -or equivalently -\begin{equation} -\label{shift1MeixnerII} -\Delta M_n(x;\beta,c)=\frac{n}{\beta}\left(\frac{c-1}{c}\right)M_{n-1}(x;\beta+1,c). -\end{equation} - -\subsection*{Backward shift operator} -\begin{equation} -\label{shift2MeixnerI} -c(\beta+x-1)M_n(x;\beta,c)-xM_n(x-1;\beta,c)=c(\beta-1)M_{n+1}(x;\beta-1,c) -\end{equation} -or equivalently -\begin{equation} -\label{shift2MeixnerII} -\nabla\left[\frac{(\beta)_xc^x}{x!}M_n(x;\beta,c)\right]= -\frac{(\beta-1)_xc^x}{x!}M_{n+1}(x;\beta-1,c). -\end{equation} - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodMeixner} -\frac{(\beta)_xc^x}{x!}M_n(x;\beta,c)=\nabla^n\left[\frac{(\beta+n)_xc^x}{x!}\right]. -\end{equation} - -\subsection*{Generating functions} -\begin{equation} -\label{GenMeixner1} -\left(1-\frac{t}{c}\right)^x(1-t)^{-x-\beta}= -\sum_{n=0}^{\infty}\frac{(\beta)_n}{n!}M_n(x;\beta,c)t^n. -\end{equation} - -\begin{equation} -\label{GenMeixner2} -\e^t\,\hyp{1}{1}{-x}{\beta}{\left(\frac{1-c}{c}\right)t}= -\sum_{n=0}^{\infty}\frac{M_n(x;\beta,c)}{n!}t^n. -\end{equation} - -\begin{equation} -\label{GenMeixner3} -(1-t)^{-\gamma}\,\hyp{2}{1}{\gamma,-x}{\beta}{\frac{(1-c)t}{c(1-t)}} -=\sum_{n=0}^{\infty}\frac{(\gamma)_n}{n!}M_n(x;\beta,c)t^n, -\quad\textrm{$\gamma$ arbitrary}. -\end{equation} - -\subsection*{Limit relations} - -\subsubsection*{Hahn $\rightarrow$ Meixner} -If we take $\alpha=b-1$, $\beta=N(1-c)c^{-1}$ in the definition (\ref{DefHahn}) -of the Hahn polynomials and let $N\rightarrow\infty$ we find the Meixner polynomials: -$$\lim_{N\rightarrow\infty} -Q_n(x;b-1,N(1-c)c^{-1},N)=M_n(x;b,c).$$ - -\subsubsection*{Dual Hahn $\rightarrow$ Meixner} -To obtain the Meixner polynomials from the dual Hahn polynomials we have to take -$\gamma=\beta-1$ and $\delta=N(1-c)c^{-1}$ in the definition (\ref{DefDualHahn}) of -the dual Hahn polynomials and let $N\rightarrow\infty$: -$$\lim_{N\rightarrow\infty} -R_n(\lambda(x);\beta-1,N(1-c)c^{-1},N)=M_n(x;\beta,c).$$ - -\subsubsection*{Meixner $\rightarrow$ Laguerre} -The Laguerre polynomials given by (\ref{DefLaguerre}) are obtained from the Meixner polynomials -if we take $\beta=\alpha+1$ and $x\rightarrow (1-c)^{-1}x$ and let $c\rightarrow 1$: -\begin{equation} -\lim_{c\rightarrow 1} -M_n((1-c)^{-1}x;\alpha+1,c)=\frac{L_n^{(\alpha)}(x)}{L_n^{(\alpha)}(0)}. -\end{equation} - -\subsubsection*{Meixner $\rightarrow$ Charlier} -The Charlier polynomials given by (\ref{DefCharlier}) are obtained from the Meixner polynomials -if we take $c=(a+\beta)^{-1}a$ and let $\beta\rightarrow\infty$: -\begin{equation} -\lim_{\beta\rightarrow\infty} -M_n(x;\beta,(a+\beta)^{-1}a)=C_n(x;a). -\end{equation} - -\subsection*{Remarks} -The Meixner polynomials are related to the Jacobi polynomials given by (\ref{DefJacobi}) -in the following way: -$$\frac{(\beta)_n}{n!}M_n(x;\beta,c)=P_n^{(\beta-1,-n-\beta-x)}((2-c)c^{-1}).$$ - -\noindent -The Meixner polynomials are also related to the Krawtchouk polynomials given by -(\ref{DefKrawtchouk}) in the following way: -\subsection*{9.11 Krawtchouk} -\label{sec9.11} -% -\paragraph{Special values} -By (9.11.1) and the binomial formula: -\begin{equation} -K_n(0;p,N)=1,\qquad -K_n(N;p,N)=(1-p^{-1})^n. -\label{9} -\end{equation} -The self-duality (p.240, Remarks, first formula) -\begin{equation} -K_n(x;p,N)=K_x(n;p,N)\qquad (n,x\in \{0,1,\ldots,N\}) -\label{147} -\end{equation} -combined with \eqref{9} yields: -\begin{equation} -K_N(x;p,N)=(1-p^{-1})^x\qquad(x\in\{0,1,\ldots,N\}). -\label{148} -\end{equation} -% -\paragraph{Symmetry} -By the orthogonality relation (9.11.2): -\begin{equation} -\frac{K_n(N-x;p,N)}{K_n(N;p,N)}=K_n(x;1-p,N). -\label{10} -\end{equation} -By \eqref{10} and \eqref{147} we have also -\begin{equation} -\frac{K_{N-n}(x;p,N)}{K_N(x;p,N)}=K_n(x;1-p,N) -\qquad(n,x\in\{0,1,\ldots,N\}), -\label{149} -\end{equation} -and, by \eqref{149}, \eqref{10} and \eqref{9}, -\begin{equation} -K_{N-n}(N-x;p,N)=\left(\frac p{p-1}\right)^{n+x-N}K_n(x;p,N) -\qquad(n,x\in\{0,1,\ldots,N\}). -\label{150} -\end{equation} -A particular case of \eqref{10} is: -\begin{equation} -K_n(N-x;\thalf,N)=(-1)^n K_n(x;\thalf,N). -\label{11} -\end{equation} -Hence -\begin{equation} -K_{2m+1}(N;\thalf,2N)=0. -\label{12} -\end{equation} -From (9.11.11): -\begin{equation} -K_{2m}(N;\thalf,2N)=\frac{(\thalf)_m}{(-N+\thalf)_m}\,. -\label{13} -\end{equation} -% -\paragraph{Quadratic transformations} -\begin{align} -K_{2m}(x+N;\thalf,2N)&=\frac{(\thalf)_m}{(-N+\thalf)_m}\, -R_m(x^2;-\thalf,-\thalf,N), -\label{31}\\ -K_{2m+1}(x+N;\thalf,2N)&=-\,\frac{(\tfrac32)_m}{N\,(-N+\thalf)_m}\, -x\,R_m(x^2-1;\thalf,\thalf,N-1), -\label{33}\\ -K_{2m}(x+N+1;\thalf,2N+1)&=\frac{(\tfrac12)_m}{(-N-\thalf)_m}\, -R_m(x(x+1);-\thalf,\thalf,N), -\label{32}\\ -K_{2m+1}(x+N+1;\thalf,2N+1)&=\frac{(\tfrac32)_m}{(-N-\thalf)_{m+1}}\, -(x+\thalf)\,R_m(x(x+1);\thalf,-\thalf,N), -\label{34} -\end{align} -where $R_m$ is a dual Hahn polynomial (9.6.1). For the proofs use -(9.6.2), (9.11.2), (9.6.4) and (9.11.4). -% -\paragraph{Generating functions} -\begin{multline} -\sum_{x=0}^N\binom Nx K_m(x;p,N)K_n(x;q,N)z^x\\ -=\left(\frac{p-z+pz}p\right)^m -\left(\frac{q-z+qz}q\right)^n -(1+z)^{N-m-n} -K_m\left(n;-\,\frac{(p-z+pz)(q-z+qz)}z,N\right). -\label{107} -\end{multline} -This follows immediately from Rosengren \cite[(3.5)]{K8}, which goes back -to Meixner \cite{K9}. -$$K_n(x;p,N)=M_n(x;-N,(p-1)^{-1}p).$$ - -\subsection*{References} -\cite{Allaway76}, \cite{NAlSalam66}, \cite{AlSalam90}, \cite{AlSalamChihara76}, -\cite{AlSalamIsmail76}, \cite{Alvarez+}, \cite{AndrewsAskey85}, \cite{Area+II}, \cite{Askey75}, -\cite{Askey89I}, \cite{Askey2005}, \cite{AskeyGasper77}, \cite{AskeyIsmail76}, -\cite{AskeyWilson85}, \cite{AtakRahmanSuslov}, \cite{AtakSuslov88}, \cite{Bavinck98}, -\cite{BavinckHaeringen}, \cite{Campigotto+}, \cite{Chihara78}, \cite{Cooper+}, -\cite{Erdelyi+}, \cite{FoataLabelle}, \cite{Gabutti}, \cite{GabuttiMathis}, \cite{Gasper73I}, -\cite{Gasper74}, \cite{HoareRahman}, \cite{Ismail2005II}, \cite{IsmailLetVal88}, -\cite{IsmailLi}, \cite{IsmailMuldoon}, \cite{IsmailStanton97}, \cite{JinWong}, -\cite{Karlin58}, \cite{Koekoek2000}, \cite{Koorn88}, \cite{LabelleYehI}, \cite{LabelleYehII}, -\cite{Lesky89}, \cite{Lesky94I}, \cite{Lesky95II}, \cite{LewanowiczII}, \cite{Meixner}, -\cite{Nikiforov+}, \cite{NikiforovUvarov}, \cite{Rahman78I}, \cite{ValentAssche}, -\cite{Viennot}, \cite{Zarzo+}, \cite{Zeng90}. - - -\section{Krawtchouk}\index{Krawtchouk polynomials} - -\par\setcounter{equation}{0} - -\subsection*{Hypergeometric representation} -\begin{equation} -\label{DefKrawtchouk} -K_n(x;p,N)=\hyp{2}{1}{-n,-x}{-N}{\frac{1}{p}},\quad n=0,1,2,\ldots,N. -\end{equation} - -\subsection*{Orthogonality relation} -\begin{eqnarray} -\label{OrtKrawtchouk} -& &\sum_{x=0}^N\binom{N}{x}p^x(1-p)^{N-x} K_m(x;p,N)K_n(x;p,N)\nonumber\\ -& &{}=\frac{(-1)^nn!}{(-N)_n}\left(\frac{1-p}{p}\right)^n\,\delta_{mn},\quad0 < p < 1. -\end{eqnarray} - -\subsection*{Recurrence relation} -\begin{eqnarray} -\label{RecKrawtchouk} --xK_n(x;p,N)&=&p(N-n)K_{n+1}(x;p,N)\nonumber\\ -& &{}\mathindent{}-\left[p(N-n)+n(1-p)\right]K_n(x;p,N)\nonumber\\ -& &{}\mathindent\mathindent{}+n(1-p)K_{n-1}(x;p,N). -\end{eqnarray} - -\subsection*{Normalized recurrence relation} -\begin{eqnarray} -\label{NormRecKrawtchouk} -xp_n(x)&=&p_{n+1}(x)+\left[p(N-n)+n(1-p)\right]p_n(x)\nonumber\\ -& &{}\mathindent{}+np(1-p)(N+1-n)p_{n-1}(x), -\end{eqnarray} -where -$$K_n(x;p,N)=\frac{1}{(-N)_np^n}p_n(x).$$ - -\subsection*{Difference equation} -\begin{eqnarray} -\label{dvKrawtchouk} --ny(x)&=&p(N-x)y(x+1)\nonumber\\ -& &{}\mathindent{}-\left[p(N-x)+x(1-p)\right]y(x)+x(1-p)y(x-1), -\end{eqnarray} -where -$$y(x)=K_n(x;p,N).$$ - -\subsection*{Forward shift operator} -\begin{equation} -\label{shift1KrawtchoukI} -K_n(x+1;p,N)-K_n(x;p,N)=-\frac{n}{Np}K_{n-1}(x;p,N-1) -\end{equation} -or equivalently -\begin{equation} -\label{shift1KrawtchoukII} -\Delta K_n(x;p,N)=-\frac{n}{Np}K_{n-1}(x;p,N-1). -\end{equation} - -\subsection*{Backward shift operator} -\begin{eqnarray} -\label{shift2KrawtchoukI} -& &(N+1-x)K_n(x;p,N)-x\left(\frac{1-p}{p}\right)K_n(x-1;p,N)\nonumber\\ -& &{}=(N+1)K_{n+1}(x;p,N+1) -\end{eqnarray} -or equivalently -\begin{equation} -\label{shift2KrawtchoukII} -\nabla\left[\binom{N}{x}\left(\frac{p}{1-p}\right)^xK_n(x;p,N)\right]= -\binom{N+1}{x}\left(\frac{p}{1-p}\right)^xK_{n+1}(x;p,N+1). -\end{equation} - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodKrawtchouk} -\binom{N}{x}\left(\frac{p}{1-p}\right)^xK_n(x;p,N)= -\nabla^n\left[\binom{N-n}{x}\left(\frac{p}{1-p}\right)^x\right]. -\end{equation} - -\subsection*{Generating functions} -For $x=0,1,2,\ldots,N$ we have -\begin{equation} -\label{GenKrawtchouk1} -\left(1-\frac{(1-p)}{p}t\right)^x(1+t)^{N-x}= -\sum_{n=0}^N\binom{N}{n}K_n(x;p,N)t^n. -\end{equation} - -\begin{equation} -\label{GenKrawtchouk2} -\left[\expe^t\,\hyp{1}{1}{-x}{-N}{-\frac{t}{p}}\right]_N= -\sum_{n=0}^N\frac{K_n(x;p,N)}{n!}t^n. -\end{equation} - -\begin{eqnarray} -\label{GenKrawtchouk3} -& &\left[(1-t)^{-\gamma}\,\hyp{2}{1}{\gamma,-x}{-N}{\frac{t}{p(t-1)}}\right]_N\nonumber\\ -& &{}=\sum_{n=0}^N\frac{(\gamma)_n}{n!}K_n(x;p,N)t^n, -\quad\textrm{$\gamma$ arbitrary}. -\end{eqnarray} - -\subsection*{Limit relations} - -\subsubsection*{Hahn $\rightarrow$ Krawtchouk} -If we take $\alpha=pt$ and $\beta=(1-p)t$ in the definition (\ref{DefHahn}) of the Hahn -polynomials and let $t\rightarrow\infty$ we obtain the Krawtchouk polynomials: -$$\lim_{t\rightarrow\infty}Q_n(x;pt,(1-p)t,N)=K_n(x;p,N).$$ - -\subsubsection*{Dual Hahn $\rightarrow$ Krawtchouk} -The Krawtchouk polynomials follow from the dual Hahn polynomials given by -(\ref{DefDualHahn}) if we set $\gamma=pt$, $\delta=(1-p)t$ and let $t\rightarrow\infty$: -$$\lim_{t\rightarrow\infty}R_n(\lambda(x);pt,(1-p)t,N)=K_n(x;p,N).$$ - -\subsubsection*{Krawtchouk $\rightarrow$ Charlier} -The Charlier polynomials given by (\ref{DefCharlier}) can be found from the Krawtchouk -polynomials by taking $p=N^{-1}a$ and letting $N\rightarrow\infty$: -\begin{equation} -\lim_{N\rightarrow\infty}K_n(x;N^{-1}a,N)=C_n(x;a). -\end{equation} - -\subsubsection*{Krawtchouk $\rightarrow$ Hermite} -The Hermite polynomials given by (\ref{DefHermite}) follow from the Krawtchouk polynomials -by setting $x\rightarrow pN+x\sqrt{2p(1-p)N}$ and then letting $N\rightarrow\infty$: -\begin{equation} -\lim_{N\rightarrow\infty} -\sqrt{\binom{N}{n}}K_n(pN+x\sqrt{2p(1-p)N};p,N) -=\frac{\displaystyle (-1)^nH_n(x)}{\displaystyle\sqrt{2^nn!\left(\frac{p}{1-p}\right)^n}}. -\end{equation} - -\subsection*{Remarks} -The Krawtchouk polynomials are self-dual, which means that -$$K_n(x;p,N)=K_x(n;p,N),\quad n,x\in\{0,1,2,\ldots,N\}.$$ -By using this relation we easily obtain the so-called dual orthogonality -relation from the orthogonality relation (\ref{OrtKrawtchouk}): -$$\sum_{n=0}^N\binom{N}{n}p^n(1-p)^{N-n} K_n(x;p,N)K_n(y;p,N)= -\frac{\displaystyle\left(\frac{1-p}{p}\right)^x}{\dbinom{N}{x}}\delta_{xy},$$ -where $0 < p < 1$ and $x,y\in\{0,1,2,\ldots,N\}$. - -\noindent -The Krawtchouk polynomials are related to the Meixner polynomials given by (\ref{DefMeixner}) -in the following way: -\subsection*{9.12 Laguerre} -\label{sec9.12} -\paragraph{Notation} -Here the Laguerre polynomial is denoted by $L_n^\al$ instead of -$L_n^{(\al)}$. -% -\paragraph{Hypergeometric representation} -\begin{align} -L_n^\al(x)&= -\frac{(\al+1)_n}{n!}\,\hyp11{-n}{\al+1}x -\label{182}\\ -&=\frac{(-x)^n}{n!} \hyp20{-n,-n-\al}-{-\,\frac1x} -\label{183}\\ -&=\frac{(-x)^n}{n!}\,C_n(n+\al;x), -\label{184} -\end{align} -where $C_n$ in \eqref{184} is a -\hyperref[sec9.14]{Charlier polynomial}. -Formula \eqref{182} is (9.12.1). Then \eqref{183} follows by reversal -of summation. Finally \eqref{184} follows by \eqref{183} and \eqref{179}. -It is also the remark on top of p.244 in \mycite{KLS}, and it is essentially -\myciteKLS{416}{(2.7.10)}. -% -\paragraph{Uniqueness of orthogonality measure} -The coefficient of $p_{n-1}(x)$ in (9.12.4) behaves as $O(n^2)$ as $n\to\iy$. -Hence \eqref{93} holds, by which the orthogonality measure is unique. -% -\paragraph{Special value} -\begin{equation} -L_n^{\al}(0)=\frac{(\al+1)_n}{n!}\,. -\label{53} -\end{equation} -Use (9.12.1) or see \mycite{DLMF}{18.6.1)}. -% -\paragraph{Quadratic transformations} -\begin{align} -H_{2n}(x)&=(-1)^n\,2^{2n}\,n!\,L_n^{-1/2}(x^2), -\label{54}\\ -H_{2n+1}(x)&=(-1)^n\,2^{2n+1}\,n!\,x\,L_n^{1/2}(x^2). -\label{55} -\end{align} -See p.244, Remarks, last two formulas. -Or see \mycite{DLMF}{(18.7.19), (18.7.20)}. -% -\paragraph{Fourier transform} -\begin{equation} -\frac1{\Ga(\al+1)}\,\int_0^\iy \frac{L_n^\al(y)}{L_n^\al(0)}\, -e^{-y}\,y^\al\,e^{ixy}\,dy= -i^n\,\frac{y^n}{(iy+1)^{n+\al+1}}\,, -\label{14} -\end{equation} -see \mycite{DLMF}{(18.17.34)}. -% -\paragraph{Differentiation formulas} -Each differentiation formula is given in two equivalent forms. -\begin{equation} -\frac d{dx}\left(x^\al L_n^\al(x)\right)= -(n+\al)\,x^{\al-1} L_n^{\al-1}(x),\qquad -\left(x\frac d{dx}+\al\right)L_n^\al(x)= -(n+\al)\,L_n^{\al-1}(x). -\label{76} -\end{equation} -% -\begin{equation} -\frac d{dx}\left(e^{-x} L_n^\al(x)\right)= --e^{-x} L_n^{\al+1}(x),\qquad -\left(\frac d{dx}-1\right)L_n^\al(x)= --L_n^{\al+1}(x). -\label{77} -\end{equation} -% -Formulas \eqref{76} and \eqref{77} follow from -\mycite{DLMF}{(13.3.18), (13.3.20)} -together with (9.12.1). -% -\paragraph{Generalized Hermite polynomials} -See \myciteKLS{146}{p.156}, \cite[Section 1.5.1]{K26}. -These are defined by -\begin{equation} -H_{2m}^\mu(x):=\const L_m^{\mu-\half}(x^2),\qquad -H_{2m+1}^\mu(x):=\const x\,L_m^{\mu+\half}(x^2). -\label{78} -\end{equation} -Then for $\mu>-\thalf$ we have orthogonality relation -\begin{equation} -\int_{-\iy}^{\iy} H_m^\mu(x)\,H_n^\mu(x)\,|x|^{2\mu}e^{-x^2}\,dx -=0\qquad(m\ne n). -\label{79} -\end{equation} -Let the Dunkl operator $T_\mu$ be defined by \eqref{72}. -If we choose the constants in \eqref{78} as -\begin{equation} -H_{2m}^\mu(x)=\frac{(-1)^m(2m)!}{(\mu+\thalf)_m}\,L_m^{\mu-\half}(x^2),\qquad -H_{2m+1}^\mu(x)=\frac{(-1)^m(2m+1)!}{(\mu+\thalf)_{m+1}}\, - x\,L_m^{\mu+\half}(x^2) - \label{80} -\end{equation} -then (see \cite[(1.6)]{K5}) -\begin{equation} -T_\mu H_n^\mu=2n\,H_{n-1}^\mu. -\label{81} -\end{equation} -Formula \eqref{81} with \eqref{80} substituted gives rise to two -differentiation formulas involving Laguerre polynomials which are equivalent to -(9.12.6) and \eqref{76}. - -Composition of \eqref{81} with itself gives -\[ -T_\mu^2 H_n^\mu=4n(n-1)\,H_{n-2}^\mu, -\] -which is equivalent to the composition of (9.12.6) and \eqref{76}: -\begin{equation} -\left(\frac{d^2}{dx^2}+\frac{2\al+1}x\,\frac d{dx}\right)L_n^\al(x^2) -=-4(n+\al)\,L_{n-1}^\al(x^2). -\label{82} -\end{equation} -$$K_n(x;p,N)=M_n(x;-N,(p-1)^{-1}p).$$ - -\subsection*{References} -\cite{AlSalam90}, \cite{AndrewsAskey85}, \cite{Area+II}, \cite{Askey75}, \cite{Askey89I}, -\cite{AskeyGasper77}, \cite{AskeyWilson85}, \cite{AtakRahmanSuslov}, \cite{Campigotto+}, -\cite{LChiharaStanton}, \cite{Chihara78}, \cite{Dette95}, \cite{Dominici}, \cite{Dunkl76}, -\cite{Dunkl84}, \cite{DunklRamirez}, \cite{Erdelyi+}, \cite{FeinsilverSchott}, -\cite{Gasper73I}, \cite{Gasper74}, \cite{HoareRahman}, \cite{Ismail2005II}, \cite{Karlin58}, -\cite{Koorn82}, \cite{Koorn88}, \cite{LabelleYehI}, \cite{LabelleYehII}, \cite{Lesky62}, -\cite{Lesky89}, \cite{Lesky94I}, \cite{Lesky95II}, \cite{LewanowiczII}, \cite{Nikiforov+}, -\cite{NikiforovUvarov}, \cite{Qiu}, \cite{Rahman78I}, \cite{Rahman79}, \cite{Stanton84}, -\cite{Stanton90}, \cite{Szego75}, \cite{Zarzo+}, \cite{Zeng90}. - - -\section{Laguerre}\index{Laguerre polynomials} - -\par\setcounter{equation}{0} - -\subsection*{Hypergeometric representation} -\begin{equation} -\label{DefLaguerre} -L_n^{(\alpha)}(x)=\frac{(\alpha+1)_n}{n!}\,\hyp{1}{1}{-n}{\alpha+1}{x}. -\end{equation} - -\subsection*{Orthogonality relation} -\begin{equation} -\label{OrtLaguerre} -\int_0^{\infty}\expe^{-x}x^{\alpha}L_m^{(\alpha)}(x)L_n^{(\alpha)}(x)\,dx= -\frac{\Gamma(n+\alpha+1)}{n!}\,\delta_{mn},\quad\alpha > -1. -\end{equation} - -\subsection*{Recurrence relation} -\begin{equation} -\label{RecLaguerre} -(n+1)L_{n+1}^{(\alpha)}(x)-(2n+\alpha+1-x)L_n^{(\alpha)}(x)+(n+\alpha)L_{n-1}^{(\alpha)}(x)=0. -\end{equation} - -\subsection*{Normalized recurrence relation} -\begin{equation} -\label{NormRecLaguerre} -xp_n(x)=p_{n+1}(x)+(2n+\alpha+1)p_n(x)+n(n+\alpha)p_{n-1}(x), -\end{equation} -where -$$L_n^{(\alpha)}(x)=\frac{(-1)^n}{n!}p_n(x).$$ - -\subsection*{Differential equation} -\begin{equation} -\label{dvLaguerre} -xy''(x)+(\alpha+1-x)y'(x)+ny(x)=0,\quad y(x)=L_n^{(\alpha)}(x). -\end{equation} - -\newpage - -\subsection*{Forward shift operator} -\begin{equation} -\label{shift1Laguerre} -\frac{d}{dx}L_n^{(\alpha)}(x)=-L_{n-1}^{(\alpha+1)}(x). -\end{equation} - -\subsection*{Backward shift operator} -\begin{equation} -\label{shift2LaguerreI} -x\frac{d}{dx}L_n^{(\alpha)}(x)+(\alpha-x)L_n^{(\alpha)}(x)=(n+1)L_{n+1}^{(\alpha-1)}(x) -\end{equation} -or equivalently -\begin{equation} -\label{shift2LaguerreII} -\frac{d}{dx}\left[\expe^{-x}x^{\alpha}L_n^{(\alpha)}(x)\right]=(n+1)\expe^{-x}x^{\alpha-1}L^{(\alpha-1)}_{n+1}(x). -\end{equation} - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodLaguerre} -\e^{-x}x^{\alpha}L_n^{(\alpha)}(x)=\frac{1}{n!}\left(\frac{d}{dx}\right)^n\left[\expe^{-x}x^{n+\alpha}\right]. -\end{equation} - -\subsection*{Generating functions} -\begin{equation} -\label{GenLaguerre1} -(1-t)^{-\alpha-1}\exp\left(\frac{xt}{t-1}\right)= -\sum_{n=0}^{\infty}L_n^{(\alpha)}(x)t^n. -\end{equation} - -\begin{equation} -\label{GenLaguerre2} -\e^t\,\hyp{0}{1}{-}{\alpha+1}{-xt} -=\sum_{n=0}^{\infty}\frac{L_n^{(\alpha)}(x)}{(\alpha+1)_n}t^n. -\end{equation} - -\begin{equation} -\label{GenLaguerre3} -(1-t)^{-\gamma}\,\hyp{1}{1}{\gamma}{\alpha+1}{\frac{xt}{t-1}} -=\sum_{n=0}^{\infty}\frac{(\gamma)_n}{(\alpha+1)_n}L_n^{(\alpha)}(x)t^n, -\quad\textrm{$\gamma$ arbitrary}. -\end{equation} - -\subsection*{Limit relations} - -\subsubsection*{Meixner-Pollaczek $\rightarrow$ Laguerre} -The Laguerre polynomials can be obtained from the Meixner-Pollaczek polynomials given by -(\ref{DefMP}) by the substitution $\lambda=\frac{1}{2}(\alpha+1)$, -$x\rightarrow -\frac{1}{2}\phi^{-1}x$ and the limit $\phi\rightarrow 0$: -$$\lim_{\phi\rightarrow 0} -P_n^{(\frac{1}{2}\alpha+\frac{1}{2})}(-\textstyle\frac{1}{2}\phi^{-1}x;\phi)=L_n^{(\alpha)}(x).$$ - -\subsubsection*{Jacobi $\rightarrow$ Laguerre} -The Laguerre polynomials are obtained from the Jacobi polynomials given by (\ref{DefJacobi}) -if we set $x\rightarrow 1-2\beta^{-1}x$ and then take the limit $\beta\rightarrow\infty$: -$$\lim_{\beta\rightarrow\infty} -P_n^{(\alpha,\beta)}(1-2\beta^{-1}x)=L_n^{(\alpha)}(x).$$ - -\subsubsection*{Meixner $\rightarrow$ Laguerre} -If we take $\beta=\alpha+1$ and $x\rightarrow (1-c)^{-1}x$ in the definition -(\ref{DefMeixner}) of the Meixner polynomials and let $c\rightarrow 1$ we obtain -the Laguerre polynomials: -$$\lim_{c\rightarrow 1} -M_n((1-c)^{-1}x;\alpha+1,c)=\frac{L_n^{(\alpha)}(x)}{L_n^{(\alpha)}(0)}.$$ - -\subsubsection*{Laguerre $\rightarrow$ Hermite} -The Hermite polynomials given by (\ref{DefHermite}) can be obtained from the Laguerre -polynomials by taking the limit $\alpha\rightarrow\infty$ in the following way: -\begin{equation} -\lim_{\alpha\rightarrow\infty} -\left(\frac{2}{\alpha}\right)^{\frac{1}{2}n} -L_n^{(\alpha)}((2\alpha)^{\frac{1}{2}}x+\alpha)=\frac{(-1)^n}{n!}H_n(x). -\end{equation} - -\subsection*{Remarks} -The definition (\ref{DefLaguerre}) of the Laguerre polynomials can also be -written as: -$$L_n^{(\alpha)}(x)=\frac{1}{n!}\sum_{k=0}^n\frac{(-n)_k}{k!}(\alpha+k+1)_{n-k}x^k.$$ -In this way the Laguerre polynomials can also be seen as polynomials in the parameter $\alpha$. -Therefore they can be defined for all $\alpha$. - -\noindent -The Laguerre polynomials are related to the Bessel polynomials given by (\ref{DefBessel}) -in the following way: -$$L_n^{(\alpha)}(x)=\frac{(-x)^n}{n!}y_n(2x^{-1};-2n-\alpha-1).$$ - -\noindent -The Laguerre polynomials are related to the Charlier polynomials given by (\ref{DefCharlier}) -in the following way: -$$\frac{(-a)^n}{n!}C_n(x;a)=L_n^{(x-n)}(a).$$ - -\noindent -The Laguerre polynomials and the Hermite polynomials given by (\ref{DefHermite}) are also -connected by the following quadratic transformations: -$$H_{2n}(x)=(-1)^nn!\,2^{2n}L_n^{(-\frac{1}{2})}(x^2)$$ -and -$$H_{2n+1}(x)=(-1)^nn!\,2^{2n+1}xL_n^{(\frac{1}{2})}(x^2).$$ - -\noindent -In combinatorics the Laguerre polynomials with $\alpha=0$ are often called Rook -\subsection*{9.14 Charlier} -\label{sec9.14} -% -\paragraph{Hypergeometric representation} -\begin{align} -C_n(x;a)&=\hyp20{-n,-x}-{-\,\frac1a} -\label{179}\\ -&=\frac{(-x)_n}{a^n} \hyp11{-n}{x-n+1}a -\label{180}\\ -&=\frac{n!}{(-a)^n}\,L_n^{x-n}(a), -\label{181} -\end{align} -where $L_n^\al(x)$ is a -\hyperref[sec9.12]{Laguerre polynomial}. -Formula \eqref{179} is (9.14.1). Then \eqref{180} follows by reversal -of the summation. Finally \eqref{181} follows by \eqref{180} and -(9.12.1). It is also the Remark on p.249 of \mycite{KLS}, and it -was earlier given in \myciteKLS{416}{(2.7.10)}. -% -\paragraph{Uniqueness of orthogonality measure} -The coefficient of $p_{n-1}(x)$ in (9.14.4) behaves as $O(n)$ as $n\to\iy$. -Hence \eqref{93} holds, by which the orthogonality measure is unique. -polynomials. - -\subsection*{References} -\cite{Abdul}, \cite{Abram}, \cite{Ahmed+82}, \cite{Allaway76}, \cite{Allaway91}, -\cite{NAlSalam66}, \cite{AlSalam64}, \cite{AlSalam90}, \cite{AlSalamChihara72}, -\cite{AlSalamChihara76}, \cite{AndrewsAskey85}, \cite{AndrewsAskeyRoy}, \cite{Askey68}, -\cite{Askey75}, \cite{Askey89I}, \cite{Askey2005}, \cite{AskeyGasper76}, \cite{AskeyGasper77}, -\cite{AskeyIsmail76}, \cite{AskeyIsmailKoorn}, \cite{AskeyWilson85}, \cite{Barrett}, -\cite{Bavinck96}, \cite{Brafman51}, \cite{Brafman57II}, \cite{Brenke}, \cite{Brown}, -\cite{BustozSavage79}, \cite{BustozSavage80}, \cite{Carlitz60}, \cite{Carlitz61I}, -\cite{Carlitz61II}, \cite{Carlitz62}, \cite{Carlitz68}, \cite{ChenSrivastava}, -\cite{ChenIsmail91}, \cite{Chihara68II}, \cite{Chihara78}, \cite{Cohen}, \cite{Cooper+}, -\cite{Dattoli2001}, \cite{DetteStudden92}, \cite{DetteStudden95}, \cite{Dimitrov2003}, -\cite{Doha2003II}, \cite{ElbertLaforgia87I}, \cite{Erdelyi}, \cite{Erdelyi+}, \cite{Exton98}, -\cite{Faldey}, \cite{Gasper73II}, \cite{Gasper77}, \cite{Gatteschi2002}, \cite{Gawronski87}, -\cite{Gawronski93}, \cite{Gillis}, \cite{GillisIsmailOffer}, \cite{Godoy+}, \cite{Grad}, -\cite{Hajmirzaahmad95}, \cite{Ismail74}, \cite{Ismail77}, \cite{Ismail2005II}, -\cite{IsmailLetVal88}, \cite{IsmailLi}, \cite{IsmailMassonRahman}, \cite{IsmailMuldoon}, -\cite{IsmailStanton97}, \cite{IsmailTamhankar}, \cite{Jackson}, \cite{Karlin58}, -\cite{Kochneff95}, \cite{Kochneff97II}, \cite{Koekoek2000}, \cite{Koorn77I}, \cite{Koorn78}, -\cite{Koorn85}, \cite{Koorn88}, \cite{Krasikov2003}, \cite{Krasikov2005}, -\cite{KuijlaarsMcLaughlin}, \cite{KwonLittle}, \cite{LabelleYehI}, \cite{LabelleYehII}, -\cite{LabelleYehIII}, \cite{Lee97I}, \cite{Lee97II}, \cite{Lee97III}, \cite{Lesky95II}, -\cite{Lesky96}, \cite{LewanowiczI}, \cite{LopezTemme2004}, \cite{Mathai}, \cite{Meixner}, -\cite{Nikiforov+}, \cite{NikiforovUvarov}, \cite{Olver}, \cite{PerezPinar}, \cite{Pittaluga}, -\cite{Rainville}, \cite{SainteViennot}, \cite{Srivastava66}, \cite{Srivastava69I}, -\cite{Srivastava69II}, \cite{Srivastava70}, \cite{Srivastava71}, \cite{Szego75}, \cite{Temme}, -\cite{Trickovic}, \cite{Trivedi}, \cite{Viennot}. - - -\section{Bessel}\index{Bessel polynomials} - -\par\setcounter{equation}{0} - -\subsection*{Hypergeometric representation} -\begin{eqnarray} -\label{DefBessel} -y_n(x;a)&=&\hyp{2}{0}{-n,n+a+1}{-}{-\frac{x}{2}}\\ -&=&(n+a+1)_n\left(\frac{x}{2}\right)^n\,\hyp{1}{1}{-n}{-2n-a}{\frac{2}{x}}, -\quad n=0,1,2,\ldots,N.\nonumber -\end{eqnarray} - -\subsection*{Orthogonality relation} -\begin{eqnarray} -\label{OrtBessel} -& &\int_0^{\infty}x^a\e^{-\frac{2}{x}}y_m(x;a)y_n(x;a)\,dx\nonumber\\ -& &=-\frac{2^{a+1}}{2n+a+1}\Gamma(-n-a)n!\,\delta_{mn},\quad a<-2N-1. -\end{eqnarray} - -\subsection*{Recurrence relation} -\begin{eqnarray} -\label{RecBessel} -& &2(n+a+1)(2n+a)y_{n+1}(x;a)\nonumber\\ -& &{}=(2n+a+1)\left[2a+(2n+a)(2n+a+2)x\right]y_n(x;a)\nonumber\\ -& &{}\mathindent{}+2n(2n+a+2)y_{n-1}(x;a). -\end{eqnarray} - -\subsection*{Normalized recurrence relation} -\begin{eqnarray} -\label{NormRecBessel} -xp_n(x)&=&p_{n+1}(x)-\frac{2a}{(2n+a)(2n+a+2)}p_n(x)\nonumber\\ -& &{}\mathindent{}-\frac{4n(n+a)}{(2n+a-1)(2n+a)^2(2n+a+1)}p_{n-1}(x), -\end{eqnarray} -where -$$y_n(x;a)=\frac{(n+a+1)_n}{2^n}p_n(x).$$ - -\subsection*{Differential equation} -\begin{equation} -\label{dvBessel} -x^2y''(x)+\left[(a+2)x+2\right]y'(x)-n(n+a+1)y(x)=0,\quad y(x)=y_n(x;a). -\end{equation} - -\subsection*{Forward shift operator} -\begin{equation} -\label{shift1Bessel} -\frac{d}{dx}y_n(x;a)=\frac{n(n+a+1)}{2}y_{n-1}(x;a+2). -\end{equation} - -\subsection*{Backward shift operator} -\begin{equation} -\label{shift2BesselI} -x^2\frac{d}{dx}y_n(x;a)+(ax+2)y_n(x;a)=2y_{n+1}(x;a-2) -\end{equation} -or equivalently -\begin{equation} -\label{shift2BesselII} -\frac{d}{dx}\left[x^a\expe^{-\frac{2}{x}}y_n(x;a)\right]=2x^{a-2}\expe^{-\frac{2}{x}}y_{n+1}(x;a-2). -\end{equation} - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodBessel} -y_n(x;a)=2^{-n}x^{-a}\expe^{\frac{2}{x}}D^n\left(x^{2n+a}\expe^{-\frac{2}{x}}\right). -\end{equation} - -\subsection*{Generating function} -\begin{equation} -\label{GenBessel} -\left(1-2xt\right)^{-\frac{1}{2}}\left(\frac{2}{1+\sqrt{1-2xt}}\right)^a -\exp\left(\frac{2t}{1+\sqrt{1-2xt}}\right)=\sum_{n=0}^{\infty}y_n(x;a)\frac{t^n}{n!}. -\end{equation} - -\subsection*{Limit relation} - -\subsubsection*{Jacobi $\rightarrow$ Bessel} -If we take $\beta=a-\alpha$ in the definition (\ref{DefJacobi}) of the Jacobi polynomials -and let $\alpha\rightarrow -\infty$ we find the Bessel polynomials: -$$\lim_{\alpha\rightarrow -\infty} -\frac{P_n^{(\alpha,a-\alpha)}(1+\alpha x)}{P_n^{(\alpha,a-\alpha)}(1)}=y_n(x;a).$$ - -\subsection*{Remarks} -The following notations are also used for the Bessel polynomials: -$$y_n(x;a,b)=y_n(2b^{-1}x;a)\quad\textrm{and}\quad\theta_n(x;a,b)=x^ny_n(x^{-1};a,b).$$ -However, the Bessel polynomials essentially depend on only one parameter. - -\noindent -The Bessel polynomials are related to the Laguerre polynomials given by (\ref{DefLaguerre}) -in the following way: -$$L_n^{(\alpha)}(x)=\frac{(-x)^n}{n!}y_n(2x^{-1};-2n-\alpha-1).$$ - -\noindent -The special case $a=-2N-2$ of the Bessel polynomials can be obtained from the pseudo Jacobi -polynomials by setting $x\rightarrow\nu x$ in the definition (\ref{DefPseudoJacobi}) of the -pseudo Jacobi polynomials and taking the limit $\nu\rightarrow\infty$ in the following way: -$$\lim\limits_{\nu\rightarrow\infty}\frac{P_n(\nu x;\nu,N)}{\nu^n} -\subsection*{9.15 Hermite} -\label{sec9.15} -% -\paragraph{Uniqueness of orthogonality measure} -The coefficient of $p_{n-1}(x)$ in (9.15.4) behaves as $O(n)$ as $n\to\iy$. -Hence \eqref{93} holds, by which the orthogonality measure is unique. -% -\paragraph{Fourier transforms} -\begin{equation} -\frac1{\sqrt{2\pi}}\,\int_{-\iy}^\iy H_n(y)\,e^{-\half y^2}\,e^{ixy}\,dy= -i^n\,H_n(x)\,e^{-\half x^2}, -\label{15} -\end{equation} -see \mycite{AAR}{(6.1.15) and Exercise 6.11}. -\begin{equation} -\frac1{\sqrt\pi}\,\int_{-\iy}^\iy H_n(y)\,e^{-y^2}\,e^{ixy}\,dy= -i^n\,x^n\,e^{-\frac14 x^2}, -\label{16} -\end{equation} -see \mycite{DLMF}{(18.17.35)}. -\begin{equation} -\frac{i^n}{2\sqrt\pi}\,\int_{-\iy}^\iy y^n\,e^{-\frac14 y^2}\,e^{-ixy}\,dy= -H_n(x)\,e^{-x^2}, -\label{17} -\end{equation} -see \mycite{AAR}{(6.1.4)}. -=\frac{2^n}{(n-2N-1)_n}y_n(x;-2N-2).$$ - -\subsection*{References} -\cite{Andrade}, \cite{BergVignat}, \cite{Carlitz57I}, \cite{Dattoli2003}, -\cite{DohaAhmed2004}, \cite{DohaAhmed2006}, \cite{Grosswald}, \cite{Ismail2005II}, -\cite{KrallFrink}, \cite{Lesky98}, \cite{NikiforovUvarov}. - - -\section{Charlier}\index{Charlier polynomials} - -\par\setcounter{equation}{0} - -\subsection*{Hypergeometric representation} -\begin{equation} -\label{DefCharlier} -C_n(x;a)=\hyp{2}{0}{-n,-x}{-}{-\frac{1}{a}}. -\end{equation} - -\subsection*{Orthogonality relation} -\begin{equation} -\label{OrtCharlier} -\sum_{x=0}^{\infty}\frac{a^x}{x!}C_m(x;a)C_n(x;a)= -a^{-n}\expe^an!\,\delta_{mn},\quad a > 0. -\end{equation} - -\subsection*{Recurrence relation} -\begin{equation} -\label{RecCharlier} --xC_n(x;a)=aC_{n+1}(x;a)-(n+a)C_n(x;a)+nC_{n-1}(x;a). -\end{equation} - -\subsection*{Normalized recurrence relation} -\begin{equation} -\label{NormRecCharlier} -xp_n(x)=p_{n+1}(x)+(n+a)p_n(x)+nap_{n-1}(x), -\end{equation} -where -$$C_n(x;a)=\left(-\frac{1}{a}\right)^np_n(x).$$ - -\subsection*{Difference equation} -\begin{equation} -\label{dvCharlier} --ny(x)=ay(x+1)-(x+a)y(x)+xy(x-1),\quad y(x)=C_n(x;a). -\end{equation} - -\subsection*{Forward shift operator} -\begin{equation} -\label{shift1CharlierI} -C_n(x+1;a)-C_n(x;a)=-\frac{n}{a}C_{n-1}(x;a) -\end{equation} -or equivalently -\begin{equation} -\label{shift1CharlierII} -\Delta C_n(x;a)=-\frac{n}{a}C_{n-1}(x;a). -\end{equation} - -\subsection*{Backward shift operator} -\begin{equation} -\label{shift2CharlierI} -C_n(x;a)-\frac{x}{a}C_n(x-1;a)=C_{n+1}(x;a) -\end{equation} -or equivalently -\begin{equation} -\label{shift2CharlierII} -\nabla\left[\frac{a^x}{x!}C_n(x;a)\right]=\frac{a^x}{x!}C_{n+1}(x;a). -\end{equation} - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodCharlier} -\frac{a^x}{x!}C_n(x;a)=\nabla^n\left[\frac{a^x}{x!}\right]. -\end{equation} - -\subsection*{Generating function} -\begin{equation} -\label{GenCharlier} -\e^t\left(1-\frac{t}{a}\right)^x=\sum_{n=0}^{\infty}\frac{C_n(x;a)}{n!}t^n. -\end{equation} - -\subsection*{Limit relations} - -\subsubsection*{Meixner $\rightarrow$ Charlier} -If we take $c=(a+\beta)^{-1}a$ in the definition (\ref{DefMeixner}) of the Meixner polynomials -and let $\beta\rightarrow\infty$ we find the Charlier polynomials: -$$\lim_{\beta\rightarrow\infty}M_n(x;\beta,(a+\beta)^{-1}a)=C_n(x;a).$$ - -\subsubsection*{Krawtchouk $\rightarrow$ Charlier} -The Charlier polynomials can be found from the Krawtchouk polynomials given by -(\ref{DefKrawtchouk}) by taking $p=N^{-1}a$ and letting $N\rightarrow\infty$: -$$\lim_{N\rightarrow\infty}K_n(x;N^{-1}a,N)=C_n(x;a).$$ - -\subsubsection*{Charlier $\rightarrow$ Hermite} -The Hermite polynomials given by (\ref{DefHermite}) are obtained from the Charlier polynomials -if we set $x\rightarrow (2a)^{1/2}x+a$ and let $a\rightarrow\infty$. In fact we have -\begin{equation} -\lim_{a\rightarrow\infty} -(2a)^{\frac{1}{2}n}C_n((2a)^{\frac{1}{2}}x+a;a)=(-1)^nH_n(x). -\end{equation} - -\subsection*{Remark} -The Charlier polynomials are related to the Laguerre polynomials given by (\ref{DefLaguerre}) -in the following way: -\subsection*{14.1 Askey-Wilson} -\label{sec14.1} -% -\paragraph{Symmetry} -The Askey-Wilson polynomials $p_n(x;a,b,c,d\,|\,q)$ are \,|\,symmetric -in $a,b,c,d$. -\sLP -This follows from the orthogonality relation (14.1.2) -together with the value of its coefficient of $x^n$ given in (14.1.5b). -Alternatively, combine (14.1.1) with \mycite{GR}{(III.15)}.\\ -As a consequence, it is sufficient to give generating function (14.1.13). Then the generating -functions (14.1.14), (14.1.15) will follow by symmetry in the parameters. -% -\paragraph{Basic hypergeometric representation} -In addition to (14.1.1) we have (in notation \eqref{111}): -\begin{multline} -p_n(\cos\tha;a,b,c,d\,|\, q) -=\frac{(ae^{-i\tha},be^{-i\tha},ce^{-i\tha},de^{-i\tha};q)_n} -{(e^{-2i\tha};q)_n}\,e^{in\tha}\\ -\times {}_8W_7\big(q^{-n}e^{2i\tha};ae^{i\tha},be^{i\tha}, -ce^{i\tha},de^{i\tha},q^{-n};q,q^{2-n}/(abcd)\big). -\label{113} -\end{multline} -This follows from (14.1.1) by combining (III.15) and (III.19) in -\mycite{GR}. -It is also given in \myciteKLS{513}{(4.2)}, but be aware for some slight errors. -The symmetry in $a,b,c,d$ is evident from \eqref{113}. -% -\paragraph{Special value} -\begin{equation} -p_n\big(\thalf(a+a^{-1});a,b,c,d\,|\, q\big)=a^{-n}\,(ab,ac,ad;q)_n\,, -\label{40} -\end{equation} -and similarly for arguments $\thalf(b+b^{-1})$, $\thalf(c+c^{-1})$ and -$\thalf(d+d^{-1})$ by symmetry of $p_n$ in $a,b,c,d$. -% -\paragraph{Trivial symmetry} -\begin{equation} -p_n(-x;a,b,c,d\,|\, q)=(-1)^n p_n(x;-a,-b,-c,-d\,|\, q). -\label{41} -\end{equation} -Both \eqref{40} and \eqref{41} are obtained from (14.1.1). -% -\paragraph{Re: (14.1.5)} -Let -\begin{equation} -p_n(x):=\frac{p_n(x;a,b,c,d\,|\, q)}{2^n(abcdq^{n-1};q)_n}=x^n+\wt k_n x^{n-1} -+\cdots\;. -\label{18} -\end{equation} -Then -\begin{equation} -\wt k_n=-\frac{(1-q^n)(a+b+c+d-(abc+abd+acd+bcd)q^{n-1})} -{2(1-q)(1-abcdq^{2n-2})}\,. -\label{19} -\end{equation} -This follows because $\tilde k_n-\tilde k_{n+1}$ equals the coefficient -$\thalf\bigl(a+a^{-1}-(A_n+C_n)\bigr)$ of $p_n(x)$ in (14.1.5). -% -\paragraph{Generating functions} -Rahman \myciteKLS{449}{(4.1), (4.9)} gives: -\begin{align} -&\sum_{n=0}^\iy \frac{(abcdq^{-1};q)_n a^n}{(ab,ac,ad,q;q)_n}\,t^n\, -p_n(\cos\tha;a,b,c,d\,|\,q) -\nonumber\\ -&=\frac{(abcdtq^{-1};q)_\iy}{(t;q)_\iy}\, -\qhyp65{(abcdq^{-1})^\half,-(abcdq^{-1})^\half,(abcd)^\half, --(abcd)^\half,a e^{i\tha},a e^{-i\tha}} -{ab,ac,ad,abcdtq^{-1},qt^{-1}}{q,q} -\nonumber\\ -&+\frac{(abcdq^{-1},abt,act,adt,ae^{i\tha},ae^{-i\tha};q)_\iy} -{(ab,ac,ad,t^{-1},ate^{i\tha},ate^{-i\tha};q)_\iy} -\nonumber\\ -&\times\qhyp65{t(abcdq^{-1})^\half,-t(abcdq^{-1})^\half,t(abcd)^\half, --t(abcd)^\half,at e^{i\tha},at e^{-i\tha}} -{abt,act,adt,abcdt^2q^{-1},qt}{q,q}\quad(|t|<1). -\label{185} -\end{align} -In the limit \eqref{109} the first term on the \RHS\ of \eqref{185} -tends to the \LHS\ of (9.1.15), while the second term tends formally -to 0. The special case $ad=bc$ of \eqref{185} was earlier given in -\myciteKLS{236}{(4.1), (4.6)}. -% -\paragraph{Limit relations}\quad\sLP -{\bf Askey-Wilson $\longrightarrow$ Wilson}\\ -Instead of (14.1.21) we can keep a polynomial of degree $n$ while the limit is approached: -\begin{equation} -\lim_{q\to1}\frac{p_n(1-\thalf x(1-q)^2;q^a,q^b,q^c,q^d\,|\, q)}{(1-q)^{3n}} -=W_n(x;a,b,c,d). -\label{109} -\end{equation} -For the proof first derive the corresponding limit for the monic polynomials by comparing -(14.1.5) with (9.4.4). -\bLP -{\bf Askey-Wilson $\longrightarrow$ Continuous Hahn}\\ -Instead of (14.4.15) we can keep a polynomial of degree $n$ while the limit is approached: -\begin{multline} -\lim_{q\uparrow1} -\frac{p_n\big(\cos\phi-x(1-q)\sin\phi;q^a e^{i\phi},q^b e^{i\phi},q^{\overline a} e^{-i\phi}, -q^{\overline b} e^{-i\phi}\,|\, q\big)} -{(1-q)^{2n}}\\ -=(-2\sin\phi)^n\,n!\,p_n(x;a,b,\overline a,\overline b)\qquad -(0<\phi<\pi). -\label{177} -\end{multline} -Here the \RHS\ has a continuous Hahn polynomial (9.4.1). -For the proof first derive the corresponding limit for the monic polynomials by comparing -(14.1.5) with (9.1.5). -In fact, define the monic polynomial -\[ -\wt p_n(x):= -\frac{p_n\big(\cos\phi-x(1-q)\sin\phi;q^a e^{i\phi},q^b e^{i\phi},q^{\overline a} e^{-i\phi}, -q^{\overline b} e^{-i\phi}\,|\, q\big)} -{(-2(1-q)\sin\phi)^n\,(abcdq^{n-1};q)_n}\,. -\] -Then it follows from (14.1.5) that -\begin{equation*} -x\,\wt p_n(x)=\wt p_{n+1}(x) -+\frac{(1-q^a)e^{i\phi}+(1-q^{-a})e^{-i\phi}+\wt A_n+\wt C_n}{2(1-q)\sin\phi}\,\wt p_n(x)\\ -+\frac{\wt A_{n-1} \wt C_n}{(1-q)^2 \sin^2\phi}\,\wt p_{n-1}(x), -\end{equation*} -where $\wt A_n$ and $\wt C_n$ are as given after (14.1.3) with $a,b,c,d$ replaced by -$q^a e^{i\phi},q^b e^{i\phi},q^{\overline a} e^{-i\phi},q^{\overline b} e^{-i\phi}$. -Then the recurrence equation for $\wt p_n(x)$ tends for $q\uparrow 1$ to -the recurrence equation (9.4.4) with $c=\overline a$, $d=\overline b$. -\bLP -{\bf Askey-Wilson $\longrightarrow$ Meixner-Pollaczek}\\ -Instead of (14.9.15) we can keep a polynomial of degree $n$ while the limit is approached: -\begin{equation} -\lim_{q\uparrow1} -\frac{p_n\big(\cos\phi-x(1-q)\sin\phi; -q^\la e^{i\phi},0,q^\la e^{-i\phi},0\,|\, q\big)}{(1-q)^n} -=n!\,P_n^{(\la)}(x;\pi-\phi)\quad -(0<\phi<\pi). -\label{178} -\end{equation} -Here the \RHS\ has a Meixner-Pollaczek polynomial (9.7.1). -For the proof first derive the corresponding limit for the monic polynomials by comparing -(14.1.5) with (9.7.4). -In fact, define the monic polynomial -\[ -\wt p_n(x):= -\frac{p_n\big(\cos\phi-x(1-q)\sin\phi; -q^\la e^{i\phi},0,q^\la e^{-i\phi},0\,|\, q\big)}{(-2(1-q)\sin\phi)^n}\,. -\] -Then it follows from (14.1.5) that -\begin{equation*} -x\,\wt p_n(x)=\wt p_{n+1}(x) -+\frac{(1-q^\la)e^{i\phi}+(1-q^{-\la})e^{-i\phi}+\wt A_n+\wt C_n} -{2(1-q)\sin\phi}\,\wt p_n(x)\\ -+\frac{\wt A_{n-1} \wt C_n}{(1-q)^2 \sin^2\phi}\,\wt p_{n-1}(x), -\end{equation*} -where $\wt A_n$ and $\wt C_n$ are as given after (14.1.3) with $a,b,c,d$ replaced by -$q^\la e^{i\phi},0,q^\la e^{-i\phi},0$. -Then the recurrence equation for $\wt p_n(x)$ tends for $q\uparrow 1$ to -the recurrence equation (9.7.4). -% -\paragraph{References} -See also Koornwinder \cite{K7}. -$$\frac{(-a)^n}{n!}C_n(x;a)=L_n^{(x-n)}(a).$$ - -\subsection*{References} -\cite{Allaway76}, \cite{NAlSalam66}, \cite{AlSalam90}, \cite{AlSalamChihara76}, -\cite{AlSalamIsmail76}, \cite{AndrewsAskey85}, \cite{Area+II}, \cite{Askey75}, -\cite{AskeyGasper77}, \cite{AskeyWilson85}, \cite{AtakRahmanSuslov}, \cite{Bavinck98}, -\cite{BavinckKoekoek}, \cite{Chihara78}, \cite{Chihara79}, \cite{Dunkl76}, \cite{Erdelyi+}, -\cite{Gasper73I}, \cite{Gasper74}, \cite{Goh}, \cite{HoareRahman}, \cite{IsmailLetVal88}, -\cite{Koekoek2000}, \cite{Koorn88}, \cite{Krasikov2002}, \cite{LabelleYehI}, -\cite{LabelleYehII}, \cite{LabelleYehIII}, \cite{Lesky62}, \cite{Lesky89}, \cite{Lesky94I}, -\cite{Lesky95II}, \cite{LewanowiczII}, \cite{LopezTemme2004}, \cite{Meixner}, \cite{Nikiforov+}, -\cite{NikiforovUvarov}, \cite{Szafraniec}, \cite{Szego75}, \cite{Viennot}, \cite{Zarzo+}, -\cite{Zeng90}. - - -\section{Hermite}\index{Hermite polynomials} - -\par\setcounter{equation}{0} - -\subsection*{Hypergeometric representation} -\begin{equation} -\label{DefHermite} -H_n(x)=(2x)^n\,\hyp{2}{0}{-n/2,-(n-1)/2}{-}{-\frac{1}{x^2}}. -\end{equation} - -\subsection*{Orthogonality relation} -\begin{equation} -\label{OrtHermite} -\frac{1}{\sqrt{\cpi}}\int_{-\infty}^{\infty}\expe^{-x^2}H_m(x)H_n(x)\,dx -=2^nn!\,\delta_{mn}. -\end{equation} - -\subsection*{Recurrence relation} -\begin{equation} -\label{RecHermite} -H_{n+1}(x)-2xH_n(x)+2nH_{n-1}(x)=0. -\end{equation} - -\subsection*{Normalized recurrence relation} -\begin{equation} -\label{NormRecHermite} -xp_n(x)=p_{n+1}(x)+\frac{n}{2}p_{n-1}(x), -\end{equation} -where -$$H_n(x)=2^np_n(x).$$ - -\subsection*{Differential equation} -\begin{equation} -\label{dvHermite} -y''(x)-2xy'(x)+2ny(x)=0,\quad y(x)=H_n(x). -\end{equation} - -\subsection*{Forward shift operator} -\begin{equation} -\label{shift1Hermite} -\frac{d}{dx}H_n(x)=2nH_{n-1}(x). -\end{equation} - -\subsection*{Backward shift operator} -\begin{equation} -\label{shift2HermiteI} -\frac{d}{dx}H_n(x)-2xH_n(x)=-H_{n+1}(x) -\end{equation} -or equivalently -\begin{equation} -\label{shift2HermiteII} -\frac{d}{dx}\left[\expe^{-x^2}H_n(x)\right]=-\expe^{-x^2}H_{n+1}(x). -\end{equation} - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodHermite} -\e^{-x^2}H_n(x)=(-1)^n\left(\frac{d}{dx}\right)^n\left[\expe^{-x^2}\right]. -\end{equation} - -\subsection*{Generating functions} -\begin{equation} -\label{GenHermite1} -\exp\left(2xt-t^2\right)=\sum_{n=0}^{\infty}\frac{H_n(x)}{n!}t^n. -\end{equation} - -\begin{equation} -\label{GenHermite2} -\left\{\begin{array}{l} -\displaystyle\expe^t\cos(2x\sqrt{t})=\sum_{n=0}^{\infty} -\frac{(-1)^n}{(2n)!}H_{2n}(x)t^n\\[5mm] -\displaystyle\frac{\expe^t}{\sqrt{t}}\sin(2x\sqrt{t})=\sum_{n=0}^{\infty} -\frac{(-1)^n}{(2n+1)!}H_{2n+1}(x)t^n. -\end{array}\right. -\end{equation} - -\begin{equation} -\label{GenHermite3} -\left\{\begin{array}{l} -\displaystyle \expe^{-t^2}\cosh(2xt)=\sum_{n=0}^{\infty} -\frac{H_{2n}(x)}{(2n)!}t^{2n}\\[5mm] -\displaystyle\expe^{-t^2}\sinh(2xt)=\sum_{n=0}^{\infty} -\frac{H_{2n+1}(x)}{(2n+1)!}t^{2n+1}. -\end{array}\right. -\end{equation} - -\begin{equation} -\label{GenHermite4} -\left\{\begin{array}{l} -\displaystyle (1+t^2)^{-\gamma}\,\hyp{1}{1}{\gamma}{\frac{1}{2}}{\frac{x^2t^2}{1+t^2}}= -\sum_{n=0}^{\infty}\frac{(\gamma)_n}{(2n)!}H_{2n}(x)t^{2n}\\[5mm] -\displaystyle\frac{xt}{\sqrt{1+t^2}}\, -\hyp{1}{1}{\gamma+\frac{1}{2}}{\frac{3}{2}}{\frac{x^2t^2}{1+t^2}} -=\sum_{n=0}^{\infty}\frac{(\gamma+\frac{1}{2})_n}{(2n+1)!}H_{2n+1}(x)t^{2n+1} -\end{array}\right. -\end{equation} -with $\gamma$ arbitrary. - -\begin{equation} -\label{GenHermite5} -\frac{1+2xt+4t^2}{(1+4t^2)^{\frac{3}{2}}}\exp\left(\frac{4x^2t^2}{1+4t^2}\right) -=\sum_{n=0}^{\infty}\frac{H_n(x)}{\lfloor n/2\rfloor\,!}t^n, -\end{equation} -where $\lfloor n/2\rfloor$ denotes the largest integer smaller than or equal to $n/2$. - -\subsection*{Limit relations} - -\subsubsection*{Meixner-Pollaczek $\rightarrow$ Hermite} -If we take $x\rightarrow (\sin\phi)^{-1}(x\sqrt{\lambda}-\lambda\cos\phi)$ -in the definition (\ref{DefMP}) of the Meixner-Pollaczek polynomials and -then let $\lambda\rightarrow\infty$ we obtain the Hermite polynomials: -$$\lim_{\lambda\rightarrow\infty} -\lambda^{-\frac{1}{2}n}P_n^{(\lambda)} -((\sin\phi)^{-1}(x\sqrt{\lambda}-\lambda\cos\phi);\phi)=\frac{H_n(x)}{n!}.$$ - -\subsubsection*{Jacobi $\rightarrow$ Hermite} -The Hermite polynomials follow from the Jacobi polynomials given by (\ref{DefJacobi}) by -taking $\beta=\alpha$ and letting $\alpha\rightarrow\infty$ in the following way: -$$\lim_{\alpha\rightarrow\infty} -\alpha^{-\frac{1}{2}n}P_n^{(\alpha,\alpha)}(\alpha^{-\frac{1}{2}}x)=\frac{H_n(x)}{2^nn!}.$$ - -\subsubsection*{Gegenbauer / Ultraspherical $\rightarrow$ Hermite} -The Hermite polynomials follow from the Gegenbauer (or ultraspherical) polynomials given by -(\ref{DefGegenbauer}) by taking $\lambda=\alpha+\frac{1}{2}$ and letting -$\alpha\rightarrow\infty$ in the following way: -$$\lim_{\alpha\rightarrow\infty} -\alpha^{-\frac{1}{2}n}C_n^{(\alpha+\frac{1}{2})}(\alpha^{-\frac{1}{2}}x)=\frac{H_n(x)}{n!}.$$ - -\subsubsection*{Krawtchouk $\rightarrow$ Hermite} -The Hermite polynomials follow from the Krawtchouk polynomials given by (\ref{DefKrawtchouk}) -by setting $x\rightarrow pN+x\sqrt{2p(1-p)N}$ and then letting $N\rightarrow\infty$: -$$\lim_{N\rightarrow\infty} -\sqrt{\binom{N}{n}}K_n(pN+x\sqrt{2p(1-p)N};p,N) -=\frac{\displaystyle (-1)^nH_n(x)}{\displaystyle\sqrt{2^nn!\left(\frac{p}{1-p}\right)^n}}.$$ - -\subsubsection*{Laguerre $\rightarrow$ Hermite} -The Hermite polynomials can be obtained from the Laguerre polynomials given by -(\ref{DefLaguerre}) by taking the limit $\alpha\rightarrow\infty$ in the following way: -$$\lim_{\alpha\rightarrow\infty} -\left(\frac{2}{\alpha}\right)^{\frac{1}{2}n} -L_n^{(\alpha)}((2\alpha)^{\frac{1}{2}}x+\alpha)=\frac{(-1)^n}{n!}H_n(x).$$ - -\subsubsection*{Charlier $\rightarrow$ Hermite} -If we set $x\rightarrow (2a)^{1/2}x+a$ in the definition (\ref{DefCharlier}) -of the Charlier polynomials and let $a\rightarrow\infty$ we find the Hermite -polynomials. In fact we have -$$\lim_{a\rightarrow\infty} -(2a)^{\frac{1}{2}n}C_n((2a)^{\frac{1}{2}}x+a;a)=(-1)^nH_n(x).$$ - -\subsection*{Remarks} -The Hermite polynomials can also be written as: -$$\frac{H_n(x)}{n!}=\sum_{k=0}^{\lfloor n/2\rfloor} -\frac{(-1)^k(2x)^{n-2k}}{k!\,(n-2k)!},$$ -where $\lfloor n/2\rfloor$ denotes the largest integer smaller than or equal to $n/2$. - -\noindent -The Hermite polynomials and the Laguerre polynomials given by (\ref{DefLaguerre}) are also -connected by the following quadratic transformations: -$$H_{2n}(x)=(-1)^nn!\,2^{2n}L_n^{(-\frac{1}{2})}(x^2)$$ -and -\subsection*{14.2 $q$-Racah} -\label{sec14.2} -\paragraph{Symmetry} -\begin{equation} -R_n(x;\al,\be,q^{-N-1},\de\,|\, q) -=\frac{(\be q,\al\de^{-1}q;q)_n}{(\al q,\be\de q;q)_n}\,\de^n\, -R_n(\de^{-1}x;\be,\al,q^{-N-1},\de^{-1}\,|\, q). -\label{84} -\end{equation} -This follows from (14.2.1) combined with \mycite{GR}{(III.15)}. -\sLP -In particular, -\begin{equation} -R_n(x;\al,\be,q^{-N-1},-1\,|\, q) -=\frac{(\be q,-\al q;q)_n}{(\al q,-\be q;q)_n}\,(-1)^n\, -R_n(-x;\be,\al,q^{-N-1},-1\,|\, q), -\label{85} -\end{equation} -and -\begin{equation} -R_n(x;\al,\al,q^{-N-1},-1\,|\, q) -=(-1)^n\,R_n(-x;\al,\al,q^{-N-1},-1\,|\, q), -\label{86} -\end{equation} - -\paragraph{Trivial symmetry} -Clearly from (14.2.1): -\begin{equation} -R_n(x;\al,\be,\ga,\de\,|\, q)=R_n(x;\be\de,\al\de^{-1},\ga,\de\,|\, q) -=R_n(x;\ga,\al\be\ga^{-1},\al,\ga\de\al^{-1}\,|\, q). -\label{83} -\end{equation} -For $\al=q^{-N-1}$ this shows that the three cases -$\al q=q^{-N}$ or $\be\de q=q^{-N}$ or $\ga q=q^{-N}$ of (14.2.1) -are not essentially different. -% -\paragraph{Duality} -It follows from (14.2.1) that -\begin{equation} -R_n(q^{-y}+\ga\de q^{y+1};q^{-N-1},\be,\ga,\de\,|\, q) -=R_y(q^{-n}+\be q^{n-N};\ga,\de,q^{-N-1},\be\,|\, q)\quad -(n,y=0,1,\ldots,N). -\end{equation} -$$H_{2n+1}(x)=(-1)^nn!\,2^{2n+1}xL_n^{(\frac{1}{2})}(x^2).$$ - -\subsection*{14.3 Continuous dual $q$-Hahn} -\label{sec14.3} -The continuous dual $q$-Hahn polynomials are the special case $d=0$ of the -Askey-Wilson polynomials: -\[ -p_n(x;a,b,c\,|\, q):=p_n(x;a,b,c,0\,|\, q). -\] -Hence all formulas in \S14.3 are specializations for $d=0$ of formulas in \S14.1. -\subsection*{References} -\cite{Abram}, \cite{NAlSalam66}, \cite{AlSalam90}, \cite{AlSalamChihara72}, -\cite{AlSalamChihara76}, \cite{AndrewsAskey85}, \cite{AndrewsAskeyRoy}, \cite{Area+I}, -\cite{Askey68}, \cite{Askey75}, \cite{Askey89I}, \cite{AskeyGasper76}, \cite{AskeyWilson85}, -\cite{Azor}, \cite{Berg}, \cite{BilodeauII}, \cite{Brafman51}, \cite{Brafman57II}, -\cite{Brenke}, \cite{CarlitzSrivastava}, \cite{ChenSrivastava}, \cite{Chihara78}, -\cite{Cohen}, \cite{Danese}, \cite{DetteStudden92}, \cite{DetteStudden95}, \cite{Doha2004I}, -\cite{Erdelyi+}, \cite{Faldey}, \cite{Gawronski87}, \cite{Gawronski93}, \cite{Gillis+}, -\cite{Godoy+}, \cite{Grad}, \cite{Ismail74}, \cite{Ismail2005II}, \cite{IsmailStanton97}, -\cite{Koekoek2000}, \cite{Koorn88}, \cite{Krasikov2004}, \cite{Kwon+}, \cite{LabelleYehIII}, -\cite{Lesky95II}, \cite{Lesky96}, \cite{LewanowiczI}, \cite{LopezTemme99}, \cite{Mathai}, -\cite{Meixner}, \cite{Nikiforov+}, \cite{NikiforovUvarov}, \cite{Olver}, \cite{Pittaluga}, -\cite{Rainville}, \cite{SainteViennot}, \cite{Srivastava71}, \cite{SrivastavaMathur}, -\cite{Szego75}, \cite{Temme}, \cite{TemmeLopez2000}, \cite{Trickovic}, \cite{Viennot}, -\cite{Weisner59}, \cite{Wyman}. - -\end{document} diff --git a/KLSadd_insertion/updated9.tex b/KLSadd_insertion/updated9.tex deleted file mode 100644 index e1b34b5..0000000 --- a/KLSadd_insertion/updated9.tex +++ /dev/null @@ -1,4162 +0,0 @@ -\documentclass[envcountchap,graybox]{svmono} - -\addtolength{\textwidth}{1mm} - -\usepackage{amsmath,amssymb} - -\usepackage{amsfonts} -%\usepackage{breqn} -\usepackage{DLMFmath} -\usepackage{DRMFfcns} - -\usepackage{mathptmx} -\usepackage{helvet} -\usepackage{courier} - -\usepackage{makeidx} -\usepackage{graphicx} - -\usepackage{multicol} -\usepackage[bottom]{footmisc} - -\makeindex - -\def\bibname{Bibliography} -\def\refname{Bibliography} - -\def\theequation{\thesection.\arabic{equation}} - -\smartqed - -\let\corollary=\undefined -\spnewtheorem{corollary}[theorem]{Corollary}{\bfseries}{\itshape} - -\newcounter{rom} - -\newcommand{\hyp}[5]{\mbox{}_{#1}{F}_{#2} -\left(\genfrac{}{}{0pt}{}{#3}{#4}\,;\,#5\right)} - -\newcommand{\qhyp}[5]{\mbox{}_{#1}{\phi}_{#2} -\left(\genfrac{}{}{0pt}{}{#3}{#4}\,;\,q,\,#5\right)} - -\newcommand{\mathindent}{\hspace{7.5mm}} - -\newcommand{\e}{\textrm{e}} - -\renewcommand{\E}{\textrm{E}} - -\renewcommand{\textfraction}{-1} - -\renewcommand{\Gamma}{\varGamma} - -\renewcommand{\leftlegendglue}{\hfil} - -\settowidth{\tocchpnum}{14\enspace} -\settowidth{\tocsecnum}{14.30\enspace} -\settowidth{\tocsubsecnum}{14.12.1\enspace} - -\makeatletter -\def\cleartoversopage{\clearpage\if@twoside\ifodd\c@page - \hbox{}\newpage\if@twocolumn\hbox{}\newpage\fi - \else\fi\fi} - -\newcommand{\clearemptyversopage}{ - \clearpage{\pagestyle{empty}\cleartoversopage}} -\makeatother - -\oddsidemargin -1.5cm -\topmargin -2.0cm -\textwidth 16.3cm -\textheight 25cm - -\begin{document} - -\author{Roelof Koekoek\\[2.5mm]Peter A. Lesky\\[2.5mm]Ren\'e F. Swarttouw} -\title{Hypergeometric orthogonal polynomials and their\\$q$-analogues} -\subtitle{-- Monograph --} -\maketitle - -\frontmatter - -\large - -\addtocounter{chapter}{8} -\pagenumbering{roman} -\chapter{Hypergeometric orthogonal polynomials} -\label{HyperOrtPol} - - -In this chapter we deal with all families of hypergeometric orthogonal polynomials -appearing in the Askey scheme on page~\pageref{scheme}. For each family of orthogonal -polynomials we state the most important properties such as a representation as a -hypergeometric function, orthogonality relation(s), the three-term recurrence relation, -the second-order differential or difference equation, the forward shift (or degree lowering) -and backward shift (or degree raising) operator, a Rodrigues-type formula and some generating -functions. In each case we use the notation which seems to be most common in the literature. -Moreover, in each case we mention the connection between various families by stating the -appropriate limit relations. See also \cite{Terwilliger2006} for an algebraic approach of -this Askey scheme and \cite{TemmeLopez2001} for a view from asymptotic analysis. -For notations the reader is referred to chapter~\ref{Definitions}. - -\section{Wilson}\index{Wilson polynomials} - -\par - -\subsection*{Hypergeometric representation} -\begin{eqnarray} -\label{DefWilson} -& &\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\nonumber\\ -& &{}=\hyp{4}{3}{-n,n+a+b+c+d-1,a+ix,a-ix}{a+b,a+c,a+d}{1}. -\end{eqnarray} - -\newpage - -\subsection*{Orthogonality relation} -If $Re(a,b,c,d)>0$ and non-real parameters occur in conjugate pairs, then -\begin{eqnarray} -\label{OrthIWilson} -& &\frac{1}{2\pi}\int_0^{\infty} -\left|\frac{\Gamma(a+ix)\Gamma(b+ix)\Gamma(c+ix)\Gamma(d+ix)}{\Gamma(2ix)}\right|^2\nonumber\\ -& &{}\mathindent\times W_m(x^2;a,b,c,d)W_n(x^2;a,b,c,d)\,dx\nonumber\\ -& &{}=\frac{\Gamma(n+a+b)\cdots\Gamma(n+c+d)}{\Gamma(2n+a+b+c+d)}(n+a+b+c+d-1)_nn!\,\delta_{mn}, -\end{eqnarray} -where -\begin{eqnarray*} -& &\Gamma(n+a+b)\cdots\Gamma(n+c+d)\\ -& &{}=\Gamma(n+a+b)\Gamma(n+a+c)\Gamma(n+a+d)\Gamma(n+b+c)\Gamma(n+b+d)\Gamma(n+c+d). -\end{eqnarray*} -If $a<0$ and $a+b$, $a+c$, $a+d$ are positive or a pair of complex conjugates -occur with positive real parts, then -\begin{eqnarray} -\label{OrtIIWilson} -& &\frac{1}{2\pi}\int_0^{\infty}\left|\frac{\Gamma(a+ix)\Gamma(b+ix)\Gamma(c+ix)\Gamma(d+ix)}{\Gamma(2ix)}\right|^2\nonumber\\ -& &{}\mathindent\mathindent\times W_m(x^2;a,b,c,d)W_n(x^2;a,b,c,d)\,dx\nonumber\\ -& &{}\mathindent{}+\frac{\Gamma(a+b)\Gamma(a+c)\Gamma(a+d)\Gamma(b-a)\Gamma(c-a)\Gamma(d-a)}{\Gamma(-2a)}\nonumber\\ -& &{}\mathindent\mathindent{}\times\sum_{\begin{array}{c} -{\scriptstyle k=0,1,2\ldots}\\{\scriptstyle a+k<0}\end{array}} -\frac{(2a)_k(a+1)_k(a+b)_k(a+c)_k(a+d)_k}{(a)_k(a-b+1)_k(a-c+1)_k(a-d+1)_kk!}\nonumber\\ -& &{}\mathindent\mathindent\mathindent\mathindent{}\times W_m(-(a+k)^2;a,b,c,d)W_n(-(a+k)^2;a,b,c,d)\nonumber\\ -& &{}=\frac{\Gamma(n+a+b)\cdots\Gamma(n+c+d)}{\Gamma(2n+a+b+c+d)}(n+a+b+c+d-1)_nn!\,\delta_{mn}. -\end{eqnarray} - -\subsection*{Recurrence relation} -\begin{equation} -\label{RecWilson} --\left(a^2+x^2\right){\tilde{W}}_n(x^2) -=A_n{\tilde{W}}_{n+1}(x^2)-\left(A_n+C_n\right){\tilde{W}}_n(x^2)+C_n{\tilde{W}}_{n-1}(x^2), -\end{equation} -where -$${\tilde{W}}_n(x^2):={\tilde{W}}_n(x^2;a,b,c,d)=\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}$$ -and -$$\left\{\begin{array}{l} -\displaystyle A_n=\frac{(n+a+b+c+d-1)(n+a+b)(n+a+c)(n+a+d)}{(2n+a+b+c+d-1)(2n+a+b+c+d)}\\[5mm] -\displaystyle C_n=\frac{n(n+b+c-1)(n+b+d-1)(n+c+d-1)}{(2n+a+b+c+d-2)(2n+a+b+c+d-1)}. -\end{array}\right.$$ - -\subsection*{Normalized recurrence relation} -\begin{equation} -\label{NormRecWilson} -xp_n(x)=p_{n+1}(x)+(A_n+C_n-a^2)p_n(x)+A_{n-1}C_np_{n-1}(x), -\end{equation} -where -$$W_n(x^2;a,b,c,d)=(-1)^n(n+a+b+c+d-1)_np_n(x^2).$$ - -\subsection*{Difference equation} -\begin{eqnarray} -\label{dvWilson} -& &n(n+a+b+c+d-1)y(x)\nonumber\\ -& &{}=B(x)y(x+i)-\left[B(x)+D(x)\right]y(x)+D(x)y(x-i), -\end{eqnarray} -where -$$y(x)=W_n(x^2;a,b,c,d)$$ -and -$$\left\{\begin{array}{l} -\displaystyle B(x)=\frac{(a-ix)(b-ix)(c-ix)(d-ix)}{2ix(2ix-1)}\\[5mm] -\displaystyle D(x)=\frac{(a+ix)(b+ix)(c+ix)(d+ix)}{2ix(2ix+1)}. -\end{array}\right.$$ - -\subsection*{Forward shift operator} -\begin{eqnarray} -\label{shift1WilsonI} -& &W_n((x+\textstyle\frac{1}{2}i)^2;a,b,c,d) --W_n((x-\textstyle\frac{1}{2}i)^2;a,b,c,d)\nonumber\\ -& &{}=-2inx(n+a+b+c+d-1)W_{n-1}(x^2;a+\textstyle\frac{1}{2},b+\textstyle\frac{1}{2}, -c+\textstyle\frac{1}{2},d+\textstyle\frac{1}{2}) -\end{eqnarray} -or equivalently -\begin{eqnarray} -\label{shift1WilsonII} -& &\frac{\delta W_n(x^2;a,b,c,d)}{\delta x^2}\nonumber\\ -& &{}=-n(n+a+b+c+d-1)W_{n-1}(x^2;a+\textstyle\frac{1}{2},b+\textstyle\frac{1}{2}, -c+\textstyle\frac{1}{2},d+\textstyle\frac{1}{2}). -\end{eqnarray} - -\newpage - -\subsection*{Backward shift operator} -\begin{eqnarray} -\label{shift2WilsonI} -& &(a-\textstyle\frac{1}{2}-ix)(b-\textstyle\frac{1}{2}-ix) -(c-\textstyle\frac{1}{2}-ix)(d-\textstyle\frac{1}{2}-ix) -W_n((x+\textstyle\frac{1}{2}i)^2;a,b,c,d)\nonumber\\ -& &{}\mathindent{}-(a-\textstyle\frac{1}{2}+ix)(b-\textstyle\frac{1}{2}+ix) -(c-\textstyle\frac{1}{2}+ix)(d-\textstyle\frac{1}{2}+ix) -W_n((x-\textstyle\frac{1}{2}i)^2;a,b,c,d)\nonumber\\ -& &{}=-2ix W_{n+1}(x^2;a-\textstyle\frac{1}{2},b-\textstyle\frac{1}{2}, -c-\textstyle\frac{1}{2},d-\textstyle\frac{1}{2}) -\end{eqnarray} -or equivalently -\begin{eqnarray} -\label{shift2WilsonII} -& &\frac{\delta\left[\omega(x;a,b,c,d)W_n(x^2;a,b,c,d)\right]}{\delta x^2}\nonumber\\ -& &{}=\omega(x;a-\textstyle\frac{1}{2},b-\textstyle\frac{1}{2}, -c-\textstyle\frac{1}{2},d-\textstyle\frac{1}{2}) -W_{n+1}(x^2;a-\textstyle\frac{1}{2},b-\textstyle\frac{1}{2},c-\textstyle\frac{1}{2},d-\textstyle\frac{1}{2}), -\end{eqnarray} -where -$$\omega(x;a,b,c,d):=\frac{1}{2ix}\left|\frac{\Gamma(a+ix)\Gamma(b+ix)\Gamma(c+ix)\Gamma(d+ix)}{\Gamma(2ix)}\right|^2.$$ - -\subsection*{Rodrigues-type formula} -\begin{eqnarray} -\label{RodWilson} -& &\omega(x;a,b,c,d)W_n(x^2;a,b,c,d)\nonumber\\ -& &{}=\left(\frac{\delta}{\delta x^2}\right)^n -\left[\omega(x;a+\textstyle\frac{1}{2}n,b+\textstyle\frac{1}{2}n, -c+\textstyle\frac{1}{2}n,d+\textstyle\frac{1}{2}n)\right]. -\end{eqnarray} - -\subsection*{Generating functions} -\begin{equation} -\label{GenWilson1} -\hyp{2}{1}{a+ix,b+ix}{a+b}{t}\,\hyp{2}{1}{c-ix,d-ix}{c+d}{t}= -\sum_{n=0}^{\infty}\frac{W_n(x^2;a,b,c,d)t^n}{(a+b)_n(c+d)_nn!}. -\end{equation} - -\begin{equation} -\label{GenWilson2} -\hyp{2}{1}{a+ix,c+ix}{a+c}{t}\,\hyp{2}{1}{b-ix,d-ix}{b+d}{t}= -\sum_{n=0}^{\infty}\frac{W_n(x^2;a,b,c,d)t^n}{(a+c)_n(b+d)_nn!}. -\end{equation} - -\begin{equation} -\label{GenWilson3} -\hyp{2}{1}{a+ix,d+ix}{a+d}{t}\,\hyp{2}{1}{b-ix,c-ix}{b+c}{t}= -\sum_{n=0}^{\infty}\frac{W_n(x^2;a,b,c,d)t^n}{(a+d)_n(b+c)_nn!}. -\end{equation} - -\begin{eqnarray} -\label{GenWilson4} -& &(1-t)^{1-a-b-c-d}\nonumber\\ -& &{}\mathindent\times\hyp{4}{3}{\frac{1}{2}(a+b+c+d-1),\frac{1}{2}(a+b+c+d),a+ix,a-ix} -{a+b,a+c,a+d}{-\frac{4t}{(1-t)^2}}\nonumber\\ -& &{}=\sum_{n=0}^{\infty}\frac{(a+b+c+d-1)_n}{(a+b)_n(a+c)_n(a+d)_nn!}W_n(x^2;a,b,c,d)t^n. -\end{eqnarray} - -\subsection*{Limit relations} - -\subsubsection*{Wilson $\rightarrow$ Continuous dual Hahn} -The continuous dual Hahn polynomials given by (\ref{DefContDualHahn}) can be found from the -Wilson polynomials by dividing by $(a+d)_n$ and letting $d\rightarrow\infty$: -\begin{equation} -\lim_{d\rightarrow\infty}\frac{W_n(x^2;a,b,c,d)}{(a+d)_n}=S_n(x^2;a,b,c). -\end{equation} - -\subsubsection*{Wilson $\rightarrow$ Continuous Hahn} -The continuous Hahn polynomials given by (\ref{DefContHahn}) are obtained from the Wilson -polynomials by the substitutions $a\rightarrow a-it$, $b\rightarrow b-it$, $c\rightarrow c+it$, -$d\rightarrow d+it$ and $x\rightarrow x+t$ and the limit $t\rightarrow\infty$ in the following -way: -\begin{equation} -\lim_{t\rightarrow\infty} -\frac{W_n((x+t)^2;a-it,b-it,c+it,d+it)}{(-2t)^nn!}=p_n(x;a,b,c,d). -\end{equation} - -\subsubsection*{Wilson $\rightarrow$ Jacobi} -The Jacobi polynomials given by (\ref{DefJacobi}) can be found from the Wilson -polynomials by substituting $a=b=\frac{1}{2}(\alpha+1)$, $c=\frac{1}{2}(\beta+1)+it$, -$d=\frac{1}{2}(\beta+1)-it$ and $x\rightarrow t\sqrt{\frac{1}{2}(1-x)}$ in the -definition (\ref{DefWilson}) of the Wilson polynomials and taking the limit -$t\rightarrow\infty$. In fact we have -\begin{eqnarray} -& &\lim_{t\rightarrow\infty}\frac{W_n(\frac{1}{2}(1-x)t^2;\frac{1}{2}(\alpha+1), -\frac{1}{2}(\alpha+1),\frac{1}{2}(\beta+1)+it,\frac{1}{2}(\beta+1)-it)}{t^{2n}n!}\nonumber\\ -& &{}=P_n^{(\alpha,\beta)}(x). -\end{eqnarray} - -\subsection*{Remarks} -Note that for $k0$, $c=\bar{a}$ and $d=\bar{b}$, then -\begin{eqnarray} -\label{OrtContHahn} -& &\frac{1}{2\pi}\int_{-\infty}^{\infty}\Gamma(a+ix)\Gamma(b+ix)\Gamma(c-ix)\Gamma(d-ix) -p_m(x;a,b,c,d)p_n(x;a,b,c,d)\,dx\nonumber\\ -& &{}=\frac{\Gamma(n+a+c)\Gamma(n+a+d)\Gamma(n+b+c)\Gamma(n+b+d)} -{(2n+a+b+c+d-1)\Gamma(n+a+b+c+d-1)n!}\,\delta_{mn}. -\end{eqnarray} - -\subsection*{Recurrence relation} -\begin{equation} -\label{RecContHahn} -(a+ix)\tilde{p}_n(x)=A_n\tilde{p}_{n+1}(x)-\left(A_n+C_n\right)\tilde{p}_n(x)+C_n\tilde{p}_{n-1}(x), -\end{equation} -where -$$\tilde{p}_n(x):=\tilde{p}_n(x;a,b,c,d)=\frac{n!}{i^n(a+c)_n(a+d)_n}p_n(x;a,b,c,d)$$ -and -$$\left\{\begin{array}{l} -\displaystyle A_n=-\frac{(n+a+b+c+d-1)(n+a+c)(n+a+d)}{(2n+a+b+c+d-1)(2n+a+b+c+d)}\\[5mm] -\displaystyle C_n=\frac{n(n+b+c-1)(n+b+d-1)}{(2n+a+b+c+d-2)(2n+a+b+c+d-1)}. -\end{array}\right.$$ - -\subsection*{Normalized recurrence relation} -\begin{equation} -\label{NormRecContHahn} -xp_n(x)=p_{n+1}(x)+i(A_n+C_n+a)p_n(x)-A_{n-1}C_np_{n-1}(x), -\end{equation} -where -$$p_n(x;a,b,c,d)=\frac{(n+a+b+c+d-1)_n}{n!}p_n(x).$$ - -\subsection*{Difference equation} -\begin{eqnarray} -\label{dvContHahn} -& &n(n+a+b+c+d-1)y(x)\nonumber\\ -& &{}=B(x)y(x+i)-\left[B(x)+D(x)\right]y(x)+D(x)y(x-i), -\end{eqnarray} -where -$$y(x)=p_n(x;a,b,c,d)$$ -and -$$\left\{\begin{array}{l} -\displaystyle B(x)=(c-ix)(d-ix)\\[5mm] -\displaystyle D(x)=(a+ix)(b+ix). -\end{array}\right.$$ - -\subsection*{Forward shift operator} -\begin{eqnarray} -\label{shift1ContHahnI} -& &p_n(x+\textstyle\frac{1}{2}i;a,b,c,d)-p_n(x-\textstyle\frac{1}{2}i;a,b,c,d)\nonumber\\ -& &{}=i(n+a+b+c+d-1)p_{n-1}(x;a+\textstyle\frac{1}{2}, -b+\textstyle\frac{1}{2},c+\textstyle\frac{1}{2},d+\textstyle\frac{1}{2}) -\end{eqnarray} -or equivalently -\begin{equation} -\label{shift1ContHahnII} -\frac{\delta p_n(x;a,b,c,d)}{\delta x}=(n+a+b+c+d-1) -p_{n-1}(x;a+\textstyle\frac{1}{2},b+\textstyle\frac{1}{2}, -c+\textstyle\frac{1}{2},d+\textstyle\frac{1}{2}). -\end{equation} - -\subsection*{Backward shift operator} -\begin{eqnarray} -\label{shift2ContHahnI} -& &(c-\textstyle\frac{1}{2}-ix)(d-\textstyle\frac{1}{2}-ix) -p_n(x+\textstyle\frac{1}{2}i;a,b,c,d)\nonumber\\ -& &{}\mathindent{}-(a-\textstyle\frac{1}{2}+ix)(b-\textstyle\frac{1}{2}+ix) -p_n(x-\textstyle\frac{1}{2}i;a,b,c,d)\nonumber\\ -& &{}=\frac{n+1}{i}p_{n+1}(x;a-\textstyle\frac{1}{2}, -b-\textstyle\frac{1}{2},c-\textstyle\frac{1}{2},d-\textstyle\frac{1}{2}) -\end{eqnarray} -or equivalently -\begin{eqnarray} -\label{shift2ContHahnII} -& &\frac{\delta\left[\omega(x;a,b,c,d)p_n(x;a,b,c,d)\right]}{\delta x}\nonumber\\ -& &{}=-(n+1)\omega(x;a-\textstyle\frac{1}{2},b-\textstyle\frac{1}{2}, -c-\textstyle\frac{1}{2},d-\textstyle\frac{1}{2})\nonumber\\ -& &{}\mathindent\times p_{n+1}(x;a-\textstyle\frac{1}{2},b-\textstyle\frac{1}{2},c-\textstyle\frac{1}{2},d-\textstyle\frac{1}{2}), -\end{eqnarray} -where -$$\omega(x;a,b,c,d)=\Gamma(a+ix)\Gamma(b+ix)\Gamma(c-ix)\Gamma(d-ix).$$ - -\subsection*{Rodrigues-type formula} -\begin{eqnarray} -\label{RodContHahn} -& &\omega(x;a,b,c,d)p_n(x;a,b,c,d)\nonumber\\ -& &{}=\frac{(-1)^n}{n!}\left(\frac{\delta}{\delta x}\right)^n -\left[\omega(x;a+\textstyle\frac{1}{2}n,b+\textstyle\frac{1}{2}n, -c+\textstyle\frac{1}{2}n,d+\textstyle\frac{1}{2}n)\right]. -\end{eqnarray} - -\subsection*{Generating functions} -\begin{equation} -\label{GenContHahn1} -\hyp{1}{1}{a+ix}{a+c}{-it}\,\hyp{1}{1}{d-ix}{b+d}{it}= -\sum_{n=0}^{\infty}\frac{p_n(x;a,b,c,d)}{(a+c)_n(b+d)_n}t^n. -\end{equation} - -\begin{equation} -\label{GenContHahn2} -\hyp{1}{1}{a+ix}{a+d}{-it}\,\hyp{1}{1}{c-ix}{b+c}{it}= -\sum_{n=0}^{\infty}\frac{p_n(x;a,b,c,d)}{(a+d)_n(b+c)_n}t^n. -\end{equation} - -\begin{eqnarray} -\label{GenContHahn3} -& &(1-t)^{1-a-b-c-d}\,\hyp{3}{2}{\frac{1}{2}(a+b+c+d-1),\frac{1}{2}(a+b+c+d),a+ix} -{a+c,a+d}{-\frac{4t}{(1-t)^2}}\nonumber\\ -& &{}=\sum_{n=0}^{\infty} -\frac{(a+b+c+d-1)_n}{(a+c)_n(a+d)_ni^n}p_n(x;a,b,c,d)t^n. -\end{eqnarray} - -\subsection*{Limit relations} - -\subsubsection*{Wilson $\rightarrow$ Continuous Hahn} -The continuous Hahn polynomials are obtained from the Wilson polynomials given by -(\ref{DefWilson}) by the substitution $a\rightarrow a-it$, $b\rightarrow b-it$, -$c\rightarrow c+it$, $d\rightarrow d+it$ and $x\rightarrow x+t$ and the limit -$t\rightarrow\infty$ in the following way: -$$\lim_{t\rightarrow\infty} -\frac{W_n((x+t)^2;a-it,b-it,c+it,d+it)}{(-2t)^nn!}=p_n(x;a,b,c,d).$$ - -\subsubsection*{Continuous Hahn $\rightarrow$ Meixner-Pollaczek} -The Meixner-Pollaczek polynomials given by (\ref{DefMP}) can be obtained from the -continuous Hahn polynomials by setting $x\rightarrow x+t$, $a=\lambda-it$, $c=\lambda+it$ -and $b=d=t\tan\phi$ and taking the limit $t\rightarrow\infty$: -\begin{equation} -\lim_{t\rightarrow\infty}\frac{p_n(x+t;\lambda-it,t\tan\phi,\lambda+it,t\tan\phi)}{t^n} -=\frac{P_n^{(\lambda)}(x;\phi)}{(\cos\phi)^n}. -\end{equation} - -\subsubsection*{Continuous Hahn $\rightarrow$ Jacobi} -The Jacobi polynomials given by (\ref{DefJacobi}) follow from the continuous Hahn polynomials -by the substitution $x\rightarrow \frac{1}{2}xt$, $a=\frac{1}{2}(\alpha+1-it)$, -$b=\frac{1}{2}(\beta+1+it)$, $c=\frac{1}{2}(\alpha+1+it)$ and $d=\frac{1}{2}(\beta+1-it)$, -division by $t^n$ and the limit $t\rightarrow\infty$: -\begin{eqnarray} -& &\lim_{t\rightarrow\infty} -\frac{p_n(\frac{1}{2}xt;\frac{1}{2}(\alpha+1-it),\frac{1}{2}(\beta+1+it), -\frac{1}{2}(\alpha+1+it),\frac{1}{2}(\beta+1-it))}{t^n}\nonumber\\ -& &{}=P_n^{(\alpha,\beta)}(x). -\end{eqnarray} - -\subsubsection*{Continuous Hahn $\rightarrow$ Pseudo Jacobi} -The pseudo Jacobi polynomials given by (\ref{DefPseudoJacobi}) follow from the continuous -Hahn polynomials by the substitution $x\rightarrow xt$, $a=\frac{1}{2}(-N+i\nu-2t)$, -$b=\frac{1}{2}(-N-i\nu+2t)$, $c=\frac{1}{2}(-N-i\nu-2t)$ and $d=\frac{1}{2}(-N+i\nu+2t)$, -division by $t^n$ and the limit $t\rightarrow\infty$: -\begin{eqnarray} -& &\lim_{t\rightarrow\infty}\frac{p_n(xt;\frac{1}{2}(-N+i\nu-2t),\frac{1}{2}(-N-i\nu+2t), -\frac{1}{2}(-N+i\nu-2t),\frac{1}{2}(-N-i\nu+2t))}{t^n}\nonumber\\ -& &{}=\frac{(n-2N-1)_n}{n!}P_n(x;\nu,N). -\end{eqnarray} - -\subsection*{Remark} -Since we have for $k0$ and $(c,d)=(\overline a,\overline b)$ or $(\overline b,\overline a)$.\\ -Thus, under these assumptions, the continuous Hahn polynomial -$p_n(x;a,b,c,d)$ -is symmetric in $a,b$ and in $c,d$. -This follows from the orthogonality relation (9.4.2) -together with the value of its coefficient of $x^n$ given in (9.4.4b).\\ -As a consequence, it is sufficient to give generating function (9.4.11). Then the generating -function (9.4.12) will follow by symmetry in the parameters. -% -\paragraph{Uniqueness of orthogonality measure} -The coefficient of $p_{n-1}(x)$ in (9.4.4) behaves as $O(n^2)$ as $n\to\iy$. -Hence \eqref{93} holds, by which the orthogonality measure is unique. -% -\paragraph{Special cases} -In the following special case there is a reduction to -Meixner-Pollaczek: -\begin{equation} -p_n(x;a,a+\thalf,a,a+\thalf)= -\frac{(2a)_n (2a+\thalf)_n}{(4a)_n}\,P_n^{(2a)}(2x;\thalf\pi). -\end{equation} -See \myciteKLS{342}{(2.6)} (note that in \myciteKLS{342}{(2.3)} the -Meixner-Pollaczek polynonmials are defined different from (9.7.1), -without a constant factor in front). - -For $0-1$ and $\beta>-1$, or for $\alpha<-N$ and $\beta<-N$, we have -\begin{eqnarray} -\label{OrtHahn} -& &\sum_{x=0}^N\binom{\alpha +x}{x}\binom{\beta+N-x}{N-x}Q_m(x;\alpha,\beta,N)Q_n(x;\alpha,\beta,N)\nonumber\\ -& &{}=\frac{(-1)^n(n+\alpha+\beta+1)_{N+1}(\beta+1)_nn!}{(2n+\alpha+\beta+1)(\alpha+1)_n(-N)_nN!}\,\delta_{mn}. -\end{eqnarray} - -\subsection*{Recurrence relation} -\begin{equation} -\label{RecHahn} --xQ_n(x)=A_nQ_{n+1}(x)-\left(A_n+C_n\right)Q_n(x)+C_nQ_{n-1}(x), -\end{equation} -where -$$Q_n(x):=Q_n(x;\alpha,\beta,N)$$ -and -$$\left\{\begin{array}{l} -\displaystyle A_n=\frac{(n+\alpha+\beta+1)(n+\alpha+1)(N-n)}{(2n+\alpha+\beta+1)(2n+\alpha+\beta+2)}\\[5mm] -\displaystyle C_n=\frac{n(n+\alpha+\beta+N+1)(n+\beta)}{(2n+\alpha+\beta)(2n+\alpha+\beta+1)}. -\end{array}\right.$$ - -\subsection*{Normalized recurrence relation} -\begin{equation} -\label{NormRecHahn} -xp_n(x)=p_{n+1}(x)+\left(A_n+C_n\right)p_n(x)+A_{n-1}C_np_{n-1}(x), -\end{equation} -where -$$Q_n(x;\alpha,\beta,N)=\frac{(n+\alpha+\beta+1)_n}{(\alpha+1)_n(-N)_n}p_n(x).$$ - -\subsection*{Difference equation} -\begin{equation} -\label{dvHahn} -n(n+\alpha+\beta+1)y(x)=B(x)y(x+1)-\left[B(x)+D(x)\right]y(x)+D(x)y(x-1), -\end{equation} -where -$$y(x)=Q_n(x;\alpha,\beta,N)$$ -and -$$\left\{\begin{array}{l} -\displaystyle B(x)=(x+\alpha+1)(x-N)\\[5mm] -\displaystyle D(x)=x(x-\beta-N-1). -\end{array}\right.$$ - -\subsection*{Forward shift operator} -\begin{eqnarray} -\label{shift1HahnI} -& &Q_n(x+1;\alpha,\beta,N)-Q_n(x;\alpha,\beta,N)\nonumber\\ -& &{}=-\frac{n(n+\alpha+\beta+1)}{(\alpha+1)N}Q_{n-1}(x;\alpha+1,\beta+1,N-1) -\end{eqnarray} -or equivalently -\begin{equation} -\label{shift1HahnII} -\Delta Q_n(x;\alpha,\beta,N)=-\frac{n(n+\alpha+\beta+1)}{(\alpha+1)N}Q_{n-1}(x;\alpha+1,\beta+1,N-1). -\end{equation} - -\subsection*{Backward shift operator} -\begin{eqnarray} -\label{shift2HahnI} -& &(x+\alpha)(N+1-x)Q_n(x;\alpha,\beta,N)-x(\beta+N+1-x)Q_n(x-1;\alpha,\beta,N)\nonumber\\ -& &{}=\alpha(N+1)Q_{n+1}(x;\alpha-1,\beta-1,N+1) -\end{eqnarray} -or equivalently -\begin{eqnarray} -\label{shift2HahnII} -& &\nabla\left[\omega(x;\alpha,\beta,N)Q_n(x;\alpha,\beta,N)\right]\nonumber\\ -& &{}=\frac{N+1}{\beta}\omega(x;\alpha-1,\beta-1,N+1)Q_{n+1}(x;\alpha-1,\beta-1,N+1), -\end{eqnarray} -where -$$\omega(x;\alpha,\beta,N)=\binom{\alpha+x}{x}\binom{\beta+N-x}{N-x}.$$ - -\subsection*{Rodrigues-type formula} -\begin{eqnarray} -\label{RodHahn} -& &\omega(x;\alpha,\beta,N)Q_n(x;\alpha,\beta,N)\nonumber\\ -& &{}=\frac{(-1)^n(\beta+1)_n}{(-N)_n}\nabla^n\left[\omega(x;\alpha+n,\beta+n,N-n)\right]. -\end{eqnarray} - -\subsection*{Generating functions} -For $x=0,1,2,\ldots,N$ we have -\begin{equation} -\label{GenHahn1} -\hyp{1}{1}{-x}{\alpha+1}{-t}\,\hyp{1}{1}{x-N}{\beta+1}{t} -=\sum_{n=0}^N\frac{(-N)_n}{(\beta+1)_nn!}Q_n(x;\alpha,\beta,N)t^n. -\end{equation} - -\begin{eqnarray} -\label{GenHahn2} -& &\hyp{2}{0}{-x,-x+\beta+N+1}{-}{-t}\,\hyp{2}{0}{x-N,x+\alpha+1}{-}{t}\nonumber\\ -& &{}=\sum_{n=0}^N\frac{(-N)_n(\alpha+1)_n}{n!}Q_n(x;\alpha,\beta,N)t^n. -\end{eqnarray} - -\begin{eqnarray} -\label{GenHahn3} -& &\left[(1-t)^{-\alpha-\beta-1}\,\hyp{3}{2}{\frac{1}{2}(\alpha+\beta+1),\frac{1}{2}(\alpha+\beta+2),-x} -{\alpha+1,-N}{-\frac{4t}{(1-t)^2}}\right]_N\nonumber\\ -& &{}=\sum_{n=0}^N\frac{(\alpha+\beta+1)_n}{n!}Q_n(x;\alpha,\beta,N)t^n. -\end{eqnarray} - -\subsection*{Limit relations} - -\subsubsection*{Racah $\rightarrow$ Hahn} -If we take $\gamma+1=-N$ and let $\delta\rightarrow\infty$ in the definition (\ref{DefRacah}) -of the Racah polynomials, we obtain the Hahn polynomials. Hence -$$\lim_{\delta\rightarrow\infty} -R_n(\lambda(x);\alpha,\beta,-N-1,\delta)=Q_n(x;\alpha,\beta,N).$$ -And if we take $\delta=-\beta-N-1$ and let $\gamma\rightarrow\infty$ in the definition (\ref{DefRacah}) -of the Racah polynomials, we also obtain the Hahn polynomials: -$$\lim_{\gamma\rightarrow\infty} -R_n(\lambda(x);\alpha,\beta,\gamma,-\beta-N-1)=Q_n(x;\alpha,\beta,N).$$ -Another way to do this is to take $\alpha+1=-N$ and $\beta\rightarrow\beta+\gamma+N+1$ in -the definition (\ref{DefRacah}) of the Racah polynomials and then take the limit -$\delta\rightarrow\infty$. In that case we obtain the Hahn polynomials in the following way: -$$\lim_{\delta\rightarrow\infty} -R_n(\lambda(x);-N-1,\beta+\gamma+N+1,\gamma,\delta)=Q_n(x;\gamma,\beta,N).$$ - -\subsubsection*{Hahn $\rightarrow$ Jacobi} -To find the Jacobi polynomials given by (\ref{DefJacobi}) from the Hahn polynomials we take -$x\rightarrow Nx$ and let $N\rightarrow\infty$. In fact we have -\begin{equation} -\lim_{N\rightarrow\infty} -Q_n(Nx;\alpha,\beta,N)=\frac{P_n^{(\alpha,\beta)}(1-2x)}{P_n^{(\alpha,\beta)}(1)}. -\end{equation} - -\subsubsection*{Hahn $\rightarrow$ Meixner} -The Meixner polynomials given by (\ref{DefMeixner}) can be obtained from the Hahn polynomials -by taking $\alpha=b-1$, $\beta=N(1-c)c^{-1}$ and letting $N\rightarrow\infty$: -\begin{equation} -\lim_{N\rightarrow\infty} -Q_n(x;b-1,N(1-c)c^{-1},N)=M_n(x;b,c). -\end{equation} - -\subsubsection*{Hahn $\rightarrow$ Krawtchouk} -The Krawtchouk polynomials given by (\ref{DefKrawtchouk}) are obtained from the Hahn polynomials -if we take $\alpha=pt$ and $\beta=(1-p)t$ and let $t\rightarrow\infty$: -\begin{equation} -\lim_{t\rightarrow\infty}Q_n(x;pt,(1-p)t,N)=K_n(x;p,N). -\end{equation} - -\subsection*{Remark} -If we interchange the role of $x$ and $n$ in (\ref{DefHahn}) we obtain the -dual Hahn polynomials given by (\ref{DefDualHahn}). - -\noindent -Since -$$Q_n(x;\alpha,\beta,N)=R_x(\lambda(n);\alpha,\beta,N)$$ -we obtain the dual orthogonality relation for the Hahn polynomials from the -orthogonality relation (\ref{OrtDualHahn}) of the dual Hahn polynomials: -\begin{eqnarray*} -& &\sum_{n=0}^N\frac{(2n+\alpha+\beta+1)(\alpha+1)_n(-N)_nN!}{(-1)^n(n+\alpha+\beta+1)_{N+1}(\beta+1)_nn!} -Q_n(x;\alpha,\beta,N)Q_n(y;\alpha,\beta,N)\\ -& &{}=\frac{\delta_{xy}}{\dbinom{\alpha+x}{x}\dbinom{\beta+N-x}{N-x}},\quad x,y \in \{0,1,2,\ldots,N\}. -\subsection*{9.5 Hahn} -\label{sec9.5} -% -\paragraph{Special values} -\begin{equation} -Q_n(0;\al,\be,N)=1,\quad -Q_n(N;\al,\be,N)=\frac{(-1)^n(\be+1)_n}{(\al+1)_n}\,. -\label{95} -\end{equation} -Use (9.5.1) and compare with (9.8.1) and \eqref{50}. - -From (9.5.3) and \eqref{1} it follows that -\begin{equation} -Q_{2n}(N;\al,\al,2N)=\frac{(\thalf)_n(N+\al+1)_n}{(-N+\thalf)_n(\al+1)_n}\,. -\label{30} -\end{equation} -From (9.5.1) and \mycite{DLMF}{(15.4.24)} it follows that -\begin{equation} -Q_N(x;\al,\be,N)=\frac{(-N-\be)_x}{(\al+1)_x}\qquad(x=0,1,\ldots,N). -\label{44} -\end{equation} -% -\paragraph{Symmetries} -By the orthogonality relation (9.5.2): -\begin{equation} -\frac{Q_n(N-x;\al,\be,N)}{Q_n(N;\al,\be,N)}=Q_n(x;\be,\al,N), -\label{96} -\end{equation} -It follows from \eqref{97} and \eqref{45} that -\begin{equation} -\frac{Q_{N-n}(x;\al,\be,N)}{Q_N(x;\al,\be,N)} -=Q_n(x;-N-\be-1,-N-\al-1,N) -\qquad(x=0,1,\ldots,N). -\label{100} -\end{equation} -% -\paragraph{Duality} -The Remark on p.208 gives the duality between Hahn and dual Hahn polynomials: -% -\begin{equation} -Q_n(x;\al,\be,N)=R_x(n(n+\al+\be+1);\al,\be,N)\quad(n,x\in\{0,1,\ldots N\}). -\label{45} -\end{equation} -\end{eqnarray*} - -\subsection*{References} -\cite{AlSalam90}, \cite{AndrewsAskey85}, \cite{Area+II}, \cite{Askey75}, -\cite{Askey89I}, \cite{Askey2005}, \cite{AskeyGasper77}, \cite{AskeyWilson85}, -\cite{AtakRahmanSuslov}, \cite{AtakSuslov88}, \cite{Chihara78}, -\cite{Ciesielski}, \cite{Cooper+}, \cite{Dette95}, \cite{Dunkl76}, -\cite{Dunkl78I}, \cite{Gasper73I}, \cite{Gasper74}, \cite{HoareRahman}, -\cite{Ismail77}, \cite{Ismail2005II}, \cite{Karlin61}, \cite{Koorn81}, \cite{Koorn88}, -\cite{LabelleYehI}, \cite{LabelleYehII}, \cite{Laine}, \cite{Lesky62}, -\cite{Lesky88}, \cite{Lesky89}, \cite{Lesky94I}, \cite{Lesky95II}, -\cite{LewanowiczII}, \cite{Neuman}, \cite{Nikiforov+}, \cite{NikiforovUvarov}, -\cite{Rahman76III}, \cite{Rahman78I}, \cite{Rahman78II}, \cite{Rahman81III}, -\cite{Sablonniere}, \cite{Stanton84}, \cite{Stanton90}, \cite{Wilson80}, \cite{Wilson70II}, -\cite{Zarzo+}. - - -\section{Dual Hahn}\index{Dual Hahn polynomials}\index{Hahn polynomials!Dual} - -\par\setcounter{equation}{0} - -\subsection*{Hypergeometric representation} -\begin{equation} -\label{DefDualHahn} -R_n(\lambda(x);\gamma,\delta,N)= -\hyp{3}{2}{-n,-x,x+\gamma+\delta+1}{\gamma+1,-N}{1},\quad n=0,1,2,\ldots,N, -\end{equation} -where -$$\lambda(x)=x(x+\gamma+\delta+1).$$ - -\subsection*{Orthogonality relation} -For $\gamma>-1$ and $\delta>-1$, or for $\gamma<-N$ and $\delta<-N$, we have -\begin{eqnarray} -\label{OrtDualHahn} -& &\sum_{x=0}^N\frac{(2x+\gamma+\delta+1)(\gamma+1)_x(-N)_xN!}{(-1)^x(x+\gamma+\delta+1)_{N+1}(\delta+1)_xx!} -R_m(\lambda(x);\gamma,\delta,N)R_n(\lambda(x);\gamma,\delta,N)\nonumber\\ -& &{}=\frac{\,\delta_{mn}}{\dbinom{\gamma+n}{n}\dbinom{\delta+N-n}{N-n}}. -\end{eqnarray} - -\subsection*{Recurrence relation} -\begin{equation} -\label{RecDualHahn} -\lambda(x)R_n(\lambda(x)) -=A_nR_{n+1}(\lambda(x))-\left(A_n+C_n\right)R_n(\lambda(x))+C_nR_{n-1}(\lambda(x)), -\end{equation} -where -$$R_n(\lambda(x)):=R_n(\lambda(x);\gamma,\delta,N)$$ -and -$$\left\{\begin{array}{l} -\displaystyle A_n=(n+\gamma+1)(n-N)\\[5mm] -\displaystyle C_n=n(n-\delta-N-1). -\end{array}\right.$$ - -\subsection*{Normalized recurrence relation} -\begin{equation} -\label{NormRecDualHahn} -xp_n(x)=p_{n+1}(x)-(A_n+C_n)p_n(x)+A_{n-1}C_np_{n-1}(x), -\end{equation} -where -$$R_n(\lambda(x);\gamma,\delta,N)=\frac{1}{(\gamma+1)_n(-N)_n}p_n(\lambda(x)).$$ - -\subsection*{Difference equation} -\begin{equation} -\label{dvDualHahn} --ny(x)=B(x)y(x+1)-\left[B(x)+D(x)\right]y(x)+D(x)y(x-1), -\end{equation} -where -$$y(x)=R_n(\lambda(x);\gamma,\delta,N)$$ -and -$$\left\{\begin{array}{l} -\displaystyle B(x)=\frac{(x+\gamma+1)(x+\gamma+\delta+1)(N-x)}{(2x+\gamma+\delta+1)(2x+\gamma+\delta+2)}\\[5mm] -\displaystyle D(x)=\frac{x(x+\gamma+\delta+N+1)(x+\delta)}{(2x+\gamma+\delta)(2x+\gamma+\delta+1)}. -\end{array}\right.$$ - -\subsection*{Forward shift operator} -\begin{eqnarray} -\label{shift1DualHahnI} -& &R_n(\lambda(x+1);\gamma,\delta,N)-R_n(\lambda(x);\gamma,\delta,N)\nonumber\\ -& &{}=-\frac{n(2x+\gamma+\delta+2)}{(\gamma+1)N}R_{n-1}(\lambda(x);\gamma+1,\delta,N-1) -\end{eqnarray} -or equivalently -\begin{equation} -\label{shift1DualHahnII} -\frac{\Delta R_n(\lambda(x);\gamma,\delta,N)}{\Delta\lambda(x)}= --\frac{n}{(\gamma+1)N}R_{n-1}(\lambda(x);\gamma+1,\delta,N-1). -\end{equation} - -\subsection*{Backward shift operator} -\begin{eqnarray} -\label{shift2DualHahnI} -& &(x+\gamma)(x+\gamma+\delta)(N+1-x)R_n(\lambda(x);\gamma,\delta,N)\nonumber\\ -& &{}\mathindent{}-x(x+\gamma+\delta+N+1)(x+\delta)R_n(\lambda(x-1);\gamma,\delta,N)\nonumber\\ -& &{}=\gamma(N+1)(2x+\gamma+\delta)R_{n+1}(\lambda(x);\gamma-1,\delta,N+1) -\end{eqnarray} -or equivalently -\begin{eqnarray} -\label{shift2DualHahnII} -& &\frac{\nabla\left[\omega(x;\gamma,\delta,N)R_n(\lambda(x);\gamma,\delta,N)\right]}{\nabla\lambda(x)}\nonumber\\ -& &{}=\frac{1}{\gamma+\delta}\omega(x;\gamma-1,\delta,N+1)R_{n+1}(\lambda(x);\gamma-1,\delta,N+1), -\end{eqnarray} -where -$$\omega(x;\gamma,\delta,N)=\frac{(-1)^x(\gamma+1)_x(\gamma+\delta+1)_x(-N)_x} -{(\gamma+\delta+N+2)_x(\delta+1)_xx!}.$$ - -\newpage - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodDualHahn} -\omega(x;\gamma,\delta,N)R_n(\lambda(x);\gamma,\delta,N) -=(\gamma+\delta+1)_n\left(\nabla_{\lambda}\right)^n\left[\omega(x;\gamma+n,\delta,N-n)\right], -\end{equation} -where -$$\nabla_{\lambda}:=\frac{\nabla}{\nabla\lambda(x)}.$$ - -\subsection*{Generating functions} -For $x=0,1,2,\ldots,N$ we have -\begin{equation} -\label{GenDualHahn1} -(1-t)^{N-x}\,\hyp{2}{1}{-x,-x-\delta}{\gamma+1}{t} -=\sum_{n=0}^N\frac{(-N)_n}{n!}R_n(\lambda(x);\gamma,\delta,N)t^n. -\end{equation} - -\begin{eqnarray} -\label{GenDualHahn2} -& &(1-t)^x\,\hyp{2}{1}{x-N,x+\gamma+1}{-\delta-N}{t}\nonumber\\ -& &=\sum_{n=0}^N\frac{(\gamma+1)_n(-N)_n}{(-\delta-N)_nn!}R_n(\lambda(x);\gamma,\delta,N)t^n. -\end{eqnarray} - -\begin{equation} -\label{GenDualHahn3} -\left[\expe^t\,\hyp{2}{2}{-x,x+\gamma+\delta+1}{\gamma+1,-N}{-t}\right]_N=\sum_{n=0}^N -\frac{R_n(\lambda(x);\gamma,\delta,N)}{n!}t^n. -\end{equation} - -\begin{eqnarray} -\label{GenDualHahn4} -& &\left[(1-t)^{-\epsilon}\,\hyp{3}{2}{\epsilon,-x,x+\gamma+\delta+1}{\gamma+1,-N}{\frac{t}{t-1}}\right]_N\nonumber\\ -& &{}=\sum_{n=0}^N\frac{(\epsilon)_n}{n!}R_n(\lambda(x);\gamma,\delta,N)t^n, -\quad\textrm{$\epsilon$ arbitrary}. -\end{eqnarray} - -\subsection*{Limit relations} - -\subsubsection*{Racah $\rightarrow$ Dual Hahn} -If we take $\alpha+1=-N$ and let $\beta\rightarrow\infty$ in the definition (\ref{DefRacah}) -of the Racah polynomials, then we obtain the dual Hahn polynomials: -$$\lim_{\beta\rightarrow\infty} -R_n(\lambda(x);-N-1,\beta,\gamma,\delta)=R_n(\lambda(x);\gamma,\delta,N).$$ -And if we take $\beta=-\delta-N-1$ and let $\alpha\rightarrow\infty$ in the definition (\ref{DefRacah}) -of the Racah polynomials, then we also obtain the dual Hahn polynomials: -$$\lim_{\alpha\rightarrow\infty} -R_n(\lambda(x);\alpha,-\delta-N-1,\gamma,\delta)=R_n(\lambda(x);\gamma,\delta,N).$$ -Finally, if we take $\gamma+1=-N$ and $\delta\rightarrow\alpha+\delta+N+1$ in the definition -(\ref{DefRacah}) of the Racah polynomials and take the limit -$\beta\rightarrow\infty$ we find the dual Hahn polynomials in the following way: -$$\lim_{\beta\rightarrow\infty} -R_n(\lambda(x);\alpha,\beta,-N-1,\alpha+\delta+N+1)=R_n(\lambda(x);\alpha,\delta,N).$$ - -\subsubsection*{Dual Hahn $\rightarrow$ Meixner} -The Meixner polynomials given by (\ref{DefMeixner}) are obtained from the dual Hahn polynomials -if we take $\gamma=\beta-1$ and $\delta=N(1-c)c^{-1}$ and let $N\rightarrow\infty$: -\begin{equation} -\lim_{N\rightarrow\infty} -R_n(\lambda(x);\beta-1,N(1-c)c^{-1},N)=M_n(x;\beta,c). -\end{equation} - -\subsubsection*{Dual Hahn $\rightarrow$ Krawtchouk} -The Krawtchouk polynomials given by (\ref{DefKrawtchouk}) can be obtained from the dual -Hahn polynomials by setting $\gamma=pt$, $\delta=(1-p)t$ and letting $t\rightarrow\infty$: -\begin{equation} -\lim_{t\rightarrow\infty}R_n(\lambda(x);pt,(1-p)t,N)=K_n(x;p,N). -\end{equation} - -\subsection*{Remark} -If we interchange the role of $x$ and $n$ in the definition (\ref{DefDualHahn}) -of the dual Hahn polynomials we obtain the Hahn polynomials given by (\ref{DefHahn}). - -\noindent -Since -$$R_n(\lambda(x);\gamma,\delta,N)=Q_x(n;\gamma,\delta,N)$$ -we obtain the dual orthogonality relation for the dual Hahn polynomials -from the orthogonality relation (\ref{OrtHahn}) for the Hahn polynomials: -\begin{eqnarray*} -& &\sum_{n=0}^N\binom{\gamma+n}{n}\binom{\delta+N-n}{N-n} R_n(\lambda(x);\gamma,\delta,N)R_n(\lambda(y);\gamma,\delta,N)\\ -& &{}=\frac{(-1)^x(x+\gamma+\delta+1)_{N+1}(\delta+1)_xx!} -{(2x+\gamma+\delta+1)(\gamma+1)_x(-N)_xN!}\delta_{xy},\quad x,y \in \{0,1,2,\ldots,N\}. -\subsection*{9.6 Dual Hahn} -\label{sec9.6} -% -\paragraph{Special values} -By \eqref{44} and \eqref{45} we have -\begin{equation} -R_n(N(N+\ga+\de+1);\ga,\de,N)=\frac{(-N-\de)_n}{(\ga+1)_n}\,. -\label{47} -\end{equation} -It follows from \eqref{95} and \eqref{45} that -\begin{equation} -R_N(x(x+\ga+\de+1);\ga,\de,N) -=\frac{(-1)^x(\de+1)_x}{(\ga+1)_x}\qquad(x=0,1,\ldots,N). -\label{101} -\end{equation} -% -\paragraph{Symmetries} -Write the weight in (9.6.2) as -\begin{equation} -w_x(\al,\be,N):=N!\,\frac{2x+\ga+\de+1}{(x+\ga+\de+1)_{N+1}}\, -\frac{(\ga+1)_x}{(\de+1)_x}\,\binom Nx. -\label{98} -\end{equation} -Then -\begin{equation} -(\de+1)_N\,w_{N-x}(\ga,\de,N)= -(-\ga-N)_N\,w_x(-\de-N-1,-\ga-N-1,N). -\label{99} -\end{equation} -Hence, by (9.6.2), -\begin{equation} -\frac{R_n((N-x)(N-x+\ga+\de+1);\ga,\de,N)}{R_n(N(N+\ga+\de+1);\ga,\de,N)} -=R_n(x(x-2N-\ga-\de-1);-N-\de-1,-N-\ga-1,N). -\label{97} -\end{equation} -Alternatively, \eqref{97} follows from (9.6.1) and -\mycite{DLMF}{(16.4.11)}. - -It follows from \eqref{96} and \eqref{45} that -\begin{equation} -\frac{R_{N-n}(x(x+\ga+\de+1);\ga,\de,N)} -{R_N(x(x+\ga+\de+1);\ga,\de,N)} -=R_n(x(x+\ga+\de+1);\de,\ga,N)\qquad(x=0,1,\ldots,N). -\label{102} -\end{equation} -% -\paragraph{Re: (9.6.11).} -The generating function (9.6.11) can be written in a more conceptual way as -\begin{equation} -(1-t)^x\,\hyp21{x-N,x+\ga+1}{-\de-N}t=\frac{N!}{(\de+1)_N}\, -\sum_{n=0}^N \om_n\,R_n(\la(x);\ga,\de,N)\,t^n, -\label{2} -\end{equation} -where -\begin{equation} -\om_n:=\binom{\ga+n}n \binom{\de+N-n}{N-n}, -\label{3} -\end{equation} -i.e., the denominator on the \RHS\ of (9.6.2). -By the duality between Hahn polynomials and dual Hahn polynomials (see \eqref{45}) the above generating function can be rewritten in -terms of Hahn polynomials: -\begin{equation} -(1-t)^n\,\hyp21{n-N,n+\al+1}{-\be-N}t=\frac{N!}{(\be+1)_N}\, -\sum_{x=0}^N w_x\,Q_n(x;\al,\be,N)\,t^x, -\label{4} -\end{equation} -where -\begin{equation} -w_x:=\binom{\al+x}x \binom{\be+N-x}{N-x}, -\label{5} -\end{equation} -i.e., the weight occurring in the orthogonality relation (9.5.2) -for Hahn polynomials. -\paragraph{Re: (9.6.15).} -There should be a closing bracket before the equality sign. -\end{eqnarray*} - -\subsection*{References} -\cite{Askey2005}, \cite{AskeyWilson85}, \cite{AtakRahmanSuslov}, \cite{AtakSuslov88}, -\cite{Ismail2005II}, \cite{Karlin61}, \cite{Koorn81}, \cite{Koorn88}, \cite{Lesky93}, -\cite{Lesky94I}, \cite{Lesky95I}, \cite{Lesky95II}, \cite{Nikiforov+}, \cite{NikiforovUvarov}, -\cite{Rahman81II}, \cite{Stanton84}, \cite{Wilson80}. - - -\section{Meixner-Pollaczek}\index{Meixner-Pollaczek polynomials} - -\par\setcounter{equation}{0} - -\subsection*{Hypergeometric representation} -\begin{equation} -\label{DefMP} -P_n^{(\lambda)}(x;\phi)=\frac{(2\lambda)_n}{n!}\expe^{in\phi}\, -\hyp{2}{1}{-n,\lambda+ix}{2\lambda}{1-\expe^{-2i\phi}}. -\end{equation} - -\subsection*{Orthogonality relation} -\begin{equation} -\label{OrtMP} -\frac{1}{2\pi}\int_{-\infty}^{\infty} -\e^{(2\phi-\pi)x}\left|\Gamma(\lambda+ix)\right|^2 -P_m^{(\lambda)}(x;\phi)P_n^{(\lambda)}(x;\phi)\,dx -{}=\frac{\Gamma(n+2\lambda)}{(2\sin\phi)^{2\lambda}n!}\,\delta_{mn}, -% \constraint{ -% $\lambda > 0$ & -% $0 < \phi < \pi$ } -\end{equation} - -\subsection*{Recurrence relation} -\begin{eqnarray} -\label{RecMP} -& &(n+1)P_{n+1}^{(\lambda)}(x;\phi)-2\left[x\sin\phi+(n+\lambda)\cos\phi\right] -P_n^{(\lambda)}(x;\phi)\nonumber\\ -& &{}\mathindent{}+(n+2\lambda-1)P_{n-1}^{(\lambda)}(x;\phi)=0. -\end{eqnarray} - -\subsection*{Normalized recurrence relation} -\begin{equation} -\label{NormRecMP} -xp_n(x)=p_{n+1}(x)-\left(\frac{n+\lambda}{\tan\phi}\right)p_n(x)+ -\frac{n(n+2\lambda-1)}{4\sin^2\phi}p_{n-1}(x), -\end{equation} -where -$$P_n^{(\lambda)}(x;\phi)=\frac{(2\sin\phi)^n}{n!}p_n(x).$$ - -\subsection*{Difference equation} -\begin{eqnarray} -\label{dvMP} -& &\e^{i\phi}(\lambda-ix)y(x+i)+2i\left[x\cos\phi-(n+\lambda)\sin\phi\right]y(x)\nonumber\\ -& &{}\mathindent{}-\e^{-i\phi}(\lambda+ix)y(x-i)=0,\quad y(x)=P_n^{(\lambda)}(x;\phi). -\end{eqnarray} - -\subsection*{Forward shift operator} -\begin{equation} -\label{shift1MPI} -P_n^{(\lambda)}(x+\textstyle\frac{1}{2}i;\phi)- -P_n^{(\lambda)}(x-\textstyle\frac{1}{2}i;\phi)=(\expe^{i\phi}-\expe^{-i\phi}) -P_{n-1}^{(\lambda+\frac{1}{2})}(x;\phi) -\end{equation} -or equivalently -\begin{equation} -\label{shift1MPII} -\frac{\delta P_n^{(\lambda)}(x;\phi)}{\delta x} -=2\sin\phi\,P_{n-1}^{(\lambda+\frac{1}{2})}(x;\phi). -\end{equation} - -\subsection*{Backward shift operator} -\begin{eqnarray} -\label{shift2MPI} -& &\e^{i\phi}(\lambda-\textstyle\frac{1}{2}-ix) -P_n^{(\lambda)}(x+\textstyle\frac{1}{2}i;\phi)+ -\e^{-i\phi}(\lambda-\textstyle\frac{1}{2}+ix) -P_n^{(\lambda)}(x-\textstyle\frac{1}{2}i;\phi)\nonumber\\ -& &{}=(n+1)P_{n+1}^{(\lambda-\frac{1}{2})}(x;\phi) -\end{eqnarray} -or equivalently -\begin{equation} -\label{shift2MPII} -\frac{\delta\left[\omega(x;\lambda,\phi)P_n^{(\lambda)}(x;\phi)\right]}{\delta x}= --(n+1)\omega(x;\lambda-\textstyle\frac{1}{2},\phi) -P_{n+1}^{(\lambda-\frac{1}{2})}(x;\phi), -\end{equation} -where -$$\omega(x;\lambda,\phi)=\Gamma(\lambda+ix)\Gamma(\lambda-ix)\e^{(2\phi-\pi)x}.$$ - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodMP} -\omega(x;\lambda,\phi)P_n^{(\lambda)}(x;\phi)=\frac{(-1)^n}{n!} -\left(\frac{\delta}{\delta x}\right)^n\left[\omega(x;\lambda+\textstyle\frac{1}{2}n,\phi)\right]. -\end{equation} - -\newpage - -\subsection*{Generating functions} -\begin{equation} -\label{GenMP1} -(1-\expe^{i\phi}t)^{-\lambda+ix}(1-\expe^{-i\phi}t)^{-\lambda-ix}= -\sum_{n=0}^{\infty}P_n^{(\lambda)}(x;\phi)t^n. -\end{equation} - -\begin{equation} -\label{GenMP2} -\e^t\,\hyp{1}{1}{\lambda+ix}{2\lambda}{(\expe^{-2i\phi}-1)t}= -\sum_{n=0}^{\infty}\frac{P_n^{(\lambda)}(x;\phi)}{(2\lambda)_n\expe^{in\phi}}t^n. -\end{equation} - -\begin{eqnarray} -\label{GenMP3} -& &(1-t)^{-\gamma}\,\hyp{2}{1}{\gamma,\lambda+ix}{2\lambda}{\frac{(1-\e^{-2i\phi})t}{t-1}}\nonumber\\ -& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n}{(2\lambda)_n}\frac{P_n^{(\lambda)}(x;\phi)}{\e^{in\phi}}t^n, -\quad\textrm{$\gamma$ arbitrary}. -\end{eqnarray} - -\subsection*{Limit relations} - -\subsubsection*{Continuous dual Hahn $\rightarrow$ Meixner-Pollaczek} -The Meixner-Pollaczek polynomials can be obtained from the continuous dual Hahn polynomials -given by (\ref{DefContDualHahn}) by the substitutions $x\rightarrow x-t$, $a=\lambda+it$, -$b=\lambda-it$ and $c=t\cot\phi$ and the limit $t\rightarrow\infty$: -$$\lim_{t\rightarrow\infty}\frac{S_n((x-t)^2;\lambda+it,\lambda-it,t\cot\phi)}{t^nn!} -=\frac{P_n^{(\lambda)}(x;\phi)}{(\sin\phi)^n}.$$ - -\subsubsection*{Continuous Hahn $\rightarrow$ Meixner-Pollaczek} -By setting $x\rightarrow x+t$, $a=\lambda-it$, $c=\lambda+it$ and $b=d=t\tan\phi$ in the -definition (\ref{DefContHahn}) of the continuous Hahn polynomials and taking the limit -$t\rightarrow\infty$ we obtain the Meixner-Pollaczek polynomials: -$$\lim_{t\rightarrow\infty}\frac{p_n(x+t;\lambda-it,t\tan\phi,\lambda+it,t\tan\phi)}{t^nn!} -=\frac{P_n^{(\lambda)}(x;\phi)}{(\cos\phi)^n}.$$ - -\newpage - -\subsubsection*{Meixner-Pollaczek $\rightarrow$ Laguerre} -The Laguerre polynomials given by (\ref{DefLaguerre}) can be obtained from the -Meixner-Pollaczek polynomials by the substitution $\lambda=\frac{1}{2}(\alpha+1)$, -$x\rightarrow -\frac{1}{2}\phi^{-1}x$ and the limit $\phi\rightarrow 0$: -\begin{equation} -\lim_{\phi\rightarrow 0} -P_n^{(\frac{1}{2}\alpha+\frac{1}{2})}(-\textstyle\frac{1}{2}\phi^{-1}x;\phi)=L_n^{(\alpha)}(x). -\end{equation} - -\subsubsection*{Meixner-Pollaczek $\rightarrow$ Hermite} -The Hermite polynomials given by (\ref{DefHermite}) are obtained from the Meixner-Pollaczek -polynomials if we substitute $x\rightarrow (\sin\phi)^{-1}(x\sqrt{\lambda}-\lambda\cos\phi)$ -and then let $\lambda\rightarrow\infty$: -\begin{equation} -\lim_{\lambda\rightarrow\infty} -\lambda^{-\frac{1}{2}n}P_n^{(\lambda)} -((\sin\phi)^{-1}(x\sqrt{\lambda}-\lambda\cos\phi);\phi)=\frac{H_n(x)}{n!}. -\end{equation} - -\subsection*{Remark} -Since we have for $k-1$ and $\beta>-1$ we have -\begin{eqnarray} -\label{OrtJacobi1} -& &\int_{-1}^1(1-x)^{\alpha}(1+x)^{\beta}P_m^{(\alpha,\beta)}(x)P_n^{(\alpha,\beta)}(x)\,dx\nonumber\\ -& &{}=\frac{2^{\alpha+\beta+1}}{2n+\alpha+\beta+1}\frac{\Gamma(n+\alpha+1)\Gamma(n+\beta+1)}{\Gamma(n+\alpha+\beta+1)n!}\,\delta_{mn}. -\end{eqnarray} -For $\alpha+\beta<-2N-1$, $\beta>-1$ and $m,n\in\{0,1,2,\ldots,N\}$ we also have -\begin{eqnarray} -\label{OrtJacobi2} -& &\int_1^{\infty}(x+1)^{\alpha}(x-1)^{\beta}P_m^{(\alpha,\beta)}(-x)P_n^{(\alpha,\beta)}(-x)\,dx\nonumber\\ -& &{}=-\frac{2^{\alpha+\beta+1}}{2n+\alpha+\beta+1}\frac{\Gamma(-n-\alpha-\beta)\Gamma(n+\alpha+\beta+1)}{\Gamma(-n-\alpha)n!}\,\delta_{mn}. -\end{eqnarray} - -\subsection*{Recurrence relation} -\begin{eqnarray} -\label{RecJacobi} -xP_n^{(\alpha,\beta)}(x)&=&\frac{2(n+1)(n+\alpha+\beta+1)}{(2n+\alpha+\beta+1)(2n+\alpha+\beta+2)}P_{n+1}^{(\alpha,\beta)}(x)\nonumber\\ -& &{}\mathindent{}+\frac{\beta^2-\alpha^2}{(2n+\alpha+\beta)(2n+\alpha+\beta+2)}P_n^{(\alpha,\beta)}(x)\nonumber\\ -& &{}\mathindent\mathindent{}+\frac{2(n+\alpha)(n+\beta)}{(2n+\alpha+\beta)(2n+\alpha+\beta+1)}P_{n-1}^{(\alpha,\beta)}(x). -\end{eqnarray} - -\subsection*{Normalized recurrence relation} -\begin{eqnarray} -\label{NormRecJacobi} -xp_n(x)&=&p_{n+1}(x)+\frac{\beta^2-\alpha^2}{(2n+\alpha+\beta)(2n+\alpha+\beta+2)}p_n(x)\nonumber\\ -& &{}\mathindent{}+\frac{4n(n+\alpha)(n+\beta)(n+\alpha+\beta)} -{(2n+\alpha+\beta-1)(2n+\alpha+\beta)^2(2n+\alpha+\beta+1)}p_{n-1}(x) -\end{eqnarray} -where -$$P_n^{(\alpha,\beta)}(x)=\frac{(n+\alpha+\beta+1)_n}{2^nn!}p_n(x).$$ - -\subsection*{Differential equation} -\begin{eqnarray} -\label{dvJacobi} -& &(1-x^2)y''(x)+\left[\beta-\alpha-(\alpha+\beta+2)x\right]y'(x)\nonumber\\ -& &{}\mathindent{}+n(n+\alpha+\beta+1)y(x)=0,\quad y(x)=P_n^{(\alpha,\beta)}(x). -\end{eqnarray} - -\subsection*{Forward shift operator} -\begin{equation} -\label{shift1Jacobi} -\frac{d}{dx}P_n^{(\alpha,\beta)}(x)=\frac{n+\alpha+\beta+1}{2}P_{n-1}^{(\alpha+1,\beta+1)}(x). -\end{equation} - -\subsection*{Backward shift operator} -\begin{eqnarray} -\label{shift2JacobiI} -& &(1-x^2)\frac{d}{dx}P_n^{(\alpha,\beta)}(x)+ -\left[(\beta-\alpha)-(\alpha+\beta)x\right]P_n^{(\alpha,\beta)}(x)\nonumber\\ -& &{}=-2(n+1)P_{n+1}^{(\alpha-1,\beta-1)}(x) -\end{eqnarray} -or equivalently -\begin{eqnarray} -\label{shift2JacobiII} -& &\frac{d}{dx}\left[(1-x)^\alpha(1+x)^\beta P_n^{(\alpha,\beta)}(x)\right]\nonumber\\ -& &{}=-2(n+1)(1-x)^{\alpha-1}(1+x)^{\beta-1}P_{n+1}^{(\alpha-1,\beta-1)}(x). -\end{eqnarray} - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodJacobi} -(1-x)^{\alpha}(1+x)^{\beta}P_n^{(\alpha,\beta)}(x)= -\frac{(-1)^n}{2^nn!}\left(\frac{d}{dx}\right)^n -\left[(1-x)^{n+\alpha}(1+x)^{n+\beta}\right]. -\end{equation} - -\subsection*{Generating functions} -\begin{equation} -\label{GenJacobi1} -\frac{2^{\alpha+\beta}}{R(1+R-t)^{\alpha}(1+R+t)^{\beta}}= -\sum_{n=0}^{\infty}P_n^{(\alpha,\beta)}(x)t^n,\quad R=\sqrt{1-2xt+t^2}. -\end{equation} - -\begin{eqnarray} -\label{GenJacobi2} -& &\hyp{0}{1}{-}{\alpha+1}{\frac{(x-1)t}{2}}\,\hyp{0}{1}{-}{\beta+1}{\frac{(x+1)t}{2}}\nonumber\\ -& &{}=\sum_{n=0}^{\infty}\frac{P_n^{(\alpha,\beta)}(x)}{(\alpha+1)_n(\beta+1)_n}t^n. -\end{eqnarray} - -\begin{eqnarray} -\label{GenJacobi3} -& &(1-t)^{-\alpha-\beta-1}\,\hyp{2}{1}{\frac{1}{2}(\alpha+\beta+1),\frac{1}{2}(\alpha+\beta+2)} -{\alpha+1}{\frac{2(x-1)t}{(1-t)^2}}\nonumber\\ -& &{}=\sum_{n=0}^{\infty}\frac{(\alpha+\beta+1)_n}{(\alpha+1)_n}P_n^{(\alpha,\beta)}(x)t^n. -\end{eqnarray} - -\begin{eqnarray} -\label{GenJacobi4} -& &(1+t)^{-\alpha-\beta-1}\,\hyp{2}{1}{\frac{1}{2}(\alpha+\beta+1),\frac{1}{2}(\alpha+\beta+2)} -{\beta+1}{\frac{2(x+1)t}{(1+t)^2}}\nonumber\\ -& &{}=\sum_{n=0}^{\infty}\frac{(\alpha+\beta+1)_n}{(\beta+1)_n}P_n^{(\alpha,\beta)}(x)t^n. -\end{eqnarray} - -\begin{eqnarray} -\label{GenJacobi5} -& &\hyp{2}{1}{\gamma,\alpha+\beta+1-\gamma}{\alpha+1}{\frac{1-R-t}{2}}\, -\hyp{2}{1}{\gamma,\alpha+\beta+1-\gamma}{\beta+1}{\frac{1-R+t}{2}}\nonumber\\ -& &{}=\sum_{n=0}^{\infty} -\frac{(\gamma)_n(\alpha+\beta+1-\gamma)_n}{(\alpha+1)_n(\beta+1)_n}P_n^{(\alpha,\beta)}(x)t^n, -\quad R=\sqrt{1-2xt+t^2} -\end{eqnarray} -with $\gamma$ arbitrary. - -\subsection*{Limit relations} - -\subsubsection*{Wilson $\rightarrow$ Jacobi} -The Jacobi polynomials can be found from the Wilson polynomials given by (\ref{DefWilson}) by -substituting $a=b=\frac{1}{2}(\alpha+1)$, $c=\frac{1}{2}(\beta+1)+it$, -$d=\frac{1}{2}(\beta+1)-it$ and $x\rightarrow t\sqrt{\frac{1}{2}(1-x)}$ in the -definition (\ref{DefWilson}) of the Wilson polynomials and taking the limit -$t\rightarrow\infty$. In fact we have -$$\lim_{t\rightarrow\infty} -\frac{W_n(\frac{1}{2}(1-x)t^2;\frac{1}{2}(\alpha+1), -\frac{1}{2}(\alpha+1),\frac{1}{2}(\beta+1)+it,\frac{1}{2}(\beta+1)-it)} -{t^{2n}n!}=P_n^{(\alpha,\beta)}(x).$$ - -\subsubsection*{Continuous Hahn $\rightarrow$ Jacobi} -The Jacobi polynomials follow from the continuous Hahn polynomials given by (\ref{DefContHahn}) -by using the substitution $x\rightarrow \frac{1}{2}xt$, $a=\frac{1}{2}(\alpha+1-it)$, -$b=\frac{1}{2}(\beta+1+it)$, $c=\frac{1}{2}(\alpha+1+it)$ and $d=\frac{1}{2}(\beta+1-it)$ -in (\ref{DefContHahn}), division by $t^n$ and the limit $t\rightarrow\infty$: -$$\lim_{t\rightarrow\infty} -\frac{p_n(\frac{1}{2}xt;\frac{1}{2}(\alpha+1-it),\frac{1}{2}(\beta+1+it), -\frac{1}{2}(\alpha+1+it),\frac{1}{2}(\beta+1-it))}{t^n}=P_n^{(\alpha,\beta)}(x).$$ - -\subsubsection*{Hahn $\rightarrow$ Jacobi} -To find the Jacobi polynomials from the Hahn polynomials given by (\ref{DefHahn}) we take -$x\rightarrow Nx$ in (\ref{DefHahn}) and let $N\rightarrow\infty$. In fact we have -$$\lim_{N\rightarrow\infty} -Q_n(Nx;\alpha,\beta,N)=\frac{P_n^{(\alpha,\beta)}(1-2x)}{P_n^{(\alpha,\beta)}(1)}.$$ - -\subsubsection*{Jacobi $\rightarrow$ Laguerre} -The Laguerre polynomials given by (\ref{DefLaguerre}) can be obtained from the Jacobi -polynomials by setting $x\rightarrow 1-2\beta^{-1}x$ and then the limit $\beta\rightarrow\infty$: -\begin{equation} -\lim_{\beta\rightarrow\infty} -P_n^{(\alpha,\beta)}(1-2\beta^{-1}x)=L_n^{(\alpha)}(x). -\end{equation} - -\subsubsection*{Jacobi $\rightarrow$ Bessel} -The Bessel polynomials given by (\ref{DefBessel}) are obtained from the Jacobi polynomials -if we take $\beta=a-\alpha$ and let $\alpha\rightarrow-\infty$: -\begin{equation} -\lim_{\alpha\rightarrow-\infty} -\frac{P_n^{(\alpha,a-\alpha)}(1+\alpha x)}{P_n^{(\alpha,a-\alpha)}(1)}=y_n(x;a). -\end{equation} - -\subsubsection*{Jacobi $\rightarrow$ Hermite} -The Hermite polynomials given by (\ref{DefHermite}) follow from the Jacobi polynomials -by taking $\beta=\alpha$ and letting $\alpha\rightarrow\infty$ in the following way: -\begin{equation} -\lim_{\alpha\rightarrow\infty} -\alpha^{-\frac{1}{2}n}P_n^{(\alpha,\alpha)}(\alpha^{-\frac{1}{2}}x)=\frac{H_n(x)}{2^nn!}. -\end{equation} - -\subsection*{Remarks} -The definition (\ref{DefJacobi}) of the Jacobi polynomials can also be written as: -$$P_n^{(\alpha,\beta)}(x)=\frac{1}{n!}\sum_{k=0}^n\frac{(-n)_k}{k!} -(n+\alpha+\beta+1)_k(\alpha+k+1)_{n-k}\left(\frac{1-x}{2}\right)^k.$$ -In this way the Jacobi polynomials can also be seen as polynomials in the parameters $\alpha$ -and $\beta$. Therefore they can be defined for all $\alpha$ and $\beta$. Then we have the -following connection with the Meixner polynomials given by (\ref{DefMeixner}): -$$\frac{(\beta)_n}{n!}M_n(x;\beta,c)=P_n^{(\beta-1,-n-\beta-x)}((2-c)c^{-1}).$$ - -\noindent -The Jacobi polynomials are related to the pseudo Jacobi polynomials defined by -(\ref{DefPseudoJacobi}) in the following way: -$$P_n(x;\nu,N)=\frac{(-2i)^nn!}{(n-2N-1)_n}P_n^{(-N-1+i\nu,-N-1-i\nu)}(ix).$$ - -\noindent -The Jacobi polynomials are also related to the Gegenbauer (or ultraspherical) polynomials -given by (\ref{DefGegenbauer}) by the quadratic transformations: -$$C_{2n}^{(\lambda)}(x)=\frac{(\lambda)_n}{(\frac{1}{2})_n} -P_n^{(\lambda-\frac{1}{2},-\frac{1}{2})}(2x^2-1)$$ -and -$$C_{2n+1}^{(\lambda)}(x)=\frac{(\lambda)_{n+1}}{(\frac{1}{2})_{n+1}} -\subsection*{9.8 Jacobi} -\label{sec9.8} -% -\paragraph{Orthogonality relation} -Write the \RHS\ of (9.8.2) as $h_n\,\de_{m,n}$. Then -\begin{equation} -\begin{split} -&\frac{h_n}{h_0}= -\frac{n+\al+\be+1}{2n+\al+\be+1}\, -\frac{(\al+1)_n(\be+1)_n}{(\al+\be+2)_n\,n!}\,,\quad -h_0=\frac{2^{\al+\be+1}\Ga(\al+1)\Ga(\be+1)}{\Ga(\al+\be+2)}\,,\sLP -&\frac{h_n}{h_0\,(P_n^{(\al,\be)}(1))^2}= -\frac{n+\al+\be+1}{2n+\al+\be+1}\, -\frac{(\be+1)_n\,n!}{(\al+1)_n\,(\al+\be+2)_n}\,. -\end{split} -\label{60} -\end{equation} - -In (9.8.3) the numerator factor $\Ga(n+\al+\be+1)$ in the last line should be -$\Ga(\be+1)$. When thus corrected, (9.8.3) can be rewritten as: -\begin{equation} -\begin{split} -&\int_1^\iy P_m^{(\al,\be)}(x)\,P_n^{(\al,\be)}(x)\,(x-1)^\al (x+1)^\be\,dx=h_n\,\de_{m,n}\,,\\ -&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad-1-\be>\al>-1,\quad m,n<-\thalf(\al+\be+1),\\ -&\frac{h_n}{h_0}= -\frac{n+\al+\be+1}{2n+\al+\be+1}\, -\frac{(\al+1)_n(\be+1)_n}{(\al+\be+2)_n\,n!}\,,\quad -h_0=\frac{2^{\al+\be+1}\Ga(\al+1)\Ga(-\al-\be-1)}{\Ga(-\be)}\,. -\end{split} -\label{122} -\end{equation} - -% -\paragraph{Symmetry} -\begin{equation} -P_n^{(\al,\be)}(-x)=(-1)^n\,P_n^{(\be,\al)}(x). -\label{48} -\end{equation} -Use (9.8.2) and (9.8.5b) or see \mycite{DLMF}{Table 18.6.1}. -% -\paragraph{Special values} -\begin{equation} -P_n^{(\al,\be)}(1)=\frac{(\al+1)_n}{n!}\,,\quad -P_n^{(\al,\be)}(-1)=\frac{(-1)^n(\be+1)_n}{n!}\,,\quad -\frac{P_n^{(\al,\be)}(-1)}{P_n^{(\al,\be)}(1)}=\frac{(-1)^n(\be+1)_n}{(\al+1)_n}\,. -\label{50} -\end{equation} -Use (9.8.1) and \eqref{48} or see \mycite{DLMF}{Table 18.6.1}. -% -\paragraph{Generating functions} -Formula (9.8.15) was first obtained by Brafman \myciteKLS{109}. -% -\paragraph{Bilateral generating functions} -For $0\le r<1$ and $x,y\in[-1,1]$ we have in terms of $F_4$ (see~\eqref{62}): -\begin{align} -&\sum_{n=0}^\iy\frac{(\al+\be+1)_n\,n!}{(\al+1)_n(\be+1)_n}\,r^n\, -P_n^{(\al,\be)}(x)\,P_n^{(\al,\be)}(y) -=\frac1{(1+r)^{\al+\be+1}} -\nonumber\\ -&\qquad\quad\times F_4\Big(\thalf(\al+\be+1),\thalf(\al+\be+2);\al+1,\be+1; -\frac{r(1-x)(1-y)}{(1+r)^2},\frac{r(1+x)(1+y)}{(1+r)^2}\Big), -\label{58}\sLP -&\sum_{n=0}^\iy\frac{2n+\al+\be+1}{n+\al+\be+1} -\frac{(\al+\be+2)_n\,n!}{(\al+1)_n(\be+1)_n}\,r^n\, -P_n^{(\al,\be)}(x)\,P_n^{(\al,\be)}(y) -=\frac{1-r}{(1+r)^{\al+\be+2}}\nonumber\\ -&\qquad\quad\times F_4\Big(\thalf(\al+\be+2),\thalf(\al+\be+3);\al+1,\be+1; -\frac{r(1-x)(1-y)}{(1+r)^2},\frac{r(1+x)(1+y)}{(1+r)^2}\Big). -\label{59} -\end{align} -Formulas \eqref{58} and \eqref{59} were first -given by Bailey \myciteKLS{91}{(2.1), (2.3)}. -See Stanton \myciteKLS{485} for a shorter proof. -(However, in the second line of -\myciteKLS{485}{(1)} $z$ and $Z$ should be interchanged.)$\;$ -As observed in Bailey \myciteKLS{91}{p.10}, \eqref{59} follows -from \eqref{58} -by applying the operator $r^{-\half(\al+\be-1)}\,\frac d{dr}\circ r^{\half(\al+\be+1)}$ -to both sides of \eqref{58}. -In view of \eqref{60}, formula \eqref{59} is the Poisson kernel for Jacobi -polynomials. The \RHS\ of \eqref{59} makes clear that this kernel is positive. -See also the discussion in Askey \myciteKLS{46}{following (2.32)}. -% -\paragraph{Quadratic transformations} -\begin{align} -\frac{C_{2n}^{(\al+\half)}(x)}{C_{2n}^{(\al+\half)}(1)} -=\frac{P_{2n}^{(\al,\al)}(x)}{P_{2n}^{(\al,\al)}(1)} -&=\frac{P_n^{(\al,-\half)}(2x^2-1)}{P_n^{(\al,-\half)}(1)}\,, -\label{51}\\ -\frac{C_{2n+1}^{(\al+\half)}(x)}{C_{2n+1}^{(\al+\half)}(1)} -=\frac{P_{2n+1}^{(\al,\al)}(x)}{P_{2n+1}^{(\al,\al)}(1)} -&=\frac{x\,P_n^{(\al,\half)}(2x^2-1)}{P_n^{(\al,\half)}(1)}\,. -\label{52} -\end{align} -See p.221, Remarks, last two formulas together with \eqref{50} and \eqref{49}. -Or see \mycite{DLMF}{(18.7.13), (18.7.14)}. -% -\paragraph{Differentiation formulas} -Each differentiation formula is given in two equivalent forms. -\begin{equation} -\begin{split} -\frac d{dx}\left((1-x)^\al P_n^{(\al,\be)}(x)\right)&= --(n+\al)\,(1-x)^{\al-1} P_n^{(\al-1,\be+1)}(x),\\ -\left((1-x)\frac d{dx}-\al\right)P_n^{(\al,\be)}(x)&= --(n+\al)\,P_n^{(\al-1,\be+1)}(x). -\end{split} -\label{68} -\end{equation} -% -\begin{equation} -\begin{split} -\frac d{dx}\left((1+x)^\be P_n^{(\al,\be)}(x)\right)&= -(n+\be)\,(1+x)^{\be-1} P_n^{(\al+1,\be-1)}(x),\\ -\left((1+x)\frac d{dx}+\be\right)P_n^{(\al,\be)}(x)&= -(n+\be)\,P_n^{(\al+1,\be-1)}(x). -\end{split} -\label{69} -\end{equation} -Formulas \eqref{68} and \eqref{69} follow from -\mycite{DLMF}{(15.5.4), (15.5.6)} -together with (9.8.1). They also follow from each other by \eqref{48}. -% -\paragraph{Generalized Gegenbauer polynomials} -These are defined by -\begin{equation} -S_{2m}^{(\al,\be)}(x):=\const P_m^{(\al,\be)}(2x^2-1),\qquad -S_{2m+1}^{(\al,\be)}(x):=\const x\,P_m^{(\al,\be+1)}(2x^2-1) -\label{70} -\end{equation} -in the notation of \myciteKLS{146}{p.156} -(see also \cite{K27}), while \cite[Section 1.5.2]{K26} -has $C_n^{(\la,\mu)}(x)=\const\allowbreak\times S_n^{(\la-\half,\mu-\half)}(x)$. -For $\al,\be>-1$ we have the orthogonality relation -\begin{equation} -\int_{-1}^1 S_m^{(\al,\be)}(x)\,S_n^{(\al,\be)}(x)\,|x|^{2\be+1}(1-x^2)^\al\,dx -=0\qquad(m\ne n). -\label{71} -\end{equation} -For $\be=\al-1$ generalized Gegenbauer polynomials are limit cases of -continuous $q$-ultraspherical polynomials, see \eqref{176}. - -If we define the {\em Dunkl operator} $T_\mu$ by -\begin{equation} -(T_\mu f)(x):=f'(x)+\mu\,\frac{f(x)-f(-x)}x -\label{72} -\end{equation} -and if we choose the constants in \eqref{70} as -\begin{equation} -S_{2m}^{(\al,\be)}(x)=\frac{(\al+\be+1)_m}{(\be+1)_m}\, P_m^{(\al,\be)}(2x^2-1),\quad -S_{2m+1}^{(\al,\be)}(x)=\frac{(\al+\be+1)_{m+1}}{(\be+1)_{m+1}}\, -x\,P_m^{(\al,\be+1)}(2x^2-1) -\label{73} -\end{equation} -then (see \cite[(1.6)]{K5}) -\begin{equation} -T_{\be+\half}S_n^{(\al,\be)}=2(\al+\be+1)\,S_{n-1}^{(\al+1,\be)}. -\label{74} -\end{equation} -Formula \eqref{74} with \eqref{73} substituted gives rise to two -differentiation formulas involving Jacobi polynomials which are equivalent to -(9.8.7) and \eqref{69}. - -Composition of \eqref{74} with itself gives -\[ -T_{\be+\half}^2S_n^{(\al,\be)}=4(\al+\be+1)(\al+\be+2)\,S_{n-2}^{(\al+2,\be)}, -\] -which is equivalent to the composition of (9.8.7) and \eqref{69}: -\begin{equation} -\left(\frac{d^2}{dx^2}+\frac{2\be+1}x\,\frac d{dx}\right)P_n^{(\al,\be)}(2x^2-1) -=4(n+\al+\be+1)(n+\be)\,P_{n-1}^{(\al+2,\be)}(2x^2-1). -\label{75} -\end{equation} -Formula \eqref{75} was also given in \myciteKLS{322}{(2.4)}. -xP_n^{(\lambda-\frac{1}{2},\frac{1}{2})}(2x^2-1).$$ - -\subsection*{References} -\cite{Abram}, \cite{Ahmed+82}, \cite{Allaway89}, \cite{NAlSalam66}, \cite{AlSalam64}, -\cite{AlSalamChihara72}, \cite{AndrewsAskey85}, \cite{AndrewsAskeyRoy}, \cite{Askey68}, -\cite{Askey70}, \cite{Askey72}, \cite{Askey73}, \cite{Askey74}, \cite{Askey75}, \cite{Askey78}, -\cite{Askey89I}, \cite{Askey2005}, \cite{AskeyFitch}, \cite{AskeyGasper71I}, \cite{AskeyGasper71II}, -\cite{AskeyGasper76}, \cite{AskeyWainger}, \cite{AskeyWilson85}, \cite{Bailey38}, \cite{Brafman51}, -\cite{Brown}, \cite{Carlitz61II}, \cite{Carlitz67}, \cite{ChenIsmail91}, \cite{Chihara78}, -\cite{Chow+}, \cite{Ciesielski}, \cite{Cooper+}, \cite{DetteStudden92}, \cite{DetteStudden95}, -\cite{DijksmaKoorn}, \cite{DimitrovRafaeli}, \cite{DimitrovRodrigues}, \cite{Doha2002I}, -\cite{Doha2003I}, \cite{Doha2004II}, \cite{Dunkl84}, \cite{ElbertLaforgia87II}, -\cite{ElbertLaforgiaRodono}, \cite{Erdelyi+}, \cite{Faldey}, \cite{FoataLeroux}, -\cite{Gasper69}, \cite{Gasper70I}, \cite{Gasper70II}, \cite{Gasper71I}, \cite{Gasper71II}, -\cite{Gasper72I}, \cite{Gasper72II}, \cite{Gasper73I}, \cite{Gasper73II}, \cite{Gasper74}, -\cite{Gasper77}, \cite{Gatteschi87}, \cite{Gautschi2008}, \cite{Gautschi2009I}, -\cite{Gautschi2009II}, \cite{GautschiLeopardi}, \cite{GawronskiShawyer}, \cite{Godoy+}, -\cite{Grad}, \cite{Hajmirzaahmad94}, \cite{HartmannStephan}, \cite{Horton}, \cite{Ismail74}, -\cite{Ismail77}, \cite{Ismail96}, \cite{Ismail2005II}, \cite{IsmailLi}, -\cite{IsmailMassonRahman}, \cite{Kochneff97I}, \cite{Koekoek99}, \cite{Koekoek2000}, -\cite{Koelink96II}, \cite{Koorn73}, \cite{Koorn74}, \cite{Koorn75}, \cite{Koorn77II}, -\cite{Koorn78}, \cite{Koorn72}, \cite{Koorn85}, \cite{Koorn88}, \cite{KuijlaarsMartinez}, -\cite{Kuijlaars+}, \cite{LabelleYehI}, \cite{LabelleYehII}, \cite{Laine}, \cite{Lesky95II}, -\cite{Lesky96}, \cite{Li96}, \cite{Li97}, \cite{LopezTemme2004}, \cite{Luke}, \cite{Martinez}, -\cite{Mathai}, \cite{Meijer}, \cite{Miller89}, \cite{MoakSaffVarga}, \cite{Nikiforov+}, -\cite{NikiforovUvarov}, \cite{Olver}, \cite{Prasad}, \cite{Rahman76I}, \cite{Rahman76II}, -\cite{Rahman77}, \cite{Rahman81I}, \cite{RahmanShah}, \cite{Rainville}, \cite{Rusev}, -\cite{Shi}, \cite{Srivastava69II}, \cite{Srivastava71}, \cite{Srivastava82}, -\cite{SrivastavaSinghal}, \cite{Stanton80I}, \cite{Stanton90}, \cite{Szego75}, \cite{Temme}, -\cite{Vertesi}, \cite{Wimp87}, \cite{WongZhang94}, \cite{WongZhang2006}, \cite{Zarzo+}, -\cite{Zayed}. - -\section*{Special cases} - -\subsection{Gegenbauer / Ultraspherical}\index{Gegenbauer polynomials} -\index{Ultraspherical polynomials} - -\par - -\subsection*{Hypergeometric representation} -The Gegenbauer (or ultraspherical) polynomials are Jacobi polynomials with -$\alpha=\beta=\lambda-\frac{1}{2}$ and another normalization: -\begin{eqnarray} -\label{DefGegenbauer} -C_n^{(\lambda)}(x)&=&\frac{(2\lambda)_n}{(\lambda+\frac{1}{2})_n} -P_n^{(\lambda-\frac{1}{2},\lambda-\frac{1}{2})}(x)\nonumber\\ -&=&\frac{(2\lambda)_n}{n!}\,\hyp{2}{1}{-n,n+2\lambda} -{\lambda+\frac{1}{2}}{\frac{1-x}{2}},\quad\lambda\neq 0. -\end{eqnarray} - -\subsection*{Orthogonality relation} -\begin{eqnarray} -\label{OrtGegenbauer} -& &\int_{-1}^1(1-x^2)^{\lambda-\frac{1}{2}}C_m^{(\lambda)}(x)C_n^{(\lambda)}(x)\,dx\nonumber\\ -& &{}=\frac{\pi\Gamma(n+2\lambda)2^{1-2\lambda}}{\left\{\Gamma(\lambda)\right\}^2(n+\lambda)n!}\,\delta_{mn}, -\quad\lambda>-\frac{1}{2}\quad\lambda\neq 0. -\end{eqnarray} - -\subsection*{Recurrence relation} -\begin{equation} -\label{RecGegenbauer} -2(n+\lambda)xC_n^{(\lambda)}(x)=(n+1)C_{n+1}^{(\lambda)}(x)+(n+2\lambda-1)C_{n-1}^{(\lambda)}(x). -\end{equation} - -\subsection*{Normalized recurrence relation} -\begin{equation} -\label{NormRecGegenbauer} -xp_n(x)=p_{n+1}(x)+\frac{n(n+2\lambda-1)}{4(n+\lambda-1)(n+\lambda)}p_{n-1}(x), -\end{equation} -where -$$C_n^{(\lambda)}(x)=\frac{2^n(\lambda)_n}{n!}p_n(x).$$ - -\subsection*{Differential equation} -\begin{equation} -\label{dvGegenbauer} -(1-x^2)y''(x)-(2\lambda+1)xy'(x)+n(n+2\lambda)y(x)=0,\quad y(x)=C_n^{(\lambda)}(x). -\end{equation} - -\subsection*{Forward shift operator} -\begin{equation} -\label{shift1Gegenbauer} -\frac{d}{dx}C_n^{(\lambda)}(x)=2\lambda C_{n-1}^{(\lambda+1)}(x). -\end{equation} - -\subsection*{Backward shift operator} -\begin{equation} -\label{shift2GegenbauerI} -(1-x^2)\frac{d}{dx}C_n^{(\lambda)}(x)+(1-2\lambda)xC_n^{(\lambda)}(x)= --\frac{(n+1)(2\lambda+n-1)}{2(\lambda-1)} C_{n+1}^{(\lambda-1)}(x) -\end{equation} -or equivalently -\begin{eqnarray} -\label{shift2GegenbauerII} -& &\frac{d}{dx}\left[(1-x^2)^{\lambda-\textstyle\frac{1}{2}}C_n^{(\lambda)}(x)\right]\nonumber\\ -& &{}=-\frac{(n+1)(2\lambda+n-1)}{2(\lambda-1)}(1-x^2)^{\lambda-\textstyle\frac{3}{2}}C_{n+1}^{(\lambda-1)}(x). -\end{eqnarray} - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodGegenbauer} -(1-x^2)^{\lambda-\frac{1}{2}}C_n^{(\lambda)}(x)= -\frac{(2\lambda)_n(-1)^n}{(\lambda+\frac{1}{2})_n2^nn!}\left(\frac{d}{dx}\right)^n -\left[(1-x^2)^{\lambda+n-\frac{1}{2}}\right]. -\end{equation} - -\subsection*{Generating functions} -\begin{equation} -\label{GenGegenbauer1} -(1-2xt+t^2)^{-\lambda}=\sum_{n=0}^{\infty}C_n^{(\lambda)}(x)t^n. -\end{equation} - -\begin{equation} -\label{GenGegenbauer2} -R^{-1}\left(\frac{1+R-xt}{2}\right)^{\frac{1}{2}-\lambda}=\sum_{n=0}^{\infty} -\frac{(\lambda+\frac{1}{2})_n}{(2\lambda)_n}C_n^{(\lambda)}(x)t^n, -\quad R=\sqrt{1-2xt+t^2}. -\end{equation} - -\begin{equation} -\label{GenGegenbauer3} -\hyp{0}{1}{-}{\lambda+\frac{1}{2}}{\frac{(x-1)t}{2}}\, -\hyp{0}{1}{-}{\lambda+\frac{1}{2}}{\frac{(x+1)t}{2}} -=\sum_{n=0}^{\infty}\frac{C_n^{(\lambda)}(x)} -{(2\lambda)_n(\lambda+\frac{1}{2})_n}t^n. -\end{equation} - -\begin{equation} -\label{GenGegenbauer4} -\e^{xt}\,\hyp{0}{1}{-}{\lambda+\frac{1}{2}}{\frac{(x^2-1)t^2}{4}}= -\sum_{n=0}^{\infty}\frac{C_n^{(\lambda)}(x)}{(2\lambda)_n}t^n. -\end{equation} - -\begin{eqnarray} -\label{GenGegenbauer5} -& &\hyp{2}{1}{\gamma,2\lambda-\gamma}{\lambda+\frac{1}{2}}{\frac{1-R-t}{2}}\, -\hyp{2}{1}{\gamma,2\lambda-\gamma}{\lambda+\frac{1}{2}}{\frac{1-R+t}{2}}\nonumber\\ -& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n(2\lambda-\gamma)_n} -{(2\lambda)_n(\lambda+\frac{1}{2})_n}C_n^{(\lambda)}(x)t^n, -\quad R=\sqrt{1-2xt+t^2},\quad\textrm{$\gamma$ arbitrary}. -\end{eqnarray} - -\begin{eqnarray} -\label{GenGegenbauer6} -& &(1-xt)^{-\gamma}\,\hyp{2}{1}{\frac{1}{2}\gamma,\frac{1}{2}\gamma+\frac{1}{2}} -{\lambda+\frac{1}{2}}{\frac{(x^2-1)t^2}{(1-xt)^2}}\nonumber\\ -& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n}{(2\lambda)_n}C_n^{(\lambda)}(x)t^n, -\quad\textrm{$\gamma$ arbitrary}. -\end{eqnarray} - -\subsection*{Limit relation} - -\subsubsection*{Gegenbauer / Ultraspherical $\rightarrow$ Hermite} -The Hermite polynomials given by (\ref{DefHermite}) follow from the Gegenbauer (or -ultraspherical) polynomials by taking $\lambda=\alpha+\frac{1}{2}$ and letting -$\alpha\rightarrow\infty$ in the following way: -\begin{equation} -\lim_{\alpha\rightarrow\infty} -\alpha^{-\frac{1}{2}n}C_n^{(\alpha+\frac{1}{2})}(\alpha^{-\frac{1}{2}}x)=\frac{H_n(x)}{n!}. -\end{equation} - -\subsection*{Remarks} -The case $\lambda=0$ needs another normalization. In that case we have the -Chebyshev polynomials of the first kind described in the next subsection. - -\noindent -The Gegenbauer (or ultraspherical) polynomials are related to the Jacobi polynomials -given by (\ref{DefJacobi}) by the quadratic transformations: -$$C_{2n}^{(\lambda)}(x)=\frac{(\lambda)_n}{(\frac{1}{2})_n} -P_n^{(\lambda-\frac{1}{2},-\frac{1}{2})}(2x^2-1)$$ -and -$$C_{2n+1}^{(\lambda)}(x)=\frac{(\lambda)_{n+1}}{(\frac{1}{2})_{n+1}} -\subsection*{9.8.1 Gegenbauer / Ultraspherical} -\label{sec9.8.1} -% -\paragraph{Notation} -Here the Gegenbauer polynomial is denoted by $C_n^\la$ instead of $C_n^{(\la)}$. -% -\paragraph{Orthogonality relation} -Write the \RHS\ of (9.8.20) as $h_n\,\de_{m,n}$. Then -\begin{equation} -\frac{h_n}{h_0}= -\frac\la{\la+n}\,\frac{(2\la)_n}{n!}\,,\quad -h_0=\frac{\pi^\half\,\Ga(\la+\thalf)}{\Ga(\la+1)},\quad -\frac{h_n}{h_0\,(C_n^\la(1))^2}= -\frac\la{\la+n}\,\frac{n!}{(2\la)_n}\,. -\label{61} -\end{equation} -% -\paragraph{Hypergeometric representation} -Beside (9.8.19) we have also -\begin{equation} -C_n^\lambda(x)=\sum_{\ell=0}^{\lfloor n/2\rfloor}\frac{(-1)^{\ell}(\lambda)_{n-\ell}} -{\ell!\;(n-2\ell)!}\,(2x)^{n-2\ell} -=(2x)^{n}\,\frac{(\lambda)_{n}}{n!}\, -\hyp21{-\thalf n,-\thalf n+\thalf}{1-\la-n}{\frac1{x^2}}. -\label{57} -\end{equation} -See \mycite{DLMF}{(18.5.10)}. -% -\paragraph{Special value} -\begin{equation} -C_n^{\la}(1)=\frac{(2\la)_n}{n!}\,. -\label{49} -\end{equation} -Use (9.8.19) or see \mycite{DLMF}{Table 18.6.1}. -% -\paragraph{Expression in terms of Jacobi} -% -\begin{equation} -\frac{C_n^\la(x)}{C_n^\la(1)}= -\frac{P_n^{(\la-\half,\la-\half)}(x)}{P_n^{(\la-\half,\la-\half)}(1)}\,,\qquad -C_n^\la(x)=\frac{(2\la)_n}{(\la+\thalf)_n}\,P_n^{(\la-\half,\la-\half)}(x). -\label{65} -\end{equation} -% -\paragraph{Re: (9.8.21)} -By iteration of recurrence relation (9.8.21): -\begin{multline} -x^2 C_n^\la(x)= -\frac{(n+1)(n+2)}{4(n+\la)(n+\la+1)}\,C_{n+2}^\la(x)+ -\frac{n^2+2n\la+\la-1}{2(n+\la-1)(n+\la+1)}\,C_n^\la(x)\\ -+\frac{(n+2\la-1)(n+2\la-2)}{4(n+\la)(n+\la-1)}\,C_{n-2}^\la(x). -\label{6} -\end{multline} -% -\paragraph{Bilateral generating functions} -\begin{multline} -\sum_{n=0}^\iy\frac{n!}{(2\la)_n}\,r^n\,C_n^\la(x)\,C_n^\la(y) -=\frac1{(1-2rxy+r^2)^\la}\,\hyp21{\thalf\la,\thalf(\la+1)}{\la+\thalf} -{\frac{4r^2(1-x^2)(1-y^2)}{(1-2rxy+r^2)^2}}\\ -(r\in(-1,1),\;x,y\in[-1,1]). -\label{66} -\end{multline} -For the proof put $\be:=\al$ in \eqref{58}, then use \eqref{63} and \eqref{65}. -The Poisson kernel for Gegenbauer polynomials can be derived in a similar way -from \eqref{59}, or alternatively by applying the operator -$r^{-\la+1}\frac d{dr}\circ r^\la$ to both sides of \eqref{66}: -\begin{multline} -\sum_{n=0}^\iy\frac{\la+n}\la\,\frac{n!}{(2\la)_n}\,r^n\,C_n^\la(x)\,C_n^\la(y) -=\frac{1-r^2}{(1-2rxy+r^2)^{\la+1}}\\ -\times\hyp21{\thalf(\la+1),\thalf(\la+2)}{\la+\thalf} -{\frac{4r^2(1-x^2)(1-y^2)}{(1-2rxy+r^2)^2}}\qquad -(r\in(-1,1),\;x,y\in[-1,1]). -\label{67} -\end{multline} -Formula \eqref{67} was obtained by Gasper \& Rahman \myciteKLS{234}{(4.4)} -as a limit case of their formula for the Poisson kernel for continuous -$q$-ultraspherical polynomials. -% -\paragraph{Trigonometric expansions} -By \mycite{DLMF}{(18.5.11), (15.8.1)}: -\begin{align} -C_n^{\la}(\cos\tha) -&=\sum_{k=0}^n\frac{(\la)_k(\la)_{n-k}}{k!\,(n-k)!}\,e^{i(n-2k)\tha} -=e^{in\tha}\frac{(\la)_n}{n!}\, -\hyp21{-n,\la}{1-\la-n}{e^{-2i\tha}}\label{103}\\ -&=\frac{(\la)_n}{2^\la n!}\, -e^{-\half i\la\pi}e^{i(n+\la)\tha}\,(\sin\tha)^{-\la}\, -\hyp21{\la,1-\la}{1-\la-n}{\frac{i e^{-i\tha}}{2\sin\tha}}\label{104}\\ -&=\frac{(\la)_n}{n!}\,\sum_{k=0}^\iy\frac{(\la)_k(1-\la)_k}{(1-\la-n)_k k!}\, -\frac{\cos((n-k+\la)\tha+\thalf(k-\la)\pi)}{(2\sin\tha)^{k+\la}}\,.\label{105} -\end{align} -In \eqref{104} and \eqref{105} we require that -$\tfrac16\pi<\tha<\tfrac56\pi$. Then the convergence is absolute for $\la>\thalf$ -and conditional for $0<\la\le\thalf$. - -By \mycite{DLMF}{(14.13.1), (14.3.21), (15.8.1)]}: -\begin{align} -C_n^\la(\cos\tha)&=\frac{2\Ga(\la+\thalf)}{\pi^\half\Ga(\la+1)}\, -\frac{(2\la)_n}{(\la+1)_n}\,(\sin\tha)^{1-2\la}\, -\sum_{k=0}^\iy\frac{(1-\la)_k(n+1)_k}{(n+\la+1)_k k!}\, -\sin\big((2k+n+1)\tha\big) -\label{7}\\ -&=\frac{2\Ga(\la+\thalf)}{\pi^\half\Ga(\la+1)}\, -\frac{(2\la)_n}{(\la+1)_n}\,(\sin\tha)^{1-2\la}\, -\Im\!\!\left(e^{i(n+1)\tha}\,\hyp21{1-\la,n+1}{n+\la+1}{e^{2i\tha}}\right)\nonumber\\ -&=\frac{2^\la\Ga(\la+\thalf)}{\pi^\half\Ga(\la+1)}\, -\frac{(2\la)_n}{(\la+1)_n}\,(\sin\tha)^{-\la}\, -\Re\!\!\left(e^{-\thalf i\la\pi}e^{i(n+\la)\tha}\, -\hyp21{\la,1-\la}{1+\la+n}{\frac{e^{i\tha}}{2i\sin\tha}}\right)\nonumber\\ -&=\frac{2^{2\la}\Ga(\la+\thalf)}{\pi^\half\Ga(\la+1)}\,\frac{(2\la)_n}{(\la+1)_n}\, -\sum_{k=0}^\iy\frac{(\la)_k(1-\la)_k}{(1+\la+n)_k k!}\, -\frac{\cos((n+k+\la)\tha-\thalf(k+\la)\pi)}{(2\sin\tha)^{k+\la}}\,. -\label{106} -\end{align} -We require that $0<\tha<\pi$ in \eqref{7} and $\tfrac16\pi<\tha<\tfrac56\pi$ in -\eqref{106} The convergence is absolute for $\la>\thalf$ and conditional for -$0<\la\le\thalf$. -For $\la\in\Zpos$ the above series terminate after the term with -$k=\la-1$. -Formulas \eqref{7} and \eqref{106} are also given in -\mycite{Sz}{(4.9.22), (4.9.25)}. -% -\paragraph{Fourier transform} -\begin{equation} -\frac{\Ga(\la+1)}{\Ga(\la+\thalf)\,\Ga(\thalf)}\, -\int_{-1}^1 \frac{C_n^\la(y)}{C_n^\la(1)}\,(1-y^2)^{\la-\half}\, -e^{ixy}\,dy -=i^n\,2^\la\,\Ga(\la+1)\,x^{-\la}\,J_{\la+n}(x). -\label{8} -\end{equation} -See \mycite{DLMF}{(18.17.17) and (18.17.18)}. -% -\paragraph{Laplace transforms} -\begin{equation} -\frac2{n!\,\Ga(\la)}\, -\int_0^\iy H_n(tx)\,t^{n+2\la-1}\,e^{-t^2}\,dt=C_n^\la(x). -\label{56} -\end{equation} -See Nielsen \cite[p.48, (4) with p.47, (1) and p.28, (10)]{K4} (1918) -or Feldheim \cite[(28)]{K3} (1942). -\begin{equation} -\frac2{\Ga(\la+\thalf)}\,\int_0^1 \frac{C_n^\la(t)}{C_n^\la(1)}\, -(1-t^2)^{\la-\half}\,t^{-1}\,(x/t)^{n+2\la+1}\,e^{-x^2/t^2}\,dt -=2^{-n}\,H_n(x)\,e^{-x^2}\quad(\la>-\thalf). -\label{46} -\end{equation} -Use Askey \& Fitch \cite[(3.29)]{K2} for $\al=\pm\thalf$ together with -\eqref{48}, \eqref{51}, \eqref{52}, \eqref{54} and \eqref{55}. -\paragraph{Addition formula} (see \mycite{AAR}{(9.8.5$'$)]}) -\begin{multline} -R_n^{(\al,\al)}\big(xy+(1-x^2)^\half(1-y^2)^\half t\big) -=\sum_{k=0}^n \frac{(-1)^k(-n)_k\,(n+2\al+1)_k}{2^{2k}((\al+1)_k)^2}\\ -\times(1-x^2)^{k/2} R_{n-k}^{(\al+k,\al+k)}(x)\,(1-y^2)^{k/2} R_{n-k}^{(\al+k,\al+k)}(y)\, -\om_k^{(\al-\half,\al-\half)}\,R_k^{(\al-\half,\al-\half)}(t), -\label{108} -\end{multline} -where -\[ -R_n^{(\al,\be)}(x):=P_n^{(\al,\be)}(x)/P_n^{(\al,\be)}(1),\quad -\om_n^{(\al,\be)}:=\frac{\int_{-1}^1 (1-x)^\al(1+x)^\be\,dx} -{\int_{-1}^1 (R_n^{(\al,\be)}(x))^2\,(1-x)^\al(1+x)^\be\,dx}\,. -\] -xP_n^{(\lambda-\frac{1}{2},\frac{1}{2})}(2x^2-1).$$ - -\subsection*{References} -\cite{Abram}, \cite{Ahmed+86}, \cite{AndrewsAskeyRoy}, \cite{Area+I}, \cite{Askey67}, \cite{Askey74}, \cite{Askey75}, -\cite{Askey89I}, \cite{AskeyFitch}, \cite{AskeyKoornRahman}, \cite{Berg}, \cite{BilodeauI}, -\cite{BojanovNikolov}, \cite{Brafman51}, \cite{Brafman57I}, \cite{Brown}, -\cite{BustozIsmail82}, \cite{BustozIsmail83}, \cite{BustozIsmail89}, \cite{BustozSavage79}, -\cite{BustozSavage80}, \cite{Carlitz61II}, \cite{Chihara78}, \cite{Common}, \cite{Danese}, -\cite{Dette94}, \cite{DijksmaKoorn}, \cite{DilcherStolarsky}, \cite{Dimitrov96}, -\cite{Dimitrov2003}, \cite{Doha2002II}, \cite{Driver}, \cite{ElbertLaforgia86I}, -\cite{ElbertLaforgia86II}, \cite{ElbertLaforgia90}, \cite{Erdelyi+}, \cite{Exton96}, -\cite{Gasper69}, \cite{Gasper72II}, \cite{Gasper85}, \cite{Grad}, \cite{Ismail74}, -\cite{Ismail2005II}, \cite{Koekoek2000}, \cite{Koorn88}, \cite{Laforgia}, \cite{LewanowiczI}, -\cite{Lorch}, \cite{Mathai}, \cite{Nagel}, \cite{Nikiforov+}, \cite{NikiforovUvarov}, -\cite{RahmanShah}, \cite{Rainville}, \cite{Reimer}, \cite{Sartoretto}, \cite{Srivastava71}, -\cite{Szego75}, \cite{Temme}, \cite{Viswanathan}, \cite{Zayed}. - -\subsection{Chebyshev}\index{Chebyshev polynomials} - -\par - -\subsection*{Hypergeometric representation} -The Chebyshev polynomials of the first kind can be obtained from the Jacobi -polynomials by taking $\alpha=\beta=-\frac{1}{2}$: -\begin{equation} -\label{DefChebyshevI} -T_n(x)=\frac{P_n^{(-\frac{1}{2},-\frac{1}{2})}(x)}{P_n^{(-\frac{1}{2},-\frac{1}{2})}(1)} -=\hyp{2}{1}{-n,n}{\frac{1}{2}}{\frac{1-x}{2}} -\end{equation} -and the Chebyshev polynomials of the second kind can be obtained from the -Jacobi polynomials by taking $\alpha=\beta=\frac{1}{2}$: -\begin{equation} -\label{DefChebyshevII} -U_n(x)=(n+1)\frac{P_n^{(\frac{1}{2},\frac{1}{2})}(x)}{P_n^{(\frac{1}{2},\frac{1}{2})}(1)} -=(n+1)\,\hyp{2}{1}{-n,n+2}{\frac{3}{2}}{\frac{1-x}{2}}. -\end{equation} - -\subsection*{Orthogonality relation} -\begin{equation} -\label{OrtChebyshevI} -\int_{-1}^1(1-x^2)^{-\frac{1}{2}}T_m(x)T_n(x)\,dx= -\left\{\begin{array}{ll} -\displaystyle\frac{\cpi}{2}\,\delta_{mn}, & n\neq 0\\[5mm] -\cpi\,\delta_{mn}, & n=0. -\end{array}\right. -\end{equation} - -\begin{equation} -\label{OrtChebyshevII} -\int_{-1}^1(1-x^2)^{\frac{1}{2}}U_m(x)U_n(x)\,dx=\frac{\cpi}{2}\,\delta_{mn}. -\end{equation} - -\subsection*{Recurrence relations} -\begin{equation} -\label{RecChebyshevI} -2xT_n(x)=T_{n+1}(x)+T_{n-1}(x),\quad T_{0}(x)=1\quad\textrm{and}\quad T_1(x)=x. -\end{equation} - -\begin{equation} -\label{RecChebyshevII} -2xU_n(x)=U_{n+1}(x)+U_{n-1}(x),\quad U_{0}(x)=1\quad\textrm{and}\quad U_1(x)=2x. -\end{equation} - -\subsection*{Normalized recurrence relations} -\begin{equation} -\label{NormRecChebyshevI} -xp_n(x)=p_{n+1}(x)+\frac{1}{4}p_{n-1}(x), -\end{equation} -where -$$T_1(x)=p_1(x)=x\quad\textrm{and}\quad T_n(x)=2^np_n(x),\quad n\neq 1.$$ - -\begin{equation} -\label{NormRecChebyshevII} -xp_n(x)=p_{n+1}(x)+\frac{1}{4}p_{n-1}(x), -\end{equation} -where -$$U_n(x)=2^np_n(x).$$ - -\subsection*{Differential equations} -\begin{equation} -\label{dvChebyshevI} -(1-x^2)y''(x)-xy'(x)+n^2y(x)=0,\quad y(x)=T_n(x). -\end{equation} - -\begin{equation} -\label{dvChebyshevII} -(1-x^2)y''(x)-3xy'(x)+n(n+2)y(x)=0,\quad y(x)=U_n(x). -\end{equation} - -\subsection*{Forward shift operator} -\begin{equation} -\label{shift1Chebyshev} -\frac{d}{dx}T_n(x)=nU_{n-1}(x). -\end{equation} - -\subsection*{Backward shift operator} -\begin{equation} -\label{shift2ChebyshevI} -(1-x^2)\frac{d}{dx}U_n(x)-xU_n(x)=-(n+1)T_{n+1}(x) -\end{equation} -or equivalently -\begin{equation} -\label{shift2ChebyshevII} -\frac{d}{dx}\left[\left(1-x^2\right)^{\frac{1}{2}}U_n(x)\right] -=-(n+1)\left(1-x^2\right)^{-\frac{1}{2}}T_{n+1}(x). -\end{equation} - -\subsection*{Rodrigues-type formulas} -\begin{equation} -\label{RodChebyshevI} -(1-x^2)^{-\frac{1}{2}}T_n(x)=\frac{(-1)^n}{(\frac{1}{2})_n2^n} -\left(\frac{d}{dx}\right)^n\left[(1-x^2)^{n-\frac{1}{2}}\right]. -\end{equation} - -\begin{equation} -\label{RodChebyshevII} -(1-x^2)^{\frac{1}{2}}U_n(x)=\frac{(n+1)(-1)^n}{(\frac{3}{2})_n2^n} -\left(\frac{d}{dx}\right)^n\left[(1-x^2)^{n+\frac{1}{2}}\right]. -\end{equation} - -\subsection*{Generating functions} -\begin{equation} -\label{GenChebyshevI1} -\frac{1-xt}{1-2xt+t^2}=\sum_{n=0}^{\infty}T_n(x)t^n. -\end{equation} - -\begin{equation} -\label{GenChebyshevI2} -R^{-1}\sqrt{\frac{1}{2}(1+R-xt)}=\sum_{n=0}^{\infty} -\frac{\left(\frac{1}{2}\right)_n}{n!}T_n(x)t^n,\quad R=\sqrt{1-2xt+t^2}. -\end{equation} - -\begin{equation} -\label{GenChebyshevI3} -\hyp{0}{1}{-}{\frac{1}{2}}{\frac{(x-1)t}{2}}\, -\hyp{0}{1}{-}{\frac{1}{2}}{\frac{(x+1)t}{2}}= -\sum_{n=0}^{\infty}\frac{T_n(x)}{\left(\frac{1}{2}\right)_nn!}t^n. -\end{equation} - -\begin{equation} -\label{GenChebyshevI4} -\e^{xt}\,\hyp{0}{1}{-}{\frac{1}{2}}{\frac{(x^2-1)t^2}{4}}= -\sum_{n=0}^{\infty}\frac{T_n(x)}{n!}t^n. -\end{equation} - -\begin{eqnarray} -\label{GenChebyshevI5} -& &\hyp{2}{1}{\gamma,-\gamma}{\frac{1}{2}}{\frac{1-R-t}{2}}\, -\hyp{2}{1}{\gamma,-\gamma}{\frac{1}{2}}{\frac{1-R+t}{2}}\nonumber\\ -& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n(-\gamma)_n}{\left(\frac{1}{2}\right)_nn!}T_n(x)t^n, -\quad R=\sqrt{1-2xt+t^2},\quad\textrm{$\gamma$ arbitrary}. -\end{eqnarray} - -\begin{eqnarray} -\label{GenChebyshevI6} -& &(1-xt)^{-\gamma}\,\hyp{2}{1}{\frac{1}{2}\gamma,\frac{1}{2}\gamma+\frac{1}{2}}{\frac{1}{2}} -{\frac{(x^2-1)t^2}{(1-xt)^2}}\nonumber\\ -& &=\sum_{n=0}^{\infty}\frac{(\gamma)_n}{n!}T_n(x)t^n,\quad\textrm{$\gamma$ arbitrary}. -\end{eqnarray} - -\begin{equation} -\label{GenChebyshevII1} -\frac{1}{1-2xt+t^2}=\sum_{n=0}^{\infty}U_n(x)t^n. -\end{equation} - -\begin{equation} -\label{GenChebyshevII2} -\frac{1}{R\sqrt{\frac{1}{2}(1+R-xt)}}=\sum_{n=0}^{\infty} -\frac{\left(\frac{3}{2}\right)_n}{(n+1)!}U_n(x)t^n,\quad R=\sqrt{1-2xt+t^2}. -\end{equation} - -\begin{equation} -\label{GenChebyshevII3} -\hyp{0}{1}{-}{\frac{3}{2}}{\frac{(x-1)t}{2}}\, -\hyp{0}{1}{-}{\frac{3}{2}}{\frac{(x+1)t}{2}}= -\sum_{n=0}^{\infty}\frac{U_n(x)}{\left(\frac{3}{2}\right)_n(n+1)!}t^n. -\end{equation} - -\begin{equation} -\label{GenChebyshevII4} -\e^{xt}\,\hyp{0}{1}{-}{\frac{3}{2}}{\frac{(x^2-1)t^2}{4}}= -\sum_{n=0}^{\infty}\frac{U_n(x)}{(n+1)!}t^n. -\end{equation} - -\begin{eqnarray} -\label{GenChebyshevII5} -& &\hyp{2}{1}{\gamma,2-\gamma}{\frac{3}{2}}{\frac{1-R-t}{2}}\, -\hyp{2}{1}{\gamma,2-\gamma}{\frac{3}{2}}{\frac{1-R+t}{2}}\nonumber\\ -& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n(2-\gamma)_n}{\left(\frac{3}{2}\right)_n(n+1)!}U_n(x)t^n, -\quad R=\sqrt{1-2xt+t^2},\quad\textrm{$\gamma$ arbitrary}. -\end{eqnarray} - -\begin{eqnarray} -\label{GenChebyshevII6} -& &(1-xt)^{-\gamma}\,\hyp{2}{1}{\frac{1}{2}\gamma,\frac{1}{2}\gamma+\frac{1}{2}}{\frac{3}{2}} -{\frac{(x^2-1)t^2}{(1-xt)^2}}\nonumber\\ -& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n}{(n+1)!}U_n(x)t^n,\quad\textrm{$\gamma$ arbitrary}. -\end{eqnarray} - -\subsection*{Remarks} -The Chebyshev polynomials can also be written as: -$$T_n(x)=\cos(n\theta),\quad x=\cos\theta$$ -and -$$U_n(x)=\frac{\sin (n+1)\theta}{\sin\theta},\quad x=\cos\theta.$$ -Further we have -$$U_n(x)=C_n^{(1)}(x)$$ -where $C_n^{(\lambda)}(x)$ denotes the Gegenbauer (or ultraspherical) -\subsection*{9.8.2 Chebyshev} -\label{sec9.8.2} -In addition to the Chebyshev polynomials $T_n$ of the first kind (9.8.35) -and $U_n$ of the second kind (9.8.36), -\begin{align} -T_n(x)&:=\frac{P_n^{(-\half,-\half)}(x)}{P_n^{(-\half,-\half)}(1)} -=\cos(n\tha),\quad x=\cos\tha,\\ -U_n(x)&:=(n+1)\,\frac{P_n^{(\half,\half)}(x)}{P_n^{(\half,\half)}(1)} -=\frac{\sin((n+1)\tha)}{\sin\tha}\,,\quad x=\cos\tha, -\end{align} -we have Chebyshev polynomials $V_n$ {\em of the third kind} -and $W_n$ {\em of the fourth kind}, -\begin{align} -V_n(x)&:=\frac{P_n^{(-\half,\half)}(x)}{P_n^{(-\half,\half)}(1)} -=\frac{\cos((n+\thalf)\tha)}{\cos(\thalf\tha)}\,,\quad x=\cos\tha,\\ -W_n(x)&:=(2n+1)\,\frac{P_n^{(\half,-\half)}(x)}{P_n^{(\half,-\half)}(1)} -=\frac{\sin((n+\thalf)\tha)}{\sin(\thalf\tha)}\,,\quad x=\cos\tha, -\end{align} -see \cite[Section 1.2.3]{K20}. Then there is the symmetry -\begin{equation} -V_n(-x)=(-1)^n W_n(x). -\label{140} -\end{equation} - -The names of Chebyshev polynomials of the third and fourth kind -and the notation $V_n(x)$ are due to Gautschi \cite{K21}. -The notation $W_n(x)$ was first used by Mason \cite{K22}. -Names and notations for Chebyshev polynomials of the third and fourth -kind are interchanged in \mycite{AAR}{Remark 2.5.3} and -\mycite{DLMF}{Table 18.3.1}. -polynomial given by (\ref{DefGegenbauer}) in the preceding subsection. - -\subsection*{References} -\cite{Abram}, \cite{AskeyFitch}, \cite{AskeyGasperHarris}, \cite{AskeyIsmail76}, -\cite{Bavinck95}, \cite{Chihara78}, \cite{Danese}, \cite{DilcherStolarsky}, -\cite{Erdelyi+}, \cite{Grad}, \cite{HartmannStephan}, \cite{Ismail2005II}, \cite{Koekoek2000}, -\cite{Luke}, \cite{Mathai}, \cite{Nikiforov+}, \cite{NikiforovUvarov}, \cite{Rainville}, -\cite{Rayes+}, \cite{Rivlin}, \cite{SainteViennot}, \cite{Szego75}, \cite{Temme}, -\cite{Wilson70I}, \cite{Zayed}, \cite{Zhang}, \cite{ZhangWang}. - -\subsection{Legendre / Spherical}\index{Legendre polynomials} -\index{Spherical polynomials} - -\par - -\subsection*{Hypergeometric representation} -The Legendre (or spherical) polynomials are Jacobi polynomials with $\alpha=\beta=0$: -\begin{equation} -\label{DefLegendre} -P_n(x)=P_n^{(0,0)}(x)=\hyp{2}{1}{-n,n+1}{1}{\frac{1-x}{2}}. -\end{equation} - -\subsection*{Orthogonality relation} -\begin{equation} -\label{OrtLegendre} -\int_{-1}^1P_m(x)P_n(x)\,dx=\frac{2}{2n+1}\,\delta_{mn}. -\end{equation} - -\subsection*{Recurrence relation} -\begin{equation} -\label{RecLegendre} -(2n+1)xP_n(x)=(n+1)P_{n+1}(x)+nP_{n-1}(x). -\end{equation} - -\subsection*{Normalized recurrence relation} -\begin{equation} -\label{NormRecLegendre} -xp_n(x)=p_{n+1}(x)+\frac{n^2}{(2n-1)(2n+1)}p_{n-1}(x), -\end{equation} -where -$$P_n(x)=\binom{2n}{n}\frac{1}{2^n}p_n(x).$$ - -\subsection*{Differential equation} -\begin{equation} -\label{dvLegendre} -(1-x^2)y''(x)-2xy'(x)+n(n+1)y(x)=0,\quad y(x)=P_n(x). -\end{equation} - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodLegendre} -P_n(x)=\frac{(-1)^n}{2^nn!}\left(\frac{d}{dx}\right)^n\left[(1-x^2)^n\right]. -\end{equation} - -\subsection*{Generating functions} -\begin{equation} -\label{GenLegendre1} -\frac{1}{\sqrt{1-2xt+t^2}}=\sum_{n=0}^{\infty}P_n(x)t^n. -\end{equation} - -\begin{equation} -\label{GenLegendre2} -\hyp{0}{1}{-}{1}{\frac{(x-1)t}{2}}\,\hyp{0}{1}{-}{1}{\frac{(x+1)t}{2}}= -\sum_{n=0}^{\infty}\frac{P_n(x)}{(n!)^2}t^n. -\end{equation} - -\begin{equation} -\label{GenLegendre3} -\e^{xt}\,\hyp{0}{1}{-}{1}{\frac{(x^2-1)t^2}{4}}= -\sum_{n=0}^{\infty}\frac{P_n(x)}{n!}t^n. -\end{equation} - -\begin{eqnarray} -\label{GenLegendre4} -& &\hyp{2}{1}{\gamma,1-\gamma}{1}{\frac{1-R-t}{2}}\, -\hyp{2}{1}{\gamma,1-\gamma}{1}{\frac{1-R+t}{2}}\nonumber\\ -& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n(1-\gamma)_n}{(n!)^2}P_n(x)t^n, -\quad R=\sqrt{1-2xt+t^2},\quad\textrm{$\gamma$ arbitrary}. -\end{eqnarray} - -\begin{eqnarray} -\label{GenLegendre5} -& &(1-xt)^{-\gamma}\,\hyp{2}{1}{\frac{1}{2}\gamma,\frac{1}{2}\gamma+\frac{1}{2}}{1} -{\frac{(x^2-1)t^2}{(1-xt)^2}}\nonumber\\ -& &{}=\sum_{n=0}^{\infty}\frac{(\gamma)_n}{n!}P_n(x)t^n,\quad\textrm{$\gamma$ arbitrary}. -\subsection*{9.9 Pseudo Jacobi (or Routh-Romanovski)} -\label{sec9.9} -In this section in \mycite{KLS} the pseudo Jacobi polynomial $P_n(x;\nu,N)$ in (9.9.1) -is considered -for $N\in\ZZ_{\ge0}$ and $n=0,1,\ldots,n$. However, we can more generally take -$-\thalf\be>0\;\;{\rm or}\;\;\al<\be<0). -\] -Then $P_n$ can be expressed as a Meixner polynomial: -\[ -P_n(x)=(-k_2(\al\be)^{-1})_n\,\be^n\, -M_n\left(-\,\frac{x+k_2\al^{-1}}{\al-\be},-k_2(\al\be)^{-1},\be\al^{-1}\right). -\] - -In 1938 Gottlieb \cite[\S2]{K1} introduces polynomials $l_n$ ``of Laguerre type'' -which turn out to be special Meixner polynomials: -$l_n(x)=e^{-n\la} M_n(x;1,e^{-\la})$. -% -\paragraph{Uniqueness of orthogonality measure} -The coefficient of $p_{n-1}(x)$ in (9.10.4) behaves as $O(n^2)$ as $n\to\iy$. -Hence \eqref{93} holds, by which the orthogonality measure is unique. -=\frac{2^n}{(n-2N-1)_n}y_n(x;-2N-2).$$ - -\subsection*{References} -\cite{Askey87}, \cite{BorodinOlshanski}, \cite{Lesky96}. - - -\section{Meixner}\index{Meixner polynomials} - -\par\setcounter{equation}{0} - -\subsection*{Hypergeometric representation} -\begin{equation} -\label{DefMeixner} -M_n(x;\beta,c)=\hyp{2}{1}{-n,-x}{\beta}{1-\frac{1}{c}}. -\end{equation} - -\subsection*{Orthogonality relation} -\begin{equation} -\label{OrtMeixner} -\sum_{x=0}^{\infty}\frac{(\beta)_x}{x!}c^xM_m(x;\beta,c)M_n(x;\beta,c) -{}=\frac{c^{-n}n!}{(\beta)_n(1-c)^{\beta}}\,\delta_{mn} -% \constraint{ -% $\beta > 0$ & -% $0 < c < 1$ } -\end{equation} - -\subsection*{Recurrence relation} -\begin{eqnarray} -\label{RecMeixner} -(c-1)xM_n(x;\beta,c)&=&c(n+\beta)M_{n+1}(x;\beta,c)\nonumber\\ -& &{}\mathindent{}-\left[n+(n+\beta)c\right]M_n(x;\beta,c)+nM_{n-1}(x;\beta,c). -\end{eqnarray} - -\subsection*{Normalized recurrence relation} -\begin{equation} -\label{NormRecMeixner} -xp_n(x)=p_{n+1}(x)+\frac{n+(n+\beta)c}{1-c}p_n(x)+ -\frac{n(n+\beta-1)c}{(1-c)^2}p_{n-1}(x), -\end{equation} -where -$$M_n(x;\beta,c)=\frac{1}{(\beta)_n}\left(\frac{c-1}{c}\right)^np_n(x).$$ - -\subsection*{Difference equation} -\begin{equation} -\label{dvMeixner} -n(c-1)y(x)=c(x+\beta)y(x+1)-\left[x+(x+\beta)c\right]y(x)+xy(x-1), -\end{equation} -where -$$y(x)=M_n(x;\beta,c).$$ - -\subsection*{Forward shift operator} -\begin{equation} -\label{shift1MeixnerI} -M_n(x+1;\beta,c)-M_n(x;\beta,c)= -\frac{n}{\beta}\left(\frac{c-1}{c}\right)M_{n-1}(x;\beta+1,c) -\end{equation} -or equivalently -\begin{equation} -\label{shift1MeixnerII} -\Delta M_n(x;\beta,c)=\frac{n}{\beta}\left(\frac{c-1}{c}\right)M_{n-1}(x;\beta+1,c). -\end{equation} - -\subsection*{Backward shift operator} -\begin{equation} -\label{shift2MeixnerI} -c(\beta+x-1)M_n(x;\beta,c)-xM_n(x-1;\beta,c)=c(\beta-1)M_{n+1}(x;\beta-1,c) -\end{equation} -or equivalently -\begin{equation} -\label{shift2MeixnerII} -\nabla\left[\frac{(\beta)_xc^x}{x!}M_n(x;\beta,c)\right]= -\frac{(\beta-1)_xc^x}{x!}M_{n+1}(x;\beta-1,c). -\end{equation} - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodMeixner} -\frac{(\beta)_xc^x}{x!}M_n(x;\beta,c)=\nabla^n\left[\frac{(\beta+n)_xc^x}{x!}\right]. -\end{equation} - -\subsection*{Generating functions} -\begin{equation} -\label{GenMeixner1} -\left(1-\frac{t}{c}\right)^x(1-t)^{-x-\beta}= -\sum_{n=0}^{\infty}\frac{(\beta)_n}{n!}M_n(x;\beta,c)t^n. -\end{equation} - -\begin{equation} -\label{GenMeixner2} -\e^t\,\hyp{1}{1}{-x}{\beta}{\left(\frac{1-c}{c}\right)t}= -\sum_{n=0}^{\infty}\frac{M_n(x;\beta,c)}{n!}t^n. -\end{equation} - -\begin{equation} -\label{GenMeixner3} -(1-t)^{-\gamma}\,\hyp{2}{1}{\gamma,-x}{\beta}{\frac{(1-c)t}{c(1-t)}} -=\sum_{n=0}^{\infty}\frac{(\gamma)_n}{n!}M_n(x;\beta,c)t^n, -\quad\textrm{$\gamma$ arbitrary}. -\end{equation} - -\subsection*{Limit relations} - -\subsubsection*{Hahn $\rightarrow$ Meixner} -If we take $\alpha=b-1$, $\beta=N(1-c)c^{-1}$ in the definition (\ref{DefHahn}) -of the Hahn polynomials and let $N\rightarrow\infty$ we find the Meixner polynomials: -$$\lim_{N\rightarrow\infty} -Q_n(x;b-1,N(1-c)c^{-1},N)=M_n(x;b,c).$$ - -\subsubsection*{Dual Hahn $\rightarrow$ Meixner} -To obtain the Meixner polynomials from the dual Hahn polynomials we have to take -$\gamma=\beta-1$ and $\delta=N(1-c)c^{-1}$ in the definition (\ref{DefDualHahn}) of -the dual Hahn polynomials and let $N\rightarrow\infty$: -$$\lim_{N\rightarrow\infty} -R_n(\lambda(x);\beta-1,N(1-c)c^{-1},N)=M_n(x;\beta,c).$$ - -\subsubsection*{Meixner $\rightarrow$ Laguerre} -The Laguerre polynomials given by (\ref{DefLaguerre}) are obtained from the Meixner polynomials -if we take $\beta=\alpha+1$ and $x\rightarrow (1-c)^{-1}x$ and let $c\rightarrow 1$: -\begin{equation} -\lim_{c\rightarrow 1} -M_n((1-c)^{-1}x;\alpha+1,c)=\frac{L_n^{(\alpha)}(x)}{L_n^{(\alpha)}(0)}. -\end{equation} - -\subsubsection*{Meixner $\rightarrow$ Charlier} -The Charlier polynomials given by (\ref{DefCharlier}) are obtained from the Meixner polynomials -if we take $c=(a+\beta)^{-1}a$ and let $\beta\rightarrow\infty$: -\begin{equation} -\lim_{\beta\rightarrow\infty} -M_n(x;\beta,(a+\beta)^{-1}a)=C_n(x;a). -\end{equation} - -\subsection*{Remarks} -The Meixner polynomials are related to the Jacobi polynomials given by (\ref{DefJacobi}) -in the following way: -$$\frac{(\beta)_n}{n!}M_n(x;\beta,c)=P_n^{(\beta-1,-n-\beta-x)}((2-c)c^{-1}).$$ - -\noindent -The Meixner polynomials are also related to the Krawtchouk polynomials given by -(\ref{DefKrawtchouk}) in the following way: -\subsection*{9.11 Krawtchouk} -\label{sec9.11} -% -\paragraph{Special values} -By (9.11.1) and the binomial formula: -\begin{equation} -K_n(0;p,N)=1,\qquad -K_n(N;p,N)=(1-p^{-1})^n. -\label{9} -\end{equation} -The self-duality (p.240, Remarks, first formula) -\begin{equation} -K_n(x;p,N)=K_x(n;p,N)\qquad (n,x\in \{0,1,\ldots,N\}) -\label{147} -\end{equation} -combined with \eqref{9} yields: -\begin{equation} -K_N(x;p,N)=(1-p^{-1})^x\qquad(x\in\{0,1,\ldots,N\}). -\label{148} -\end{equation} -% -\paragraph{Symmetry} -By the orthogonality relation (9.11.2): -\begin{equation} -\frac{K_n(N-x;p,N)}{K_n(N;p,N)}=K_n(x;1-p,N). -\label{10} -\end{equation} -By \eqref{10} and \eqref{147} we have also -\begin{equation} -\frac{K_{N-n}(x;p,N)}{K_N(x;p,N)}=K_n(x;1-p,N) -\qquad(n,x\in\{0,1,\ldots,N\}), -\label{149} -\end{equation} -and, by \eqref{149}, \eqref{10} and \eqref{9}, -\begin{equation} -K_{N-n}(N-x;p,N)=\left(\frac p{p-1}\right)^{n+x-N}K_n(x;p,N) -\qquad(n,x\in\{0,1,\ldots,N\}). -\label{150} -\end{equation} -A particular case of \eqref{10} is: -\begin{equation} -K_n(N-x;\thalf,N)=(-1)^n K_n(x;\thalf,N). -\label{11} -\end{equation} -Hence -\begin{equation} -K_{2m+1}(N;\thalf,2N)=0. -\label{12} -\end{equation} -From (9.11.11): -\begin{equation} -K_{2m}(N;\thalf,2N)=\frac{(\thalf)_m}{(-N+\thalf)_m}\,. -\label{13} -\end{equation} -% -\paragraph{Quadratic transformations} -\begin{align} -K_{2m}(x+N;\thalf,2N)&=\frac{(\thalf)_m}{(-N+\thalf)_m}\, -R_m(x^2;-\thalf,-\thalf,N), -\label{31}\\ -K_{2m+1}(x+N;\thalf,2N)&=-\,\frac{(\tfrac32)_m}{N\,(-N+\thalf)_m}\, -x\,R_m(x^2-1;\thalf,\thalf,N-1), -\label{33}\\ -K_{2m}(x+N+1;\thalf,2N+1)&=\frac{(\tfrac12)_m}{(-N-\thalf)_m}\, -R_m(x(x+1);-\thalf,\thalf,N), -\label{32}\\ -K_{2m+1}(x+N+1;\thalf,2N+1)&=\frac{(\tfrac32)_m}{(-N-\thalf)_{m+1}}\, -(x+\thalf)\,R_m(x(x+1);\thalf,-\thalf,N), -\label{34} -\end{align} -where $R_m$ is a dual Hahn polynomial (9.6.1). For the proofs use -(9.6.2), (9.11.2), (9.6.4) and (9.11.4). -% -\paragraph{Generating functions} -\begin{multline} -\sum_{x=0}^N\binom Nx K_m(x;p,N)K_n(x;q,N)z^x\\ -=\left(\frac{p-z+pz}p\right)^m -\left(\frac{q-z+qz}q\right)^n -(1+z)^{N-m-n} -K_m\left(n;-\,\frac{(p-z+pz)(q-z+qz)}z,N\right). -\label{107} -\end{multline} -This follows immediately from Rosengren \cite[(3.5)]{K8}, which goes back -to Meixner \cite{K9}. -$$K_n(x;p,N)=M_n(x;-N,(p-1)^{-1}p).$$ - -\subsection*{References} -\cite{Allaway76}, \cite{NAlSalam66}, \cite{AlSalam90}, \cite{AlSalamChihara76}, -\cite{AlSalamIsmail76}, \cite{Alvarez+}, \cite{AndrewsAskey85}, \cite{Area+II}, \cite{Askey75}, -\cite{Askey89I}, \cite{Askey2005}, \cite{AskeyGasper77}, \cite{AskeyIsmail76}, -\cite{AskeyWilson85}, \cite{AtakRahmanSuslov}, \cite{AtakSuslov88}, \cite{Bavinck98}, -\cite{BavinckHaeringen}, \cite{Campigotto+}, \cite{Chihara78}, \cite{Cooper+}, -\cite{Erdelyi+}, \cite{FoataLabelle}, \cite{Gabutti}, \cite{GabuttiMathis}, \cite{Gasper73I}, -\cite{Gasper74}, \cite{HoareRahman}, \cite{Ismail2005II}, \cite{IsmailLetVal88}, -\cite{IsmailLi}, \cite{IsmailMuldoon}, \cite{IsmailStanton97}, \cite{JinWong}, -\cite{Karlin58}, \cite{Koekoek2000}, \cite{Koorn88}, \cite{LabelleYehI}, \cite{LabelleYehII}, -\cite{Lesky89}, \cite{Lesky94I}, \cite{Lesky95II}, \cite{LewanowiczII}, \cite{Meixner}, -\cite{Nikiforov+}, \cite{NikiforovUvarov}, \cite{Rahman78I}, \cite{ValentAssche}, -\cite{Viennot}, \cite{Zarzo+}, \cite{Zeng90}. - - -\section{Krawtchouk}\index{Krawtchouk polynomials} - -\par\setcounter{equation}{0} - -\subsection*{Hypergeometric representation} -\begin{equation} -\label{DefKrawtchouk} -K_n(x;p,N)=\hyp{2}{1}{-n,-x}{-N}{\frac{1}{p}},\quad n=0,1,2,\ldots,N. -\end{equation} - -\subsection*{Orthogonality relation} -\begin{eqnarray} -\label{OrtKrawtchouk} -& &\sum_{x=0}^N\binom{N}{x}p^x(1-p)^{N-x} K_m(x;p,N)K_n(x;p,N)\nonumber\\ -& &{}=\frac{(-1)^nn!}{(-N)_n}\left(\frac{1-p}{p}\right)^n\,\delta_{mn},\quad0 < p < 1. -\end{eqnarray} - -\subsection*{Recurrence relation} -\begin{eqnarray} -\label{RecKrawtchouk} --xK_n(x;p,N)&=&p(N-n)K_{n+1}(x;p,N)\nonumber\\ -& &{}\mathindent{}-\left[p(N-n)+n(1-p)\right]K_n(x;p,N)\nonumber\\ -& &{}\mathindent\mathindent{}+n(1-p)K_{n-1}(x;p,N). -\end{eqnarray} - -\subsection*{Normalized recurrence relation} -\begin{eqnarray} -\label{NormRecKrawtchouk} -xp_n(x)&=&p_{n+1}(x)+\left[p(N-n)+n(1-p)\right]p_n(x)\nonumber\\ -& &{}\mathindent{}+np(1-p)(N+1-n)p_{n-1}(x), -\end{eqnarray} -where -$$K_n(x;p,N)=\frac{1}{(-N)_np^n}p_n(x).$$ - -\subsection*{Difference equation} -\begin{eqnarray} -\label{dvKrawtchouk} --ny(x)&=&p(N-x)y(x+1)\nonumber\\ -& &{}\mathindent{}-\left[p(N-x)+x(1-p)\right]y(x)+x(1-p)y(x-1), -\end{eqnarray} -where -$$y(x)=K_n(x;p,N).$$ - -\subsection*{Forward shift operator} -\begin{equation} -\label{shift1KrawtchoukI} -K_n(x+1;p,N)-K_n(x;p,N)=-\frac{n}{Np}K_{n-1}(x;p,N-1) -\end{equation} -or equivalently -\begin{equation} -\label{shift1KrawtchoukII} -\Delta K_n(x;p,N)=-\frac{n}{Np}K_{n-1}(x;p,N-1). -\end{equation} - -\subsection*{Backward shift operator} -\begin{eqnarray} -\label{shift2KrawtchoukI} -& &(N+1-x)K_n(x;p,N)-x\left(\frac{1-p}{p}\right)K_n(x-1;p,N)\nonumber\\ -& &{}=(N+1)K_{n+1}(x;p,N+1) -\end{eqnarray} -or equivalently -\begin{equation} -\label{shift2KrawtchoukII} -\nabla\left[\binom{N}{x}\left(\frac{p}{1-p}\right)^xK_n(x;p,N)\right]= -\binom{N+1}{x}\left(\frac{p}{1-p}\right)^xK_{n+1}(x;p,N+1). -\end{equation} - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodKrawtchouk} -\binom{N}{x}\left(\frac{p}{1-p}\right)^xK_n(x;p,N)= -\nabla^n\left[\binom{N-n}{x}\left(\frac{p}{1-p}\right)^x\right]. -\end{equation} - -\subsection*{Generating functions} -For $x=0,1,2,\ldots,N$ we have -\begin{equation} -\label{GenKrawtchouk1} -\left(1-\frac{(1-p)}{p}t\right)^x(1+t)^{N-x}= -\sum_{n=0}^N\binom{N}{n}K_n(x;p,N)t^n. -\end{equation} - -\begin{equation} -\label{GenKrawtchouk2} -\left[\expe^t\,\hyp{1}{1}{-x}{-N}{-\frac{t}{p}}\right]_N= -\sum_{n=0}^N\frac{K_n(x;p,N)}{n!}t^n. -\end{equation} - -\begin{eqnarray} -\label{GenKrawtchouk3} -& &\left[(1-t)^{-\gamma}\,\hyp{2}{1}{\gamma,-x}{-N}{\frac{t}{p(t-1)}}\right]_N\nonumber\\ -& &{}=\sum_{n=0}^N\frac{(\gamma)_n}{n!}K_n(x;p,N)t^n, -\quad\textrm{$\gamma$ arbitrary}. -\end{eqnarray} - -\subsection*{Limit relations} - -\subsubsection*{Hahn $\rightarrow$ Krawtchouk} -If we take $\alpha=pt$ and $\beta=(1-p)t$ in the definition (\ref{DefHahn}) of the Hahn -polynomials and let $t\rightarrow\infty$ we obtain the Krawtchouk polynomials: -$$\lim_{t\rightarrow\infty}Q_n(x;pt,(1-p)t,N)=K_n(x;p,N).$$ - -\subsubsection*{Dual Hahn $\rightarrow$ Krawtchouk} -The Krawtchouk polynomials follow from the dual Hahn polynomials given by -(\ref{DefDualHahn}) if we set $\gamma=pt$, $\delta=(1-p)t$ and let $t\rightarrow\infty$: -$$\lim_{t\rightarrow\infty}R_n(\lambda(x);pt,(1-p)t,N)=K_n(x;p,N).$$ - -\subsubsection*{Krawtchouk $\rightarrow$ Charlier} -The Charlier polynomials given by (\ref{DefCharlier}) can be found from the Krawtchouk -polynomials by taking $p=N^{-1}a$ and letting $N\rightarrow\infty$: -\begin{equation} -\lim_{N\rightarrow\infty}K_n(x;N^{-1}a,N)=C_n(x;a). -\end{equation} - -\subsubsection*{Krawtchouk $\rightarrow$ Hermite} -The Hermite polynomials given by (\ref{DefHermite}) follow from the Krawtchouk polynomials -by setting $x\rightarrow pN+x\sqrt{2p(1-p)N}$ and then letting $N\rightarrow\infty$: -\begin{equation} -\lim_{N\rightarrow\infty} -\sqrt{\binom{N}{n}}K_n(pN+x\sqrt{2p(1-p)N};p,N) -=\frac{\displaystyle (-1)^nH_n(x)}{\displaystyle\sqrt{2^nn!\left(\frac{p}{1-p}\right)^n}}. -\end{equation} - -\subsection*{Remarks} -The Krawtchouk polynomials are self-dual, which means that -$$K_n(x;p,N)=K_x(n;p,N),\quad n,x\in\{0,1,2,\ldots,N\}.$$ -By using this relation we easily obtain the so-called dual orthogonality -relation from the orthogonality relation (\ref{OrtKrawtchouk}): -$$\sum_{n=0}^N\binom{N}{n}p^n(1-p)^{N-n} K_n(x;p,N)K_n(y;p,N)= -\frac{\displaystyle\left(\frac{1-p}{p}\right)^x}{\dbinom{N}{x}}\delta_{xy},$$ -where $0 < p < 1$ and $x,y\in\{0,1,2,\ldots,N\}$. - -\noindent -The Krawtchouk polynomials are related to the Meixner polynomials given by (\ref{DefMeixner}) -in the following way: -\subsection*{9.12 Laguerre} -\label{sec9.12} -\paragraph{Notation} -Here the Laguerre polynomial is denoted by $L_n^\al$ instead of -$L_n^{(\al)}$. -% -\paragraph{Hypergeometric representation} -\begin{align} -L_n^\al(x)&= -\frac{(\al+1)_n}{n!}\,\hyp11{-n}{\al+1}x -\label{182}\\ -&=\frac{(-x)^n}{n!} \hyp20{-n,-n-\al}-{-\,\frac1x} -\label{183}\\ -&=\frac{(-x)^n}{n!}\,C_n(n+\al;x), -\label{184} -\end{align} -where $C_n$ in \eqref{184} is a -\hyperref[sec9.14]{Charlier polynomial}. -Formula \eqref{182} is (9.12.1). Then \eqref{183} follows by reversal -of summation. Finally \eqref{184} follows by \eqref{183} and \eqref{179}. -It is also the remark on top of p.244 in \mycite{KLS}, and it is essentially -\myciteKLS{416}{(2.7.10)}. -% -\paragraph{Uniqueness of orthogonality measure} -The coefficient of $p_{n-1}(x)$ in (9.12.4) behaves as $O(n^2)$ as $n\to\iy$. -Hence \eqref{93} holds, by which the orthogonality measure is unique. -% -\paragraph{Special value} -\begin{equation} -L_n^{\al}(0)=\frac{(\al+1)_n}{n!}\,. -\label{53} -\end{equation} -Use (9.12.1) or see \mycite{DLMF}{18.6.1)}. -% -\paragraph{Quadratic transformations} -\begin{align} -H_{2n}(x)&=(-1)^n\,2^{2n}\,n!\,L_n^{-1/2}(x^2), -\label{54}\\ -H_{2n+1}(x)&=(-1)^n\,2^{2n+1}\,n!\,x\,L_n^{1/2}(x^2). -\label{55} -\end{align} -See p.244, Remarks, last two formulas. -Or see \mycite{DLMF}{(18.7.19), (18.7.20)}. -% -\paragraph{Fourier transform} -\begin{equation} -\frac1{\Ga(\al+1)}\,\int_0^\iy \frac{L_n^\al(y)}{L_n^\al(0)}\, -e^{-y}\,y^\al\,e^{ixy}\,dy= -i^n\,\frac{y^n}{(iy+1)^{n+\al+1}}\,, -\label{14} -\end{equation} -see \mycite{DLMF}{(18.17.34)}. -% -\paragraph{Differentiation formulas} -Each differentiation formula is given in two equivalent forms. -\begin{equation} -\frac d{dx}\left(x^\al L_n^\al(x)\right)= -(n+\al)\,x^{\al-1} L_n^{\al-1}(x),\qquad -\left(x\frac d{dx}+\al\right)L_n^\al(x)= -(n+\al)\,L_n^{\al-1}(x). -\label{76} -\end{equation} -% -\begin{equation} -\frac d{dx}\left(e^{-x} L_n^\al(x)\right)= --e^{-x} L_n^{\al+1}(x),\qquad -\left(\frac d{dx}-1\right)L_n^\al(x)= --L_n^{\al+1}(x). -\label{77} -\end{equation} -% -Formulas \eqref{76} and \eqref{77} follow from -\mycite{DLMF}{(13.3.18), (13.3.20)} -together with (9.12.1). -% -\paragraph{Generalized Hermite polynomials} -See \myciteKLS{146}{p.156}, \cite[Section 1.5.1]{K26}. -These are defined by -\begin{equation} -H_{2m}^\mu(x):=\const L_m^{\mu-\half}(x^2),\qquad -H_{2m+1}^\mu(x):=\const x\,L_m^{\mu+\half}(x^2). -\label{78} -\end{equation} -Then for $\mu>-\thalf$ we have orthogonality relation -\begin{equation} -\int_{-\iy}^{\iy} H_m^\mu(x)\,H_n^\mu(x)\,|x|^{2\mu}e^{-x^2}\,dx -=0\qquad(m\ne n). -\label{79} -\end{equation} -Let the Dunkl operator $T_\mu$ be defined by \eqref{72}. -If we choose the constants in \eqref{78} as -\begin{equation} -H_{2m}^\mu(x)=\frac{(-1)^m(2m)!}{(\mu+\thalf)_m}\,L_m^{\mu-\half}(x^2),\qquad -H_{2m+1}^\mu(x)=\frac{(-1)^m(2m+1)!}{(\mu+\thalf)_{m+1}}\, - x\,L_m^{\mu+\half}(x^2) - \label{80} -\end{equation} -then (see \cite[(1.6)]{K5}) -\begin{equation} -T_\mu H_n^\mu=2n\,H_{n-1}^\mu. -\label{81} -\end{equation} -Formula \eqref{81} with \eqref{80} substituted gives rise to two -differentiation formulas involving Laguerre polynomials which are equivalent to -(9.12.6) and \eqref{76}. - -Composition of \eqref{81} with itself gives -\[ -T_\mu^2 H_n^\mu=4n(n-1)\,H_{n-2}^\mu, -\] -which is equivalent to the composition of (9.12.6) and \eqref{76}: -\begin{equation} -\left(\frac{d^2}{dx^2}+\frac{2\al+1}x\,\frac d{dx}\right)L_n^\al(x^2) -=-4(n+\al)\,L_{n-1}^\al(x^2). -\label{82} -\end{equation} -$$K_n(x;p,N)=M_n(x;-N,(p-1)^{-1}p).$$ - -\subsection*{References} -\cite{AlSalam90}, \cite{AndrewsAskey85}, \cite{Area+II}, \cite{Askey75}, \cite{Askey89I}, -\cite{AskeyGasper77}, \cite{AskeyWilson85}, \cite{AtakRahmanSuslov}, \cite{Campigotto+}, -\cite{LChiharaStanton}, \cite{Chihara78}, \cite{Dette95}, \cite{Dominici}, \cite{Dunkl76}, -\cite{Dunkl84}, \cite{DunklRamirez}, \cite{Erdelyi+}, \cite{FeinsilverSchott}, -\cite{Gasper73I}, \cite{Gasper74}, \cite{HoareRahman}, \cite{Ismail2005II}, \cite{Karlin58}, -\cite{Koorn82}, \cite{Koorn88}, \cite{LabelleYehI}, \cite{LabelleYehII}, \cite{Lesky62}, -\cite{Lesky89}, \cite{Lesky94I}, \cite{Lesky95II}, \cite{LewanowiczII}, \cite{Nikiforov+}, -\cite{NikiforovUvarov}, \cite{Qiu}, \cite{Rahman78I}, \cite{Rahman79}, \cite{Stanton84}, -\cite{Stanton90}, \cite{Szego75}, \cite{Zarzo+}, \cite{Zeng90}. - - -\section{Laguerre}\index{Laguerre polynomials} - -\par\setcounter{equation}{0} - -\subsection*{Hypergeometric representation} -\begin{equation} -\label{DefLaguerre} -L_n^{(\alpha)}(x)=\frac{(\alpha+1)_n}{n!}\,\hyp{1}{1}{-n}{\alpha+1}{x}. -\end{equation} - -\subsection*{Orthogonality relation} -\begin{equation} -\label{OrtLaguerre} -\int_0^{\infty}\expe^{-x}x^{\alpha}L_m^{(\alpha)}(x)L_n^{(\alpha)}(x)\,dx= -\frac{\Gamma(n+\alpha+1)}{n!}\,\delta_{mn},\quad\alpha > -1. -\end{equation} - -\subsection*{Recurrence relation} -\begin{equation} -\label{RecLaguerre} -(n+1)L_{n+1}^{(\alpha)}(x)-(2n+\alpha+1-x)L_n^{(\alpha)}(x)+(n+\alpha)L_{n-1}^{(\alpha)}(x)=0. -\end{equation} - -\subsection*{Normalized recurrence relation} -\begin{equation} -\label{NormRecLaguerre} -xp_n(x)=p_{n+1}(x)+(2n+\alpha+1)p_n(x)+n(n+\alpha)p_{n-1}(x), -\end{equation} -where -$$L_n^{(\alpha)}(x)=\frac{(-1)^n}{n!}p_n(x).$$ - -\subsection*{Differential equation} -\begin{equation} -\label{dvLaguerre} -xy''(x)+(\alpha+1-x)y'(x)+ny(x)=0,\quad y(x)=L_n^{(\alpha)}(x). -\end{equation} - -\newpage - -\subsection*{Forward shift operator} -\begin{equation} -\label{shift1Laguerre} -\frac{d}{dx}L_n^{(\alpha)}(x)=-L_{n-1}^{(\alpha+1)}(x). -\end{equation} - -\subsection*{Backward shift operator} -\begin{equation} -\label{shift2LaguerreI} -x\frac{d}{dx}L_n^{(\alpha)}(x)+(\alpha-x)L_n^{(\alpha)}(x)=(n+1)L_{n+1}^{(\alpha-1)}(x) -\end{equation} -or equivalently -\begin{equation} -\label{shift2LaguerreII} -\frac{d}{dx}\left[\expe^{-x}x^{\alpha}L_n^{(\alpha)}(x)\right]=(n+1)\expe^{-x}x^{\alpha-1}L^{(\alpha-1)}_{n+1}(x). -\end{equation} - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodLaguerre} -\e^{-x}x^{\alpha}L_n^{(\alpha)}(x)=\frac{1}{n!}\left(\frac{d}{dx}\right)^n\left[\expe^{-x}x^{n+\alpha}\right]. -\end{equation} - -\subsection*{Generating functions} -\begin{equation} -\label{GenLaguerre1} -(1-t)^{-\alpha-1}\exp\left(\frac{xt}{t-1}\right)= -\sum_{n=0}^{\infty}L_n^{(\alpha)}(x)t^n. -\end{equation} - -\begin{equation} -\label{GenLaguerre2} -\e^t\,\hyp{0}{1}{-}{\alpha+1}{-xt} -=\sum_{n=0}^{\infty}\frac{L_n^{(\alpha)}(x)}{(\alpha+1)_n}t^n. -\end{equation} - -\begin{equation} -\label{GenLaguerre3} -(1-t)^{-\gamma}\,\hyp{1}{1}{\gamma}{\alpha+1}{\frac{xt}{t-1}} -=\sum_{n=0}^{\infty}\frac{(\gamma)_n}{(\alpha+1)_n}L_n^{(\alpha)}(x)t^n, -\quad\textrm{$\gamma$ arbitrary}. -\end{equation} - -\subsection*{Limit relations} - -\subsubsection*{Meixner-Pollaczek $\rightarrow$ Laguerre} -The Laguerre polynomials can be obtained from the Meixner-Pollaczek polynomials given by -(\ref{DefMP}) by the substitution $\lambda=\frac{1}{2}(\alpha+1)$, -$x\rightarrow -\frac{1}{2}\phi^{-1}x$ and the limit $\phi\rightarrow 0$: -$$\lim_{\phi\rightarrow 0} -P_n^{(\frac{1}{2}\alpha+\frac{1}{2})}(-\textstyle\frac{1}{2}\phi^{-1}x;\phi)=L_n^{(\alpha)}(x).$$ - -\subsubsection*{Jacobi $\rightarrow$ Laguerre} -The Laguerre polynomials are obtained from the Jacobi polynomials given by (\ref{DefJacobi}) -if we set $x\rightarrow 1-2\beta^{-1}x$ and then take the limit $\beta\rightarrow\infty$: -$$\lim_{\beta\rightarrow\infty} -P_n^{(\alpha,\beta)}(1-2\beta^{-1}x)=L_n^{(\alpha)}(x).$$ - -\subsubsection*{Meixner $\rightarrow$ Laguerre} -If we take $\beta=\alpha+1$ and $x\rightarrow (1-c)^{-1}x$ in the definition -(\ref{DefMeixner}) of the Meixner polynomials and let $c\rightarrow 1$ we obtain -the Laguerre polynomials: -$$\lim_{c\rightarrow 1} -M_n((1-c)^{-1}x;\alpha+1,c)=\frac{L_n^{(\alpha)}(x)}{L_n^{(\alpha)}(0)}.$$ - -\subsubsection*{Laguerre $\rightarrow$ Hermite} -The Hermite polynomials given by (\ref{DefHermite}) can be obtained from the Laguerre -polynomials by taking the limit $\alpha\rightarrow\infty$ in the following way: -\begin{equation} -\lim_{\alpha\rightarrow\infty} -\left(\frac{2}{\alpha}\right)^{\frac{1}{2}n} -L_n^{(\alpha)}((2\alpha)^{\frac{1}{2}}x+\alpha)=\frac{(-1)^n}{n!}H_n(x). -\end{equation} - -\subsection*{Remarks} -The definition (\ref{DefLaguerre}) of the Laguerre polynomials can also be -written as: -$$L_n^{(\alpha)}(x)=\frac{1}{n!}\sum_{k=0}^n\frac{(-n)_k}{k!}(\alpha+k+1)_{n-k}x^k.$$ -In this way the Laguerre polynomials can also be seen as polynomials in the parameter $\alpha$. -Therefore they can be defined for all $\alpha$. - -\noindent -The Laguerre polynomials are related to the Bessel polynomials given by (\ref{DefBessel}) -in the following way: -$$L_n^{(\alpha)}(x)=\frac{(-x)^n}{n!}y_n(2x^{-1};-2n-\alpha-1).$$ - -\noindent -The Laguerre polynomials are related to the Charlier polynomials given by (\ref{DefCharlier}) -in the following way: -$$\frac{(-a)^n}{n!}C_n(x;a)=L_n^{(x-n)}(a).$$ - -\noindent -The Laguerre polynomials and the Hermite polynomials given by (\ref{DefHermite}) are also -connected by the following quadratic transformations: -$$H_{2n}(x)=(-1)^nn!\,2^{2n}L_n^{(-\frac{1}{2})}(x^2)$$ -and -$$H_{2n+1}(x)=(-1)^nn!\,2^{2n+1}xL_n^{(\frac{1}{2})}(x^2).$$ - -\noindent -In combinatorics the Laguerre polynomials with $\alpha=0$ are often called Rook -\subsection*{9.14 Charlier} -\label{sec9.14} -% -\paragraph{Hypergeometric representation} -\begin{align} -C_n(x;a)&=\hyp20{-n,-x}-{-\,\frac1a} -\label{179}\\ -&=\frac{(-x)_n}{a^n} \hyp11{-n}{x-n+1}a -\label{180}\\ -&=\frac{n!}{(-a)^n}\,L_n^{x-n}(a), -\label{181} -\end{align} -where $L_n^\al(x)$ is a -\hyperref[sec9.12]{Laguerre polynomial}. -Formula \eqref{179} is (9.14.1). Then \eqref{180} follows by reversal -of the summation. Finally \eqref{181} follows by \eqref{180} and -(9.12.1). It is also the Remark on p.249 of \mycite{KLS}, and it -was earlier given in \myciteKLS{416}{(2.7.10)}. -% -\paragraph{Uniqueness of orthogonality measure} -The coefficient of $p_{n-1}(x)$ in (9.14.4) behaves as $O(n)$ as $n\to\iy$. -Hence \eqref{93} holds, by which the orthogonality measure is unique. -polynomials. - -\subsection*{References} -\cite{Abdul}, \cite{Abram}, \cite{Ahmed+82}, \cite{Allaway76}, \cite{Allaway91}, -\cite{NAlSalam66}, \cite{AlSalam64}, \cite{AlSalam90}, \cite{AlSalamChihara72}, -\cite{AlSalamChihara76}, \cite{AndrewsAskey85}, \cite{AndrewsAskeyRoy}, \cite{Askey68}, -\cite{Askey75}, \cite{Askey89I}, \cite{Askey2005}, \cite{AskeyGasper76}, \cite{AskeyGasper77}, -\cite{AskeyIsmail76}, \cite{AskeyIsmailKoorn}, \cite{AskeyWilson85}, \cite{Barrett}, -\cite{Bavinck96}, \cite{Brafman51}, \cite{Brafman57II}, \cite{Brenke}, \cite{Brown}, -\cite{BustozSavage79}, \cite{BustozSavage80}, \cite{Carlitz60}, \cite{Carlitz61I}, -\cite{Carlitz61II}, \cite{Carlitz62}, \cite{Carlitz68}, \cite{ChenSrivastava}, -\cite{ChenIsmail91}, \cite{Chihara68II}, \cite{Chihara78}, \cite{Cohen}, \cite{Cooper+}, -\cite{Dattoli2001}, \cite{DetteStudden92}, \cite{DetteStudden95}, \cite{Dimitrov2003}, -\cite{Doha2003II}, \cite{ElbertLaforgia87I}, \cite{Erdelyi}, \cite{Erdelyi+}, \cite{Exton98}, -\cite{Faldey}, \cite{Gasper73II}, \cite{Gasper77}, \cite{Gatteschi2002}, \cite{Gawronski87}, -\cite{Gawronski93}, \cite{Gillis}, \cite{GillisIsmailOffer}, \cite{Godoy+}, \cite{Grad}, -\cite{Hajmirzaahmad95}, \cite{Ismail74}, \cite{Ismail77}, \cite{Ismail2005II}, -\cite{IsmailLetVal88}, \cite{IsmailLi}, \cite{IsmailMassonRahman}, \cite{IsmailMuldoon}, -\cite{IsmailStanton97}, \cite{IsmailTamhankar}, \cite{Jackson}, \cite{Karlin58}, -\cite{Kochneff95}, \cite{Kochneff97II}, \cite{Koekoek2000}, \cite{Koorn77I}, \cite{Koorn78}, -\cite{Koorn85}, \cite{Koorn88}, \cite{Krasikov2003}, \cite{Krasikov2005}, -\cite{KuijlaarsMcLaughlin}, \cite{KwonLittle}, \cite{LabelleYehI}, \cite{LabelleYehII}, -\cite{LabelleYehIII}, \cite{Lee97I}, \cite{Lee97II}, \cite{Lee97III}, \cite{Lesky95II}, -\cite{Lesky96}, \cite{LewanowiczI}, \cite{LopezTemme2004}, \cite{Mathai}, \cite{Meixner}, -\cite{Nikiforov+}, \cite{NikiforovUvarov}, \cite{Olver}, \cite{PerezPinar}, \cite{Pittaluga}, -\cite{Rainville}, \cite{SainteViennot}, \cite{Srivastava66}, \cite{Srivastava69I}, -\cite{Srivastava69II}, \cite{Srivastava70}, \cite{Srivastava71}, \cite{Szego75}, \cite{Temme}, -\cite{Trickovic}, \cite{Trivedi}, \cite{Viennot}. - - -\section{Bessel}\index{Bessel polynomials} - -\par\setcounter{equation}{0} - -\subsection*{Hypergeometric representation} -\begin{eqnarray} -\label{DefBessel} -y_n(x;a)&=&\hyp{2}{0}{-n,n+a+1}{-}{-\frac{x}{2}}\\ -&=&(n+a+1)_n\left(\frac{x}{2}\right)^n\,\hyp{1}{1}{-n}{-2n-a}{\frac{2}{x}}, -\quad n=0,1,2,\ldots,N.\nonumber -\end{eqnarray} - -\subsection*{Orthogonality relation} -\begin{eqnarray} -\label{OrtBessel} -& &\int_0^{\infty}x^a\e^{-\frac{2}{x}}y_m(x;a)y_n(x;a)\,dx\nonumber\\ -& &=-\frac{2^{a+1}}{2n+a+1}\Gamma(-n-a)n!\,\delta_{mn},\quad a<-2N-1. -\end{eqnarray} - -\subsection*{Recurrence relation} -\begin{eqnarray} -\label{RecBessel} -& &2(n+a+1)(2n+a)y_{n+1}(x;a)\nonumber\\ -& &{}=(2n+a+1)\left[2a+(2n+a)(2n+a+2)x\right]y_n(x;a)\nonumber\\ -& &{}\mathindent{}+2n(2n+a+2)y_{n-1}(x;a). -\end{eqnarray} - -\subsection*{Normalized recurrence relation} -\begin{eqnarray} -\label{NormRecBessel} -xp_n(x)&=&p_{n+1}(x)-\frac{2a}{(2n+a)(2n+a+2)}p_n(x)\nonumber\\ -& &{}\mathindent{}-\frac{4n(n+a)}{(2n+a-1)(2n+a)^2(2n+a+1)}p_{n-1}(x), -\end{eqnarray} -where -$$y_n(x;a)=\frac{(n+a+1)_n}{2^n}p_n(x).$$ - -\subsection*{Differential equation} -\begin{equation} -\label{dvBessel} -x^2y''(x)+\left[(a+2)x+2\right]y'(x)-n(n+a+1)y(x)=0,\quad y(x)=y_n(x;a). -\end{equation} - -\subsection*{Forward shift operator} -\begin{equation} -\label{shift1Bessel} -\frac{d}{dx}y_n(x;a)=\frac{n(n+a+1)}{2}y_{n-1}(x;a+2). -\end{equation} - -\subsection*{Backward shift operator} -\begin{equation} -\label{shift2BesselI} -x^2\frac{d}{dx}y_n(x;a)+(ax+2)y_n(x;a)=2y_{n+1}(x;a-2) -\end{equation} -or equivalently -\begin{equation} -\label{shift2BesselII} -\frac{d}{dx}\left[x^a\expe^{-\frac{2}{x}}y_n(x;a)\right]=2x^{a-2}\expe^{-\frac{2}{x}}y_{n+1}(x;a-2). -\end{equation} - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodBessel} -y_n(x;a)=2^{-n}x^{-a}\expe^{\frac{2}{x}}D^n\left(x^{2n+a}\expe^{-\frac{2}{x}}\right). -\end{equation} - -\subsection*{Generating function} -\begin{equation} -\label{GenBessel} -\left(1-2xt\right)^{-\frac{1}{2}}\left(\frac{2}{1+\sqrt{1-2xt}}\right)^a -\exp\left(\frac{2t}{1+\sqrt{1-2xt}}\right)=\sum_{n=0}^{\infty}y_n(x;a)\frac{t^n}{n!}. -\end{equation} - -\subsection*{Limit relation} - -\subsubsection*{Jacobi $\rightarrow$ Bessel} -If we take $\beta=a-\alpha$ in the definition (\ref{DefJacobi}) of the Jacobi polynomials -and let $\alpha\rightarrow -\infty$ we find the Bessel polynomials: -$$\lim_{\alpha\rightarrow -\infty} -\frac{P_n^{(\alpha,a-\alpha)}(1+\alpha x)}{P_n^{(\alpha,a-\alpha)}(1)}=y_n(x;a).$$ - -\subsection*{Remarks} -The following notations are also used for the Bessel polynomials: -$$y_n(x;a,b)=y_n(2b^{-1}x;a)\quad\textrm{and}\quad\theta_n(x;a,b)=x^ny_n(x^{-1};a,b).$$ -However, the Bessel polynomials essentially depend on only one parameter. - -\noindent -The Bessel polynomials are related to the Laguerre polynomials given by (\ref{DefLaguerre}) -in the following way: -$$L_n^{(\alpha)}(x)=\frac{(-x)^n}{n!}y_n(2x^{-1};-2n-\alpha-1).$$ - -\noindent -The special case $a=-2N-2$ of the Bessel polynomials can be obtained from the pseudo Jacobi -polynomials by setting $x\rightarrow\nu x$ in the definition (\ref{DefPseudoJacobi}) of the -pseudo Jacobi polynomials and taking the limit $\nu\rightarrow\infty$ in the following way: -$$\lim\limits_{\nu\rightarrow\infty}\frac{P_n(\nu x;\nu,N)}{\nu^n} -\subsection*{9.15 Hermite} -\label{sec9.15} -% -\paragraph{Uniqueness of orthogonality measure} -The coefficient of $p_{n-1}(x)$ in (9.15.4) behaves as $O(n)$ as $n\to\iy$. -Hence \eqref{93} holds, by which the orthogonality measure is unique. -% -\paragraph{Fourier transforms} -\begin{equation} -\frac1{\sqrt{2\pi}}\,\int_{-\iy}^\iy H_n(y)\,e^{-\half y^2}\,e^{ixy}\,dy= -i^n\,H_n(x)\,e^{-\half x^2}, -\label{15} -\end{equation} -see \mycite{AAR}{(6.1.15) and Exercise 6.11}. -\begin{equation} -\frac1{\sqrt\pi}\,\int_{-\iy}^\iy H_n(y)\,e^{-y^2}\,e^{ixy}\,dy= -i^n\,x^n\,e^{-\frac14 x^2}, -\label{16} -\end{equation} -see \mycite{DLMF}{(18.17.35)}. -\begin{equation} -\frac{i^n}{2\sqrt\pi}\,\int_{-\iy}^\iy y^n\,e^{-\frac14 y^2}\,e^{-ixy}\,dy= -H_n(x)\,e^{-x^2}, -\label{17} -\end{equation} -see \mycite{AAR}{(6.1.4)}. -=\frac{2^n}{(n-2N-1)_n}y_n(x;-2N-2).$$ - -\subsection*{References} -\cite{Andrade}, \cite{BergVignat}, \cite{Carlitz57I}, \cite{Dattoli2003}, -\cite{DohaAhmed2004}, \cite{DohaAhmed2006}, \cite{Grosswald}, \cite{Ismail2005II}, -\cite{KrallFrink}, \cite{Lesky98}, \cite{NikiforovUvarov}. - - -\section{Charlier}\index{Charlier polynomials} - -\par\setcounter{equation}{0} - -\subsection*{Hypergeometric representation} -\begin{equation} -\label{DefCharlier} -C_n(x;a)=\hyp{2}{0}{-n,-x}{-}{-\frac{1}{a}}. -\end{equation} - -\subsection*{Orthogonality relation} -\begin{equation} -\label{OrtCharlier} -\sum_{x=0}^{\infty}\frac{a^x}{x!}C_m(x;a)C_n(x;a)= -a^{-n}\expe^an!\,\delta_{mn},\quad a > 0. -\end{equation} - -\subsection*{Recurrence relation} -\begin{equation} -\label{RecCharlier} --xC_n(x;a)=aC_{n+1}(x;a)-(n+a)C_n(x;a)+nC_{n-1}(x;a). -\end{equation} - -\subsection*{Normalized recurrence relation} -\begin{equation} -\label{NormRecCharlier} -xp_n(x)=p_{n+1}(x)+(n+a)p_n(x)+nap_{n-1}(x), -\end{equation} -where -$$C_n(x;a)=\left(-\frac{1}{a}\right)^np_n(x).$$ - -\subsection*{Difference equation} -\begin{equation} -\label{dvCharlier} --ny(x)=ay(x+1)-(x+a)y(x)+xy(x-1),\quad y(x)=C_n(x;a). -\end{equation} - -\subsection*{Forward shift operator} -\begin{equation} -\label{shift1CharlierI} -C_n(x+1;a)-C_n(x;a)=-\frac{n}{a}C_{n-1}(x;a) -\end{equation} -or equivalently -\begin{equation} -\label{shift1CharlierII} -\Delta C_n(x;a)=-\frac{n}{a}C_{n-1}(x;a). -\end{equation} - -\subsection*{Backward shift operator} -\begin{equation} -\label{shift2CharlierI} -C_n(x;a)-\frac{x}{a}C_n(x-1;a)=C_{n+1}(x;a) -\end{equation} -or equivalently -\begin{equation} -\label{shift2CharlierII} -\nabla\left[\frac{a^x}{x!}C_n(x;a)\right]=\frac{a^x}{x!}C_{n+1}(x;a). -\end{equation} - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodCharlier} -\frac{a^x}{x!}C_n(x;a)=\nabla^n\left[\frac{a^x}{x!}\right]. -\end{equation} - -\subsection*{Generating function} -\begin{equation} -\label{GenCharlier} -\e^t\left(1-\frac{t}{a}\right)^x=\sum_{n=0}^{\infty}\frac{C_n(x;a)}{n!}t^n. -\end{equation} - -\subsection*{Limit relations} - -\subsubsection*{Meixner $\rightarrow$ Charlier} -If we take $c=(a+\beta)^{-1}a$ in the definition (\ref{DefMeixner}) of the Meixner polynomials -and let $\beta\rightarrow\infty$ we find the Charlier polynomials: -$$\lim_{\beta\rightarrow\infty}M_n(x;\beta,(a+\beta)^{-1}a)=C_n(x;a).$$ - -\subsubsection*{Krawtchouk $\rightarrow$ Charlier} -The Charlier polynomials can be found from the Krawtchouk polynomials given by -(\ref{DefKrawtchouk}) by taking $p=N^{-1}a$ and letting $N\rightarrow\infty$: -$$\lim_{N\rightarrow\infty}K_n(x;N^{-1}a,N)=C_n(x;a).$$ - -\subsubsection*{Charlier $\rightarrow$ Hermite} -The Hermite polynomials given by (\ref{DefHermite}) are obtained from the Charlier polynomials -if we set $x\rightarrow (2a)^{1/2}x+a$ and let $a\rightarrow\infty$. In fact we have -\begin{equation} -\lim_{a\rightarrow\infty} -(2a)^{\frac{1}{2}n}C_n((2a)^{\frac{1}{2}}x+a;a)=(-1)^nH_n(x). -\end{equation} - -\subsection*{Remark} -The Charlier polynomials are related to the Laguerre polynomials given by (\ref{DefLaguerre}) -in the following way: -\subsection*{14.1 Askey-Wilson} -\label{sec14.1} -% -\paragraph{Symmetry} -The Askey-Wilson polynomials $p_n(x;a,b,c,d\,|\,q)$ are \,|\,symmetric -in $a,b,c,d$. -\sLP -This follows from the orthogonality relation (14.1.2) -together with the value of its coefficient of $x^n$ given in (14.1.5b). -Alternatively, combine (14.1.1) with \mycite{GR}{(III.15)}.\\ -As a consequence, it is sufficient to give generating function (14.1.13). Then the generating -functions (14.1.14), (14.1.15) will follow by symmetry in the parameters. -% -\paragraph{Basic hypergeometric representation} -In addition to (14.1.1) we have (in notation \eqref{111}): -\begin{multline} -p_n(\cos\tha;a,b,c,d\,|\, q) -=\frac{(ae^{-i\tha},be^{-i\tha},ce^{-i\tha},de^{-i\tha};q)_n} -{(e^{-2i\tha};q)_n}\,e^{in\tha}\\ -\times {}_8W_7\big(q^{-n}e^{2i\tha};ae^{i\tha},be^{i\tha}, -ce^{i\tha},de^{i\tha},q^{-n};q,q^{2-n}/(abcd)\big). -\label{113} -\end{multline} -This follows from (14.1.1) by combining (III.15) and (III.19) in -\mycite{GR}. -It is also given in \myciteKLS{513}{(4.2)}, but be aware for some slight errors. -The symmetry in $a,b,c,d$ is evident from \eqref{113}. -% -\paragraph{Special value} -\begin{equation} -p_n\big(\thalf(a+a^{-1});a,b,c,d\,|\, q\big)=a^{-n}\,(ab,ac,ad;q)_n\,, -\label{40} -\end{equation} -and similarly for arguments $\thalf(b+b^{-1})$, $\thalf(c+c^{-1})$ and -$\thalf(d+d^{-1})$ by symmetry of $p_n$ in $a,b,c,d$. -% -\paragraph{Trivial symmetry} -\begin{equation} -p_n(-x;a,b,c,d\,|\, q)=(-1)^n p_n(x;-a,-b,-c,-d\,|\, q). -\label{41} -\end{equation} -Both \eqref{40} and \eqref{41} are obtained from (14.1.1). -% -\paragraph{Re: (14.1.5)} -Let -\begin{equation} -p_n(x):=\frac{p_n(x;a,b,c,d\,|\, q)}{2^n(abcdq^{n-1};q)_n}=x^n+\wt k_n x^{n-1} -+\cdots\;. -\label{18} -\end{equation} -Then -\begin{equation} -\wt k_n=-\frac{(1-q^n)(a+b+c+d-(abc+abd+acd+bcd)q^{n-1})} -{2(1-q)(1-abcdq^{2n-2})}\,. -\label{19} -\end{equation} -This follows because $\tilde k_n-\tilde k_{n+1}$ equals the coefficient -$\thalf\bigl(a+a^{-1}-(A_n+C_n)\bigr)$ of $p_n(x)$ in (14.1.5). -% -\paragraph{Generating functions} -Rahman \myciteKLS{449}{(4.1), (4.9)} gives: -\begin{align} -&\sum_{n=0}^\iy \frac{(abcdq^{-1};q)_n a^n}{(ab,ac,ad,q;q)_n}\,t^n\, -p_n(\cos\tha;a,b,c,d\,|\,q) -\nonumber\\ -&=\frac{(abcdtq^{-1};q)_\iy}{(t;q)_\iy}\, -\qhyp65{(abcdq^{-1})^\half,-(abcdq^{-1})^\half,(abcd)^\half, --(abcd)^\half,a e^{i\tha},a e^{-i\tha}} -{ab,ac,ad,abcdtq^{-1},qt^{-1}}{q,q} -\nonumber\\ -&+\frac{(abcdq^{-1},abt,act,adt,ae^{i\tha},ae^{-i\tha};q)_\iy} -{(ab,ac,ad,t^{-1},ate^{i\tha},ate^{-i\tha};q)_\iy} -\nonumber\\ -&\times\qhyp65{t(abcdq^{-1})^\half,-t(abcdq^{-1})^\half,t(abcd)^\half, --t(abcd)^\half,at e^{i\tha},at e^{-i\tha}} -{abt,act,adt,abcdt^2q^{-1},qt}{q,q}\quad(|t|<1). -\label{185} -\end{align} -In the limit \eqref{109} the first term on the \RHS\ of \eqref{185} -tends to the \LHS\ of (9.1.15), while the second term tends formally -to 0. The special case $ad=bc$ of \eqref{185} was earlier given in -\myciteKLS{236}{(4.1), (4.6)}. -% -\paragraph{Limit relations}\quad\sLP -{\bf Askey-Wilson $\longrightarrow$ Wilson}\\ -Instead of (14.1.21) we can keep a polynomial of degree $n$ while the limit is approached: -\begin{equation} -\lim_{q\to1}\frac{p_n(1-\thalf x(1-q)^2;q^a,q^b,q^c,q^d\,|\, q)}{(1-q)^{3n}} -=W_n(x;a,b,c,d). -\label{109} -\end{equation} -For the proof first derive the corresponding limit for the monic polynomials by comparing -(14.1.5) with (9.4.4). -\bLP -{\bf Askey-Wilson $\longrightarrow$ Continuous Hahn}\\ -Instead of (14.4.15) we can keep a polynomial of degree $n$ while the limit is approached: -\begin{multline} -\lim_{q\uparrow1} -\frac{p_n\big(\cos\phi-x(1-q)\sin\phi;q^a e^{i\phi},q^b e^{i\phi},q^{\overline a} e^{-i\phi}, -q^{\overline b} e^{-i\phi}\,|\, q\big)} -{(1-q)^{2n}}\\ -=(-2\sin\phi)^n\,n!\,p_n(x;a,b,\overline a,\overline b)\qquad -(0<\phi<\pi). -\label{177} -\end{multline} -Here the \RHS\ has a continuous Hahn polynomial (9.4.1). -For the proof first derive the corresponding limit for the monic polynomials by comparing -(14.1.5) with (9.1.5). -In fact, define the monic polynomial -\[ -\wt p_n(x):= -\frac{p_n\big(\cos\phi-x(1-q)\sin\phi;q^a e^{i\phi},q^b e^{i\phi},q^{\overline a} e^{-i\phi}, -q^{\overline b} e^{-i\phi}\,|\, q\big)} -{(-2(1-q)\sin\phi)^n\,(abcdq^{n-1};q)_n}\,. -\] -Then it follows from (14.1.5) that -\begin{equation*} -x\,\wt p_n(x)=\wt p_{n+1}(x) -+\frac{(1-q^a)e^{i\phi}+(1-q^{-a})e^{-i\phi}+\wt A_n+\wt C_n}{2(1-q)\sin\phi}\,\wt p_n(x)\\ -+\frac{\wt A_{n-1} \wt C_n}{(1-q)^2 \sin^2\phi}\,\wt p_{n-1}(x), -\end{equation*} -where $\wt A_n$ and $\wt C_n$ are as given after (14.1.3) with $a,b,c,d$ replaced by -$q^a e^{i\phi},q^b e^{i\phi},q^{\overline a} e^{-i\phi},q^{\overline b} e^{-i\phi}$. -Then the recurrence equation for $\wt p_n(x)$ tends for $q\uparrow 1$ to -the recurrence equation (9.4.4) with $c=\overline a$, $d=\overline b$. -\bLP -{\bf Askey-Wilson $\longrightarrow$ Meixner-Pollaczek}\\ -Instead of (14.9.15) we can keep a polynomial of degree $n$ while the limit is approached: -\begin{equation} -\lim_{q\uparrow1} -\frac{p_n\big(\cos\phi-x(1-q)\sin\phi; -q^\la e^{i\phi},0,q^\la e^{-i\phi},0\,|\, q\big)}{(1-q)^n} -=n!\,P_n^{(\la)}(x;\pi-\phi)\quad -(0<\phi<\pi). -\label{178} -\end{equation} -Here the \RHS\ has a Meixner-Pollaczek polynomial (9.7.1). -For the proof first derive the corresponding limit for the monic polynomials by comparing -(14.1.5) with (9.7.4). -In fact, define the monic polynomial -\[ -\wt p_n(x):= -\frac{p_n\big(\cos\phi-x(1-q)\sin\phi; -q^\la e^{i\phi},0,q^\la e^{-i\phi},0\,|\, q\big)}{(-2(1-q)\sin\phi)^n}\,. -\] -Then it follows from (14.1.5) that -\begin{equation*} -x\,\wt p_n(x)=\wt p_{n+1}(x) -+\frac{(1-q^\la)e^{i\phi}+(1-q^{-\la})e^{-i\phi}+\wt A_n+\wt C_n} -{2(1-q)\sin\phi}\,\wt p_n(x)\\ -+\frac{\wt A_{n-1} \wt C_n}{(1-q)^2 \sin^2\phi}\,\wt p_{n-1}(x), -\end{equation*} -where $\wt A_n$ and $\wt C_n$ are as given after (14.1.3) with $a,b,c,d$ replaced by -$q^\la e^{i\phi},0,q^\la e^{-i\phi},0$. -Then the recurrence equation for $\wt p_n(x)$ tends for $q\uparrow 1$ to -the recurrence equation (9.7.4). -% -\paragraph{References} -See also Koornwinder \cite{K7}. -$$\frac{(-a)^n}{n!}C_n(x;a)=L_n^{(x-n)}(a).$$ - -\subsection*{References} -\cite{Allaway76}, \cite{NAlSalam66}, \cite{AlSalam90}, \cite{AlSalamChihara76}, -\cite{AlSalamIsmail76}, \cite{AndrewsAskey85}, \cite{Area+II}, \cite{Askey75}, -\cite{AskeyGasper77}, \cite{AskeyWilson85}, \cite{AtakRahmanSuslov}, \cite{Bavinck98}, -\cite{BavinckKoekoek}, \cite{Chihara78}, \cite{Chihara79}, \cite{Dunkl76}, \cite{Erdelyi+}, -\cite{Gasper73I}, \cite{Gasper74}, \cite{Goh}, \cite{HoareRahman}, \cite{IsmailLetVal88}, -\cite{Koekoek2000}, \cite{Koorn88}, \cite{Krasikov2002}, \cite{LabelleYehI}, -\cite{LabelleYehII}, \cite{LabelleYehIII}, \cite{Lesky62}, \cite{Lesky89}, \cite{Lesky94I}, -\cite{Lesky95II}, \cite{LewanowiczII}, \cite{LopezTemme2004}, \cite{Meixner}, \cite{Nikiforov+}, -\cite{NikiforovUvarov}, \cite{Szafraniec}, \cite{Szego75}, \cite{Viennot}, \cite{Zarzo+}, -\cite{Zeng90}. - - -\section{Hermite}\index{Hermite polynomials} - -\par\setcounter{equation}{0} - -\subsection*{Hypergeometric representation} -\begin{equation} -\label{DefHermite} -H_n(x)=(2x)^n\,\hyp{2}{0}{-n/2,-(n-1)/2}{-}{-\frac{1}{x^2}}. -\end{equation} - -\subsection*{Orthogonality relation} -\begin{equation} -\label{OrtHermite} -\frac{1}{\sqrt{\cpi}}\int_{-\infty}^{\infty}\expe^{-x^2}H_m(x)H_n(x)\,dx -=2^nn!\,\delta_{mn}. -\end{equation} - -\subsection*{Recurrence relation} -\begin{equation} -\label{RecHermite} -H_{n+1}(x)-2xH_n(x)+2nH_{n-1}(x)=0. -\end{equation} - -\subsection*{Normalized recurrence relation} -\begin{equation} -\label{NormRecHermite} -xp_n(x)=p_{n+1}(x)+\frac{n}{2}p_{n-1}(x), -\end{equation} -where -$$H_n(x)=2^np_n(x).$$ - -\subsection*{Differential equation} -\begin{equation} -\label{dvHermite} -y''(x)-2xy'(x)+2ny(x)=0,\quad y(x)=H_n(x). -\end{equation} - -\subsection*{Forward shift operator} -\begin{equation} -\label{shift1Hermite} -\frac{d}{dx}H_n(x)=2nH_{n-1}(x). -\end{equation} - -\subsection*{Backward shift operator} -\begin{equation} -\label{shift2HermiteI} -\frac{d}{dx}H_n(x)-2xH_n(x)=-H_{n+1}(x) -\end{equation} -or equivalently -\begin{equation} -\label{shift2HermiteII} -\frac{d}{dx}\left[\expe^{-x^2}H_n(x)\right]=-\expe^{-x^2}H_{n+1}(x). -\end{equation} - -\subsection*{Rodrigues-type formula} -\begin{equation} -\label{RodHermite} -\e^{-x^2}H_n(x)=(-1)^n\left(\frac{d}{dx}\right)^n\left[\expe^{-x^2}\right]. -\end{equation} - -\subsection*{Generating functions} -\begin{equation} -\label{GenHermite1} -\exp\left(2xt-t^2\right)=\sum_{n=0}^{\infty}\frac{H_n(x)}{n!}t^n. -\end{equation} - -\begin{equation} -\label{GenHermite2} -\left\{\begin{array}{l} -\displaystyle\expe^t\cos(2x\sqrt{t})=\sum_{n=0}^{\infty} -\frac{(-1)^n}{(2n)!}H_{2n}(x)t^n\\[5mm] -\displaystyle\frac{\expe^t}{\sqrt{t}}\sin(2x\sqrt{t})=\sum_{n=0}^{\infty} -\frac{(-1)^n}{(2n+1)!}H_{2n+1}(x)t^n. -\end{array}\right. -\end{equation} - -\begin{equation} -\label{GenHermite3} -\left\{\begin{array}{l} -\displaystyle \expe^{-t^2}\cosh(2xt)=\sum_{n=0}^{\infty} -\frac{H_{2n}(x)}{(2n)!}t^{2n}\\[5mm] -\displaystyle\expe^{-t^2}\sinh(2xt)=\sum_{n=0}^{\infty} -\frac{H_{2n+1}(x)}{(2n+1)!}t^{2n+1}. -\end{array}\right. -\end{equation} - -\begin{equation} -\label{GenHermite4} -\left\{\begin{array}{l} -\displaystyle (1+t^2)^{-\gamma}\,\hyp{1}{1}{\gamma}{\frac{1}{2}}{\frac{x^2t^2}{1+t^2}}= -\sum_{n=0}^{\infty}\frac{(\gamma)_n}{(2n)!}H_{2n}(x)t^{2n}\\[5mm] -\displaystyle\frac{xt}{\sqrt{1+t^2}}\, -\hyp{1}{1}{\gamma+\frac{1}{2}}{\frac{3}{2}}{\frac{x^2t^2}{1+t^2}} -=\sum_{n=0}^{\infty}\frac{(\gamma+\frac{1}{2})_n}{(2n+1)!}H_{2n+1}(x)t^{2n+1} -\end{array}\right. -\end{equation} -with $\gamma$ arbitrary. - -\begin{equation} -\label{GenHermite5} -\frac{1+2xt+4t^2}{(1+4t^2)^{\frac{3}{2}}}\exp\left(\frac{4x^2t^2}{1+4t^2}\right) -=\sum_{n=0}^{\infty}\frac{H_n(x)}{\lfloor n/2\rfloor\,!}t^n, -\end{equation} -where $\lfloor n/2\rfloor$ denotes the largest integer smaller than or equal to $n/2$. - -\subsection*{Limit relations} - -\subsubsection*{Meixner-Pollaczek $\rightarrow$ Hermite} -If we take $x\rightarrow (\sin\phi)^{-1}(x\sqrt{\lambda}-\lambda\cos\phi)$ -in the definition (\ref{DefMP}) of the Meixner-Pollaczek polynomials and -then let $\lambda\rightarrow\infty$ we obtain the Hermite polynomials: -$$\lim_{\lambda\rightarrow\infty} -\lambda^{-\frac{1}{2}n}P_n^{(\lambda)} -((\sin\phi)^{-1}(x\sqrt{\lambda}-\lambda\cos\phi);\phi)=\frac{H_n(x)}{n!}.$$ - -\subsubsection*{Jacobi $\rightarrow$ Hermite} -The Hermite polynomials follow from the Jacobi polynomials given by (\ref{DefJacobi}) by -taking $\beta=\alpha$ and letting $\alpha\rightarrow\infty$ in the following way: -$$\lim_{\alpha\rightarrow\infty} -\alpha^{-\frac{1}{2}n}P_n^{(\alpha,\alpha)}(\alpha^{-\frac{1}{2}}x)=\frac{H_n(x)}{2^nn!}.$$ - -\subsubsection*{Gegenbauer / Ultraspherical $\rightarrow$ Hermite} -The Hermite polynomials follow from the Gegenbauer (or ultraspherical) polynomials given by -(\ref{DefGegenbauer}) by taking $\lambda=\alpha+\frac{1}{2}$ and letting -$\alpha\rightarrow\infty$ in the following way: -$$\lim_{\alpha\rightarrow\infty} -\alpha^{-\frac{1}{2}n}C_n^{(\alpha+\frac{1}{2})}(\alpha^{-\frac{1}{2}}x)=\frac{H_n(x)}{n!}.$$ - -\subsubsection*{Krawtchouk $\rightarrow$ Hermite} -The Hermite polynomials follow from the Krawtchouk polynomials given by (\ref{DefKrawtchouk}) -by setting $x\rightarrow pN+x\sqrt{2p(1-p)N}$ and then letting $N\rightarrow\infty$: -$$\lim_{N\rightarrow\infty} -\sqrt{\binom{N}{n}}K_n(pN+x\sqrt{2p(1-p)N};p,N) -=\frac{\displaystyle (-1)^nH_n(x)}{\displaystyle\sqrt{2^nn!\left(\frac{p}{1-p}\right)^n}}.$$ - -\subsubsection*{Laguerre $\rightarrow$ Hermite} -The Hermite polynomials can be obtained from the Laguerre polynomials given by -(\ref{DefLaguerre}) by taking the limit $\alpha\rightarrow\infty$ in the following way: -$$\lim_{\alpha\rightarrow\infty} -\left(\frac{2}{\alpha}\right)^{\frac{1}{2}n} -L_n^{(\alpha)}((2\alpha)^{\frac{1}{2}}x+\alpha)=\frac{(-1)^n}{n!}H_n(x).$$ - -\subsubsection*{Charlier $\rightarrow$ Hermite} -If we set $x\rightarrow (2a)^{1/2}x+a$ in the definition (\ref{DefCharlier}) -of the Charlier polynomials and let $a\rightarrow\infty$ we find the Hermite -polynomials. In fact we have -$$\lim_{a\rightarrow\infty} -(2a)^{\frac{1}{2}n}C_n((2a)^{\frac{1}{2}}x+a;a)=(-1)^nH_n(x).$$ - -\subsection*{Remarks} -The Hermite polynomials can also be written as: -$$\frac{H_n(x)}{n!}=\sum_{k=0}^{\lfloor n/2\rfloor} -\frac{(-1)^k(2x)^{n-2k}}{k!\,(n-2k)!},$$ -where $\lfloor n/2\rfloor$ denotes the largest integer smaller than or equal to $n/2$. - -\noindent -The Hermite polynomials and the Laguerre polynomials given by (\ref{DefLaguerre}) are also -connected by the following quadratic transformations: -$$H_{2n}(x)=(-1)^nn!\,2^{2n}L_n^{(-\frac{1}{2})}(x^2)$$ -and -\subsection*{14.2 $q$-Racah} -\label{sec14.2} -\paragraph{Symmetry} -\begin{equation} -R_n(x;\al,\be,q^{-N-1},\de\,|\, q) -=\frac{(\be q,\al\de^{-1}q;q)_n}{(\al q,\be\de q;q)_n}\,\de^n\, -R_n(\de^{-1}x;\be,\al,q^{-N-1},\de^{-1}\,|\, q). -\label{84} -\end{equation} -This follows from (14.2.1) combined with \mycite{GR}{(III.15)}. -\sLP -In particular, -\begin{equation} -R_n(x;\al,\be,q^{-N-1},-1\,|\, q) -=\frac{(\be q,-\al q;q)_n}{(\al q,-\be q;q)_n}\,(-1)^n\, -R_n(-x;\be,\al,q^{-N-1},-1\,|\, q), -\label{85} -\end{equation} -and -\begin{equation} -R_n(x;\al,\al,q^{-N-1},-1\,|\, q) -=(-1)^n\,R_n(-x;\al,\al,q^{-N-1},-1\,|\, q), -\label{86} -\end{equation} - -\paragraph{Trivial symmetry} -Clearly from (14.2.1): -\begin{equation} -R_n(x;\al,\be,\ga,\de\,|\, q)=R_n(x;\be\de,\al\de^{-1},\ga,\de\,|\, q) -=R_n(x;\ga,\al\be\ga^{-1},\al,\ga\de\al^{-1}\,|\, q). -\label{83} -\end{equation} -For $\al=q^{-N-1}$ this shows that the three cases -$\al q=q^{-N}$ or $\be\de q=q^{-N}$ or $\ga q=q^{-N}$ of (14.2.1) -are not essentially different. -% -\paragraph{Duality} -It follows from (14.2.1) that -\begin{equation} -R_n(q^{-y}+\ga\de q^{y+1};q^{-N-1},\be,\ga,\de\,|\, q) -=R_y(q^{-n}+\be q^{n-N};\ga,\de,q^{-N-1},\be\,|\, q)\quad -(n,y=0,1,\ldots,N). -\end{equation} -$$H_{2n+1}(x)=(-1)^nn!\,2^{2n+1}xL_n^{(\frac{1}{2})}(x^2).$$ - -\subsection*{14.3 Continuous dual $q$-Hahn} -\label{sec14.3} -The continuous dual $q$-Hahn polynomials are the special case $d=0$ of the -Askey-Wilson polynomials: -\[ -p_n(x;a,b,c\,|\, q):=p_n(x;a,b,c,0\,|\, q). -\] -Hence all formulas in \S14.3 are specializations for $d=0$ of formulas in \S14.1. -\subsection*{References} -\cite{Abram}, \cite{NAlSalam66}, \cite{AlSalam90}, \cite{AlSalamChihara72}, -\cite{AlSalamChihara76}, \cite{AndrewsAskey85}, \cite{AndrewsAskeyRoy}, \cite{Area+I}, -\cite{Askey68}, \cite{Askey75}, \cite{Askey89I}, \cite{AskeyGasper76}, \cite{AskeyWilson85}, -\cite{Azor}, \cite{Berg}, \cite{BilodeauII}, \cite{Brafman51}, \cite{Brafman57II}, -\cite{Brenke}, \cite{CarlitzSrivastava}, \cite{ChenSrivastava}, \cite{Chihara78}, -\cite{Cohen}, \cite{Danese}, \cite{DetteStudden92}, \cite{DetteStudden95}, \cite{Doha2004I}, -\cite{Erdelyi+}, \cite{Faldey}, \cite{Gawronski87}, \cite{Gawronski93}, \cite{Gillis+}, -\cite{Godoy+}, \cite{Grad}, \cite{Ismail74}, \cite{Ismail2005II}, \cite{IsmailStanton97}, -\cite{Koekoek2000}, \cite{Koorn88}, \cite{Krasikov2004}, \cite{Kwon+}, \cite{LabelleYehIII}, -\cite{Lesky95II}, \cite{Lesky96}, \cite{LewanowiczI}, \cite{LopezTemme99}, \cite{Mathai}, -\cite{Meixner}, \cite{Nikiforov+}, \cite{NikiforovUvarov}, \cite{Olver}, \cite{Pittaluga}, -\cite{Rainville}, \cite{SainteViennot}, \cite{Srivastava71}, \cite{SrivastavaMathur}, -\cite{Szego75}, \cite{Temme}, \cite{TemmeLopez2000}, \cite{Trickovic}, \cite{Viennot}, -\cite{Weisner59}, \cite{Wyman}. - -\end{document} diff --git a/README.md b/README.md index b9ec267..34dc88c 100644 --- a/README.md +++ b/README.md @@ -29,7 +29,10 @@ To achieve the desired result, one should execute the progams in the following o ## KLSadd Insertion Project linetest.py and updateChapters (mostly) written by Rahul Shah (ILIKEFUUD) -Edits the DRMF chapter files to include the relevant KLS addendum additions. Additions are (currently)only being added right before the "References" paragraphs in each section. The linetest.py file is the first working model of the code, but it is very messy and esoteric. Comments have been made to aid interpretation. The new project file, updateChapters.py is a more readable version, but not quite up to date. + +Edits the DRMF chapter files to include the relevant KLS addendum additions. Additions are (currently) only being added right before the "References" paragraphs in each section. The linetest.py file is the first working model of the code, but it is very messy and esoteric. Comments have been made to aid interpretation. The new project file, updateChapters.py is a more readable version, but not quite up to date. Left to do in updateChapters.py: import over the necessary files to make the pdf work, change all instances of "section" to "paragraph" and add "\large\bf" to make font large and bold like the other chapter headings. Also necessary to add initials of programmer wherever the new additions were added (ex. "%RS added, %RS end") + +Also: rewrite code for "smarter" edits (ex. add new limit relations straight to the From 6f5180bf32c6f9e08cbcbd93b6f76a4ace832cd1 Mon Sep 17 00:00:00 2001 From: Rahul Shah Date: Wed, 17 Feb 2016 16:59:00 -0500 Subject: [PATCH 008/402] Updated README for new insertion tasks --- README.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/README.md b/README.md index 34dc88c..e580ece 100644 --- a/README.md +++ b/README.md @@ -35,4 +35,4 @@ Edits the DRMF chapter files to include the relevant KLS addendum additions. Add Left to do in updateChapters.py: import over the necessary files to make the pdf work, change all instances of "section" to "paragraph" and add "\large\bf" to make font large and bold like the other chapter headings. Also necessary to add initials of programmer wherever the new additions were added (ex. "%RS added, %RS end") -Also: rewrite code for "smarter" edits (ex. add new limit relations straight to the +Also: rewrite code for "smarter" edits (ex. add new limit relations straight to the) From 603372c34c461b673bd76d97f98f7206e615302c Mon Sep 17 00:00:00 2001 From: Rahul Shah Date: Wed, 17 Feb 2016 17:02:22 -0500 Subject: [PATCH 009/402] Updated README for new insertion tasks --- README.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/README.md b/README.md index e580ece..917598a 100644 --- a/README.md +++ b/README.md @@ -35,4 +35,4 @@ Edits the DRMF chapter files to include the relevant KLS addendum additions. Add Left to do in updateChapters.py: import over the necessary files to make the pdf work, change all instances of "section" to "paragraph" and add "\large\bf" to make font large and bold like the other chapter headings. Also necessary to add initials of programmer wherever the new additions were added (ex. "%RS added, %RS end") -Also: rewrite code for "smarter" edits (ex. add new limit relations straight to the) +Also: rewrite code for "smarter" edits (ex. add new limit relations straight to the limit relations section in the section itself, not at the end) From 359c1e31f0b57242481fce0dcc7a67bdae81f6cf Mon Sep 17 00:00:00 2001 From: Rahul Shah Date: Wed, 17 Feb 2016 17:11:06 -0500 Subject: [PATCH 010/402] Added README for KLS Addendum Insertion --- KLSadd_insertion/README.md | 11 +++++++++++ 1 file changed, 11 insertions(+) create mode 100644 KLSadd_insertion/README.md diff --git a/KLSadd_insertion/README.md b/KLSadd_insertion/README.md new file mode 100644 index 0000000..ec95792 --- /dev/null +++ b/KLSadd_insertion/README.md @@ -0,0 +1,11 @@ +##KLS Addendum Insertion Project + +This project uses python and string manipulation to insert sections of the KLSadd.tex file into the appropriate DRMF chapter sections. The updateChapters.py program is the most recently updated and cleanest version, however it is not completed. Should be run with a simple call to + +python updateChapters.py + +The linetest.py program is the original program file fully updated. It is much harder to read and should only be used as a reference to update the updateChapters.py program. + +NOTE: Both the updateChapters.py and linetest.py lack a pdf version of their chapter 14 output. One must be generated to view results. + +NOTE: The KLSadd.tex file only deals with chapters 9 and 14, as stated in the document itself. From 2a69face761558a0eb0e8960bf14b7eae6fd4547 Mon Sep 17 00:00:00 2001 From: Rahul Shah Date: Wed, 17 Feb 2016 17:13:32 -0500 Subject: [PATCH 011/402] Updated README formatting --- KLSadd_insertion/README.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/KLSadd_insertion/README.md b/KLSadd_insertion/README.md index ec95792..ae09455 100644 --- a/KLSadd_insertion/README.md +++ b/KLSadd_insertion/README.md @@ -1,9 +1,9 @@ ##KLS Addendum Insertion Project This project uses python and string manipulation to insert sections of the KLSadd.tex file into the appropriate DRMF chapter sections. The updateChapters.py program is the most recently updated and cleanest version, however it is not completed. Should be run with a simple call to - +``` python updateChapters.py - +``` The linetest.py program is the original program file fully updated. It is much harder to read and should only be used as a reference to update the updateChapters.py program. NOTE: Both the updateChapters.py and linetest.py lack a pdf version of their chapter 14 output. One must be generated to view results. From 0a7c2d29d8fb59e17e1228ff543ded4329fc9bb0 Mon Sep 17 00:00:00 2001 From: Rahul Shah Date: Thu, 18 Feb 2016 15:14:30 -0500 Subject: [PATCH 012/402] Updated README with information about program input and output and sensitive data. --- KLSadd_insertion/README.md | 4 ++++ 1 file changed, 4 insertions(+) diff --git a/KLSadd_insertion/README.md b/KLSadd_insertion/README.md index ae09455..84b8b12 100644 --- a/KLSadd_insertion/README.md +++ b/KLSadd_insertion/README.md @@ -9,3 +9,7 @@ The linetest.py program is the original program file fully updated. It is much h NOTE: Both the updateChapters.py and linetest.py lack a pdf version of their chapter 14 output. One must be generated to view results. NOTE: The KLSadd.tex file only deals with chapters 9 and 14, as stated in the document itself. + +NOTE: when you run updateChapters.py or linetest.py you need the KLSadd.tex file, tempchap9.tex, and tempchap14.tex in order to run the program. The program *generates* updated9.tex and updated14.tex files. + +**THIS MEANS YOU SHOULD NOT RUN updateChapters.py OR linetest.py FROM YOUR GIT DIRECTORY. THE CHAPTER FILES ARE NOT PUBLIC DOMAIN, DO NOT COMMIT THEM! COPY THE FILES OVER INTO A SEPERATE DIRECTORY AND THEN COPY THEM BACK OVER WHEN YOU ARE READY TO COMMIT** From 8eb9cf9bae5657ac01072c9b59b5b277bd29e292 Mon Sep 17 00:00:00 2001 From: Rahul Shah Date: Thu, 18 Feb 2016 15:54:08 -0500 Subject: [PATCH 013/402] Now inserts relevant usepackages from KLSadd --- KLSadd_insertion/updateChapters.py | 22 +++++++++++++++++++--- 1 file changed, 19 insertions(+), 3 deletions(-) diff --git a/KLSadd_insertion/updateChapters.py b/KLSadd_insertion/updateChapters.py index f7f40fe..c320327 100644 --- a/KLSadd_insertion/updateChapters.py +++ b/KLSadd_insertion/updateChapters.py @@ -9,7 +9,7 @@ 9 and 14 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -NOTE! AS OF 2/8/16 THIS FILE IS NOT YET UPDATED TO THE FULL CAPACITY OF THE PREVIOUS FILE. IF FURTHER DEVELOPMENT IS NEEDED, REFER TO THE +NOTE! AS OF 2/18/16 THIS FILE IS NOT YET UPDATED TO THE FULL CAPACITY OF THE PREVIOUS FILE. IF FURTHER DEVELOPMENT IS NEEDED, REFER TO THE linetest.py FILE IF THIS FILE DOES NOT ADDRESS THE NEW/EXTRA GOALS. This means that XCITE PARSE etc HAS NOT BEEN ADDED Current update status: incomplete @@ -28,6 +28,22 @@ paras = [] mathPeople = [] + +#2/18/16 this method addresses the goal of hardcoding in the necessary packages to let the chapter files run as pdf's. Currently only works with chapter 9, ask Dr. Cohl to help port your chapter 14 output file into a pdf + +def prepareForPDF(str): + footmiscIndex = 0 + index = 0 + for word in str: + index+=1 + if("footmisc" in word): + footmiscIndex+= index + #edits the chapter string sent to include hyperref, xparse, and cite packages + str[footmiscIndex] += "\\usepackage[pdftex]{hyperref} \n\\usepackage {xparse} \n\\usepackage{cite} \n" + + #not sure if needed, but I passed anyway + pass + #method to find the indices of the reference paragraphs def findReferences(str): references = [] @@ -48,7 +64,7 @@ def findReferences(str): #method to change file string(actually a list right now), returns string to be written to file def fixChapter(str, references, p): #str is the file string, references is the specific references for the file, and p is the paras variable(not sure if needed) - #TODO: OPTIMIZE + #TODO: OPTIMIZE(?) count = 0 #count is used to represent the values in count for i in references: #add @@ -106,7 +122,7 @@ def fixChapter(str, references, p): str9 = ''.join(fixChapter(entire9, references9, paras)) str14 = ''.join(fixChapter(entire14, references14, paras)) - #write to file + #write to files #new output files for safety with open("updated9.tex","w") as temp9: temp9.write(str9) From 2cb1881f86d0f0c9b5ea9e4d42646b853daba873 Mon Sep 17 00:00:00 2001 From: Rahul Shah Date: Thu, 18 Feb 2016 16:50:20 -0500 Subject: [PATCH 014/402] Added method to insert KLS commands --- KLSadd_insertion/updateChapters.py | 30 +++++++++++++++++++++++++++++- 1 file changed, 29 insertions(+), 1 deletion(-) diff --git a/KLSadd_insertion/updateChapters.py b/KLSadd_insertion/updateChapters.py index c320327..3f98a88 100644 --- a/KLSadd_insertion/updateChapters.py +++ b/KLSadd_insertion/updateChapters.py @@ -12,6 +12,14 @@ NOTE! AS OF 2/18/16 THIS FILE IS NOT YET UPDATED TO THE FULL CAPACITY OF THE PREVIOUS FILE. IF FURTHER DEVELOPMENT IS NEEDED, REFER TO THE linetest.py FILE IF THIS FILE DOES NOT ADDRESS THE NEW/EXTRA GOALS. This means that XCITE PARSE etc HAS NOT BEEN ADDED +Additional goals (already addressed in linetest.py): +-insert correct usepackages +-insert correct commands found in KLSadd.tex to chapter files +-insert editor's initials into sections (ie %RS: begin addition, %RS: end addition) + +Additional goals (not already addressed in linetest.py): +-rewrite for "smarter" edits (ex. add new limit relations straight to the limit relations section in the section itself, not at the end) + Current update status: incomplete Goals:change the book chapter files to include paragraphs from the addendum, and after that insert some edits so they are intelligently integrated into the chapter @@ -28,7 +36,6 @@ paras = [] mathPeople = [] - #2/18/16 this method addresses the goal of hardcoding in the necessary packages to let the chapter files run as pdf's. Currently only works with chapter 9, ask Dr. Cohl to help port your chapter 14 output file into a pdf def prepareForPDF(str): @@ -44,6 +51,27 @@ def prepareForPDF(str): #not sure if needed, but I passed anyway pass +#2/18/16 this method reads in relevant commands that are in KLSadd.tex and returns them as a list and also adds + +def getCommands(kls): + newCommands = [] + for word in kls: + index+=1 + if("smallskipamount" in word): + newCommands.append(index-1) + if("mybibitem[1]" in word): + newCommands.append(index) + + #add \large\bf KLSadd: to make KLSadd additions appear like the other paragraphs + if("paragraph{" in word): + temp = word[0:word.find("{")+ 1] + "\large\\bf KLSadd: " + word[word.find("{")+ 1: word.find("}")+1] + +#2/18/16 this method addresses the goal of hardcoding in the necessary commands to let the chapter files run as pdf's. Currently only works with chapter 9 + +def insertCommands(kls, str): + #reads in from KLSadd.tex, finds appropriate commands, and puts them in str + + #method to find the indices of the reference paragraphs def findReferences(str): references = [] From d619e2be56d1f3c01f849db1adf047b817c022e0 Mon Sep 17 00:00:00 2001 From: Rahul Shah Date: Fri, 19 Feb 2016 16:30:00 -0500 Subject: [PATCH 015/402] Added new method to complete new tasks --- KLSadd_insertion/updateChapters.py | 21 ++++++++++++++++----- 1 file changed, 16 insertions(+), 5 deletions(-) diff --git a/KLSadd_insertion/updateChapters.py b/KLSadd_insertion/updateChapters.py index 3f98a88..98644ec 100644 --- a/KLSadd_insertion/updateChapters.py +++ b/KLSadd_insertion/updateChapters.py @@ -9,7 +9,7 @@ 9 and 14 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -NOTE! AS OF 2/18/16 THIS FILE IS NOT YET UPDATED TO THE FULL CAPACITY OF THE PREVIOUS FILE. IF FURTHER DEVELOPMENT IS NEEDED, REFER TO THE +NOTE! AS OF 2/19/16 THIS FILE IS NOT YET UPDATED TO THE FULL CAPACITY OF THE PREVIOUS FILE. IF FURTHER DEVELOPMENT IS NEEDED, REFER TO THE linetest.py FILE IF THIS FILE DOES NOT ADDRESS THE NEW/EXTRA GOALS. This means that XCITE PARSE etc HAS NOT BEEN ADDED Additional goals (already addressed in linetest.py): @@ -35,7 +35,7 @@ chapNums = [] paras = [] mathPeople = [] - +newCommands = [] #used to hold extra commands that need to be inserted from KLSadd to the chapter files #2/18/16 this method addresses the goal of hardcoding in the necessary packages to let the chapter files run as pdf's. Currently only works with chapter 9, ask Dr. Cohl to help port your chapter 14 output file into a pdf def prepareForPDF(str): @@ -54,7 +54,6 @@ def prepareForPDF(str): #2/18/16 this method reads in relevant commands that are in KLSadd.tex and returns them as a list and also adds def getCommands(kls): - newCommands = [] for word in kls: index+=1 if("smallskipamount" in word): @@ -65,12 +64,22 @@ def getCommands(kls): #add \large\bf KLSadd: to make KLSadd additions appear like the other paragraphs if("paragraph{" in word): temp = word[0:word.find("{")+ 1] + "\large\\bf KLSadd: " + word[word.find("{")+ 1: word.find("}")+1] - + #pretty sure I had to do something here but I forgot, so pass? + pass + #2/18/16 this method addresses the goal of hardcoding in the necessary commands to let the chapter files run as pdf's. Currently only works with chapter 9 def insertCommands(kls, str): - #reads in from KLSadd.tex, finds appropriate commands, and puts them in str + #reads in the newCommands[] and puts them in str + beginIndex = -1 #the index of the "begin document" keyphrase, this is where the new commands need to be inserted. + #find index of begin document in KLSadd + for w in kls: + index+=1 + if("begin{document}" in word): + beginIndex += index + str[beginIndex] += newCommands + return str #method to find the indices of the reference paragraphs def findReferences(str): @@ -145,6 +154,8 @@ def fixChapter(str, references, p): #two variables for the references lists one for chapter 9 one for chapter 14 references9 = findReferences(entire9) references14 = findReferences(entire14) + + #call the fixChapter method to get a list with the addendum paragraphs added in str9 = ''.join(fixChapter(entire9, references9, paras)) From d0e96e19599e727d83b157846488855e3bef19a0 Mon Sep 17 00:00:00 2001 From: Rahul Shah Date: Fri, 19 Feb 2016 16:55:59 -0500 Subject: [PATCH 016/402] Edward, check here for tasks --- KLSadd_insertion/EdwardTasks.txt | 21 +++++++++++++++++++++ 1 file changed, 21 insertions(+) create mode 100644 KLSadd_insertion/EdwardTasks.txt diff --git a/KLSadd_insertion/EdwardTasks.txt b/KLSadd_insertion/EdwardTasks.txt new file mode 100644 index 0000000..951ac3c --- /dev/null +++ b/KLSadd_insertion/EdwardTasks.txt @@ -0,0 +1,21 @@ +Fix your github; + +If I'm not here Monday, https://help.github.com/categories/ssh/ +go there and try to work out your issue, rm your git directory and try git clone again. + +This is what updateChapters.py currently does: + +-rips paragraphs from KLSadd and puts them after the References paragraphs. +TESTED: WORKING + +This is what has been written but not implemented/tested: + +-method to add relevant usepackages from KLSadd (cite, hyperref, xparse) +-method to add relevant new commands from KLSadd +-method to add "\large\bf KLSadd:" after every "paragraph{" inserted from KLSadd so everything looks neat + +This is what needs to be written: +-method that inserts limit relations straight into the limit relations paragraph in relevant sections instead of just at the end of the section + +keep checking back here every time I've commited to see what's left, and you can change stuff around here just add a comment saying what you did and the date you did it + From bbafdd7dd20b04a8baaab6d90a728243e21c0aa9 Mon Sep 17 00:00:00 2001 From: Rahul Shah Date: Fri, 19 Feb 2016 16:59:46 -0500 Subject: [PATCH 017/402] Added instructions for EdwardTasks.txt --- KLSadd_insertion/README.md | 4 ++++ 1 file changed, 4 insertions(+) diff --git a/KLSadd_insertion/README.md b/KLSadd_insertion/README.md index 84b8b12..5c3febb 100644 --- a/KLSadd_insertion/README.md +++ b/KLSadd_insertion/README.md @@ -13,3 +13,7 @@ NOTE: The KLSadd.tex file only deals with chapters 9 and 14, as stated in the do NOTE: when you run updateChapters.py or linetest.py you need the KLSadd.tex file, tempchap9.tex, and tempchap14.tex in order to run the program. The program *generates* updated9.tex and updated14.tex files. **THIS MEANS YOU SHOULD NOT RUN updateChapters.py OR linetest.py FROM YOUR GIT DIRECTORY. THE CHAPTER FILES ARE NOT PUBLIC DOMAIN, DO NOT COMMIT THEM! COPY THE FILES OVER INTO A SEPERATE DIRECTORY AND THEN COPY THEM BACK OVER WHEN YOU ARE READY TO COMMIT** + + +**Edward, check the EdwardTasks.txt file to check what's done and what needs work, if your github is still down just try to write out what the method should be like in a seperate file** + From beddbd2464aabfbc29ee591532c1a1fa7e518a34 Mon Sep 17 00:00:00 2001 From: Rahul Shah Date: Mon, 22 Feb 2016 15:31:59 -0500 Subject: [PATCH 018/402] Updated for 2/22/16 --- KLSadd_insertion/EdwardTasks.txt | 5 ++++- 1 file changed, 4 insertions(+), 1 deletion(-) diff --git a/KLSadd_insertion/EdwardTasks.txt b/KLSadd_insertion/EdwardTasks.txt index 951ac3c..2e78c55 100644 --- a/KLSadd_insertion/EdwardTasks.txt +++ b/KLSadd_insertion/EdwardTasks.txt @@ -1,8 +1,11 @@ Fix your github; -If I'm not here Monday, https://help.github.com/categories/ssh/ +https://help.github.com/categories/ssh/ go there and try to work out your issue, rm your git directory and try git clone again. + +If all else fails, try ssh -vT git@github.com + This is what updateChapters.py currently does: -rips paragraphs from KLSadd and puts them after the References paragraphs. From 200a6d0273c671b39de76fdfde8e2902b3d39484 Mon Sep 17 00:00:00 2001 From: Rahul Shah Date: Mon, 22 Feb 2016 16:01:29 -0500 Subject: [PATCH 019/402] added method calls to updateCHapters.py testing needed --- KLSadd_insertion/updateChapters.py | 30 ++++++++++++++++++++---------- 1 file changed, 20 insertions(+), 10 deletions(-) diff --git a/KLSadd_insertion/updateChapters.py b/KLSadd_insertion/updateChapters.py index 98644ec..7cd7dcb 100644 --- a/KLSadd_insertion/updateChapters.py +++ b/KLSadd_insertion/updateChapters.py @@ -9,7 +9,7 @@ 9 and 14 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -NOTE! AS OF 2/19/16 THIS FILE IS NOT YET UPDATED TO THE FULL CAPACITY OF THE PREVIOUS FILE. IF FURTHER DEVELOPMENT IS NEEDED, REFER TO THE +NOTE! AS OF 2/22/16 THIS FILE IS NOT YET UPDATED TO THE FULL CAPACITY OF THE PREVIOUS FILE. IF FURTHER DEVELOPMENT IS NEEDED, REFER TO THE linetest.py FILE IF THIS FILE DOES NOT ADDRESS THE NEW/EXTRA GOALS. This means that XCITE PARSE etc HAS NOT BEEN ADDED Additional goals (already addressed in linetest.py): @@ -36,7 +36,8 @@ paras = [] mathPeople = [] newCommands = [] #used to hold extra commands that need to be inserted from KLSadd to the chapter files -#2/18/16 this method addresses the goal of hardcoding in the necessary packages to let the chapter files run as pdf's. Currently only works with chapter 9, ask Dr. Cohl to help port your chapter 14 output file into a pdf +#2/18/16 this method addresses the goal of hardcoding in the necessary packages to let the chapter files run as pdf's. +#Currently only works with chapter 9, ask Dr. Cohl to help port your chapter 14 output file into a pdf def prepareForPDF(str): footmiscIndex = 0 @@ -47,9 +48,8 @@ def prepareForPDF(str): footmiscIndex+= index #edits the chapter string sent to include hyperref, xparse, and cite packages str[footmiscIndex] += "\\usepackage[pdftex]{hyperref} \n\\usepackage {xparse} \n\\usepackage{cite} \n" - - #not sure if needed, but I passed anyway - pass + + return str #2/18/16 this method reads in relevant commands that are in KLSadd.tex and returns them as a list and also adds @@ -99,16 +99,22 @@ def findReferences(str): return references #method to change file string(actually a list right now), returns string to be written to file -def fixChapter(str, references, p): - #str is the file string, references is the specific references for the file, and p is the paras variable(not sure if needed) +#If you write a method that changes something, it is preffered that you call the method in here +def fixChapter(str, references, p, kls): + #str is the file string(actually a list), references is the specific references for the file, and p is the paras variable(not sure if needed) kls is the KLSadd.tex as a list #TODO: OPTIMIZE(?) count = 0 #count is used to represent the values in count for i in references: - #add + #add the paragraphs before the Reference paragraphs start str[i-3] += p[count] count+=1 - #I actually don't remember why I had to do this but I think you need it str[i-1] += p[count] + + #I actually don't remember why I had to do this but I think you need it ^^^^^^^^^^^^^^^^^^^ + str = prepareForPDF(str) + getCommands(kls) + str = insertCommands(kls,str) + #probably won't work because I don't know how anything works return str #open the KLSadd file to do things with @@ -155,12 +161,16 @@ def fixChapter(str, references, p): references9 = findReferences(entire9) references14 = findReferences(entire14) - #call the fixChapter method to get a list with the addendum paragraphs added in str9 = ''.join(fixChapter(entire9, references9, paras)) str14 = ''.join(fixChapter(entire14, references14, paras)) +""" +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +If you are writing something that will make a change to the chapter files, write it BEFORE this line, this part is where the lists representing the words/strings in the chapter are joined together and updated as a string! +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +""" #write to files #new output files for safety with open("updated9.tex","w") as temp9: From 38a731c4dc993dd90e6471dda98b18efa354fdd3 Mon Sep 17 00:00:00 2001 From: Rahul Shah Date: Mon, 22 Feb 2016 16:45:35 -0500 Subject: [PATCH 020/402] Added original file of updateChapters as a backup --- KLSadd_insertion/updateChaptersBACKUP1.py | 117 ++++++++++++++++++++++ 1 file changed, 117 insertions(+) create mode 100644 KLSadd_insertion/updateChaptersBACKUP1.py diff --git a/KLSadd_insertion/updateChaptersBACKUP1.py b/KLSadd_insertion/updateChaptersBACKUP1.py new file mode 100644 index 0000000..a738d07 --- /dev/null +++ b/KLSadd_insertion/updateChaptersBACKUP1.py @@ -0,0 +1,117 @@ +""" +Rahul Shah +self-taught python don't kill me I know its bad +2/5/16 +Re-write of linetest.py so people other than me can read it + +This project was/is being written to streamline the update process of the book used +for the online repository, updated via the KLSadd addendum file which only affects chapters +9 and 14 + +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +NOTE! AS OF 2/18/16 THIS FILE IS NOT YET UPDATED TO THE FULL CAPACITY OF THE PREVIOUS FILE. IF FURTHER DEVELOPMENT IS NEEDED, REFER TO THE +linetest.py FILE IF THIS FILE DOES NOT ADDRESS THE NEW/EXTRA GOALS. This means that XCITE PARSE etc HAS NOT BEEN ADDED + +Current update status: incomplete + +Goals:change the book chapter files to include paragraphs from the addendum, and after that insert some edits so they are intelligently integrated into the chapter +""" + +#start out by reading KLSadd.tex to get all of the paragraphs that must be added to the chapter files +#also keep track of which chapter each one is in + +#variables: chapNums for the chapter number each section belongs to paras will hold the sections that will be copied over +#chapNums is needed to know which file to open (9 or 14) +#mathPeople is just the name for the sections that are used, like Wilson, Racah, etc. I know some are not people, its just a var name +#yes I like lists +chapNums = [] +paras = [] +mathPeople = [] + +#method to find the indices of the reference paragraphs +def findReferences(str): + references = [] + index = -1 + canAdd = False + for word in str: + index+=1 + #check sections and subsections + if("section{" in word or "subsection*{" in word): + w = word[word.find("{")+1: word.find("}")] + if(w not in mathPeople): + canAdd = True + if("\\subsection*{References}" in word and canAdd == True): + references.append(index) + canAdd = False + return references + +#method to change file string(actually a list right now), returns string to be written to file +def fixChapter(str, references, p): + #str is the file string, references is the specific references for the file, and p is the paras variable(not sure if needed) + #TODO: OPTIMIZE + count = 0 #count is used to represent the values in count + for i in references: + #add + str[i-3] += p[count] + count+=1 + #I actually don't remember why I had to do this but I think you need it + str[i-1] += p[count] + return str + +#open the KLSadd file to do things with +with open("KLSadd.tex", "r") as add: + #store the file as a string + addendum = add.readlines() + index = 0 + indexes = [] + for word in addendum: + index+=1 + if("." in word and "\\subsection*{" in word): + name = word[word.find(" "): word.find("}")] + mathPeople.append(name) + if("9." in word): + chapNums.append(9) + if("14." in word): + chapNums.append(9) + #get the index + indexes.append(index-1) + #now indexes holds all of the places there is a section + #using these indexes, get all of the words in between and add that to the paras[] + for i in range(len(indexes)-1): + str = ''.join(addendum[indexes[i]: indexes[i+1]-1]) + paras.append(str) + + #paras now holds the paragraphs that need to go into the chapter files, but they need to go in the appropriate + #section(like Wilson, Racah, Hahn, etc.) so we use the mathPeople variable + #we can use the section names to place the relevant paragraphs in the right place + + #as of 2/8/16 the paragraphs will go before the References paragraph of the relevant section + #parse both files 9 and 14 as strings + + #chapter 9 + with open("tempchap9.tex", "r") as ch9: + entire9 = ch9.readlines() #reads in as a list of strings + ch9.close() + + #chapter 14 + with open("tempchap14.tex", "r") as ch14: + entire14 = ch14.readlines() + ch14.close() + #call the findReferences method to find the index of the References paragraph in the two file strings + #two variables for the references lists one for chapter 9 one for chapter 14 + references9 = findReferences(entire9) + references14 = findReferences(entire14) + + #call the fixChapter method to get a list with the addendum paragraphs added in + str9 = ''.join(fixChapter(entire9, references9, paras)) + str14 = ''.join(fixChapter(entire14, references14, paras)) + + #write to file + #new output files for safety + with open("updated9.tex","w") as temp9: + temp9.write(str9) + temp9.close() + + with open("updated14.tex", "w") as temp14: + temp14.write(str9) + From 4178fe1ba94e83bc9b3cab80cd78f16861c7c605 Mon Sep 17 00:00:00 2001 From: Edward Bian Date: Mon, 22 Feb 2016 16:53:57 -0500 Subject: [PATCH 021/402] edward bian test commit --- KLSadd_insertion/updateChapters.py | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/KLSadd_insertion/updateChapters.py b/KLSadd_insertion/updateChapters.py index 7cd7dcb..04c197c 100644 --- a/KLSadd_insertion/updateChapters.py +++ b/KLSadd_insertion/updateChapters.py @@ -7,7 +7,7 @@ This project was/is being written to streamline the update process of the book used for the online repository, updated via the KLSadd addendum file which only affects chapters 9 and 14 - +#Edward test commit ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ NOTE! AS OF 2/22/16 THIS FILE IS NOT YET UPDATED TO THE FULL CAPACITY OF THE PREVIOUS FILE. IF FURTHER DEVELOPMENT IS NEEDED, REFER TO THE linetest.py FILE IF THIS FILE DOES NOT ADDRESS THE NEW/EXTRA GOALS. This means that XCITE PARSE etc HAS NOT BEEN ADDED From ec0635f8477e2adf632cd0688694e7fa0dceaac8 Mon Sep 17 00:00:00 2001 From: Edward Bian Date: Mon, 22 Feb 2016 17:02:03 -0500 Subject: [PATCH 022/402] more testing edward bian --- KLSadd_insertion/updateChapters.py | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/KLSadd_insertion/updateChapters.py b/KLSadd_insertion/updateChapters.py index 04c197c..1bbca08 100644 --- a/KLSadd_insertion/updateChapters.py +++ b/KLSadd_insertion/updateChapters.py @@ -7,7 +7,7 @@ This project was/is being written to streamline the update process of the book used for the online repository, updated via the KLSadd addendum file which only affects chapters 9 and 14 -#Edward test commit +#Edward test 2 commit ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ NOTE! AS OF 2/22/16 THIS FILE IS NOT YET UPDATED TO THE FULL CAPACITY OF THE PREVIOUS FILE. IF FURTHER DEVELOPMENT IS NEEDED, REFER TO THE linetest.py FILE IF THIS FILE DOES NOT ADDRESS THE NEW/EXTRA GOALS. This means that XCITE PARSE etc HAS NOT BEEN ADDED From 4f586f2898174090aaa775dc4c815689db88e01c Mon Sep 17 00:00:00 2001 From: Edward Bian Date: Mon, 22 Feb 2016 17:16:43 -0500 Subject: [PATCH 023/402] more practice with git(just added some text) --- KLSadd_insertion/EdwardTasks.txt | 2 ++ 1 file changed, 2 insertions(+) diff --git a/KLSadd_insertion/EdwardTasks.txt b/KLSadd_insertion/EdwardTasks.txt index 2e78c55..cd4bdcf 100644 --- a/KLSadd_insertion/EdwardTasks.txt +++ b/KLSadd_insertion/EdwardTasks.txt @@ -5,6 +5,8 @@ go there and try to work out your issue, rm your git directory and try git clone If all else fails, try ssh -vT git@github.com + ^ +ABOVE IS DONE | This is what updateChapters.py currently does: From 5241bb1d524c9fe317eeedf8a5969fc97852e310 Mon Sep 17 00:00:00 2001 From: Edward Bian Date: Fri, 26 Feb 2016 12:41:44 -0500 Subject: [PATCH 024/402] Fixing Stuff for myself(Edward) --- KLSadd_insertion/EdwardTasks.txt | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/KLSadd_insertion/EdwardTasks.txt b/KLSadd_insertion/EdwardTasks.txt index cd4bdcf..0f5a079 100644 --- a/KLSadd_insertion/EdwardTasks.txt +++ b/KLSadd_insertion/EdwardTasks.txt @@ -1,4 +1,4 @@ -Fix your github; +Fix your github; hopefully fixed now (fixed if this message shows up on github) https://help.github.com/categories/ssh/ go there and try to work out your issue, rm your git directory and try git clone again. From c41d33be4782997c632d7571e0bff042741355fb Mon Sep 17 00:00:00 2001 From: Rahul Shah Date: Fri, 26 Feb 2016 17:16:25 -0500 Subject: [PATCH 025/402] Fixed some errors, incomplete --- KLSadd_insertion/updateChapters.py | 68 +++++++++++++++++------------- 1 file changed, 38 insertions(+), 30 deletions(-) diff --git a/KLSadd_insertion/updateChapters.py b/KLSadd_insertion/updateChapters.py index 1bbca08..35ed303 100644 --- a/KLSadd_insertion/updateChapters.py +++ b/KLSadd_insertion/updateChapters.py @@ -9,7 +9,7 @@ 9 and 14 #Edward test 2 commit ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -NOTE! AS OF 2/22/16 THIS FILE IS NOT YET UPDATED TO THE FULL CAPACITY OF THE PREVIOUS FILE. IF FURTHER DEVELOPMENT IS NEEDED, REFER TO THE +NOTE! AS OF 2/26/16 THIS FILE IS NOT YET UPDATED TO THE FULL CAPACITY OF THE PREVIOUS FILE. IF FURTHER DEVELOPMENT IS NEEDED, REFER TO THE linetest.py FILE IF THIS FILE DOES NOT ADDRESS THE NEW/EXTRA GOALS. This means that XCITE PARSE etc HAS NOT BEEN ADDED Additional goals (already addressed in linetest.py): @@ -39,47 +39,53 @@ #2/18/16 this method addresses the goal of hardcoding in the necessary packages to let the chapter files run as pdf's. #Currently only works with chapter 9, ask Dr. Cohl to help port your chapter 14 output file into a pdf -def prepareForPDF(str): +def prepareForPDF(chap): + print(type(chap)) footmiscIndex = 0 index = 0 - for word in str: + for word in chap: index+=1 if("footmisc" in word): footmiscIndex+= index #edits the chapter string sent to include hyperref, xparse, and cite packages - str[footmiscIndex] += "\\usepackage[pdftex]{hyperref} \n\\usepackage {xparse} \n\\usepackage{cite} \n" + #str[footmiscIndex] += "\\usepackage[pdftex]{hyperref} \n\\usepackage {xparse} \n\\usepackage{cite} \n" + chap.insert(footmiscIndex, "\\usepackage[pdftex]{hyperref} \n\\usepackage {xparse} \n\\usepackage{cite} \n") - return str + return chap #2/18/16 this method reads in relevant commands that are in KLSadd.tex and returns them as a list and also adds def getCommands(kls): + index = 0 for word in kls: index+=1 if("smallskipamount" in word): newCommands.append(index-1) if("mybibitem[1]" in word): newCommands.append(index) - #add \large\bf KLSadd: to make KLSadd additions appear like the other paragraphs if("paragraph{" in word): + #not sure if this works, needs testing! temp = word[0:word.find("{")+ 1] + "\large\\bf KLSadd: " + word[word.find("{")+ 1: word.find("}")+1] - #pretty sure I had to do something here but I forgot, so pass? - pass + + #pretty sure I had to do something here but I forgot, so pass? + pass #2/18/16 this method addresses the goal of hardcoding in the necessary commands to let the chapter files run as pdf's. Currently only works with chapter 9 -def insertCommands(kls, str): - #reads in the newCommands[] and puts them in str +def insertCommands(kls, chap): + #reads in the newCommands[] and puts them in chap beginIndex = -1 #the index of the "begin document" keyphrase, this is where the new commands need to be inserted. #find index of begin document in KLSadd + index = 0 for w in kls: index+=1 if("begin{document}" in word): beginIndex += index - str[beginIndex] += newCommands - return str + #chap[beginIndex] += newCommands + chap.insert(beginIndex, newCommands) + return chap #method to find the indices of the reference paragraphs def findReferences(str): @@ -100,22 +106,23 @@ def findReferences(str): #method to change file string(actually a list right now), returns string to be written to file #If you write a method that changes something, it is preffered that you call the method in here -def fixChapter(str, references, p, kls): - #str is the file string(actually a list), references is the specific references for the file, and p is the paras variable(not sure if needed) kls is the KLSadd.tex as a list +def fixChapter(chap, references, p, kls): + #chap is the file string(actually a list), references is the specific references for the file, + #and p is the paras variable(not sure if needed) kls is the KLSadd.tex as a list #TODO: OPTIMIZE(?) count = 0 #count is used to represent the values in count for i in references: #add the paragraphs before the Reference paragraphs start - str[i-3] += p[count] + chap[i-3] += p[count] count+=1 - str[i-1] += p[count] + chap[i-1] += p[count] #I actually don't remember why I had to do this but I think you need it ^^^^^^^^^^^^^^^^^^^ - str = prepareForPDF(str) + chap = prepareForPDF(chap) getCommands(kls) - str = insertCommands(kls,str) + chap = insertCommands(kls,chap) #probably won't work because I don't know how anything works - return str + return chap #open the KLSadd file to do things with with open("KLSadd.tex", "r") as add: @@ -163,20 +170,21 @@ def fixChapter(str, references, p, kls): #call the fixChapter method to get a list with the addendum paragraphs added in - str9 = ''.join(fixChapter(entire9, references9, paras)) - str14 = ''.join(fixChapter(entire14, references14, paras)) + str9 = ''.join(fixChapter(entire9, references9, paras, addendum)) + str14 = ''.join(fixChapter(entire14, references14, paras, addendum)) """ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -If you are writing something that will make a change to the chapter files, write it BEFORE this line, this part is where the lists representing the words/strings in the chapter are joined together and updated as a string! +If you are writing something that will make a change to the chapter files, write it BEFORE this line, this part +is where the lists representing the words/strings in the chapter are joined together and updated as a string! ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ """ - #write to files - #new output files for safety - with open("updated9.tex","w") as temp9: - temp9.write(str9) - temp9.close() - - with open("updated14.tex", "w") as temp14: - temp14.write(str9) + + +#write to files +#new output files for safety +with open("updated9.tex","w") as temp9: + temp9.write(str9) +with open("updated14.tex", "w") as temp14: + temp14.write(str9) From e5dbb2d60ff3262ee6cf96fd79a9c325e5164d17 Mon Sep 17 00:00:00 2001 From: Rahul Shah Date: Fri, 26 Feb 2016 17:30:41 -0500 Subject: [PATCH 026/402] Fixed some more bugs with list and types --- KLSadd_insertion/updateChapters.py | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/KLSadd_insertion/updateChapters.py b/KLSadd_insertion/updateChapters.py index 35ed303..a7cb7bc 100644 --- a/KLSadd_insertion/updateChapters.py +++ b/KLSadd_insertion/updateChapters.py @@ -7,7 +7,7 @@ This project was/is being written to streamline the update process of the book used for the online repository, updated via the KLSadd addendum file which only affects chapters 9 and 14 -#Edward test 2 commit + ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ NOTE! AS OF 2/26/16 THIS FILE IS NOT YET UPDATED TO THE FULL CAPACITY OF THE PREVIOUS FILE. IF FURTHER DEVELOPMENT IS NEEDED, REFER TO THE linetest.py FILE IF THIS FILE DOES NOT ADDRESS THE NEW/EXTRA GOALS. This means that XCITE PARSE etc HAS NOT BEEN ADDED @@ -40,7 +40,6 @@ #Currently only works with chapter 9, ask Dr. Cohl to help port your chapter 14 output file into a pdf def prepareForPDF(chap): - print(type(chap)) footmiscIndex = 0 index = 0 for word in chap: @@ -168,7 +167,8 @@ def fixChapter(chap, references, p, kls): references9 = findReferences(entire9) references14 = findReferences(entire14) - + #ERROR! entire9 sometimes contains lists, must only contain strings. Should be fixed by next week. Check the + #getCommands method, I suspect that is the problem #call the fixChapter method to get a list with the addendum paragraphs added in str9 = ''.join(fixChapter(entire9, references9, paras, addendum)) str14 = ''.join(fixChapter(entire14, references14, paras, addendum)) From da1bd90125221368b9c6f06d0006dd54948376a8 Mon Sep 17 00:00:00 2001 From: physikerwelt Date: Sat, 27 Feb 2016 15:09:18 +0100 Subject: [PATCH 027/402] Reformat code * fix typo * follow IDE suggestions --- AlexDanoff/remove_excess.py | 209 +++++++++++++++++------------------- 1 file changed, 98 insertions(+), 111 deletions(-) diff --git a/AlexDanoff/remove_excess.py b/AlexDanoff/remove_excess.py index c520105..18c16b2 100644 --- a/AlexDanoff/remove_excess.py +++ b/AlexDanoff/remove_excess.py @@ -1,11 +1,11 @@ -"""This program begins with an unprocessed LaTeX file and removes parts that are unnecesssary for the DLMF.""" +"""This program begins with an unprocessed LaTeX file and removes parts that are unnecessary for the DLMF.""" import re import sys import parentheses from utilities import (writeout, readin, get_line_lengths, - find_line) + find_line) EQ_START = r'\begin{equation}' EQ_END = r'\end{equation}' @@ -18,65 +18,70 @@ STD_REGEX = r'.*?###open_(\d+)###.*?###close_\1###' -def main(): +def main(): if len(sys.argv) != 3: fname = "ZE.tex" - dot_ind = fname.index(".") + 1 + dot_ind = fname.index(".") + 1 ofname = fname[:dot_ind] + "1." + fname[dot_ind:] else: fname = sys.argv[1] - ofname = sys.argv[2] + ofname = sys.argv[2] writeout(ofname, remove_excess(readin(fname))) -def remove_group(name, content, skip_lines): - """Removes the group bounded by the group name and then curly braces.""" - - str_pat = name + STD_REGEX - pattern = re.compile(str_pat, re.DOTALL) - - return pattern.sub(r'', content) def remove_section(start, end, content): - """Removes the section or part bounded by start and end.""" - + """Removes the section or part bounded by start and end. + :param start: + :param end: + :param content: + """ + str_pat = start + r'.*?(' + end + r')' - pattern = re.compile(str_pat, re.DOTALL) + pattern = re.compile(str_pat, re.DOTALL) return pattern.sub(r'\1', content) + def remove_begin_end(name, content): - """Removes a group bound by \\begin{name} and \\end{name}.""" + """Removes a group bound by \\begin{name} and \\end{name}. + :param name: + :param content: + """ str_pat = r'\\begin{' + name + r'}.*?\\end{' + name + r'}' pattern = re.compile(str_pat, re.DOTALL) return pattern.sub(r'', content) + def remove_excess(content): - """Removes the excess pieces from the given content and returns the updated version as a string.""" + """Removes the excess pieces from the given content and returns the updated version as a string. + :param content: + """ oldest = content updated = _get_preamble() old = content - #edit single lines - content = re.sub(r'\\bibliography{\.\./bib/DLMF}', r'\\bibliographystyle{plain}' + "\n" + r'\\bibliography{/home/hcohl/DRMF/DLMF/DLMF.bib}', content) - content = re.sub(r'\\author\[.*?\]{(.*?)}', r'\\author{\1}', content) - content = re.sub(r'{math}', r'{equation}', content) - - #modify labels, etc - content = re.sub(r'\\begin{equation}\[.*?\]+', r'\\begin{equation}', content, re.DOTALL) + # edit single lines + content = re.sub(r'\\bibliography{\.\./bib/DLMF}', + r'\\bibliographystyle{plain}' + "\n" + r'\\bibliography{/home/hcohl/DRMF/DLMF/DLMF.bib}', content) + content = re.sub(r'\\author\[.*?\]{(.*?)}', r'\\author{\1}', content) + content = re.sub(r'\{math\}', r'{equation}', content) + + # modify labels, etc + content = re.sub(r'\\begin\{equation\}\[.*?\]+', r'\\begin{equation}', content, re.DOTALL) - print(len(old) -len(content)) + print(len(old) - len(content)) old = content - - #remove from something to the start of the next section/part/etc + + # remove from something to the start of the next section/part/etc content = remove_section(r'\\section{Special Notation}', r'\\part', content) content = remove_section(r'\\part{Computation}', r'\\bibliographystyle{plain}', content) content = remove_section(r'\\part{References}', r'\\bibliographystyle{plain}', content) @@ -85,17 +90,16 @@ def remove_excess(content): content = remove_section(r'\\subsection{Graphics}', r'\\subsection', content) content = remove_section(r'\\section{Integrals}', r'\\section', content) content = remove_section(r'\\subsection{Integrals}', r'\\subsection', content) - content = remove_section(r'\\section{Physical Applications}', r'\\bibliographystyle{plain}', content) + content = remove_section(r'\\section{Physical Applications}', r'\\bibliographystyle{plain}', content) print(len(old) - len(content)) - old = content to_join = [] - #takes out comment lines first + # takes out comment lines first for line in content.split("\n"): - - #if line only consists of % replace with nothing, otherwise don't add back + + # if line only consists of % replace with nothing, otherwise don't add back if line.lstrip().startswith("%"): if line.rstrip().endswith("%"): line = "" @@ -108,153 +112,150 @@ def remove_excess(content): skip_lines = set() - #remove begin/end groups + # remove begin/end groups content = remove_begin_end("figuregroup", content) content = remove_begin_end("comment", content) content = remove_begin_end("figure", content) content = remove_begin_end("errata", content) - content = parentheses.remove(content, curly=True) + content = parentheses.remove(content, curly=True) to_remove = [ r'\\citet', - r'\\lxDeclare\[.*?\]', + r'\\lxDeclare\[.*?\]', r'\\note', r'\\origref', r'\\lxRefDeclaration', r'\\MarkDefn.*?###open_(\d+)###.*?###close_\1###', - r'\\indexdefn', + r'\\indexdefn', r'\\MarkNotation.*?###open_(\d+)###.*?###close_\1###', - r'\\affiliation', + r'\\affiliation', ] lengths = get_line_lengths(content) - #go through each group and remove it + # go through each group and remove it for name in to_remove: - + str_regex = name + STD_REGEX pattern = re.compile(str_regex, re.DOTALL) - #go through every match, finding start and end + # go through every match, finding start and end for match in pattern.finditer(content): - - group = match.group() - start = find_line(match.start(), lengths) - end = find_line(match.end(), lengths) + end = find_line(match.end(), lengths) - #print("{0}: {1} - {2}".format(group, start, end)) + # print("{0}: {1} - {2}".format(group, start, end)) - skip_lines.update(range(start, end + 1)) + skip_lines.update(range(start, end + 1)) - #define items that cannot be at the beginning of any line or be contined in any line - illegal_starts = [r'\documentclass{DLMF}', r'\thischapter', r'\part{Notation}', r'\begin{equationgroup', r'\end{equationgroup', r'\begin{onecolumn', r'\end{onecolumn'] + # define items that cannot be at the beginning of any line or be contined in any line + illegal_starts = [r'\documentclass{DLMF}', r'\thischapter', r'\part{Notation}', r'\begin{equationgroup', + r'\end{equationgroup', r'\begin{onecolumn', r'\end{onecolumn'] illegal_elements = [r'TwoToOneRule', r'OneToTwoRule', r'\citet'] lines = content.split("\n") - #various flags that will be useful when going through the lines + # various flags that will be useful when going through the lines in_eq = False - in_eqmix = False in_const = False in_cases = False eqmix_label = "" const_str = "" - #remove trailing % and whitespace and add to updated - also remove lines that should be skipped + # remove trailing % and whitespace and add to updated - also remove lines that should be skipped for lnum, line in enumerate(lines): lnum += 1 - #don't add line back if it should be skipped + # don't add line back if it should be skipped if lnum in skip_lines: continue - + line = parentheses.insert(line, curly=True) - line_checks = [line.lstrip().startswith(start) for start in illegal_starts] + [element in line for element in illegal_elements] - - #skip current line if it starts with or contains an illegal element + line_checks = [line.lstrip().startswith(start) for start in illegal_starts] + [element in line for element in + illegal_elements] + + # skip current line if it starts with or contains an illegal element if any(line_checks): - continue + continue cleaned = line.rstrip().rstrip("%").rstrip() - #line marks the start of an equationmix, set the flag and remove the line + # line marks the start of an equationmix, set the flag and remove the line if EQMIX_START in cleaned: - in_eqmix = True - eqmix_label = cleaned[cleaned.index(r'\label'):] + eqmix_label = cleaned[cleaned.index(r'\label'):] continue - #line marks the end of an equationmix, set the flag and remove the line + # line marks the end of an equationmix, set the flag and remove the line if EQMIX_END in cleaned: - in_eqmix = False eqmix_label = "" continue - #if this line marks the start of an equation, set the flag + # if this line marks the start of an equation, set the flag if EQ_START in cleaned: - in_eq = True + in_eq = True cleaned = cleaned + eqmix_label - - #if this line marks the end of an equation, set the flag + + # if this line marks the end of an equation, set the flag if EQ_END in cleaned: in_eq = False - #comment out constraints and replace commas with double commas; we need to build the entire constraint string and then add it back together + # comment out constraints and replace commas with double commas; we need to build the entire constraint + # string and then add it back together if cleaned.lstrip().startswith(r"\constraint"): in_const = True - #starting a cases statement + # starting a cases statement if CASES_START in cleaned: in_cases = True - #ending a cases statement + # ending a cases statement if CASES_END in cleaned: in_cases = False - #remove commas, periods, colons, and semi-colons from the end of equations + # remove commas, periods, colons, and semi-colons from the end of equations if in_eq and not in_const: to_strip = ":;," - - #don't take off trailing commas when in a cases block + + # don't take off trailing commas when in a cases block if in_cases: to_strip = to_strip[:-1] - + cleaned = cleaned.rstrip(to_strip) - #we're still in a constraint, replace commas and comment out + # we're still in a constraint, replace commas and comment out if in_const: const_str += cleaned - #we're done with the constraint, make substitutions + # we're done with the constraint, make substitutions if cleaned.rstrip().rstrip(".,").endswith("}"): - const_str = re.sub(r'\$[;,](\s*(?:\$|or))', r'$ &\1', const_str) + const_str = re.sub(r'\$[;,](\s*(?:\$|or))', r'$ &\1', const_str) in_const = False - - #constraint ends with two }, put one on next line + + # constraint ends with two }, put one on next line if cleaned.rstrip().endswith("}}"): const_str = const_str[:-1] + "\n}" - + const_lines = [] split = const_str.split("\n") - + offset = 1 - #account for the case when there are no newlines present + # account for the case when there are no newlines present if len(split) == 1: - offset = 0 + offset = 0 - #split multiline constraints back into multiple lines + # split multiline constraints back into multiple lines for const_line in split[offset:]: - - #if line is not just a bracket, comment it out + + # if line is not just a bracket, comment it out if const_line.strip() != "}": const_line = "%" + const_line - + const_lines.append(const_line) updated.extend(const_lines) @@ -268,44 +269,30 @@ def remove_excess(content): content = '\n'.join(updated) - #remove blank lines around begin and end equation - space_pat = re.compile(r'\\begin{equation}(.*?)$\s+$', re.MULTILINE) + # remove blank lines around begin and end equation + space_pat = re.compile(r'\\begin\{equation\}(.*?)$\s+$', re.MULTILINE) content = space_pat.sub(r'\\begin{equation}\1', content) - space_pat = re.compile(r'^\s*$\n\\end{equation}', re.MULTILINE) + space_pat = re.compile(r'^\s*$\n\\end\{equation\}', re.MULTILINE) content = space_pat.sub(r'\\end{equation}', content) - #remove consecutive blank lines + # remove consecutive blank lines content = re.sub(r'(\n){3,}', '\n\n', content) print("Final: {0}".format(len(oldest) - len(content))) return content -#returns the preamble as a list of it's lines -def _get_preamble(): - preamble = [] - - preamble.append('\\documentclass{article}') - preamble.append('\\usepackage{amsmath}') - preamble.append('\\usepackage{amsfonts}') - preamble.append('\\usepackage{breqn}') - preamble.append('\\usepackage{DLMFmath}') - preamble.append('\\usepackage{DRMFfcns}') - preamble.append('') - preamble.append('\\oddsidemargin -0.7cm') - preamble.append('\\textwidth 18.3cm') - preamble.append('\\textheight 26.0cm') - preamble.append('\\topmargin -2.0cm') - preamble.append('') - preamble.append('% \constraint{') - preamble.append('% \substitution{') - preamble.append('% \drmfnote{') - preamble.append('% \drmfname{') - preamble.append('') +# returns the preamble as a list of it's lines +def _get_preamble(): + preamble = ['\\documentclass{article}', '\\usepackage{amsmath}', '\\usepackage{amsfonts}', '\\usepackage{breqn}', + '\\usepackage{DLMFmath}', '\\usepackage{DRMFfcns}', '', '\\oddsidemargin -0.7cm', '\\textwidth 18.3cm', + '\\textheight 26.0cm', '\\topmargin -2.0cm', '', '% \constraint{', '% \substitution{', + '% \drmfnote{', '% \drmfname{', ''] return preamble + if __name__ == "__main__": main() From c7f8c57ce2fc5f08fa303af8a2ff7f19167e9fbb Mon Sep 17 00:00:00 2001 From: physikerwelt Date: Sat, 27 Feb 2016 16:55:33 +0100 Subject: [PATCH 028/402] Add travis integration --- .travis.yml | 2 ++ README.md | 2 ++ 2 files changed, 4 insertions(+) create mode 100644 .travis.yml diff --git a/.travis.yml b/.travis.yml new file mode 100644 index 0000000..b85e205 --- /dev/null +++ b/.travis.yml @@ -0,0 +1,2 @@ +language: python +script: nosetests -w AlexDanoff \ No newline at end of file diff --git a/README.md b/README.md index 917598a..bd507b5 100644 --- a/README.md +++ b/README.md @@ -1,5 +1,7 @@ # DRMF-Seeding-Project +[![Build Status](https://travis-ci.org/DRMF/DRMF-Seeding-Project.svg?branch=master)](https://travis-ci.org/DRMF/DRMF-Seeding-Project) + This project is meant to convert different designated source formats to semantic LaTeX for the DRMF project. One major aspect of this conversion is the inclusion, insertion, and replacement of LaTeX semantic macros. From 4315dbedbb80e752e8274c4e6e6ba7a62e368022 Mon Sep 17 00:00:00 2001 From: physikerwelt Date: Sat, 27 Feb 2016 17:20:48 +0100 Subject: [PATCH 029/402] Reformat code * prevent data exploits by moving the private data into a directroy that is ignored by .gitignore --- AlexDanoff/prepare_annotations.py | 83 +++++++++-------- AlexDanoff/remove_excess.py | 2 +- AlexDanoff/replace_special.py | 144 +++++++++++++++--------------- 3 files changed, 119 insertions(+), 110 deletions(-) diff --git a/AlexDanoff/prepare_annotations.py b/AlexDanoff/prepare_annotations.py index 818eede..5f22cac 100644 --- a/AlexDanoff/prepare_annotations.py +++ b/AlexDanoff/prepare_annotations.py @@ -19,7 +19,7 @@ import parentheses from utilities import (readin, writeout, remove_inner_whitespace, - get_line_lengths, find_line) + get_line_lengths, find_line) FLUSH_START = r'\begin{flushright}' FLUSH_END = r'\end{flushright}' @@ -27,25 +27,27 @@ ANNOTATION_STR = r'\s*%\s*\\(?:constraint|substitution|drmf(?:note|name)|proof)' ANNOTATION_PAT = re.compile(ANNOTATION_STR, re.MULTILINE) -BRACE_PAT = re.compile(r'(?P' + ANNOTATION_STR + r')###open_(\d+)###(?![$ ])(?P.*?)(?![$ ])###close_\2###', re.DOTALL) +BRACE_PAT = re.compile( + r'(?P' + ANNOTATION_STR + r')###open_(\d+)###(?![$ ])(?P.*?)(?![$ ])###close_\2###', re.DOTALL) MEASURE_PAT = re.compile(r'(\S)\\\\\[0.2cm\]') -def main(): - #get the names of the input and output files +def main(): + # get the names of the input and output files if len(sys.argv) != 3: - fname = "ZE.3.tex" - ofname = "ZE.4.tex" + fname = "data/ZE.2.tex" + ofname = "data/ZE.3.tex" else: fname = sys.argv[1] - ofname = sys.argv[2] + ofname = sys.argv[2] writeout(ofname, prepare_annotations(readin(fname))) + def prepare_annotations(content): """ Prepares annotations in `content` to be compatible with WikiText. @@ -55,14 +57,15 @@ def prepare_annotations(content): * special formatting * etc. """ - + docstart = content.find(r'\begin{document}') - eq_comment_pattern = re.compile(r'\\begin{equation}(?P\\label{(?P.*?)}.*?\n^[^%]*$)(?P\n^ *%[\w\s\\{}()<>[\]&=@%,.:;|\'~_^$/+-]*?\n)\\end{equation}', re.MULTILINE) + eq_comment_pattern = re.compile( + r'\\begin{equation}(?P\\label{(?P.*?)}.*?\n^[^%]*$)(?P\n^ *%[\w\s\\{}()<>[\]&=@%,.:;|\'~_^$/+-]*?\n)\\end{equation}', + re.MULTILINE) content = content[:docstart] + eq_comment_pattern.sub(replace_comments, content[docstart:]) - special_annotations = { r'\\constraint': "C", r'\\substitution': "S", @@ -71,7 +74,7 @@ def prepare_annotations(content): r'\\proof': "PROOF" } - #replace the annotations with their "code" + # replace the annotations with their "code" for annotation, code in special_annotations.iteritems(): content = content[:docstart] + re.sub(annotation, r'{\\bf ' + code + '}:~', content[docstart:]) @@ -82,7 +85,8 @@ def prepare_annotations(content): return content -#moves annotations from inside equation blocks to the end of them and wraps them in a flushright block + +# moves annotations from inside equation blocks to the end of them and wraps them in a flushright block def replace_comments(match): """ Processes each equation block that contains comments. @@ -110,7 +114,7 @@ def replace_comments(match): before = match.group("before") - #print("Equation label: {0}".format(match.group("eq_label"))) + # print("Equation label: {0}".format(match.group("eq_label"))) replacement = r'\begin{equation}' + before @@ -128,26 +132,26 @@ def replace_comments(match): comment = BRACE_PAT.sub(r'\g\g', comment) comment = parentheses.insert(comment, curly=True) - #make sure every line in the comment is actually a comment + # make sure every line in the comment is actually a comment for comment_line in comment.split("\n"): stripped = comment_line.lstrip() - #line isn't really a comment, add it back to the equation + # line isn't really a comment, add it back to the equation if not stripped.startswith("%"): replacement += comment_line + "\n" - - else: #otherwise, process it (take off % and add in \displaystyle) and add it to the comment + + else: # otherwise, process it (take off % and add in \displaystyle) and add it to the comment num_annotations = len(ANNOTATION_PAT.findall(comment_line)) - #add measurement to the previous line if the current line is starting a new annotation + # add measurement to the previous line if the current line is starting a new annotation add_measure = num_annotations and annotations_left != total_annotations annotations_left -= num_annotations - + leading_whitespace = len(comment_line) - len(stripped) + 1 - comment_line = " " * leading_whitespace + stripped.lstrip("%") + comment_line = " " * leading_whitespace + stripped.lstrip("%") old_len = len(comment_line) @@ -158,22 +162,22 @@ def replace_comments(match): first_brack = comment_line.find("{") - #if we're going to go out of bounds, nothing to do on this line + # if we're going to go out of bounds, nothing to do on this line if first_brack + 1 < len(comment_line): after_first = comment_line[first_brack + 1] else: continue - + dollar_locs = _get_dollar_locs(comment_line) offset = 0 - #there's at least one $ in the line + # there's at least one $ in the line while dollar_locs: - - #the $ are balanced + + # the $ are balanced if len(dollar_locs) % 2 == 0: - + dollar_iter = iter(dollar_locs) start, end = (dollar_iter.next(), dollar_iter.next()) @@ -181,15 +185,17 @@ def replace_comments(match): previous = "" disp_string = '${' + r'\displaystyle ' - #only try to look behind the $ if it isn't the first character in the line + # only try to look behind the $ if it isn't the first character in the line if start != 0: - previous = comment_line[start-1] - - #we may need to add a space before displaystyle + previous = comment_line[start - 1] + + # we may need to add a space before displaystyle if previous == " ": disp_string = " " + disp_string - comment_line = comment_line[:start] + " " * extra_whitespace + disp_string + comment_line[start + 1:end] + '}' + comment_line[end:] + comment_line = comment_line[:start] + " " * extra_whitespace + disp_string + comment_line[ + start + 1:end] + '}' + comment_line[ + end:] comment_line = comment_line.rstrip().rstrip("}") + " " dollar_locs = _get_dollar_locs(comment_line) @@ -197,9 +203,9 @@ def replace_comments(match): dollar_locs = dollar_locs[offset:] - #add the measurement at the end + # add the measurement at the end if add_measure: - last = updated[-1] + last = updated[-1] last += r'\\[0.2cm]' last = MEASURE_PAT.sub(r'\1 \\\\[0.2cm]', last) updated[-1] = last @@ -222,19 +228,19 @@ def replace_comments(match): replacement += r'\end{flushright}' - #print("") + # print("") return replacement -#returns a list of the locations of the $ in the given string -def _get_dollar_locs(words): +# returns a list of the locations of the $ in the given string +def _get_dollar_locs(words): locs = [] start = 0 next_loc = words.find("$", start) - #keep adding to the list as long as more $ remain + # keep adding to the list as long as more $ remain while next_loc != -1: locs.append(next_loc) start = next_loc + 1 @@ -242,6 +248,7 @@ def _get_dollar_locs(words): return locs -#call main function + +# call main function if __name__ == "__main__": main() diff --git a/AlexDanoff/remove_excess.py b/AlexDanoff/remove_excess.py index 18c16b2..90eb3fd 100644 --- a/AlexDanoff/remove_excess.py +++ b/AlexDanoff/remove_excess.py @@ -22,7 +22,7 @@ def main(): if len(sys.argv) != 3: - fname = "ZE.tex" + fname = "data/ZE.tex" dot_ind = fname.index(".") + 1 ofname = fname[:dot_ind] + "1." + fname[dot_ind:] diff --git a/AlexDanoff/replace_special.py b/AlexDanoff/replace_special.py index 83e1a73..4407413 100644 --- a/AlexDanoff/replace_special.py +++ b/AlexDanoff/replace_special.py @@ -3,7 +3,7 @@ import re import sys -#for compatibility with Python 3 +# for compatibility with Python 3 try: from itertools import izip except ImportError: @@ -29,26 +29,27 @@ STD_REGEX = r'.*?###open_(\d+)###.*?###close_\1###' -def main(): +def main(): if len(sys.argv) != 3: - fname = "ZE.2.tex" - ofname = "ZE.3.tex" + fname = "data/ZE.1.tex" + ofname = "data/ZE.2.tex" else: fname = sys.argv[1] - ofname = sys.argv[2] + ofname = sys.argv[2] writeout(ofname, remove_special(readin(fname))) - + + def remove_special(content): """Removes the excess pieces from the given content and returns the updated version as a string.""" lines = content.split("\n") - - #various flags that will help us keep track of what elements we are inside of currently + + # various flags that will help us keep track of what elements we are inside of currently inside = { "constraint": [r'\constraint{', False], "substitution": [r'\substitution{', False], @@ -64,7 +65,7 @@ def remove_special(content): pi_pat = re.compile(r'(\s*)\\pi(\s*\b|[aeiou])') expe_pat = re.compile(r'\b([^\\]?\W*)\s*e\s*\^') - spaces_pat = re.compile(r' {2,}') + spaces_pat = re.compile(r' {2,}') paren_pat = re.compile(r'\(\s*(.*?)\s*\)') dollar_pat = re.compile(r'(? Date: Sat, 27 Feb 2016 18:51:15 +0100 Subject: [PATCH 030/402] reorganized the structure --- .../src}/conversion_ecf.py | 0 AlexDanoff/{ => src}/parentheses.py | 0 AlexDanoff/{ => src}/prepare_annotations.py | 0 AlexDanoff/{ => src}/remove_excess.py | 0 AlexDanoff/{ => src}/replace_special.py | 0 AlexDanoff/{ => src}/utilities.py | 0 AlexDanoff/test/test_remove_excess.py | 10 ++++++++++ 7 files changed, 10 insertions(+) rename {eCF-Seeding-Project => AlexDanoff/src}/conversion_ecf.py (100%) rename AlexDanoff/{ => src}/parentheses.py (100%) rename AlexDanoff/{ => src}/prepare_annotations.py (100%) rename AlexDanoff/{ => src}/remove_excess.py (100%) rename AlexDanoff/{ => src}/replace_special.py (100%) rename AlexDanoff/{ => src}/utilities.py (100%) create mode 100644 AlexDanoff/test/test_remove_excess.py diff --git a/eCF-Seeding-Project/conversion_ecf.py b/AlexDanoff/src/conversion_ecf.py similarity index 100% rename from eCF-Seeding-Project/conversion_ecf.py rename to AlexDanoff/src/conversion_ecf.py diff --git a/AlexDanoff/parentheses.py b/AlexDanoff/src/parentheses.py similarity index 100% rename from AlexDanoff/parentheses.py rename to AlexDanoff/src/parentheses.py diff --git a/AlexDanoff/prepare_annotations.py b/AlexDanoff/src/prepare_annotations.py similarity index 100% rename from AlexDanoff/prepare_annotations.py rename to AlexDanoff/src/prepare_annotations.py diff --git a/AlexDanoff/remove_excess.py b/AlexDanoff/src/remove_excess.py similarity index 100% rename from AlexDanoff/remove_excess.py rename to AlexDanoff/src/remove_excess.py diff --git a/AlexDanoff/replace_special.py b/AlexDanoff/src/replace_special.py similarity index 100% rename from AlexDanoff/replace_special.py rename to AlexDanoff/src/replace_special.py diff --git a/AlexDanoff/utilities.py b/AlexDanoff/src/utilities.py similarity index 100% rename from AlexDanoff/utilities.py rename to AlexDanoff/src/utilities.py diff --git a/AlexDanoff/test/test_remove_excess.py b/AlexDanoff/test/test_remove_excess.py new file mode 100644 index 0000000..a070edd --- /dev/null +++ b/AlexDanoff/test/test_remove_excess.py @@ -0,0 +1,10 @@ +from unittest import TestCase +from remove_excess import remove_section + + +class TestRemoveExcess(TestCase): + def test_remove_section(self): + content = 'AA{Begin} This should be removed {End}DDBB' + result = remove_section(r'{Begin}', r'{End}', content) + self.assertEqual('AA{End}DDBB', result) + From 06ea2c398a5f04284220ac4db199b3563f659812 Mon Sep 17 00:00:00 2001 From: Azeem Mohammed Date: Mon, 29 Feb 2016 12:14:46 +0100 Subject: [PATCH 031/402] Add Azeems programs * Add azeems contributions * Move data directory --- .gitignore | 506 ++++++------ AlexDanoff/src/prepare_annotations.py | 4 +- AlexDanoff/src/remove_excess.py | 2 +- AlexDanoff/src/replace_special.py | 4 +- Azeem/OrthogonalPolynomials.mmd | 81 ++ Azeem/Zeta.py | 7 + Azeem/new.Glossary.csv | 691 ++++++++++++++++ Azeem/tex2Wiki.py | 1069 +++++++++++++++++++++++++ 8 files changed, 2106 insertions(+), 258 deletions(-) create mode 100644 Azeem/OrthogonalPolynomials.mmd create mode 100644 Azeem/Zeta.py create mode 100644 Azeem/new.Glossary.csv create mode 100644 Azeem/tex2Wiki.py diff --git a/.gitignore b/.gitignore index c4436c3..a1bd6a8 100644 --- a/.gitignore +++ b/.gitignore @@ -1,253 +1,253 @@ -/data -Macro-Replacement/sources/ -KLSadd_insertion/chapterFilesIgnore - -### Python template -# Byte-compiled / optimized / DLL files -__pycache__/ -*.py[cod] -*$py.class - -# C extensions -*.so - -# Distribution / packaging -.Python -env/ -build/ -develop-eggs/ -dist/ -downloads/ -eggs/ -.eggs/ -lib/ -lib64/ -parts/ -sdist/ -var/ -*.egg-info/ -/KLSadd_insertion/hsc09.pdf -/KLSadd_insertion/hsc09.tex -/KLSadd_insertion/tempchap9.pdf -.installed.cfg -*.egg - -# PyInstaller -# Usually these files are written by a python script from a template -# before PyInstaller builds the exe, so as to inject date/other infos into it. -*.manifest -*.spec - -# Installer logs -pip-log.txt -pip-delete-this-directory.txt - -# Unit test / coverage reports -htmlcov/ -.tox/ -.coverage -.coverage.* -.cache -nosetests.xml -coverage.xml -*,cover - -# Translations -*.mo -*.pot - -# Django stuff: -*.log - -# Sphinx documentation -docs/_build/ - -# PyBuilder -target/ - -### JetBrains template -# Covers JetBrains IDEs: IntelliJ, RubyMine, PhpStorm, AppCode, PyCharm, CLion, Android Studio - -*.iml - -## Directory-based project format: -.idea/ -# if you remove the above rule, at least ignore the following: - -# User-specific stuff: -# .idea/workspace.xml -# .idea/tasks.xml -# .idea/dictionaries - -# Sensitive or high-churn files: -# .idea/dataSources.ids -# .idea/dataSources.xml -# .idea/sqlDataSources.xml -# .idea/dynamic.xml -# .idea/uiDesigner.xml - -# Gradle: -# .idea/gradle.xml -# .idea/libraries - -# Mongo Explorer plugin: -# .idea/mongoSettings.xml - -## File-based project format: -*.ipr -*.iws - -## Plugin-specific files: - -# IntelliJ -/out/ - -# mpeltonen/sbt-idea plugin -.idea_modules/ - -# JIRA plugin -atlassian-ide-plugin.xml - -# Crashlytics plugin (for Android Studio and IntelliJ) -com_crashlytics_export_strings.xml -crashlytics.properties -crashlytics-build.properties - -### TeX template -## Core latex/pdflatex auxiliary files: -*.aux -*.lof -*.log -*.lot -*.fls -*.out -*.toc - -## Intermediate documents: -*.dvi -*-converted-to.* -# these rules might exclude image files for figures etc. -# *.ps -# *.eps -# *.pdf - -## Bibliography auxiliary files (bibtex/biblatex/biber): -*.bbl -*.bcf -*.blg -*-blx.aux -*-blx.bib -*.brf -*.run.xml - -## Build tool auxiliary files: -*.fdb_latexmk -*.synctex -*.synctex.gz -*.synctex.gz(busy) -*.pdfsync - -## Auxiliary and intermediate files from other packages: - - -# algorithms -*.alg -*.loa - -# achemso -acs-*.bib - -# amsthm -*.thm - -# beamer -*.nav -*.snm -*.vrb - -#(e)ledmac/(e)ledpar -*.end -*.[1-9] -*.[1-9][0-9] -*.[1-9][0-9][0-9] -*.[1-9]R -*.[1-9][0-9]R -*.[1-9][0-9][0-9]R -*.eledsec[1-9] -*.eledsec[1-9]R -*.eledsec[1-9][0-9] -*.eledsec[1-9][0-9]R -*.eledsec[1-9][0-9][0-9] -*.eledsec[1-9][0-9][0-9]R - -# glossaries -*.acn -*.acr -*.glg -*.glo -*.gls - -# gnuplottex -*-gnuplottex-* - -# hyperref -*.brf - -# knitr -*-concordance.tex -*.tikz -*-tikzDictionary - -# listings -*.lol - -# makeidx -*.idx -*.ilg -*.ind -*.ist - -# minitoc -*.maf -*.mtc -*.mtc[0-9] -*.mtc[1-9][0-9] - -# minted -_minted* -*.pyg - -# morewrites -*.mw - -# mylatexformat -*.fmt - -# nomencl -*.nlo - -# sagetex -*.sagetex.sage -*.sagetex.py -*.sagetex.scmd - -# sympy -*.sout -*.sympy -sympy-plots-for-*.tex/ - -# TikZ & PGF -*.dpth -*.md5 -*.auxlock - -# todonotes -*.tdo - -# xindy -*.xdy - -# WinEdt -*.bak -*.sav - +/data +Macro-Replacement/sources/ +KLSadd_insertion/chapterFilesIgnore + +### Python template +# Byte-compiled / optimized / DLL files +__pycache__/ +*.py[cod] +*$py.class + +# C extensions +*.so + +# Distribution / packaging +.Python +env/ +build/ +develop-eggs/ +dist/ +downloads/ +eggs/ +.eggs/ +lib/ +lib64/ +parts/ +sdist/ +var/ +*.egg-info/ +/KLSadd_insertion/hsc09.pdf +/KLSadd_insertion/hsc09.tex +/KLSadd_insertion/tempchap9.pdf +.installed.cfg +*.egg + +# PyInstaller +# Usually these files are written by a python script from a template +# before PyInstaller builds the exe, so as to inject date/other infos into it. +*.manifest +*.spec + +# Installer logs +pip-log.txt +pip-delete-this-directory.txt + +# Unit test / coverage reports +htmlcov/ +.tox/ +.coverage +.coverage.* +.cache +nosetests.xml +coverage.xml +*,cover + +# Translations +*.mo +*.pot + +# Django stuff: +*.log + +# Sphinx documentation +docs/_build/ + +# PyBuilder +target/ + +### JetBrains template +# Covers JetBrains IDEs: IntelliJ, RubyMine, PhpStorm, AppCode, PyCharm, CLion, Android Studio + +*.iml + +## Directory-based project format: +.idea/ +# if you remove the above rule, at least ignore the following: + +# User-specific stuff: +# .idea/workspace.xml +# .idea/tasks.xml +# .idea/dictionaries + +# Sensitive or high-churn files: +# .idea/dataSources.ids +# .idea/dataSources.xml +# .idea/sqlDataSources.xml +# .idea/dynamic.xml +# .idea/uiDesigner.xml + +# Gradle: +# .idea/gradle.xml +# .idea/libraries + +# Mongo Explorer plugin: +# .idea/mongoSettings.xml + +## File-based project format: +*.ipr +*.iws + +## Plugin-specific files: + +# IntelliJ +/out/ + +# mpeltonen/sbt-idea plugin +.idea_modules/ + +# JIRA plugin +atlassian-ide-plugin.xml + +# Crashlytics plugin (for Android Studio and IntelliJ) +com_crashlytics_export_strings.xml +crashlytics.properties +crashlytics-build.properties + +### TeX template +## Core latex/pdflatex auxiliary files: +*.aux +*.lof +*.log +*.lot +*.fls +*.out +*.toc + +## Intermediate documents: +*.dvi +*-converted-to.* +# these rules might exclude image files for figures etc. +# *.ps +# *.eps +# *.pdf + +## Bibliography auxiliary files (bibtex/biblatex/biber): +*.bbl +*.bcf +*.blg +*-blx.aux +*-blx.bib +*.brf +*.run.xml + +## Build tool auxiliary files: +*.fdb_latexmk +*.synctex +*.synctex.gz +*.synctex.gz(busy) +*.pdfsync + +## Auxiliary and intermediate files from other packages: + + +# algorithms +*.alg +*.loa + +# achemso +acs-*.bib + +# amsthm +*.thm + +# beamer +*.nav +*.snm +*.vrb + +#(e)ledmac/(e)ledpar +*.end +*.[1-9] +*.[1-9][0-9] +*.[1-9][0-9][0-9] +*.[1-9]R +*.[1-9][0-9]R +*.[1-9][0-9][0-9]R +*.eledsec[1-9] +*.eledsec[1-9]R +*.eledsec[1-9][0-9] +*.eledsec[1-9][0-9]R +*.eledsec[1-9][0-9][0-9] +*.eledsec[1-9][0-9][0-9]R + +# glossaries +*.acn +*.acr +*.glg +*.glo +*.gls + +# gnuplottex +*-gnuplottex-* + +# hyperref +*.brf + +# knitr +*-concordance.tex +*.tikz +*-tikzDictionary + +# listings +*.lol + +# makeidx +*.idx +*.ilg +*.ind +*.ist + +# minitoc +*.maf +*.mtc +*.mtc[0-9] +*.mtc[1-9][0-9] + +# minted +_minted* +*.pyg + +# morewrites +*.mw + +# mylatexformat +*.fmt + +# nomencl +*.nlo + +# sagetex +*.sagetex.sage +*.sagetex.py +*.sagetex.scmd + +# sympy +*.sout +*.sympy +sympy-plots-for-*.tex/ + +# TikZ & PGF +*.dpth +*.md5 +*.auxlock + +# todonotes +*.tdo + +# xindy +*.xdy + +# WinEdt +*.bak +*.sav + diff --git a/AlexDanoff/src/prepare_annotations.py b/AlexDanoff/src/prepare_annotations.py index 5f22cac..4e10ab2 100644 --- a/AlexDanoff/src/prepare_annotations.py +++ b/AlexDanoff/src/prepare_annotations.py @@ -37,8 +37,8 @@ def main(): # get the names of the input and output files if len(sys.argv) != 3: - fname = "data/ZE.2.tex" - ofname = "data/ZE.3.tex" + fname = "../../data/ZE.2.tex" + ofname = "../../data/ZE.3.tex" else: diff --git a/AlexDanoff/src/remove_excess.py b/AlexDanoff/src/remove_excess.py index 90eb3fd..9bc28e9 100644 --- a/AlexDanoff/src/remove_excess.py +++ b/AlexDanoff/src/remove_excess.py @@ -22,7 +22,7 @@ def main(): if len(sys.argv) != 3: - fname = "data/ZE.tex" + fname = "../../data/ZE.tex" dot_ind = fname.index(".") + 1 ofname = fname[:dot_ind] + "1." + fname[dot_ind:] diff --git a/AlexDanoff/src/replace_special.py b/AlexDanoff/src/replace_special.py index 4407413..54ee6ae 100644 --- a/AlexDanoff/src/replace_special.py +++ b/AlexDanoff/src/replace_special.py @@ -33,8 +33,8 @@ def main(): if len(sys.argv) != 3: - fname = "data/ZE.1.tex" - ofname = "data/ZE.2.tex" + fname = "../../data/ZE.1.tex" + ofname = "../../data/ZE.2.tex" else: diff --git a/Azeem/OrthogonalPolynomials.mmd b/Azeem/OrthogonalPolynomials.mmd new file mode 100644 index 0000000..c841fb5 --- /dev/null +++ b/Azeem/OrthogonalPolynomials.mmd @@ -0,0 +1,81 @@ +drmf_bof +'''Orthogonal Polynomials''' +{{#set:Section=0}} +== Sections in KLS Chapter 1 == + +

+* [[Orthogonal polynomials|Orthogonal polynomials]] +* [[The gamma and beta function|The gamma and beta function]] +* [[The shifted factorial and binomial coefficients|The shifted factorial and binomial coefficients]] +* [[Hypergeometric functions|Hypergeometric functions]] +* [[The binomial theorem and other summation formulas|The binomial theorem and other summation formulas]] +* [[Some integrals|Some integrals]] +* [[Transformation formulas|Transformation formulas]] +* [[The q-shifted factorial|The ''q''-shifted factorial]] +* [[The q-gamma function and q-binomial coefficients|The ''q''-gamma function and ''q''-binomial coefficients]] +* [[Basic hypergeometric functions|Basic hypergeometric functions]] +* [[The q-binomial theorem and other summation formulas|The ''q''-binomial theorem and other summation formulas]] +* [[More integrals|More integrals]] +* [[Transformation formulas|Transformation formulas]] +* [[Some q-analogues of special functions|Some ''q''-analogues of special functions]] +* [[The q-derivative and q-integral|The ''q''-derivative and ''q''-integral]] +* [[Shift operators and Rodrigues-type formulas|Shift operators and Rodrigues-type formulas]] +
+== Sections in KLS Chapter 9 == + +
+* [[Wilson|Wilson]] +* [[Racah|Racah]] +* [[Continuous dual Hahn|Continuous dual Hahn]] +* [[Continuous Hahn|Continuous Hahn]] +* [[Hahn|Hahn]] +* [[Dual Hahn|Dual Hahn]] +* [[Meixner-Pollaczek|Meixner-Pollaczek]] +* [[Jacobi|Jacobi]] +* [[Jacobi: Special cases|Jacobi: Special cases]] +* [[Pseudo Jacobi|Pseudo Jacobi]] +* [[Meixner|Meixner]] +* [[Krawtchouk|Krawtchouk]] +* [[Laguerre|Laguerre]] +* [[Bessel|Bessel]] +* [[Charlier|Charlier]] +* [[Hermite|Hermite]] +
+== Sections in KLS Chapter 14 == + +
+* [[Askey-Wilson|Askey-Wilson]] +* [[q-Racah|''q''-Racah]] +* [[Continuous dual q-Hahn|Continuous dual ''q''-Hahn]] +* [[Continuous q-Hahn|Continuous ''q''-Hahn]] +* [[Big q-Jacobi|Big ''q''-Jacobi]] +* [[Big q-Jacobi: Special case|Big ''q''-Jacobi: Special case]] +* [[q-Hahn|''q''-Hahn]] +* [[Dual q-Hahn|Dual ''q''-Hahn]] +* [[Al-Salam-Chihara|Al-Salam-Chihara]] +* [[q-Meixner-Pollaczek|''q''-Meixner-Pollaczek]] +* [[Continuous q-Jacobi|Continuous ''q''-Jacobi]] +* [[Continuous q-Jacobi: Special cases|Continuous ''q''-Jacobi: Special cases]] +* [[Big q-Laguerre|Big ''q''-Laguerre]] +* [[Little q-Jacobi|Little ''q''-Jacobi]] +* [[Little q-Jacobi: Special case|Little ''q''-Jacobi: Special case]] +* [[q-Meixner|''q''-Meixner]] +* [[Quantum q-Krawtchouk|Quantum ''q''-Krawtchouk]] +* [[q-Krawtchouk|''q''-Krawtchouk]] +* [[Affine q-Krawtchouk|Affine ''q''-Krawtchouk]] +* [[Dual q-Krawtchouk|Dual ''q''-Krawtchouk]] +* [[Continuous big q-Hermite|Continuous big ''q''-Hermite]] +* [[Continuous q-Laguerre|Continuous ''q''-Laguerre]] +* [[Little q-Laguerre / Wall|Little ''q''-Laguerre / Wall]] +* [[q-Laguerre|''q''-Laguerre]] +* [[q-Bessel|''q''-Bessel]] +* [[q-Charlier|''q''-Charlier]] +* [[Al-Salam-Carlitz I|Al-Salam-Carlitz I]] +* [[Al-Salam-Carlitz II|Al-Salam-Carlitz II]] +* [[Continuous q-Hermite|Continuous ''q''-Hermite]] +* [[Stieltjes-Wigert|Stieltjes-Wigert]] +* [[Discrete q-Hermite I|Discrete ''q''-Hermite I]] +* [[Discrete q-Hermite II|Discrete ''q''-Hermite II]] +
+ +drmf_eof diff --git a/Azeem/Zeta.py b/Azeem/Zeta.py new file mode 100644 index 0000000..198ee3d --- /dev/null +++ b/Azeem/Zeta.py @@ -0,0 +1,7 @@ +import os +import time +inp="y" +if inp!="n": + print("Zeta and Related Functions") + os.system("python tex2Wiki.py ../data/ZE.3.tex ../data/ZE.4.mmd") +print("Done") diff --git a/Azeem/new.Glossary.csv b/Azeem/new.Glossary.csv new file mode 100644 index 0000000..8875e6b --- /dev/null +++ b/Azeem/new.Glossary.csv @@ -0,0 +1,691 @@ +\AGM@{a}{g},arithmetic-geometric mean,arithmetic-geometric-mean,F|F:P,$M$,http://dlmf.nist.gov/19.8#SS1.p1, +\AThetaFunction@{x},Jacobi theta function $\vartheta$,a-theta-function,F|F:P,$\vartheta$,http://dlmf.nist.gov/23.1, +\AbstractJacobi{p}{q},generic Jacobian elliptic function,generic-Jacobian-elliptic-function,F|F,$\mathrm{pq}$,http://dlmf.nist.gov/22.2#E10, +\Acos@@{z},general inverse cosine function,multivalued-inverse-cosine,F|F:P:nP,$\mathrm{Arccos}$,http://dlmf.nist.gov/4.23#E2, +\Acosh@@{z},general inverse hyperbolic cosine function,multivalued-hyperbolic-inverse-cosine,F|F:P:nP,$\mathrm{Arccosh}$,http://dlmf.nist.gov/4.37#E2, +\Acot@@{z},general inverse cotangent function,multivalued-inverse-cotangent,F|F:P:nP,$\mathrm{Arccot}$,http://dlmf.nist.gov/4.23#E6, +\Acoth@@{z},general inverse hyperbolic cotangent function,multivalued-hyperbolic-inverse-cotangent,F|F:P:nP,$\mathrm{Arccoth}$,http://dlmf.nist.gov/4.37#E6, +\Acsc@@{z},general inverse cosecant function,multivalued-inverse-cosecant,F|F:P:nP,$\mathrm{Arccsc}$,http://dlmf.nist.gov/4.23#E4, +\Acsch@@{z},general inverse hyperbolic cosecant function,multivalued-hyperbolic-inverse-cosecant,F|F:P:nP,$\mathrm{Arccsch}$,http://dlmf.nist.gov/4.37#E4, +\AffqKrawtchouk{n}@@{q^{-x}}{p}{N}{q},affine $q$-Krawtchouk polynomial,affine-q-Krawtchouk-polynomial-K,P|F:P:PS,$K^{\mathrm{Aff}}_{n}$,http://drmf.wmflabs.org/wiki/Definition:AffqKrawtchouk, +\AiryAi@{z},Airy function ${\rm Ai}$,Airy-Ai,F|F:P,$\mathrm{Ai}$,http://dlmf.nist.gov/9.2#i, +\AiryBi@{z},Airy function ${\rm Bi}$,Airy-Bi,F|F:P,$\mathrm{Bi}$,http://dlmf.nist.gov/9.2#i, +\AiryModulusM@{z},Airy modulus function $M$,modulus-Airy-M,F|F:P,$M$,http://dlmf.nist.gov/9.8#i, +\AiryModulusN@{z},Airy modulus function $N$,modulus-Airy-N,F|F:P,$N$,http://dlmf.nist.gov/9.8#i, +\AiryPhasePhi@{z},Airy phase function $\phi$,phase-Airy-Phi,F|F:P,$\phi$,http://dlmf.nist.gov/9.8#i, +\AiryPhaseTheta@{z},Airy phase function $\theta$,phase-Airy-Theta,F|F:P,$\theta$,http://dlmf.nist.gov/9.8#i, +\AlSalamCarlitzI{a}{n}@{x}{q},Al-Salam-Carlitz I polynomial,EMPTY,P|F:P,$U^{(n)}_{\alpha}$,http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzI, +\AlSalamCarlitzII{a}{n}@{x}{q},Al-Salam-Carlitz II polynomial,EMPTY,P|F:P,$V^{(n)}_{\alpha}$,http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzII, +\AlSalamChihara{n}@@{x}{a}{b}{q},Al-Salam-Chihara polynomial,Al-Salam-Chihara-polynomial-Q,P|F:P:PS,$Q_{n}$,http://drmf.wmflabs.org/wiki/Definition:AlSalamChihara, +\AlSalamChiharaQ{n}@{x}{a}{b}{q},Al-Salam-Chihara polynomial,AlSalam-Chihara-polynomial-Q,P|F:P,$Q_{n}$,http://dlmf.nist.gov/18.28#E7, +\AlSalamIsmail{n}@{x}{s}{q},Al-Salam Ismail polynomial,Al-Salam-Ismail-q-polynomial-U,P|P:F,$U_{n}$,http://drmf.wmflabs.org/wiki/Definition:AlSalamIsmail, +\AngerA{\nu}@{z},Anger-Weber function,associated-Anger-Weber-A,F|F:P,$\mathbf{A}_{\nu}$,http://dlmf.nist.gov/11.10#E4, +\AngerJ{\nu}@{z},Anger function,Anger-J,F|F:P,$\mathbf{J}_{\nu}$,http://dlmf.nist.gov/11.10#E1, +\AntiDer@{x}{f(x)},semantic antiderivative,antiderivative,SM|O,"$\int f(x){\mathrm d}\!x$",http://drmf.wmflabs.org/wiki/Definition:AntiDer, +\AppellFi@{\alpha}{\beta}{\beta'}{\gamma}{x}{y},Appell function $F_1$,Appell-F-1,F|F:P,${F_{1}}$,http://dlmf.nist.gov/16.13#E1, +\AppellFii@{\alpha}{\beta}{\beta'}{\gamma}{\gamma'}{x}{y},Appell function $F_2$,Appell-F-2,F|F:P,${F_{2}}$,http://dlmf.nist.gov/16.13#E2, +\AppellFiii@{\alpha}{\alpha'}{\beta}{\beta'}{\gamma}{x}{y},Appell function $F_3$,Appell-F-3,F|F:P,${F_{3}}$,http://dlmf.nist.gov/16.13#E3, +\AppellFiv@{\alpha}{\beta}{\gamma}{\gamma'}{x}{y},Appell function $F_4$,Appell-F-4,F|F:P,${F_{4}}$,http://dlmf.nist.gov/16.13#E4, +\Asec@@{z},general inverse secant function,multivalued-inverse-secant,F|F:P:nP,$\mathrm{Arcsec}$,http://dlmf.nist.gov/4.23#E5, +\Asech@@{z},general inverse hyperbolic secant function,multivalued-hyperbolic-inverse-secant,F|F:P:nP,$\mathrm{Arcsech}$,http://dlmf.nist.gov/4.37#E5, +\Asin@@{z},general inverse sine function,multivalued-inverse-sine,F|F:P:nP,$\mathrm{Arcsin}$,http://dlmf.nist.gov/4.23#E1, +\Asinh@@{z},general inverse hyperbolic sine function,multivalued-hyperbolic-inverse-sine,F|F:P:nP,$\mathrm{Arcsinh}$,http://dlmf.nist.gov/4.37#E1, +\AskeyWilson{n}@{x}{a}{b}{c}{d}{q},Askey-Wilson polynomial,Askey-Wilson-polynomial-p,P|F:P,$p_{n}$,http://dlmf.nist.gov/18.28#E1, +\AskeyWilsonp{n}@{x}{a}{b}{c}{d}{q},Askey-Wilson polynomial,Askey-Wilson-polynomial-p,P|F:P,$p_{n}$,http://dlmf.nist.gov/18.28#E1, +\AssJacobiP{\alpha}{\beta}{n}@{x}{c},associated Jacobi polynomial,associated-Jacobi-polynomial-P,P|F:P,"$P^{(\alpha,\alpha)}_{n}$",http://dlmf.nist.gov/18.30#E4, +\AssLegendrePoly{n}@{x}{c},associated Legendre polynomial,Legendre-spherical-polynomial-P,F|F:P,$P_{n}$,http://dlmf.nist.gov/18.30#E6, +\Atan@@{z},general inverse tangent function,multivalued-inverse-tangent,F|F:P:nP,$\mathrm{Arctan}$,http://dlmf.nist.gov/4.23#E3, +\Atanh@@{z},general inverse hyperbolic tangent function,multivalued-hyperbolic-inverse-tangent,F|F:P:nP,$\mathrm{Arctanh}$,http://dlmf.nist.gov/4.37#E3, +\BarnesGamma@{z},Barnes $G$-function (or double gamma function),Barnes-Gamma,F|F:P,$G$,http://dlmf.nist.gov/5.17#E1, +\BellNumber@{n},Bell number,Bell-number,I|F:P,$B$,http://dlmf.nist.gov/26.7#i, +\BernoulliB{n}@{x},Bernoulli polynomial,Bernoulli-polynomial-B,P|F:P,$B_{n}$,http://dlmf.nist.gov/24.2#i, +\BesselA{\nu}@{\Matrix{T}},Bessel function of matrix argument of the first kind,Bessel-of-matrix-A,F|F:P,$A_{\nu}$,http://dlmf.nist.gov/35.5#i, +\BesselB{\nu}@{\Matrix{T}},Bessel function of matrix argument of the second kind,Bessel-of-matrix-B,F|F:P,$B_{\nu}$,http://dlmf.nist.gov/35.5#E3, +\BesselI{\nu}@{z},modified Bessel function of the first kind,modified-Bessel-first-kind,F|F:P,$I_{\nu}$,http://dlmf.nist.gov/23.1, +\BesselItilde{\nu}@{x},modified Bessel function of the first kind of imaginary order,modified-Bessel-first-kind-imaginary-order,F|F:P,$\widetilde{I}_{\nu}$,http://dlmf.nist.gov/23.1, +\BesselJ{\nu}@{z},Bessel function of the first kind,Bessel-J,F|F:P,$J_{\nu}$,http://dlmf.nist.gov/10.2#E2, +\BesselJtilde{\nu}@{x},Bessel function of the first kind of imaginary order,Bessel-J-imaginary-order,F|F:P,$\widetilde{J}_{\nu}$,http://dlmf.nist.gov/10.24#Ex1, +\BesselK{\nu}@{z},modified Bessel function of the second kind,modified-Bessel-second-kind,F|F:P,$K_{\nu}$,http://dlmf.nist.gov/23.1, +\BesselKtilde{\nu}@{x},modified Bessel function of the second kind of imaginary order,modified-Bessel-second-kind-imaginary-order,F|F:P,$\widetilde{K}_{\nu}$,http://dlmf.nist.gov/10.45#E2, +\BesselModulusM{\nu}@{x},modulus of Bessel functions,modulus-Bessel-M,F|F:P,$M_{\nu}$,http://dlmf.nist.gov/10.18#E1, +\BesselModulusN{\nu}@{x},modulus of derivatives of Bessel functions,modulus-Bessel-N,F|F:P,$N_{\nu}$,http://dlmf.nist.gov/10.18#E2, +\BesselPhasePhi{\nu}@{x},phase of derivatives of Bessel functions,phase-Bessel-Phi,F|F:P,$\phi_{\nu}$,http://dlmf.nist.gov/10.18#E3, +\BesselPhaseTheta{\nu}@{x},phase of Bessel functions,phase-Bessel-Theta,F|F:P,$\theta_{\nu}$,http://dlmf.nist.gov/10.18#E3, +\BesselPoly{n}@{x}{a},Bessel polynomial,Bessel-polynomial-y,P|F:P,$y_{n}$,http://dlmf.nist.gov/18.34#E1, +\BesselPolyy{n}@{x}{a},Bessel polynomial,Bessel-polynomial-y,P|F:P,$y_{n}$,http://dlmf.nist.gov/18.34#E1, +\BesselY{\nu}@{z},Bessel function of the second kind,Bessel-Y-Weber,F|F:P,$Y_{\nu}$,http://dlmf.nist.gov/10.2#E3, +\BesselYtilde{\nu}@{x},Bessel function of the second kind of imaginary order,Bessel-Y-Weber-imaginary-order,F|F:P,$\widetilde{Y}_{\nu}$,http://dlmf.nist.gov/10.24#Ex1, +\BickleyKi{\alpha}@{x},Bickley function,Bickley-Ki,F|F:P,$\mathrm{Ki}_{\alpha}$,http://dlmf.nist.gov/10.43#E11, +\BigO@{x},order not exceeding,Big-O,Q|F:P,$O$,http://dlmf.nist.gov/2.1#E3, +\CanonicInt{K}@{\Vector{x}},canonical integral $\Psi_K$,canonical-integral,F|F:P,$\Psi_{K}$,http://dlmf.nist.gov/36.2#E4, +\CanonicIntU@{\Vector{x}},umbilic canonical integral $\Psi^{({\rm U})}$,canonical-umbilic-integral,F|F:P,$\Psi^{(\mathrm{U})}$,http://dlmf.nist.gov/36.2#E5, +\CanonicIntE@{\Vector{x}},elliptic umbilic canonical integral $\Psi^{({\rm E})}$,elliptic-umbilic-canonical-integral,F|F:P,$\Psi^{(\mathrm{E})}$,http://dlmf.nist.gov/36.2#E6, +\CanonicIntH@{\Vector{x}},hyperbolic umbilic canonical integral $\Psi^{({\rm H})}$,hyperbolic-umbilic-canonical-integral,F|F:P,$\Psi^{(\mathrm{H})}$,http://dlmf.nist.gov/36.2#E8, +\CatalanNumber@{n},Catalan number,Catalan-number,I|F:P,$C$,http://dlmf.nist.gov/26.5#E1, +\CatalansConstant,Catalan's constant,Catalan's-constant,C|F,$G$,http://dlmf.nist.gov/25.11.E40, +\Charlier{n}@{x}{a},Charlier polynomial,Charlier-polynomial-C,P|F:P,$C_{n}$,http://dlmf.nist.gov/18.19#T1.t1.r11, +\CharlierC{n}@{x}{a},Charlier polynomial,Charlier-polynomial-C,P|F:P,$C_{n}$,http://dlmf.nist.gov/18.19#T1.t1.r11, +\ChebyC{n}@{x},dilated Chebyshev polynomial of the first kind,dilated-Chebyshev-polynomial-second-kind-C,P|F:P,$C_{n}$,http://dlmf.nist.gov/18.1#E3, +\ChebyS{n}@{x},dilated Chebyshev polynomial of the second kind,dilated-Chebyshev-polynomial-first-kind-S,P|F:P,$S_{n}$,http://dlmf.nist.gov/18.1#E3, +\ChebyT{n}@{x},Chebyshev polynomial of the first kind,Chebyshev-polynomial-first-kind-T,P|F:P,$T_{n}$,http://dlmf.nist.gov/18.3#T1.t1.r8, +\ChebyTs{n}@{x},shifted Chebyshev polynomial of the first kind,shifted-Chebyshev-polynomial-first-kind-T-star,P|F:P,$T^{*}_{n}$,http://dlmf.nist.gov/18.3#T1.t1.r20, +\ChebyU{n}@{x},Chebyshev polynomial of the second kind,Chebyshev-polynomial-second-kind-U,P|F:P,$U_{n}$,http://dlmf.nist.gov/18.3#T1.t1.r11, +\ChebyUs{n}@{x},shifted Chebyshev polynomial of the second kind,shifted-Chebyshev-polynomial-second-kind-U-star,P|F:P,$U^{*}_{n}$,http://dlmf.nist.gov/18.3#T1.t1.r23, +\ChebyV{n}@{x},Chebyshev polynomial of the third kind,Chebyshev-polynomial-third-kind-V,P|F:P,$V_{n}$,http://dlmf.nist.gov/18.3#T1.t1.r14, +\ChebyW{n}@{x},Chebyshev polynomial of the fourth kind,Chebyshev-polynomial-fourth-kind-W,P|F:P,$W_{n}$,http://dlmf.nist.gov/18.3#T1.t1.r17, +\ChebyshevPsi@{x},Chebyshev $\ChebyshevPsi$-function,Chebyshev-psi,F|F:P,$\psi$,http://dlmf.nist.gov/25.16#E1, +\CiglerqChebyT{n}@{x}{s}{q},Cigler $q$-Chebyshev polynomial $T$,Cigler-q-Chebyshev-polynomial-T,P|P:F,$T_{n}$,http://drmf.wmflabs.org/wiki/Definition:CiglerqChebyT, +\CiglerqChebyU{n}@{x}{s}{q},Cigler $q$-Chebyshev polynomial $U$,Cigler-q-Chebyshev-polynomial-U,P|P:F,$U_{n}$,http://drmf.wmflabs.org/wiki/Definition:CiglerqChebyU, +\CompEllIntCE@@{k},Legendre's complementary complete elliptic integral of the second kind,complementary-complete-elliptic-integral-second-kind-E,F|F:P:nP,${E'}$,http://dlmf.nist.gov/19.2#E9, +\CompEllIntCK@@{k},Legendre's complementary complete elliptic integral of the first kind,complementary-complete-elliptic-integral-first-kind-K,F|F:P:nP,${K'}$,http://dlmf.nist.gov/19.2#E9, +\CompEllIntD@@{k},complete elliptic integral of Legendre's type,complete-elliptic-integral-D,F|F:P:nP,$D$,http://dlmf.nist.gov/19.2#E8, +\CompEllIntE@@{k},Legendre's complete elliptic integral of the second kind,complete-elliptic-integral-second-kind-E,F|F:P:nP,$E$,http://dlmf.nist.gov/19.2#E8, +\CompEllIntK@@{k},Legendre's complete elliptic integral of the first kind,complete-elliptic-integral-first-kind-K,F|F:P:nP,$K$,http://dlmf.nist.gov/19.2#E8, +\CompEllIntPi@{\alpha^2}{k},Legendre's complete elliptic integral of the third kind,complete-elliptic-integral-third-kind-Pi,F|F:P,$\Pi$,http://dlmf.nist.gov/19.2#E8, +\Complex,set of complex numbers,EMPTY,SN|F,$\mathbb{C}$,http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r1, +\ComplexZeroAiryBi{k},$k$\textsuperscript{th} complex zero of Airy $\AiryBi$,complex-zero-Airy-Bi,F|F,$\beta_{k}$,http://dlmf.nist.gov/9.9#SS1.p2, +\ComplexZeroAiryBiPrime{k},$k$\textsuperscript{th} complex zero of Airy $\AiryBi'$,complex-zero-derivative-Airy-Bi,F|F,$\beta'_{k}$,http://dlmf.nist.gov/9.9#SS1.p2, +\CompositionsC[m]@{n},number of compositions,number-of-compositions-C,I|F:F:P:P,$c_{m}$,http://dlmf.nist.gov/26.11#p1, +\CosInt@{z},cosine integral ${\rm Ci}$,cosine-integral,F|F:P,$\mathrm{Ci}$,http://dlmf.nist.gov/6.2#E11, +\CosIntCin@{z},cosine integral ${\rm Cin}$,cosine-integral-Cin,F|F:P,$\mathrm{Cin}$,http://dlmf.nist.gov/6.2#E12, +\CosIntg@{a}{z},generalized cosine integral ${\rm Ci}$,generalized-cosine-integral-Ci,F|F:P,$\mathrm{Ci}$,http://dlmf.nist.gov/8.21#E2, +\CoshInt@{z},hyperbolic cosine integral,hyperbolic-cosine-integral,F|F:P,$\mathrm{Chi}$,http://dlmf.nist.gov/6.2#E16, +\CoulombC{\ell}@{\eta},normalizing constant for Coulomb radial functions,Coulomb-C,F|F:P,$C_{\ell}$,http://dlmf.nist.gov/33.2#E5, +\CoulombF{\ell}@{\eta}{\rho},regular Coulomb radial function,regular-Coulomb-F,F|F:P,$F_{\ell}$,http://dlmf.nist.gov/33.2#E3, +\CoulombG{\ell}@{\eta}{\rho},irregular Coulomb radial function,irregular-Coulomb-G,F|F:P,$G_{\ell}$,http://dlmf.nist.gov/33.2#E11, +\CoulombH{\pm}{\ell}@{\eta}{\rho},irregular Coulomb radial functions,irregular-Coulomb-H,F|F:P,${H^{\pm}_{\ell}}$,http://dlmf.nist.gov/33.2#E7, +\CoulombM{\ell}@{\eta}{\rho},envelope of Coulomb functions,envelope-Coulomb-M,F|F:P,$M_{\ell}$,http://dlmf.nist.gov/33.3#E1, +\CoulombSigma{\ell}@{\eta},Coulomb phase shift,Coulomb-phase-shift,F|F:P,${\sigma_{\ell}}$,http://dlmf.nist.gov/33.2#E10, +\CoulombTheta{\ell}@{\eta}{\rho},phase of Coulomb functions,Coulomb-phase,F|F:P,${\theta_{\ell}}$,http://dlmf.nist.gov/33.2#E9, +\Coulombc@{\epsilon}{\ell}{r},irregular Coulomb function $c$,Coulomb-c,F|F:P,$c$,http://dlmf.nist.gov/23.1, +\Coulombf@{\epsilon}{\ell}{r},regular Coulomb function $f$,Coulomb-f,F|F:P,$f$,http://dlmf.nist.gov/33.14#E4, +\Coulombh@{\epsilon}{\ell}{r},irregular Coulomb function $h$,Coulomb-h,F|F:P,$h$,http://dlmf.nist.gov/33.14#E7, +\Coulombrhotp@{\eta}{\ell},outer turning point for Coulomb radial functions,Coulomb-radial-outer-turning-point,F|F:P,$\rho_{\mathrm{tp}}$,http://dlmf.nist.gov/33.2#E2, +\Coulombrtp@{\epsilon}{\ell},outer turning point for Coulomb functions,Coulomb-outer-turning-point,F|F:P,$r_{\mathrm{tp}}$,http://dlmf.nist.gov/33.14#E3, +\Coulombs@{\epsilon}{\ell}{r},regular Coulomb function $s$,Coulomb-s,F|F:P,$s$,http://dlmf.nist.gov/33.14#E9, +\CuspCat{K}@{t}{\Vector{x}},cuspoid catastrophe,cuspoid-catastrophe,F|F:P,$\Phi_{K}$,http://dlmf.nist.gov/36.2#E1, +\Cylinder{\nu}@{z},cylinder function,cylinder-function,F|F:P,$\mathscr{C}_{\nu}$,http://dlmf.nist.gov/10.2#P5.p1, +\DawsonsInt@{z},Dawson's integral,Dawsons-integral,F|F:P,$F$,http://dlmf.nist.gov/7.2#E5, +\DedekindModularEta@{\tau},Dedekind's eta function (or Dedekind modular function),Dedekind-modular-Eta,F|F:P,$\eta$,http://dlmf.nist.gov/23.15#E9,http://dlmf.nist.gov/27.14#E12 +\DiffCat{K}@{\Vector{x}}{k},diffraction catastrophe,diffraction-catastrophe,F|F:P,$\Psi_{K}$,http://dlmf.nist.gov/36.2#E10, +\DiffCatE@{\Vector{x}}{k},elliptic umbilic canonical integral function,diffraction-elliptic-umbilic-catastrophe,F|F:P,$\Psi^{(\mathrm{E})}$,http://dlmf.nist.gov/23.1, +\DiffCatH@{\Vector{x}}{k},hyperbolic umbilic canonical integral function,diffraction-hyperbolic-umbilic-catastrophe,F|F:P,$\Psi^{(\mathrm{H})}$,http://dlmf.nist.gov/36.2#E11, +\DiffCatU@{\Vector{x}}{k},umbilic canonical integral function,diffraction-umbilic-catastrophe,F|F:P,$\Psi^{(\mathrm{U})}$,http://dlmf.nist.gov/36.2#E11, +\Dilogarithm@{z},dilogarithm,dilogarithm,F|F:P,$\mathrm{Li}_2$,http://dlmf.nist.gov/25.12#E1, +\Diracdelta@{x-a},Dirac delta (or Dirac delta function),Dirac-delta,D|F:P,$\Diracdelta@{x-a}$,http://dlmf.nist.gov/1.17#SS1.p1, +\DirichletCharacter@@{n}{k},Dirichlet character,Dirichlet-character,I|F:P:nP,$\chi$,http://dlmf.nist.gov/27.8, +\DirichletL@{s}{\chi},Dirichlet $\DirichletL$-function,Dirichlet-L,F|F:P,$L$,http://dlmf.nist.gov/25.15#E1, +\DiscriminantDelta@{\tau},discriminant function,discriminant-delta,F|F:P,$\Delta$,http://dlmf.nist.gov/27.14#E16, +\DiskOP{\alpha}{m}{n}@{z},disk polynomial,disk-orthogonal-polynomial-R,P|F:P,"$R^{(\alpha)}_{m,n}$",http://dlmf.nist.gov/18.37#E1, +\Distribution{f}{g},distribution,EMPTY,D|F,"$\left\langle f,g\right\rangle$",http://dlmf.nist.gov/1.16#SS1.p5, +\DivisorFunctionD[k]@{n},divisor function,divisor-function-D,F|F:P,$d_{k}$,http://dlmf.nist.gov/27.2#E9, +\DivisorSigma{\alpha}@{n},sum of powers of divisors,divisor-sigma,I|F:P,$\sigma_{\alpha}$,http://dlmf.nist.gov/27.2#E10, +\EllIntD@{\phi}{k},incomplete elliptic integral of Legendre's type,elliptic-integral-third-kind-D,F|F:P,$D$,http://dlmf.nist.gov/19.2#E6, +\EllIntE@{\phi}{k},Legendre's incomplete elliptic integral of the second kind,elliptic-integral-second-kind-E,F|F:P,$E$,http://dlmf.nist.gov/19.2#E5, +\EllIntF@{\phi}{k},Legendre's incomplete elliptic integral of the first kind,elliptic-integral-first-kind-F,F|F:P,$F$,http://dlmf.nist.gov/19.2#E4, +\EllIntPi@{\phi}{\alpha^2}{k},Legendre's incomplete elliptic integral of the third kind,elliptic-integral-third-kind-Pi,F|F:P,$\Pi$,http://dlmf.nist.gov/19.2#E7, +\EllIntR{-a}@{\Vector{b}}{\Vector{z}},Carlson integral $R_{-a}$,Carlson-integral-R,F|F:P,$R_{-a}$,http://dlmf.nist.gov/19.16#E9, +\EllIntRC@@{x}{y},Carlson's elliptic integral $R_C$,Carlson-integral-RC,F|F:P:nP,$R_C$,http://dlmf.nist.gov/19.2#E17, +\EllIntRD@@{x}{y}{z},Carlson's elliptic integral $R_D$,Carlson-integral-RD,F|F:P:nP,$R_D$,http://dlmf.nist.gov/19.16#E5, +\EllIntRF@@{x}{y}{z},Carlson's symmetric elliptic integral of first kind,Carlson-integral-RF,F|F:P:nP,$R_F$,http://dlmf.nist.gov/19.16#E1, +\EllIntRG@@{x}{y}{z},Carlson's symmetric elliptic integral of second kind,Carlson-integral-RG,F|F:P:nP,$R_G$,http://dlmf.nist.gov/19.16#E3, +\EllIntRJ@@{x}{y}{z}{p},Carlson's symmetric elliptic integral of third kind,Carlson-integral-RJ,F|F:P:nP,$R_J$,http://dlmf.nist.gov/19.16#E2, +\EllIntcel@{k_c}{p}{a}{b},Bulirsch's complete elliptic integral,Bulirsch-integral-cel,F|F:P,$\mathrm{cel}$,http://dlmf.nist.gov/19.2#E11, +\EllIntelone@{x}{k_c},Bulirsch's incomplete elliptic integral of the first kind,Bulirsch-integral-el-1,F|F:P,$\mathrm{el1}$,http://dlmf.nist.gov/19.2#E15, +\EllIntelthree@{x}{k_c}{p},Bulirsch's incomplete elliptic integral of the third kind,Bulirsch-integral-el-3,F|F:P,$\mathrm{el3}$,http://dlmf.nist.gov/19.2#E16, +\EllInteltwo@{x}{k_c}{a}{b},Bulirsch's incomplete elliptic integral of the second kind,Bulirsch-integral-el-2,F|F:P,$\mathrm{el2}$,http://dlmf.nist.gov/19.2#E12, +\EulerBeta@{a}{b},beta function,Euler-Beta,F|F:P,$\mathrm{B}$,http://dlmf.nist.gov/5.12#E1, +\EulerConstant,Euler's (Euler-Mascheroni) constant,EMPTY,C|F,$\gamma$,http://dlmf.nist.gov/5.2#E3, +\EulerE{n}@{x},Euler polynomial,Euler-polynomial-E,P|F:P,$E_{n}$,http://dlmf.nist.gov/24.2#ii, +\EulerGamma@{z},Euler's gamma function,Euler-Gamma,F|F:P,$\Gamma$,http://dlmf.nist.gov/5.2#E1, +\EulerPhi@{x},Euler's reciprocal function,Euler-phi,F|F:P,$\mathit{f}$,http://dlmf.nist.gov/27.14#E2, +\EulerSumH@{s},Euler sums,Euler-sum-H,F|F:P,$H$,http://dlmf.nist.gov/25.16#SS2.p1, +\EulerTotientPhi[k]@{n},Euler's totient $\phi_k(n)$,Euler-totient-phi,I|F:P,$\phi_{k}$,http://dlmf.nist.gov/27.2#E6, +\EulerianNumber{n}{k},Eulerian number,Eulerian-number,I|F,$\left\langle n\atop k\right\rangle$,http://dlmf.nist.gov/26.14#SS1.p3, +\ExpInt@{z},exponential integral $E_1$,exponential-integral,F|F:P,$E_1$,http://dlmf.nist.gov/6.2#E1, +\ExpIntEin@{z},complementary exponential integral ${\rm Ein}$,complementary-exponential-integral,F|F:P,$\mathrm{Ein}$,http://dlmf.nist.gov/6.2#E3, +\ExpInti@{x},exponential integral ${\rm Ei}$,exponential-integral-Ei,F|F:P,$\mathrm{Ei}$,http://dlmf.nist.gov/6.2#E1, +\ExpIntn{p}@{z},generalized exponential integral,exponential-integral-En,F|F:P,$E_{p}$,http://dlmf.nist.gov/8.19#E1, +\FerrersHatQ[-\mu]{-\frac{1}{2}+i\tau}@{x},conical function $\widehat{\mathsf{Q}}_\nu^\mu$,Ferrers-conical-legendre-Q-hat,F|FO:PO:FnO:PnO,$\widehat{\mathsf{Q}}_\nu^\mu$,http://dlmf.nist.gov/14.20#E2, +\FerrersP[\mu]{\nu}@{x},Ferrers function of the first kind,Ferrers-Legendre-P-first-kind,F|FO:PO:FnO:PnO,$\mathsf{P}_\nu^\mu$,http://dlmf.nist.gov/14.3#E1, +\FerrersQ[\mu]{\nu}@{x},Ferrers function of the second kind,Ferrers-Legendre-Q-first-kind,F|FO:PO:FnO:PnO,$\mathsf{Q}_\nu^\mu$,http://dlmf.nist.gov/14.3#E2, +\FishersHh{n}@{z},probability function,Fishers-Hh,F|F:P,$\mathit{Hh}_{n}$,http://dlmf.nist.gov/7.18#E12, +\FresnelCos@{z},Fresnel integral $C$,Fresnel-cosine-integral,F|F:P,$C$,http://dlmf.nist.gov/7.2#E7, +\FresnelF@{z},Fresnel integral ${\mathcal F}$,Fresnel-integral,F|F:P,$\mathcal{F}$,http://dlmf.nist.gov/7.2#E6, +\FresnelSin@{z},Fresnel integral $S$,Fresnel-sine-integral,F|F:P,$S$,http://dlmf.nist.gov/7.2#E8, +\Fresnelf@{z},auxiliary function for Fresnel integrals ${\rm f}$,Fresnel-auxilliary-function-f,F|F:P,$\mathrm{f}$,http://dlmf.nist.gov/7.2#E10, +\Fresnelg@{z},auxiliary function for Fresnel integrals ${\rm g}$,Fresnel-auxilliary-function-g,F|F:P,$\mathrm{g}$,http://dlmf.nist.gov/7.2#E11, +\GammaP@{a}{z},normalized incomplete gamma function $P$,incomplete-gamma-P,F|F:P,$P$,http://dlmf.nist.gov/8.2#E4, +\GammaQ@{a}{z},normalized incomplete gamma function $Q$,incomplete-gamma-Q,F|F:P,$Q$,http://dlmf.nist.gov/8.2#E4, +\GaussSum@{n}{\DirichletCharacter},Gauss sum,Gauss-sum,F|F:P,$G$,http://dlmf.nist.gov/27.10#E9, +\GenBernoulliB{\ell}{n}@{x},generalized Bernoulli polynomial,generalized-Bernoulli-polynomial-B,P|F:P,$B^{(\ell)}_{n}$,http://dlmf.nist.gov/24.16#P1.p1, +\GenBesselFun@{\rho}{\beta}{z},generalized Bessel function,generalized-Bessel-function,F|F:P,$\phi$,http://dlmf.nist.gov/10.46#E1, +\GenEulerE{\ell}{n}@{x},generalized Euler polynomial,generalized-Euler-polynomial-E,P|F:P,$E^{(\ell)}_{n}$,http://dlmf.nist.gov/24.16#P1.p1, +\GenEulerSumH@{s}{z},generalized Euler sums,generalized-Euler-sum-H,F|F:P,$H$,http://dlmf.nist.gov/25.16#SS2.p7, +\GenGegenbauer{\alpha}{\beta}{n}@{x},Generalized Gegenbauer polynomial,generalized-Gegenbauer-polynomial-S,P|F:P,"$S^{(\alpha,\beta)}_{n}$",http://drmf.wmflabs.org/wiki/Definition:GenGegenbauer, +\GenHermite[\alpha]{n}@{x},generalized Hermite polynomial,generalized-Hermite-polynomial-H,P|FO:PO:FnO:PnO,$H_n^{\mu}$,http://drmf.wmflabs.org/wiki/Definition:GenHermite, +\GoodStat@{z},Goodwin-Staton integral,Goodwin-Staton-integral,F|F:P,$G$,http://dlmf.nist.gov/7.2#E12, +\GottliebLaguerre{n}@{x},Gottlieb-Laguerre polynomial,Gottlieb-Laguerre-polynomial-l,P|F:P,$l_{n}$,http://drmf.wmflabs.org/wiki/Definition:GottliebLaguerre, +\Gudermannian@@{x},Gudermannian function,inverse-Gudermannian,F|F:P:nP,$\mathrm{gd}$,http://dlmf.nist.gov/4.23#E39, +\Hahn{n}@{x}{\alpha}{\beta}{N},Hahn polynomial,Hahn-polynomial-Q,P|F:P,$Q_{n}$,http://dlmf.nist.gov/18.19#T1.t1.r3, +\HahnQ{n}@{x}{\alpha}{\beta}{N},Hahn polynomial,Hahn-polynomial-Q,P|F:P,$Q_{n}$,http://dlmf.nist.gov/18.19#T1.t1.r3, +\HahnR{n}@{x}{\gamma}{\delta}{N},dual Hahn polynomial,dual-Hahn-R,P|F:P,$R_{n}$,http://dlmf.nist.gov/18.25#T1.t1.r5, +\HahnS{n}@{x}{a}{b}{c},continuous dual Hahn polynomial,continuous-dual-Hahn-S,P|F:P,$S_{n}$,http://dlmf.nist.gov/18.25#T1.t1.r3, +\Hahnp{n}@{x}{a}{b}{\conj{a}}{\conj{b}},continuous Hahn polynomial,continuous-Hahn-polynomial-p,P|F:P,$p_{n}$,http://dlmf.nist.gov/18.19#P2.p1, +\HankelHi{\nu}@{z},Hankel function of the first kind,Hankel-H-1-Bessel-third-kind,F|F:P,${H^{(1)}_{\nu}}$,http://dlmf.nist.gov/10.2#E5, +\HankelHii{\nu}@{z},Hankel function of the second kind,Hankel-H-2-Bessel-third-kind,F|F:P,${H^{(2)}_{\nu}}$,http://dlmf.nist.gov/10.2#E6, +\HarmonicNumber{n},Harmonic number,Harmonic-number,I|F,$H_{n}$,http://dlmf.nist.gov/25.11#E33, +\HeavisideH@{x},Heaviside step function,Heaviside-H,F|F:P,$H$,http://dlmf.nist.gov/1.16#E13, +\Hermite{n}@{x},Hermite polynomial $H_n$,Hermite-polynomial-H,P|F:P,$H_{n}$,http://dlmf.nist.gov/18.3#T1.t1.r28, +\HermiteH{n}@{x},Hermite polynomial $H_n$,Hermite-polynomial-H,P|F:P,$H_{n}$,http://dlmf.nist.gov/18.3#T1.t1.r28, +\HermiteHe{n}@{x},Hermite polynomial $He_n$,Hermite-polynomial-He,P|F:P,$\mathit{He}_{n}$,http://dlmf.nist.gov/18.3#T1.t1.r29, +\HeunFunction{m}{a}{b}@@{a}{q}{\alpha}{\beta}{\gamma}{\delta}{z},Heun functions $(a;b)Hf_m^\nu$,Heun-function,F|FO:PO:nPO:FnO:PnO:nPnO,"$(s_1,s_2)\mathit{Hf}_{m}^\nu$",http://dlmf.nist.gov/31.4#p1, +\HeunLocal@@{a}{q}{\alpha}{\beta}{\gamma}{\delta}{z},Heun functions $Hl$,Heun-local,F|F:P:nP,$\mathit{H\!\ell}$,http://dlmf.nist.gov/31.3#E1, +\HeunPolynom{n}{m}@@{a}{q}{-n}{\beta}{\gamma}{\delta}{z},Heun polynomial,Heun-polynomial-m,P|F:P:nP,"$\mathit{Hp}_{n,m}$",http://dlmf.nist.gov/31.5#E2, +\HilbertTrans@{f}{x},Hilbert transform,Hilbert-transform,O|F:P,$\mathcal{H}$,http://dlmf.nist.gov/1.14#SS5.p1, +\HurwitzZeta@{s}{a},Hurwitz zeta function,Hurwitz-zeta,F|F:P,$\zeta$,http://dlmf.nist.gov/25.11#E1, +\HyperPsi@{a}{b}{\Matrix{T}},confluent hypergeometric function of matrix argument of the second kind,confluent-hypergeometric-of-matrix-Psi,F|F:P,$\Psi$,http://dlmf.nist.gov/35.6#E2, +\HyperboldpFq{p}{q}@@@{\Vector{a}}{\Vector{b}}{z},scaled (or Olver's) generalized hypergeometric function,hypergeometric-bold-pFq,F|F:fo1:fo2:fo3,${{}_{p}{\mathbf{F}}_{q}}$,http://dlmf.nist.gov/16.2#E5, +\HypergeoF@@@{a}{b}{c}{z},hypergeometric function,Gauss-hypergeometric-F,F|F:fo1:fo2:fo3,$F$,http://dlmf.nist.gov/15.2#E1, +\HypergeoboldF@@@{a}{b}{c}{z},Olver's hypergeometric function,scaled-hypergeometric-bold-F,F|F:fo1:fo2:fo3,$\mathbf{F}$,http://dlmf.nist.gov/15.2#E2, +"\HyperpFq{p}{q}@@@{a_1,...,a_p}{b_1,...,b_q}{z}",generalized hypergeometric function,Gauss-hypergeometric-pFq,F|F:fo1:fo2:fo3,${{}_{p}F_{q}}$,http://dlmf.nist.gov/16.2#E1, +"\HyperpHq{p}{q}@@@{{a_1,...,a_p}{b_1,...,b_q}{z}",bilateral hypergeometric function,hypergeometric-pHq,F|F:fo1:fo2:fo3,${{}_{p}H_{q}}$,http://dlmf.nist.gov/16.4#E16, +\IncBeta{x}@{a}{b},incomplete beta function,incomplete-Beta,F|F:P,$\mathrm{B}_{x}$,http://dlmf.nist.gov/8.17#E1, +\IncGamma@{a}{z},incomplete gamma function $\Gamma$,incomplete-Gamma,F|F:P,$\Gamma$,http://dlmf.nist.gov/8.2#E2, +\IncI{x}@{a}{b},regularized incomplete beta function,IncI,F|F:P,$I_{x}$,http://dlmf.nist.gov/8.17#E2, +\Int{a}{b}@{x}{f(x)},semantic definite integral,semantic-definite-integral,SM|O,"$\int_{a}^{b}f(x){\mathrm d}\!x$",http://drmf.wmflabs.org/wiki/Definition:Int, +\Integer,set of integers,EMPTY,SN|F,$\mathbb{Z}$,http://dlmf.nist.gov/front/introduction#Sx4.p2.t1.r20, +\IntgenAiryA{k}@{z}{p},generalized Airy function from integral representation $A_k$,Integral-generalized-Airy-A,F|F:P,$A_{k}$,http://dlmf.nist.gov/9.13#SS2.p1, +\IntgenAiryB{k}@{z}{p},generalized Airy function from integral representation $B_k$,Integral-generalized-Airy-B,F|F:P,$B_{k}$,http://dlmf.nist.gov/9.13#SS2.p1, +\JacksonqBesselI{\nu}@{z}{q},Jackson $q$-Bessel function 1,EMPTY,F|F:P,$J^{(1)}_{q}$,http://drmf.wmflabs.org/wiki/Definition:JacksonqBesselI, +\JacksonqBesselII{\nu}@{z}{q},Jackson $q$-Bessel function 2,EMPTY,F|F:P,$J^{(2)}_{q}$,http://drmf.wmflabs.org/wiki/Definition:JacksonqBesselII, +\JacksonqBesselIII{\nu}@{z}{q},Jackson/Hahn-Exton $q$-Bessel function 3,EMPTY,F|F:P,$J^{(3)}_{q}$,http://drmf.wmflabs.org/wiki/Definition:JacksonqBesselIII, +\Jacobi{\alpha}{\beta}{n}@{x},Jacobi polynomial,Jacobi-polynomial-P,P|F:P,"$P^{(\alpha,\beta)}_{n}$",http://dlmf.nist.gov/18.3#T1.t1.r3, +\JacobiEpsilon@{x}{k},Jacobi's epsilon function,Jacobi-Epsilon,F|F:P,$\mathcal{E}$,http://dlmf.nist.gov/22.16#E14, +\JacobiG{n}@{p}{q}{x},shifted Jacobi polynomial,shifted-Jacobi-polynomial-G,P|F:P,$G_{n}$,http://dlmf.nist.gov/18.1#E2, +\JacobiP{\alpha}{\beta}{n}@{x},Jacobi polynomial,Jacobi-polynomial-P,P|F:P,"$P^{(\alpha,\beta)}_{n}$",http://dlmf.nist.gov/18.3#T1.t1.r3, +\JacobiSymbol{n}{P},Jacobi symbol,Jacobi-symbol,I|F,$(n|P)$,http://dlmf.nist.gov/27.9#p3, +\JacobiTheta{j}@{z}{q},Jacobi theta function,Jacobi-theta,F|F:P,$\theta_{p}$,http://dlmf.nist.gov/20.2#i, +\JacobiThetaCombined{n}{m}@{z}{q},combined theta function,Jacobi-theta-combined,F|F:P,"$\varphi_{m,n}$",http://dlmf.nist.gov/20.11#SS5.p2, +\JacobiThetaTau{j}@{z}{\tau},Jacobi theta $\tau$ function $\theta_j$,Jacobi-theta-tau,F|F:P,$\theta_{p}$,http://dlmf.nist.gov/20.2#i, +\JacobiZeta@{x}{k},Jacobi's zeta function,Jacobi-Zeta,F|F:P,$\mathrm{Z}$,http://dlmf.nist.gov/22.16#E32, +\Jacobiam@@{x}{k},Jacobi's amplitude function ${\rm am}$,Jacobi-elliptic-amplitude,F|F:P:nP,$\mathrm{am}$,http://dlmf.nist.gov/23.1, +\Jacobicd@@{z}{k},Jacobian elliptic function ${\rm cd}$,Jacobi-elliptic-cd,F|F:P:nP,$\mathrm{cd}$,http://dlmf.nist.gov/22.2#E8, +\Jacobicn@@{z}{k},Jacobian elliptic function ${\rm cn}$,Jacobi-elliptic-cn,F|F:P:nP,$\mathrm{cn}$,http://dlmf.nist.gov/22.2#E5, +\Jacobics@@{z}{k},Jacobian elliptic function ${\rm cs}$,Jacobi-elliptic-cs,F|F:P:nP,$\mathrm{cs}$,http://dlmf.nist.gov/23.1, +\Jacobidc@@{z}{k},Jacobian elliptic function ${\rm dc}$,Jacobi-elliptic-dc,F|F:P:nP,$\mathrm{dc}$,http://dlmf.nist.gov/23.1, +\Jacobidn@@{z}{k},Jacobian elliptic function ${\rm dn}$,Jacobi-elliptic-dn,F|F:P:nP,$\mathrm{dn}$,http://dlmf.nist.gov/22.2#E6, +\Jacobids@@{z}{k},Jacobian elliptic function ${\rm ds}$,Jacobi-elliptic-ds,F|F:P:nP,$\mathrm{ds}$,http://dlmf.nist.gov/23.1, +\Jacobinc@@{z}{k},Jacobian elliptic function ${\rm nc}$,Jacobi-elliptic-nc,F|F:P:nP,$\mathrm{nc}$,http://dlmf.nist.gov/23.1, +\Jacobind@@{z}{k},Jacobian elliptic function ${\rm nd}$,Jacobi-elliptic-nd,F|F:P:nP,$\mathrm{nd}$,http://dlmf.nist.gov/23.1, +\Jacobins@@{z}{k},Jacobian elliptic function ${\rm ns}$,Jacobi-elliptic-ns,F|F:P:nP,$\mathrm{ns}$,http://dlmf.nist.gov/23.1, +\Jacobiphi{\alpha}{\beta}{\lambda}@{t},Jacobi function,Jacobi-hypergeometric-phi,F|F:P,"$\phi^{(\alpha,\beta)}_{\lambda}$",http://dlmf.nist.gov/15.9#E11, +\Jacobisc@@{z}{k},Jacobian elliptic function ${\rm sc}$,Jacobi-elliptic-sc,F|F:P:nP,$\mathrm{sc}$,http://dlmf.nist.gov/22.2#E9, +\Jacobisd@@{z}{k},Jacobian elliptic function ${\rm sd}$,Jacobi-elliptic-sd,F|F:P:nP,$\mathrm{sd}$,http://dlmf.nist.gov/22.2#E7, +\Jacobisn@@{z}{k},Jacobian elliptic function ${\rm sn}$,Jacobi-elliptic-sn,F|F:P:nP,$\mathrm{sn}$,http://dlmf.nist.gov/22.2#E4, +\JonquierePhi@{z}{s}=\Polylogarithm{s}@{z},Jonquiére's function,Jonquiere-phi,F|F:P,$\phi$,http://dlmf.nist.gov/23.1, +\JordanJ{k}@{n},Jordan's function,Jordan-J,F|F:P,$J_{k}$,http://dlmf.nist.gov/27.2#E11, +\Kelvinbei{\nu}@@{x},Kelvin function ${\rm bei}_\nu$,Kelvin-bei,F|F:P:nP,$\mathrm{bei}_{\nu}$,http://dlmf.nist.gov/10.61#E1, +\Kelvinber{\nu}@@{x},Kelvin function ${\rm ber}_\nu$,Kelvin-ber,F|F:P:nP,$\mathrm{ber}_{\nu}$,http://dlmf.nist.gov/10.61#E1, +\Kelvinkei{\nu}@@{x},Kelvin function ${\rm kei}_\nu$,Kelvin-kei,F|F:P:nP,$\mathrm{kei}_{\nu}$,http://dlmf.nist.gov/10.61#E2, +\Kelvinker{\nu}@@{x},Kelvin function ${\rm ker}_\nu$,Kelvin-ker,F|F:P:nP,$\mathrm{ker}_{\nu}$,http://dlmf.nist.gov/10.61#E2, +\Krawtchouk{n}@{x}{p}{N},Krawtchouk polynomial,Krawtchouk-polynomial-K,P|F:P,$K_{n}$,http://dlmf.nist.gov/18.19#T1.t1.r6, +\KrawtchoukK{n}@{x}{p}{N},Krawtchouk polynomial,Krawtchouk-polynomial-K,P|F:P,$K_{n}$,http://dlmf.nist.gov/18.19#T1.t1.r6, +\Kronecker{j}{k},Kronecker delta,EMPTY,I|F,"$\delta_{m,n}$",http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r4, +\KummerM@{a}{b}{z},Kummer confluent hypergeometric function $M$,Kummer-confluent-hypergeometric-M,F|F:P,$M$,http://dlmf.nist.gov/13.2#E2, +\KummerU@{a}{b}{z},Kummer confluent hypergeometric function $U$,Kummer-confluent-hypergeometric-U,F|F:P,$U$,http://dlmf.nist.gov/13.2#E6, +\KummerboldM@{a}{b}{z},Olver's confluent hypergeometric function,Kummer-confluent-hypergeometric-bold-M,F|F:P,${\mathbf{M}}$,http://dlmf.nist.gov/13.2#E3, +\Laguerre[\alpha]{n}@{x},Laguerre (or generalized Laguerre) polynomial,generalized-Laguerre-polynomial-L,P|FnO:PnO:FO:PO,$L_n^{(\alpha)}$,http://dlmf.nist.gov/18.3#T1.t1.r27, +\LaguerreL[\alpha]{n}@{x},Laguerre (or generalized Laguerre) polynomial,generalized-Laguerre-polynomial-L,P|FnO:PnO:FO:PO,$L_n^{(\alpha)}$,http://dlmf.nist.gov/18.3#T1.t1.r27, +\LambertW@{x},Lambert $\LambertW$-function,Lambert-W,F|F:P,$W$,http://dlmf.nist.gov/23.1, +\LambertWm@{x},non-principal branch of Lambert $\LambertW$-function,Lambert-Wm,F|F:P,$\mathrm{Wm}$,http://dlmf.nist.gov/4.13#p2, +\LambertWp@{x},principal branch of Lambert $\LambertW$-function,Lambert-Wp,F|F:P,$\mathrm{Wp}$,http://dlmf.nist.gov/23.1, +\LameEc{m}{\nu}@{z}{k^2},Lamé function $Ec_\nu^m$,Lame-Ec,F|F:P,$\mathit{Ec}^{m}_{\nu}$,http://dlmf.nist.gov/23.1, +\LameEs{m}{\nu}@{z}{k^2},Lamé function $Es_\nu^m$,Lame-Es,F|F:P,$\mathit{Es}^{m}_{\nu}$,http://dlmf.nist.gov/23.1, +\Lamea{n}{\nu}@{k^2},eigenvalues of Lamé's equation $a_\nu^m$,Lame-eigenvalue-a,F|F:P,$a^{n}_{\nu}$,http://dlmf.nist.gov/29.3#SS1.p1, +\Lameb{n}{\nu}@{k^2},eigenvalues of Lamé's equation $b_\nu^m$,Lame-eigenvalue-b,F|F:P,$b^{n}_{\nu}$,http://dlmf.nist.gov/29.3#SS1.p1, +\LamecE{m}{2n+1}@{z}{k^2},Lamé polynomial $cE_{2n+1}^m$,Lame-polynomial-cE,P|F:P,$\mathit{cE}^{m}_{\nu}$,http://dlmf.nist.gov/23.1, +\LamecdE{m}{2n+2}@{z}{k^2},Lamé polynomial $cdE_{2n+2}^m$,Lame-polynomial-cdE,P|F:P,$\mathit{cdE}^{m}_{\nu}$,http://dlmf.nist.gov/23.1, +\LamedE{m}{2n+1}@{z}{k^2},Lamé polynomial $dE_{2n+1}^m$,Lame-polynomial-dE,P|F:P,$\mathit{dE}^{m}_{\nu}$,http://dlmf.nist.gov/29.12#E4, +\LamesE{m}{2n+1}@{z}{k^2},Lamé polynomial $sE_{2n+1}^m$,Lame-polynomial-sE,P|F:P,$\mathit{sE}^{m}_{\nu}$,http://dlmf.nist.gov/29.12#E2, +\LamescE{m}{2n+2}@{z}{k^2},Lamé polynomial $scE_{2n+2}^m$,Lame-polynomial-scE,P|F:P,$\mathit{scE}^{m}_{\nu}$,http://dlmf.nist.gov/29.12#E5, +\LamescdE{m}{2n+3}@{z}{k^2},Lamé polynomial $scdE_{2n+3}^m$,Lame-polynomial-scdE,P|F:P,$\mathit{scdE}^{m}_{\nu}$,http://dlmf.nist.gov/29.12#E8, +\LamesdE{m}{2n+2}@{z}{k^2},Lamé polynomial $sdE_{2n+2}^m$,Lame-polynomial-sdE,P|F:P,$\mathit{sdE}^{m}_{\nu}$,http://dlmf.nist.gov/29.12#E6, +\LameuE{m}{2n}@{z}{k^2},Lamé polynomial $uE_{2n}^m$,Lame-polynomial-uE,P|F:P,$\mathit{uE}^{m}_{\nu}$,http://dlmf.nist.gov/29.12#E1, +\LaplaceTrans@{f}{s},Laplace transform,Laplace-transform,O|F:P,$\mathscr{L}$,http://dlmf.nist.gov/1.14#E17, +\Lattice{L},lattice in $\Complex$,EMPTY,SN|F,$\mathbb{L}$,http://dlmf.nist.gov/23.2#i, +"\LauricellaFD@{a}{b_1,...,b_n}{c}{z_1,...,z_n}",Lauricella's multivariate hypergeometric function,Lauricella-FD,F|F:P,$F_D$,http://dlmf.nist.gov/19.15#p1, +\LegendreBlackQ[\mu]{\nu}@{z},Olver's associated Legendre function,associated-Legendre-black-Q,F|FO:PO:FnO:PnO,$\boldsymbol{Q}_\nu^\mu$,http://dlmf.nist.gov/14.3#E10, +\LegendreP[\mu]{\nu}@{z},associated Legendre function of the first kind,Legendre-P-first-kind,F|FO:PO:FnO:PnO,$P_\nu^\mu$,http://dlmf.nist.gov/14.3#E6, +\LegendrePoly{n}@{x},Legendre polynomial,Legendre-spherical-polynomial,P|F:P,$P_{n}$,http://dlmf.nist.gov/18.3#T1.t1.r25, +\LegendrePolys{n}@{x},shifted Legendre polynomial,shifted-spherical-Legendre-polynomial-s,P|F:P,$P^{*}_{n}$,http://dlmf.nist.gov/18.3#T1.t1.r26, +\LegendreQ[\mu]{\nu}@{z},associated Legendre function of the second kind,Legendre-Q-second-kind,F|FO:PO:FnO:PnO,$Q_\nu^\mu$,http://dlmf.nist.gov/14.3#E7, +\LegendreSymbol{n}{p},Legendre symbol,Legendre-symbol,I|F,$(n|p)$,http://dlmf.nist.gov/27.9, +\LerchPhi@{z}{s}{a},Lerch's transcendent,Lerch-Phi,F|F:P,$\Phi$,http://dlmf.nist.gov/25.14#E1, +\LeviCivita{j}{k}{\ell},Levi-Civita symbol,EMPTY,I|F,$\epsilon_{jkl}$,http://dlmf.nist.gov/1.6#E14, +\LinBrF@{a}{u},line-broadening function,line-broadening-function,F|F:P,$H$,http://dlmf.nist.gov/7.19#E4, +\LiouvilleLambda@{n},Liouville's function,Liouville-lambda,F|F:P,$\lambda$,http://dlmf.nist.gov/27.2#E13, +\Ln@@{z},general logarithm function,multivalued-natural-logarithm,F|F:P:nP,$\mathrm{Ln}$,http://dlmf.nist.gov/4.2#E1, +\LogInt@{x},logarithmic integral,logarithmic-integral,F|F:P,$\mathrm{li}$,http://dlmf.nist.gov/6.2#E8, +\LommelS{\mu}{\nu}@{z},"Lommel function $S$",Lommel-S,F|F:P,"$S_{{\mu},{\nu}}$",http://dlmf.nist.gov/11.9#E5, +\Lommels{\mu}{\nu}@{z},"Lommel function $s$",Lommel-s,F|F:P,"$s_{{\mu},{\nu}}$",http://dlmf.nist.gov/11.9#E3, +\Longleftrightarrow,is equivalent to,equivalent-to,SM|F,$\Longleftrightarrow $,http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r13, +\Longrightarrow,implies,EMPTY,SM|F,$\Longrightarrow$,http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r12, +\MangoldtLambda@{n},Mangoldt's function,Mangoldt-Lambda,F|F:P,$\Lambda$,http://dlmf.nist.gov/27.2#E14, +\MathieuCe{\nu}@@{z}{q},modified Mathieu function ${\rm Ce}_\nu$,modified-Mathieu-Ce,F|F:P:nP,$\mathrm{Ce}_{n}$,http://dlmf.nist.gov/28.20#E3, +\MathieuD{j}@{\nu}{\mu}{z},cross-products of modified Mathieu functions and their derivatives ${\rm D}_{j}$,Mathieu-D,F|F:P,$\mathrm{D}_{j}$,http://dlmf.nist.gov/28.28#E24, +\MathieuDc{j}@{n}{m}{z},cross-products of radial Mathieu functions and their derivatives ${\rm Dc}_j$,Mathieu-Dc,F|F:P,$\mathrm{Dc}_{j}$,http://dlmf.nist.gov/28.28#E39, +\MathieuDs{j}@{n}{m}{z},cross-products of radial Mathieu functions and their derivatives ${\rm Ds}_j$,Mathieu-Ds,F|F:P,$\mathrm{Ds}_{j}$,http://dlmf.nist.gov/28.28#E35, +\MathieuDsc{j}@{n}{m}{z},cross-products of radial Mathieu functions and their derivatives ${\rm Dsc}_j$,Mathieu-Dsc,F|F:P,$\mathrm{Dsc}_{j}$,http://dlmf.nist.gov/28.28#E40, +\MathieuEigenvaluea{n}@@{q},eigenvalues of Mathieu equation $a_n(q)$,Mathieu-eigenvalue-a,F|F:P:nP,$a_{n}$,http://dlmf.nist.gov/28.2#SS5.p1, +\MathieuEigenvalueb{n}@@{q},eigenvalues of Mathieu equation $b_n(q)$,Mathieu-eigenvalue-b,F|F:P:nP,$b_{n}$,http://dlmf.nist.gov/28.2#SS5.p1, +\MathieuEigenvaluelambda{\nu+2n}@@{q},eigenvalues of Mathieu equation $\lambda_\nu(q)$,Mathieu-eigenvalue-lambda,F|F:P:nP,$\lambda_{\nu}$,http://dlmf.nist.gov/28.12#SS1.p2, +\MathieuFc{m}@{z}{h},Mathieu function ${\rm Fc}_m$,Mathieu-Fc,F|F:P,$\mathrm{Fc}_{m}$,http://dlmf.nist.gov/23.1, +\MathieuFe{n}@@{z}{q},modified Mathieu function ${\rm Fe}_n$,modified-Mathieu-Fe,F|F:P:nP,$\mathrm{Fe}_{n}$,http://dlmf.nist.gov/28.20#E6, +\MathieuFs{m}@{z}{h},Mathieu function ${\rm Fs}_m$,Mathieu-Fs,F|F:P,$\mathrm{Fs}_{m}$,http://dlmf.nist.gov/23.1, +\MathieuGc{m}@{z}{h},Mathieu function ${\rm Gc}_m$,Mathieu-Gc,F|F:P,$\mathrm{Gc}_{m}$,http://dlmf.nist.gov/23.1, +\MathieuGe{n}@@{z}{q},modified Mathieu function ${\rm Ge}_n$,modified-Mathieu-Ge,F|F:P:nP,$\mathrm{Ge}_{n}$,http://dlmf.nist.gov/28.20#E7, +\MathieuGs{m}@{z}{h},Mathieu function ${\rm Gs}_m$,Mathieu-Gs,F|F:P,$\mathrm{Gs}_{m}$,http://dlmf.nist.gov/23.1, +\MathieuIe{n}@@{z}{h},modified Mathieu function ${\rm Ie}_n$,modified-Mathieu-Ie,F|F:P:nP,$\mathrm{Ie}_{n}$,http://dlmf.nist.gov/28.20#E17, +\MathieuIo{n}@@{z}{h},modified Mathieu function ${\rm Io}_n$,modified-Mathieu-Io,F|F:P:nP,$\mathrm{Io}_{n}$,http://dlmf.nist.gov/28.20#E18, +\MathieuKe{n}@@{z}{h},modified Mathieu function ${\rm Ke}_n$,modified-Mathieu-Ke,F|F:P:nP,$\mathrm{Ke}_{n}$,http://dlmf.nist.gov/28.20#E19, +\MathieuKo{n}@@{z}{h},modified Mathieu function ${\rm Ko}_n$,modified-Mathieu-Ko,F|F:P:nP,$\mathrm{Ko}_{n}$,http://dlmf.nist.gov/28.20#E20, +\MathieuM{j}{\nu}@@{z}{h},modified Mathieu function ${\rm M}_\nu^{(j)}$,modified-Mathieu-M,F|F:P:nP,${\mathrm{M}^{(j)}_{\nu}}$,http://dlmf.nist.gov/23.1, +\MathieuMc{j}{n}@@{z}{h},radial Mathieu function ${\rm Mc}_n^{(j)}$,modified-Mathieu-Mc,F|F:P:nP,${\mathrm{Mc}^{(j)}_{\nu}}$,http://dlmf.nist.gov/28.20#E15, +\MathieuMe{\nu}@@{z}{q},modified Mathieu function ${\rm Me}_\nu$,modified-Mathieu-Me,F|F:P:nP,$\mathrm{Me}_{n}$,http://dlmf.nist.gov/28.20#E5, +\MathieuMs{j}{n}@@{z}{h},radial Mathieu function ${\rm Ms}_n^{(j)}$,modified-Mathieu-Ms,F|F:P:nP,${\mathrm{Ms}^{(j)}_{\nu}}$,http://dlmf.nist.gov/28.20#E16, +\MathieuSe{\nu}@@{z}{q},modified Mathieu function ${\rm Se}_\nu$,modified-Mathieu-Se,F|F:P:nP,$\mathrm{Se}_{n}$,http://dlmf.nist.gov/28.20#E4, +\Mathieuce{n}@@{z}{q},Mathieu function ${\rm ce}_\nu$,Mathieu-ce,F|F:P:nP,$\mathrm{ce}_{n}$,http://dlmf.nist.gov/23.1, +\Mathieufe{n}@@{z}{q},second solution of Mathieu's equation ${\rm fe}_n$,Mathieu-fe,F|F:P:nP,$\mathrm{fe}_{n}$,http://dlmf.nist.gov/28.5#E1, +\Mathieuge{n}@@{z}{q},second solution of Mathieu's equation ${\rm ge}_n$,Mathieu-ge,F|F:P:nP,$\mathrm{ge}_{n}$,http://dlmf.nist.gov/28.5#E2, +\Mathieume{n}@@{z}{q},Mathieu function ${\rm me}_\nu$,Mathieu-me,F|F:P:nP,$\mathrm{me}_{\nu}$,http://dlmf.nist.gov/23.1, +\Mathieuse{n}@@{z}{q},Mathieu function ${\rm se}_\nu$,Mathieu-se,F|F:P:nP,$\mathrm{se}_{\nu}$,http://dlmf.nist.gov/23.1, +\Matrix{I},Bessel function of matrix argument (first kind),EMPTY,F|F,$\Matrix{T}$,http://dlmf.nist.gov/front/introduction#Sx4.p2.t1.r9, +\MeijerG{m}{n}{p}{q}@@@{z}{\Vector{a}}{\Vector{b}},Meijer $G$-function,Meijer-G,F|F:fo1:fo2:fo3,"${G^{m,n}_{p,q}}$",http://dlmf.nist.gov/16.17#E1, +\Meixner{n}@{x}{\beta}{c},Meixner polynomial,Meixner-polynomial-M,P|F:P,$M_{n}$,http://dlmf.nist.gov/18.19#T1.t1.r9, +\MeixnerM{n}@{x}{\beta}{c},Meixner polynomial,Meixner-polynomial-M,P|F:P,$M_{n}$,http://dlmf.nist.gov/18.19#T1.t1.r9, +\MeixnerPollaczek{\lambda}{n}@{x}{\phi},Meixner-Pollaczek polynomial,Meixner-Pollaczek-polynomial-P,P|F:P,$P^{(\alpha)}_{n}$,http://dlmf.nist.gov/18.19#P3.p1, +\MeixnerPollaczekP{\lambda}{n}@{x}{\phi},Meixner-Pollaczek polynomial,Meixner-Pollaczek-polynomial-P,P|F:P,$P^{(\alpha)}_{n}$,http://dlmf.nist.gov/18.19#P3.p1, +\MellinTrans@{f}{s},Mellin transform,Mellin-transform,O|F:P,$\mathscr{M}$,http://dlmf.nist.gov/1.14#E32, +\Mills@{x},Mills ratio ${\sf M}$,Mills-ratio,F|F:P,$\mathsf{M}$,http://dlmf.nist.gov/7.8#E1, +\MittagLeffler{a}{b}@{z},Mittag-Leffler function,Mittag-Leffler-function,F|F:P,"$E_{a,b}$",http://dlmf.nist.gov/10.46#E3, +\ModCylinder{\nu}@{z},modified cylinder function,modified-cylinder-function,F|F:P,$\mathscr{Z}_{\nu}$,http://dlmf.nist.gov/10.25#P2.p1, +\ModularJ@{\tau},Klein's complete invariant,Kleins-invariant-modular-J,F|F:P,$J$,http://dlmf.nist.gov/23.15#E7, +\ModularLambda@{\tau},elliptic modular function,modular-Lambda,F|F:P,$\lambda$,http://dlmf.nist.gov/23.15#E6, +\MoebiusMu@{n},Möbius function,Moebius-mu,F|F:P,$\mu$,http://dlmf.nist.gov/27.2#E12, +\NatNumber,set of positive integers,EMPTY,SN|F,$\mathbb{N}$,http://dlmf.nist.gov/front/introduction#Sx4.p2.t1.r11, +\NeumannFactor{n},Neumann factor,Neumann-factor,I|F,$\epsilon_{m}$,http://drmf.wmflabs.org/wiki/Definition:NeumannFactor, +\NeumannPoly{n}@{x},Neumann's polynomial,Neumann-polynomial,P|F:P,$O_{k}$,http://dlmf.nist.gov/10.23#E12, +\NumPrimeDivNu@{n},$\nu(n)$ : number of distinct primes dividing $n$,number-of-primes-dividing-nu,I|F:P,$\nu$,http://dlmf.nist.gov/27.2#SS1.p1, +\NumPrimesLessPi@{x},number of primes not exceeding $x$,number-of-primes-less-pi,I|F:P,$\pi$,http://dlmf.nist.gov/27.2#E2, +\NumSquaresR{k}@{n},number of squares,number-of-squares-R,I|F:P,$r_{k}$,http://dlmf.nist.gov/27.13#SS4.p1, +\ODEgenAiryA{n}@{z},generalized Airy function from differential equation $A_n$,ODE-generalized-Airy-A,F|F:P,$A_{n}$,http://dlmf.nist.gov/9.13#SS1.p2, +\ODEgenAiryB{n}@{z},generalized Airy function from differential equation $B_n$,ODE-generalized-Airy-B,F|F:P,$B_{n}$,http://dlmf.nist.gov/9.13#SS1.p2, +\Pade{p}{q}{f},Padé approximant,Pade-approximant,O|F,${[p/q]_{f}}$,http://dlmf.nist.gov/3.11#SS4.p1, +\ParabolicU@{a}{z},parabolic cylinder function $U$,parabolic-U,F|F:P,$U$,http://dlmf.nist.gov/12.4.E1, +\ParabolicUbar@{a}{x},parabolic cylinder function $\overline{U}$,parabolic-U-bar,F|F:P,$\overline{U}$,http://dlmf.nist.gov/12.2.21, +\ParabolicV@{a}{z},parabolic cylinder function $V$,parabolic-V,F|F:P,$V$,http://dlmf.nist.gov/12.4.E2, +\ParabolicW@{a}{x},parabolic cylinder function $W$,parabolic-W,F|F:P,$W$,http://dlmf.nist.gov/12.14.i, +\PartitionsP[k]@{n},total number of partitions,partition-function,I|F:F:P:P,$p_{k}$,http://dlmf.nist.gov/26.2#P4.p2, +\PeriodicBernoulliB{n}@{x},periodic Bernoulli functions,periodic-Bernoulli-polynomial-B,F|F:P,$\widetilde{B}_{n}$,http://dlmf.nist.gov/24.2#iii, +\PeriodicEulerE{n}@{x},periodic Euler functions,periodic-Euler-polynomial-E,F|F:P,$\widetilde{E}_{n}$,http://dlmf.nist.gov/24.2#iii, +\PeriodicZeta@{x}{s},periodic zeta function,periodic-zeta,F|F:P,$F$,http://dlmf.nist.gov/25.13#E1, +\Permutations{n},"set of permutations of ${\bf n}$",set-of-permutations,SN|F,$\mathfrak{S}_{n}$,http://dlmf.nist.gov/23.1, +\PlanePartitionsPP@{n},number of plane partitions of $n$,plane-partitions-PP,I|F:P,$\mathit{pp}$,http://dlmf.nist.gov/26.12#SS1.p4, +\PollaczekP{\lambda}{n}@{x}{a}{b},Pollaczek polynomial,Pollaczek-polynomial-P,P|F:P,$P^{(\lambda)}_{n}$,http://dlmf.nist.gov/18.35#E4, +\Polylogarithm{s}@{z},polylogarithm,polylogarithm,F|F:P,$\mathrm{Li}_{s}$,http://dlmf.nist.gov/25.12#E10, +\Prod{n}{k1}{k2}@{f(n)},semantic product,EMPTY,SM|O:O,$\prod_{n=\alpha}^{\beta}$,http://drmf.wmflabs.org/wiki/Definition:Prod, +\Racah{n}@{x}{\alpha}{\beta}{\gamma}{\delta},Racah polynomial,Racah-polynomial-R,P|F:P,$R_{n}$,http://dlmf.nist.gov/18.25#T1.t1.r4, +\RacahR{n}@{x}{\alpha}{\beta}{\gamma}{\delta},Racah polynomial,Racah-polynomial-R,P|F:P,$R_{n}$,http://dlmf.nist.gov/18.25#T1.t1.r4, +\RamanujanSum{k}@{n},Ramanujan's sum,Ramanujan-sum,F|F:P,$c_{k}$,http://dlmf.nist.gov/27.10#E4, +\RamanujanTau@{n},Ramanujan's tau function,Ramanujan-tau,F|F:P,$\tau$,http://dlmf.nist.gov/27.14#E18, +\Rational,set of rational numbers,EMPTY,SN|F,$\mathbb{Q}$,http://dlmf.nist.gov/front/introduction#Sx4.p2.t1.r13, +\RayleighFun{n}@{\nu},Rayleigh function,Rayleigh-function,F|F:P,$\sigma_{n}$,http://dlmf.nist.gov/10.21#E55, +\Real,set of real numbers,EMPTY,SN|F,$\mathbb{R}$,http://dlmf.nist.gov/front/introduction#Sx4.p2.t1.r14, +\RepInterfc{n}@{z},repeated integrals of the complementary error function,repeated-integral-complementary-error-function,F|F:P,$\mathrm{i}^{n}\mathrm{erfc}$,http://dlmf.nist.gov/7.18#E2, +\Residue,residue,EMPTY,O|F,$\mathrm{res}$,http://dlmf.nist.gov/1.10#SS3.p5, +\RestrictedCompositionsC@{\mathrm{condition}}{n},restricted number of compositions,restricted-number-of-compositions-C,I|F:P,$c_{m}$,http://dlmf.nist.gov/26.11#p1, +\RestrictedPartitionsP[k]@{\mathrm{condition}}{n},restricted number of partitions,restricted-partitions-P,I|F:F:P:P,$p_{k}$,http://dlmf.nist.gov/26.10#i, +\RiemannP@{\begin{Bmatrix} \alpha & \beta &\gamma & \\ a_1 & b_1 & c_1 & z \\ a_2 & b_2 & c_2 & \end{Bmatrix}},Riemann-Papperitz $\RiemannP$-symbol for solutions of the generalized hypergeometric differential equation,Riemann-P-symbol,Q|F:P,$P$,http://dlmf.nist.gov/15.11#E3, +\RiemannTheta@{\Vector{z}}{\Matrix{\Omega}},Riemann theta function,Riemann-theta,F|F:P,$\theta$,http://dlmf.nist.gov/21.2#E1, +\RiemannThetaChar{\Vector{\alpha}}{\Vector{\beta}}@{\Vector{z}}{\Matrix{\Omega}},Riemann theta function with characteristics,Riemann-theta-characterstics,F|F:P,$\theta\left[\Vector{\alpha} \atop \Vector{\beta} \right]$,http://dlmf.nist.gov/21.2#E5, +\RiemannThetaHat@{\Vector{z}}{\Matrix{\Omega}},scaled Riemann theta function,Riemann-theta-hat,F|F:P,$\hat{\theta}$,http://dlmf.nist.gov/21.2#E2, +\RiemannXi@{s},Riemann's $\RiemannXi$-function,Riemann-xi,F|F:P,$\xi$,http://dlmf.nist.gov/25.4#E4, +\RiemannZeta@{s},Riemann zeta function,Riemann-zeta,F|F:P,$\zeta$,http://dlmf.nist.gov/25.2#E1, +\Schwarzian{z}{\zeta},Schwarzian derivative,EMPTY,O|F,"$\left\{f,g\right\}$",http://dlmf.nist.gov/1.13#E20, +\ScorerGi@{z},Scorer function (inhomogeneous Airy function) ${\rm Gi}$,Scorer-Gi,F|F:P,$\mathrm{Gi}$,http://dlmf.nist.gov/9.12#i, +\ScorerHi@{z},Scorer function (inhomogeneous Airy function) ${\rm Hi}$,Scorer-Hi,F|F:P,$\mathrm{Hi}$,http://dlmf.nist.gov/9.12#i, +\SinCosIntf@{z},auxiliary function for sine and cosine integrals ${\rm f}$,sine-cosine-integral-f,F|F:P,$\mathrm{f}$,http://dlmf.nist.gov/6.2#E17, +\SinCosIntg@{z},auxiliary function for sine and cosine integrals ${\rm g}$,sine-cosine-integral-g,F|F:P,$\mathrm{g}$,http://dlmf.nist.gov/6.2#E18, +\SinInt@{z},sine integral ${\rm Si}$,sine-integral,F|F:P,$\mathrm{Si}$,http://dlmf.nist.gov/6.2#E9, +\SinIntg@{a}{z},generalized sine integral ${\rm Si}$,generalized-sine-integral-Si,F|F:P,$\mathrm{Si}$,http://dlmf.nist.gov/8.21#E2, +\SinhInt@{z},hyperbolic sine integral,hyperbolic-sine-integral,F|F:P,$\mathrm{Shi}$,http://dlmf.nist.gov/6.2#E15, +\SphBesselIi{n}@{z},modified spherical Bessel function of the first kind ${\sf i}_n^{(1)}$,spherical-Bessel-I-1,F|F:P,${\mathsf{i}^{(1)}_{n}}$,http://dlmf.nist.gov/10.47#E7, +\SphBesselIii{n}@{z},modified spherical Bessel function of the first kind ${\sf i}_n^{(2)}$,spherical-Bessel-I-2,F|F:P,${\mathsf{i}^{(2)}_{n}}$,http://dlmf.nist.gov/10.47#E8, +\SphBesselJ{n}@{z},spherical Bessel function of the first kind ${\sf j}_n$,spherical-Bessel-J,F|F:P,$\mathsf{j}_{n}$,http://dlmf.nist.gov/10.47#E3, +\SphBesselK{n}@{z},modified spherical Bessel function of the second kind ${\sf k}_n$,spherical-Bessel-K,F|F:P,$\mathsf{k}_{n}$,http://dlmf.nist.gov/10.47#E9, +\SphBesselY{n}@{z},spherical Bessel function of the second kind ${\sf y}_n$,spherical-Bessel-Y,F|F:P,$\mathsf{y}_{n}$,http://dlmf.nist.gov/10.47#E4, +\SphHankelHi{n}@{z},spherical Hankel function of the first kind ${\sf h}_n^{(1)}$,spherical-Hankel-H-1-Bessel-third-kind,F|F:P,${\mathsf{h}^{(1)}_{n}}$,http://dlmf.nist.gov/10.47#E5, +\SphHankelHii{n}@{z},spherical Hankel function of the second kind ${\sf h}_n^{(2)}$,spherical-Hankel-H-2-Bessel-third-kind,F|F:P,${\mathsf{h}^{(2)}_{n}}$,http://dlmf.nist.gov/10.47#E6, +\SphericalHarmonicY{l}{m}@{\theta}{\phi},spherical harmonic,spherical-harmonic-Y,F|F:P,"$Y_{{\ell},{m}}$",http://dlmf.nist.gov/14.30#E1, +\SpheroidalEigenvalueLambda{m}{n}@{\gamma^2},eigenvalues of the spheroidal differential equation,spheroidal-eigenvalue-lambda,O|F:P,$\lambda^{m}_{n}$,http://dlmf.nist.gov/30.3#SS1.p1, +\SpheroidalOnCutPs{m}{n}@{x}{\gamma^2},spheroidal wave function of the first kind ${\sf Ps}_n^m$,spheroidal-wave-on-cut-Ps,F|F:P,$\mathsf{Ps}^{m}_{n}$,http://dlmf.nist.gov/30.4#i, +\SpheroidalOnCutQs{m}{n}@{x}{\gamma^2},spheroidal wave function of the second kind ${\sf Qs}_n^m$,spheroidal-wave-on-cut-Qs,F|F:P,$\mathsf{Qs}^{m}_{n}$,http://dlmf.nist.gov/30.5, +\SpheroidalPs{m}{n}@{z}{\gamma^2},spheroidal wave function of complex argument $Ps_n^m$,spheroidal-wave-Ps,F|F:P,$\mathit{Ps}^{m}_{n}$,http://dlmf.nist.gov/30.6#p1, +\SpheroidalQs{m}{n}@{z}{\gamma^2},spheroidal wave function of complex argument $Qs_n^m$,spheroidal-wave-Qs,F|F:P,$\mathit{Qs}^{m}_{n}$,http://dlmf.nist.gov/30.6#p1, +\SpheroidalRadialS{m}{j}{n}@{z}{\gamma},radial spheroidal wave function,radial-spheroidal-wave-S,F|F:P,$S^{m(j)}_{n}$,http://dlmf.nist.gov/30.11#E3, +\StieltjesConstants{n},Stieltjes constants,Stieltjes-constants,C|F,$\gamma_{n}$,http://drmf.wmflabs.org/wiki/Definition:StieltjesConstants, +\StieltjesTrans@{f}{s},Stieltjes transform,Stieltjes-transform,O|F:P,$\mathcal{S}$,http://dlmf.nist.gov/1.14#E47, +\StieltjesWigert{n}@@{x}{q},Stieltjes-Wigert polynomial,Stieltjes-Wigert-polynomial-S,P|F:P:PS,$S_{n}$,http://drmf.wmflabs.org/wiki/Definition:StieltjesWigert, +\StieltjesWigertS{n}@{x}{q},Stieltjes-Wigert polynomial,Stieltjes-Wigert-polynomial-S,P|F:P,$S_{n}$,http://dlmf.nist.gov/18.27#E18, +\StirlingS@{n}{k},Stirling number of the first kind,Stirling-number-first-kind-S,I|F:P,$s$,http://dlmf.nist.gov/26.8#SS1.p1, +\StirlingSS@{n}{k},Stirling number of the second kind,Stirling-number-second-kind-S,I|F:P,$S$,http://dlmf.nist.gov/26.8#SS1.p3, +\StruveH{\nu}@{z},Struve function ${\bf H}_{\nu}$,Struve-H,F|F:P,$\mathbf{H}_{\nu}$,http://dlmf.nist.gov/11.2#E1, +\StruveK{\nu}@{z},Struve function ${\bf K}_{\nu}$,associated-Struve-K,F|F:P,$\mathbf{K}_{\nu}$,http://dlmf.nist.gov/11.2#E5, +\StruveL{\nu}@{z},modified Struve function ${\bf L}_{\nu}$,modified-Struve-L,F|F:P,$\mathbf{L}_{\nu}$,http://dlmf.nist.gov/11.2#E2, +\StruveM{\nu}@{z},modified Struve function ${\bf M}_{\nu}$,associated-Struve-M,F|F:P,$\mathbf{M}_{\nu}$,http://dlmf.nist.gov/11.2#E6, +\Sum{n}{0}{k}@{f(n)},semantic sum,EMPTY,SM|F:P,$\sum_{n=k}^{N}$,http://drmf.wmflabs.org/wiki/Definition:Sum, +\SurfaceHarmonicY{l}{m}@{\theta}{\phi},surface harmonic of the first kind,surface-harmonic-Y,F|F:P,$Y_{l}^{m}$,http://dlmf.nist.gov/14.30#E2, +\TriangleOP{\alpha}{\beta}{\gamma}{m}{n}@{x}{y},triangle polynomial,triangle-orthogonal-polynomial-P,P|F:P,"$P^{\alpha,\beta,\gamma}_{m,n}$",http://dlmf.nist.gov/18.37#E7, +\Ultra{\lambda}{n}@{x},ultraspherical/Gegenbauer polynomial,ultraspherical-Gegenbauer-polynomial,P|F:P,$C^{\mu}_{n}$,http://dlmf.nist.gov/18.3#T1.t1.r5, +\Ultraspherical{\lambda}{n}@{x},ultraspherical (or Gegenbauer) polynomial,ultraspherical-Gegenbauer-polynomial,P|F:P,$C^{(\mu)}_{n}$,http://dlmf.nist.gov/18.3#T1.t1.r5, +\UmbilicCatE@{s}{t}{\Vector{x}},elliptic umbilic catastrophe,elliptic-umbilic-catastrophe,F|F:P,$\Phi^{(\mathrm{E})}$,http://dlmf.nist.gov/36.2#E2, +\UmbilicCatH@{s}{t}{\Vector{x}},hyperbolic umbilic catastrophe,hyperbolic-umbilic-catastrophe,F|F:P,$\Phi^{(\mathrm{H})}$,http://dlmf.nist.gov/36.2#E3, +\VariationalOp[a;b]@{f},total variation,variational-operator,F|FO:PO:FnO:PnO,"$\mathcal{V}_{a,b}$",http://dlmf.nist.gov/1.4#E33, +\VoigtU@{x}{t},Voigt function ${\sf U}$,Voigt-U,F|F:P,$\mathsf{U}$,http://dlmf.nist.gov/7.19#E1, +\VoigtV@{x}{t},Voigt function ${\sf V}$,Voigt-V,F|F:P,$\mathsf{V}$,http://dlmf.nist.gov/7.19#E2, +\WaringG@{k},Waring's function $G$,Waring-G,F|F:P,$G$,http://dlmf.nist.gov/27.13#SS3.p1, +\Waringg@{k},Waring's function $g$,Waring-g,F|F:P,$g$,http://dlmf.nist.gov/27.13#SS3.p1, +\WeberE{\nu}@{z},Weber function,Weber-E,F|F:P,$\mathbf{E}_{\nu}$,http://dlmf.nist.gov/11.10#E2, +\WeierPInv@@{z}{g_2}{g_3},Weierstrass $\WeierPInv$-function on invariants $\WeierPInv@{z}{g_2}{g_3}$,Weierstrass-P-on-invariants,F|F:P:nP,$\wp$,http://dlmf.nist.gov/23.1, +\WeierPLat@@{z}{\Lattice{L}},Weierstrass $\WeierPLat$-function on lattice $\WeierPLat@{z}{\Lattice{L}}$,Weierstrass-P-on-lattice,F|F:P:nP,$\wp$,http://dlmf.nist.gov/23.1, +\WeiersigmaInv@@{z}{g_2}{g_3},Weierstrass sigma function on invariants,Weierstrass-sigma-on-invariants,F|F:P:nP,$\sigma$,http://dlmf.nist.gov/23.2#E6, +\WeiersigmaLat@@{z}{\Lattice{L}},Weierstrass sigma function on lattice,Weierstrass-sigma-on-lattice,F|F:P:nP,$\sigma$,http://dlmf.nist.gov/23.2#E6, +\WeierzetaInv@@{z}{g_2}{g_3},Weierstrass zeta function on invariants,Weierstrass-zeta-on-invariants,F|F:P:nP,$\zeta$,http://dlmf.nist.gov/23.2#E5, +\WeierzetaLat@@{z}{\Lattice{L}},Weierstrass zeta function on lattice,Weierstrass-zeta-on-lattice,F|F:P:nP,$\zeta$,http://dlmf.nist.gov/23.2#E5, +\WhitD{\nu}@{z},parabolic cylinder function,Whittaker-D,F|F:P,$D_{n}$,http://dlmf.nist.gov/23.1, +\WhitM{\kappa}{\mu}@{z},Whittaker function $M_{\kappa;\mu}$,Whittaker-confluent-hypergeometric-M,F|F:P,"$M_{\kappa,\mu}$",http://dlmf.nist.gov/13.14#E2, +\WhitW{\kappa}{\mu}@{z},Whittaker function $W_{\kappa;\mu}$,Whittaker-confluent-hypergeometric-W,F|F:P,"$W_{\kappa,\mu}$",http://dlmf.nist.gov/13.14#E3, +\Wilson{n}@{x}{a}{b}{c}{d},Wilson polynomial,Wilson-polynomial-W,P|F:P,$W_{n}$,http://dlmf.nist.gov/18.25#T1.t1.r2, +\WilsonW{n}@{x}{a}{b}{c}{d},Wilson polynomial,Wilson-polynomial-W,P|F:P,$W_{n}$,http://dlmf.nist.gov/18.25#T1.t1.r2, +\Wronskian@{w_1(z);w_2(z)},Wronskian,Wronskian,F|F:P,$\mathscr{W}$,http://dlmf.nist.gov/1.13#E4, +\ZeroAiryAi{k},$k$\textsuperscript{th} zero of Airy $\AiryAi$,zero-Airy-Ai,F|F,$a_{k}$,http://dlmf.nist.gov/23.1, +\ZeroAiryAiPrime{k},$k$\textsuperscript{th} zero of Airy $\AiryAi'$,zero-derivative-Airy-Ai,F|F,$a'_{k}$,http://dlmf.nist.gov/23.1, +\ZeroAiryBi{k},$k$\textsuperscript{th} zero of Airy $\AiryBi$,zero-Airy-Bi,F|F,$b_{k}$,http://dlmf.nist.gov/23.1, +\ZeroAiryBiPrime{k},$k$\textsuperscript{th} zero of Airy $\AiryBi'$,zero-derivative-Airy-Bi,F|F,$b'_{k}$,http://dlmf.nist.gov/23.1, +\ZeroBesselJ{\nu}{m},zeros of the Bessel function $\BesselJ{\nu}@{x}$,zero-Bessel-J,F|F,"$j_{\nu,m}$",http://dlmf.nist.gov/23.1, +\ZeroBesselJPrime{\nu}{m},zeros of the Bessel function derivative $\BesselJ{\nu}'@{x}$,zero-derivative-Bessel-J,F|F,"${j'_{\nu,m}}$",http://dlmf.nist.gov/23.1, +\ZeroBesselY{\nu}{m},zeros of the Bessel function $\BesselY{\nu}@{x}$,zero-Bessel-Y-Weber,F|F,"$y_{\nu,m}$",http://dlmf.nist.gov/23.1, +\ZeroBesselYPrime{\nu}{m},zeros of the Bessel function derivative $\BesselY{\nu}'@{x}$,zero-derivative-Bessel-Y-Weber,F|F,"${y'_{\nu,m}}$",http://dlmf.nist.gov/23.1, +\ZonalPoly{\kappa}@{\Matrix{T}},zonal polynomial,zonal-polynomial,P|F:P,$Z_{\kappa}$,http://dlmf.nist.gov/35.4#SS1.p3, +\acos@@{z},inverse cosine function,inverse-cosine,F|F:P:nP,$\mathrm{arccos}$,http://dlmf.nist.gov/4.23#SS2.p1, +\acosh@@{z},inverse hyperbolic cosine function,hyperbolic-inverse-cosine,F|F:P:nP,$\mathrm{arccosh}$,http://dlmf.nist.gov/4.37#SS2.p1, +\acot@@{z},inverse cotangent function,inverse-cotangent,F|F:P:nP,$\mathrm{arccot}$,http://dlmf.nist.gov/4.23#E9, +\acoth@@{z},inverse hyperbolic cotangent function,hyperbolic-inverse-cotangent,F|F:P:nP,$\mathrm{arccoth}$,http://dlmf.nist.gov/4.37#E9, +\acsc@@{z},inverse cosecant function,inverse-cosecant,F|F:P:nP,$\mathrm{arccsc}$,http://dlmf.nist.gov/4.23#E7, +\acsch@@{z},inverse hyperbolic cosecant function,hyperbolic-inverse-cosecant,F|F:P:nP,$\mathrm{arccsch}$,http://dlmf.nist.gov/4.37#E7, +\arcGudermannian@@{x},inverse Gudermannian function,inverse-Gudermannian,F|F:P:nP,${\mathrm{gd}^{-1}}$,http://dlmf.nist.gov/4.23#E41, +\arcJacobicd@{x}{k},inverse Jacobian elliptic function ${\rm arccd}$,inverse-Jacobi-elliptic-cd,F|F:P,$\mathrm{arccd}$,http://dlmf.nist.gov/22.15#SS1.p1, +\arcJacobicn@{x}{k},inverse Jacobian elliptic function ${\rm arccn}$,inverse-Jacobi-elliptic-cn,F|F:P,$\mathrm{arccn}$,http://dlmf.nist.gov/22.15#SS1.p1, +\arcJacobics@{x}{k},inverse Jacobian elliptic function ${\rm arccs}$,inverse-Jacobi-elliptic-cs,F|F:P,$\mathrm{arccs}$,http://dlmf.nist.gov/22.15#SS1.p1, +\arcJacobidc@{x}{k},inverse Jacobian elliptic function ${\rm arcdc}$,inverse-Jacobi-elliptic-dc,F|F:P,$\mathrm{arcdc}$,http://dlmf.nist.gov/22.15#SS1.p1, +\arcJacobidn@{x}{k},inverse Jacobian elliptic function ${\rm arcdn}$,inverse-Jacobi-elliptic-dn,F|F:P,$\mathrm{arcdn}$,http://dlmf.nist.gov/22.15#SS1.p1, +\arcJacobids@{x}{k},inverse Jacobian elliptic function ${\rm arcds}$,inverse-Jacobi-elliptic-ds,F|F:P,$\mathrm{arcds}$,http://dlmf.nist.gov/22.15#SS1.p1, +\arcJacobinc@{x}{k},inverse Jacobian elliptic function ${\rm arcnc}$,inverse-Jacobi-elliptic-nc,F|F:P,$\mathrm{arcnc}$,http://dlmf.nist.gov/22.15#SS1.p1, +\arcJacobind@{x}{k},inverse Jacobian elliptic function ${\rm arcnd}$,inverse-Jacobi-elliptic-nd,F|F:P,$\mathrm{arcnd}$,http://dlmf.nist.gov/22.15#SS1.p1, +\arcJacobins@{x}{k},inverse Jacobian elliptic function ${\rm arcns}$,inverse-Jacobi-elliptic-ns,F|F:P,$\mathrm{arcns}$,http://dlmf.nist.gov/22.15#SS1.p1, +\arcJacobisc@{x}{k},inverse Jacobian elliptic function ${\rm arcsc}$,inverse-Jacobi-elliptic-sc,F|F:P,$\mathrm{arcsc}$,http://dlmf.nist.gov/22.15#SS1.p1, +\arcJacobisd@{x}{k},inverse Jacobian elliptic function ${\rm arcsd}$,inverse-Jacobi-elliptic-sd,F|F:P,$\mathrm{arcsd}$,http://dlmf.nist.gov/22.15#SS1.p1, +\arcJacobisn@{x}{k},inverse Jacobian elliptic function ${\rm arcsn}$,inverse-Jacobi-elliptic-sn,F|F:P,$\mathrm{arcsn}$,http://dlmf.nist.gov/22.15#SS1.p1, +\asec@@{z},inverse secant function,inverse-secant,F|F:P:nP,$\mathrm{arcsec}$,http://dlmf.nist.gov/4.23#E8, +\asech@@{z},inverse hyperbolic secant function,hyperbolic-inverse-secant,F|F:P:nP,$\mathrm{arcsech}$,http://dlmf.nist.gov/4.37#E8, +\asin@@{z},inverse sine function,inverse-sine,F|F:P:nP,$\mathrm{arcsin}$,http://dlmf.nist.gov/4.23#SS2.p1, +\asinh@@{z},inverse hyperbolic sine function,hyperbolic-inverse-sine,F|F:P:nP,$\mathrm{arcsinh}$,http://dlmf.nist.gov/4.37#SS2.p1, +\atan@@{z},inverse tangent function,inverse-tangent,F|F:P:nP,$\mathrm{arctan}$,http://dlmf.nist.gov/4.23#SS2.p1, +\atanh@@{z},inverse hyperbolic tangent function,hyperbolic-inverse-tangent,F|F:P:nP,$\mathrm{arctanh}$,http://dlmf.nist.gov/4.37#SS2.p1, +\bDiff[x],backward difference,EMPTY,O|F,"$\,\nabla_{x}$",http://dlmf.nist.gov/18.1#EGx2, +\bigqJacobi{n}@@{x}{a}{b}{c}{q},big $q$-Jacobi polynomial,big-q-Jacobi-polynomial-P,P|F:P:PS,$P_{n}$,http://drmf.wmflabs.org/wiki/Definition:bigqJacobi, +\bigqJacobiIVparam{n}@@{x}{a}{b}{c}{d}{q},big $q$-Jacobi polynomial with four parameters,big-q-Jacobi-polynomial-four-parameters-P,P|F:P,$P_{n}$,http://drmf.wmflabs.org/wiki/Definition:bigqJacobiIVparam, +\bigqLaguerre{n}@{x}{a}{b}{q},big $q$-Laguerre polynomial,EMPTY,P|F:P,$P_{n}$,http://drmf.wmflabs.org/wiki/Definition:bigqLaguerre, +\bigqLegendre{n}@{x}{c}{q},big $q$-Legendre polynomial,EMPTY,P|F:P,$P_{n}$,http://drmf.wmflabs.org/wiki/Definition:bigqLegendre, +\binom{n}{k},binomial coefficient,EMPTY,I|F,$\binom{n}{k}$,http://dlmf.nist.gov/1.2#E1,http://dlmf.nist.gov/26.3#SS1.p1 +\binomial{n}{k},binomial coefficient,EMPTY,I|F,$\binom{n}{k}$,http://dlmf.nist.gov/1.2#E1,http://dlmf.nist.gov/26.3#SS1.p1 +\cDiff[x],central difference,EMPTY,O|F,"$\,\delta_{x}$",http://dlmf.nist.gov/18.1#EGx3, +\card,number of elements of a finite set,EMPTY,SN|F,$\left|A\right|$,http://dlmf.nist.gov/26.1, +\cartprod,$G\cartprod H$: Cartesian product of groups $G$ and $H$,EMPTY,O|F,$\times$,http://dlmf.nist.gov/23.1, +\ceiling{x},ceiling,EMPTY,I|F,$\left\lceil a\right\rceil$,http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r17, +\conj{z},complex conjugate,EMPTY,O|F,$\overline{a}$,http://dlmf.nist.gov/1.9#E11, +\cos@@{z},cosine function,cos,F|F:P:nP,$\mathrm{cos}$,http://dlmf.nist.gov/4.14#E2, +\cosh@@{z},hyperbolic cosine function,hyperbolic-cosine,F|F:P:nP,$\mathrm{cosh}$,http://dlmf.nist.gov/4.28#E2, +\cosintg@{a}{z},generalized cosine integral ${\rm ci}$,generalized-cosine-integral-ci,F|F:P,$\mathrm{ci}$,http://dlmf.nist.gov/8.21#E1, +\cot@@{z},cotangent function,cotangent,F|F:P:nP,$\mathrm{cot}$,http://dlmf.nist.gov/4.14#E7, +\coth@@{z},hyperbolic cotangent function,hyperbolic-cotangent,F|F:P:nP,$\mathrm{coth}$,http://dlmf.nist.gov/4.28#E7, +\cpi,ratio of a circle's circumference to its diameter,ratio-of-a-circle's-circumference-to-its-diameter,C|F,$\pi$,http://dlmf.nist.gov/5.19.E4, +\csc@@{z},cosecant function,cosecant,F|F:P:nP,$\mathrm{csc}$,http://dlmf.nist.gov/4.14#E5, +\csch@@{z},hyperbolic cosecant function,hyperbolic-cosecant,F|F:P:nP,$\mathrm{csch}$,http://dlmf.nist.gov/4.28#E5, +\ctsHahn{n}@@{x}{a}{b}{c}{d},continuous Hahn polynomial,continuous-Hahn-polynomial,P|F:P:PS,$p_{n}$,http://dlmf.nist.gov/18.19#P2.p1, +\ctsbigqHermite{n}@{x}{a}{q},continuous big $q$-Hermite polynomial,EMPTY,P|F:P,$H_{n}$,http://drmf.wmflabs.org/wiki/Definition:ctsbigqHermite, +\ctsdualHahn{n}@@{x^2}{a}{b}{c},continuous dual Hahn polynomial,continuous-dual-Hahn-normalized-S,P|F:P:PS,$S_{n}$,http://dlmf.nist.gov/18.25#T1.t1.r3, +\ctsdualqHahn{n}@{x}{a}{b}{c}{q},continuous dual $q$-Hahn polynomial,EMPTY,P|F:P,$p_{n}$,http://drmf.wmflabs.org/wiki/Definition:ctsdualqHahn, +\ctsqHahn{n}@{x}{a}{b}{c}{d}{q},continuous $q$-Hahn polynomial,EMPTY,P|F:P,$p_{n}$,http://drmf.wmflabs.org/wiki/Definition:ctsqHahn, +\ctsqHermite{n}@@{x}{q},continuous $q$-Hermite polynomial,continuous-q-Hermite-polynomial-H,P|F:P:PS,$H_{n}$,http://drmf.wmflabs.org/wiki/Definition:ctsqHermite, +\ctsqJacobi{\alpha}{\beta}{m}@{x}{q},continuous $q$-Jacobi polynomial,EMPTY,P|F:P,"$P^{(\alpha,\beta)}_{n}$",http://drmf.wmflabs.org/wiki/Definition:ctsqJacobi, +\ctsqJacobiRahman{n}@@{x}{q},continuous $q$-Jacobi-Rahman-polynomial,continuous-q-Jacobi-Rahman-polynomial-P,P|F:P:PS,"$P^{(\alpha,\beta)}_{n}$",http://drmf.wmflabs.org/wiki/Definition:ctsqJacobiRahman, +\ctsqLaguerre{\alpha}{n}@{x}{q},continuous $q$-Laguerre polynomial,EMPTY,P|F:P,$P^{(n)}_{\alpha}$,http://drmf.wmflabs.org/wiki/Definition:ctsqLaguerre, +\ctsqLegendre{n}@@{x}{q},continuous $q$-Legendre polynomial,continuous-q-Legendre-polynomial-P,P|F:P:PS,$P_{n}$,http://drmf.wmflabs.org/wiki/Definition:ctsqLegendre, +\ctsqLegendreRahman{n}@@{x}{q},continuous $q$-Legendre-Rahman polynomial,continuous-q-Legendre-Rahman-polynomial-P,P|F:P:PS,$P_{n}$,http://drmf.wmflabs.org/wiki/Definition:ctsqLegendreRahman, +\ctsqUltra{n}@{x}{\beta}{q},continuous $q$-ultraspherical/Rogers polynomial,continuous-q-ultraspherical-Rogers-polynomial,P|F:P,$C_{n}$,http://dlmf.nist.gov/18.28#E13, +\ctsqUltrae{n}@{e^{i\theta}}{\beta}{q},continuous $q$-ultraspherical/Rogers polynomial exponential argument,continuous-q-ultraspherical-Rogers-polynomial-exponential-argument,P|F:P,$C_{n}$,http://drmf.wmflabs.org/wiki/Definition:ctsqUltrae, +\curl,curl of vector-valued function,EMPTY,F|F,$\nabla\times$,http://dlmf.nist.gov/1.6#E22, +\deltaDistribution,Dirac delta distribution,Dirac delta distribution,D|F,$\delta_x$,http://dlmf.nist.gov/1.17.i, +\deriv{f}{x},derivative,EMPTY,O|F,$\deriv{}{x}}$,http://dlmf.nist.gov/1.4#E4, +\det,determinant,EMPTY,L|F,$\det$,http://dlmf.nist.gov/1.3#i, +\diag,diagonal part of a matrix,EMPTY,L|F,$\diag[\Matrix{a}]$,http://dlmf.nist.gov/21.5.E7, +\diff{x},differential,EMPTY,O|F,$\mathrm{d}^nx$,http://dlmf.nist.gov/1.4#iv, +\digamma@{z},psi (or digamma) function,digamma,F|F:P,$\psi$,http://dlmf.nist.gov/5.2#E2, +\discrqHermiteI{n}@@{x}{q},discrete $q$-Hermite I polynomial,discrete-q-Hermite-polynomial-h-I,P|F:P:PS,$h_{n}$,http://drmf.wmflabs.org/wiki/Definition:discrqHermiteI, +\discrqHermiteII{n}@@{x}{q},discrete $q$-Hermite II polynomial,discrete-q-Hermite-polynomial-II-h-tilde,P|F:P:PS,$\tilde{h}_{n}$,http://drmf.wmflabs.org/wiki/Definition:discrqHermiteII, +\divergence,divergence of vector-valued function,EMPTY,F|F,$\mathrm{div}$,http://dlmf.nist.gov/1.6#E21, +\divides,divides,EMPTY,O|F,$\nabla\cdot$,http://dlmf.nist.gov/24.1, +\dualHahn{n}@@{\lambda(x)}{\gamma}{\delta}{N},dual Hahn polynomial,dual-Hahn-polynomial-R,P|F:P:PS,$R_{n}$,http://dlmf.nist.gov/18.25#T1.t1.r5, +\dualqHahn{n}@@{\mu(x)}{\gamma}{\delta}{N}{q},dual $q$-Hahn polynomial,dual-q-Hahn-R,P|F:P:PS,$R_{n}$,http://drmf.wmflabs.org/wiki/Definition:dualqHahn, +\dualqKrawtchouk{n}@@{\lambda(x)}{c}{N}{q},dual $q$-Krawtchouk polynomial,dual-q-Krawtchouk-polynomial-K,P|F:P:PS,$K_{n}$,http://drmf.wmflabs.org/wiki/Definition:dualqKrawtchouk, +\erf@@{z},error function,error-function,F|F:P:nP,$\mathrm{erf}$,http://dlmf.nist.gov/7.2#E1, +\erfc@@{z},complementary error function,repeated-integral-complementary-error-function,F|F:P:nP,$\mathrm{erfc}$,http://dlmf.nist.gov/7.2#E2, +\erfw@@{z},complementary error function $w$,error-function-w,F|F:P:nP,$w$,http://dlmf.nist.gov/7.2#E3, +\exp@@{z},exponential function,exponential,F|F:P:nP,$\mathrm{exp}$,http://dlmf.nist.gov/4.2#E19, +\expe,the base of the natural logarithm,EMPTY,F|F,$\mathrm{e}$,http://dlmf.nist.gov/4.2.E11, +\exptrace@{\Matrix{X}},exponential of trace of a matrix,exponential-trace,L|F:P,$\mathrm{etr}$,http://dlmf.nist.gov/35.1#p3.t1.r10, +\f{f}@{x},function,EMPTY,SM|F:P,${f}$,http://drmf.wmflabs.org/wiki/Definition:f, +\fDiff,forward difference operator,EMPTY,O|F,"$\,\Delta_{y}$",http://dlmf.nist.gov/3.6#SS1.p1, +\floor{x},floor,EMPTY,I|F,$\left\lfloor a\right\rfloor$,http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r16, +\forall,for every,EMPTY,Q|F,$\forall$,http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r11, +\gradient,gradient of differentiable scalar function,EMPTY,F|F,$\nabla$,http://dlmf.nist.gov/1.6#E20, +"\idem@{\chi_1}{\chi_1,...,\chi_n}",$\idem$ function,idem,F|F:P,$\mathrm{idem}$,http://dlmf.nist.gov/17.1#p3, +\ifrac,semantic slash fraction,EMPTY,SM|F,$(a)/(b)$,http://dlmf.nist.gov/4.2.E8, +\imagpart{z},imaginary part,EMPTY,O|F,"$\Im {z}$",http://dlmf.nist.gov/1.9#E2, +\in,element of,EMPTY,Q|F,$\in$,http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r9, +\incgamma@{a}{z},incomplete gamma function $\gamma$,incomplete-gamma,F|F:P,$\gamma$,http://dlmf.nist.gov/8.2#E1, +\incgammastar@{a}{z},incomplete gamma function $\gamma^{*}$,incomplete-gamma-star,F|F:P,$\gamma^{*}$,http://dlmf.nist.gov/8.2#E6, +\inf,greatest lower bound (infimum),EMPTY,Q|F,$\mathrm{inf}$,http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r24, +\int,integral,EMPTY,F|F,$\int$,http://dlmf.nist.gov/1.4#iv, +\inverf@@{x},inverse error function,inverse-error-function,F|F:P:nP,$\mathrm{inverf}$,http://dlmf.nist.gov/7.17#E1, +\inverfc@@{x},inverse complementary error function,inverse-complementary-error-function,F|F:P:nP,$\mathrm{inverfc}$,http://dlmf.nist.gov/7.17#E1, +\iunit,imaginary unit,EMPTY,C|F,$\mathrm{i}$,http://dlmf.nist.gov/1.9.i, +\liminf,least limit point,EMPTY,Q|F,$\mathrm{liminf}$,http://dlmf.nist.gov/front/introduction#Sx4.p2.t1.r4, +\littleo@{x},order less than,little-o,Q|F:P,$o$,http://dlmf.nist.gov/2.1#E2, +\littleqJacobi{n}@@{x}{a}{b}{q},little $q$-Jacobi polynomial,little-q-Jacobi-polynomial-p,P|F:P:PS,$p_{n}$,http://drmf.wmflabs.org/wiki/Definition:littleqJacobi, +\littleqLaguerre{n}@{x}{a}{q},little $q$-Laguerre / Wall polynomial,EMPTY,P|F:P,$p_{n}$,http://drmf.wmflabs.org/wiki/Definition:littleqLaguerre, +\littleqLegendre{m}@{x}{q},little $q$-Legendre polynomial,EMPTY,P|F:P,$p_{n}$,http://drmf.wmflabs.org/wiki/Definition:littleqLegendre, +\ln@@{z},principal branch of logarithm function,natural-logarithm,F|F:P:nP,$\mathrm{ln}$,http://dlmf.nist.gov/4.2#E2, +\log@@{z},principle branch of natural logarithm,logarithm,F|F:P:nP,$\mathrm{log}$,http://dlmf.nist.gov/4.2#E2, +\logb{a}@@{z},logarithm to general base,logarithm,F|F:P:nP,$\mathrm{log}_{b}$,http://dlmf.nist.gov/4.2#EGx1, +\lrselection,bracketed generalization of $\pm$,EMPTY,SM|F,$\left\{\begin{matrix}x\end{matrix}\right\}$,http://drmf.wmflabs.org/wiki/Definition:Lrselection, +\lselection,left bracketed generalization of $\pm$,EMPTY,SM|F,$\left\{\begin{matrix}x\end{matrix}\right.$,http://dlmf.nist.gov/28.4.E22, +\mEulerBeta{m}@{a}{b},multivariate beta function,multivariate-Euler-Beta,F|F:P,$\mathrm{B}_{m}$,http://dlmf.nist.gov/35.3#E3, +\mEulerGamma{m}@{a},multivariate gamma function,multivariate-Euler-Gamma,F|F:P,$\Gamma_{m}$,http://dlmf.nist.gov/35.3#i, +\mod,modulo,EMPTY,Q|F,$\mod$,http://dlmf.nist.gov/front/introduction#Sx4.p2.t1.r10, +\monicAffqKrawtchouk{n}@@{q^{-x}}{p}{N}{q},monic affine $q$-Krawtchouk polynomial,affine-q-Krawtchouk-polynomial-monic-p,P|F:P:PS,$\widehat{K}^{\mathrm{Aff}}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicAffqKrawtchouk, +\monicAlSalamCarlitzI{a}{n}@@{x}{q},monic Al-Salam-Carlitz I polynomial,q-Al-Salam-Carlitz-I-polynomial-monic-p,P|F:P:PS,${\widehat U}^{(\alpha)}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicAlSalamCarlitzI, +\monicAlSalamCarlitzII{a}{n}@@{x}{q},monic Al-Salam-Carlitz II polynomial,q-Al-Salam-Carlitz-II-polynomial-monic-p,P|F:P:PS,${\widehat V}^{(n)}_{\alpha}$,http://drmf.wmflabs.org/wiki/Definition:monicAlSalamCarlitzII, +\monicAlSalamChihara{n}@@{x}{a}{b}{q},monic Al-Salam-Chihara polynomial,Al-Salam-Chihara-polynomial-monic-p,P|F:P:PS,${\widehat Q}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicAlSalamChihara, +\monicAskeyWilson{n}@@{x}{a}{b}{c}{d}{q},monic Askey-Wilson polynomial,Askey-Wilson-polynomial-monic-p,P|F:P:PS,${\widehat p}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicAskeyWilson, +\monicBesselPoly{n}@@{x}{a},monic Bessel polynomial,Bessel-polynomial-monic-p,P|F:P:PS,${\widehat y}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicBesselPoly, +\monicCharlier{n}@@{x}{a},monic Charlier polynomial,Charlier-polynomial-monic-p,P|F:P:PS,${\widehat C}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicCharlier, +\monicChebyT{n}@@{x},monic Chebyshev polynomial of the first kind,Chebyshev-polynomial-first-kind-monic-p,P|F:P:PS,${\widehat T}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicChebyT, +\monicChebyU{n}@@{x},monic Chebyshev polynomial of the second kind,Chebyshev-polynomial-second-kind-monic-p,P|F:P:PS,${\widehat U}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicChebyU, +\monicHahn{n}@@{x}{\alpha}{\beta}{N},monic Hahn polynomial,Hahn-polynomial-monic-p,P|F:P:PS,${\widehat Q}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicHahn, +\monicHermite{n}@@{x},monic Hermite polynomial,Hermite-polynomial-monic,P|F:P:PS,${\widehat H}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicHermite, +\monicJacobi{n}@@{x},monic Jacobi polynomial,Jacobi-polynomial-monic-p,P|F:P:PS,"${\widehat P}^{(\alpha,\beta)}_{n}$",http://drmf.wmflabs.org/wiki/Definition:monicJacobi, +\monicKrawtchouk{n}@@{q^{-x}}{p}{N}{q},monic Krawtchouk polynomial,Krawtchouk-polynomial-monic-p,P|F:P:PS,${\widehat K}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicKrawtchouk, +\monicLaguerre{n}@@{x},monic Laguerre polynomial,generalized-Laguerre-polynomial-monic-p,P|F:P:PS,${\widehat L}^{\alpha}_n$,http://drmf.wmflabs.org/wiki/Definition:monicLaguerre, +\monicLegendrePoly{n}@@{x},monic Legendre polynomial,Legendre-spherical-polynomial-monic-p,P|F:P:PS,${\widehat P}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicLegendrePoly, +\monicMeixner{n}@@{x}{\beta}{c},monic Meixner polynomial,Meixner-polynomial-monic-p,P|F:P:PS,${\widehat M}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicMeixner, +\monicMeixnerPollaczek{\lambda}{n}@@{x}{\phi},monic Meixner-Pollaczek polynomial,Meixner-Pollaczek-polynomial-monic-p,P|F:P:PS,${\widehat P}^{(\alpha)}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicMeixnerPollaczek, +\monicRacah{n}@@{\lambda(x)}{\alpha}{\beta}{\gamma}{\delta},monic Racah polynomial,Racah-polynomial-monic-p,P|F:P:PS,${\widehat R}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicRacah, +\monicStieltjesWigert{n}@@{x}{q},monic Stieltjes-Wigert polynomial,Stieltjes-Wigert-polynomial-monic-p,P|F:P:PS,${\widehat S}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicStieltjesWigert, +\monicUltra{n}@@{x},monic ultraspherical/Gegenbauer polynomial,ultraspherical-Gegenbauer-polynomial-monic-p,P|F:P:PS,${\widehat C}^{\mu}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicUltra, +\monicWilson{n}@@{x^2}{a}{b}{c}{d},monic Wilson polynomial,Wilson-polynomial-monic-p,P|F:P:PS,${\widehat W}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicWilson, +\monicbigqJacobi{n}@@{x}{a}{b}{c}{q},monic big $q$-Jacobi polynomial,big-q-Jacobi-polynomial-monic-p,P|F:P:PS,${\widehat P}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicbigqJacobi, +\monicbigqLaguerre{n}@@{x}{a}{b}{q},monic big $q$-Laguerre polynomial,big-q-Laguerre-polynomial-monic-p,P|F:P:PS,${\widehat P}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicbigqLaguerre, +\monicbigqLegendre{n}@@{x}{c}{q},monic big $q$-Legendre polynomial,big-q-Legendre-polynomial-monic-p,P|F:P:PS,${\widehat P}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicbigqLegendre, +\monicctsHahn{n}@@{x}{a}{b}{c}{d},monic continuous Hahn polynomial,continuous-Hahn-polynomial-monic-p,P|F:P:PS,${\widehat p}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicctsHahn, +\monicctsbigqHermite{n}@@{x}{a}{q},monic continuous big $q$-Hermite polynomial,continuous-big-q-Hermite-polynomial-monic-p,P|F:P:PS,${\widehat H}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicctsbigqHermite, +\monicctsdualHahn{n}@@{x^2}{a}{b}{c}{d},monic continuous dual Hahn polynomial,continuous-dual-Hahn-polynomial-monic-p,P|F:P:PS,${\widehat S}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicctsdualHahn, +\monicctsdualqHahn{n}@@{x^2}{a}{b}{c},monic continuous dual $q$-Hahn polynomial,continuous-dual-q-Hahn-polynomial-monic-p,P|F:P:PS,${\widehat p}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicctsdualqHahn, +\monicctsqHahn{n}@@{x}{a}{b}{c}{d}{q},monic continuous $q$-Hahn polynomial,continuous-q-Hahn-polynomial-monic-p,P|F:P:PS,${\widehat p}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicctsqHahn, +\monicctsqHermite{n}@@{x}{q},monic continuous $q$-Hermite polynomial,continuous-q-Hermite-polynomial-monic-p,P|F:P:PS,${\widehat H}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicctsqHermite, +\monicctsqJacobi{n}@@{x}{q},monic continuous $q$-Jacobi polynomial,continuous-q-Jacobi-polynomial-monic-p,P|F:P:PS,"${\widehat P}^{(\alpha,\beta)}_{n}$",http://drmf.wmflabs.org/wiki/Definition:monicctsqJacobi, +\monicctsqLaguerre{n}@@{x}{q},monic continuous $q$-Laguerre polynomial,continuous-q-Laguerre-polynomial-monic-p,P|F:P:PS,${\widehat P}^{(\alpha)}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicctsqLaguerre, +\monicctsqLegendre{n}@@{x}{q},monic continuous $q$-Legendre polynomial,continuous-q-Legendre-polynomial-monic-p,P|F:P:PS,${\widehat P}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicctsqLegendre, +\monicctsqUltra{n}@@{x}{\beta}{q},monic continuous $q$-ultraspherical/Rogers polynomial,continuous-q-ultraspherical-Rogers-polynomial-monic-p,P|F:P:PS,${\widehat C}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicctsqUltra, +\monicdiscrqHermiteI{n}@@{x}{q},monic discrete $q$-Hermite I polynomial,discrete-q-Hermite-polynomial-I-monic-p,P|F:P:PS,${\widehat h}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicdiscrqHermiteI, +\monicdiscrqHermiteII{n}@@{x}{q},monic discrete $q$-Hermite II polynomial,discrete-q-Hermite-polynomial-II-monic-p,P|F:P:PS,${\widehat \tilde{p}_{n}}$,http://drmf.wmflabs.org/wiki/Definition:monicdiscrqHermiteII, +\monicdualHahn{n}@@{\lambda(x)}{\gamma}{\delta}{N},monic dual Hahn polynomial,dual-Hahn-polynomial-monic-p,P|F:P:PS,${\widehat R}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicdualHahn, +\monicdualqHahn{n}@@{\mu(x)}{\gamma}{\delta}{N}{q},monic dual $q$-Hahn polynomial,dual-q-Hahn-monic-p,P|F:P:PS,${\widehat R}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicdualqHahn, +\monicdualqKrawtchouk{n}@@{\lambda(x)}{c}{N}{q},monic dual $q$-Krawtchouk polynomial,dual-q-Krawtchouk-polynomial-monic-p,P|F:P:PS,${\widehat K}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicdualqKrawtchouk, +\moniclittleqJacobi{n}@@{x}{a}{b}{q},monic little $q$-Jacobi polynomial,little-q-Jacobi-polynomial-monic-p,P|F:P:PS,${\widehat p}_{n}$,http://drmf.wmflabs.org/wiki/Definition:moniclittleqJacobi, +\moniclittleqLaguerre{n}@@{x}{a}{q},monic little $q$-Laguerre/Wall polynomial,little-q-Laguerre-Wall-polynomial-monic-p,P|F:P:PS,${\widehat p}_{n}$,http://drmf.wmflabs.org/wiki/Definition:moniclittleqLaguerre, +\moniclittleqLegendre{n}@@{x}{q},monic little $q$-Legendre polynomial,little-q-Legendre-polynomial-monic-p,P|F:P:PS,${\widehat p}_{n}$,http://drmf.wmflabs.org/wiki/Definition:moniclittleqLegendre, +\monicpseudoJacobi{n}@@{x}{\nu}{N},monic pseudo-Jacobi polynomial,pseudo-Jacobi-polynomial-monic-p,P|F:P:PS,${\widehat P}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicpseudoJacobi, +\monicqBesselPoly{n}@@{x}{a}{q},monic $q$-Bessel polynomial,q-Bessel-polynomial-monic-p,P|F:P:PS,${\widehat y}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicqBessel, +\monicqCharlier{n}@@{q^{-x}}{a}{q},monic $q$-Charlier polynomial,q-Charlier-polynomial-monic-p,P|F:P:PS,${\widehat C}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicqCharlier, +\monicqHahn{n}@@{q^{-x}}{\alpha}{\beta}{N},monic $q$-Hahn polynomial,q-Hahn-polynomial-monic-p,P|F:P:PS,${\widehat Q}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicqHahn, +\monicqKrawtchouk{n}@@{q^{-x}}{p}{N}{q},monic $q$-Krawtchouk polynomial,q-Krawtchouk-polynomial-monic-p,P|F:P:PS,${\widehat K}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicqKrawtchouk, +\monicqLaguerre{n}@@{x}{q},monic $q$-Laguerre polynomial,q-Laguerre-polynomial-monic-p,P|F:P:PS,${\widehat L}^{(\alpha)}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicqLaguerre, +\monicqMeixner{n}@@{q^{-x}}{b}{c}{q},monic $q$-Meixner polynomial,q-Meixner-polynomial-monic-p,P|F:P:PS,${\widehat M}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicqMeixner, +\monicqMeixnerPollaczek{n}@@{x}{a}{q},monic $q$-Meixner-Pollaczek polynomial,q-Meixner-Pollaczek-polynomial-monic-p,P|F:P:PS,${\widehat P}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicqMeixnerPollaczek, +\monicqRacah{n}@@{\mu(x)}{\alpha}{\beta}{\gamma}{\delta}{q},monic $q$-Racah polynomial,q-Racah-polynomial-monic-R,P|F:P:PS,${\widehat R}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicqRacah, +\monicqinvAlSalamChihara{n}@{x}{a}{b}{q^{-1}},monic $q$-inverse Al-Salam-Chihara polynomial,q-inverse-AlSalam-Chihara-polynomial-monic-p,P|F:P,${\widehat Q}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicqinvAlSalamChihara, +\monicqtmqKrawtchouk{n}@@{q^{-x}}{p}{N}{q},monic quantum $q$-Krawtchouk polynomial,quantum-q-Krawtchouk-polynomial-monic-p,P|F:P:PS,${\widehat K^{\mathrm{qtm}}}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicqtmqKrawtchouk, +"\multinomial{n_1+n_2+\dots+n_k}{n_1,...,n_k}",multinomial coefficient,EMPTY,I|F,$\left(A;B\right)$,http://dlmf.nist.gov/26.4.E2, +\ninej@@{j_{11}}{j_{12}}{j_{13}}{j_{21}}{j_{22}}{j_{23}}{j_{31}}{j_{32}}{j_{33}},$\ninej$ symbol,EMPTY,F|F:P:nP,$$,http://dlmf.nist.gov/34.6#E1, +\normAskeyWilsonptilde{n}@@{x}{a}{b}{c}{d}{q},normalized Askey-Wilson polynomial ${\tilde p}$,Askey-Wilson-polynomial-normalized-p-tilde,P|F:P:PS,${\tilde p}_{n}$,http://drmf.wmflabs.org/wiki/Definition:normAskeyWilsonptilde, +\normJacobiR{\alpha}{\beta}{n}@{x},normalized Jacobi polynomial,Jacobi-polynomial-R,P|F:P,"$R^{(\alpha,\beta)}_{n}$",http://drmf.wmflabs.org/wiki/Definition:normJacobiR, +\normWilsonWtilde{n}@@{x^2}{a}{b}{c}{d},normalized Wilson polynomial ${\tilde W}$,Wilson-polynomial-normalized-W-tilde,P|F:P:PS,${\tilde W}_{n}$,http://drmf.wmflabs.org/wiki/Definition:normWilsonWtilde, +\normctsHahnptilde{n}@@{x}{a}{b}{c}{d},normalized continuous Hahn polynomial ${\tilde p}$,continuous-Hahn-polynomial-normalized-p-tilde,P|F:P:PS,${\tilde p}_{n}$,http://drmf.wmflabs.org/wiki/Definition:normctsHahnptilde, +\normctsdualHahnStilde{n}@@{x^2}{a}{b}{c}{d},normalized continuous dual Hahn polynomial ${\tilde S}$,continuous-dual-Hahn-polynomial-normalized-S-tilde,P|P:PS,${\tilde S}_{n}$,http://drmf.wmflabs.org/wiki/Definition:normctsdualHahnStilde, +\normctsdualqHahnptilde{n}@@{x}{a}{b}{c}{q},normalized continuous dual $q$-Hahn polynomial ${\tilde p}$,continuous-dual-q-Hahn-polynomial-normalized-p-tilde,P|F:P:PS,${\tilde p}_{n}$,http://drmf.wmflabs.org/wiki/Definition:normctsdualqHahnptilde, +\normctsqHahnptilde{n}@@{x}{a}{b}{c}{d}{q},normalized continuous $q$-Hahn polynomial ${\tilde p}$,continuous-q-Hahn-polynomial-normalized-p-tilde,P|F:P:PS,${\tilde p}_{n}$,http://drmf.wmflabs.org/wiki/Definition:normctsqHahnptilde, +\normctsqJacobiPtilde{n}@@{x}{q},normalized continuous $q$-Jacobi polynomial ${\tilde P}$,continuous-q-Jacobi-polynomial-P-tilde,P|F:P:PS,"${\tilde P}^{(\alpha,\beta)}_{n}$",http://drmf.wmflabs.org/wiki/Definition:normctsqJacobiPtilde, +\notin,not an element of,EMPTY,Q|F,$\notin$,http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r10, +\opminus,negative unity to an integer power,EMPTY,I|F,$(-1)$,http://dlmf.nist.gov/5.7.E7, +\pderiv{f}{x},partial derivative,EMPTY,O|F,$\frac{\partial f}{\partial x}$,http://dlmf.nist.gov/1.5#E3, +\pdiff{x},power of a partial differential,EMPTY,O|F,"$\partial^nx$",http://dlmf.nist.gov/1.5#E3, +\pgcd,greatest common divisor,EMPTY,I|F,"$\left(m,n\right)$",http://dlmf.nist.gov/27.1, +\ph@@{z},phase,phase,O|F:P:nP,$\mathrm{ph}$,http://dlmf.nist.gov/1.9#E7, +\pochhammer{a}{n},Pochhammer symbol,EMPTY,F|F,$(a)_n$,http://dlmf.nist.gov/5.2#iii, +\poly{p}{n}@{x},polynomial,EMPTY,SM|F:P,${p}_{n}$,http://drmf.wmflabs.org/wiki/Definition:poly, +\polygamma{n}@{z},polygamma functions,polygamma,F|F:P,$\psi^{(n)}$,http://dlmf.nist.gov/5.15, +\power,power function,EMPTY,F|F,$x^y$,http://dlmf.nist.gov/36.12.E9, +\prod,product,EMPTY,SM|F,$\Pi$,http://drmf.wmflabs.org/wiki/Definition:prod, +\pseudoJacobi{n}@{x}{\nu}{N},pseudo Jacobi polynomial,EMPTY,P|F:P,$P_{n}$,http://drmf.wmflabs.org/wiki/Definition:pseudoJacobi, +\psfactorial{a}{\kappa},partitional shifted factorial,EMPTY,F|F,${\left[a\right]_{\kappa}}$,http://dlmf.nist.gov/35.4#E1, +\pvint,Cauchy principal value,EMPTY,O|F,$\pvint_a^b$,http://dlmf.nist.gov/1.4#E24, +\qAppelli@{a}{b}{b'}{c}{q}{x}{y},first $q$-Appell function,q-Appell-Phi-1,F|F:P,$\Phi^{(1)}$,http://dlmf.nist.gov/17.4.E5, +\qAppellii@{a}{b}{b'}{c}{c'}{q}{x}{y},second $q$-Appell function,q-Appell-Phi-2,F|F:P,$\Phi^{(2)}$,http://dlmf.nist.gov/17.4.E6, +\qAppelliii@{a}{a'}{b}{b'}{c}{q}{x}{y},third $q$-Appell function,q-Appell-Phi-3,F|F:P,$\Phi^{(3)}$,http://dlmf.nist.gov/17.4.E7, +\qAppelliv@{a}{b}{c}{c'}{q}{x}{y},fourth $q$-Appell function,q-Appell-Phi-4,F|F:P,$\Phi^{(4)}$,http://dlmf.nist.gov/17.4.E8, +\qBernoulli{n}@{x}{q},$q$-Bernoulli polynomial,q-Bernoulli-polynomial,P|F:P,$\beta_{q}$,http://dlmf.nist.gov/17.3#E7, +\qBesselPoly{n}@{x}{b}{q},$q$-Bessel polynomial,EMPTY,P|F:P,$y_{n}$,http://drmf.wmflabs.org/wiki/Definition:qBessel, +\qBeta{q}@{a}{b},$q$-beta function,q-Beta,F|F:P,$\mathrm{B}_{q}$,http://dlmf.nist.gov/5.18#E11, +\qBinomial{n}{m}{q},$q$-binomial coefficient (or Gaussian polynomial),EMPTY,F|F,$\left[n \atop m\right]_{q}$,http://dlmf.nist.gov/17.2#E27,http://dlmf.nist.gov/26.9#SS2.p1 +\qCharlier{n}@{x}{c}{q},$q$-Charlier polynomial,EMPTY,P|F:P,$C_{n}$,http://drmf.wmflabs.org/wiki/Definition:qCharlier, +\qCos{q}@@{x},$q$-analogue of the $\cos$ function: ${\rm Cos}_q$,q-Cos,F|F:nP:P,$\mathrm{Cos}_{q}$,http://dlmf.nist.gov/17.3#E6, +\qCosKLS{q}@@{z},$q$-analogue of the $\cos$ function used in KLS: $\mathrm{Cos}_{q}$,q-Cos-KLS,P|F:nP:P,$\mathrm{Cos}_{q}$,http://drmf.wmflabs.org/wiki/Definition:qCosKLS, +\qDigamma{q}@{z},$q$-digamma function,q-digamma,F|F:P,$\psi_{q}$,http://drmf.wmflabs.org/wiki/Definition:qDigamma, +\qEuler{m}{s}@{q},$q$-Euler number,q-Euler-number,F|F:P,"$A_{m,n}$",http://dlmf.nist.gov/17.3#E8, +\qExp{q}@@{x},$q$-analogue of the $\exp$ function: $E_q$,q-Exp,F|F:nP:P,$E_{q}$,http://dlmf.nist.gov/17.3#E2, +\qExpKLS{q}@@{z},$q$-analogue of the $\exp$ function used in KLS: $\mathrm{E}_{q}$,q-Exp-KLS,F|F:nP:P,$\mathrm{E}_{q}$,http://drmf.wmflabs.org/wiki/Definition:qExpKLS, +\qFactorial{n}{q},$q$-factorial,q-factorial,F|F,$n!_q$,http://dlmf.nist.gov/5.18#E2, +\qGamma{q}@{z},$q$-gamma function,q-gamma,F|F:P,$\Gamma_{q}$,http://dlmf.nist.gov/5.18#E4, +\qHahn{n}@@{q^{-x}}{\alpha}{\beta}{N},$q$-Hahn polynomial,q-Hahn-polynomial-Q,P|P:PS,$Q_{n}$,http://drmf.wmflabs.org/wiki/Definition:qHahn, +\qHahnQ{n}@{x}{\alpha}{\beta}{N}{q},$q$-Hahn polynomial,q-Hahn-polynomial-Q,P|F:P,$Q_{n}$,http://dlmf.nist.gov/18.27#E3, +\qHermiteH{n}@{x}{q},continuous $q$-Hermite polynomial,continuous-q-Hermite-polynomial-H,P|F:P,$H_{n}$,http://dlmf.nist.gov/18.28#E16, +\qHermitehI{n}@{x}{q},discrete $q$-Hermite I polynomial,discrete-q-Hermite-polynomial-h-I,P|F:P,$h_{n}$,http://dlmf.nist.gov/18.27#E21, +\qHermitehII{n}@{x}{q},discrete $q$-Hermite II polynomial,discrete-q-Hermite-polynomial-h-II,P|F:P,$\tilde{h}_{n}$,http://dlmf.nist.gov/18.27#E23, +"\qHyperrWs{r}{r+1}@@@{a_1}{a_4,a_5,...,a_{r+1}}{q}{z}",very-well-poised basic hypergeometric (or $q$-hypergeometric) function,very-well-poised-q-hypergeometric-rWs,F|F:fo1:fo2:fo3,${{}_{r+1}W_{r}}$,http://drmf.wmflabs.org/wiki/Definition:qHyperrWs, +"\qHyperrphis{r}{s}@@@{a_1,...a_r}{b_1,...,b_s}{q}{z}",basic hypergeometric (or $q$-hypergeometric) function,q-hypergeometric-rphis,F|F:fo1:fo2:fo3,${{}_{r}\phi_{s}}$,http://dlmf.nist.gov/17.4#E1, +"\qHyperrpsis{r}{s}@@@{a_1,...,a_r}{b_1,...,b_s}{q}{z}",bilateral basic hypergeometric (or bilateral $q$-hypergeometric) function,q-hypergeometric-rpsis,F|F:fo1:fo2:fo3,${{}_{r}\psi_{s}}$,http://dlmf.nist.gov/17.4#E3, +\qJacobiP{n}@{x}{a}{b}{c}{q},"big $q$-Jacobi polynomial $P_n(x;a,b,c;q)$",q-Jacobi-polynomial-P,P|F:P,$P_{n}$,http://dlmf.nist.gov/18.27#E5, +\qJacobiPP{\alpha}{\beta}{n}@{x}{c}{d}{q},"big $q$-Jacobi polynomial $P_n^{(\alpha,\beta)}(x;c,d;q)$",big-q-Jacobi-polynomial-P-type-2,P|F:P,"$P^{(\alpha,\beta)}_{n}$",http://dlmf.nist.gov/18.27#E6, +\qJacobip{n}@{x}{a}{b}{q},little $q$-Jacobi polynomial,q-Jacobi-polynomial-p,P|F:P,$p_{n}$,http://dlmf.nist.gov/18.27#E13, +\qKrawtchouk{n}@@{q^{-x}}{p}{N}{q},$q$-Krawtchouk polynomial,q-Krawtchouk-polynomial-K,P|F:P:PS,$K_{n}$,http://drmf.wmflabs.org/wiki/Definition:qKrawtchouk, +\qLaguerre[\alpha]{n}@@{x}{q},$q$-Laguerre polynomial,q-Laguerre-polynomial-L,P|FnO:PnO:nPnO:F:P:PS,$L_n^{(\alpha)}$,http://drmf.wmflabs.org/wiki/Definition:qLaguerre, +\qLaguerreL[\alpha]{n}@{x}{q},$q$-Laguerre polynomial,q-Laguerre-polynomial-L,P|FnO:PnO:FO:PO,$L_n^{(\alpha)}$,http://dlmf.nist.gov/18.27#E15, +\qLegendre{n}@{x}{q},continuous $q$-Legendre polynomial,EMPTY,P|F:P,$\qLegendre{n}$,http://drmf.wmflabs.org/wiki/Definition:qLegendre, +\qMeixner{n}@{x}{b}{c}{q},$q$-Meixner polynomial,q-Meixner-polynomial-M,P|F:P,$M_{n}$,http://drmf.wmflabs.org/wiki/Definition:qMeixner, +\qMeixnerPollaczek{n}@{x}{a}{q},$q$-Meixner-Pollaczek polynomial,q-Meixner-Pollaczek-polynomial-M,P|F:P,$P_{n}$,http://drmf.wmflabs.org/wiki/Definition:qMeixnerPollaczek, +"\qMultiPochhammer{a_1,\ldots,a_n}{q}{n}",$q$-multi-Pochhammer symbol,q-multi-Pochhammer,F|F,"$\(a_1,\ldots,a_k;q)_n$",http://dlmf.nist.gov/17.2.E5, +"\qMultinomial{a_1+\ldots+a_n}{a_1,\ldots,a_n}{q}",$q$-multinomial,q-multinomial,F|F,"$\left[a_1+\ldots+a_n \atop a_1,\ldots,a_n\right]_{q}$",http://dlmf.nist.gov/26.16.E1, +\qPochhammer{a}{q}{n},$q$-Pochhammer symbol,EMPTY,F|F,$(a;q)_n$,http://dlmf.nist.gov/5.18#i,http://dlmf.nist.gov/17.2#SS1.p1 +\qPolygamma{n}{q}@{z},$q$-polygamma function,q-polygamma,F|F:P,$\psi_{q}^{(n)}$,http://drmf.wmflabs.org/wiki/Definition:qPolygamma, +\qRacah{n}@{x}{\alpha}{\beta}{\gamma}{\delta}{q},$q$-Racah polynomial,q-Racah-polynomial-R,P|F:P,$R_{n}$,http://dlmf.nist.gov/18.28#E19, +\qRacahR{n}@{x}{\alpha}{\beta}{\gamma}{\delta}{q},$q$-Racah polynomial,q-Racah-polynomial-R,P|F:P,$R_{n}$,http://dlmf.nist.gov/18.28#E19, +\qSin{q}@@{x},$q$-analogue of the $\sin$ function: ${\rm Sin}_q$,q-Sin,F|F:nP:P,$\mathrm{Sin}_{q}$,http://dlmf.nist.gov/17.3#E4, +\qSinKLS{q}@@{z},$q$-analogue of the $\sin$ function used in KLS: $\mathrm{Sin}_q$,q-Sin-KLS,F|F:nP:P,$\mathrm{Sin}_{q}$,http://drmf.wmflabs.org/wiki/Definition:qSinKLS, +\qStirling{m}{s}@{q},$q$-Stirling number,q-Stirling-number,F|F:P,"$a_{m,s}$",http://dlmf.nist.gov/17.3#E9, +\qUltraspherical{n}@{x}{\beta}{q},continuous $q$-ultraspherical polynomial,q-ultraspherical-polynomial,P|F:P,$C_{n}$,http://dlmf.nist.gov/18.28#E13, +\qcos{q}@@{x},$q$-analogue of the $\cos$ function: $\cos_q$,q-cos,F|F:nP:P,$\mathrm{cos}_{q}$,http://dlmf.nist.gov/17.3#E5, +\qcosKLS{q}@@{z},$q$-analogue of the $\cos$ function used in KLS: $\cos_q$,q-cos-KLS,P|F:nP:P,$\mathrm{cos}_{q}$,http://drmf.wmflabs.org/wiki/Definition:qcosKLS, +\qderiv[n]{q}@{z},$q$-derivative,q-derivative,O|FO:PO:FnO:PnO,$\mathcal{D}_q^n$,http://drmf.wmflabs.org/wiki/Definition:qderiv, +\qdiff{q}{x},$q$-differential,EMPTY,O|F,"${\mathrm d}^n_qx$",http://dlmf.nist.gov/17.2#SS5.p1, +\qexp{q}@@{x},$q$-analogue of the $\exp$ function: $e_q$,q-exp,F|F:nP:P,$e_{q}$,http://dlmf.nist.gov/17.3#E1, +\qexpKLS{q}@@{z},$q$-analogue of the $\exp$ function used in KLS: $\mathrm{e}_{q}$,q-exp-KLS,F|F:nP:P,$\mathrm{e}_{q}$,http://drmf.wmflabs.org/wiki/Definition:qexpKLS, +\qinvAlSalamChihara{n}@{x}{a}{b}{q^{-1}},$q$-inverse Al-Salam-Chihara polynomial,q-inverse-AlSalam-Chihara-polynomial-Q,P|F:P,$Q_{n}$,http://dlmf.nist.gov/23.1, +\qinvHermiteh{n}@{x}{q},continuous $q$-inverse Hermite polynomial,continuous-q-inverse-Hermite-polynomial-h,P|F:P,$h_{n}$,http://dlmf.nist.gov/23.1, +\qsin{q}@@{x},$q$-analogue of the $\sin$ function: $\sin_q$,q-sin,F|F:nP:P,$\mathrm{sin}_{q}$,http://dlmf.nist.gov/17.3#E3, +\qsinKLS{q}@@{z},$q$-analogue of the $\sin$ function used in KLS: $\sin_q$,q-sin-KLS,F|F:nP:P,$\mathrm{sin}_{q}$,http://drmf.wmflabs.org/wiki/Definition:qsinKLS, +\qtmqKrawtchouk{n}@@{q^{-x}}{p}{N}{q},quantum $q$-Krawtchouk polynomial,quantum-q-Krawtchouk-polynomial-K,P|F:P:PS,$K^{\mathrm{qtm}}_{n}$,http://drmf.wmflabs.org/wiki/Definition:qtmqKrawtchouk, +\realpart{z},real part,EMPTY,O|F,"$\Re {z}$",http://dlmf.nist.gov/1.9#E2, +\terminant{p}@{z},terminant function,terminant-function,F|F:P,$F_{p}$,http://dlmf.nist.gov/2.11#E11, +\rescaledTerminant{p}@{z},rescaled terminant functions,rescaled-terminant-function,F|F:P,$G_{p}$,http://dlmf.nist.gov/9.7#SS5.p1, +\rselection,right bracketed generalization of $\pm$,EMPTY,SM|F,$\left.\begin{matrix}x\end{matrix}\right\$,http://dlmf.nist.gov/28.4.E22, +\sec@@{z},secant function,secant,F|F:P:nP,$\mathrm{sec}$,http://dlmf.nist.gov/4.14#E6, +\sech@@{z},hyperbolic secant function,hyperbolic-secant,F|F:P:nP,$\mathrm{sech}$,http://dlmf.nist.gov/4.28#E6, +\selection,generalization of $\pm$,EMPTY,SM|F,$\;\begin{matrix}x\end{matrix}\;$,http://dlmf.nist.gov/10.23.E7, +\setminus,set subtraction,EMPTY,Q|F,$\setminus$,http://dlmf.nist.gov/front/introduction#Sx4.p2.t1.r19, +\setmod,$S_1 \setmod S_2$: set of all elements of $S_1$ modulo elements of $S_2$,EMPTY,SN|F,$/$,http://dlmf.nist.gov/21.1#p2.t1.r17, +\sign@@{x},sign function,sign,F|F:P:nP,$\mathrm{sign}$,http://dlmf.nist.gov/front/introduction#Sx4.p2.t1.r18, +\sim,asymptotic equality,EMPTY,Q|F,$\tilde$,http://dlmf.nist.gov/2.1#E1, +\sin@@{z},sine function,sin,F|F:P:nP,$\mathrm{sin}$,http://dlmf.nist.gov/4.14#E1, +\sinInt@{z},sine integral ${\rm si}$,shifted-sine-integral,F|F:P,$\mathrm{si}$,http://dlmf.nist.gov/6.2#E10, +\sinh@@{z},hyperbolic sine function,hyperbolic-sine,F|F:P:nP,$\mathrm{sinh}$,http://dlmf.nist.gov/4.28#E1, +\sinintg@{a}{z},generalized sine integral ${\rm si}$,generalized-sine-integral-si,F|F:P,$\mathrm{si}$,http://dlmf.nist.gov/8.21#E1, +\sixj@@{j_{1}}{j_{2}}{j_{3}}{l_{1}}{l_{2}}{l_{3}},$\sixj$ symbol,EMPTY,F|F:P:nP,$\left\{\begin{array}{ccc}j_1 & j_2 & j_3\\l_1 & l_2 & l_3\end{array}\right\}$,http://dlmf.nist.gov/23.1, +\subset,is contained in,EMPTY,Q|F,$\subset$,http://dlmf.nist.gov/front/introduction#Sx4.p2.t1.r2, +\subseteq,is in or is contained in,EMPTY,Q|F,$\subseteq$,http://dlmf.nist.gov/front/introduction#Sx4.p2.t1.r3, +\sum,sum,EMPTY,SM|F,$\Sigma$,http://drmf.wmflabs.org/wiki/Definition:sum, +\sup,least upper bound (supremum),EMPTY,Q|F,$\mathrm{sup}$,http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r25, +\tan@@{z},tangent function,tangent,F|F:P:nP,$\mathrm{tan}$,http://dlmf.nist.gov/4.14#E4, +\tanh@@{z},hyperbolic tangent function,hyperbolic-tangent,F|F:P:nP,$\mathrm{tanh}$,http://dlmf.nist.gov/4.28#E4, +\threej@@{j_1}{j_2}{j_3}{m_1}{m_2}{m_3},$\threej$ symbol,EMPTY,F|F:P:nP,$\left(\begin{array}{ccc}j_1 & j_2 & j_3\\m_1 & m_2 & m_3\end{array}\right)$,http://dlmf.nist.gov/34.2#E4, +\trace,trace of matrix,trace,L|F,$\mathrm{etr}$,http://dlmf.nist.gov/23.1, +\transpose,transpose of a matrix,EMPTY,L|F,$A^{\mathrm{T}}$,http://dlmf.nist.gov/1.3.E5, +\weight{w}@{x},weight function,EMPTY,SM|F:P,${w}$,http://drmf.wmflabs.org/wiki/Definition:weight, +\wigner{j_1}{m_1}{j_2}{m_2}{j_3}{-m_3},Clebsch-Gordan coefficient,EMPTY,F|F,"$\left(j_{1}\;m_{1}\;j_{2}\;m_{2} | j_{1}\;j_{2}\; j_{3}\,\,-m_{3}\right)$",http://dlmf.nist.gov/34.1#p4, +\BesselPolyIIparam{n}@{x}{a}{b},Bessel polynomial with two parameters $y_n$,Bessel-polynomial-two-parameters-y,P|F:P,$y_{n}$,http://drmf.wmflabs.org/wiki/Definition:BesselPolyIIparam, +\BesselPolyTheta{n}@{x}{a}{b},Bessel polynomial with two parameters $\theta_n$,Bessel-polynomial-two-parameters-theta,P|F:P,$\theta_{n}$,http://drmf.wmflabs.org/wiki/Definition:BesselPolyTheta, +\FourierTrans@{f}{s},Fourier transform,Fourier-transform,O|F:P,$\mathscr{F}$,http://dlmf.nist.gov/1.14#E1, +\Continuous@{A},space of continuous functions on a set $A$,set-of-continuous-functions,S|F,$C(A)$,http://dlmf.nist.gov/1.4#ii, +\SpaceTestFunTempered@{A},space of test functions for tempered distributions,space-of-test-functions-for-tempered-distributions-T,S|F,${\mathcal T}(\Real)$,http://dlmf.nist.gov/1.16#v, diff --git a/Azeem/tex2Wiki.py b/Azeem/tex2Wiki.py new file mode 100644 index 0000000..0711a0f --- /dev/null +++ b/Azeem/tex2Wiki.py @@ -0,0 +1,1069 @@ +# Version 14: Templates +# Convert tex to wikiText +import csv # imported for using csv format +import sys # imported for getting args +import os # imported for copying file + + +def isnumber(char): + return char[0] in "0123456789" + + +def getString(line): # Gets all data within curly braces on a line + # -------Initialization------- + stringWrite = "" + getStr = False + pW = "" + # ---------------------------- + for c in line: + if c == "{" or c == "}": # if there is a curly brace in the line + getStr = not (getStr) # toggle the getStr flag + if c == "}": # no more info needed + break + elif getStr: # if within curly braces + if not (pW == c and c == "-"): # if there is no double dash (makes single dash) + if c != "$": # if not a $ sign + stringWrite += c # this character is part of the data + else: # replace $ with '' + stringWrite += "\'\'" # add double ' + pW = c # change last character for finding double dashes + return (stringWrite.rstrip('\n').lstrip()) # return the data without newlines and without leading spaces + + +def getG(line): # gets equation for symbols list + start = True + final = "" + for c in line: + if c == "$" and start: + final += "{\\displaystyle " + start = False + elif c == "$": + final += "}" + start = True + else: + final += c + return (final) + + +def getEq(line): # Gets all data within constraints,substitutions + # -------Initialization------- + per = 1 + stringWrite = "" + fEq = False + count = 0 + # ---------------------------- + for c in line: # read each character + if count >= 0 and per != 0: + per += 1 + if c != " " and c != "%": + per = 0 + if c == "{" and per == 0: + if count > 0: + stringWrite += c + count += 1 + elif c == "}" and per == 0: + count -= 1 + if count > 0: + stringWrite += c + elif c == "$" and per == 0: # either begin or end equation + fEq = not (fEq) # toggle fEq flag to know if begin or end equation + if fEq: # if begin + stringWrite += "{\\displaystyle " + else: # if end + stringWrite += "}" + elif c == "\n" and per == 0: # if newline + stringWrite = stringWrite.strip() # remove all leading and trailing whitespace + + # should above be rstrip?<-------------------------------------------------------------------CHECK THIS + + stringWrite += c # add the newline character + per += 1 # watch for % signs + elif count > 0 and per == 0: # not special character + stringWrite += c # write the character + + return (stringWrite.rstrip().lstrip()) + + +def getEqP(line, Flag): # Gets all data within proofs + if Flag: + a = 1 + else: + a = 0 + per = 1 + stringWrite = "" + fEq = False + count = 0 + length = 0 + for c in line: + if fEq and c != " " and c != "$" and c != "{" and c != "}" and not c.isalpha(): + length += 1 + if count >= 0 and per != 0: + per += 1 + if c != " " and c != "%": + # if c!="%": + per = 0 + if c == "{" and per == 0: + if count > 0: + stringWrite += c + count += 1 + elif c == "}" and per == 0: + count -= 1 + if count > 0: + stringWrite += c + elif c == "$" and per == 0: + fEq = not (fEq) + if fEq: + stringWrite += "{\\displaystyle " + else: + if length < 10: + stringWrite += "}" + else: + stringWrite += "}" + "
" + length = 0 + elif c == "\n" and per == 0: + stringWrite = stringWrite.strip() + stringWrite += c + per += 1 + elif count > 0 and per == 0: + stringWrite += c + + return (stringWrite.lstrip()) + + +def getSym(line): # Gets all symbols on a line for symbols list + symList = [] + if line == "": + return symList + symbol = "" + symFlag = False + argFlag = False + cC = 0 + for i in range(0, len(line)): + # if "MeixnerPollaczek{\\lambda}{n}@{x}{\\phi}=\frac{\\pochhammer{2\\lambda}{n}}" in line: + c = line[i] + if symFlag: + if c == "{" or c == "[": + cC += 1 + argFlag = True + if c != "}" and c != "]": + if argFlag or c.isalpha(): + symbol += c + else: + symFlag = False + argFlag = False + symList.append(symbol) + symList += (getSym(symbol)) + symbol = "" + else: + cC -= 1 + symbol += c + if i + 1 == len(line): + p = "" + else: + p = line[i + 1] + if cC == 0 and p != "{" and p != "[" and p != "@": + symFlag = False + # if "monicAskeyWilson{n+1}@{x}{a}{b}{c}{d}{q}+\\frac{1}{2}" in line: + symList.append(symbol) + symList += (getSym(symbol)) + argFlag = False + symbol = "" + + elif c == "\\": + symFlag = True + elif c == "&": + symList.append("&") + + # if "MeixnerPollaczek{\\lambda}{n}@{x}{\\phi}=\frac{\\pochhammer{2\\lambda}{n}}" in line: + symList.append(symbol) + symList += getSym(symbol) + return (symList) + + +def unmodLabel(label): + label = label.replace(".0", ".") + label = label.replace(":0", ":") + return (label) + + +def secLabel(label): + return (label.replace("\'\'", "")) + + +def modLabel(label): + # label.replace("Formula:KLS:","KLS;") + isNumer = False + newlabel = "" + num = "" + for i in range(0, len(label)): + if isNumer and not isnumber(label[i]): + if len(num) > 1: + newlabel += num + isNumer = False + num = "" + else: + newlabel += "0" + str(num) + num = "" + isNumer = False + if isnumber(label[i]): + isNumer = True + num += str(label[i]) + else: + isNumer = False + newlabel += label[i] + if len(num) > 1: + newlabel += num + elif len(num) == 1: + newlabel += "0" + num + return (newlabel) + + +# try: +for jsahlfkjsd in range(0, 1): + tex = open(sys.argv[1], 'r') + wiki = open(sys.argv[2], 'w') + main = open("OrthogonalPolynomials.mmd", "r") + mainText = main.read() + mainPrepend = "" + mainWrite = open("OrthogonalPolynomials.mmd.new", "w") + tester = open("testData.txt", 'w') + # glossary=open('Glossary', 'r') + glossary = open('new.Glossary.csv', 'rb') + gCSV = csv.reader(glossary, delimiter=',', quotechar='\"') + # lLinks=open('BruceLabelLinks', 'r') + lGlos = glossary.readlines() + # lLink=lLinks.readlines() + math = False + constraint = False + substitution = False + symbols = [] + lines = tex.readlines() + refLines = [] + sections = [] + labels = [] + eqs = [] + refEqs = [] + parse = False + head = False + # chapRef=[("GA",open("GA.tex",'r')),("ZE",open("ZE.3.tex",'r'))] + refLabels = [] + '''for c in chapRef: + refLines=refLines+(c[1].readlines()) + c[1].close()''' + for i in range(0, len(refLines)): + line = refLines[i] + if "\\begin{equation}" in line: + sLabel = line.find("\\formula{") + 9 + eLabel = line.find("}", sLabel) + label = modLabel(line[sLabel:eLabel]) + '''for l in lLink: + if l.find(label)!=-1 and l[len(label)+1]=="=": + rlabel=l[l.find("=>")+3:l.find("\\n")] + rlabel=rlabel.replace("/","") + rlabel=rlabel.replace("#",":") + break''' + refLabels.append(label) + refEqs.append("") + math = True + elif math: + refEqs[len(refEqs) - 1] += line + if "\\end{equation}" in refLines[i + 1] or "\\constraint" in refLines[i + 1] or "\\substitution" in \ + refLines[i + 1] or "\\drmfn" in refLines[i + 1]: + math = False + + for i in range(0, len(lines)): + line = lines[i] + if "\\begin{document}" in line: + # wiki.write("drmf_bof\n") + parse = True + elif "\\end{document}" in line and parse: + # wiki.write("\n") + mainPrepend += "\n" + mainText = mainPrepend + mainText + mainText = mainText.replace("drmf_bof\n", "") + mainText = mainText.replace("drmf_eof\n", "") + mainText = mainText.replace("\'\'\'Orthogonal Polynomials\'\'\'\n", "") + mainText = mainText.replace("{{#set:Section=0}}\n", "") + mainText = mainText[0:mainText.rfind("== Sections ")] + mainText = "drmf_bof\n\'\'\'Orthogonal Polynomials\'\'\'\n{{#set:Section=0}}\n" + mainText + "\ndrmf_eof\n" + mainWrite.write(mainText) + mainWrite.close() + main.close() + os.system("cp -f OrthogonalPolynomials.mmd.new OrthogonalPolynomials.mmd") + # wiki.write("\ndrmf_eof\n") + parse = False + elif "\\title" in line and parse: + stringWrite = "\'\'\'" + stringWrite += getString(line) + "\'\'\'\n" + labels.append("Orthogonal Polynomials") + sections.append(["Orthogonal Polynomials", 0]) + # wiki.write(stringWrite) + # mainPrepend+=stringWrite + chapter = getString(line) + mainPrepend += ( + "\n== Sections in " + chapter + " ==\n\n
\n") + elif "\\part" in line: + if getString(line) == "BOF": + parse = False + elif getString(line) == "EOF": + parse = True + elif parse: + mainPrepend += ("\n
\n= " + getString(line) + " =\n") + head = True + elif "\\section" in line: + mainPrepend += ("* [[" + secLabel(getString(line)) + "|" + getString(line) + "]]\n") + sections.append([getString(line)]) + + secCounter = 0 + eqCounter = 0 + for i in range(0, len(lines)): + line = lines[i] + # line.replace("\\begin{equation}","$") + # line.replace("\\end{equation},"$") + '''if "\\begin{document}" in line: + wiki.write("drmf_bof\n") + parse=True + elif "\\end{document}" in line and parse: + wiki.write("\ndrmf_eof\n") + parse=False + elif "\\title" in line and parse: + stringWrite="\'\'\'" + stringWrite+=getString(line)+"\'\'\'\n" + labels.append("Orthogonal Polynomials") + wiki.write(stringWrite\n) + elif "\\part" in line: + if getString(line)=="BOF": + parse=False + elif getString(line)=="EOF": + parse=True + elif parse: + wiki.write("\n
\n= "+getString(line)+" =\n") + head=True + ''' + if "\\section" in line: + parse = True + secCounter += 1 + wiki.write("drmf_bof\n") + wiki.write("\'\'\'" + secLabel(getString(line)) + "\'\'\'\n") + wiki.write("{{DISPLAYTITLE:" + (sections[secCounter][0]) + "}}\n") + # wiki.write("
\n") + # wiki.write("
<< [["+secLabel(sections[secCounter-1][0])+"|"+secLabel(sections[secCounter-1][0])+"]]
\n") + # wiki.write("
[[Orthogonal_Polynomials#"+secLabel(sections[secCounter][0])+"|"+secLabel(sections[secCounter][0])+"]]
\n") + # wiki.write("
[["+secLabel(sections[(secCounter+1)%len(sections)][0])+"|"+secLabel(sections[(secCounter+1)%len(sections)][0])+"]] >>
\n
\n\n") + # wiki.write("{{head|pre="+secLabel(sections[secCounter-1][0])+"|cur="+secLabel(sections[secCounter][0])+"|next="+secLabel(sections[(secCounter+1)%len(sections)][0])+"}}\n") + wiki.write("{{#set:Chapter=" + chapter + "}}\n") + wiki.write("{{#set:Section=" + str(secCounter) + "}}\n") + wiki.write("{{headSection}}\n") + head = True + wiki.write("== " + getString(line) + " ==\n") + elif ("\\section" in lines[(i + 1) % len(lines)] or "\\end{document}" in lines[(i + 1) % len(lines)]) and parse: + # wiki.write("
\n") + # wiki.write("
<< [["+secLabel(sections[secCounter-1][0])+"|"+secLabel(sections[secCounter-1][0])+"]]
\n") + # wiki.write("
[[Orthogonal_Polynomials#"+secLabel(sections[secCounter][0])+"|"+secLabel(sections[secCounter][0])+"]]
\n") + # wiki.write("
[["+secLabel(sections[(secCounter+1)%len(sections)][0])+"|"+secLabel(sections[(secCounter+1)%len(sections)][0])+"]] >>
\n
\n\n") + # wiki.write("{{foot|pre="+secLabel(sections[secCounter-1][0])+"|cur="+secLabel(sections[secCounter][0])+"|next="+secLabel(sections[(secCounter+1)%len(sections)][0])+"}}\n") + wiki.write("{{footSection}}\n") + wiki.write("drmf_eof\n") + sections[secCounter].append(eqCounter) + eqCounter = 0 + + elif "\\subsection" in line and parse: + wiki.write("\n== " + getString(line) + " ==\n") + head = True + elif "\\paragraph" in line and parse: + wiki.write("\n=== " + getString(line) + " ===\n") + head = True + elif "\\subsubsection" in line and parse: + wiki.write("\n=== " + getString(line) + " ===\n") + head = True + + elif "\\begin{equation}" in line and parse: + # symLine="" + if head: + wiki.write("\n") + head = False + sLabel = line.find("\\formula{") + 9 + eLabel = line.find("}", sLabel) + label = modLabel(line[sLabel:eLabel]) + eqCounter += 1 + '''for l in lLink: + if label==l[0:l.find("=")-1]: + rlabel=l[l.find("=>")+3:l.find("\\n")] + rlabel=rlabel.replace("/","") + rlabel=rlabel.replace("#",":") + rlabel=rlabel.replace("!",":") + break''' + labels.append(label) + eqs.append("") + # wiki.write("\n\n") + wiki.write("{\displaystyle \n") + math = True + elif "\\begin{equation}" in line and not parse: + sLabel = line.find("\\formula{") + 9 + eLabel = line.find("}", sLabel) + label = modLabel(line[sLabel:eLabel]) + '''for l in lLink: + if l.find(label)!=-1: + rlabel=l[l.find("=>")+3:l.find("\\n")] + rlabel=rlabel.replace("/","") + rlabel=rlabel.replace("#",":") + break''' + labels.append("*" + label) # special marker + eqs.append("") + math = True + elif "\\end{equation}" in line: + + math = False + elif "\\constraint" in line and parse: + constraint = True + math = False + conLine = "" + elif "\\substitution" in line and parse: + substitution = True + math = False + subLine = "" + # wiki.write("
Substitution(s): "+getEq(line)+"

\n") + elif "\\proof" in line and parse: + math = False + elif "\\drmfn" in line and parse: + math = False + if "\\drmfname" in line and parse: + wiki.write("
This formula has the name: " + getString(line) + "

\n") + elif math and parse: + flagM = True + eqs[len(eqs) - 1] += line + + if "\\end{equation}" in lines[i + 1] and not "\\subsection" in lines[i + 3] and not "\\section" in lines[ + i + 3] and not "\\part" in lines[i + 3]: + u = i + flagM2 = False + while flagM: + u += 1 + if "\\begin{equation}" in lines[u] in lines[u]: + flagM = False + if "\\section" in lines[u] or "\\subsection" in lines[i] or "\\part" in lines[ + u] or "\\end{document}" in lines[u]: + flagM = False + flagM2 = True + if not (flagM2): + + wiki.write(line.rstrip("\n")) + wiki.write("\n}
\n") + else: + wiki.write(line.rstrip("\n")) + wiki.write("\n}\n") + elif "\\end{equation}" in lines[i + 1]: + wiki.write(line.rstrip("\n")) + wiki.write("\n}\n") + elif "\\constraint" in lines[i + 1] or "\\substitution" in lines[i + 1] or "\\drmfn" in lines[i + 1]: + wiki.write(line.rstrip("\n")) + wiki.write("\n}\n") + else: + wiki.write(line) + elif math and not parse: + eqs[len(eqs) - 1] += line + if "\\end{equation}" in lines[i + 1] or "\\constraint" in lines[i + 1] or "\\substitution" in lines[ + i + 1] or "\\drmfn" in lines[i + 1]: + math = False + if substitution and parse: + subLine = subLine + line.replace("&", "&
") + if "\\end{equation}" in lines[i + 1] or "\\substitution" in lines[i + 1] or "\\constraint" in lines[ + i + 1] or "\\drmfn" in lines[i + 1] or "\\proof" in lines[i + 1]: + substitution = False + wiki.write("
Substitution(s): " + getEq(subLine) + "

\n") + + if constraint and parse: + conLine = conLine + line.replace("&", "&
") + if "\\end{equation}" in lines[i + 1] or "\\substitution" in lines[i + 1] or "\\constraint" in lines[ + i + 1] or "\\drmfn" in lines[i + 1] or "\\proof" in lines[i + 1]: + constraint = False + wiki.write("
Constraint(s): " + getEq(conLine) + "

\n") + + eqCounter = 0 + endNum = len(labels) - 1 + '''for n in range(1,len(labels)): + if n+1!=len(labels) and labels[n+1][0]=="*" and labels[n][0]!="*": + labels[n]=""+labels[n] + endNum=n + elif labels[n][0]=="*": + labels[n]=""+labels[n][1:] + else: + labels[n]=""+labels[n]''' + '''for n in range(0,len(refLabels)): + refLabels[n]=""+refLabels[n]''' + parse = False + constraint = False + substitution = False + note = False + hCon = True + hSub = True + hNote = True + hProof = True + proof = False + comToWrite = "" + secCount = -1 + newSec = False + for i in range(0, len(lines)): + line = lines[i] + + if "\\section" in line: + secCount += 1 + newSec = True + eqS = 0 + + if "\\begin{equation}" in line: + symLine = line.strip("\n") + eqS += 1 + constraint = False + substitution = False + note = False + comToWrite = "" + hCon = True + hSub = True + hNote = True + hProof = True + proof = False + parse = True + symbols = [] + eqCounter += 1 + wiki.write("drmf_bof\n") + label = labels[eqCounter] + wiki.write("\'\'\'" + secLabel(label) + "\'\'\'\n") + wiki.write("{{DISPLAYTITLE:" + (labels[eqCounter]) + "}}\n") + if eqCounter < endNum: # FOR ANYTHING THAT IS NOT THE EXTRA EQUATIONS + wiki.write("
\n") + if newSec: + wiki.write("
<< [[" + secLabel(sections[secCount][0]).replace(" ", + "_") + "|" + secLabel( + sections[secCount][0]) + "]]
\n") + else: + wiki.write("
<< [[" + secLabel(labels[eqCounter - 1]).replace(" ", + "_") + "|" + secLabel( + labels[eqCounter - 1]) + "]]
\n") + wiki.write("
[[" + secLabel(sections[secCount + 1][0]).replace(" ", + "_") + "#" + secLabel( + labels[eqCounter][len("Formula:"):]) + "|formula in " + secLabel( + sections[secCount + 1][0]) + "]]
\n") + if eqS == sections[secCount][1]: + wiki.write( + "
[[" + secLabel(sections[(secCount + 1) % len(sections)][0]).replace( + " ", "_") + "|" + secLabel(sections[(secCount + 1) % len(sections)][0]) + "]] >>
\n") + else: + wiki.write( + "
[[" + secLabel(labels[(eqCounter + 1) % (endNum + 1)]).replace(" ", + "_") + "|" + secLabel( + labels[(eqCounter + 1) % (endNum + 1)]) + "]] >>
\n") + wiki.write("
\n\n") + elif eqCounter == endNum: + wiki.write("
\n") + if newSec: + newSec = False + wiki.write("
<< [[" + secLabel(sections[secCount][0]).replace(" ", + "_") + "|" + secLabel( + sections[secCount][0]) + "]]
\n") + else: + wiki.write("
<< [[" + secLabel(labels[eqCounter - 1]).replace(" ", + "_") + "|" + secLabel( + labels[eqCounter - 1]) + "]]
\n") + wiki.write("
[[" + secLabel(sections[secCount + 1][0]).replace(" ", + "_") + "#" + secLabel( + labels[eqCounter][len("Formula:"):]) + "|formula in " + secLabel( + sections[secCount + 1][0]) + "]]
\n") + wiki.write("
[[" + secLabel( + labels[(eqCounter + 1) % (endNum + 1)].replace(" ", "_")) + "|" + secLabel( + labels[(eqCounter + 1) % (endNum + 1)]) + "]]
\n") + wiki.write("
\n\n") + + wiki.write("
{\displaystyle \n") + math = True + elif "\\end{equation}" in line: + wiki.write(comToWrite) + parse = False + math = False + if hProof: + wiki.write( + "\n== Proof ==\n\nWe ask users to provide proof(s), reference(s) to proof(s), or further clarification on the proof(s) in this space.\n") + + wiki.write("\n== Symbols List ==\n\n") + newSym = [] + # if "09.07:04" in label: + for x in symbols: + flagA = True + # if x not in newSym: + cN = 0 + cC = 0 + flag = True + ArgCx = 0 + for z in x: + if z.isalpha() and flag or z == "&": + cN += 1 + else: + flag = False + if z == "{" or z == "[": + cC += 1 + if z == "}" or z == "]": + cC -= 1 + if cC == 0: + ArgCx += 1 + noA = x[:cN] + for y in newSym: + cN = 0 + cC = 0 + ArgC = 0 + flag = True + for z in y: + if z.isalpha() and flag or z == "&": + cN += 1 + else: + flag = False + if z == "{" or z == "[": + cC += 1 + if z == "}" or z == "]": + cC -= 1 + if cC == 0: + ArgC += 1 + + if y[:cN] == noA: # and ArgC==ArgCx: + flagA = False + break + if flagA: + newSym.append(x) + newSym.reverse() + symF = False + ampFlag = False + finSym = [] + for s in range(len(newSym) - 1, -1, -1): + symbolPar = "\\" + newSym[s] + ArgCx = 0 + parCx = 0 + parFlag = False + cC = 0 + for z in symbolPar: + if z == "@": + parFlag = True + elif z.isalpha() or z == "&": + cN += 1 + else: + if z == "{" or z == "[": + cC += 1 + if z == "}" or z == "]": + cC -= 1 + if cC == 0: + if parFlag: + parCx += 1 + else: + ArgCx += 1 + + if symbolPar.find("{") != -1 or symbolPar.find("[") != -1: + if symbolPar.find("[") == -1: + symbol = symbolPar[0:symbolPar.find("{")] + elif symbolPar.find("{") == -1 or symbolPar.find("[") < symbolPar.find("{"): + symbol = symbolPar[0:symbolPar.find("[")] + else: + symbol = symbolPar[0:symbolPar.find("{")] + else: + symbol = symbolPar + numArg = parCx + numPar = ArgCx + gFlag = False + checkFlag = False + get = False + gCSV = csv.reader(open('new.Glossary.csv', 'rb'), delimiter=',', quotechar='\"') + preG = "" + if symbol == "\\&": + ampFlag = True + for S in gCSV: + G = S + ArgCx = 0 + parCx = 0 + parFlag = False + cC = 0 + ind = G[0].find("@") + if ind == -1: + ind = len(G[0]) - 1 + for z in G[0]: + if z == "@": + parFlag = True + elif z.isalpha(): + cN += 1 + else: + if z == "{" or z == "[": + cC += 1 + if z == "}" or z == "]": + cC -= 1 + if cC == 0: + if parFlag: + parCx += 1 + else: + ArgCx += 1 + if G[0].find(symbol) == 0 and (len(G[0]) == len(symbol) or not G[0][ + len(symbol)].isalpha()): # and (numPar!=0 or numArg!=0): + checkFlag = True + get = True + preG = S + + + elif checkFlag: + get = True + checkFlag = False + + if (get): + if get: + G = preG + get = False + checkFlag = False + if True: + if symbolPar.find("@") != -1: + Q = symbolPar[:symbolPar.find("@")] + else: + Q = symbolPar + listArgs = [] + if len(Q) > len(symbol) and (Q[len(symbol)] == "{" or Q[len(symbol)] == "["): + ap = "" + for o in range(len(symbol), len(Q)): + if Q[o] == "{" or z == "[": + argFlag = True + elif Q[o] == "}" or z == "]": + argFlag = False + listArgs.append(ap) + ap = "" + else: + ap += Q[o] + '''websiteU=g[g.find("http://"):].strip("\n") + k=0 + websites=[] + for r in range(0,len(websiteU)): + if websiteU[r]==",": + if websiteU[k:r].find("http://")!=-1: + websites.append(websiteU[k:r+1].strip(" ")) + k=r + + + if websiteU[k:r].find("http://")!=-1: + websites.append(websiteU[k:r+1].strip(" ")) + websiteF="" + for d in websites: + websiteF=websiteF+" ["+d+" "+d+"]"''' + # websiteF=G[4].strip("\n") + websiteF = "" + web1 = G[5] + for t in range(5, len(G)): + if G[t] != "": + websiteF = websiteF + " [" + G[t] + " " + G[t] + "]" + + # p1=Q + # if Q.find("@")!=-1: + # p1=Q[:Q.find("@")] + p1 = G[4].strip("$") + p1 = "{\\displaystyle " + p1 + "}" + # if checkFlag: + new1 = "" + new2 = "" + pause = False + mathF = True + '''for k in range(0,len(p1)): + if p1[k]=="$": + if mathF: + new1+="{\\displaystyle " + else: + new1+="}" + mathF=not mathF + + elif p1[k]=="#" and p1[k+1].isdigit(): + pause=True + elif pause: + num=int(p1[k]) + #letter=chr(num+96) + letter=listArgs[num-1] + new1+=letter + pause=False + + else: + new1+=p1[k]''' + p2 = G[1] + for k in range(0, len(p2)): + if p2[k] == "$": + if mathF: + new2 += "{\\displaystyle " + else: + new2 += "}" + mathF = not mathF + else: + new2 += p2[k] + # p1=new1 + p2 = new2 + finSym.append(web1 + " " + p1 + "] : " + p2 + " :" + websiteF) + break + # gFlag=True + # if not symF: + # symF=True + # wiki.write("[") + # else: + # wiki.write("
\n") + # wiki.write("[") + # wiki.write(g[g.find("http://"):].strip("\n")) + # wiki.write(" ") + # wiki.write(getEq(g[g.find("{$"):]).strip("\n").replace("\\\\","\\")) + # wiki.write("] : ") + # wiki.write(g[g.find(" {")+2:g.find("} ",g.find(" {")+1)]) + # wiki.write(" : [") + # wiki.write(g[g.find("http://"):].strip("\n")) + # wiki.write(" ") + # wiki.write(g[g.find("http://"):].strip("\n")) + # wiki.write("] ")''' + + # preG=S + if not gFlag: + del newSym[s] + + gFlag = True + # finSym.reverse() + if ampFlag: + wiki.write("& : logical and") + gFlag = False + for y in finSym: + if y == "& : logical and": + pass + elif gFlag: + gFlag = False + wiki.write("[" + y) + else: + wiki.write("
\n[" + y) + + wiki.write("\n
\n") + + wiki.write("\n== Bibliography==\n\n") # should there be a space between bibliography and ==? + r = unmodLabel(labels[eqCounter]) + q = r.find("KLS:") + 4 + p = r.find(":", q) + section = r[q:p] + equation = r[p + 1:] + if equation.find(":") != -1: + equation = equation[0:equation.find(":")] + + wiki.write( + "[http://homepage.tudelft.nl/11r49/askey/contents.html Equation in Section " + section + "] of [[Bibliography#KLS|'''KLS''']].\n\n") # Where should it link to? + wiki.write("== URL links ==\n\nWe ask users to provide relevant URL links in this space.\n\n") + if eqCounter < endNum: + wiki.write("
\n") + if newSec: + newSec = False + wiki.write("
<< [[" + secLabel(sections[secCount][0]).replace(" ", + "_") + "|" + secLabel( + sections[secCount][0]) + "]]
\n") + else: + wiki.write("
<< [[" + secLabel(labels[eqCounter - 1]).replace(" ", + "_") + "|" + secLabel( + labels[eqCounter - 1]) + "]]
\n") + wiki.write("
[[" + secLabel(sections[secCount + 1][0]).replace(" ", + "_") + "#" + secLabel( + labels[eqCounter][len("Formula:"):]) + "|formula in " + secLabel( + sections[secCount + 1][0]) + "]]
\n") + if eqS == sections[secCount][1]: + wiki.write("
[[" + sections[(secCount + 1) % len(sections)][0].replace(" ", + "_") + "|" + + sections[(secCount + 1) % len(sections)][0] + "]] >>
\n") + else: + wiki.write( + "
[[" + secLabel(labels[(eqCounter + 1) % (endNum + 1)]).replace(" ", + "_") + "|" + secLabel( + labels[(eqCounter + 1) % (endNum + 1)]) + "]] >>
\n") + wiki.write("
\n\ndrmf_eof\n") + else: # FOR EXTRA EQUATIONS + wiki.write("
\n") + wiki.write("
<< [[" + labels[endNum - 1].replace(" ", "_") + "|" + labels[ + endNum - 1] + "]]
\n") + wiki.write("
[[" + labels[0].replace(" ", "_") + "#" + labels[endNum][ + 8:] + "|formula in " + + labels[0] + "]]
\n") + wiki.write("
[[" + labels[(0) % endNum].replace(" ", "_") + "|" + labels[ + 0 % endNum] + "]]
\n") + wiki.write("
\n\ndrmf_eof\n") + + + + elif "\\constraint" in line and parse: + # symbols=symbols+getSym(line) + symLine = line.strip("\n") + if hCon: + comToWrite = comToWrite + "\n== Constraint(s) ==\n\n" + hCon = False + constraint = True + math = False + conLine = "" + # wiki.write("
"+getEq(line)+"

\n") + elif "\\substitution" in line and parse: + # symbols=symbols+getSym(line) + symLine = line.strip("\n") + if hSub: + comToWrite = comToWrite + "\n== Substitution(s) ==\n\n" + hSub = False + # wiki.write("
"+getEq(line)+"

\n") + substitution = True + math = False + subLine = "" + elif "\\drmfname" in line and parse: + math = False + comToWrite = "\n== Name ==\n\n
" + getString(line) + "

\n" + comToWrite + + elif "\\drmfnote" in line and parse: + symbols = symbols + getSym(line) + if hNote: + comToWrite = comToWrite + "\n== Note(s) ==\n\n" + hNote = False + note = True + math = False + noteLine = "" + + elif "\\proof" in line and parse: + # symbols=symbols+getSym(line) + symLine = line.strip("\n") + if hProof: + hProof = False + comToWrite = comToWrite + "\n== Proof ==\n\nWe ask users to provide proof(s), reference(s) to proof(s), or further clarification on the proof(s) in this space. \n

\n
" + proof = True + proofLine = "" + pause = False + pauseP = False + for ind in range(0, len(line)): + if line[ind:ind + 7] == "\\eqref{": + pause = True + eqR = line[ind:line.find("}", ind) + 1] + rLab = getString(eqR) + '''for l in lLink: + if rLab==l[0:l.find("=")-1]: + rlabel=l[l.find("=>")+3:l.find("\\n")] + rlabel=rlabel.replace("/","") + rlabel=rlabel.replace("#",":") + rlabel=rlabel.replace("!",":") + break''' + + eInd = refLabels.index("" + label) + z = line[line.find("}", ind + 7) + 1] + if z == "." or z == ",": + pauseP = True + proofLine += ("
\n{\displaystyle \n" + refEqs[ + eInd] + "}" + z + "
\n") + else: + if z == "}": + proofLine += ( + "
\n{\displaystyle \n" + refEqs[eInd] + "}
") + else: + proofLine += ( + "
\n{\displaystyle \n" + refEqs[eInd] + "}
\n") + + + else: + if pause: + if line[ind] == "}": + pause = False + elif pauseP: + pauseP = False + elif line[ind] == "\n" and "\\end{equation}" in lines[i + 1]: + pass + else: + proofLine += (line[ind]) + if "\\end{equation}" in lines[i + 1]: + proof = False + # symLine+=line.strip("\n") + wiki.write(comToWrite + getEqP(proofLine, False) + "
\n
\n") + comToWrite = "" + symbols = symbols + getSym(symLine) + symLine = "" + + # wiki.write(line) + + elif proof: + symLine += line.strip("\n") + pauseP = False + # symbols=symbols+getSym(line) + for ind in range(0, len(line)): + if line[ind:ind + 7] == "\\eqref{": + pause = True + eqR = line[ind:line.find("}", ind) + 1] + rLab = getString(eqR) + '''for l in lLink: + if rLab==l[0:l.find("=")-1]: + rlabel=l[l.find("=>")+3:l.find("\\n")] + rlabel=rlabel.replace("/","") + rlabel=rlabel.replace("#",":") + rlabel=rlabel.replace("!",":") + break''' + + eInd = refLabels.index("" + label) + z = line[line.find("}", ind + 7) + 1] + if z == "." or z == ",": + pauseP = True + proofLine += ("
\n{\displaystyle \n" + refEqs[ + eInd] + "}" + z + "
\n") + else: + proofLine += ( + "
\n{\displaystyle \n" + refEqs[eInd] + "}
\n") + + else: + if pause: + if line[ind] == "}": + pause = False + elif pauseP: + pauseP = False + elif line[ind] == "\n" and "\\end{equation}" in lines[i + 1]: + pass + + else: + proofLine += (line[ind]) + if "\\end{equation}" in lines[i + 1]: + proof = False + # symLine+=line.strip("\n") + wiki.write(comToWrite + getEqP(proofLine, False).rstrip("\n") + "
\n
\n") + comToWrite = "" + symbols = symbols + getSym(symLine) + symLine = "" + + elif math: + if "\\end{equation}" in lines[i + 1] or "\\constraint" in lines[i + 1] or "\\substitution" in lines[ + i + 1] or "\\proof" in lines[i + 1] or "\\drmfnote" in lines[i + 1] or "\\drmfname" in lines[ + i + 1]: + wiki.write(line.rstrip("\n")) + symLine += line.strip("\n") + symbols = symbols + getSym(symLine) + symLine = "" + wiki.write("\n}
\n") + else: + symLine += line.strip("\n") + wiki.write(line) + if note and parse: + noteLine = noteLine + line + symbols = symbols + getSym(line) + if "\\end{equation}" in lines[i + 1] or "\\drmfn" in lines[i + 1] or "\\constraint" in lines[ + i + 1] or "\\substitution" in lines[i + 1] or "\\proof" in lines[i + 1]: + note = False + comToWrite = comToWrite + "
" + getEq(noteLine) + "

\n" + + if constraint and parse: + conLine = conLine + line.replace("&", "&
") + + symLine += line.strip("\n") + # symbols=symbols+getSym(line) + if "\\end{equation}" in lines[i + 1] or "\\drmfn" in lines[i + 1] or "\\constraint" in lines[ + i + 1] or "\\substitution" in lines[i + 1] or "\\proof" in lines[i + 1]: + constraint = False + symbols = symbols + getSym(symLine) + symLine = "" + wiki.write(comToWrite + "
" + getEq(conLine) + "

\n") + comToWrite = "" + if substitution and parse: + subLine = subLine + line.replace("&", "&
") + + symLine += line.strip("\n") + # symbols=symbols+getSym(line) + if "\\end{equation}" in lines[i + 1] or "\\drmfn" in lines[i + 1] or "\\substitution" in lines[ + i + 1] or "\\constraint" in lines[i + 1] or "\\proof" in lines[i + 1]: + substitution = False + symbols = symbols + getSym(symLine) + symLine = "" + wiki.write(comToWrite + "
" + getEq(subLine) + "

\n") + comToWrite = "" +# except Exception as detail: #If exception occured +# print("Exception",detail) #print details of error +# except: #If anythin else occured... +# print ("ERROR",sys.exc_info()[0])#ERROR with basic info From 3de50241c8fc18dda3d01d46cbe64d3cde6fae64 Mon Sep 17 00:00:00 2001 From: physikerwelt Date: Mon, 29 Feb 2016 14:41:34 +0100 Subject: [PATCH 032/402] Add test for Azeems code --- .gitignore | 3 - .travis.yml | 4 +- Azeem/{ => src}/OrthogonalPolynomials.mmd | 0 Azeem/{ => src}/Zeta.py | 0 Azeem/{ => src}/new.Glossary.csv | 0 Azeem/src/tex2Wiki.py | 1069 +++++++++++++++++++++ Azeem/test/test_isnumber.py | 8 + Azeem/tex2Wiki.py | 1069 --------------------- 8 files changed, 1079 insertions(+), 1074 deletions(-) rename Azeem/{ => src}/OrthogonalPolynomials.mmd (100%) rename Azeem/{ => src}/Zeta.py (100%) rename Azeem/{ => src}/new.Glossary.csv (100%) create mode 100644 Azeem/src/tex2Wiki.py create mode 100644 Azeem/test/test_isnumber.py delete mode 100644 Azeem/tex2Wiki.py diff --git a/.gitignore b/.gitignore index a1bd6a8..e9b040a 100644 --- a/.gitignore +++ b/.gitignore @@ -117,7 +117,6 @@ crashlytics-build.properties ## Core latex/pdflatex auxiliary files: *.aux *.lof -*.log *.lot *.fls *.out @@ -191,8 +190,6 @@ acs-*.bib *-gnuplottex-* # hyperref -*.brf - # knitr *-concordance.tex *.tikz diff --git a/.travis.yml b/.travis.yml index b85e205..14a628f 100644 --- a/.travis.yml +++ b/.travis.yml @@ -1,2 +1,2 @@ -language: python -script: nosetests -w AlexDanoff \ No newline at end of file +language: python +script: nosetests AlexDanoff Azeem \ No newline at end of file diff --git a/Azeem/OrthogonalPolynomials.mmd b/Azeem/src/OrthogonalPolynomials.mmd similarity index 100% rename from Azeem/OrthogonalPolynomials.mmd rename to Azeem/src/OrthogonalPolynomials.mmd diff --git a/Azeem/Zeta.py b/Azeem/src/Zeta.py similarity index 100% rename from Azeem/Zeta.py rename to Azeem/src/Zeta.py diff --git a/Azeem/new.Glossary.csv b/Azeem/src/new.Glossary.csv similarity index 100% rename from Azeem/new.Glossary.csv rename to Azeem/src/new.Glossary.csv diff --git a/Azeem/src/tex2Wiki.py b/Azeem/src/tex2Wiki.py new file mode 100644 index 0000000..6f6a219 --- /dev/null +++ b/Azeem/src/tex2Wiki.py @@ -0,0 +1,1069 @@ +# Version 14: Templates +# Convert tex to wikiText +import csv # imported for using csv format +import sys # imported for getting args +import os # imported for copying file + + +def isnumber(char): + return char[0] in "0123456789" + + +def getString(line): # Gets all data within curly braces on a line + # -------Initialization------- + stringWrite = "" + getStr = False + pW = "" + # ---------------------------- + for c in line: + if c == "{" or c == "}": # if there is a curly brace in the line + getStr = not (getStr) # toggle the getStr flag + if c == "}": # no more info needed + break + elif getStr: # if within curly braces + if not (pW == c and c == "-"): # if there is no double dash (makes single dash) + if c != "$": # if not a $ sign + stringWrite += c # this character is part of the data + else: # replace $ with '' + stringWrite += "\'\'" # add double ' + pW = c # change last character for finding double dashes + return (stringWrite.rstrip('\n').lstrip()) # return the data without newlines and without leading spaces + + +def getG(line): # gets equation for symbols list + start = True + final = "" + for c in line: + if c == "$" and start: + final += "{\\displaystyle " + start = False + elif c == "$": + final += "}" + start = True + else: + final += c + return (final) + + +def getEq(line): # Gets all data within constraints,substitutions + # -------Initialization------- + per = 1 + stringWrite = "" + fEq = False + count = 0 + # ---------------------------- + for c in line: # read each character + if count >= 0 and per != 0: + per += 1 + if c != " " and c != "%": + per = 0 + if c == "{" and per == 0: + if count > 0: + stringWrite += c + count += 1 + elif c == "}" and per == 0: + count -= 1 + if count > 0: + stringWrite += c + elif c == "$" and per == 0: # either begin or end equation + fEq = not (fEq) # toggle fEq flag to know if begin or end equation + if fEq: # if begin + stringWrite += "{\\displaystyle " + else: # if end + stringWrite += "}" + elif c == "\n" and per == 0: # if newline + stringWrite = stringWrite.strip() # remove all leading and trailing whitespace + + # should above be rstrip?<-------------------------------------------------------------------CHECK THIS + + stringWrite += c # add the newline character + per += 1 # watch for % signs + elif count > 0 and per == 0: # not special character + stringWrite += c # write the character + + return (stringWrite.rstrip().lstrip()) + + +def getEqP(line, Flag): # Gets all data within proofs + if Flag: + a = 1 + else: + a = 0 + per = 1 + stringWrite = "" + fEq = False + count = 0 + length = 0 + for c in line: + if fEq and c != " " and c != "$" and c != "{" and c != "}" and not c.isalpha(): + length += 1 + if count >= 0 and per != 0: + per += 1 + if c != " " and c != "%": + # if c!="%": + per = 0 + if c == "{" and per == 0: + if count > 0: + stringWrite += c + count += 1 + elif c == "}" and per == 0: + count -= 1 + if count > 0: + stringWrite += c + elif c == "$" and per == 0: + fEq = not (fEq) + if fEq: + stringWrite += "{\\displaystyle " + else: + if length < 10: + stringWrite += "}" + else: + stringWrite += "}" + "
" + length = 0 + elif c == "\n" and per == 0: + stringWrite = stringWrite.strip() + stringWrite += c + per += 1 + elif count > 0 and per == 0: + stringWrite += c + + return (stringWrite.lstrip()) + + +def getSym(line): # Gets all symbols on a line for symbols list + symList = [] + if line == "": + return symList + symbol = "" + symFlag = False + argFlag = False + cC = 0 + for i in range(0, len(line)): + # if "MeixnerPollaczek{\\lambda}{n}@{x}{\\phi}=\frac{\\pochhammer{2\\lambda}{n}}" in line: + c = line[i] + if symFlag: + if c == "{" or c == "[": + cC += 1 + argFlag = True + if c != "}" and c != "]": + if argFlag or c.isalpha(): + symbol += c + else: + symFlag = False + argFlag = False + symList.append(symbol) + symList += (getSym(symbol)) + symbol = "" + else: + cC -= 1 + symbol += c + if i + 1 == len(line): + p = "" + else: + p = line[i + 1] + if cC == 0 and p != "{" and p != "[" and p != "@": + symFlag = False + # if "monicAskeyWilson{n+1}@{x}{a}{b}{c}{d}{q}+\\frac{1}{2}" in line: + symList.append(symbol) + symList += (getSym(symbol)) + argFlag = False + symbol = "" + + elif c == "\\": + symFlag = True + elif c == "&": + symList.append("&") + + # if "MeixnerPollaczek{\\lambda}{n}@{x}{\\phi}=\frac{\\pochhammer{2\\lambda}{n}}" in line: + symList.append(symbol) + symList += getSym(symbol) + return (symList) + + +def unmodLabel(label): + label = label.replace(".0", ".") + label = label.replace(":0", ":") + return (label) + + +def secLabel(label): + return (label.replace("\'\'", "")) + + +def modLabel(label): + # label.replace("Formula:KLS:","KLS;") + isNumer = False + newlabel = "" + num = "" + for i in range(0, len(label)): + if isNumer and not isnumber(label[i]): + if len(num) > 1: + newlabel += num + isNumer = False + num = "" + else: + newlabel += "0" + str(num) + num = "" + isNumer = False + if isnumber(label[i]): + isNumer = True + num += str(label[i]) + else: + isNumer = False + newlabel += label[i] + if len(num) > 1: + newlabel += num + elif len(num) == 1: + newlabel += "0" + num + return (newlabel) + +def main(): + # try: + for jsahlfkjsd in range(0, 1): + tex = open(sys.argv[1], 'r') + wiki = open(sys.argv[2], 'w') + main = open("OrthogonalPolynomials.mmd", "r") + mainText = main.read() + mainPrepend = "" + mainWrite = open("OrthogonalPolynomials.mmd.new", "w") + tester = open("testData.txt", 'w') + # glossary=open('Glossary', 'r') + glossary = open('new.Glossary.csv', 'rb') + gCSV = csv.reader(glossary, delimiter=',', quotechar='\"') + # lLinks=open('BruceLabelLinks', 'r') + lGlos = glossary.readlines() + # lLink=lLinks.readlines() + math = False + constraint = False + substitution = False + symbols = [] + lines = tex.readlines() + refLines = [] + sections = [] + labels = [] + eqs = [] + refEqs = [] + parse = False + head = False + # chapRef=[("GA",open("GA.tex",'r')),("ZE",open("ZE.3.tex",'r'))] + refLabels = [] + '''for c in chapRef: + refLines=refLines+(c[1].readlines()) + c[1].close()''' + for i in range(0, len(refLines)): + line = refLines[i] + if "\\begin{equation}" in line: + sLabel = line.find("\\formula{") + 9 + eLabel = line.find("}", sLabel) + label = modLabel(line[sLabel:eLabel]) + '''for l in lLink: + if l.find(label)!=-1 and l[len(label)+1]=="=": + rlabel=l[l.find("=>")+3:l.find("\\n")] + rlabel=rlabel.replace("/","") + rlabel=rlabel.replace("#",":") + break''' + refLabels.append(label) + refEqs.append("") + math = True + elif math: + refEqs[len(refEqs) - 1] += line + if "\\end{equation}" in refLines[i + 1] or "\\constraint" in refLines[i + 1] or "\\substitution" in \ + refLines[i + 1] or "\\drmfn" in refLines[i + 1]: + math = False + + for i in range(0, len(lines)): + line = lines[i] + if "\\begin{document}" in line: + # wiki.write("drmf_bof\n") + parse = True + elif "\\end{document}" in line and parse: + # wiki.write("\n") + mainPrepend += "\n" + mainText = mainPrepend + mainText + mainText = mainText.replace("drmf_bof\n", "") + mainText = mainText.replace("drmf_eof\n", "") + mainText = mainText.replace("\'\'\'Orthogonal Polynomials\'\'\'\n", "") + mainText = mainText.replace("{{#set:Section=0}}\n", "") + mainText = mainText[0:mainText.rfind("== Sections ")] + mainText = "drmf_bof\n\'\'\'Orthogonal Polynomials\'\'\'\n{{#set:Section=0}}\n" + mainText + "\ndrmf_eof\n" + mainWrite.write(mainText) + mainWrite.close() + main.close() + os.system("cp -f OrthogonalPolynomials.mmd.new OrthogonalPolynomials.mmd") + # wiki.write("\ndrmf_eof\n") + parse = False + elif "\\title" in line and parse: + stringWrite = "\'\'\'" + stringWrite += getString(line) + "\'\'\'\n" + labels.append("Orthogonal Polynomials") + sections.append(["Orthogonal Polynomials", 0]) + # wiki.write(stringWrite) + # mainPrepend+=stringWrite + chapter = getString(line) + mainPrepend += ( + "\n== Sections in " + chapter + " ==\n\n
\n") + elif "\\part" in line: + if getString(line) == "BOF": + parse = False + elif getString(line) == "EOF": + parse = True + elif parse: + mainPrepend += ("\n
\n= " + getString(line) + " =\n") + head = True + elif "\\section" in line: + mainPrepend += ("* [[" + secLabel(getString(line)) + "|" + getString(line) + "]]\n") + sections.append([getString(line)]) + + secCounter = 0 + eqCounter = 0 + for i in range(0, len(lines)): + line = lines[i] + # line.replace("\\begin{equation}","$") + # line.replace("\\end{equation},"$") + '''if "\\begin{document}" in line: + wiki.write("drmf_bof\n") + parse=True + elif "\\end{document}" in line and parse: + wiki.write("\ndrmf_eof\n") + parse=False + elif "\\title" in line and parse: + stringWrite="\'\'\'" + stringWrite+=getString(line)+"\'\'\'\n" + labels.append("Orthogonal Polynomials") + wiki.write(stringWrite\n) + elif "\\part" in line: + if getString(line)=="BOF": + parse=False + elif getString(line)=="EOF": + parse=True + elif parse: + wiki.write("\n
\n= "+getString(line)+" =\n") + head=True + ''' + if "\\section" in line: + parse = True + secCounter += 1 + wiki.write("drmf_bof\n") + wiki.write("\'\'\'" + secLabel(getString(line)) + "\'\'\'\n") + wiki.write("{{DISPLAYTITLE:" + (sections[secCounter][0]) + "}}\n") + # wiki.write("
\n") + # wiki.write("
<< [["+secLabel(sections[secCounter-1][0])+"|"+secLabel(sections[secCounter-1][0])+"]]
\n") + # wiki.write("
[[Orthogonal_Polynomials#"+secLabel(sections[secCounter][0])+"|"+secLabel(sections[secCounter][0])+"]]
\n") + # wiki.write("
[["+secLabel(sections[(secCounter+1)%len(sections)][0])+"|"+secLabel(sections[(secCounter+1)%len(sections)][0])+"]] >>
\n
\n\n") + # wiki.write("{{head|pre="+secLabel(sections[secCounter-1][0])+"|cur="+secLabel(sections[secCounter][0])+"|next="+secLabel(sections[(secCounter+1)%len(sections)][0])+"}}\n") + wiki.write("{{#set:Chapter=" + chapter + "}}\n") + wiki.write("{{#set:Section=" + str(secCounter) + "}}\n") + wiki.write("{{headSection}}\n") + head = True + wiki.write("== " + getString(line) + " ==\n") + elif ("\\section" in lines[(i + 1) % len(lines)] or "\\end{document}" in lines[(i + 1) % len(lines)]) and parse: + # wiki.write("
\n") + # wiki.write("
<< [["+secLabel(sections[secCounter-1][0])+"|"+secLabel(sections[secCounter-1][0])+"]]
\n") + # wiki.write("
[[Orthogonal_Polynomials#"+secLabel(sections[secCounter][0])+"|"+secLabel(sections[secCounter][0])+"]]
\n") + # wiki.write("
[["+secLabel(sections[(secCounter+1)%len(sections)][0])+"|"+secLabel(sections[(secCounter+1)%len(sections)][0])+"]] >>
\n
\n\n") + # wiki.write("{{foot|pre="+secLabel(sections[secCounter-1][0])+"|cur="+secLabel(sections[secCounter][0])+"|next="+secLabel(sections[(secCounter+1)%len(sections)][0])+"}}\n") + wiki.write("{{footSection}}\n") + wiki.write("drmf_eof\n") + sections[secCounter].append(eqCounter) + eqCounter = 0 + + elif "\\subsection" in line and parse: + wiki.write("\n== " + getString(line) + " ==\n") + head = True + elif "\\paragraph" in line and parse: + wiki.write("\n=== " + getString(line) + " ===\n") + head = True + elif "\\subsubsection" in line and parse: + wiki.write("\n=== " + getString(line) + " ===\n") + head = True + + elif "\\begin{equation}" in line and parse: + # symLine="" + if head: + wiki.write("\n") + head = False + sLabel = line.find("\\formula{") + 9 + eLabel = line.find("}", sLabel) + label = modLabel(line[sLabel:eLabel]) + eqCounter += 1 + '''for l in lLink: + if label==l[0:l.find("=")-1]: + rlabel=l[l.find("=>")+3:l.find("\\n")] + rlabel=rlabel.replace("/","") + rlabel=rlabel.replace("#",":") + rlabel=rlabel.replace("!",":") + break''' + labels.append(label) + eqs.append("") + # wiki.write("\n\n") + wiki.write("{\displaystyle \n") + math = True + elif "\\begin{equation}" in line and not parse: + sLabel = line.find("\\formula{") + 9 + eLabel = line.find("}", sLabel) + label = modLabel(line[sLabel:eLabel]) + '''for l in lLink: + if l.find(label)!=-1: + rlabel=l[l.find("=>")+3:l.find("\\n")] + rlabel=rlabel.replace("/","") + rlabel=rlabel.replace("#",":") + break''' + labels.append("*" + label) # special marker + eqs.append("") + math = True + elif "\\end{equation}" in line: + + math = False + elif "\\constraint" in line and parse: + constraint = True + math = False + conLine = "" + elif "\\substitution" in line and parse: + substitution = True + math = False + subLine = "" + # wiki.write("
Substitution(s): "+getEq(line)+"

\n") + elif "\\proof" in line and parse: + math = False + elif "\\drmfn" in line and parse: + math = False + if "\\drmfname" in line and parse: + wiki.write("
This formula has the name: " + getString(line) + "

\n") + elif math and parse: + flagM = True + eqs[len(eqs) - 1] += line + + if "\\end{equation}" in lines[i + 1] and not "\\subsection" in lines[i + 3] and not "\\section" in lines[ + i + 3] and not "\\part" in lines[i + 3]: + u = i + flagM2 = False + while flagM: + u += 1 + if "\\begin{equation}" in lines[u] in lines[u]: + flagM = False + if "\\section" in lines[u] or "\\subsection" in lines[i] or "\\part" in lines[ + u] or "\\end{document}" in lines[u]: + flagM = False + flagM2 = True + if not (flagM2): + + wiki.write(line.rstrip("\n")) + wiki.write("\n}
\n") + else: + wiki.write(line.rstrip("\n")) + wiki.write("\n}\n") + elif "\\end{equation}" in lines[i + 1]: + wiki.write(line.rstrip("\n")) + wiki.write("\n}\n") + elif "\\constraint" in lines[i + 1] or "\\substitution" in lines[i + 1] or "\\drmfn" in lines[i + 1]: + wiki.write(line.rstrip("\n")) + wiki.write("\n}\n") + else: + wiki.write(line) + elif math and not parse: + eqs[len(eqs) - 1] += line + if "\\end{equation}" in lines[i + 1] or "\\constraint" in lines[i + 1] or "\\substitution" in lines[ + i + 1] or "\\drmfn" in lines[i + 1]: + math = False + if substitution and parse: + subLine = subLine + line.replace("&", "&
") + if "\\end{equation}" in lines[i + 1] or "\\substitution" in lines[i + 1] or "\\constraint" in lines[ + i + 1] or "\\drmfn" in lines[i + 1] or "\\proof" in lines[i + 1]: + substitution = False + wiki.write("
Substitution(s): " + getEq(subLine) + "

\n") + + if constraint and parse: + conLine = conLine + line.replace("&", "&
") + if "\\end{equation}" in lines[i + 1] or "\\substitution" in lines[i + 1] or "\\constraint" in lines[ + i + 1] or "\\drmfn" in lines[i + 1] or "\\proof" in lines[i + 1]: + constraint = False + wiki.write("
Constraint(s): " + getEq(conLine) + "

\n") + + eqCounter = 0 + endNum = len(labels) - 1 + '''for n in range(1,len(labels)): + if n+1!=len(labels) and labels[n+1][0]=="*" and labels[n][0]!="*": + labels[n]=""+labels[n] + endNum=n + elif labels[n][0]=="*": + labels[n]=""+labels[n][1:] + else: + labels[n]=""+labels[n]''' + '''for n in range(0,len(refLabels)): + refLabels[n]=""+refLabels[n]''' + parse = False + constraint = False + substitution = False + note = False + hCon = True + hSub = True + hNote = True + hProof = True + proof = False + comToWrite = "" + secCount = -1 + newSec = False + for i in range(0, len(lines)): + line = lines[i] + + if "\\section" in line: + secCount += 1 + newSec = True + eqS = 0 + + if "\\begin{equation}" in line: + symLine = line.strip("\n") + eqS += 1 + constraint = False + substitution = False + note = False + comToWrite = "" + hCon = True + hSub = True + hNote = True + hProof = True + proof = False + parse = True + symbols = [] + eqCounter += 1 + wiki.write("drmf_bof\n") + label = labels[eqCounter] + wiki.write("\'\'\'" + secLabel(label) + "\'\'\'\n") + wiki.write("{{DISPLAYTITLE:" + (labels[eqCounter]) + "}}\n") + if eqCounter < endNum: # FOR ANYTHING THAT IS NOT THE EXTRA EQUATIONS + wiki.write("
\n") + if newSec: + wiki.write("
<< [[" + secLabel(sections[secCount][0]).replace(" ", + "_") + "|" + secLabel( + sections[secCount][0]) + "]]
\n") + else: + wiki.write("
<< [[" + secLabel(labels[eqCounter - 1]).replace(" ", + "_") + "|" + secLabel( + labels[eqCounter - 1]) + "]]
\n") + wiki.write("
[[" + secLabel(sections[secCount + 1][0]).replace(" ", + "_") + "#" + secLabel( + labels[eqCounter][len("Formula:"):]) + "|formula in " + secLabel( + sections[secCount + 1][0]) + "]]
\n") + if eqS == sections[secCount][1]: + wiki.write( + "
[[" + secLabel(sections[(secCount + 1) % len(sections)][0]).replace( + " ", "_") + "|" + secLabel(sections[(secCount + 1) % len(sections)][0]) + "]] >>
\n") + else: + wiki.write( + "
[[" + secLabel(labels[(eqCounter + 1) % (endNum + 1)]).replace(" ", + "_") + "|" + secLabel( + labels[(eqCounter + 1) % (endNum + 1)]) + "]] >>
\n") + wiki.write("
\n\n") + elif eqCounter == endNum: + wiki.write("
\n") + if newSec: + newSec = False + wiki.write("
<< [[" + secLabel(sections[secCount][0]).replace(" ", + "_") + "|" + secLabel( + sections[secCount][0]) + "]]
\n") + else: + wiki.write("
<< [[" + secLabel(labels[eqCounter - 1]).replace(" ", + "_") + "|" + secLabel( + labels[eqCounter - 1]) + "]]
\n") + wiki.write("
[[" + secLabel(sections[secCount + 1][0]).replace(" ", + "_") + "#" + secLabel( + labels[eqCounter][len("Formula:"):]) + "|formula in " + secLabel( + sections[secCount + 1][0]) + "]]
\n") + wiki.write("
[[" + secLabel( + labels[(eqCounter + 1) % (endNum + 1)].replace(" ", "_")) + "|" + secLabel( + labels[(eqCounter + 1) % (endNum + 1)]) + "]]
\n") + wiki.write("
\n\n") + + wiki.write("
{\displaystyle \n") + math = True + elif "\\end{equation}" in line: + wiki.write(comToWrite) + parse = False + math = False + if hProof: + wiki.write( + "\n== Proof ==\n\nWe ask users to provide proof(s), reference(s) to proof(s), or further clarification on the proof(s) in this space.\n") + + wiki.write("\n== Symbols List ==\n\n") + newSym = [] + # if "09.07:04" in label: + for x in symbols: + flagA = True + # if x not in newSym: + cN = 0 + cC = 0 + flag = True + ArgCx = 0 + for z in x: + if z.isalpha() and flag or z == "&": + cN += 1 + else: + flag = False + if z == "{" or z == "[": + cC += 1 + if z == "}" or z == "]": + cC -= 1 + if cC == 0: + ArgCx += 1 + noA = x[:cN] + for y in newSym: + cN = 0 + cC = 0 + ArgC = 0 + flag = True + for z in y: + if z.isalpha() and flag or z == "&": + cN += 1 + else: + flag = False + if z == "{" or z == "[": + cC += 1 + if z == "}" or z == "]": + cC -= 1 + if cC == 0: + ArgC += 1 + + if y[:cN] == noA: # and ArgC==ArgCx: + flagA = False + break + if flagA: + newSym.append(x) + newSym.reverse() + symF = False + ampFlag = False + finSym = [] + for s in range(len(newSym) - 1, -1, -1): + symbolPar = "\\" + newSym[s] + ArgCx = 0 + parCx = 0 + parFlag = False + cC = 0 + for z in symbolPar: + if z == "@": + parFlag = True + elif z.isalpha() or z == "&": + cN += 1 + else: + if z == "{" or z == "[": + cC += 1 + if z == "}" or z == "]": + cC -= 1 + if cC == 0: + if parFlag: + parCx += 1 + else: + ArgCx += 1 + + if symbolPar.find("{") != -1 or symbolPar.find("[") != -1: + if symbolPar.find("[") == -1: + symbol = symbolPar[0:symbolPar.find("{")] + elif symbolPar.find("{") == -1 or symbolPar.find("[") < symbolPar.find("{"): + symbol = symbolPar[0:symbolPar.find("[")] + else: + symbol = symbolPar[0:symbolPar.find("{")] + else: + symbol = symbolPar + numArg = parCx + numPar = ArgCx + gFlag = False + checkFlag = False + get = False + gCSV = csv.reader(open('new.Glossary.csv', 'rb'), delimiter=',', quotechar='\"') + preG = "" + if symbol == "\\&": + ampFlag = True + for S in gCSV: + G = S + ArgCx = 0 + parCx = 0 + parFlag = False + cC = 0 + ind = G[0].find("@") + if ind == -1: + ind = len(G[0]) - 1 + for z in G[0]: + if z == "@": + parFlag = True + elif z.isalpha(): + cN += 1 + else: + if z == "{" or z == "[": + cC += 1 + if z == "}" or z == "]": + cC -= 1 + if cC == 0: + if parFlag: + parCx += 1 + else: + ArgCx += 1 + if G[0].find(symbol) == 0 and (len(G[0]) == len(symbol) or not G[0][ + len(symbol)].isalpha()): # and (numPar!=0 or numArg!=0): + checkFlag = True + get = True + preG = S + + + elif checkFlag: + get = True + checkFlag = False + + if (get): + if get: + G = preG + get = False + checkFlag = False + if True: + if symbolPar.find("@") != -1: + Q = symbolPar[:symbolPar.find("@")] + else: + Q = symbolPar + listArgs = [] + if len(Q) > len(symbol) and (Q[len(symbol)] == "{" or Q[len(symbol)] == "["): + ap = "" + for o in range(len(symbol), len(Q)): + if Q[o] == "{" or z == "[": + argFlag = True + elif Q[o] == "}" or z == "]": + argFlag = False + listArgs.append(ap) + ap = "" + else: + ap += Q[o] + '''websiteU=g[g.find("http://"):].strip("\n") + k=0 + websites=[] + for r in range(0,len(websiteU)): + if websiteU[r]==",": + if websiteU[k:r].find("http://")!=-1: + websites.append(websiteU[k:r+1].strip(" ")) + k=r + + + if websiteU[k:r].find("http://")!=-1: + websites.append(websiteU[k:r+1].strip(" ")) + websiteF="" + for d in websites: + websiteF=websiteF+" ["+d+" "+d+"]"''' + # websiteF=G[4].strip("\n") + websiteF = "" + web1 = G[5] + for t in range(5, len(G)): + if G[t] != "": + websiteF = websiteF + " [" + G[t] + " " + G[t] + "]" + + # p1=Q + # if Q.find("@")!=-1: + # p1=Q[:Q.find("@")] + p1 = G[4].strip("$") + p1 = "{\\displaystyle " + p1 + "}" + # if checkFlag: + new1 = "" + new2 = "" + pause = False + mathF = True + '''for k in range(0,len(p1)): + if p1[k]=="$": + if mathF: + new1+="{\\displaystyle " + else: + new1+="}" + mathF=not mathF + + elif p1[k]=="#" and p1[k+1].isdigit(): + pause=True + elif pause: + num=int(p1[k]) + #letter=chr(num+96) + letter=listArgs[num-1] + new1+=letter + pause=False + + else: + new1+=p1[k]''' + p2 = G[1] + for k in range(0, len(p2)): + if p2[k] == "$": + if mathF: + new2 += "{\\displaystyle " + else: + new2 += "}" + mathF = not mathF + else: + new2 += p2[k] + # p1=new1 + p2 = new2 + finSym.append(web1 + " " + p1 + "] : " + p2 + " :" + websiteF) + break + # gFlag=True + # if not symF: + # symF=True + # wiki.write("[") + # else: + # wiki.write("
\n") + # wiki.write("[") + # wiki.write(g[g.find("http://"):].strip("\n")) + # wiki.write(" ") + # wiki.write(getEq(g[g.find("{$"):]).strip("\n").replace("\\\\","\\")) + # wiki.write("] : ") + # wiki.write(g[g.find(" {")+2:g.find("} ",g.find(" {")+1)]) + # wiki.write(" : [") + # wiki.write(g[g.find("http://"):].strip("\n")) + # wiki.write(" ") + # wiki.write(g[g.find("http://"):].strip("\n")) + # wiki.write("] ")''' + + # preG=S + if not gFlag: + del newSym[s] + + gFlag = True + # finSym.reverse() + if ampFlag: + wiki.write("& : logical and") + gFlag = False + for y in finSym: + if y == "& : logical and": + pass + elif gFlag: + gFlag = False + wiki.write("[" + y) + else: + wiki.write("
\n[" + y) + + wiki.write("\n
\n") + + wiki.write("\n== Bibliography==\n\n") # should there be a space between bibliography and ==? + r = unmodLabel(labels[eqCounter]) + q = r.find("KLS:") + 4 + p = r.find(":", q) + section = r[q:p] + equation = r[p + 1:] + if equation.find(":") != -1: + equation = equation[0:equation.find(":")] + + wiki.write( + "[http://homepage.tudelft.nl/11r49/askey/contents.html Equation in Section " + section + "] of [[Bibliography#KLS|'''KLS''']].\n\n") # Where should it link to? + wiki.write("== URL links ==\n\nWe ask users to provide relevant URL links in this space.\n\n") + if eqCounter < endNum: + wiki.write("
\n") + if newSec: + newSec = False + wiki.write("
<< [[" + secLabel(sections[secCount][0]).replace(" ", + "_") + "|" + secLabel( + sections[secCount][0]) + "]]
\n") + else: + wiki.write("
<< [[" + secLabel(labels[eqCounter - 1]).replace(" ", + "_") + "|" + secLabel( + labels[eqCounter - 1]) + "]]
\n") + wiki.write("
[[" + secLabel(sections[secCount + 1][0]).replace(" ", + "_") + "#" + secLabel( + labels[eqCounter][len("Formula:"):]) + "|formula in " + secLabel( + sections[secCount + 1][0]) + "]]
\n") + if eqS == sections[secCount][1]: + wiki.write("
[[" + sections[(secCount + 1) % len(sections)][0].replace(" ", + "_") + "|" + + sections[(secCount + 1) % len(sections)][0] + "]] >>
\n") + else: + wiki.write( + "
[[" + secLabel(labels[(eqCounter + 1) % (endNum + 1)]).replace(" ", + "_") + "|" + secLabel( + labels[(eqCounter + 1) % (endNum + 1)]) + "]] >>
\n") + wiki.write("
\n\ndrmf_eof\n") + else: # FOR EXTRA EQUATIONS + wiki.write("
\n") + wiki.write("
<< [[" + labels[endNum - 1].replace(" ", "_") + "|" + labels[ + endNum - 1] + "]]
\n") + wiki.write("
[[" + labels[0].replace(" ", "_") + "#" + labels[endNum][ + 8:] + "|formula in " + + labels[0] + "]]
\n") + wiki.write("
[[" + labels[(0) % endNum].replace(" ", "_") + "|" + labels[ + 0 % endNum] + "]]
\n") + wiki.write("
\n\ndrmf_eof\n") + + + + elif "\\constraint" in line and parse: + # symbols=symbols+getSym(line) + symLine = line.strip("\n") + if hCon: + comToWrite = comToWrite + "\n== Constraint(s) ==\n\n" + hCon = False + constraint = True + math = False + conLine = "" + # wiki.write("
"+getEq(line)+"

\n") + elif "\\substitution" in line and parse: + # symbols=symbols+getSym(line) + symLine = line.strip("\n") + if hSub: + comToWrite = comToWrite + "\n== Substitution(s) ==\n\n" + hSub = False + # wiki.write("
"+getEq(line)+"

\n") + substitution = True + math = False + subLine = "" + elif "\\drmfname" in line and parse: + math = False + comToWrite = "\n== Name ==\n\n
" + getString(line) + "

\n" + comToWrite + + elif "\\drmfnote" in line and parse: + symbols = symbols + getSym(line) + if hNote: + comToWrite = comToWrite + "\n== Note(s) ==\n\n" + hNote = False + note = True + math = False + noteLine = "" + + elif "\\proof" in line and parse: + # symbols=symbols+getSym(line) + symLine = line.strip("\n") + if hProof: + hProof = False + comToWrite = comToWrite + "\n== Proof ==\n\nWe ask users to provide proof(s), reference(s) to proof(s), or further clarification on the proof(s) in this space. \n

\n
" + proof = True + proofLine = "" + pause = False + pauseP = False + for ind in range(0, len(line)): + if line[ind:ind + 7] == "\\eqref{": + pause = True + eqR = line[ind:line.find("}", ind) + 1] + rLab = getString(eqR) + '''for l in lLink: + if rLab==l[0:l.find("=")-1]: + rlabel=l[l.find("=>")+3:l.find("\\n")] + rlabel=rlabel.replace("/","") + rlabel=rlabel.replace("#",":") + rlabel=rlabel.replace("!",":") + break''' + + eInd = refLabels.index("" + label) + z = line[line.find("}", ind + 7) + 1] + if z == "." or z == ",": + pauseP = True + proofLine += ("
\n{\displaystyle \n" + refEqs[ + eInd] + "}" + z + "
\n") + else: + if z == "}": + proofLine += ( + "
\n{\displaystyle \n" + refEqs[eInd] + "}
") + else: + proofLine += ( + "
\n{\displaystyle \n" + refEqs[eInd] + "}
\n") + + + else: + if pause: + if line[ind] == "}": + pause = False + elif pauseP: + pauseP = False + elif line[ind] == "\n" and "\\end{equation}" in lines[i + 1]: + pass + else: + proofLine += (line[ind]) + if "\\end{equation}" in lines[i + 1]: + proof = False + # symLine+=line.strip("\n") + wiki.write(comToWrite + getEqP(proofLine, False) + "
\n
\n") + comToWrite = "" + symbols = symbols + getSym(symLine) + symLine = "" + + # wiki.write(line) + + elif proof: + symLine += line.strip("\n") + pauseP = False + # symbols=symbols+getSym(line) + for ind in range(0, len(line)): + if line[ind:ind + 7] == "\\eqref{": + pause = True + eqR = line[ind:line.find("}", ind) + 1] + rLab = getString(eqR) + '''for l in lLink: + if rLab==l[0:l.find("=")-1]: + rlabel=l[l.find("=>")+3:l.find("\\n")] + rlabel=rlabel.replace("/","") + rlabel=rlabel.replace("#",":") + rlabel=rlabel.replace("!",":") + break''' + + eInd = refLabels.index("" + label) + z = line[line.find("}", ind + 7) + 1] + if z == "." or z == ",": + pauseP = True + proofLine += ("
\n{\displaystyle \n" + refEqs[ + eInd] + "}" + z + "
\n") + else: + proofLine += ( + "
\n{\displaystyle \n" + refEqs[eInd] + "}
\n") + + else: + if pause: + if line[ind] == "}": + pause = False + elif pauseP: + pauseP = False + elif line[ind] == "\n" and "\\end{equation}" in lines[i + 1]: + pass + + else: + proofLine += (line[ind]) + if "\\end{equation}" in lines[i + 1]: + proof = False + # symLine+=line.strip("\n") + wiki.write(comToWrite + getEqP(proofLine, False).rstrip("\n") + "
\n
\n") + comToWrite = "" + symbols = symbols + getSym(symLine) + symLine = "" + + elif math: + if "\\end{equation}" in lines[i + 1] or "\\constraint" in lines[i + 1] or "\\substitution" in lines[ + i + 1] or "\\proof" in lines[i + 1] or "\\drmfnote" in lines[i + 1] or "\\drmfname" in lines[ + i + 1]: + wiki.write(line.rstrip("\n")) + symLine += line.strip("\n") + symbols = symbols + getSym(symLine) + symLine = "" + wiki.write("\n}
\n") + else: + symLine += line.strip("\n") + wiki.write(line) + if note and parse: + noteLine = noteLine + line + symbols = symbols + getSym(line) + if "\\end{equation}" in lines[i + 1] or "\\drmfn" in lines[i + 1] or "\\constraint" in lines[ + i + 1] or "\\substitution" in lines[i + 1] or "\\proof" in lines[i + 1]: + note = False + comToWrite = comToWrite + "
" + getEq(noteLine) + "

\n" + + if constraint and parse: + conLine = conLine + line.replace("&", "&
") + + symLine += line.strip("\n") + # symbols=symbols+getSym(line) + if "\\end{equation}" in lines[i + 1] or "\\drmfn" in lines[i + 1] or "\\constraint" in lines[ + i + 1] or "\\substitution" in lines[i + 1] or "\\proof" in lines[i + 1]: + constraint = False + symbols = symbols + getSym(symLine) + symLine = "" + wiki.write(comToWrite + "
" + getEq(conLine) + "

\n") + comToWrite = "" + if substitution and parse: + subLine = subLine + line.replace("&", "&
") + + symLine += line.strip("\n") + # symbols=symbols+getSym(line) + if "\\end{equation}" in lines[i + 1] or "\\drmfn" in lines[i + 1] or "\\substitution" in lines[ + i + 1] or "\\constraint" in lines[i + 1] or "\\proof" in lines[i + 1]: + substitution = False + symbols = symbols + getSym(symLine) + symLine = "" + wiki.write(comToWrite + "
" + getEq(subLine) + "

\n") + comToWrite = "" + # except Exception as detail: #If exception occured + # print("Exception",detail) #print details of error + # except: #If anythin else occured... + # print ("ERROR",sys.exc_info()[0])#ERROR with basic info diff --git a/Azeem/test/test_isnumber.py b/Azeem/test/test_isnumber.py new file mode 100644 index 0000000..84ae1c7 --- /dev/null +++ b/Azeem/test/test_isnumber.py @@ -0,0 +1,8 @@ +from unittest import TestCase +from tex2Wiki import isnumber + + +class TestIsnumber(TestCase): + def test_isnumber(self): + self.assertEqual(True, isnumber("1")) + self.assertEqual(False, isnumber("a")) diff --git a/Azeem/tex2Wiki.py b/Azeem/tex2Wiki.py deleted file mode 100644 index 0711a0f..0000000 --- a/Azeem/tex2Wiki.py +++ /dev/null @@ -1,1069 +0,0 @@ -# Version 14: Templates -# Convert tex to wikiText -import csv # imported for using csv format -import sys # imported for getting args -import os # imported for copying file - - -def isnumber(char): - return char[0] in "0123456789" - - -def getString(line): # Gets all data within curly braces on a line - # -------Initialization------- - stringWrite = "" - getStr = False - pW = "" - # ---------------------------- - for c in line: - if c == "{" or c == "}": # if there is a curly brace in the line - getStr = not (getStr) # toggle the getStr flag - if c == "}": # no more info needed - break - elif getStr: # if within curly braces - if not (pW == c and c == "-"): # if there is no double dash (makes single dash) - if c != "$": # if not a $ sign - stringWrite += c # this character is part of the data - else: # replace $ with '' - stringWrite += "\'\'" # add double ' - pW = c # change last character for finding double dashes - return (stringWrite.rstrip('\n').lstrip()) # return the data without newlines and without leading spaces - - -def getG(line): # gets equation for symbols list - start = True - final = "" - for c in line: - if c == "$" and start: - final += "{\\displaystyle " - start = False - elif c == "$": - final += "}" - start = True - else: - final += c - return (final) - - -def getEq(line): # Gets all data within constraints,substitutions - # -------Initialization------- - per = 1 - stringWrite = "" - fEq = False - count = 0 - # ---------------------------- - for c in line: # read each character - if count >= 0 and per != 0: - per += 1 - if c != " " and c != "%": - per = 0 - if c == "{" and per == 0: - if count > 0: - stringWrite += c - count += 1 - elif c == "}" and per == 0: - count -= 1 - if count > 0: - stringWrite += c - elif c == "$" and per == 0: # either begin or end equation - fEq = not (fEq) # toggle fEq flag to know if begin or end equation - if fEq: # if begin - stringWrite += "{\\displaystyle " - else: # if end - stringWrite += "}" - elif c == "\n" and per == 0: # if newline - stringWrite = stringWrite.strip() # remove all leading and trailing whitespace - - # should above be rstrip?<-------------------------------------------------------------------CHECK THIS - - stringWrite += c # add the newline character - per += 1 # watch for % signs - elif count > 0 and per == 0: # not special character - stringWrite += c # write the character - - return (stringWrite.rstrip().lstrip()) - - -def getEqP(line, Flag): # Gets all data within proofs - if Flag: - a = 1 - else: - a = 0 - per = 1 - stringWrite = "" - fEq = False - count = 0 - length = 0 - for c in line: - if fEq and c != " " and c != "$" and c != "{" and c != "}" and not c.isalpha(): - length += 1 - if count >= 0 and per != 0: - per += 1 - if c != " " and c != "%": - # if c!="%": - per = 0 - if c == "{" and per == 0: - if count > 0: - stringWrite += c - count += 1 - elif c == "}" and per == 0: - count -= 1 - if count > 0: - stringWrite += c - elif c == "$" and per == 0: - fEq = not (fEq) - if fEq: - stringWrite += "{\\displaystyle " - else: - if length < 10: - stringWrite += "}" - else: - stringWrite += "}" + "
" - length = 0 - elif c == "\n" and per == 0: - stringWrite = stringWrite.strip() - stringWrite += c - per += 1 - elif count > 0 and per == 0: - stringWrite += c - - return (stringWrite.lstrip()) - - -def getSym(line): # Gets all symbols on a line for symbols list - symList = [] - if line == "": - return symList - symbol = "" - symFlag = False - argFlag = False - cC = 0 - for i in range(0, len(line)): - # if "MeixnerPollaczek{\\lambda}{n}@{x}{\\phi}=\frac{\\pochhammer{2\\lambda}{n}}" in line: - c = line[i] - if symFlag: - if c == "{" or c == "[": - cC += 1 - argFlag = True - if c != "}" and c != "]": - if argFlag or c.isalpha(): - symbol += c - else: - symFlag = False - argFlag = False - symList.append(symbol) - symList += (getSym(symbol)) - symbol = "" - else: - cC -= 1 - symbol += c - if i + 1 == len(line): - p = "" - else: - p = line[i + 1] - if cC == 0 and p != "{" and p != "[" and p != "@": - symFlag = False - # if "monicAskeyWilson{n+1}@{x}{a}{b}{c}{d}{q}+\\frac{1}{2}" in line: - symList.append(symbol) - symList += (getSym(symbol)) - argFlag = False - symbol = "" - - elif c == "\\": - symFlag = True - elif c == "&": - symList.append("&") - - # if "MeixnerPollaczek{\\lambda}{n}@{x}{\\phi}=\frac{\\pochhammer{2\\lambda}{n}}" in line: - symList.append(symbol) - symList += getSym(symbol) - return (symList) - - -def unmodLabel(label): - label = label.replace(".0", ".") - label = label.replace(":0", ":") - return (label) - - -def secLabel(label): - return (label.replace("\'\'", "")) - - -def modLabel(label): - # label.replace("Formula:KLS:","KLS;") - isNumer = False - newlabel = "" - num = "" - for i in range(0, len(label)): - if isNumer and not isnumber(label[i]): - if len(num) > 1: - newlabel += num - isNumer = False - num = "" - else: - newlabel += "0" + str(num) - num = "" - isNumer = False - if isnumber(label[i]): - isNumer = True - num += str(label[i]) - else: - isNumer = False - newlabel += label[i] - if len(num) > 1: - newlabel += num - elif len(num) == 1: - newlabel += "0" + num - return (newlabel) - - -# try: -for jsahlfkjsd in range(0, 1): - tex = open(sys.argv[1], 'r') - wiki = open(sys.argv[2], 'w') - main = open("OrthogonalPolynomials.mmd", "r") - mainText = main.read() - mainPrepend = "" - mainWrite = open("OrthogonalPolynomials.mmd.new", "w") - tester = open("testData.txt", 'w') - # glossary=open('Glossary', 'r') - glossary = open('new.Glossary.csv', 'rb') - gCSV = csv.reader(glossary, delimiter=',', quotechar='\"') - # lLinks=open('BruceLabelLinks', 'r') - lGlos = glossary.readlines() - # lLink=lLinks.readlines() - math = False - constraint = False - substitution = False - symbols = [] - lines = tex.readlines() - refLines = [] - sections = [] - labels = [] - eqs = [] - refEqs = [] - parse = False - head = False - # chapRef=[("GA",open("GA.tex",'r')),("ZE",open("ZE.3.tex",'r'))] - refLabels = [] - '''for c in chapRef: - refLines=refLines+(c[1].readlines()) - c[1].close()''' - for i in range(0, len(refLines)): - line = refLines[i] - if "\\begin{equation}" in line: - sLabel = line.find("\\formula{") + 9 - eLabel = line.find("}", sLabel) - label = modLabel(line[sLabel:eLabel]) - '''for l in lLink: - if l.find(label)!=-1 and l[len(label)+1]=="=": - rlabel=l[l.find("=>")+3:l.find("\\n")] - rlabel=rlabel.replace("/","") - rlabel=rlabel.replace("#",":") - break''' - refLabels.append(label) - refEqs.append("") - math = True - elif math: - refEqs[len(refEqs) - 1] += line - if "\\end{equation}" in refLines[i + 1] or "\\constraint" in refLines[i + 1] or "\\substitution" in \ - refLines[i + 1] or "\\drmfn" in refLines[i + 1]: - math = False - - for i in range(0, len(lines)): - line = lines[i] - if "\\begin{document}" in line: - # wiki.write("drmf_bof\n") - parse = True - elif "\\end{document}" in line and parse: - # wiki.write("\n") - mainPrepend += "\n" - mainText = mainPrepend + mainText - mainText = mainText.replace("drmf_bof\n", "") - mainText = mainText.replace("drmf_eof\n", "") - mainText = mainText.replace("\'\'\'Orthogonal Polynomials\'\'\'\n", "") - mainText = mainText.replace("{{#set:Section=0}}\n", "") - mainText = mainText[0:mainText.rfind("== Sections ")] - mainText = "drmf_bof\n\'\'\'Orthogonal Polynomials\'\'\'\n{{#set:Section=0}}\n" + mainText + "\ndrmf_eof\n" - mainWrite.write(mainText) - mainWrite.close() - main.close() - os.system("cp -f OrthogonalPolynomials.mmd.new OrthogonalPolynomials.mmd") - # wiki.write("\ndrmf_eof\n") - parse = False - elif "\\title" in line and parse: - stringWrite = "\'\'\'" - stringWrite += getString(line) + "\'\'\'\n" - labels.append("Orthogonal Polynomials") - sections.append(["Orthogonal Polynomials", 0]) - # wiki.write(stringWrite) - # mainPrepend+=stringWrite - chapter = getString(line) - mainPrepend += ( - "\n== Sections in " + chapter + " ==\n\n
\n") - elif "\\part" in line: - if getString(line) == "BOF": - parse = False - elif getString(line) == "EOF": - parse = True - elif parse: - mainPrepend += ("\n
\n= " + getString(line) + " =\n") - head = True - elif "\\section" in line: - mainPrepend += ("* [[" + secLabel(getString(line)) + "|" + getString(line) + "]]\n") - sections.append([getString(line)]) - - secCounter = 0 - eqCounter = 0 - for i in range(0, len(lines)): - line = lines[i] - # line.replace("\\begin{equation}","$") - # line.replace("\\end{equation},"$") - '''if "\\begin{document}" in line: - wiki.write("drmf_bof\n") - parse=True - elif "\\end{document}" in line and parse: - wiki.write("\ndrmf_eof\n") - parse=False - elif "\\title" in line and parse: - stringWrite="\'\'\'" - stringWrite+=getString(line)+"\'\'\'\n" - labels.append("Orthogonal Polynomials") - wiki.write(stringWrite\n) - elif "\\part" in line: - if getString(line)=="BOF": - parse=False - elif getString(line)=="EOF": - parse=True - elif parse: - wiki.write("\n
\n= "+getString(line)+" =\n") - head=True - ''' - if "\\section" in line: - parse = True - secCounter += 1 - wiki.write("drmf_bof\n") - wiki.write("\'\'\'" + secLabel(getString(line)) + "\'\'\'\n") - wiki.write("{{DISPLAYTITLE:" + (sections[secCounter][0]) + "}}\n") - # wiki.write("
\n") - # wiki.write("
<< [["+secLabel(sections[secCounter-1][0])+"|"+secLabel(sections[secCounter-1][0])+"]]
\n") - # wiki.write("
[[Orthogonal_Polynomials#"+secLabel(sections[secCounter][0])+"|"+secLabel(sections[secCounter][0])+"]]
\n") - # wiki.write("
[["+secLabel(sections[(secCounter+1)%len(sections)][0])+"|"+secLabel(sections[(secCounter+1)%len(sections)][0])+"]] >>
\n
\n\n") - # wiki.write("{{head|pre="+secLabel(sections[secCounter-1][0])+"|cur="+secLabel(sections[secCounter][0])+"|next="+secLabel(sections[(secCounter+1)%len(sections)][0])+"}}\n") - wiki.write("{{#set:Chapter=" + chapter + "}}\n") - wiki.write("{{#set:Section=" + str(secCounter) + "}}\n") - wiki.write("{{headSection}}\n") - head = True - wiki.write("== " + getString(line) + " ==\n") - elif ("\\section" in lines[(i + 1) % len(lines)] or "\\end{document}" in lines[(i + 1) % len(lines)]) and parse: - # wiki.write("
\n") - # wiki.write("
<< [["+secLabel(sections[secCounter-1][0])+"|"+secLabel(sections[secCounter-1][0])+"]]
\n") - # wiki.write("
[[Orthogonal_Polynomials#"+secLabel(sections[secCounter][0])+"|"+secLabel(sections[secCounter][0])+"]]
\n") - # wiki.write("
[["+secLabel(sections[(secCounter+1)%len(sections)][0])+"|"+secLabel(sections[(secCounter+1)%len(sections)][0])+"]] >>
\n
\n\n") - # wiki.write("{{foot|pre="+secLabel(sections[secCounter-1][0])+"|cur="+secLabel(sections[secCounter][0])+"|next="+secLabel(sections[(secCounter+1)%len(sections)][0])+"}}\n") - wiki.write("{{footSection}}\n") - wiki.write("drmf_eof\n") - sections[secCounter].append(eqCounter) - eqCounter = 0 - - elif "\\subsection" in line and parse: - wiki.write("\n== " + getString(line) + " ==\n") - head = True - elif "\\paragraph" in line and parse: - wiki.write("\n=== " + getString(line) + " ===\n") - head = True - elif "\\subsubsection" in line and parse: - wiki.write("\n=== " + getString(line) + " ===\n") - head = True - - elif "\\begin{equation}" in line and parse: - # symLine="" - if head: - wiki.write("\n") - head = False - sLabel = line.find("\\formula{") + 9 - eLabel = line.find("}", sLabel) - label = modLabel(line[sLabel:eLabel]) - eqCounter += 1 - '''for l in lLink: - if label==l[0:l.find("=")-1]: - rlabel=l[l.find("=>")+3:l.find("\\n")] - rlabel=rlabel.replace("/","") - rlabel=rlabel.replace("#",":") - rlabel=rlabel.replace("!",":") - break''' - labels.append(label) - eqs.append("") - # wiki.write("\n\n") - wiki.write("{\displaystyle \n") - math = True - elif "\\begin{equation}" in line and not parse: - sLabel = line.find("\\formula{") + 9 - eLabel = line.find("}", sLabel) - label = modLabel(line[sLabel:eLabel]) - '''for l in lLink: - if l.find(label)!=-1: - rlabel=l[l.find("=>")+3:l.find("\\n")] - rlabel=rlabel.replace("/","") - rlabel=rlabel.replace("#",":") - break''' - labels.append("*" + label) # special marker - eqs.append("") - math = True - elif "\\end{equation}" in line: - - math = False - elif "\\constraint" in line and parse: - constraint = True - math = False - conLine = "" - elif "\\substitution" in line and parse: - substitution = True - math = False - subLine = "" - # wiki.write("
Substitution(s): "+getEq(line)+"

\n") - elif "\\proof" in line and parse: - math = False - elif "\\drmfn" in line and parse: - math = False - if "\\drmfname" in line and parse: - wiki.write("
This formula has the name: " + getString(line) + "

\n") - elif math and parse: - flagM = True - eqs[len(eqs) - 1] += line - - if "\\end{equation}" in lines[i + 1] and not "\\subsection" in lines[i + 3] and not "\\section" in lines[ - i + 3] and not "\\part" in lines[i + 3]: - u = i - flagM2 = False - while flagM: - u += 1 - if "\\begin{equation}" in lines[u] in lines[u]: - flagM = False - if "\\section" in lines[u] or "\\subsection" in lines[i] or "\\part" in lines[ - u] or "\\end{document}" in lines[u]: - flagM = False - flagM2 = True - if not (flagM2): - - wiki.write(line.rstrip("\n")) - wiki.write("\n}
\n") - else: - wiki.write(line.rstrip("\n")) - wiki.write("\n}\n") - elif "\\end{equation}" in lines[i + 1]: - wiki.write(line.rstrip("\n")) - wiki.write("\n}\n") - elif "\\constraint" in lines[i + 1] or "\\substitution" in lines[i + 1] or "\\drmfn" in lines[i + 1]: - wiki.write(line.rstrip("\n")) - wiki.write("\n}\n") - else: - wiki.write(line) - elif math and not parse: - eqs[len(eqs) - 1] += line - if "\\end{equation}" in lines[i + 1] or "\\constraint" in lines[i + 1] or "\\substitution" in lines[ - i + 1] or "\\drmfn" in lines[i + 1]: - math = False - if substitution and parse: - subLine = subLine + line.replace("&", "&
") - if "\\end{equation}" in lines[i + 1] or "\\substitution" in lines[i + 1] or "\\constraint" in lines[ - i + 1] or "\\drmfn" in lines[i + 1] or "\\proof" in lines[i + 1]: - substitution = False - wiki.write("
Substitution(s): " + getEq(subLine) + "

\n") - - if constraint and parse: - conLine = conLine + line.replace("&", "&
") - if "\\end{equation}" in lines[i + 1] or "\\substitution" in lines[i + 1] or "\\constraint" in lines[ - i + 1] or "\\drmfn" in lines[i + 1] or "\\proof" in lines[i + 1]: - constraint = False - wiki.write("
Constraint(s): " + getEq(conLine) + "

\n") - - eqCounter = 0 - endNum = len(labels) - 1 - '''for n in range(1,len(labels)): - if n+1!=len(labels) and labels[n+1][0]=="*" and labels[n][0]!="*": - labels[n]=""+labels[n] - endNum=n - elif labels[n][0]=="*": - labels[n]=""+labels[n][1:] - else: - labels[n]=""+labels[n]''' - '''for n in range(0,len(refLabels)): - refLabels[n]=""+refLabels[n]''' - parse = False - constraint = False - substitution = False - note = False - hCon = True - hSub = True - hNote = True - hProof = True - proof = False - comToWrite = "" - secCount = -1 - newSec = False - for i in range(0, len(lines)): - line = lines[i] - - if "\\section" in line: - secCount += 1 - newSec = True - eqS = 0 - - if "\\begin{equation}" in line: - symLine = line.strip("\n") - eqS += 1 - constraint = False - substitution = False - note = False - comToWrite = "" - hCon = True - hSub = True - hNote = True - hProof = True - proof = False - parse = True - symbols = [] - eqCounter += 1 - wiki.write("drmf_bof\n") - label = labels[eqCounter] - wiki.write("\'\'\'" + secLabel(label) + "\'\'\'\n") - wiki.write("{{DISPLAYTITLE:" + (labels[eqCounter]) + "}}\n") - if eqCounter < endNum: # FOR ANYTHING THAT IS NOT THE EXTRA EQUATIONS - wiki.write("
\n") - if newSec: - wiki.write("
<< [[" + secLabel(sections[secCount][0]).replace(" ", - "_") + "|" + secLabel( - sections[secCount][0]) + "]]
\n") - else: - wiki.write("
<< [[" + secLabel(labels[eqCounter - 1]).replace(" ", - "_") + "|" + secLabel( - labels[eqCounter - 1]) + "]]
\n") - wiki.write("
[[" + secLabel(sections[secCount + 1][0]).replace(" ", - "_") + "#" + secLabel( - labels[eqCounter][len("Formula:"):]) + "|formula in " + secLabel( - sections[secCount + 1][0]) + "]]
\n") - if eqS == sections[secCount][1]: - wiki.write( - "
[[" + secLabel(sections[(secCount + 1) % len(sections)][0]).replace( - " ", "_") + "|" + secLabel(sections[(secCount + 1) % len(sections)][0]) + "]] >>
\n") - else: - wiki.write( - "
[[" + secLabel(labels[(eqCounter + 1) % (endNum + 1)]).replace(" ", - "_") + "|" + secLabel( - labels[(eqCounter + 1) % (endNum + 1)]) + "]] >>
\n") - wiki.write("
\n\n") - elif eqCounter == endNum: - wiki.write("
\n") - if newSec: - newSec = False - wiki.write("
<< [[" + secLabel(sections[secCount][0]).replace(" ", - "_") + "|" + secLabel( - sections[secCount][0]) + "]]
\n") - else: - wiki.write("
<< [[" + secLabel(labels[eqCounter - 1]).replace(" ", - "_") + "|" + secLabel( - labels[eqCounter - 1]) + "]]
\n") - wiki.write("
[[" + secLabel(sections[secCount + 1][0]).replace(" ", - "_") + "#" + secLabel( - labels[eqCounter][len("Formula:"):]) + "|formula in " + secLabel( - sections[secCount + 1][0]) + "]]
\n") - wiki.write("
[[" + secLabel( - labels[(eqCounter + 1) % (endNum + 1)].replace(" ", "_")) + "|" + secLabel( - labels[(eqCounter + 1) % (endNum + 1)]) + "]]
\n") - wiki.write("
\n\n") - - wiki.write("
{\displaystyle \n") - math = True - elif "\\end{equation}" in line: - wiki.write(comToWrite) - parse = False - math = False - if hProof: - wiki.write( - "\n== Proof ==\n\nWe ask users to provide proof(s), reference(s) to proof(s), or further clarification on the proof(s) in this space.\n") - - wiki.write("\n== Symbols List ==\n\n") - newSym = [] - # if "09.07:04" in label: - for x in symbols: - flagA = True - # if x not in newSym: - cN = 0 - cC = 0 - flag = True - ArgCx = 0 - for z in x: - if z.isalpha() and flag or z == "&": - cN += 1 - else: - flag = False - if z == "{" or z == "[": - cC += 1 - if z == "}" or z == "]": - cC -= 1 - if cC == 0: - ArgCx += 1 - noA = x[:cN] - for y in newSym: - cN = 0 - cC = 0 - ArgC = 0 - flag = True - for z in y: - if z.isalpha() and flag or z == "&": - cN += 1 - else: - flag = False - if z == "{" or z == "[": - cC += 1 - if z == "}" or z == "]": - cC -= 1 - if cC == 0: - ArgC += 1 - - if y[:cN] == noA: # and ArgC==ArgCx: - flagA = False - break - if flagA: - newSym.append(x) - newSym.reverse() - symF = False - ampFlag = False - finSym = [] - for s in range(len(newSym) - 1, -1, -1): - symbolPar = "\\" + newSym[s] - ArgCx = 0 - parCx = 0 - parFlag = False - cC = 0 - for z in symbolPar: - if z == "@": - parFlag = True - elif z.isalpha() or z == "&": - cN += 1 - else: - if z == "{" or z == "[": - cC += 1 - if z == "}" or z == "]": - cC -= 1 - if cC == 0: - if parFlag: - parCx += 1 - else: - ArgCx += 1 - - if symbolPar.find("{") != -1 or symbolPar.find("[") != -1: - if symbolPar.find("[") == -1: - symbol = symbolPar[0:symbolPar.find("{")] - elif symbolPar.find("{") == -1 or symbolPar.find("[") < symbolPar.find("{"): - symbol = symbolPar[0:symbolPar.find("[")] - else: - symbol = symbolPar[0:symbolPar.find("{")] - else: - symbol = symbolPar - numArg = parCx - numPar = ArgCx - gFlag = False - checkFlag = False - get = False - gCSV = csv.reader(open('new.Glossary.csv', 'rb'), delimiter=',', quotechar='\"') - preG = "" - if symbol == "\\&": - ampFlag = True - for S in gCSV: - G = S - ArgCx = 0 - parCx = 0 - parFlag = False - cC = 0 - ind = G[0].find("@") - if ind == -1: - ind = len(G[0]) - 1 - for z in G[0]: - if z == "@": - parFlag = True - elif z.isalpha(): - cN += 1 - else: - if z == "{" or z == "[": - cC += 1 - if z == "}" or z == "]": - cC -= 1 - if cC == 0: - if parFlag: - parCx += 1 - else: - ArgCx += 1 - if G[0].find(symbol) == 0 and (len(G[0]) == len(symbol) or not G[0][ - len(symbol)].isalpha()): # and (numPar!=0 or numArg!=0): - checkFlag = True - get = True - preG = S - - - elif checkFlag: - get = True - checkFlag = False - - if (get): - if get: - G = preG - get = False - checkFlag = False - if True: - if symbolPar.find("@") != -1: - Q = symbolPar[:symbolPar.find("@")] - else: - Q = symbolPar - listArgs = [] - if len(Q) > len(symbol) and (Q[len(symbol)] == "{" or Q[len(symbol)] == "["): - ap = "" - for o in range(len(symbol), len(Q)): - if Q[o] == "{" or z == "[": - argFlag = True - elif Q[o] == "}" or z == "]": - argFlag = False - listArgs.append(ap) - ap = "" - else: - ap += Q[o] - '''websiteU=g[g.find("http://"):].strip("\n") - k=0 - websites=[] - for r in range(0,len(websiteU)): - if websiteU[r]==",": - if websiteU[k:r].find("http://")!=-1: - websites.append(websiteU[k:r+1].strip(" ")) - k=r - - - if websiteU[k:r].find("http://")!=-1: - websites.append(websiteU[k:r+1].strip(" ")) - websiteF="" - for d in websites: - websiteF=websiteF+" ["+d+" "+d+"]"''' - # websiteF=G[4].strip("\n") - websiteF = "" - web1 = G[5] - for t in range(5, len(G)): - if G[t] != "": - websiteF = websiteF + " [" + G[t] + " " + G[t] + "]" - - # p1=Q - # if Q.find("@")!=-1: - # p1=Q[:Q.find("@")] - p1 = G[4].strip("$") - p1 = "{\\displaystyle " + p1 + "}" - # if checkFlag: - new1 = "" - new2 = "" - pause = False - mathF = True - '''for k in range(0,len(p1)): - if p1[k]=="$": - if mathF: - new1+="{\\displaystyle " - else: - new1+="}" - mathF=not mathF - - elif p1[k]=="#" and p1[k+1].isdigit(): - pause=True - elif pause: - num=int(p1[k]) - #letter=chr(num+96) - letter=listArgs[num-1] - new1+=letter - pause=False - - else: - new1+=p1[k]''' - p2 = G[1] - for k in range(0, len(p2)): - if p2[k] == "$": - if mathF: - new2 += "{\\displaystyle " - else: - new2 += "}" - mathF = not mathF - else: - new2 += p2[k] - # p1=new1 - p2 = new2 - finSym.append(web1 + " " + p1 + "] : " + p2 + " :" + websiteF) - break - # gFlag=True - # if not symF: - # symF=True - # wiki.write("[") - # else: - # wiki.write("
\n") - # wiki.write("[") - # wiki.write(g[g.find("http://"):].strip("\n")) - # wiki.write(" ") - # wiki.write(getEq(g[g.find("{$"):]).strip("\n").replace("\\\\","\\")) - # wiki.write("] : ") - # wiki.write(g[g.find(" {")+2:g.find("} ",g.find(" {")+1)]) - # wiki.write(" : [") - # wiki.write(g[g.find("http://"):].strip("\n")) - # wiki.write(" ") - # wiki.write(g[g.find("http://"):].strip("\n")) - # wiki.write("] ")''' - - # preG=S - if not gFlag: - del newSym[s] - - gFlag = True - # finSym.reverse() - if ampFlag: - wiki.write("& : logical and") - gFlag = False - for y in finSym: - if y == "& : logical and": - pass - elif gFlag: - gFlag = False - wiki.write("[" + y) - else: - wiki.write("
\n[" + y) - - wiki.write("\n
\n") - - wiki.write("\n== Bibliography==\n\n") # should there be a space between bibliography and ==? - r = unmodLabel(labels[eqCounter]) - q = r.find("KLS:") + 4 - p = r.find(":", q) - section = r[q:p] - equation = r[p + 1:] - if equation.find(":") != -1: - equation = equation[0:equation.find(":")] - - wiki.write( - "[http://homepage.tudelft.nl/11r49/askey/contents.html Equation in Section " + section + "] of [[Bibliography#KLS|'''KLS''']].\n\n") # Where should it link to? - wiki.write("== URL links ==\n\nWe ask users to provide relevant URL links in this space.\n\n") - if eqCounter < endNum: - wiki.write("
\n") - if newSec: - newSec = False - wiki.write("
<< [[" + secLabel(sections[secCount][0]).replace(" ", - "_") + "|" + secLabel( - sections[secCount][0]) + "]]
\n") - else: - wiki.write("
<< [[" + secLabel(labels[eqCounter - 1]).replace(" ", - "_") + "|" + secLabel( - labels[eqCounter - 1]) + "]]
\n") - wiki.write("
[[" + secLabel(sections[secCount + 1][0]).replace(" ", - "_") + "#" + secLabel( - labels[eqCounter][len("Formula:"):]) + "|formula in " + secLabel( - sections[secCount + 1][0]) + "]]
\n") - if eqS == sections[secCount][1]: - wiki.write("
[[" + sections[(secCount + 1) % len(sections)][0].replace(" ", - "_") + "|" + - sections[(secCount + 1) % len(sections)][0] + "]] >>
\n") - else: - wiki.write( - "
[[" + secLabel(labels[(eqCounter + 1) % (endNum + 1)]).replace(" ", - "_") + "|" + secLabel( - labels[(eqCounter + 1) % (endNum + 1)]) + "]] >>
\n") - wiki.write("
\n\ndrmf_eof\n") - else: # FOR EXTRA EQUATIONS - wiki.write("
\n") - wiki.write("
<< [[" + labels[endNum - 1].replace(" ", "_") + "|" + labels[ - endNum - 1] + "]]
\n") - wiki.write("
[[" + labels[0].replace(" ", "_") + "#" + labels[endNum][ - 8:] + "|formula in " + - labels[0] + "]]
\n") - wiki.write("
[[" + labels[(0) % endNum].replace(" ", "_") + "|" + labels[ - 0 % endNum] + "]]
\n") - wiki.write("
\n\ndrmf_eof\n") - - - - elif "\\constraint" in line and parse: - # symbols=symbols+getSym(line) - symLine = line.strip("\n") - if hCon: - comToWrite = comToWrite + "\n== Constraint(s) ==\n\n" - hCon = False - constraint = True - math = False - conLine = "" - # wiki.write("
"+getEq(line)+"

\n") - elif "\\substitution" in line and parse: - # symbols=symbols+getSym(line) - symLine = line.strip("\n") - if hSub: - comToWrite = comToWrite + "\n== Substitution(s) ==\n\n" - hSub = False - # wiki.write("
"+getEq(line)+"

\n") - substitution = True - math = False - subLine = "" - elif "\\drmfname" in line and parse: - math = False - comToWrite = "\n== Name ==\n\n
" + getString(line) + "

\n" + comToWrite - - elif "\\drmfnote" in line and parse: - symbols = symbols + getSym(line) - if hNote: - comToWrite = comToWrite + "\n== Note(s) ==\n\n" - hNote = False - note = True - math = False - noteLine = "" - - elif "\\proof" in line and parse: - # symbols=symbols+getSym(line) - symLine = line.strip("\n") - if hProof: - hProof = False - comToWrite = comToWrite + "\n== Proof ==\n\nWe ask users to provide proof(s), reference(s) to proof(s), or further clarification on the proof(s) in this space. \n

\n
" - proof = True - proofLine = "" - pause = False - pauseP = False - for ind in range(0, len(line)): - if line[ind:ind + 7] == "\\eqref{": - pause = True - eqR = line[ind:line.find("}", ind) + 1] - rLab = getString(eqR) - '''for l in lLink: - if rLab==l[0:l.find("=")-1]: - rlabel=l[l.find("=>")+3:l.find("\\n")] - rlabel=rlabel.replace("/","") - rlabel=rlabel.replace("#",":") - rlabel=rlabel.replace("!",":") - break''' - - eInd = refLabels.index("" + label) - z = line[line.find("}", ind + 7) + 1] - if z == "." or z == ",": - pauseP = True - proofLine += ("
\n{\displaystyle \n" + refEqs[ - eInd] + "}" + z + "
\n") - else: - if z == "}": - proofLine += ( - "
\n{\displaystyle \n" + refEqs[eInd] + "}
") - else: - proofLine += ( - "
\n{\displaystyle \n" + refEqs[eInd] + "}
\n") - - - else: - if pause: - if line[ind] == "}": - pause = False - elif pauseP: - pauseP = False - elif line[ind] == "\n" and "\\end{equation}" in lines[i + 1]: - pass - else: - proofLine += (line[ind]) - if "\\end{equation}" in lines[i + 1]: - proof = False - # symLine+=line.strip("\n") - wiki.write(comToWrite + getEqP(proofLine, False) + "
\n
\n") - comToWrite = "" - symbols = symbols + getSym(symLine) - symLine = "" - - # wiki.write(line) - - elif proof: - symLine += line.strip("\n") - pauseP = False - # symbols=symbols+getSym(line) - for ind in range(0, len(line)): - if line[ind:ind + 7] == "\\eqref{": - pause = True - eqR = line[ind:line.find("}", ind) + 1] - rLab = getString(eqR) - '''for l in lLink: - if rLab==l[0:l.find("=")-1]: - rlabel=l[l.find("=>")+3:l.find("\\n")] - rlabel=rlabel.replace("/","") - rlabel=rlabel.replace("#",":") - rlabel=rlabel.replace("!",":") - break''' - - eInd = refLabels.index("" + label) - z = line[line.find("}", ind + 7) + 1] - if z == "." or z == ",": - pauseP = True - proofLine += ("
\n{\displaystyle \n" + refEqs[ - eInd] + "}" + z + "
\n") - else: - proofLine += ( - "
\n{\displaystyle \n" + refEqs[eInd] + "}
\n") - - else: - if pause: - if line[ind] == "}": - pause = False - elif pauseP: - pauseP = False - elif line[ind] == "\n" and "\\end{equation}" in lines[i + 1]: - pass - - else: - proofLine += (line[ind]) - if "\\end{equation}" in lines[i + 1]: - proof = False - # symLine+=line.strip("\n") - wiki.write(comToWrite + getEqP(proofLine, False).rstrip("\n") + "
\n
\n") - comToWrite = "" - symbols = symbols + getSym(symLine) - symLine = "" - - elif math: - if "\\end{equation}" in lines[i + 1] or "\\constraint" in lines[i + 1] or "\\substitution" in lines[ - i + 1] or "\\proof" in lines[i + 1] or "\\drmfnote" in lines[i + 1] or "\\drmfname" in lines[ - i + 1]: - wiki.write(line.rstrip("\n")) - symLine += line.strip("\n") - symbols = symbols + getSym(symLine) - symLine = "" - wiki.write("\n}
\n") - else: - symLine += line.strip("\n") - wiki.write(line) - if note and parse: - noteLine = noteLine + line - symbols = symbols + getSym(line) - if "\\end{equation}" in lines[i + 1] or "\\drmfn" in lines[i + 1] or "\\constraint" in lines[ - i + 1] or "\\substitution" in lines[i + 1] or "\\proof" in lines[i + 1]: - note = False - comToWrite = comToWrite + "
" + getEq(noteLine) + "

\n" - - if constraint and parse: - conLine = conLine + line.replace("&", "&
") - - symLine += line.strip("\n") - # symbols=symbols+getSym(line) - if "\\end{equation}" in lines[i + 1] or "\\drmfn" in lines[i + 1] or "\\constraint" in lines[ - i + 1] or "\\substitution" in lines[i + 1] or "\\proof" in lines[i + 1]: - constraint = False - symbols = symbols + getSym(symLine) - symLine = "" - wiki.write(comToWrite + "
" + getEq(conLine) + "

\n") - comToWrite = "" - if substitution and parse: - subLine = subLine + line.replace("&", "&
") - - symLine += line.strip("\n") - # symbols=symbols+getSym(line) - if "\\end{equation}" in lines[i + 1] or "\\drmfn" in lines[i + 1] or "\\substitution" in lines[ - i + 1] or "\\constraint" in lines[i + 1] or "\\proof" in lines[i + 1]: - substitution = False - symbols = symbols + getSym(symLine) - symLine = "" - wiki.write(comToWrite + "
" + getEq(subLine) + "

\n") - comToWrite = "" -# except Exception as detail: #If exception occured -# print("Exception",detail) #print details of error -# except: #If anythin else occured... -# print ("ERROR",sys.exc_info()[0])#ERROR with basic info From f93c1562110ac366348a68c95b5c27281d568358 Mon Sep 17 00:00:00 2001 From: HowardCohl Date: Mon, 29 Feb 2016 11:03:41 -0500 Subject: [PATCH 033/402] Update README.md --- AlexDanoff/README.md | 14 -------------- 1 file changed, 14 deletions(-) diff --git a/AlexDanoff/README.md b/AlexDanoff/README.md index fc8395b..49dea05 100644 --- a/AlexDanoff/README.md +++ b/AlexDanoff/README.md @@ -1,9 +1,5 @@ # DRMF-Seeding-Project -This project is meant to convert different designated source formats to MediaWiki WikiText. - -## MediaWiki Wikitext Generation - ## Alex Danoff Seeding Source Code To achieve the desired result, one should execute the progams in the following order (using the output of the previous program as input for the next one): @@ -11,13 +7,3 @@ To achieve the desired result, one should execute the progams in the following o 1. remove_excess.py 2. replace_special.py 3. prepare_annotations.py - -## KLS Seeding Project - -## DLMF Seeding Project - -## eCF Seeding Project - -## BMP Seeding Project - - From a54331e451af5320c51ba6c7a825cc1a6e29cb8f Mon Sep 17 00:00:00 2001 From: physikerwelt Date: Mon, 29 Feb 2016 22:18:38 +0100 Subject: [PATCH 034/402] Experiment with MediaWiki dump format output --- AlexDanoff/src/remove_excess.py | 2 +- Azeem/experiments/xml_output.py | 16 ++++++ Azeem/src/Zeta.py | 2 +- Azeem/src/tex2Wiki.py | 86 +++++++++++++++++++-------------- 4 files changed, 69 insertions(+), 37 deletions(-) create mode 100644 Azeem/experiments/xml_output.py diff --git a/AlexDanoff/src/remove_excess.py b/AlexDanoff/src/remove_excess.py index 9bc28e9..183712c 100644 --- a/AlexDanoff/src/remove_excess.py +++ b/AlexDanoff/src/remove_excess.py @@ -23,7 +23,7 @@ def main(): if len(sys.argv) != 3: fname = "../../data/ZE.tex" - dot_ind = fname.index(".") + 1 + dot_ind = fname.rfind(".") + 1 ofname = fname[:dot_ind] + "1." + fname[dot_ind:] else: diff --git a/Azeem/experiments/xml_output.py b/Azeem/experiments/xml_output.py new file mode 100644 index 0000000..97e8fe5 --- /dev/null +++ b/Azeem/experiments/xml_output.py @@ -0,0 +1,16 @@ +import xml.etree.ElementTree as ET + + +def main(): + ET.register_namespace('', 'http://www.mediawiki.org/xml/export-0.10/') + root = ET.Element('{http://www.mediawiki.org/xml/export-0.10/}mediawiki') + page = ET.SubElement(root, 'page') + title = ET.SubElement(page, 'title') + title.text = 'the Title & stuff' + ET.dump(root) + tree = ET.ElementTree(root) + tree.write("page.xhtml", xml_declaration=True, encoding='utf-8', method='xml') + + +if __name__ == "__main__": + main() diff --git a/Azeem/src/Zeta.py b/Azeem/src/Zeta.py index 198ee3d..830722c 100644 --- a/Azeem/src/Zeta.py +++ b/Azeem/src/Zeta.py @@ -3,5 +3,5 @@ inp="y" if inp!="n": print("Zeta and Related Functions") - os.system("python tex2Wiki.py ../data/ZE.3.tex ../data/ZE.4.mmd") + os.system("python tex2Wiki.py ../../data/ZE.3.tex ../../data/ZE.4.mmd") print("Done") diff --git a/Azeem/src/tex2Wiki.py b/Azeem/src/tex2Wiki.py index 6f6a219..8eb35f7 100644 --- a/Azeem/src/tex2Wiki.py +++ b/Azeem/src/tex2Wiki.py @@ -4,6 +4,7 @@ import sys # imported for getting args import os # imported for copying file +global wiki def isnumber(char): return char[0] in "0123456789" @@ -217,6 +218,9 @@ def modLabel(label): newlabel += "0" + num return (newlabel) +def appendText(text): + wiki.write(text) + def main(): # try: for jsahlfkjsd in range(0, 1): @@ -301,7 +305,7 @@ def main(): # mainPrepend+=stringWrite chapter = getString(line) mainPrepend += ( - "\n== Sections in " + chapter + " ==\n\n
\n") + "\n== Sections in " + chapter + " ==\n\n
\n") elif "\\part" in line: if getString(line) == "BOF": parse = False @@ -356,7 +360,8 @@ def main(): wiki.write("{{headSection}}\n") head = True wiki.write("== " + getString(line) + " ==\n") - elif ("\\section" in lines[(i + 1) % len(lines)] or "\\end{document}" in lines[(i + 1) % len(lines)]) and parse: + elif ("\\section" in lines[(i + 1) % len(lines)] or "\\end{document}" in lines[ + (i + 1) % len(lines)]) and parse: # wiki.write("
\n") # wiki.write("
<< [["+secLabel(sections[secCounter-1][0])+"|"+secLabel(sections[secCounter-1][0])+"]]
\n") # wiki.write("
[[Orthogonal_Polynomials#"+secLabel(sections[secCounter][0])+"|"+secLabel(sections[secCounter][0])+"]]
\n") @@ -433,8 +438,9 @@ def main(): flagM = True eqs[len(eqs) - 1] += line - if "\\end{equation}" in lines[i + 1] and not "\\subsection" in lines[i + 3] and not "\\section" in lines[ - i + 3] and not "\\part" in lines[i + 3]: + if "\\end{equation}" in lines[i + 1] and not "\\subsection" in lines[i + 3] and not "\\section" in \ + lines[ + i + 3] and not "\\part" in lines[i + 3]: u = i flagM2 = False while flagM: @@ -546,8 +552,10 @@ def main(): sections[secCount + 1][0]) + "]]
\n") if eqS == sections[secCount][1]: wiki.write( - "
[[" + secLabel(sections[(secCount + 1) % len(sections)][0]).replace( - " ", "_") + "|" + secLabel(sections[(secCount + 1) % len(sections)][0]) + "]] >>
\n") + "
[[" + secLabel( + sections[(secCount + 1) % len(sections)][0]).replace( + " ", "_") + "|" + secLabel( + sections[(secCount + 1) % len(sections)][0]) + "]] >>
\n") else: wiki.write( "
[[" + secLabel(labels[(eqCounter + 1) % (endNum + 1)]).replace(" ", @@ -796,21 +804,21 @@ def main(): break # gFlag=True # if not symF: - # symF=True - # wiki.write("[") - # else: - # wiki.write("
\n") - # wiki.write("[") - # wiki.write(g[g.find("http://"):].strip("\n")) - # wiki.write(" ") - # wiki.write(getEq(g[g.find("{$"):]).strip("\n").replace("\\\\","\\")) - # wiki.write("] : ") - # wiki.write(g[g.find(" {")+2:g.find("} ",g.find(" {")+1)]) - # wiki.write(" : [") - # wiki.write(g[g.find("http://"):].strip("\n")) - # wiki.write(" ") - # wiki.write(g[g.find("http://"):].strip("\n")) - # wiki.write("] ")''' + # symF=True + # wiki.write("[") + # else: + # wiki.write("
\n") + # wiki.write("[") + # wiki.write(g[g.find("http://"):].strip("\n")) + # wiki.write(" ") + # wiki.write(getEq(g[g.find("{$"):]).strip("\n").replace("\\\\","\\")) + # wiki.write("] : ") + # wiki.write(g[g.find(" {")+2:g.find("} ",g.find(" {")+1)]) + # wiki.write(" : [") + # wiki.write(g[g.find("http://"):].strip("\n")) + # wiki.write(" ") + # wiki.write(g[g.find("http://"):].strip("\n")) + # wiki.write("] ")''' # preG=S if not gFlag: @@ -860,9 +868,10 @@ def main(): labels[eqCounter][len("Formula:"):]) + "|formula in " + secLabel( sections[secCount + 1][0]) + "]]
\n") if eqS == sections[secCount][1]: - wiki.write("
[[" + sections[(secCount + 1) % len(sections)][0].replace(" ", - "_") + "|" + - sections[(secCount + 1) % len(sections)][0] + "]] >>
\n") + wiki.write( + "
[[" + sections[(secCount + 1) % len(sections)][0].replace(" ", + "_") + "|" + + sections[(secCount + 1) % len(sections)][0] + "]] >>
\n") else: wiki.write( "
[[" + secLabel(labels[(eqCounter + 1) % (endNum + 1)]).replace(" ", @@ -891,7 +900,7 @@ def main(): constraint = True math = False conLine = "" - # wiki.write("
"+getEq(line)+"

\n") + # wiki.write("
"+getEq(line)+"

\n") elif "\\substitution" in line and parse: # symbols=symbols+getSym(line) symLine = line.strip("\n") @@ -947,10 +956,12 @@ def main(): else: if z == "}": proofLine += ( - "
\n{\displaystyle \n" + refEqs[eInd] + "}
") + "
\n{\displaystyle \n" + refEqs[ + eInd] + "}
") else: proofLine += ( - "
\n{\displaystyle \n" + refEqs[eInd] + "}
\n") + "
\n{\displaystyle \n" + refEqs[ + eInd] + "}
\n") else: @@ -971,7 +982,7 @@ def main(): symbols = symbols + getSym(symLine) symLine = "" - # wiki.write(line) + # wiki.write(line) elif proof: symLine += line.strip("\n") @@ -998,7 +1009,8 @@ def main(): eInd] + "}" + z + "
\n") else: proofLine += ( - "
\n{\displaystyle \n" + refEqs[eInd] + "}
\n") + "
\n{\displaystyle \n" + refEqs[ + eInd] + "}
\n") else: if pause: @@ -1021,8 +1033,9 @@ def main(): elif math: if "\\end{equation}" in lines[i + 1] or "\\constraint" in lines[i + 1] or "\\substitution" in lines[ - i + 1] or "\\proof" in lines[i + 1] or "\\drmfnote" in lines[i + 1] or "\\drmfname" in lines[ - i + 1]: + i + 1] or "\\proof" in lines[i + 1] or "\\drmfnote" in lines[i + 1] or "\\drmfname" in \ + lines[ + i + 1]: wiki.write(line.rstrip("\n")) symLine += line.strip("\n") symbols = symbols + getSym(symLine) @@ -1063,7 +1076,10 @@ def main(): symLine = "" wiki.write(comToWrite + "
" + getEq(subLine) + "

\n") comToWrite = "" - # except Exception as detail: #If exception occured - # print("Exception",detail) #print details of error - # except: #If anythin else occured... - # print ("ERROR",sys.exc_info()[0])#ERROR with basic info + # except Exception as detail: #If exception occured + # print("Exception",detail) #print details of error + # except: #If anythin else occured... + # print ("ERROR",sys.exc_info()[0])#ERROR with basic info + +if __name__ == "__main__": + main() \ No newline at end of file From de810a04b92cd04b00fa0a23708a975a20bab06a Mon Sep 17 00:00:00 2001 From: physikerwelt Date: Mon, 29 Feb 2016 22:29:21 +0100 Subject: [PATCH 035/402] fix copy problem within windows --- Azeem/src/tex2Wiki.py | 241 +++++++++++++++++++++--------------------- 1 file changed, 119 insertions(+), 122 deletions(-) diff --git a/Azeem/src/tex2Wiki.py b/Azeem/src/tex2Wiki.py index 8eb35f7..da30863 100644 --- a/Azeem/src/tex2Wiki.py +++ b/Azeem/src/tex2Wiki.py @@ -3,9 +3,11 @@ import csv # imported for using csv format import sys # imported for getting args import os # imported for copying file +from shutil import copyfile global wiki + def isnumber(char): return char[0] in "0123456789" @@ -218,12 +220,16 @@ def modLabel(label): newlabel += "0" + num return (newlabel) -def appendText(text): + +def append_text(text): + global wiki wiki.write(text) + def main(): # try: for jsahlfkjsd in range(0, 1): + global wiki tex = open(sys.argv[1], 'r') wiki = open(sys.argv[2], 'w') main = open("OrthogonalPolynomials.mmd", "r") @@ -278,10 +284,10 @@ def main(): for i in range(0, len(lines)): line = lines[i] if "\\begin{document}" in line: - # wiki.write("drmf_bof\n") + # append_text("drmf_bof\n") parse = True elif "\\end{document}" in line and parse: - # wiki.write("
\n") + # append_text("
\n") mainPrepend += "
\n" mainText = mainPrepend + mainText mainText = mainText.replace("drmf_bof\n", "") @@ -293,15 +299,15 @@ def main(): mainWrite.write(mainText) mainWrite.close() main.close() - os.system("cp -f OrthogonalPolynomials.mmd.new OrthogonalPolynomials.mmd") - # wiki.write("\ndrmf_eof\n") + copyfile('OrthogonalPolynomials.mmd', 'OrthogonalPolynomials.mmd.new') + # append_text("\ndrmf_eof\n") parse = False elif "\\title" in line and parse: stringWrite = "\'\'\'" stringWrite += getString(line) + "\'\'\'\n" labels.append("Orthogonal Polynomials") sections.append(["Orthogonal Polynomials", 0]) - # wiki.write(stringWrite) + # append_text(stringWrite) # mainPrepend+=stringWrite chapter = getString(line) mainPrepend += ( @@ -325,67 +331,57 @@ def main(): # line.replace("\\begin{equation}","$") # line.replace("\\end{equation},"$") '''if "\\begin{document}" in line: - wiki.write("drmf_bof\n") + append_text("drmf_bof\n") parse=True elif "\\end{document}" in line and parse: - wiki.write("\ndrmf_eof\n") + append_text("\ndrmf_eof\n") parse=False elif "\\title" in line and parse: stringWrite="\'\'\'" stringWrite+=getString(line)+"\'\'\'\n" labels.append("Orthogonal Polynomials") - wiki.write(stringWrite\n) + append_text(stringWrite\n) elif "\\part" in line: if getString(line)=="BOF": parse=False elif getString(line)=="EOF": parse=True elif parse: - wiki.write("\n
\n= "+getString(line)+" =\n") + append_text("\n
\n= "+getString(line)+" =\n") head=True ''' if "\\section" in line: parse = True secCounter += 1 - wiki.write("drmf_bof\n") - wiki.write("\'\'\'" + secLabel(getString(line)) + "\'\'\'\n") - wiki.write("{{DISPLAYTITLE:" + (sections[secCounter][0]) + "}}\n") - # wiki.write("
\n") - # wiki.write("
<< [["+secLabel(sections[secCounter-1][0])+"|"+secLabel(sections[secCounter-1][0])+"]]
\n") - # wiki.write("
[[Orthogonal_Polynomials#"+secLabel(sections[secCounter][0])+"|"+secLabel(sections[secCounter][0])+"]]
\n") - # wiki.write("
[["+secLabel(sections[(secCounter+1)%len(sections)][0])+"|"+secLabel(sections[(secCounter+1)%len(sections)][0])+"]] >>
\n
\n\n") - # wiki.write("{{head|pre="+secLabel(sections[secCounter-1][0])+"|cur="+secLabel(sections[secCounter][0])+"|next="+secLabel(sections[(secCounter+1)%len(sections)][0])+"}}\n") - wiki.write("{{#set:Chapter=" + chapter + "}}\n") - wiki.write("{{#set:Section=" + str(secCounter) + "}}\n") - wiki.write("{{headSection}}\n") + append_text("drmf_bof\n") + append_text("\'\'\'" + secLabel(getString(line)) + "\'\'\'\n") + append_text("{{DISPLAYTITLE:" + (sections[secCounter][0]) + "}}\n") + append_text("{{#set:Chapter=" + chapter + "}}\n") + append_text("{{#set:Section=" + str(secCounter) + "}}\n") + append_text("{{headSection}}\n") head = True - wiki.write("== " + getString(line) + " ==\n") + append_text("== " + getString(line) + " ==\n") elif ("\\section" in lines[(i + 1) % len(lines)] or "\\end{document}" in lines[ (i + 1) % len(lines)]) and parse: - # wiki.write("
\n") - # wiki.write("
<< [["+secLabel(sections[secCounter-1][0])+"|"+secLabel(sections[secCounter-1][0])+"]]
\n") - # wiki.write("
[[Orthogonal_Polynomials#"+secLabel(sections[secCounter][0])+"|"+secLabel(sections[secCounter][0])+"]]
\n") - # wiki.write("
[["+secLabel(sections[(secCounter+1)%len(sections)][0])+"|"+secLabel(sections[(secCounter+1)%len(sections)][0])+"]] >>
\n
\n\n") - # wiki.write("{{foot|pre="+secLabel(sections[secCounter-1][0])+"|cur="+secLabel(sections[secCounter][0])+"|next="+secLabel(sections[(secCounter+1)%len(sections)][0])+"}}\n") - wiki.write("{{footSection}}\n") - wiki.write("drmf_eof\n") + append_text("{{footSection}}\n") + append_text("drmf_eof\n") sections[secCounter].append(eqCounter) eqCounter = 0 elif "\\subsection" in line and parse: - wiki.write("\n== " + getString(line) + " ==\n") + append_text("\n== " + getString(line) + " ==\n") head = True elif "\\paragraph" in line and parse: - wiki.write("\n=== " + getString(line) + " ===\n") + append_text("\n=== " + getString(line) + " ===\n") head = True elif "\\subsubsection" in line and parse: - wiki.write("\n=== " + getString(line) + " ===\n") + append_text("\n=== " + getString(line) + " ===\n") head = True elif "\\begin{equation}" in line and parse: # symLine="" if head: - wiki.write("\n") + append_text("\n") head = False sLabel = line.find("\\formula{") + 9 eLabel = line.find("}", sLabel) @@ -400,8 +396,8 @@ def main(): break''' labels.append(label) eqs.append("") - # wiki.write("\n\n") - wiki.write("{\displaystyle \n") + # append_text("\n\n") + append_text("{\displaystyle \n") math = True elif "\\begin{equation}" in line and not parse: sLabel = line.find("\\formula{") + 9 @@ -427,13 +423,13 @@ def main(): substitution = True math = False subLine = "" - # wiki.write("
Substitution(s): "+getEq(line)+"

\n") + # append_text("
Substitution(s): "+getEq(line)+"

\n") elif "\\proof" in line and parse: math = False elif "\\drmfn" in line and parse: math = False if "\\drmfname" in line and parse: - wiki.write("
This formula has the name: " + getString(line) + "

\n") + append_text("
This formula has the name: " + getString(line) + "

\n") elif math and parse: flagM = True eqs[len(eqs) - 1] += line @@ -453,19 +449,19 @@ def main(): flagM2 = True if not (flagM2): - wiki.write(line.rstrip("\n")) - wiki.write("\n}
\n") + append_text(line.rstrip("\n")) + append_text("\n}
\n") else: - wiki.write(line.rstrip("\n")) - wiki.write("\n}\n") + append_text(line.rstrip("\n")) + append_text("\n}\n") elif "\\end{equation}" in lines[i + 1]: - wiki.write(line.rstrip("\n")) - wiki.write("\n}\n") + append_text(line.rstrip("\n")) + append_text("\n}\n") elif "\\constraint" in lines[i + 1] or "\\substitution" in lines[i + 1] or "\\drmfn" in lines[i + 1]: - wiki.write(line.rstrip("\n")) - wiki.write("\n}\n") + append_text(line.rstrip("\n")) + append_text("\n}\n") else: - wiki.write(line) + append_text(line) elif math and not parse: eqs[len(eqs) - 1] += line if "\\end{equation}" in lines[i + 1] or "\\constraint" in lines[i + 1] or "\\substitution" in lines[ @@ -476,14 +472,14 @@ def main(): if "\\end{equation}" in lines[i + 1] or "\\substitution" in lines[i + 1] or "\\constraint" in lines[ i + 1] or "\\drmfn" in lines[i + 1] or "\\proof" in lines[i + 1]: substitution = False - wiki.write("
Substitution(s): " + getEq(subLine) + "

\n") + append_text("
Substitution(s): " + getEq(subLine) + "

\n") if constraint and parse: conLine = conLine + line.replace("&", "&
") if "\\end{equation}" in lines[i + 1] or "\\substitution" in lines[i + 1] or "\\constraint" in lines[ i + 1] or "\\drmfn" in lines[i + 1] or "\\proof" in lines[i + 1]: constraint = False - wiki.write("
Constraint(s): " + getEq(conLine) + "

\n") + append_text("
Constraint(s): " + getEq(conLine) + "

\n") eqCounter = 0 endNum = len(labels) - 1 @@ -532,67 +528,67 @@ def main(): parse = True symbols = [] eqCounter += 1 - wiki.write("drmf_bof\n") + append_text("drmf_bof\n") label = labels[eqCounter] - wiki.write("\'\'\'" + secLabel(label) + "\'\'\'\n") - wiki.write("{{DISPLAYTITLE:" + (labels[eqCounter]) + "}}\n") + append_text("\'\'\'" + secLabel(label) + "\'\'\'\n") + append_text("{{DISPLAYTITLE:" + (labels[eqCounter]) + "}}\n") if eqCounter < endNum: # FOR ANYTHING THAT IS NOT THE EXTRA EQUATIONS - wiki.write("
\n") + append_text("
\n") if newSec: - wiki.write("
<< [[" + secLabel(sections[secCount][0]).replace(" ", - "_") + "|" + secLabel( + append_text("
<< [[" + secLabel(sections[secCount][0]).replace(" ", + "_") + "|" + secLabel( sections[secCount][0]) + "]]
\n") else: - wiki.write("
<< [[" + secLabel(labels[eqCounter - 1]).replace(" ", - "_") + "|" + secLabel( + append_text("
<< [[" + secLabel(labels[eqCounter - 1]).replace(" ", + "_") + "|" + secLabel( labels[eqCounter - 1]) + "]]
\n") - wiki.write("
[[" + secLabel(sections[secCount + 1][0]).replace(" ", - "_") + "#" + secLabel( + append_text("
[[" + secLabel(sections[secCount + 1][0]).replace(" ", + "_") + "#" + secLabel( labels[eqCounter][len("Formula:"):]) + "|formula in " + secLabel( sections[secCount + 1][0]) + "]]
\n") if eqS == sections[secCount][1]: - wiki.write( + append_text( "
[[" + secLabel( sections[(secCount + 1) % len(sections)][0]).replace( " ", "_") + "|" + secLabel( sections[(secCount + 1) % len(sections)][0]) + "]] >>
\n") else: - wiki.write( + append_text( "
[[" + secLabel(labels[(eqCounter + 1) % (endNum + 1)]).replace(" ", "_") + "|" + secLabel( labels[(eqCounter + 1) % (endNum + 1)]) + "]] >>
\n") - wiki.write("
\n\n") + append_text("
\n\n") elif eqCounter == endNum: - wiki.write("
\n") + append_text("
\n") if newSec: newSec = False - wiki.write("
<< [[" + secLabel(sections[secCount][0]).replace(" ", - "_") + "|" + secLabel( + append_text("
<< [[" + secLabel(sections[secCount][0]).replace(" ", + "_") + "|" + secLabel( sections[secCount][0]) + "]]
\n") else: - wiki.write("
<< [[" + secLabel(labels[eqCounter - 1]).replace(" ", - "_") + "|" + secLabel( + append_text("
<< [[" + secLabel(labels[eqCounter - 1]).replace(" ", + "_") + "|" + secLabel( labels[eqCounter - 1]) + "]]
\n") - wiki.write("
[[" + secLabel(sections[secCount + 1][0]).replace(" ", - "_") + "#" + secLabel( + append_text("
[[" + secLabel(sections[secCount + 1][0]).replace(" ", + "_") + "#" + secLabel( labels[eqCounter][len("Formula:"):]) + "|formula in " + secLabel( sections[secCount + 1][0]) + "]]
\n") - wiki.write("
[[" + secLabel( + append_text("
[[" + secLabel( labels[(eqCounter + 1) % (endNum + 1)].replace(" ", "_")) + "|" + secLabel( labels[(eqCounter + 1) % (endNum + 1)]) + "]]
\n") - wiki.write("
\n\n") + append_text("
\n\n") - wiki.write("
{\displaystyle \n") + append_text("
{\displaystyle \n") math = True elif "\\end{equation}" in line: - wiki.write(comToWrite) + append_text(comToWrite) parse = False math = False if hProof: - wiki.write( + append_text( "\n== Proof ==\n\nWe ask users to provide proof(s), reference(s) to proof(s), or further clarification on the proof(s) in this space.\n") - wiki.write("\n== Symbols List ==\n\n") + append_text("\n== Symbols List ==\n\n") newSym = [] # if "09.07:04" in label: for x in symbols: @@ -805,20 +801,20 @@ def main(): # gFlag=True # if not symF: # symF=True - # wiki.write("[") + # append_text("[") # else: - # wiki.write("
\n") - # wiki.write("[") - # wiki.write(g[g.find("http://"):].strip("\n")) - # wiki.write(" ") - # wiki.write(getEq(g[g.find("{$"):]).strip("\n").replace("\\\\","\\")) - # wiki.write("] : ") - # wiki.write(g[g.find(" {")+2:g.find("} ",g.find(" {")+1)]) - # wiki.write(" : [") - # wiki.write(g[g.find("http://"):].strip("\n")) - # wiki.write(" ") - # wiki.write(g[g.find("http://"):].strip("\n")) - # wiki.write("] ")''' + # append_text("
\n") + # append_text("[") + # append_text(g[g.find("http://"):].strip("\n")) + # append_text(" ") + # append_text(getEq(g[g.find("{$"):]).strip("\n").replace("\\\\","\\")) + # append_text("] : ") + # append_text(g[g.find(" {")+2:g.find("} ",g.find(" {")+1)]) + # append_text(" : [") + # append_text(g[g.find("http://"):].strip("\n")) + # append_text(" ") + # append_text(g[g.find("http://"):].strip("\n")) + # append_text("] ")''' # preG=S if not gFlag: @@ -827,20 +823,20 @@ def main(): gFlag = True # finSym.reverse() if ampFlag: - wiki.write("& : logical and") + append_text("& : logical and") gFlag = False for y in finSym: if y == "& : logical and": pass elif gFlag: gFlag = False - wiki.write("[" + y) + append_text("[" + y) else: - wiki.write("
\n[" + y) + append_text("
\n[" + y) - wiki.write("\n
\n") + append_text("\n
\n") - wiki.write("\n== Bibliography==\n\n") # should there be a space between bibliography and ==? + append_text("\n== Bibliography==\n\n") # should there be a space between bibliography and ==? r = unmodLabel(labels[eqCounter]) q = r.find("KLS:") + 4 p = r.find(":", q) @@ -849,45 +845,45 @@ def main(): if equation.find(":") != -1: equation = equation[0:equation.find(":")] - wiki.write( + append_text( "[http://homepage.tudelft.nl/11r49/askey/contents.html Equation in Section " + section + "] of [[Bibliography#KLS|'''KLS''']].\n\n") # Where should it link to? - wiki.write("== URL links ==\n\nWe ask users to provide relevant URL links in this space.\n\n") + append_text("== URL links ==\n\nWe ask users to provide relevant URL links in this space.\n\n") if eqCounter < endNum: - wiki.write("
\n") + append_text("
\n") if newSec: newSec = False - wiki.write("
<< [[" + secLabel(sections[secCount][0]).replace(" ", - "_") + "|" + secLabel( + append_text("
<< [[" + secLabel(sections[secCount][0]).replace(" ", + "_") + "|" + secLabel( sections[secCount][0]) + "]]
\n") else: - wiki.write("
<< [[" + secLabel(labels[eqCounter - 1]).replace(" ", - "_") + "|" + secLabel( + append_text("
<< [[" + secLabel(labels[eqCounter - 1]).replace(" ", + "_") + "|" + secLabel( labels[eqCounter - 1]) + "]]
\n") - wiki.write("
[[" + secLabel(sections[secCount + 1][0]).replace(" ", - "_") + "#" + secLabel( + append_text("
[[" + secLabel(sections[secCount + 1][0]).replace(" ", + "_") + "#" + secLabel( labels[eqCounter][len("Formula:"):]) + "|formula in " + secLabel( sections[secCount + 1][0]) + "]]
\n") if eqS == sections[secCount][1]: - wiki.write( + append_text( "
[[" + sections[(secCount + 1) % len(sections)][0].replace(" ", "_") + "|" + sections[(secCount + 1) % len(sections)][0] + "]] >>
\n") else: - wiki.write( + append_text( "
[[" + secLabel(labels[(eqCounter + 1) % (endNum + 1)]).replace(" ", "_") + "|" + secLabel( labels[(eqCounter + 1) % (endNum + 1)]) + "]] >>
\n") - wiki.write("
\n\ndrmf_eof\n") + append_text("
\n\ndrmf_eof\n") else: # FOR EXTRA EQUATIONS - wiki.write("
\n") - wiki.write("
<< [[" + labels[endNum - 1].replace(" ", "_") + "|" + labels[ + append_text("
\n") + append_text("
<< [[" + labels[endNum - 1].replace(" ", "_") + "|" + labels[ endNum - 1] + "]]
\n") - wiki.write("
[[" + labels[0].replace(" ", "_") + "#" + labels[endNum][ - 8:] + "|formula in " + - labels[0] + "]]
\n") - wiki.write("
[[" + labels[(0) % endNum].replace(" ", "_") + "|" + labels[ + append_text("
[[" + labels[0].replace(" ", "_") + "#" + labels[endNum][ + 8:] + "|formula in " + + labels[0] + "]]
\n") + append_text("
[[" + labels[(0) % endNum].replace(" ", "_") + "|" + labels[ 0 % endNum] + "]]
\n") - wiki.write("
\n\ndrmf_eof\n") + append_text("
\n\ndrmf_eof\n") @@ -900,14 +896,14 @@ def main(): constraint = True math = False conLine = "" - # wiki.write("
"+getEq(line)+"

\n") + # append_text("
"+getEq(line)+"

\n") elif "\\substitution" in line and parse: # symbols=symbols+getSym(line) symLine = line.strip("\n") if hSub: comToWrite = comToWrite + "\n== Substitution(s) ==\n\n" hSub = False - # wiki.write("
"+getEq(line)+"

\n") + # append_text("
"+getEq(line)+"

\n") substitution = True math = False subLine = "" @@ -977,12 +973,12 @@ def main(): if "\\end{equation}" in lines[i + 1]: proof = False # symLine+=line.strip("\n") - wiki.write(comToWrite + getEqP(proofLine, False) + "
\n
\n") + append_text(comToWrite + getEqP(proofLine, False) + "
\n
\n") comToWrite = "" symbols = symbols + getSym(symLine) symLine = "" - # wiki.write(line) + # append_text(line) elif proof: symLine += line.strip("\n") @@ -1026,7 +1022,7 @@ def main(): if "\\end{equation}" in lines[i + 1]: proof = False # symLine+=line.strip("\n") - wiki.write(comToWrite + getEqP(proofLine, False).rstrip("\n") + "
\n
\n") + append_text(comToWrite + getEqP(proofLine, False).rstrip("\n") + "
\n
\n") comToWrite = "" symbols = symbols + getSym(symLine) symLine = "" @@ -1036,14 +1032,14 @@ def main(): i + 1] or "\\proof" in lines[i + 1] or "\\drmfnote" in lines[i + 1] or "\\drmfname" in \ lines[ i + 1]: - wiki.write(line.rstrip("\n")) + append_text(line.rstrip("\n")) symLine += line.strip("\n") symbols = symbols + getSym(symLine) symLine = "" - wiki.write("\n}
\n") + append_text("\n}
\n") else: symLine += line.strip("\n") - wiki.write(line) + append_text(line) if note and parse: noteLine = noteLine + line symbols = symbols + getSym(line) @@ -1062,7 +1058,7 @@ def main(): constraint = False symbols = symbols + getSym(symLine) symLine = "" - wiki.write(comToWrite + "
" + getEq(conLine) + "

\n") + append_text(comToWrite + "
" + getEq(conLine) + "

\n") comToWrite = "" if substitution and parse: subLine = subLine + line.replace("&", "&
") @@ -1074,12 +1070,13 @@ def main(): substitution = False symbols = symbols + getSym(symLine) symLine = "" - wiki.write(comToWrite + "
" + getEq(subLine) + "

\n") + append_text(comToWrite + "
" + getEq(subLine) + "

\n") comToWrite = "" # except Exception as detail: #If exception occured # print("Exception",detail) #print details of error # except: #If anythin else occured... # print ("ERROR",sys.exc_info()[0])#ERROR with basic info + if __name__ == "__main__": - main() \ No newline at end of file + main() From e3fffff2fce6a8fd8356fe639e5824e8a959b338 Mon Sep 17 00:00:00 2001 From: ILIKEFUUD Date: Mon, 29 Feb 2016 16:34:35 -0500 Subject: [PATCH 036/402] Deleted for redundancy All taken care of and redundant with the README anyway --- KLSadd_insertion/EdwardTasks.txt | 26 -------------------------- 1 file changed, 26 deletions(-) delete mode 100644 KLSadd_insertion/EdwardTasks.txt diff --git a/KLSadd_insertion/EdwardTasks.txt b/KLSadd_insertion/EdwardTasks.txt deleted file mode 100644 index 0f5a079..0000000 --- a/KLSadd_insertion/EdwardTasks.txt +++ /dev/null @@ -1,26 +0,0 @@ -Fix your github; hopefully fixed now (fixed if this message shows up on github) - -https://help.github.com/categories/ssh/ -go there and try to work out your issue, rm your git directory and try git clone again. - - -If all else fails, try ssh -vT git@github.com - ^ -ABOVE IS DONE | - -This is what updateChapters.py currently does: - --rips paragraphs from KLSadd and puts them after the References paragraphs. -TESTED: WORKING - -This is what has been written but not implemented/tested: - --method to add relevant usepackages from KLSadd (cite, hyperref, xparse) --method to add relevant new commands from KLSadd --method to add "\large\bf KLSadd:" after every "paragraph{" inserted from KLSadd so everything looks neat - -This is what needs to be written: --method that inserts limit relations straight into the limit relations paragraph in relevant sections instead of just at the end of the section - -keep checking back here every time I've commited to see what's left, and you can change stuff around here just add a comment saying what you did and the date you did it - From 9928601d99435163f7329e6195cc318256a80dd8 Mon Sep 17 00:00:00 2001 From: Rahul Shah Date: Mon, 29 Feb 2016 16:44:42 -0500 Subject: [PATCH 037/402] Updated for redundancy --- KLSadd_insertion/README.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/KLSadd_insertion/README.md b/KLSadd_insertion/README.md index 5c3febb..71ad1ef 100644 --- a/KLSadd_insertion/README.md +++ b/KLSadd_insertion/README.md @@ -4,6 +4,7 @@ This project uses python and string manipulation to insert sections of the KLSad ``` python updateChapters.py ``` +The program must have a tempchap9.tex and a tempchap14.tex as well as the KLSadd.tex file in the same directory! The linetest.py program is the original program file fully updated. It is much harder to read and should only be used as a reference to update the updateChapters.py program. NOTE: Both the updateChapters.py and linetest.py lack a pdf version of their chapter 14 output. One must be generated to view results. @@ -15,5 +16,4 @@ NOTE: when you run updateChapters.py or linetest.py you need the KLSadd.tex file **THIS MEANS YOU SHOULD NOT RUN updateChapters.py OR linetest.py FROM YOUR GIT DIRECTORY. THE CHAPTER FILES ARE NOT PUBLIC DOMAIN, DO NOT COMMIT THEM! COPY THE FILES OVER INTO A SEPERATE DIRECTORY AND THEN COPY THEM BACK OVER WHEN YOU ARE READY TO COMMIT** -**Edward, check the EdwardTasks.txt file to check what's done and what needs work, if your github is still down just try to write out what the method should be like in a seperate file** From 75ee246d8373b2a054de83d3fe1fb087b8cdc406 Mon Sep 17 00:00:00 2001 From: physikerwelt Date: Mon, 29 Feb 2016 22:49:09 +0100 Subject: [PATCH 038/402] remove superfluous displaystyle in math tags --- Azeem/src/tex2Wiki.py | 59 +++++++++++++++++++++---------------------- 1 file changed, 29 insertions(+), 30 deletions(-) diff --git a/Azeem/src/tex2Wiki.py b/Azeem/src/tex2Wiki.py index da30863..7643694 100644 --- a/Azeem/src/tex2Wiki.py +++ b/Azeem/src/tex2Wiki.py @@ -2,7 +2,6 @@ # Convert tex to wikiText import csv # imported for using csv format import sys # imported for getting args -import os # imported for copying file from shutil import copyfile global wiki @@ -38,10 +37,10 @@ def getG(line): # gets equation for symbols list final = "" for c in line: if c == "$" and start: - final += "{\\displaystyle " + final += " " start = False elif c == "$": - final += "}" + final += "" start = True else: final += c @@ -71,9 +70,9 @@ def getEq(line): # Gets all data within constraints,substitutions elif c == "$" and per == 0: # either begin or end equation fEq = not (fEq) # toggle fEq flag to know if begin or end equation if fEq: # if begin - stringWrite += "{\\displaystyle " + stringWrite += "" else: # if end - stringWrite += "}" + stringWrite += "" elif c == "\n" and per == 0: # if newline stringWrite = stringWrite.strip() # remove all leading and trailing whitespace @@ -116,12 +115,12 @@ def getEqP(line, Flag): # Gets all data within proofs elif c == "$" and per == 0: fEq = not (fEq) if fEq: - stringWrite += "{\\displaystyle " + stringWrite += " " else: if length < 10: - stringWrite += "}" + stringWrite += "" else: - stringWrite += "}" + "
" + stringWrite += "" + "
" length = 0 elif c == "\n" and per == 0: stringWrite = stringWrite.strip() @@ -397,7 +396,7 @@ def main(): labels.append(label) eqs.append("") # append_text("\n\n") - append_text("{\displaystyle \n") + append_text("") math = True elif "\\begin{equation}" in line and not parse: sLabel = line.find("\\formula{") + 9 @@ -450,16 +449,16 @@ def main(): if not (flagM2): append_text(line.rstrip("\n")) - append_text("\n}
\n") + append_text("\n
\n") else: append_text(line.rstrip("\n")) - append_text("\n}\n") + append_text("\n\n") elif "\\end{equation}" in lines[i + 1]: append_text(line.rstrip("\n")) - append_text("\n}\n") + append_text("\n\n") elif "\\constraint" in lines[i + 1] or "\\substitution" in lines[i + 1] or "\\drmfn" in lines[i + 1]: append_text(line.rstrip("\n")) - append_text("\n}\n") + append_text("\n\n") else: append_text(line) elif math and not parse: @@ -578,7 +577,7 @@ def main(): labels[(eqCounter + 1) % (endNum + 1)]) + "]]
\n") append_text("
\n\n") - append_text("
{\displaystyle \n") + append_text("
\n") math = True elif "\\end{equation}" in line: append_text(comToWrite) @@ -759,7 +758,7 @@ def main(): # if Q.find("@")!=-1: # p1=Q[:Q.find("@")] p1 = G[4].strip("$") - p1 = "{\\displaystyle " + p1 + "}" + p1 = "" + p1 + "" # if checkFlag: new1 = "" new2 = "" @@ -768,9 +767,9 @@ def main(): '''for k in range(0,len(p1)): if p1[k]=="$": if mathF: - new1+="{\\displaystyle " + new1+="" else: - new1+="}" + new1+="" mathF=not mathF elif p1[k]=="#" and p1[k+1].isdigit(): @@ -788,9 +787,9 @@ def main(): for k in range(0, len(p2)): if p2[k] == "$": if mathF: - new2 += "{\\displaystyle " + new2 += " " else: - new2 += "}" + new2 += "" mathF = not mathF else: new2 += p2[k] @@ -947,17 +946,17 @@ def main(): z = line[line.find("}", ind + 7) + 1] if z == "." or z == ",": pauseP = True - proofLine += ("
\n{\displaystyle \n" + refEqs[ - eInd] + "}" + z + "
\n") + proofLine += ("
\n" + refEqs[ + eInd] + "" + z + "
\n") else: if z == "}": proofLine += ( - "
\n{\displaystyle \n" + refEqs[ - eInd] + "}
") + "
\n" + refEqs[ + eInd] + "
") else: proofLine += ( - "
\n{\displaystyle \n" + refEqs[ - eInd] + "}
\n") + "
\n \n" + refEqs[ + eInd] + "
\n") else: @@ -1001,12 +1000,12 @@ def main(): z = line[line.find("}", ind + 7) + 1] if z == "." or z == ",": pauseP = True - proofLine += ("
\n{\displaystyle \n" + refEqs[ - eInd] + "}" + z + "
\n") + proofLine += ("
\n" + refEqs[ + eInd] + "" + z + "
\n") else: proofLine += ( - "
\n{\displaystyle \n" + refEqs[ - eInd] + "}
\n") + "
\n" + refEqs[ + eInd] + "
\n") else: if pause: @@ -1036,7 +1035,7 @@ def main(): symLine += line.strip("\n") symbols = symbols + getSym(symLine) symLine = "" - append_text("\n}
\n") + append_text("\n
\n") else: symLine += line.strip("\n") append_text(line) From a28006da2a4ea328722853d64dbbbd8f4accbdea Mon Sep 17 00:00:00 2001 From: Rahul Shah Date: Mon, 29 Feb 2016 17:19:28 -0500 Subject: [PATCH 039/402] Fixed some bugs with KLSadd command insertion --- KLSadd_insertion/updateChapters.py | 20 +++++++++++++------- 1 file changed, 13 insertions(+), 7 deletions(-) diff --git a/KLSadd_insertion/updateChapters.py b/KLSadd_insertion/updateChapters.py index a7cb7bc..a3d6ac1 100644 --- a/KLSadd_insertion/updateChapters.py +++ b/KLSadd_insertion/updateChapters.py @@ -35,7 +35,8 @@ chapNums = [] paras = [] mathPeople = [] -newCommands = [] #used to hold extra commands that need to be inserted from KLSadd to the chapter files +newCommands = [] #used to hold the indexes of the commands +comms = "" #holds the ACTUAL STRINGS of the commands #2/18/16 this method addresses the goal of hardcoding in the necessary packages to let the chapter files run as pdf's. #Currently only works with chapter 9, ask Dr. Cohl to help port your chapter 14 output file into a pdf @@ -68,11 +69,13 @@ def getCommands(kls): temp = word[0:word.find("{")+ 1] + "\large\\bf KLSadd: " + word[word.find("{")+ 1: word.find("}")+1] #pretty sure I had to do something here but I forgot, so pass? - pass + #duh, obviously I need to store the commands somewhere! + comms = kls[newCommands[0]:newCommands[1]] + return comms #2/18/16 this method addresses the goal of hardcoding in the necessary commands to let the chapter files run as pdf's. Currently only works with chapter 9 -def insertCommands(kls, chap): +def insertCommands(kls, chap, cms): #reads in the newCommands[] and puts them in chap beginIndex = -1 #the index of the "begin document" keyphrase, this is where the new commands need to be inserted. @@ -82,8 +85,11 @@ def insertCommands(kls, chap): index+=1 if("begin{document}" in word): beginIndex += index - #chap[beginIndex] += newCommands - chap.insert(beginIndex, newCommands) + tempIndex = 0 + #D'OH FOUND THE PROBLEM newCommands stores just the integer value of the indexes of the commands. we need to get the actual commands. easy fix, do next time + for i in cms: + chap.insert(beginIndex+tempIndex,i) + tempIndex +=1 return chap #method to find the indices of the reference paragraphs @@ -118,8 +124,8 @@ def fixChapter(chap, references, p, kls): #I actually don't remember why I had to do this but I think you need it ^^^^^^^^^^^^^^^^^^^ chap = prepareForPDF(chap) - getCommands(kls) - chap = insertCommands(kls,chap) + cms = getCommands(kls) + chap = insertCommands(kls,chap, cms) #probably won't work because I don't know how anything works return chap From ac051e773bae3e9da895a059b3e147099db5b5b6 Mon Sep 17 00:00:00 2001 From: physikerwelt Date: Tue, 1 Mar 2016 11:33:59 +0100 Subject: [PATCH 040/402] Output XML directly * DRMF used an intermediate mmd file. That file was converted to a wikidump using an IDL program. * This change writes to the wikidump format directly --- Azeem/src/Zeta.py | 7 ------ Azeem/src/tex2Wiki.py | 50 ++++++++++++++++++++++++++++++++++++++----- 2 files changed, 45 insertions(+), 12 deletions(-) delete mode 100644 Azeem/src/Zeta.py diff --git a/Azeem/src/Zeta.py b/Azeem/src/Zeta.py deleted file mode 100644 index 830722c..0000000 --- a/Azeem/src/Zeta.py +++ /dev/null @@ -1,7 +0,0 @@ -import os -import time -inp="y" -if inp!="n": - print("Zeta and Related Functions") - os.system("python tex2Wiki.py ../../data/ZE.3.tex ../../data/ZE.4.mmd") -print("Done") diff --git a/Azeem/src/tex2Wiki.py b/Azeem/src/tex2Wiki.py index 7643694..a0143b3 100644 --- a/Azeem/src/tex2Wiki.py +++ b/Azeem/src/tex2Wiki.py @@ -3,8 +3,12 @@ import csv # imported for using csv format import sys # imported for getting args from shutil import copyfile +import xml.etree.ElementTree as ET +import datetime -global wiki +wiki = '' +ET.register_namespace('', 'http://www.mediawiki.org/xml/export-0.10/') +root = ET.Element('{http://www.mediawiki.org/xml/export-0.10/}mediawiki') def isnumber(char): @@ -222,15 +226,49 @@ def modLabel(label): def append_text(text): global wiki - wiki.write(text) + wiki.text = wiki.text + text + + +def append_revision(param): + global wiki + page = ET.SubElement(root, 'page') + title = ET.SubElement(page, 'title') + title.text = param + revision = ET.SubElement(page, 'revision') + timestamp = ET.SubElement(revision, 'timestamp') + timestamp.text = str(datetime.datetime.utcnow()) + contributor = ET.SubElement(revision, 'contributor') + username = ET.SubElement(contributor, 'username') + username.text = 'SeedBot' + wiki = ET.SubElement(revision, 'text') + wiki.text = '' + + +def writeout(ofname): + tree = ET.ElementTree(root) + tree.write(ofname, xml_declaration=True, encoding='utf-8', method='xml') def main(): + if len(sys.argv) != 3: + + fname = "../../data/ZE.3.tex" + ofname = "../../data/ZE.4.xml" + + else: + + fname = sys.argv[1] + ofname = sys.argv[2] + + readin(fname) + writeout(ofname) + + +def readin(ofname): # try: for jsahlfkjsd in range(0, 1): global wiki - tex = open(sys.argv[1], 'r') - wiki = open(sys.argv[2], 'w') + tex = open(ofname, 'r') main = open("OrthogonalPolynomials.mmd", "r") mainText = main.read() mainPrepend = "" @@ -352,6 +390,7 @@ def main(): if "\\section" in line: parse = True secCounter += 1 + append_revision(secLabel(getString(line))) append_text("drmf_bof\n") append_text("\'\'\'" + secLabel(getString(line)) + "\'\'\'\n") append_text("{{DISPLAYTITLE:" + (sections[secCounter][0]) + "}}\n") @@ -527,8 +566,9 @@ def main(): parse = True symbols = [] eqCounter += 1 - append_text("drmf_bof\n") label = labels[eqCounter] + append_revision(secLabel(label)) + append_text("drmf_bof\n") append_text("\'\'\'" + secLabel(label) + "\'\'\'\n") append_text("{{DISPLAYTITLE:" + (labels[eqCounter]) + "}}\n") if eqCounter < endNum: # FOR ANYTHING THAT IS NOT THE EXTRA EQUATIONS From 82a4c7fbf2839f5044ea6a6bd1dd6566eaa0c02f Mon Sep 17 00:00:00 2001 From: physikerwelt Date: Tue, 1 Mar 2016 12:11:47 +0100 Subject: [PATCH 041/402] tex2Wiki should parse DLMF and KLS sources * tex to wiki should find formula names in both \formula and \label commands * this change also uses the DLMF format, where the \label tag was used --- Azeem/src/tex2Wiki.py | 40 ++++++++++------------------------------ 1 file changed, 10 insertions(+), 30 deletions(-) diff --git a/Azeem/src/tex2Wiki.py b/Azeem/src/tex2Wiki.py index a0143b3..cef5170 100644 --- a/Azeem/src/tex2Wiki.py +++ b/Azeem/src/tex2Wiki.py @@ -196,8 +196,13 @@ def secLabel(label): return (label.replace("\'\'", "")) -def modLabel(label): +def modLabel(line): # label.replace("Formula:KLS:","KLS;") + start_label = line.find("\\formula{") + 9 + if start_label == 8: + start_label = line.find("\\label{") + 7 + end_label = line.find("}", start_label) + label = line[start_label:end_label] isNumer = False newlabel = "" num = "" @@ -221,7 +226,7 @@ def modLabel(label): newlabel += num elif len(num) == 1: newlabel += "0" + num - return (newlabel) + return newlabel def append_text(text): @@ -300,15 +305,7 @@ def readin(ofname): for i in range(0, len(refLines)): line = refLines[i] if "\\begin{equation}" in line: - sLabel = line.find("\\formula{") + 9 - eLabel = line.find("}", sLabel) - label = modLabel(line[sLabel:eLabel]) - '''for l in lLink: - if l.find(label)!=-1 and l[len(label)+1]=="=": - rlabel=l[l.find("=>")+3:l.find("\\n")] - rlabel=rlabel.replace("/","") - rlabel=rlabel.replace("#",":") - break''' + label = modLabel(line) refLabels.append(label) refEqs.append("") math = True @@ -421,32 +418,15 @@ def readin(ofname): if head: append_text("\n") head = False - sLabel = line.find("\\formula{") + 9 - eLabel = line.find("}", sLabel) - label = modLabel(line[sLabel:eLabel]) + label = modLabel(line) eqCounter += 1 - '''for l in lLink: - if label==l[0:l.find("=")-1]: - rlabel=l[l.find("=>")+3:l.find("\\n")] - rlabel=rlabel.replace("/","") - rlabel=rlabel.replace("#",":") - rlabel=rlabel.replace("!",":") - break''' labels.append(label) eqs.append("") # append_text("\n\n") append_text("") math = True elif "\\begin{equation}" in line and not parse: - sLabel = line.find("\\formula{") + 9 - eLabel = line.find("}", sLabel) - label = modLabel(line[sLabel:eLabel]) - '''for l in lLink: - if l.find(label)!=-1: - rlabel=l[l.find("=>")+3:l.find("\\n")] - rlabel=rlabel.replace("/","") - rlabel=rlabel.replace("#",":") - break''' + label = modLabel(line) labels.append("*" + label) # special marker eqs.append("") math = True From 58a80e683aa224cf80ce36cb15f30a6a7d95bfb4 Mon Sep 17 00:00:00 2001 From: physikerwelt Date: Tue, 1 Mar 2016 12:27:11 +0100 Subject: [PATCH 042/402] Enable coversall --- .travis.yml | 7 ++++++- README.md | 1 + 2 files changed, 7 insertions(+), 1 deletion(-) diff --git a/.travis.yml b/.travis.yml index 14a628f..8344e4d 100644 --- a/.travis.yml +++ b/.travis.yml @@ -1,2 +1,7 @@ language: python -script: nosetests AlexDanoff Azeem \ No newline at end of file +install: + - pip install coveralls +script: + nosetests --with-coverage AlexDanoff Azeem +after_success: + coveralls \ No newline at end of file diff --git a/README.md b/README.md index bd507b5..ef118d7 100644 --- a/README.md +++ b/README.md @@ -1,6 +1,7 @@ # DRMF-Seeding-Project [![Build Status](https://travis-ci.org/DRMF/DRMF-Seeding-Project.svg?branch=master)](https://travis-ci.org/DRMF/DRMF-Seeding-Project) +[![Coverage Status](https://coveralls.io/repos/github/DRMF/DRMF-Seeding-Project/badge.svg?branch=master)](https://coveralls.io/github/DRMF/DRMF-Seeding-Project?branch=master) This project is meant to convert different designated source formats to semantic LaTeX for the DRMF project. One major aspect of this conversion is the inclusion, insertion, and replacement From 25065f6c26fc1c8e1df9d5e7fff08d85d854fd42 Mon Sep 17 00:00:00 2001 From: physikerwelt Date: Tue, 1 Mar 2016 12:59:12 +0100 Subject: [PATCH 043/402] Only coverage src files --- .coveragerc | 5 +++++ 1 file changed, 5 insertions(+) create mode 100644 .coveragerc diff --git a/.coveragerc b/.coveragerc new file mode 100644 index 0000000..378def2 --- /dev/null +++ b/.coveragerc @@ -0,0 +1,5 @@ +[report] +omit = + */python?.?/* + */test/* + */site-packages/nose/* From 81a480695dc4d65ce08ae5effbb192991df79b24 Mon Sep 17 00:00:00 2001 From: physikerwelt Date: Tue, 1 Mar 2016 17:43:07 +0100 Subject: [PATCH 044/402] Remove drmf BOF and EOF markers The markers are no longer required since the all data is inserted into the XML dump directly --- Azeem/src/tex2Wiki.py | 38 ++++---------------------------------- 1 file changed, 4 insertions(+), 34 deletions(-) diff --git a/Azeem/src/tex2Wiki.py b/Azeem/src/tex2Wiki.py index cef5170..9003bfe 100644 --- a/Azeem/src/tex2Wiki.py +++ b/Azeem/src/tex2Wiki.py @@ -318,23 +318,20 @@ def readin(ofname): for i in range(0, len(lines)): line = lines[i] if "\\begin{document}" in line: - # append_text("drmf_bof\n") parse = True elif "\\end{document}" in line and parse: # append_text("
\n") mainPrepend += "
\n" mainText = mainPrepend + mainText - mainText = mainText.replace("drmf_bof\n", "") - mainText = mainText.replace("drmf_eof\n", "") mainText = mainText.replace("\'\'\'Orthogonal Polynomials\'\'\'\n", "") mainText = mainText.replace("{{#set:Section=0}}\n", "") mainText = mainText[0:mainText.rfind("== Sections ")] - mainText = "drmf_bof\n\'\'\'Orthogonal Polynomials\'\'\'\n{{#set:Section=0}}\n" + mainText + "\ndrmf_eof\n" + append_revision('Orthogonal Polynomials') + mainText = "{{#set:Section=0}}\n" + mainText mainWrite.write(mainText) mainWrite.close() main.close() copyfile('OrthogonalPolynomials.mmd', 'OrthogonalPolynomials.mmd.new') - # append_text("\ndrmf_eof\n") parse = False elif "\\title" in line and parse: stringWrite = "\'\'\'" @@ -362,34 +359,10 @@ def readin(ofname): eqCounter = 0 for i in range(0, len(lines)): line = lines[i] - # line.replace("\\begin{equation}","$") - # line.replace("\\end{equation},"$") - '''if "\\begin{document}" in line: - append_text("drmf_bof\n") - parse=True - elif "\\end{document}" in line and parse: - append_text("\ndrmf_eof\n") - parse=False - elif "\\title" in line and parse: - stringWrite="\'\'\'" - stringWrite+=getString(line)+"\'\'\'\n" - labels.append("Orthogonal Polynomials") - append_text(stringWrite\n) - elif "\\part" in line: - if getString(line)=="BOF": - parse=False - elif getString(line)=="EOF": - parse=True - elif parse: - append_text("\n
\n= "+getString(line)+" =\n") - head=True - ''' if "\\section" in line: parse = True secCounter += 1 append_revision(secLabel(getString(line))) - append_text("drmf_bof\n") - append_text("\'\'\'" + secLabel(getString(line)) + "\'\'\'\n") append_text("{{DISPLAYTITLE:" + (sections[secCounter][0]) + "}}\n") append_text("{{#set:Chapter=" + chapter + "}}\n") append_text("{{#set:Section=" + str(secCounter) + "}}\n") @@ -399,7 +372,6 @@ def readin(ofname): elif ("\\section" in lines[(i + 1) % len(lines)] or "\\end{document}" in lines[ (i + 1) % len(lines)]) and parse: append_text("{{footSection}}\n") - append_text("drmf_eof\n") sections[secCounter].append(eqCounter) eqCounter = 0 @@ -548,8 +520,6 @@ def readin(ofname): eqCounter += 1 label = labels[eqCounter] append_revision(secLabel(label)) - append_text("drmf_bof\n") - append_text("\'\'\'" + secLabel(label) + "\'\'\'\n") append_text("{{DISPLAYTITLE:" + (labels[eqCounter]) + "}}\n") if eqCounter < endNum: # FOR ANYTHING THAT IS NOT THE EXTRA EQUATIONS append_text("
\n") @@ -892,7 +862,7 @@ def readin(ofname): "
[[" + secLabel(labels[(eqCounter + 1) % (endNum + 1)]).replace(" ", "_") + "|" + secLabel( labels[(eqCounter + 1) % (endNum + 1)]) + "]] >>
\n") - append_text("
\n\ndrmf_eof\n") + append_text("
\n") else: # FOR EXTRA EQUATIONS append_text("
\n") append_text("
<< [[" + labels[endNum - 1].replace(" ", "_") + "|" + labels[ @@ -902,7 +872,7 @@ def readin(ofname): labels[0] + "]]
\n") append_text("
[[" + labels[(0) % endNum].replace(" ", "_") + "|" + labels[ 0 % endNum] + "]]
\n") - append_text("
\n\ndrmf_eof\n") + append_text("
\n") From 6fffbf62d4aad99d35bef1e02b7f79fee7cd11e6 Mon Sep 17 00:00:00 2001 From: physikerwelt Date: Wed, 2 Mar 2016 16:15:51 +0100 Subject: [PATCH 045/402] Provide script to test the whole pipeline --- Pipeline/src/OrthogonalPolynomials.mmd | 81 +++ Pipeline/src/new.Glossary.csv | 691 +++++++++++++++++++++++++ Pipeline/src/pipeline.py | 31 ++ 3 files changed, 803 insertions(+) create mode 100644 Pipeline/src/OrthogonalPolynomials.mmd create mode 100644 Pipeline/src/new.Glossary.csv create mode 100644 Pipeline/src/pipeline.py diff --git a/Pipeline/src/OrthogonalPolynomials.mmd b/Pipeline/src/OrthogonalPolynomials.mmd new file mode 100644 index 0000000..c841fb5 --- /dev/null +++ b/Pipeline/src/OrthogonalPolynomials.mmd @@ -0,0 +1,81 @@ +drmf_bof +'''Orthogonal Polynomials''' +{{#set:Section=0}} +== Sections in KLS Chapter 1 == + +
+* [[Orthogonal polynomials|Orthogonal polynomials]] +* [[The gamma and beta function|The gamma and beta function]] +* [[The shifted factorial and binomial coefficients|The shifted factorial and binomial coefficients]] +* [[Hypergeometric functions|Hypergeometric functions]] +* [[The binomial theorem and other summation formulas|The binomial theorem and other summation formulas]] +* [[Some integrals|Some integrals]] +* [[Transformation formulas|Transformation formulas]] +* [[The q-shifted factorial|The ''q''-shifted factorial]] +* [[The q-gamma function and q-binomial coefficients|The ''q''-gamma function and ''q''-binomial coefficients]] +* [[Basic hypergeometric functions|Basic hypergeometric functions]] +* [[The q-binomial theorem and other summation formulas|The ''q''-binomial theorem and other summation formulas]] +* [[More integrals|More integrals]] +* [[Transformation formulas|Transformation formulas]] +* [[Some q-analogues of special functions|Some ''q''-analogues of special functions]] +* [[The q-derivative and q-integral|The ''q''-derivative and ''q''-integral]] +* [[Shift operators and Rodrigues-type formulas|Shift operators and Rodrigues-type formulas]] +
+== Sections in KLS Chapter 9 == + +
+* [[Wilson|Wilson]] +* [[Racah|Racah]] +* [[Continuous dual Hahn|Continuous dual Hahn]] +* [[Continuous Hahn|Continuous Hahn]] +* [[Hahn|Hahn]] +* [[Dual Hahn|Dual Hahn]] +* [[Meixner-Pollaczek|Meixner-Pollaczek]] +* [[Jacobi|Jacobi]] +* [[Jacobi: Special cases|Jacobi: Special cases]] +* [[Pseudo Jacobi|Pseudo Jacobi]] +* [[Meixner|Meixner]] +* [[Krawtchouk|Krawtchouk]] +* [[Laguerre|Laguerre]] +* [[Bessel|Bessel]] +* [[Charlier|Charlier]] +* [[Hermite|Hermite]] +
+== Sections in KLS Chapter 14 == + +
+* [[Askey-Wilson|Askey-Wilson]] +* [[q-Racah|''q''-Racah]] +* [[Continuous dual q-Hahn|Continuous dual ''q''-Hahn]] +* [[Continuous q-Hahn|Continuous ''q''-Hahn]] +* [[Big q-Jacobi|Big ''q''-Jacobi]] +* [[Big q-Jacobi: Special case|Big ''q''-Jacobi: Special case]] +* [[q-Hahn|''q''-Hahn]] +* [[Dual q-Hahn|Dual ''q''-Hahn]] +* [[Al-Salam-Chihara|Al-Salam-Chihara]] +* [[q-Meixner-Pollaczek|''q''-Meixner-Pollaczek]] +* [[Continuous q-Jacobi|Continuous ''q''-Jacobi]] +* [[Continuous q-Jacobi: Special cases|Continuous ''q''-Jacobi: Special cases]] +* [[Big q-Laguerre|Big ''q''-Laguerre]] +* [[Little q-Jacobi|Little ''q''-Jacobi]] +* [[Little q-Jacobi: Special case|Little ''q''-Jacobi: Special case]] +* [[q-Meixner|''q''-Meixner]] +* [[Quantum q-Krawtchouk|Quantum ''q''-Krawtchouk]] +* [[q-Krawtchouk|''q''-Krawtchouk]] +* [[Affine q-Krawtchouk|Affine ''q''-Krawtchouk]] +* [[Dual q-Krawtchouk|Dual ''q''-Krawtchouk]] +* [[Continuous big q-Hermite|Continuous big ''q''-Hermite]] +* [[Continuous q-Laguerre|Continuous ''q''-Laguerre]] +* [[Little q-Laguerre / Wall|Little ''q''-Laguerre / Wall]] +* [[q-Laguerre|''q''-Laguerre]] +* [[q-Bessel|''q''-Bessel]] +* [[q-Charlier|''q''-Charlier]] +* [[Al-Salam-Carlitz I|Al-Salam-Carlitz I]] +* [[Al-Salam-Carlitz II|Al-Salam-Carlitz II]] +* [[Continuous q-Hermite|Continuous ''q''-Hermite]] +* [[Stieltjes-Wigert|Stieltjes-Wigert]] +* [[Discrete q-Hermite I|Discrete ''q''-Hermite I]] +* [[Discrete q-Hermite II|Discrete ''q''-Hermite II]] +
+ +drmf_eof diff --git a/Pipeline/src/new.Glossary.csv b/Pipeline/src/new.Glossary.csv new file mode 100644 index 0000000..8875e6b --- /dev/null +++ b/Pipeline/src/new.Glossary.csv @@ -0,0 +1,691 @@ +\AGM@{a}{g},arithmetic-geometric mean,arithmetic-geometric-mean,F|F:P,$M$,http://dlmf.nist.gov/19.8#SS1.p1, +\AThetaFunction@{x},Jacobi theta function $\vartheta$,a-theta-function,F|F:P,$\vartheta$,http://dlmf.nist.gov/23.1, +\AbstractJacobi{p}{q},generic Jacobian elliptic function,generic-Jacobian-elliptic-function,F|F,$\mathrm{pq}$,http://dlmf.nist.gov/22.2#E10, +\Acos@@{z},general inverse cosine function,multivalued-inverse-cosine,F|F:P:nP,$\mathrm{Arccos}$,http://dlmf.nist.gov/4.23#E2, +\Acosh@@{z},general inverse hyperbolic cosine function,multivalued-hyperbolic-inverse-cosine,F|F:P:nP,$\mathrm{Arccosh}$,http://dlmf.nist.gov/4.37#E2, +\Acot@@{z},general inverse cotangent function,multivalued-inverse-cotangent,F|F:P:nP,$\mathrm{Arccot}$,http://dlmf.nist.gov/4.23#E6, +\Acoth@@{z},general inverse hyperbolic cotangent function,multivalued-hyperbolic-inverse-cotangent,F|F:P:nP,$\mathrm{Arccoth}$,http://dlmf.nist.gov/4.37#E6, +\Acsc@@{z},general inverse cosecant function,multivalued-inverse-cosecant,F|F:P:nP,$\mathrm{Arccsc}$,http://dlmf.nist.gov/4.23#E4, +\Acsch@@{z},general inverse hyperbolic cosecant function,multivalued-hyperbolic-inverse-cosecant,F|F:P:nP,$\mathrm{Arccsch}$,http://dlmf.nist.gov/4.37#E4, +\AffqKrawtchouk{n}@@{q^{-x}}{p}{N}{q},affine $q$-Krawtchouk polynomial,affine-q-Krawtchouk-polynomial-K,P|F:P:PS,$K^{\mathrm{Aff}}_{n}$,http://drmf.wmflabs.org/wiki/Definition:AffqKrawtchouk, +\AiryAi@{z},Airy function ${\rm Ai}$,Airy-Ai,F|F:P,$\mathrm{Ai}$,http://dlmf.nist.gov/9.2#i, +\AiryBi@{z},Airy function ${\rm Bi}$,Airy-Bi,F|F:P,$\mathrm{Bi}$,http://dlmf.nist.gov/9.2#i, +\AiryModulusM@{z},Airy modulus function $M$,modulus-Airy-M,F|F:P,$M$,http://dlmf.nist.gov/9.8#i, +\AiryModulusN@{z},Airy modulus function $N$,modulus-Airy-N,F|F:P,$N$,http://dlmf.nist.gov/9.8#i, +\AiryPhasePhi@{z},Airy phase function $\phi$,phase-Airy-Phi,F|F:P,$\phi$,http://dlmf.nist.gov/9.8#i, +\AiryPhaseTheta@{z},Airy phase function $\theta$,phase-Airy-Theta,F|F:P,$\theta$,http://dlmf.nist.gov/9.8#i, +\AlSalamCarlitzI{a}{n}@{x}{q},Al-Salam-Carlitz I polynomial,EMPTY,P|F:P,$U^{(n)}_{\alpha}$,http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzI, +\AlSalamCarlitzII{a}{n}@{x}{q},Al-Salam-Carlitz II polynomial,EMPTY,P|F:P,$V^{(n)}_{\alpha}$,http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzII, +\AlSalamChihara{n}@@{x}{a}{b}{q},Al-Salam-Chihara polynomial,Al-Salam-Chihara-polynomial-Q,P|F:P:PS,$Q_{n}$,http://drmf.wmflabs.org/wiki/Definition:AlSalamChihara, +\AlSalamChiharaQ{n}@{x}{a}{b}{q},Al-Salam-Chihara polynomial,AlSalam-Chihara-polynomial-Q,P|F:P,$Q_{n}$,http://dlmf.nist.gov/18.28#E7, +\AlSalamIsmail{n}@{x}{s}{q},Al-Salam Ismail polynomial,Al-Salam-Ismail-q-polynomial-U,P|P:F,$U_{n}$,http://drmf.wmflabs.org/wiki/Definition:AlSalamIsmail, +\AngerA{\nu}@{z},Anger-Weber function,associated-Anger-Weber-A,F|F:P,$\mathbf{A}_{\nu}$,http://dlmf.nist.gov/11.10#E4, +\AngerJ{\nu}@{z},Anger function,Anger-J,F|F:P,$\mathbf{J}_{\nu}$,http://dlmf.nist.gov/11.10#E1, +\AntiDer@{x}{f(x)},semantic antiderivative,antiderivative,SM|O,"$\int f(x){\mathrm d}\!x$",http://drmf.wmflabs.org/wiki/Definition:AntiDer, +\AppellFi@{\alpha}{\beta}{\beta'}{\gamma}{x}{y},Appell function $F_1$,Appell-F-1,F|F:P,${F_{1}}$,http://dlmf.nist.gov/16.13#E1, +\AppellFii@{\alpha}{\beta}{\beta'}{\gamma}{\gamma'}{x}{y},Appell function $F_2$,Appell-F-2,F|F:P,${F_{2}}$,http://dlmf.nist.gov/16.13#E2, +\AppellFiii@{\alpha}{\alpha'}{\beta}{\beta'}{\gamma}{x}{y},Appell function $F_3$,Appell-F-3,F|F:P,${F_{3}}$,http://dlmf.nist.gov/16.13#E3, +\AppellFiv@{\alpha}{\beta}{\gamma}{\gamma'}{x}{y},Appell function $F_4$,Appell-F-4,F|F:P,${F_{4}}$,http://dlmf.nist.gov/16.13#E4, +\Asec@@{z},general inverse secant function,multivalued-inverse-secant,F|F:P:nP,$\mathrm{Arcsec}$,http://dlmf.nist.gov/4.23#E5, +\Asech@@{z},general inverse hyperbolic secant function,multivalued-hyperbolic-inverse-secant,F|F:P:nP,$\mathrm{Arcsech}$,http://dlmf.nist.gov/4.37#E5, +\Asin@@{z},general inverse sine function,multivalued-inverse-sine,F|F:P:nP,$\mathrm{Arcsin}$,http://dlmf.nist.gov/4.23#E1, +\Asinh@@{z},general inverse hyperbolic sine function,multivalued-hyperbolic-inverse-sine,F|F:P:nP,$\mathrm{Arcsinh}$,http://dlmf.nist.gov/4.37#E1, +\AskeyWilson{n}@{x}{a}{b}{c}{d}{q},Askey-Wilson polynomial,Askey-Wilson-polynomial-p,P|F:P,$p_{n}$,http://dlmf.nist.gov/18.28#E1, +\AskeyWilsonp{n}@{x}{a}{b}{c}{d}{q},Askey-Wilson polynomial,Askey-Wilson-polynomial-p,P|F:P,$p_{n}$,http://dlmf.nist.gov/18.28#E1, +\AssJacobiP{\alpha}{\beta}{n}@{x}{c},associated Jacobi polynomial,associated-Jacobi-polynomial-P,P|F:P,"$P^{(\alpha,\alpha)}_{n}$",http://dlmf.nist.gov/18.30#E4, +\AssLegendrePoly{n}@{x}{c},associated Legendre polynomial,Legendre-spherical-polynomial-P,F|F:P,$P_{n}$,http://dlmf.nist.gov/18.30#E6, +\Atan@@{z},general inverse tangent function,multivalued-inverse-tangent,F|F:P:nP,$\mathrm{Arctan}$,http://dlmf.nist.gov/4.23#E3, +\Atanh@@{z},general inverse hyperbolic tangent function,multivalued-hyperbolic-inverse-tangent,F|F:P:nP,$\mathrm{Arctanh}$,http://dlmf.nist.gov/4.37#E3, +\BarnesGamma@{z},Barnes $G$-function (or double gamma function),Barnes-Gamma,F|F:P,$G$,http://dlmf.nist.gov/5.17#E1, +\BellNumber@{n},Bell number,Bell-number,I|F:P,$B$,http://dlmf.nist.gov/26.7#i, +\BernoulliB{n}@{x},Bernoulli polynomial,Bernoulli-polynomial-B,P|F:P,$B_{n}$,http://dlmf.nist.gov/24.2#i, +\BesselA{\nu}@{\Matrix{T}},Bessel function of matrix argument of the first kind,Bessel-of-matrix-A,F|F:P,$A_{\nu}$,http://dlmf.nist.gov/35.5#i, +\BesselB{\nu}@{\Matrix{T}},Bessel function of matrix argument of the second kind,Bessel-of-matrix-B,F|F:P,$B_{\nu}$,http://dlmf.nist.gov/35.5#E3, +\BesselI{\nu}@{z},modified Bessel function of the first kind,modified-Bessel-first-kind,F|F:P,$I_{\nu}$,http://dlmf.nist.gov/23.1, +\BesselItilde{\nu}@{x},modified Bessel function of the first kind of imaginary order,modified-Bessel-first-kind-imaginary-order,F|F:P,$\widetilde{I}_{\nu}$,http://dlmf.nist.gov/23.1, +\BesselJ{\nu}@{z},Bessel function of the first kind,Bessel-J,F|F:P,$J_{\nu}$,http://dlmf.nist.gov/10.2#E2, +\BesselJtilde{\nu}@{x},Bessel function of the first kind of imaginary order,Bessel-J-imaginary-order,F|F:P,$\widetilde{J}_{\nu}$,http://dlmf.nist.gov/10.24#Ex1, +\BesselK{\nu}@{z},modified Bessel function of the second kind,modified-Bessel-second-kind,F|F:P,$K_{\nu}$,http://dlmf.nist.gov/23.1, +\BesselKtilde{\nu}@{x},modified Bessel function of the second kind of imaginary order,modified-Bessel-second-kind-imaginary-order,F|F:P,$\widetilde{K}_{\nu}$,http://dlmf.nist.gov/10.45#E2, +\BesselModulusM{\nu}@{x},modulus of Bessel functions,modulus-Bessel-M,F|F:P,$M_{\nu}$,http://dlmf.nist.gov/10.18#E1, +\BesselModulusN{\nu}@{x},modulus of derivatives of Bessel functions,modulus-Bessel-N,F|F:P,$N_{\nu}$,http://dlmf.nist.gov/10.18#E2, +\BesselPhasePhi{\nu}@{x},phase of derivatives of Bessel functions,phase-Bessel-Phi,F|F:P,$\phi_{\nu}$,http://dlmf.nist.gov/10.18#E3, +\BesselPhaseTheta{\nu}@{x},phase of Bessel functions,phase-Bessel-Theta,F|F:P,$\theta_{\nu}$,http://dlmf.nist.gov/10.18#E3, +\BesselPoly{n}@{x}{a},Bessel polynomial,Bessel-polynomial-y,P|F:P,$y_{n}$,http://dlmf.nist.gov/18.34#E1, +\BesselPolyy{n}@{x}{a},Bessel polynomial,Bessel-polynomial-y,P|F:P,$y_{n}$,http://dlmf.nist.gov/18.34#E1, +\BesselY{\nu}@{z},Bessel function of the second kind,Bessel-Y-Weber,F|F:P,$Y_{\nu}$,http://dlmf.nist.gov/10.2#E3, +\BesselYtilde{\nu}@{x},Bessel function of the second kind of imaginary order,Bessel-Y-Weber-imaginary-order,F|F:P,$\widetilde{Y}_{\nu}$,http://dlmf.nist.gov/10.24#Ex1, +\BickleyKi{\alpha}@{x},Bickley function,Bickley-Ki,F|F:P,$\mathrm{Ki}_{\alpha}$,http://dlmf.nist.gov/10.43#E11, +\BigO@{x},order not exceeding,Big-O,Q|F:P,$O$,http://dlmf.nist.gov/2.1#E3, +\CanonicInt{K}@{\Vector{x}},canonical integral $\Psi_K$,canonical-integral,F|F:P,$\Psi_{K}$,http://dlmf.nist.gov/36.2#E4, +\CanonicIntU@{\Vector{x}},umbilic canonical integral $\Psi^{({\rm U})}$,canonical-umbilic-integral,F|F:P,$\Psi^{(\mathrm{U})}$,http://dlmf.nist.gov/36.2#E5, +\CanonicIntE@{\Vector{x}},elliptic umbilic canonical integral $\Psi^{({\rm E})}$,elliptic-umbilic-canonical-integral,F|F:P,$\Psi^{(\mathrm{E})}$,http://dlmf.nist.gov/36.2#E6, +\CanonicIntH@{\Vector{x}},hyperbolic umbilic canonical integral $\Psi^{({\rm H})}$,hyperbolic-umbilic-canonical-integral,F|F:P,$\Psi^{(\mathrm{H})}$,http://dlmf.nist.gov/36.2#E8, +\CatalanNumber@{n},Catalan number,Catalan-number,I|F:P,$C$,http://dlmf.nist.gov/26.5#E1, +\CatalansConstant,Catalan's constant,Catalan's-constant,C|F,$G$,http://dlmf.nist.gov/25.11.E40, +\Charlier{n}@{x}{a},Charlier polynomial,Charlier-polynomial-C,P|F:P,$C_{n}$,http://dlmf.nist.gov/18.19#T1.t1.r11, +\CharlierC{n}@{x}{a},Charlier polynomial,Charlier-polynomial-C,P|F:P,$C_{n}$,http://dlmf.nist.gov/18.19#T1.t1.r11, +\ChebyC{n}@{x},dilated Chebyshev polynomial of the first kind,dilated-Chebyshev-polynomial-second-kind-C,P|F:P,$C_{n}$,http://dlmf.nist.gov/18.1#E3, +\ChebyS{n}@{x},dilated Chebyshev polynomial of the second kind,dilated-Chebyshev-polynomial-first-kind-S,P|F:P,$S_{n}$,http://dlmf.nist.gov/18.1#E3, +\ChebyT{n}@{x},Chebyshev polynomial of the first kind,Chebyshev-polynomial-first-kind-T,P|F:P,$T_{n}$,http://dlmf.nist.gov/18.3#T1.t1.r8, +\ChebyTs{n}@{x},shifted Chebyshev polynomial of the first kind,shifted-Chebyshev-polynomial-first-kind-T-star,P|F:P,$T^{*}_{n}$,http://dlmf.nist.gov/18.3#T1.t1.r20, +\ChebyU{n}@{x},Chebyshev polynomial of the second kind,Chebyshev-polynomial-second-kind-U,P|F:P,$U_{n}$,http://dlmf.nist.gov/18.3#T1.t1.r11, +\ChebyUs{n}@{x},shifted Chebyshev polynomial of the second kind,shifted-Chebyshev-polynomial-second-kind-U-star,P|F:P,$U^{*}_{n}$,http://dlmf.nist.gov/18.3#T1.t1.r23, +\ChebyV{n}@{x},Chebyshev polynomial of the third kind,Chebyshev-polynomial-third-kind-V,P|F:P,$V_{n}$,http://dlmf.nist.gov/18.3#T1.t1.r14, +\ChebyW{n}@{x},Chebyshev polynomial of the fourth kind,Chebyshev-polynomial-fourth-kind-W,P|F:P,$W_{n}$,http://dlmf.nist.gov/18.3#T1.t1.r17, +\ChebyshevPsi@{x},Chebyshev $\ChebyshevPsi$-function,Chebyshev-psi,F|F:P,$\psi$,http://dlmf.nist.gov/25.16#E1, +\CiglerqChebyT{n}@{x}{s}{q},Cigler $q$-Chebyshev polynomial $T$,Cigler-q-Chebyshev-polynomial-T,P|P:F,$T_{n}$,http://drmf.wmflabs.org/wiki/Definition:CiglerqChebyT, +\CiglerqChebyU{n}@{x}{s}{q},Cigler $q$-Chebyshev polynomial $U$,Cigler-q-Chebyshev-polynomial-U,P|P:F,$U_{n}$,http://drmf.wmflabs.org/wiki/Definition:CiglerqChebyU, +\CompEllIntCE@@{k},Legendre's complementary complete elliptic integral of the second kind,complementary-complete-elliptic-integral-second-kind-E,F|F:P:nP,${E'}$,http://dlmf.nist.gov/19.2#E9, +\CompEllIntCK@@{k},Legendre's complementary complete elliptic integral of the first kind,complementary-complete-elliptic-integral-first-kind-K,F|F:P:nP,${K'}$,http://dlmf.nist.gov/19.2#E9, +\CompEllIntD@@{k},complete elliptic integral of Legendre's type,complete-elliptic-integral-D,F|F:P:nP,$D$,http://dlmf.nist.gov/19.2#E8, +\CompEllIntE@@{k},Legendre's complete elliptic integral of the second kind,complete-elliptic-integral-second-kind-E,F|F:P:nP,$E$,http://dlmf.nist.gov/19.2#E8, +\CompEllIntK@@{k},Legendre's complete elliptic integral of the first kind,complete-elliptic-integral-first-kind-K,F|F:P:nP,$K$,http://dlmf.nist.gov/19.2#E8, +\CompEllIntPi@{\alpha^2}{k},Legendre's complete elliptic integral of the third kind,complete-elliptic-integral-third-kind-Pi,F|F:P,$\Pi$,http://dlmf.nist.gov/19.2#E8, +\Complex,set of complex numbers,EMPTY,SN|F,$\mathbb{C}$,http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r1, +\ComplexZeroAiryBi{k},$k$\textsuperscript{th} complex zero of Airy $\AiryBi$,complex-zero-Airy-Bi,F|F,$\beta_{k}$,http://dlmf.nist.gov/9.9#SS1.p2, +\ComplexZeroAiryBiPrime{k},$k$\textsuperscript{th} complex zero of Airy $\AiryBi'$,complex-zero-derivative-Airy-Bi,F|F,$\beta'_{k}$,http://dlmf.nist.gov/9.9#SS1.p2, +\CompositionsC[m]@{n},number of compositions,number-of-compositions-C,I|F:F:P:P,$c_{m}$,http://dlmf.nist.gov/26.11#p1, +\CosInt@{z},cosine integral ${\rm Ci}$,cosine-integral,F|F:P,$\mathrm{Ci}$,http://dlmf.nist.gov/6.2#E11, +\CosIntCin@{z},cosine integral ${\rm Cin}$,cosine-integral-Cin,F|F:P,$\mathrm{Cin}$,http://dlmf.nist.gov/6.2#E12, +\CosIntg@{a}{z},generalized cosine integral ${\rm Ci}$,generalized-cosine-integral-Ci,F|F:P,$\mathrm{Ci}$,http://dlmf.nist.gov/8.21#E2, +\CoshInt@{z},hyperbolic cosine integral,hyperbolic-cosine-integral,F|F:P,$\mathrm{Chi}$,http://dlmf.nist.gov/6.2#E16, +\CoulombC{\ell}@{\eta},normalizing constant for Coulomb radial functions,Coulomb-C,F|F:P,$C_{\ell}$,http://dlmf.nist.gov/33.2#E5, +\CoulombF{\ell}@{\eta}{\rho},regular Coulomb radial function,regular-Coulomb-F,F|F:P,$F_{\ell}$,http://dlmf.nist.gov/33.2#E3, +\CoulombG{\ell}@{\eta}{\rho},irregular Coulomb radial function,irregular-Coulomb-G,F|F:P,$G_{\ell}$,http://dlmf.nist.gov/33.2#E11, +\CoulombH{\pm}{\ell}@{\eta}{\rho},irregular Coulomb radial functions,irregular-Coulomb-H,F|F:P,${H^{\pm}_{\ell}}$,http://dlmf.nist.gov/33.2#E7, +\CoulombM{\ell}@{\eta}{\rho},envelope of Coulomb functions,envelope-Coulomb-M,F|F:P,$M_{\ell}$,http://dlmf.nist.gov/33.3#E1, +\CoulombSigma{\ell}@{\eta},Coulomb phase shift,Coulomb-phase-shift,F|F:P,${\sigma_{\ell}}$,http://dlmf.nist.gov/33.2#E10, +\CoulombTheta{\ell}@{\eta}{\rho},phase of Coulomb functions,Coulomb-phase,F|F:P,${\theta_{\ell}}$,http://dlmf.nist.gov/33.2#E9, +\Coulombc@{\epsilon}{\ell}{r},irregular Coulomb function $c$,Coulomb-c,F|F:P,$c$,http://dlmf.nist.gov/23.1, +\Coulombf@{\epsilon}{\ell}{r},regular Coulomb function $f$,Coulomb-f,F|F:P,$f$,http://dlmf.nist.gov/33.14#E4, +\Coulombh@{\epsilon}{\ell}{r},irregular Coulomb function $h$,Coulomb-h,F|F:P,$h$,http://dlmf.nist.gov/33.14#E7, +\Coulombrhotp@{\eta}{\ell},outer turning point for Coulomb radial functions,Coulomb-radial-outer-turning-point,F|F:P,$\rho_{\mathrm{tp}}$,http://dlmf.nist.gov/33.2#E2, +\Coulombrtp@{\epsilon}{\ell},outer turning point for Coulomb functions,Coulomb-outer-turning-point,F|F:P,$r_{\mathrm{tp}}$,http://dlmf.nist.gov/33.14#E3, +\Coulombs@{\epsilon}{\ell}{r},regular Coulomb function $s$,Coulomb-s,F|F:P,$s$,http://dlmf.nist.gov/33.14#E9, +\CuspCat{K}@{t}{\Vector{x}},cuspoid catastrophe,cuspoid-catastrophe,F|F:P,$\Phi_{K}$,http://dlmf.nist.gov/36.2#E1, +\Cylinder{\nu}@{z},cylinder function,cylinder-function,F|F:P,$\mathscr{C}_{\nu}$,http://dlmf.nist.gov/10.2#P5.p1, +\DawsonsInt@{z},Dawson's integral,Dawsons-integral,F|F:P,$F$,http://dlmf.nist.gov/7.2#E5, +\DedekindModularEta@{\tau},Dedekind's eta function (or Dedekind modular function),Dedekind-modular-Eta,F|F:P,$\eta$,http://dlmf.nist.gov/23.15#E9,http://dlmf.nist.gov/27.14#E12 +\DiffCat{K}@{\Vector{x}}{k},diffraction catastrophe,diffraction-catastrophe,F|F:P,$\Psi_{K}$,http://dlmf.nist.gov/36.2#E10, +\DiffCatE@{\Vector{x}}{k},elliptic umbilic canonical integral function,diffraction-elliptic-umbilic-catastrophe,F|F:P,$\Psi^{(\mathrm{E})}$,http://dlmf.nist.gov/23.1, +\DiffCatH@{\Vector{x}}{k},hyperbolic umbilic canonical integral function,diffraction-hyperbolic-umbilic-catastrophe,F|F:P,$\Psi^{(\mathrm{H})}$,http://dlmf.nist.gov/36.2#E11, +\DiffCatU@{\Vector{x}}{k},umbilic canonical integral function,diffraction-umbilic-catastrophe,F|F:P,$\Psi^{(\mathrm{U})}$,http://dlmf.nist.gov/36.2#E11, +\Dilogarithm@{z},dilogarithm,dilogarithm,F|F:P,$\mathrm{Li}_2$,http://dlmf.nist.gov/25.12#E1, +\Diracdelta@{x-a},Dirac delta (or Dirac delta function),Dirac-delta,D|F:P,$\Diracdelta@{x-a}$,http://dlmf.nist.gov/1.17#SS1.p1, +\DirichletCharacter@@{n}{k},Dirichlet character,Dirichlet-character,I|F:P:nP,$\chi$,http://dlmf.nist.gov/27.8, +\DirichletL@{s}{\chi},Dirichlet $\DirichletL$-function,Dirichlet-L,F|F:P,$L$,http://dlmf.nist.gov/25.15#E1, +\DiscriminantDelta@{\tau},discriminant function,discriminant-delta,F|F:P,$\Delta$,http://dlmf.nist.gov/27.14#E16, +\DiskOP{\alpha}{m}{n}@{z},disk polynomial,disk-orthogonal-polynomial-R,P|F:P,"$R^{(\alpha)}_{m,n}$",http://dlmf.nist.gov/18.37#E1, +\Distribution{f}{g},distribution,EMPTY,D|F,"$\left\langle f,g\right\rangle$",http://dlmf.nist.gov/1.16#SS1.p5, +\DivisorFunctionD[k]@{n},divisor function,divisor-function-D,F|F:P,$d_{k}$,http://dlmf.nist.gov/27.2#E9, +\DivisorSigma{\alpha}@{n},sum of powers of divisors,divisor-sigma,I|F:P,$\sigma_{\alpha}$,http://dlmf.nist.gov/27.2#E10, +\EllIntD@{\phi}{k},incomplete elliptic integral of Legendre's type,elliptic-integral-third-kind-D,F|F:P,$D$,http://dlmf.nist.gov/19.2#E6, +\EllIntE@{\phi}{k},Legendre's incomplete elliptic integral of the second kind,elliptic-integral-second-kind-E,F|F:P,$E$,http://dlmf.nist.gov/19.2#E5, +\EllIntF@{\phi}{k},Legendre's incomplete elliptic integral of the first kind,elliptic-integral-first-kind-F,F|F:P,$F$,http://dlmf.nist.gov/19.2#E4, +\EllIntPi@{\phi}{\alpha^2}{k},Legendre's incomplete elliptic integral of the third kind,elliptic-integral-third-kind-Pi,F|F:P,$\Pi$,http://dlmf.nist.gov/19.2#E7, +\EllIntR{-a}@{\Vector{b}}{\Vector{z}},Carlson integral $R_{-a}$,Carlson-integral-R,F|F:P,$R_{-a}$,http://dlmf.nist.gov/19.16#E9, +\EllIntRC@@{x}{y},Carlson's elliptic integral $R_C$,Carlson-integral-RC,F|F:P:nP,$R_C$,http://dlmf.nist.gov/19.2#E17, +\EllIntRD@@{x}{y}{z},Carlson's elliptic integral $R_D$,Carlson-integral-RD,F|F:P:nP,$R_D$,http://dlmf.nist.gov/19.16#E5, +\EllIntRF@@{x}{y}{z},Carlson's symmetric elliptic integral of first kind,Carlson-integral-RF,F|F:P:nP,$R_F$,http://dlmf.nist.gov/19.16#E1, +\EllIntRG@@{x}{y}{z},Carlson's symmetric elliptic integral of second kind,Carlson-integral-RG,F|F:P:nP,$R_G$,http://dlmf.nist.gov/19.16#E3, +\EllIntRJ@@{x}{y}{z}{p},Carlson's symmetric elliptic integral of third kind,Carlson-integral-RJ,F|F:P:nP,$R_J$,http://dlmf.nist.gov/19.16#E2, +\EllIntcel@{k_c}{p}{a}{b},Bulirsch's complete elliptic integral,Bulirsch-integral-cel,F|F:P,$\mathrm{cel}$,http://dlmf.nist.gov/19.2#E11, +\EllIntelone@{x}{k_c},Bulirsch's incomplete elliptic integral of the first kind,Bulirsch-integral-el-1,F|F:P,$\mathrm{el1}$,http://dlmf.nist.gov/19.2#E15, +\EllIntelthree@{x}{k_c}{p},Bulirsch's incomplete elliptic integral of the third kind,Bulirsch-integral-el-3,F|F:P,$\mathrm{el3}$,http://dlmf.nist.gov/19.2#E16, +\EllInteltwo@{x}{k_c}{a}{b},Bulirsch's incomplete elliptic integral of the second kind,Bulirsch-integral-el-2,F|F:P,$\mathrm{el2}$,http://dlmf.nist.gov/19.2#E12, +\EulerBeta@{a}{b},beta function,Euler-Beta,F|F:P,$\mathrm{B}$,http://dlmf.nist.gov/5.12#E1, +\EulerConstant,Euler's (Euler-Mascheroni) constant,EMPTY,C|F,$\gamma$,http://dlmf.nist.gov/5.2#E3, +\EulerE{n}@{x},Euler polynomial,Euler-polynomial-E,P|F:P,$E_{n}$,http://dlmf.nist.gov/24.2#ii, +\EulerGamma@{z},Euler's gamma function,Euler-Gamma,F|F:P,$\Gamma$,http://dlmf.nist.gov/5.2#E1, +\EulerPhi@{x},Euler's reciprocal function,Euler-phi,F|F:P,$\mathit{f}$,http://dlmf.nist.gov/27.14#E2, +\EulerSumH@{s},Euler sums,Euler-sum-H,F|F:P,$H$,http://dlmf.nist.gov/25.16#SS2.p1, +\EulerTotientPhi[k]@{n},Euler's totient $\phi_k(n)$,Euler-totient-phi,I|F:P,$\phi_{k}$,http://dlmf.nist.gov/27.2#E6, +\EulerianNumber{n}{k},Eulerian number,Eulerian-number,I|F,$\left\langle n\atop k\right\rangle$,http://dlmf.nist.gov/26.14#SS1.p3, +\ExpInt@{z},exponential integral $E_1$,exponential-integral,F|F:P,$E_1$,http://dlmf.nist.gov/6.2#E1, +\ExpIntEin@{z},complementary exponential integral ${\rm Ein}$,complementary-exponential-integral,F|F:P,$\mathrm{Ein}$,http://dlmf.nist.gov/6.2#E3, +\ExpInti@{x},exponential integral ${\rm Ei}$,exponential-integral-Ei,F|F:P,$\mathrm{Ei}$,http://dlmf.nist.gov/6.2#E1, +\ExpIntn{p}@{z},generalized exponential integral,exponential-integral-En,F|F:P,$E_{p}$,http://dlmf.nist.gov/8.19#E1, +\FerrersHatQ[-\mu]{-\frac{1}{2}+i\tau}@{x},conical function $\widehat{\mathsf{Q}}_\nu^\mu$,Ferrers-conical-legendre-Q-hat,F|FO:PO:FnO:PnO,$\widehat{\mathsf{Q}}_\nu^\mu$,http://dlmf.nist.gov/14.20#E2, +\FerrersP[\mu]{\nu}@{x},Ferrers function of the first kind,Ferrers-Legendre-P-first-kind,F|FO:PO:FnO:PnO,$\mathsf{P}_\nu^\mu$,http://dlmf.nist.gov/14.3#E1, +\FerrersQ[\mu]{\nu}@{x},Ferrers function of the second kind,Ferrers-Legendre-Q-first-kind,F|FO:PO:FnO:PnO,$\mathsf{Q}_\nu^\mu$,http://dlmf.nist.gov/14.3#E2, +\FishersHh{n}@{z},probability function,Fishers-Hh,F|F:P,$\mathit{Hh}_{n}$,http://dlmf.nist.gov/7.18#E12, +\FresnelCos@{z},Fresnel integral $C$,Fresnel-cosine-integral,F|F:P,$C$,http://dlmf.nist.gov/7.2#E7, +\FresnelF@{z},Fresnel integral ${\mathcal F}$,Fresnel-integral,F|F:P,$\mathcal{F}$,http://dlmf.nist.gov/7.2#E6, +\FresnelSin@{z},Fresnel integral $S$,Fresnel-sine-integral,F|F:P,$S$,http://dlmf.nist.gov/7.2#E8, +\Fresnelf@{z},auxiliary function for Fresnel integrals ${\rm f}$,Fresnel-auxilliary-function-f,F|F:P,$\mathrm{f}$,http://dlmf.nist.gov/7.2#E10, +\Fresnelg@{z},auxiliary function for Fresnel integrals ${\rm g}$,Fresnel-auxilliary-function-g,F|F:P,$\mathrm{g}$,http://dlmf.nist.gov/7.2#E11, +\GammaP@{a}{z},normalized incomplete gamma function $P$,incomplete-gamma-P,F|F:P,$P$,http://dlmf.nist.gov/8.2#E4, +\GammaQ@{a}{z},normalized incomplete gamma function $Q$,incomplete-gamma-Q,F|F:P,$Q$,http://dlmf.nist.gov/8.2#E4, +\GaussSum@{n}{\DirichletCharacter},Gauss sum,Gauss-sum,F|F:P,$G$,http://dlmf.nist.gov/27.10#E9, +\GenBernoulliB{\ell}{n}@{x},generalized Bernoulli polynomial,generalized-Bernoulli-polynomial-B,P|F:P,$B^{(\ell)}_{n}$,http://dlmf.nist.gov/24.16#P1.p1, +\GenBesselFun@{\rho}{\beta}{z},generalized Bessel function,generalized-Bessel-function,F|F:P,$\phi$,http://dlmf.nist.gov/10.46#E1, +\GenEulerE{\ell}{n}@{x},generalized Euler polynomial,generalized-Euler-polynomial-E,P|F:P,$E^{(\ell)}_{n}$,http://dlmf.nist.gov/24.16#P1.p1, +\GenEulerSumH@{s}{z},generalized Euler sums,generalized-Euler-sum-H,F|F:P,$H$,http://dlmf.nist.gov/25.16#SS2.p7, +\GenGegenbauer{\alpha}{\beta}{n}@{x},Generalized Gegenbauer polynomial,generalized-Gegenbauer-polynomial-S,P|F:P,"$S^{(\alpha,\beta)}_{n}$",http://drmf.wmflabs.org/wiki/Definition:GenGegenbauer, +\GenHermite[\alpha]{n}@{x},generalized Hermite polynomial,generalized-Hermite-polynomial-H,P|FO:PO:FnO:PnO,$H_n^{\mu}$,http://drmf.wmflabs.org/wiki/Definition:GenHermite, +\GoodStat@{z},Goodwin-Staton integral,Goodwin-Staton-integral,F|F:P,$G$,http://dlmf.nist.gov/7.2#E12, +\GottliebLaguerre{n}@{x},Gottlieb-Laguerre polynomial,Gottlieb-Laguerre-polynomial-l,P|F:P,$l_{n}$,http://drmf.wmflabs.org/wiki/Definition:GottliebLaguerre, +\Gudermannian@@{x},Gudermannian function,inverse-Gudermannian,F|F:P:nP,$\mathrm{gd}$,http://dlmf.nist.gov/4.23#E39, +\Hahn{n}@{x}{\alpha}{\beta}{N},Hahn polynomial,Hahn-polynomial-Q,P|F:P,$Q_{n}$,http://dlmf.nist.gov/18.19#T1.t1.r3, +\HahnQ{n}@{x}{\alpha}{\beta}{N},Hahn polynomial,Hahn-polynomial-Q,P|F:P,$Q_{n}$,http://dlmf.nist.gov/18.19#T1.t1.r3, +\HahnR{n}@{x}{\gamma}{\delta}{N},dual Hahn polynomial,dual-Hahn-R,P|F:P,$R_{n}$,http://dlmf.nist.gov/18.25#T1.t1.r5, +\HahnS{n}@{x}{a}{b}{c},continuous dual Hahn polynomial,continuous-dual-Hahn-S,P|F:P,$S_{n}$,http://dlmf.nist.gov/18.25#T1.t1.r3, +\Hahnp{n}@{x}{a}{b}{\conj{a}}{\conj{b}},continuous Hahn polynomial,continuous-Hahn-polynomial-p,P|F:P,$p_{n}$,http://dlmf.nist.gov/18.19#P2.p1, +\HankelHi{\nu}@{z},Hankel function of the first kind,Hankel-H-1-Bessel-third-kind,F|F:P,${H^{(1)}_{\nu}}$,http://dlmf.nist.gov/10.2#E5, +\HankelHii{\nu}@{z},Hankel function of the second kind,Hankel-H-2-Bessel-third-kind,F|F:P,${H^{(2)}_{\nu}}$,http://dlmf.nist.gov/10.2#E6, +\HarmonicNumber{n},Harmonic number,Harmonic-number,I|F,$H_{n}$,http://dlmf.nist.gov/25.11#E33, +\HeavisideH@{x},Heaviside step function,Heaviside-H,F|F:P,$H$,http://dlmf.nist.gov/1.16#E13, +\Hermite{n}@{x},Hermite polynomial $H_n$,Hermite-polynomial-H,P|F:P,$H_{n}$,http://dlmf.nist.gov/18.3#T1.t1.r28, +\HermiteH{n}@{x},Hermite polynomial $H_n$,Hermite-polynomial-H,P|F:P,$H_{n}$,http://dlmf.nist.gov/18.3#T1.t1.r28, +\HermiteHe{n}@{x},Hermite polynomial $He_n$,Hermite-polynomial-He,P|F:P,$\mathit{He}_{n}$,http://dlmf.nist.gov/18.3#T1.t1.r29, +\HeunFunction{m}{a}{b}@@{a}{q}{\alpha}{\beta}{\gamma}{\delta}{z},Heun functions $(a;b)Hf_m^\nu$,Heun-function,F|FO:PO:nPO:FnO:PnO:nPnO,"$(s_1,s_2)\mathit{Hf}_{m}^\nu$",http://dlmf.nist.gov/31.4#p1, +\HeunLocal@@{a}{q}{\alpha}{\beta}{\gamma}{\delta}{z},Heun functions $Hl$,Heun-local,F|F:P:nP,$\mathit{H\!\ell}$,http://dlmf.nist.gov/31.3#E1, +\HeunPolynom{n}{m}@@{a}{q}{-n}{\beta}{\gamma}{\delta}{z},Heun polynomial,Heun-polynomial-m,P|F:P:nP,"$\mathit{Hp}_{n,m}$",http://dlmf.nist.gov/31.5#E2, +\HilbertTrans@{f}{x},Hilbert transform,Hilbert-transform,O|F:P,$\mathcal{H}$,http://dlmf.nist.gov/1.14#SS5.p1, +\HurwitzZeta@{s}{a},Hurwitz zeta function,Hurwitz-zeta,F|F:P,$\zeta$,http://dlmf.nist.gov/25.11#E1, +\HyperPsi@{a}{b}{\Matrix{T}},confluent hypergeometric function of matrix argument of the second kind,confluent-hypergeometric-of-matrix-Psi,F|F:P,$\Psi$,http://dlmf.nist.gov/35.6#E2, +\HyperboldpFq{p}{q}@@@{\Vector{a}}{\Vector{b}}{z},scaled (or Olver's) generalized hypergeometric function,hypergeometric-bold-pFq,F|F:fo1:fo2:fo3,${{}_{p}{\mathbf{F}}_{q}}$,http://dlmf.nist.gov/16.2#E5, +\HypergeoF@@@{a}{b}{c}{z},hypergeometric function,Gauss-hypergeometric-F,F|F:fo1:fo2:fo3,$F$,http://dlmf.nist.gov/15.2#E1, +\HypergeoboldF@@@{a}{b}{c}{z},Olver's hypergeometric function,scaled-hypergeometric-bold-F,F|F:fo1:fo2:fo3,$\mathbf{F}$,http://dlmf.nist.gov/15.2#E2, +"\HyperpFq{p}{q}@@@{a_1,...,a_p}{b_1,...,b_q}{z}",generalized hypergeometric function,Gauss-hypergeometric-pFq,F|F:fo1:fo2:fo3,${{}_{p}F_{q}}$,http://dlmf.nist.gov/16.2#E1, +"\HyperpHq{p}{q}@@@{{a_1,...,a_p}{b_1,...,b_q}{z}",bilateral hypergeometric function,hypergeometric-pHq,F|F:fo1:fo2:fo3,${{}_{p}H_{q}}$,http://dlmf.nist.gov/16.4#E16, +\IncBeta{x}@{a}{b},incomplete beta function,incomplete-Beta,F|F:P,$\mathrm{B}_{x}$,http://dlmf.nist.gov/8.17#E1, +\IncGamma@{a}{z},incomplete gamma function $\Gamma$,incomplete-Gamma,F|F:P,$\Gamma$,http://dlmf.nist.gov/8.2#E2, +\IncI{x}@{a}{b},regularized incomplete beta function,IncI,F|F:P,$I_{x}$,http://dlmf.nist.gov/8.17#E2, +\Int{a}{b}@{x}{f(x)},semantic definite integral,semantic-definite-integral,SM|O,"$\int_{a}^{b}f(x){\mathrm d}\!x$",http://drmf.wmflabs.org/wiki/Definition:Int, +\Integer,set of integers,EMPTY,SN|F,$\mathbb{Z}$,http://dlmf.nist.gov/front/introduction#Sx4.p2.t1.r20, +\IntgenAiryA{k}@{z}{p},generalized Airy function from integral representation $A_k$,Integral-generalized-Airy-A,F|F:P,$A_{k}$,http://dlmf.nist.gov/9.13#SS2.p1, +\IntgenAiryB{k}@{z}{p},generalized Airy function from integral representation $B_k$,Integral-generalized-Airy-B,F|F:P,$B_{k}$,http://dlmf.nist.gov/9.13#SS2.p1, +\JacksonqBesselI{\nu}@{z}{q},Jackson $q$-Bessel function 1,EMPTY,F|F:P,$J^{(1)}_{q}$,http://drmf.wmflabs.org/wiki/Definition:JacksonqBesselI, +\JacksonqBesselII{\nu}@{z}{q},Jackson $q$-Bessel function 2,EMPTY,F|F:P,$J^{(2)}_{q}$,http://drmf.wmflabs.org/wiki/Definition:JacksonqBesselII, +\JacksonqBesselIII{\nu}@{z}{q},Jackson/Hahn-Exton $q$-Bessel function 3,EMPTY,F|F:P,$J^{(3)}_{q}$,http://drmf.wmflabs.org/wiki/Definition:JacksonqBesselIII, +\Jacobi{\alpha}{\beta}{n}@{x},Jacobi polynomial,Jacobi-polynomial-P,P|F:P,"$P^{(\alpha,\beta)}_{n}$",http://dlmf.nist.gov/18.3#T1.t1.r3, +\JacobiEpsilon@{x}{k},Jacobi's epsilon function,Jacobi-Epsilon,F|F:P,$\mathcal{E}$,http://dlmf.nist.gov/22.16#E14, +\JacobiG{n}@{p}{q}{x},shifted Jacobi polynomial,shifted-Jacobi-polynomial-G,P|F:P,$G_{n}$,http://dlmf.nist.gov/18.1#E2, +\JacobiP{\alpha}{\beta}{n}@{x},Jacobi polynomial,Jacobi-polynomial-P,P|F:P,"$P^{(\alpha,\beta)}_{n}$",http://dlmf.nist.gov/18.3#T1.t1.r3, +\JacobiSymbol{n}{P},Jacobi symbol,Jacobi-symbol,I|F,$(n|P)$,http://dlmf.nist.gov/27.9#p3, +\JacobiTheta{j}@{z}{q},Jacobi theta function,Jacobi-theta,F|F:P,$\theta_{p}$,http://dlmf.nist.gov/20.2#i, +\JacobiThetaCombined{n}{m}@{z}{q},combined theta function,Jacobi-theta-combined,F|F:P,"$\varphi_{m,n}$",http://dlmf.nist.gov/20.11#SS5.p2, +\JacobiThetaTau{j}@{z}{\tau},Jacobi theta $\tau$ function $\theta_j$,Jacobi-theta-tau,F|F:P,$\theta_{p}$,http://dlmf.nist.gov/20.2#i, +\JacobiZeta@{x}{k},Jacobi's zeta function,Jacobi-Zeta,F|F:P,$\mathrm{Z}$,http://dlmf.nist.gov/22.16#E32, +\Jacobiam@@{x}{k},Jacobi's amplitude function ${\rm am}$,Jacobi-elliptic-amplitude,F|F:P:nP,$\mathrm{am}$,http://dlmf.nist.gov/23.1, +\Jacobicd@@{z}{k},Jacobian elliptic function ${\rm cd}$,Jacobi-elliptic-cd,F|F:P:nP,$\mathrm{cd}$,http://dlmf.nist.gov/22.2#E8, +\Jacobicn@@{z}{k},Jacobian elliptic function ${\rm cn}$,Jacobi-elliptic-cn,F|F:P:nP,$\mathrm{cn}$,http://dlmf.nist.gov/22.2#E5, +\Jacobics@@{z}{k},Jacobian elliptic function ${\rm cs}$,Jacobi-elliptic-cs,F|F:P:nP,$\mathrm{cs}$,http://dlmf.nist.gov/23.1, +\Jacobidc@@{z}{k},Jacobian elliptic function ${\rm dc}$,Jacobi-elliptic-dc,F|F:P:nP,$\mathrm{dc}$,http://dlmf.nist.gov/23.1, +\Jacobidn@@{z}{k},Jacobian elliptic function ${\rm dn}$,Jacobi-elliptic-dn,F|F:P:nP,$\mathrm{dn}$,http://dlmf.nist.gov/22.2#E6, +\Jacobids@@{z}{k},Jacobian elliptic function ${\rm ds}$,Jacobi-elliptic-ds,F|F:P:nP,$\mathrm{ds}$,http://dlmf.nist.gov/23.1, +\Jacobinc@@{z}{k},Jacobian elliptic function ${\rm nc}$,Jacobi-elliptic-nc,F|F:P:nP,$\mathrm{nc}$,http://dlmf.nist.gov/23.1, +\Jacobind@@{z}{k},Jacobian elliptic function ${\rm nd}$,Jacobi-elliptic-nd,F|F:P:nP,$\mathrm{nd}$,http://dlmf.nist.gov/23.1, +\Jacobins@@{z}{k},Jacobian elliptic function ${\rm ns}$,Jacobi-elliptic-ns,F|F:P:nP,$\mathrm{ns}$,http://dlmf.nist.gov/23.1, +\Jacobiphi{\alpha}{\beta}{\lambda}@{t},Jacobi function,Jacobi-hypergeometric-phi,F|F:P,"$\phi^{(\alpha,\beta)}_{\lambda}$",http://dlmf.nist.gov/15.9#E11, +\Jacobisc@@{z}{k},Jacobian elliptic function ${\rm sc}$,Jacobi-elliptic-sc,F|F:P:nP,$\mathrm{sc}$,http://dlmf.nist.gov/22.2#E9, +\Jacobisd@@{z}{k},Jacobian elliptic function ${\rm sd}$,Jacobi-elliptic-sd,F|F:P:nP,$\mathrm{sd}$,http://dlmf.nist.gov/22.2#E7, +\Jacobisn@@{z}{k},Jacobian elliptic function ${\rm sn}$,Jacobi-elliptic-sn,F|F:P:nP,$\mathrm{sn}$,http://dlmf.nist.gov/22.2#E4, +\JonquierePhi@{z}{s}=\Polylogarithm{s}@{z},Jonquiére's function,Jonquiere-phi,F|F:P,$\phi$,http://dlmf.nist.gov/23.1, +\JordanJ{k}@{n},Jordan's function,Jordan-J,F|F:P,$J_{k}$,http://dlmf.nist.gov/27.2#E11, +\Kelvinbei{\nu}@@{x},Kelvin function ${\rm bei}_\nu$,Kelvin-bei,F|F:P:nP,$\mathrm{bei}_{\nu}$,http://dlmf.nist.gov/10.61#E1, +\Kelvinber{\nu}@@{x},Kelvin function ${\rm ber}_\nu$,Kelvin-ber,F|F:P:nP,$\mathrm{ber}_{\nu}$,http://dlmf.nist.gov/10.61#E1, +\Kelvinkei{\nu}@@{x},Kelvin function ${\rm kei}_\nu$,Kelvin-kei,F|F:P:nP,$\mathrm{kei}_{\nu}$,http://dlmf.nist.gov/10.61#E2, +\Kelvinker{\nu}@@{x},Kelvin function ${\rm ker}_\nu$,Kelvin-ker,F|F:P:nP,$\mathrm{ker}_{\nu}$,http://dlmf.nist.gov/10.61#E2, +\Krawtchouk{n}@{x}{p}{N},Krawtchouk polynomial,Krawtchouk-polynomial-K,P|F:P,$K_{n}$,http://dlmf.nist.gov/18.19#T1.t1.r6, +\KrawtchoukK{n}@{x}{p}{N},Krawtchouk polynomial,Krawtchouk-polynomial-K,P|F:P,$K_{n}$,http://dlmf.nist.gov/18.19#T1.t1.r6, +\Kronecker{j}{k},Kronecker delta,EMPTY,I|F,"$\delta_{m,n}$",http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r4, +\KummerM@{a}{b}{z},Kummer confluent hypergeometric function $M$,Kummer-confluent-hypergeometric-M,F|F:P,$M$,http://dlmf.nist.gov/13.2#E2, +\KummerU@{a}{b}{z},Kummer confluent hypergeometric function $U$,Kummer-confluent-hypergeometric-U,F|F:P,$U$,http://dlmf.nist.gov/13.2#E6, +\KummerboldM@{a}{b}{z},Olver's confluent hypergeometric function,Kummer-confluent-hypergeometric-bold-M,F|F:P,${\mathbf{M}}$,http://dlmf.nist.gov/13.2#E3, +\Laguerre[\alpha]{n}@{x},Laguerre (or generalized Laguerre) polynomial,generalized-Laguerre-polynomial-L,P|FnO:PnO:FO:PO,$L_n^{(\alpha)}$,http://dlmf.nist.gov/18.3#T1.t1.r27, +\LaguerreL[\alpha]{n}@{x},Laguerre (or generalized Laguerre) polynomial,generalized-Laguerre-polynomial-L,P|FnO:PnO:FO:PO,$L_n^{(\alpha)}$,http://dlmf.nist.gov/18.3#T1.t1.r27, +\LambertW@{x},Lambert $\LambertW$-function,Lambert-W,F|F:P,$W$,http://dlmf.nist.gov/23.1, +\LambertWm@{x},non-principal branch of Lambert $\LambertW$-function,Lambert-Wm,F|F:P,$\mathrm{Wm}$,http://dlmf.nist.gov/4.13#p2, +\LambertWp@{x},principal branch of Lambert $\LambertW$-function,Lambert-Wp,F|F:P,$\mathrm{Wp}$,http://dlmf.nist.gov/23.1, +\LameEc{m}{\nu}@{z}{k^2},Lamé function $Ec_\nu^m$,Lame-Ec,F|F:P,$\mathit{Ec}^{m}_{\nu}$,http://dlmf.nist.gov/23.1, +\LameEs{m}{\nu}@{z}{k^2},Lamé function $Es_\nu^m$,Lame-Es,F|F:P,$\mathit{Es}^{m}_{\nu}$,http://dlmf.nist.gov/23.1, +\Lamea{n}{\nu}@{k^2},eigenvalues of Lamé's equation $a_\nu^m$,Lame-eigenvalue-a,F|F:P,$a^{n}_{\nu}$,http://dlmf.nist.gov/29.3#SS1.p1, +\Lameb{n}{\nu}@{k^2},eigenvalues of Lamé's equation $b_\nu^m$,Lame-eigenvalue-b,F|F:P,$b^{n}_{\nu}$,http://dlmf.nist.gov/29.3#SS1.p1, +\LamecE{m}{2n+1}@{z}{k^2},Lamé polynomial $cE_{2n+1}^m$,Lame-polynomial-cE,P|F:P,$\mathit{cE}^{m}_{\nu}$,http://dlmf.nist.gov/23.1, +\LamecdE{m}{2n+2}@{z}{k^2},Lamé polynomial $cdE_{2n+2}^m$,Lame-polynomial-cdE,P|F:P,$\mathit{cdE}^{m}_{\nu}$,http://dlmf.nist.gov/23.1, +\LamedE{m}{2n+1}@{z}{k^2},Lamé polynomial $dE_{2n+1}^m$,Lame-polynomial-dE,P|F:P,$\mathit{dE}^{m}_{\nu}$,http://dlmf.nist.gov/29.12#E4, +\LamesE{m}{2n+1}@{z}{k^2},Lamé polynomial $sE_{2n+1}^m$,Lame-polynomial-sE,P|F:P,$\mathit{sE}^{m}_{\nu}$,http://dlmf.nist.gov/29.12#E2, +\LamescE{m}{2n+2}@{z}{k^2},Lamé polynomial $scE_{2n+2}^m$,Lame-polynomial-scE,P|F:P,$\mathit{scE}^{m}_{\nu}$,http://dlmf.nist.gov/29.12#E5, +\LamescdE{m}{2n+3}@{z}{k^2},Lamé polynomial $scdE_{2n+3}^m$,Lame-polynomial-scdE,P|F:P,$\mathit{scdE}^{m}_{\nu}$,http://dlmf.nist.gov/29.12#E8, +\LamesdE{m}{2n+2}@{z}{k^2},Lamé polynomial $sdE_{2n+2}^m$,Lame-polynomial-sdE,P|F:P,$\mathit{sdE}^{m}_{\nu}$,http://dlmf.nist.gov/29.12#E6, +\LameuE{m}{2n}@{z}{k^2},Lamé polynomial $uE_{2n}^m$,Lame-polynomial-uE,P|F:P,$\mathit{uE}^{m}_{\nu}$,http://dlmf.nist.gov/29.12#E1, +\LaplaceTrans@{f}{s},Laplace transform,Laplace-transform,O|F:P,$\mathscr{L}$,http://dlmf.nist.gov/1.14#E17, +\Lattice{L},lattice in $\Complex$,EMPTY,SN|F,$\mathbb{L}$,http://dlmf.nist.gov/23.2#i, +"\LauricellaFD@{a}{b_1,...,b_n}{c}{z_1,...,z_n}",Lauricella's multivariate hypergeometric function,Lauricella-FD,F|F:P,$F_D$,http://dlmf.nist.gov/19.15#p1, +\LegendreBlackQ[\mu]{\nu}@{z},Olver's associated Legendre function,associated-Legendre-black-Q,F|FO:PO:FnO:PnO,$\boldsymbol{Q}_\nu^\mu$,http://dlmf.nist.gov/14.3#E10, +\LegendreP[\mu]{\nu}@{z},associated Legendre function of the first kind,Legendre-P-first-kind,F|FO:PO:FnO:PnO,$P_\nu^\mu$,http://dlmf.nist.gov/14.3#E6, +\LegendrePoly{n}@{x},Legendre polynomial,Legendre-spherical-polynomial,P|F:P,$P_{n}$,http://dlmf.nist.gov/18.3#T1.t1.r25, +\LegendrePolys{n}@{x},shifted Legendre polynomial,shifted-spherical-Legendre-polynomial-s,P|F:P,$P^{*}_{n}$,http://dlmf.nist.gov/18.3#T1.t1.r26, +\LegendreQ[\mu]{\nu}@{z},associated Legendre function of the second kind,Legendre-Q-second-kind,F|FO:PO:FnO:PnO,$Q_\nu^\mu$,http://dlmf.nist.gov/14.3#E7, +\LegendreSymbol{n}{p},Legendre symbol,Legendre-symbol,I|F,$(n|p)$,http://dlmf.nist.gov/27.9, +\LerchPhi@{z}{s}{a},Lerch's transcendent,Lerch-Phi,F|F:P,$\Phi$,http://dlmf.nist.gov/25.14#E1, +\LeviCivita{j}{k}{\ell},Levi-Civita symbol,EMPTY,I|F,$\epsilon_{jkl}$,http://dlmf.nist.gov/1.6#E14, +\LinBrF@{a}{u},line-broadening function,line-broadening-function,F|F:P,$H$,http://dlmf.nist.gov/7.19#E4, +\LiouvilleLambda@{n},Liouville's function,Liouville-lambda,F|F:P,$\lambda$,http://dlmf.nist.gov/27.2#E13, +\Ln@@{z},general logarithm function,multivalued-natural-logarithm,F|F:P:nP,$\mathrm{Ln}$,http://dlmf.nist.gov/4.2#E1, +\LogInt@{x},logarithmic integral,logarithmic-integral,F|F:P,$\mathrm{li}$,http://dlmf.nist.gov/6.2#E8, +\LommelS{\mu}{\nu}@{z},"Lommel function $S$",Lommel-S,F|F:P,"$S_{{\mu},{\nu}}$",http://dlmf.nist.gov/11.9#E5, +\Lommels{\mu}{\nu}@{z},"Lommel function $s$",Lommel-s,F|F:P,"$s_{{\mu},{\nu}}$",http://dlmf.nist.gov/11.9#E3, +\Longleftrightarrow,is equivalent to,equivalent-to,SM|F,$\Longleftrightarrow $,http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r13, +\Longrightarrow,implies,EMPTY,SM|F,$\Longrightarrow$,http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r12, +\MangoldtLambda@{n},Mangoldt's function,Mangoldt-Lambda,F|F:P,$\Lambda$,http://dlmf.nist.gov/27.2#E14, +\MathieuCe{\nu}@@{z}{q},modified Mathieu function ${\rm Ce}_\nu$,modified-Mathieu-Ce,F|F:P:nP,$\mathrm{Ce}_{n}$,http://dlmf.nist.gov/28.20#E3, +\MathieuD{j}@{\nu}{\mu}{z},cross-products of modified Mathieu functions and their derivatives ${\rm D}_{j}$,Mathieu-D,F|F:P,$\mathrm{D}_{j}$,http://dlmf.nist.gov/28.28#E24, +\MathieuDc{j}@{n}{m}{z},cross-products of radial Mathieu functions and their derivatives ${\rm Dc}_j$,Mathieu-Dc,F|F:P,$\mathrm{Dc}_{j}$,http://dlmf.nist.gov/28.28#E39, +\MathieuDs{j}@{n}{m}{z},cross-products of radial Mathieu functions and their derivatives ${\rm Ds}_j$,Mathieu-Ds,F|F:P,$\mathrm{Ds}_{j}$,http://dlmf.nist.gov/28.28#E35, +\MathieuDsc{j}@{n}{m}{z},cross-products of radial Mathieu functions and their derivatives ${\rm Dsc}_j$,Mathieu-Dsc,F|F:P,$\mathrm{Dsc}_{j}$,http://dlmf.nist.gov/28.28#E40, +\MathieuEigenvaluea{n}@@{q},eigenvalues of Mathieu equation $a_n(q)$,Mathieu-eigenvalue-a,F|F:P:nP,$a_{n}$,http://dlmf.nist.gov/28.2#SS5.p1, +\MathieuEigenvalueb{n}@@{q},eigenvalues of Mathieu equation $b_n(q)$,Mathieu-eigenvalue-b,F|F:P:nP,$b_{n}$,http://dlmf.nist.gov/28.2#SS5.p1, +\MathieuEigenvaluelambda{\nu+2n}@@{q},eigenvalues of Mathieu equation $\lambda_\nu(q)$,Mathieu-eigenvalue-lambda,F|F:P:nP,$\lambda_{\nu}$,http://dlmf.nist.gov/28.12#SS1.p2, +\MathieuFc{m}@{z}{h},Mathieu function ${\rm Fc}_m$,Mathieu-Fc,F|F:P,$\mathrm{Fc}_{m}$,http://dlmf.nist.gov/23.1, +\MathieuFe{n}@@{z}{q},modified Mathieu function ${\rm Fe}_n$,modified-Mathieu-Fe,F|F:P:nP,$\mathrm{Fe}_{n}$,http://dlmf.nist.gov/28.20#E6, +\MathieuFs{m}@{z}{h},Mathieu function ${\rm Fs}_m$,Mathieu-Fs,F|F:P,$\mathrm{Fs}_{m}$,http://dlmf.nist.gov/23.1, +\MathieuGc{m}@{z}{h},Mathieu function ${\rm Gc}_m$,Mathieu-Gc,F|F:P,$\mathrm{Gc}_{m}$,http://dlmf.nist.gov/23.1, +\MathieuGe{n}@@{z}{q},modified Mathieu function ${\rm Ge}_n$,modified-Mathieu-Ge,F|F:P:nP,$\mathrm{Ge}_{n}$,http://dlmf.nist.gov/28.20#E7, +\MathieuGs{m}@{z}{h},Mathieu function ${\rm Gs}_m$,Mathieu-Gs,F|F:P,$\mathrm{Gs}_{m}$,http://dlmf.nist.gov/23.1, +\MathieuIe{n}@@{z}{h},modified Mathieu function ${\rm Ie}_n$,modified-Mathieu-Ie,F|F:P:nP,$\mathrm{Ie}_{n}$,http://dlmf.nist.gov/28.20#E17, +\MathieuIo{n}@@{z}{h},modified Mathieu function ${\rm Io}_n$,modified-Mathieu-Io,F|F:P:nP,$\mathrm{Io}_{n}$,http://dlmf.nist.gov/28.20#E18, +\MathieuKe{n}@@{z}{h},modified Mathieu function ${\rm Ke}_n$,modified-Mathieu-Ke,F|F:P:nP,$\mathrm{Ke}_{n}$,http://dlmf.nist.gov/28.20#E19, +\MathieuKo{n}@@{z}{h},modified Mathieu function ${\rm Ko}_n$,modified-Mathieu-Ko,F|F:P:nP,$\mathrm{Ko}_{n}$,http://dlmf.nist.gov/28.20#E20, +\MathieuM{j}{\nu}@@{z}{h},modified Mathieu function ${\rm M}_\nu^{(j)}$,modified-Mathieu-M,F|F:P:nP,${\mathrm{M}^{(j)}_{\nu}}$,http://dlmf.nist.gov/23.1, +\MathieuMc{j}{n}@@{z}{h},radial Mathieu function ${\rm Mc}_n^{(j)}$,modified-Mathieu-Mc,F|F:P:nP,${\mathrm{Mc}^{(j)}_{\nu}}$,http://dlmf.nist.gov/28.20#E15, +\MathieuMe{\nu}@@{z}{q},modified Mathieu function ${\rm Me}_\nu$,modified-Mathieu-Me,F|F:P:nP,$\mathrm{Me}_{n}$,http://dlmf.nist.gov/28.20#E5, +\MathieuMs{j}{n}@@{z}{h},radial Mathieu function ${\rm Ms}_n^{(j)}$,modified-Mathieu-Ms,F|F:P:nP,${\mathrm{Ms}^{(j)}_{\nu}}$,http://dlmf.nist.gov/28.20#E16, +\MathieuSe{\nu}@@{z}{q},modified Mathieu function ${\rm Se}_\nu$,modified-Mathieu-Se,F|F:P:nP,$\mathrm{Se}_{n}$,http://dlmf.nist.gov/28.20#E4, +\Mathieuce{n}@@{z}{q},Mathieu function ${\rm ce}_\nu$,Mathieu-ce,F|F:P:nP,$\mathrm{ce}_{n}$,http://dlmf.nist.gov/23.1, +\Mathieufe{n}@@{z}{q},second solution of Mathieu's equation ${\rm fe}_n$,Mathieu-fe,F|F:P:nP,$\mathrm{fe}_{n}$,http://dlmf.nist.gov/28.5#E1, +\Mathieuge{n}@@{z}{q},second solution of Mathieu's equation ${\rm ge}_n$,Mathieu-ge,F|F:P:nP,$\mathrm{ge}_{n}$,http://dlmf.nist.gov/28.5#E2, +\Mathieume{n}@@{z}{q},Mathieu function ${\rm me}_\nu$,Mathieu-me,F|F:P:nP,$\mathrm{me}_{\nu}$,http://dlmf.nist.gov/23.1, +\Mathieuse{n}@@{z}{q},Mathieu function ${\rm se}_\nu$,Mathieu-se,F|F:P:nP,$\mathrm{se}_{\nu}$,http://dlmf.nist.gov/23.1, +\Matrix{I},Bessel function of matrix argument (first kind),EMPTY,F|F,$\Matrix{T}$,http://dlmf.nist.gov/front/introduction#Sx4.p2.t1.r9, +\MeijerG{m}{n}{p}{q}@@@{z}{\Vector{a}}{\Vector{b}},Meijer $G$-function,Meijer-G,F|F:fo1:fo2:fo3,"${G^{m,n}_{p,q}}$",http://dlmf.nist.gov/16.17#E1, +\Meixner{n}@{x}{\beta}{c},Meixner polynomial,Meixner-polynomial-M,P|F:P,$M_{n}$,http://dlmf.nist.gov/18.19#T1.t1.r9, +\MeixnerM{n}@{x}{\beta}{c},Meixner polynomial,Meixner-polynomial-M,P|F:P,$M_{n}$,http://dlmf.nist.gov/18.19#T1.t1.r9, +\MeixnerPollaczek{\lambda}{n}@{x}{\phi},Meixner-Pollaczek polynomial,Meixner-Pollaczek-polynomial-P,P|F:P,$P^{(\alpha)}_{n}$,http://dlmf.nist.gov/18.19#P3.p1, +\MeixnerPollaczekP{\lambda}{n}@{x}{\phi},Meixner-Pollaczek polynomial,Meixner-Pollaczek-polynomial-P,P|F:P,$P^{(\alpha)}_{n}$,http://dlmf.nist.gov/18.19#P3.p1, +\MellinTrans@{f}{s},Mellin transform,Mellin-transform,O|F:P,$\mathscr{M}$,http://dlmf.nist.gov/1.14#E32, +\Mills@{x},Mills ratio ${\sf M}$,Mills-ratio,F|F:P,$\mathsf{M}$,http://dlmf.nist.gov/7.8#E1, +\MittagLeffler{a}{b}@{z},Mittag-Leffler function,Mittag-Leffler-function,F|F:P,"$E_{a,b}$",http://dlmf.nist.gov/10.46#E3, +\ModCylinder{\nu}@{z},modified cylinder function,modified-cylinder-function,F|F:P,$\mathscr{Z}_{\nu}$,http://dlmf.nist.gov/10.25#P2.p1, +\ModularJ@{\tau},Klein's complete invariant,Kleins-invariant-modular-J,F|F:P,$J$,http://dlmf.nist.gov/23.15#E7, +\ModularLambda@{\tau},elliptic modular function,modular-Lambda,F|F:P,$\lambda$,http://dlmf.nist.gov/23.15#E6, +\MoebiusMu@{n},Möbius function,Moebius-mu,F|F:P,$\mu$,http://dlmf.nist.gov/27.2#E12, +\NatNumber,set of positive integers,EMPTY,SN|F,$\mathbb{N}$,http://dlmf.nist.gov/front/introduction#Sx4.p2.t1.r11, +\NeumannFactor{n},Neumann factor,Neumann-factor,I|F,$\epsilon_{m}$,http://drmf.wmflabs.org/wiki/Definition:NeumannFactor, +\NeumannPoly{n}@{x},Neumann's polynomial,Neumann-polynomial,P|F:P,$O_{k}$,http://dlmf.nist.gov/10.23#E12, +\NumPrimeDivNu@{n},$\nu(n)$ : number of distinct primes dividing $n$,number-of-primes-dividing-nu,I|F:P,$\nu$,http://dlmf.nist.gov/27.2#SS1.p1, +\NumPrimesLessPi@{x},number of primes not exceeding $x$,number-of-primes-less-pi,I|F:P,$\pi$,http://dlmf.nist.gov/27.2#E2, +\NumSquaresR{k}@{n},number of squares,number-of-squares-R,I|F:P,$r_{k}$,http://dlmf.nist.gov/27.13#SS4.p1, +\ODEgenAiryA{n}@{z},generalized Airy function from differential equation $A_n$,ODE-generalized-Airy-A,F|F:P,$A_{n}$,http://dlmf.nist.gov/9.13#SS1.p2, +\ODEgenAiryB{n}@{z},generalized Airy function from differential equation $B_n$,ODE-generalized-Airy-B,F|F:P,$B_{n}$,http://dlmf.nist.gov/9.13#SS1.p2, +\Pade{p}{q}{f},Padé approximant,Pade-approximant,O|F,${[p/q]_{f}}$,http://dlmf.nist.gov/3.11#SS4.p1, +\ParabolicU@{a}{z},parabolic cylinder function $U$,parabolic-U,F|F:P,$U$,http://dlmf.nist.gov/12.4.E1, +\ParabolicUbar@{a}{x},parabolic cylinder function $\overline{U}$,parabolic-U-bar,F|F:P,$\overline{U}$,http://dlmf.nist.gov/12.2.21, +\ParabolicV@{a}{z},parabolic cylinder function $V$,parabolic-V,F|F:P,$V$,http://dlmf.nist.gov/12.4.E2, +\ParabolicW@{a}{x},parabolic cylinder function $W$,parabolic-W,F|F:P,$W$,http://dlmf.nist.gov/12.14.i, +\PartitionsP[k]@{n},total number of partitions,partition-function,I|F:F:P:P,$p_{k}$,http://dlmf.nist.gov/26.2#P4.p2, +\PeriodicBernoulliB{n}@{x},periodic Bernoulli functions,periodic-Bernoulli-polynomial-B,F|F:P,$\widetilde{B}_{n}$,http://dlmf.nist.gov/24.2#iii, +\PeriodicEulerE{n}@{x},periodic Euler functions,periodic-Euler-polynomial-E,F|F:P,$\widetilde{E}_{n}$,http://dlmf.nist.gov/24.2#iii, +\PeriodicZeta@{x}{s},periodic zeta function,periodic-zeta,F|F:P,$F$,http://dlmf.nist.gov/25.13#E1, +\Permutations{n},"set of permutations of ${\bf n}$",set-of-permutations,SN|F,$\mathfrak{S}_{n}$,http://dlmf.nist.gov/23.1, +\PlanePartitionsPP@{n},number of plane partitions of $n$,plane-partitions-PP,I|F:P,$\mathit{pp}$,http://dlmf.nist.gov/26.12#SS1.p4, +\PollaczekP{\lambda}{n}@{x}{a}{b},Pollaczek polynomial,Pollaczek-polynomial-P,P|F:P,$P^{(\lambda)}_{n}$,http://dlmf.nist.gov/18.35#E4, +\Polylogarithm{s}@{z},polylogarithm,polylogarithm,F|F:P,$\mathrm{Li}_{s}$,http://dlmf.nist.gov/25.12#E10, +\Prod{n}{k1}{k2}@{f(n)},semantic product,EMPTY,SM|O:O,$\prod_{n=\alpha}^{\beta}$,http://drmf.wmflabs.org/wiki/Definition:Prod, +\Racah{n}@{x}{\alpha}{\beta}{\gamma}{\delta},Racah polynomial,Racah-polynomial-R,P|F:P,$R_{n}$,http://dlmf.nist.gov/18.25#T1.t1.r4, +\RacahR{n}@{x}{\alpha}{\beta}{\gamma}{\delta},Racah polynomial,Racah-polynomial-R,P|F:P,$R_{n}$,http://dlmf.nist.gov/18.25#T1.t1.r4, +\RamanujanSum{k}@{n},Ramanujan's sum,Ramanujan-sum,F|F:P,$c_{k}$,http://dlmf.nist.gov/27.10#E4, +\RamanujanTau@{n},Ramanujan's tau function,Ramanujan-tau,F|F:P,$\tau$,http://dlmf.nist.gov/27.14#E18, +\Rational,set of rational numbers,EMPTY,SN|F,$\mathbb{Q}$,http://dlmf.nist.gov/front/introduction#Sx4.p2.t1.r13, +\RayleighFun{n}@{\nu},Rayleigh function,Rayleigh-function,F|F:P,$\sigma_{n}$,http://dlmf.nist.gov/10.21#E55, +\Real,set of real numbers,EMPTY,SN|F,$\mathbb{R}$,http://dlmf.nist.gov/front/introduction#Sx4.p2.t1.r14, +\RepInterfc{n}@{z},repeated integrals of the complementary error function,repeated-integral-complementary-error-function,F|F:P,$\mathrm{i}^{n}\mathrm{erfc}$,http://dlmf.nist.gov/7.18#E2, +\Residue,residue,EMPTY,O|F,$\mathrm{res}$,http://dlmf.nist.gov/1.10#SS3.p5, +\RestrictedCompositionsC@{\mathrm{condition}}{n},restricted number of compositions,restricted-number-of-compositions-C,I|F:P,$c_{m}$,http://dlmf.nist.gov/26.11#p1, +\RestrictedPartitionsP[k]@{\mathrm{condition}}{n},restricted number of partitions,restricted-partitions-P,I|F:F:P:P,$p_{k}$,http://dlmf.nist.gov/26.10#i, +\RiemannP@{\begin{Bmatrix} \alpha & \beta &\gamma & \\ a_1 & b_1 & c_1 & z \\ a_2 & b_2 & c_2 & \end{Bmatrix}},Riemann-Papperitz $\RiemannP$-symbol for solutions of the generalized hypergeometric differential equation,Riemann-P-symbol,Q|F:P,$P$,http://dlmf.nist.gov/15.11#E3, +\RiemannTheta@{\Vector{z}}{\Matrix{\Omega}},Riemann theta function,Riemann-theta,F|F:P,$\theta$,http://dlmf.nist.gov/21.2#E1, +\RiemannThetaChar{\Vector{\alpha}}{\Vector{\beta}}@{\Vector{z}}{\Matrix{\Omega}},Riemann theta function with characteristics,Riemann-theta-characterstics,F|F:P,$\theta\left[\Vector{\alpha} \atop \Vector{\beta} \right]$,http://dlmf.nist.gov/21.2#E5, +\RiemannThetaHat@{\Vector{z}}{\Matrix{\Omega}},scaled Riemann theta function,Riemann-theta-hat,F|F:P,$\hat{\theta}$,http://dlmf.nist.gov/21.2#E2, +\RiemannXi@{s},Riemann's $\RiemannXi$-function,Riemann-xi,F|F:P,$\xi$,http://dlmf.nist.gov/25.4#E4, +\RiemannZeta@{s},Riemann zeta function,Riemann-zeta,F|F:P,$\zeta$,http://dlmf.nist.gov/25.2#E1, +\Schwarzian{z}{\zeta},Schwarzian derivative,EMPTY,O|F,"$\left\{f,g\right\}$",http://dlmf.nist.gov/1.13#E20, +\ScorerGi@{z},Scorer function (inhomogeneous Airy function) ${\rm Gi}$,Scorer-Gi,F|F:P,$\mathrm{Gi}$,http://dlmf.nist.gov/9.12#i, +\ScorerHi@{z},Scorer function (inhomogeneous Airy function) ${\rm Hi}$,Scorer-Hi,F|F:P,$\mathrm{Hi}$,http://dlmf.nist.gov/9.12#i, +\SinCosIntf@{z},auxiliary function for sine and cosine integrals ${\rm f}$,sine-cosine-integral-f,F|F:P,$\mathrm{f}$,http://dlmf.nist.gov/6.2#E17, +\SinCosIntg@{z},auxiliary function for sine and cosine integrals ${\rm g}$,sine-cosine-integral-g,F|F:P,$\mathrm{g}$,http://dlmf.nist.gov/6.2#E18, +\SinInt@{z},sine integral ${\rm Si}$,sine-integral,F|F:P,$\mathrm{Si}$,http://dlmf.nist.gov/6.2#E9, +\SinIntg@{a}{z},generalized sine integral ${\rm Si}$,generalized-sine-integral-Si,F|F:P,$\mathrm{Si}$,http://dlmf.nist.gov/8.21#E2, +\SinhInt@{z},hyperbolic sine integral,hyperbolic-sine-integral,F|F:P,$\mathrm{Shi}$,http://dlmf.nist.gov/6.2#E15, +\SphBesselIi{n}@{z},modified spherical Bessel function of the first kind ${\sf i}_n^{(1)}$,spherical-Bessel-I-1,F|F:P,${\mathsf{i}^{(1)}_{n}}$,http://dlmf.nist.gov/10.47#E7, +\SphBesselIii{n}@{z},modified spherical Bessel function of the first kind ${\sf i}_n^{(2)}$,spherical-Bessel-I-2,F|F:P,${\mathsf{i}^{(2)}_{n}}$,http://dlmf.nist.gov/10.47#E8, +\SphBesselJ{n}@{z},spherical Bessel function of the first kind ${\sf j}_n$,spherical-Bessel-J,F|F:P,$\mathsf{j}_{n}$,http://dlmf.nist.gov/10.47#E3, +\SphBesselK{n}@{z},modified spherical Bessel function of the second kind ${\sf k}_n$,spherical-Bessel-K,F|F:P,$\mathsf{k}_{n}$,http://dlmf.nist.gov/10.47#E9, +\SphBesselY{n}@{z},spherical Bessel function of the second kind ${\sf y}_n$,spherical-Bessel-Y,F|F:P,$\mathsf{y}_{n}$,http://dlmf.nist.gov/10.47#E4, +\SphHankelHi{n}@{z},spherical Hankel function of the first kind ${\sf h}_n^{(1)}$,spherical-Hankel-H-1-Bessel-third-kind,F|F:P,${\mathsf{h}^{(1)}_{n}}$,http://dlmf.nist.gov/10.47#E5, +\SphHankelHii{n}@{z},spherical Hankel function of the second kind ${\sf h}_n^{(2)}$,spherical-Hankel-H-2-Bessel-third-kind,F|F:P,${\mathsf{h}^{(2)}_{n}}$,http://dlmf.nist.gov/10.47#E6, +\SphericalHarmonicY{l}{m}@{\theta}{\phi},spherical harmonic,spherical-harmonic-Y,F|F:P,"$Y_{{\ell},{m}}$",http://dlmf.nist.gov/14.30#E1, +\SpheroidalEigenvalueLambda{m}{n}@{\gamma^2},eigenvalues of the spheroidal differential equation,spheroidal-eigenvalue-lambda,O|F:P,$\lambda^{m}_{n}$,http://dlmf.nist.gov/30.3#SS1.p1, +\SpheroidalOnCutPs{m}{n}@{x}{\gamma^2},spheroidal wave function of the first kind ${\sf Ps}_n^m$,spheroidal-wave-on-cut-Ps,F|F:P,$\mathsf{Ps}^{m}_{n}$,http://dlmf.nist.gov/30.4#i, +\SpheroidalOnCutQs{m}{n}@{x}{\gamma^2},spheroidal wave function of the second kind ${\sf Qs}_n^m$,spheroidal-wave-on-cut-Qs,F|F:P,$\mathsf{Qs}^{m}_{n}$,http://dlmf.nist.gov/30.5, +\SpheroidalPs{m}{n}@{z}{\gamma^2},spheroidal wave function of complex argument $Ps_n^m$,spheroidal-wave-Ps,F|F:P,$\mathit{Ps}^{m}_{n}$,http://dlmf.nist.gov/30.6#p1, +\SpheroidalQs{m}{n}@{z}{\gamma^2},spheroidal wave function of complex argument $Qs_n^m$,spheroidal-wave-Qs,F|F:P,$\mathit{Qs}^{m}_{n}$,http://dlmf.nist.gov/30.6#p1, +\SpheroidalRadialS{m}{j}{n}@{z}{\gamma},radial spheroidal wave function,radial-spheroidal-wave-S,F|F:P,$S^{m(j)}_{n}$,http://dlmf.nist.gov/30.11#E3, +\StieltjesConstants{n},Stieltjes constants,Stieltjes-constants,C|F,$\gamma_{n}$,http://drmf.wmflabs.org/wiki/Definition:StieltjesConstants, +\StieltjesTrans@{f}{s},Stieltjes transform,Stieltjes-transform,O|F:P,$\mathcal{S}$,http://dlmf.nist.gov/1.14#E47, +\StieltjesWigert{n}@@{x}{q},Stieltjes-Wigert polynomial,Stieltjes-Wigert-polynomial-S,P|F:P:PS,$S_{n}$,http://drmf.wmflabs.org/wiki/Definition:StieltjesWigert, +\StieltjesWigertS{n}@{x}{q},Stieltjes-Wigert polynomial,Stieltjes-Wigert-polynomial-S,P|F:P,$S_{n}$,http://dlmf.nist.gov/18.27#E18, +\StirlingS@{n}{k},Stirling number of the first kind,Stirling-number-first-kind-S,I|F:P,$s$,http://dlmf.nist.gov/26.8#SS1.p1, +\StirlingSS@{n}{k},Stirling number of the second kind,Stirling-number-second-kind-S,I|F:P,$S$,http://dlmf.nist.gov/26.8#SS1.p3, +\StruveH{\nu}@{z},Struve function ${\bf H}_{\nu}$,Struve-H,F|F:P,$\mathbf{H}_{\nu}$,http://dlmf.nist.gov/11.2#E1, +\StruveK{\nu}@{z},Struve function ${\bf K}_{\nu}$,associated-Struve-K,F|F:P,$\mathbf{K}_{\nu}$,http://dlmf.nist.gov/11.2#E5, +\StruveL{\nu}@{z},modified Struve function ${\bf L}_{\nu}$,modified-Struve-L,F|F:P,$\mathbf{L}_{\nu}$,http://dlmf.nist.gov/11.2#E2, +\StruveM{\nu}@{z},modified Struve function ${\bf M}_{\nu}$,associated-Struve-M,F|F:P,$\mathbf{M}_{\nu}$,http://dlmf.nist.gov/11.2#E6, +\Sum{n}{0}{k}@{f(n)},semantic sum,EMPTY,SM|F:P,$\sum_{n=k}^{N}$,http://drmf.wmflabs.org/wiki/Definition:Sum, +\SurfaceHarmonicY{l}{m}@{\theta}{\phi},surface harmonic of the first kind,surface-harmonic-Y,F|F:P,$Y_{l}^{m}$,http://dlmf.nist.gov/14.30#E2, +\TriangleOP{\alpha}{\beta}{\gamma}{m}{n}@{x}{y},triangle polynomial,triangle-orthogonal-polynomial-P,P|F:P,"$P^{\alpha,\beta,\gamma}_{m,n}$",http://dlmf.nist.gov/18.37#E7, +\Ultra{\lambda}{n}@{x},ultraspherical/Gegenbauer polynomial,ultraspherical-Gegenbauer-polynomial,P|F:P,$C^{\mu}_{n}$,http://dlmf.nist.gov/18.3#T1.t1.r5, +\Ultraspherical{\lambda}{n}@{x},ultraspherical (or Gegenbauer) polynomial,ultraspherical-Gegenbauer-polynomial,P|F:P,$C^{(\mu)}_{n}$,http://dlmf.nist.gov/18.3#T1.t1.r5, +\UmbilicCatE@{s}{t}{\Vector{x}},elliptic umbilic catastrophe,elliptic-umbilic-catastrophe,F|F:P,$\Phi^{(\mathrm{E})}$,http://dlmf.nist.gov/36.2#E2, +\UmbilicCatH@{s}{t}{\Vector{x}},hyperbolic umbilic catastrophe,hyperbolic-umbilic-catastrophe,F|F:P,$\Phi^{(\mathrm{H})}$,http://dlmf.nist.gov/36.2#E3, +\VariationalOp[a;b]@{f},total variation,variational-operator,F|FO:PO:FnO:PnO,"$\mathcal{V}_{a,b}$",http://dlmf.nist.gov/1.4#E33, +\VoigtU@{x}{t},Voigt function ${\sf U}$,Voigt-U,F|F:P,$\mathsf{U}$,http://dlmf.nist.gov/7.19#E1, +\VoigtV@{x}{t},Voigt function ${\sf V}$,Voigt-V,F|F:P,$\mathsf{V}$,http://dlmf.nist.gov/7.19#E2, +\WaringG@{k},Waring's function $G$,Waring-G,F|F:P,$G$,http://dlmf.nist.gov/27.13#SS3.p1, +\Waringg@{k},Waring's function $g$,Waring-g,F|F:P,$g$,http://dlmf.nist.gov/27.13#SS3.p1, +\WeberE{\nu}@{z},Weber function,Weber-E,F|F:P,$\mathbf{E}_{\nu}$,http://dlmf.nist.gov/11.10#E2, +\WeierPInv@@{z}{g_2}{g_3},Weierstrass $\WeierPInv$-function on invariants $\WeierPInv@{z}{g_2}{g_3}$,Weierstrass-P-on-invariants,F|F:P:nP,$\wp$,http://dlmf.nist.gov/23.1, +\WeierPLat@@{z}{\Lattice{L}},Weierstrass $\WeierPLat$-function on lattice $\WeierPLat@{z}{\Lattice{L}}$,Weierstrass-P-on-lattice,F|F:P:nP,$\wp$,http://dlmf.nist.gov/23.1, +\WeiersigmaInv@@{z}{g_2}{g_3},Weierstrass sigma function on invariants,Weierstrass-sigma-on-invariants,F|F:P:nP,$\sigma$,http://dlmf.nist.gov/23.2#E6, +\WeiersigmaLat@@{z}{\Lattice{L}},Weierstrass sigma function on lattice,Weierstrass-sigma-on-lattice,F|F:P:nP,$\sigma$,http://dlmf.nist.gov/23.2#E6, +\WeierzetaInv@@{z}{g_2}{g_3},Weierstrass zeta function on invariants,Weierstrass-zeta-on-invariants,F|F:P:nP,$\zeta$,http://dlmf.nist.gov/23.2#E5, +\WeierzetaLat@@{z}{\Lattice{L}},Weierstrass zeta function on lattice,Weierstrass-zeta-on-lattice,F|F:P:nP,$\zeta$,http://dlmf.nist.gov/23.2#E5, +\WhitD{\nu}@{z},parabolic cylinder function,Whittaker-D,F|F:P,$D_{n}$,http://dlmf.nist.gov/23.1, +\WhitM{\kappa}{\mu}@{z},Whittaker function $M_{\kappa;\mu}$,Whittaker-confluent-hypergeometric-M,F|F:P,"$M_{\kappa,\mu}$",http://dlmf.nist.gov/13.14#E2, +\WhitW{\kappa}{\mu}@{z},Whittaker function $W_{\kappa;\mu}$,Whittaker-confluent-hypergeometric-W,F|F:P,"$W_{\kappa,\mu}$",http://dlmf.nist.gov/13.14#E3, +\Wilson{n}@{x}{a}{b}{c}{d},Wilson polynomial,Wilson-polynomial-W,P|F:P,$W_{n}$,http://dlmf.nist.gov/18.25#T1.t1.r2, +\WilsonW{n}@{x}{a}{b}{c}{d},Wilson polynomial,Wilson-polynomial-W,P|F:P,$W_{n}$,http://dlmf.nist.gov/18.25#T1.t1.r2, +\Wronskian@{w_1(z);w_2(z)},Wronskian,Wronskian,F|F:P,$\mathscr{W}$,http://dlmf.nist.gov/1.13#E4, +\ZeroAiryAi{k},$k$\textsuperscript{th} zero of Airy $\AiryAi$,zero-Airy-Ai,F|F,$a_{k}$,http://dlmf.nist.gov/23.1, +\ZeroAiryAiPrime{k},$k$\textsuperscript{th} zero of Airy $\AiryAi'$,zero-derivative-Airy-Ai,F|F,$a'_{k}$,http://dlmf.nist.gov/23.1, +\ZeroAiryBi{k},$k$\textsuperscript{th} zero of Airy $\AiryBi$,zero-Airy-Bi,F|F,$b_{k}$,http://dlmf.nist.gov/23.1, +\ZeroAiryBiPrime{k},$k$\textsuperscript{th} zero of Airy $\AiryBi'$,zero-derivative-Airy-Bi,F|F,$b'_{k}$,http://dlmf.nist.gov/23.1, +\ZeroBesselJ{\nu}{m},zeros of the Bessel function $\BesselJ{\nu}@{x}$,zero-Bessel-J,F|F,"$j_{\nu,m}$",http://dlmf.nist.gov/23.1, +\ZeroBesselJPrime{\nu}{m},zeros of the Bessel function derivative $\BesselJ{\nu}'@{x}$,zero-derivative-Bessel-J,F|F,"${j'_{\nu,m}}$",http://dlmf.nist.gov/23.1, +\ZeroBesselY{\nu}{m},zeros of the Bessel function $\BesselY{\nu}@{x}$,zero-Bessel-Y-Weber,F|F,"$y_{\nu,m}$",http://dlmf.nist.gov/23.1, +\ZeroBesselYPrime{\nu}{m},zeros of the Bessel function derivative $\BesselY{\nu}'@{x}$,zero-derivative-Bessel-Y-Weber,F|F,"${y'_{\nu,m}}$",http://dlmf.nist.gov/23.1, +\ZonalPoly{\kappa}@{\Matrix{T}},zonal polynomial,zonal-polynomial,P|F:P,$Z_{\kappa}$,http://dlmf.nist.gov/35.4#SS1.p3, +\acos@@{z},inverse cosine function,inverse-cosine,F|F:P:nP,$\mathrm{arccos}$,http://dlmf.nist.gov/4.23#SS2.p1, +\acosh@@{z},inverse hyperbolic cosine function,hyperbolic-inverse-cosine,F|F:P:nP,$\mathrm{arccosh}$,http://dlmf.nist.gov/4.37#SS2.p1, +\acot@@{z},inverse cotangent function,inverse-cotangent,F|F:P:nP,$\mathrm{arccot}$,http://dlmf.nist.gov/4.23#E9, +\acoth@@{z},inverse hyperbolic cotangent function,hyperbolic-inverse-cotangent,F|F:P:nP,$\mathrm{arccoth}$,http://dlmf.nist.gov/4.37#E9, +\acsc@@{z},inverse cosecant function,inverse-cosecant,F|F:P:nP,$\mathrm{arccsc}$,http://dlmf.nist.gov/4.23#E7, +\acsch@@{z},inverse hyperbolic cosecant function,hyperbolic-inverse-cosecant,F|F:P:nP,$\mathrm{arccsch}$,http://dlmf.nist.gov/4.37#E7, +\arcGudermannian@@{x},inverse Gudermannian function,inverse-Gudermannian,F|F:P:nP,${\mathrm{gd}^{-1}}$,http://dlmf.nist.gov/4.23#E41, +\arcJacobicd@{x}{k},inverse Jacobian elliptic function ${\rm arccd}$,inverse-Jacobi-elliptic-cd,F|F:P,$\mathrm{arccd}$,http://dlmf.nist.gov/22.15#SS1.p1, +\arcJacobicn@{x}{k},inverse Jacobian elliptic function ${\rm arccn}$,inverse-Jacobi-elliptic-cn,F|F:P,$\mathrm{arccn}$,http://dlmf.nist.gov/22.15#SS1.p1, +\arcJacobics@{x}{k},inverse Jacobian elliptic function ${\rm arccs}$,inverse-Jacobi-elliptic-cs,F|F:P,$\mathrm{arccs}$,http://dlmf.nist.gov/22.15#SS1.p1, +\arcJacobidc@{x}{k},inverse Jacobian elliptic function ${\rm arcdc}$,inverse-Jacobi-elliptic-dc,F|F:P,$\mathrm{arcdc}$,http://dlmf.nist.gov/22.15#SS1.p1, +\arcJacobidn@{x}{k},inverse Jacobian elliptic function ${\rm arcdn}$,inverse-Jacobi-elliptic-dn,F|F:P,$\mathrm{arcdn}$,http://dlmf.nist.gov/22.15#SS1.p1, +\arcJacobids@{x}{k},inverse Jacobian elliptic function ${\rm arcds}$,inverse-Jacobi-elliptic-ds,F|F:P,$\mathrm{arcds}$,http://dlmf.nist.gov/22.15#SS1.p1, +\arcJacobinc@{x}{k},inverse Jacobian elliptic function ${\rm arcnc}$,inverse-Jacobi-elliptic-nc,F|F:P,$\mathrm{arcnc}$,http://dlmf.nist.gov/22.15#SS1.p1, +\arcJacobind@{x}{k},inverse Jacobian elliptic function ${\rm arcnd}$,inverse-Jacobi-elliptic-nd,F|F:P,$\mathrm{arcnd}$,http://dlmf.nist.gov/22.15#SS1.p1, +\arcJacobins@{x}{k},inverse Jacobian elliptic function ${\rm arcns}$,inverse-Jacobi-elliptic-ns,F|F:P,$\mathrm{arcns}$,http://dlmf.nist.gov/22.15#SS1.p1, +\arcJacobisc@{x}{k},inverse Jacobian elliptic function ${\rm arcsc}$,inverse-Jacobi-elliptic-sc,F|F:P,$\mathrm{arcsc}$,http://dlmf.nist.gov/22.15#SS1.p1, +\arcJacobisd@{x}{k},inverse Jacobian elliptic function ${\rm arcsd}$,inverse-Jacobi-elliptic-sd,F|F:P,$\mathrm{arcsd}$,http://dlmf.nist.gov/22.15#SS1.p1, +\arcJacobisn@{x}{k},inverse Jacobian elliptic function ${\rm arcsn}$,inverse-Jacobi-elliptic-sn,F|F:P,$\mathrm{arcsn}$,http://dlmf.nist.gov/22.15#SS1.p1, +\asec@@{z},inverse secant function,inverse-secant,F|F:P:nP,$\mathrm{arcsec}$,http://dlmf.nist.gov/4.23#E8, +\asech@@{z},inverse hyperbolic secant function,hyperbolic-inverse-secant,F|F:P:nP,$\mathrm{arcsech}$,http://dlmf.nist.gov/4.37#E8, +\asin@@{z},inverse sine function,inverse-sine,F|F:P:nP,$\mathrm{arcsin}$,http://dlmf.nist.gov/4.23#SS2.p1, +\asinh@@{z},inverse hyperbolic sine function,hyperbolic-inverse-sine,F|F:P:nP,$\mathrm{arcsinh}$,http://dlmf.nist.gov/4.37#SS2.p1, +\atan@@{z},inverse tangent function,inverse-tangent,F|F:P:nP,$\mathrm{arctan}$,http://dlmf.nist.gov/4.23#SS2.p1, +\atanh@@{z},inverse hyperbolic tangent function,hyperbolic-inverse-tangent,F|F:P:nP,$\mathrm{arctanh}$,http://dlmf.nist.gov/4.37#SS2.p1, +\bDiff[x],backward difference,EMPTY,O|F,"$\,\nabla_{x}$",http://dlmf.nist.gov/18.1#EGx2, +\bigqJacobi{n}@@{x}{a}{b}{c}{q},big $q$-Jacobi polynomial,big-q-Jacobi-polynomial-P,P|F:P:PS,$P_{n}$,http://drmf.wmflabs.org/wiki/Definition:bigqJacobi, +\bigqJacobiIVparam{n}@@{x}{a}{b}{c}{d}{q},big $q$-Jacobi polynomial with four parameters,big-q-Jacobi-polynomial-four-parameters-P,P|F:P,$P_{n}$,http://drmf.wmflabs.org/wiki/Definition:bigqJacobiIVparam, +\bigqLaguerre{n}@{x}{a}{b}{q},big $q$-Laguerre polynomial,EMPTY,P|F:P,$P_{n}$,http://drmf.wmflabs.org/wiki/Definition:bigqLaguerre, +\bigqLegendre{n}@{x}{c}{q},big $q$-Legendre polynomial,EMPTY,P|F:P,$P_{n}$,http://drmf.wmflabs.org/wiki/Definition:bigqLegendre, +\binom{n}{k},binomial coefficient,EMPTY,I|F,$\binom{n}{k}$,http://dlmf.nist.gov/1.2#E1,http://dlmf.nist.gov/26.3#SS1.p1 +\binomial{n}{k},binomial coefficient,EMPTY,I|F,$\binom{n}{k}$,http://dlmf.nist.gov/1.2#E1,http://dlmf.nist.gov/26.3#SS1.p1 +\cDiff[x],central difference,EMPTY,O|F,"$\,\delta_{x}$",http://dlmf.nist.gov/18.1#EGx3, +\card,number of elements of a finite set,EMPTY,SN|F,$\left|A\right|$,http://dlmf.nist.gov/26.1, +\cartprod,$G\cartprod H$: Cartesian product of groups $G$ and $H$,EMPTY,O|F,$\times$,http://dlmf.nist.gov/23.1, +\ceiling{x},ceiling,EMPTY,I|F,$\left\lceil a\right\rceil$,http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r17, +\conj{z},complex conjugate,EMPTY,O|F,$\overline{a}$,http://dlmf.nist.gov/1.9#E11, +\cos@@{z},cosine function,cos,F|F:P:nP,$\mathrm{cos}$,http://dlmf.nist.gov/4.14#E2, +\cosh@@{z},hyperbolic cosine function,hyperbolic-cosine,F|F:P:nP,$\mathrm{cosh}$,http://dlmf.nist.gov/4.28#E2, +\cosintg@{a}{z},generalized cosine integral ${\rm ci}$,generalized-cosine-integral-ci,F|F:P,$\mathrm{ci}$,http://dlmf.nist.gov/8.21#E1, +\cot@@{z},cotangent function,cotangent,F|F:P:nP,$\mathrm{cot}$,http://dlmf.nist.gov/4.14#E7, +\coth@@{z},hyperbolic cotangent function,hyperbolic-cotangent,F|F:P:nP,$\mathrm{coth}$,http://dlmf.nist.gov/4.28#E7, +\cpi,ratio of a circle's circumference to its diameter,ratio-of-a-circle's-circumference-to-its-diameter,C|F,$\pi$,http://dlmf.nist.gov/5.19.E4, +\csc@@{z},cosecant function,cosecant,F|F:P:nP,$\mathrm{csc}$,http://dlmf.nist.gov/4.14#E5, +\csch@@{z},hyperbolic cosecant function,hyperbolic-cosecant,F|F:P:nP,$\mathrm{csch}$,http://dlmf.nist.gov/4.28#E5, +\ctsHahn{n}@@{x}{a}{b}{c}{d},continuous Hahn polynomial,continuous-Hahn-polynomial,P|F:P:PS,$p_{n}$,http://dlmf.nist.gov/18.19#P2.p1, +\ctsbigqHermite{n}@{x}{a}{q},continuous big $q$-Hermite polynomial,EMPTY,P|F:P,$H_{n}$,http://drmf.wmflabs.org/wiki/Definition:ctsbigqHermite, +\ctsdualHahn{n}@@{x^2}{a}{b}{c},continuous dual Hahn polynomial,continuous-dual-Hahn-normalized-S,P|F:P:PS,$S_{n}$,http://dlmf.nist.gov/18.25#T1.t1.r3, +\ctsdualqHahn{n}@{x}{a}{b}{c}{q},continuous dual $q$-Hahn polynomial,EMPTY,P|F:P,$p_{n}$,http://drmf.wmflabs.org/wiki/Definition:ctsdualqHahn, +\ctsqHahn{n}@{x}{a}{b}{c}{d}{q},continuous $q$-Hahn polynomial,EMPTY,P|F:P,$p_{n}$,http://drmf.wmflabs.org/wiki/Definition:ctsqHahn, +\ctsqHermite{n}@@{x}{q},continuous $q$-Hermite polynomial,continuous-q-Hermite-polynomial-H,P|F:P:PS,$H_{n}$,http://drmf.wmflabs.org/wiki/Definition:ctsqHermite, +\ctsqJacobi{\alpha}{\beta}{m}@{x}{q},continuous $q$-Jacobi polynomial,EMPTY,P|F:P,"$P^{(\alpha,\beta)}_{n}$",http://drmf.wmflabs.org/wiki/Definition:ctsqJacobi, +\ctsqJacobiRahman{n}@@{x}{q},continuous $q$-Jacobi-Rahman-polynomial,continuous-q-Jacobi-Rahman-polynomial-P,P|F:P:PS,"$P^{(\alpha,\beta)}_{n}$",http://drmf.wmflabs.org/wiki/Definition:ctsqJacobiRahman, +\ctsqLaguerre{\alpha}{n}@{x}{q},continuous $q$-Laguerre polynomial,EMPTY,P|F:P,$P^{(n)}_{\alpha}$,http://drmf.wmflabs.org/wiki/Definition:ctsqLaguerre, +\ctsqLegendre{n}@@{x}{q},continuous $q$-Legendre polynomial,continuous-q-Legendre-polynomial-P,P|F:P:PS,$P_{n}$,http://drmf.wmflabs.org/wiki/Definition:ctsqLegendre, +\ctsqLegendreRahman{n}@@{x}{q},continuous $q$-Legendre-Rahman polynomial,continuous-q-Legendre-Rahman-polynomial-P,P|F:P:PS,$P_{n}$,http://drmf.wmflabs.org/wiki/Definition:ctsqLegendreRahman, +\ctsqUltra{n}@{x}{\beta}{q},continuous $q$-ultraspherical/Rogers polynomial,continuous-q-ultraspherical-Rogers-polynomial,P|F:P,$C_{n}$,http://dlmf.nist.gov/18.28#E13, +\ctsqUltrae{n}@{e^{i\theta}}{\beta}{q},continuous $q$-ultraspherical/Rogers polynomial exponential argument,continuous-q-ultraspherical-Rogers-polynomial-exponential-argument,P|F:P,$C_{n}$,http://drmf.wmflabs.org/wiki/Definition:ctsqUltrae, +\curl,curl of vector-valued function,EMPTY,F|F,$\nabla\times$,http://dlmf.nist.gov/1.6#E22, +\deltaDistribution,Dirac delta distribution,Dirac delta distribution,D|F,$\delta_x$,http://dlmf.nist.gov/1.17.i, +\deriv{f}{x},derivative,EMPTY,O|F,$\deriv{}{x}}$,http://dlmf.nist.gov/1.4#E4, +\det,determinant,EMPTY,L|F,$\det$,http://dlmf.nist.gov/1.3#i, +\diag,diagonal part of a matrix,EMPTY,L|F,$\diag[\Matrix{a}]$,http://dlmf.nist.gov/21.5.E7, +\diff{x},differential,EMPTY,O|F,$\mathrm{d}^nx$,http://dlmf.nist.gov/1.4#iv, +\digamma@{z},psi (or digamma) function,digamma,F|F:P,$\psi$,http://dlmf.nist.gov/5.2#E2, +\discrqHermiteI{n}@@{x}{q},discrete $q$-Hermite I polynomial,discrete-q-Hermite-polynomial-h-I,P|F:P:PS,$h_{n}$,http://drmf.wmflabs.org/wiki/Definition:discrqHermiteI, +\discrqHermiteII{n}@@{x}{q},discrete $q$-Hermite II polynomial,discrete-q-Hermite-polynomial-II-h-tilde,P|F:P:PS,$\tilde{h}_{n}$,http://drmf.wmflabs.org/wiki/Definition:discrqHermiteII, +\divergence,divergence of vector-valued function,EMPTY,F|F,$\mathrm{div}$,http://dlmf.nist.gov/1.6#E21, +\divides,divides,EMPTY,O|F,$\nabla\cdot$,http://dlmf.nist.gov/24.1, +\dualHahn{n}@@{\lambda(x)}{\gamma}{\delta}{N},dual Hahn polynomial,dual-Hahn-polynomial-R,P|F:P:PS,$R_{n}$,http://dlmf.nist.gov/18.25#T1.t1.r5, +\dualqHahn{n}@@{\mu(x)}{\gamma}{\delta}{N}{q},dual $q$-Hahn polynomial,dual-q-Hahn-R,P|F:P:PS,$R_{n}$,http://drmf.wmflabs.org/wiki/Definition:dualqHahn, +\dualqKrawtchouk{n}@@{\lambda(x)}{c}{N}{q},dual $q$-Krawtchouk polynomial,dual-q-Krawtchouk-polynomial-K,P|F:P:PS,$K_{n}$,http://drmf.wmflabs.org/wiki/Definition:dualqKrawtchouk, +\erf@@{z},error function,error-function,F|F:P:nP,$\mathrm{erf}$,http://dlmf.nist.gov/7.2#E1, +\erfc@@{z},complementary error function,repeated-integral-complementary-error-function,F|F:P:nP,$\mathrm{erfc}$,http://dlmf.nist.gov/7.2#E2, +\erfw@@{z},complementary error function $w$,error-function-w,F|F:P:nP,$w$,http://dlmf.nist.gov/7.2#E3, +\exp@@{z},exponential function,exponential,F|F:P:nP,$\mathrm{exp}$,http://dlmf.nist.gov/4.2#E19, +\expe,the base of the natural logarithm,EMPTY,F|F,$\mathrm{e}$,http://dlmf.nist.gov/4.2.E11, +\exptrace@{\Matrix{X}},exponential of trace of a matrix,exponential-trace,L|F:P,$\mathrm{etr}$,http://dlmf.nist.gov/35.1#p3.t1.r10, +\f{f}@{x},function,EMPTY,SM|F:P,${f}$,http://drmf.wmflabs.org/wiki/Definition:f, +\fDiff,forward difference operator,EMPTY,O|F,"$\,\Delta_{y}$",http://dlmf.nist.gov/3.6#SS1.p1, +\floor{x},floor,EMPTY,I|F,$\left\lfloor a\right\rfloor$,http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r16, +\forall,for every,EMPTY,Q|F,$\forall$,http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r11, +\gradient,gradient of differentiable scalar function,EMPTY,F|F,$\nabla$,http://dlmf.nist.gov/1.6#E20, +"\idem@{\chi_1}{\chi_1,...,\chi_n}",$\idem$ function,idem,F|F:P,$\mathrm{idem}$,http://dlmf.nist.gov/17.1#p3, +\ifrac,semantic slash fraction,EMPTY,SM|F,$(a)/(b)$,http://dlmf.nist.gov/4.2.E8, +\imagpart{z},imaginary part,EMPTY,O|F,"$\Im {z}$",http://dlmf.nist.gov/1.9#E2, +\in,element of,EMPTY,Q|F,$\in$,http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r9, +\incgamma@{a}{z},incomplete gamma function $\gamma$,incomplete-gamma,F|F:P,$\gamma$,http://dlmf.nist.gov/8.2#E1, +\incgammastar@{a}{z},incomplete gamma function $\gamma^{*}$,incomplete-gamma-star,F|F:P,$\gamma^{*}$,http://dlmf.nist.gov/8.2#E6, +\inf,greatest lower bound (infimum),EMPTY,Q|F,$\mathrm{inf}$,http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r24, +\int,integral,EMPTY,F|F,$\int$,http://dlmf.nist.gov/1.4#iv, +\inverf@@{x},inverse error function,inverse-error-function,F|F:P:nP,$\mathrm{inverf}$,http://dlmf.nist.gov/7.17#E1, +\inverfc@@{x},inverse complementary error function,inverse-complementary-error-function,F|F:P:nP,$\mathrm{inverfc}$,http://dlmf.nist.gov/7.17#E1, +\iunit,imaginary unit,EMPTY,C|F,$\mathrm{i}$,http://dlmf.nist.gov/1.9.i, +\liminf,least limit point,EMPTY,Q|F,$\mathrm{liminf}$,http://dlmf.nist.gov/front/introduction#Sx4.p2.t1.r4, +\littleo@{x},order less than,little-o,Q|F:P,$o$,http://dlmf.nist.gov/2.1#E2, +\littleqJacobi{n}@@{x}{a}{b}{q},little $q$-Jacobi polynomial,little-q-Jacobi-polynomial-p,P|F:P:PS,$p_{n}$,http://drmf.wmflabs.org/wiki/Definition:littleqJacobi, +\littleqLaguerre{n}@{x}{a}{q},little $q$-Laguerre / Wall polynomial,EMPTY,P|F:P,$p_{n}$,http://drmf.wmflabs.org/wiki/Definition:littleqLaguerre, +\littleqLegendre{m}@{x}{q},little $q$-Legendre polynomial,EMPTY,P|F:P,$p_{n}$,http://drmf.wmflabs.org/wiki/Definition:littleqLegendre, +\ln@@{z},principal branch of logarithm function,natural-logarithm,F|F:P:nP,$\mathrm{ln}$,http://dlmf.nist.gov/4.2#E2, +\log@@{z},principle branch of natural logarithm,logarithm,F|F:P:nP,$\mathrm{log}$,http://dlmf.nist.gov/4.2#E2, +\logb{a}@@{z},logarithm to general base,logarithm,F|F:P:nP,$\mathrm{log}_{b}$,http://dlmf.nist.gov/4.2#EGx1, +\lrselection,bracketed generalization of $\pm$,EMPTY,SM|F,$\left\{\begin{matrix}x\end{matrix}\right\}$,http://drmf.wmflabs.org/wiki/Definition:Lrselection, +\lselection,left bracketed generalization of $\pm$,EMPTY,SM|F,$\left\{\begin{matrix}x\end{matrix}\right.$,http://dlmf.nist.gov/28.4.E22, +\mEulerBeta{m}@{a}{b},multivariate beta function,multivariate-Euler-Beta,F|F:P,$\mathrm{B}_{m}$,http://dlmf.nist.gov/35.3#E3, +\mEulerGamma{m}@{a},multivariate gamma function,multivariate-Euler-Gamma,F|F:P,$\Gamma_{m}$,http://dlmf.nist.gov/35.3#i, +\mod,modulo,EMPTY,Q|F,$\mod$,http://dlmf.nist.gov/front/introduction#Sx4.p2.t1.r10, +\monicAffqKrawtchouk{n}@@{q^{-x}}{p}{N}{q},monic affine $q$-Krawtchouk polynomial,affine-q-Krawtchouk-polynomial-monic-p,P|F:P:PS,$\widehat{K}^{\mathrm{Aff}}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicAffqKrawtchouk, +\monicAlSalamCarlitzI{a}{n}@@{x}{q},monic Al-Salam-Carlitz I polynomial,q-Al-Salam-Carlitz-I-polynomial-monic-p,P|F:P:PS,${\widehat U}^{(\alpha)}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicAlSalamCarlitzI, +\monicAlSalamCarlitzII{a}{n}@@{x}{q},monic Al-Salam-Carlitz II polynomial,q-Al-Salam-Carlitz-II-polynomial-monic-p,P|F:P:PS,${\widehat V}^{(n)}_{\alpha}$,http://drmf.wmflabs.org/wiki/Definition:monicAlSalamCarlitzII, +\monicAlSalamChihara{n}@@{x}{a}{b}{q},monic Al-Salam-Chihara polynomial,Al-Salam-Chihara-polynomial-monic-p,P|F:P:PS,${\widehat Q}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicAlSalamChihara, +\monicAskeyWilson{n}@@{x}{a}{b}{c}{d}{q},monic Askey-Wilson polynomial,Askey-Wilson-polynomial-monic-p,P|F:P:PS,${\widehat p}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicAskeyWilson, +\monicBesselPoly{n}@@{x}{a},monic Bessel polynomial,Bessel-polynomial-monic-p,P|F:P:PS,${\widehat y}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicBesselPoly, +\monicCharlier{n}@@{x}{a},monic Charlier polynomial,Charlier-polynomial-monic-p,P|F:P:PS,${\widehat C}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicCharlier, +\monicChebyT{n}@@{x},monic Chebyshev polynomial of the first kind,Chebyshev-polynomial-first-kind-monic-p,P|F:P:PS,${\widehat T}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicChebyT, +\monicChebyU{n}@@{x},monic Chebyshev polynomial of the second kind,Chebyshev-polynomial-second-kind-monic-p,P|F:P:PS,${\widehat U}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicChebyU, +\monicHahn{n}@@{x}{\alpha}{\beta}{N},monic Hahn polynomial,Hahn-polynomial-monic-p,P|F:P:PS,${\widehat Q}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicHahn, +\monicHermite{n}@@{x},monic Hermite polynomial,Hermite-polynomial-monic,P|F:P:PS,${\widehat H}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicHermite, +\monicJacobi{n}@@{x},monic Jacobi polynomial,Jacobi-polynomial-monic-p,P|F:P:PS,"${\widehat P}^{(\alpha,\beta)}_{n}$",http://drmf.wmflabs.org/wiki/Definition:monicJacobi, +\monicKrawtchouk{n}@@{q^{-x}}{p}{N}{q},monic Krawtchouk polynomial,Krawtchouk-polynomial-monic-p,P|F:P:PS,${\widehat K}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicKrawtchouk, +\monicLaguerre{n}@@{x},monic Laguerre polynomial,generalized-Laguerre-polynomial-monic-p,P|F:P:PS,${\widehat L}^{\alpha}_n$,http://drmf.wmflabs.org/wiki/Definition:monicLaguerre, +\monicLegendrePoly{n}@@{x},monic Legendre polynomial,Legendre-spherical-polynomial-monic-p,P|F:P:PS,${\widehat P}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicLegendrePoly, +\monicMeixner{n}@@{x}{\beta}{c},monic Meixner polynomial,Meixner-polynomial-monic-p,P|F:P:PS,${\widehat M}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicMeixner, +\monicMeixnerPollaczek{\lambda}{n}@@{x}{\phi},monic Meixner-Pollaczek polynomial,Meixner-Pollaczek-polynomial-monic-p,P|F:P:PS,${\widehat P}^{(\alpha)}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicMeixnerPollaczek, +\monicRacah{n}@@{\lambda(x)}{\alpha}{\beta}{\gamma}{\delta},monic Racah polynomial,Racah-polynomial-monic-p,P|F:P:PS,${\widehat R}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicRacah, +\monicStieltjesWigert{n}@@{x}{q},monic Stieltjes-Wigert polynomial,Stieltjes-Wigert-polynomial-monic-p,P|F:P:PS,${\widehat S}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicStieltjesWigert, +\monicUltra{n}@@{x},monic ultraspherical/Gegenbauer polynomial,ultraspherical-Gegenbauer-polynomial-monic-p,P|F:P:PS,${\widehat C}^{\mu}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicUltra, +\monicWilson{n}@@{x^2}{a}{b}{c}{d},monic Wilson polynomial,Wilson-polynomial-monic-p,P|F:P:PS,${\widehat W}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicWilson, +\monicbigqJacobi{n}@@{x}{a}{b}{c}{q},monic big $q$-Jacobi polynomial,big-q-Jacobi-polynomial-monic-p,P|F:P:PS,${\widehat P}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicbigqJacobi, +\monicbigqLaguerre{n}@@{x}{a}{b}{q},monic big $q$-Laguerre polynomial,big-q-Laguerre-polynomial-monic-p,P|F:P:PS,${\widehat P}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicbigqLaguerre, +\monicbigqLegendre{n}@@{x}{c}{q},monic big $q$-Legendre polynomial,big-q-Legendre-polynomial-monic-p,P|F:P:PS,${\widehat P}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicbigqLegendre, +\monicctsHahn{n}@@{x}{a}{b}{c}{d},monic continuous Hahn polynomial,continuous-Hahn-polynomial-monic-p,P|F:P:PS,${\widehat p}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicctsHahn, +\monicctsbigqHermite{n}@@{x}{a}{q},monic continuous big $q$-Hermite polynomial,continuous-big-q-Hermite-polynomial-monic-p,P|F:P:PS,${\widehat H}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicctsbigqHermite, +\monicctsdualHahn{n}@@{x^2}{a}{b}{c}{d},monic continuous dual Hahn polynomial,continuous-dual-Hahn-polynomial-monic-p,P|F:P:PS,${\widehat S}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicctsdualHahn, +\monicctsdualqHahn{n}@@{x^2}{a}{b}{c},monic continuous dual $q$-Hahn polynomial,continuous-dual-q-Hahn-polynomial-monic-p,P|F:P:PS,${\widehat p}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicctsdualqHahn, +\monicctsqHahn{n}@@{x}{a}{b}{c}{d}{q},monic continuous $q$-Hahn polynomial,continuous-q-Hahn-polynomial-monic-p,P|F:P:PS,${\widehat p}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicctsqHahn, +\monicctsqHermite{n}@@{x}{q},monic continuous $q$-Hermite polynomial,continuous-q-Hermite-polynomial-monic-p,P|F:P:PS,${\widehat H}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicctsqHermite, +\monicctsqJacobi{n}@@{x}{q},monic continuous $q$-Jacobi polynomial,continuous-q-Jacobi-polynomial-monic-p,P|F:P:PS,"${\widehat P}^{(\alpha,\beta)}_{n}$",http://drmf.wmflabs.org/wiki/Definition:monicctsqJacobi, +\monicctsqLaguerre{n}@@{x}{q},monic continuous $q$-Laguerre polynomial,continuous-q-Laguerre-polynomial-monic-p,P|F:P:PS,${\widehat P}^{(\alpha)}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicctsqLaguerre, +\monicctsqLegendre{n}@@{x}{q},monic continuous $q$-Legendre polynomial,continuous-q-Legendre-polynomial-monic-p,P|F:P:PS,${\widehat P}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicctsqLegendre, +\monicctsqUltra{n}@@{x}{\beta}{q},monic continuous $q$-ultraspherical/Rogers polynomial,continuous-q-ultraspherical-Rogers-polynomial-monic-p,P|F:P:PS,${\widehat C}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicctsqUltra, +\monicdiscrqHermiteI{n}@@{x}{q},monic discrete $q$-Hermite I polynomial,discrete-q-Hermite-polynomial-I-monic-p,P|F:P:PS,${\widehat h}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicdiscrqHermiteI, +\monicdiscrqHermiteII{n}@@{x}{q},monic discrete $q$-Hermite II polynomial,discrete-q-Hermite-polynomial-II-monic-p,P|F:P:PS,${\widehat \tilde{p}_{n}}$,http://drmf.wmflabs.org/wiki/Definition:monicdiscrqHermiteII, +\monicdualHahn{n}@@{\lambda(x)}{\gamma}{\delta}{N},monic dual Hahn polynomial,dual-Hahn-polynomial-monic-p,P|F:P:PS,${\widehat R}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicdualHahn, +\monicdualqHahn{n}@@{\mu(x)}{\gamma}{\delta}{N}{q},monic dual $q$-Hahn polynomial,dual-q-Hahn-monic-p,P|F:P:PS,${\widehat R}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicdualqHahn, +\monicdualqKrawtchouk{n}@@{\lambda(x)}{c}{N}{q},monic dual $q$-Krawtchouk polynomial,dual-q-Krawtchouk-polynomial-monic-p,P|F:P:PS,${\widehat K}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicdualqKrawtchouk, +\moniclittleqJacobi{n}@@{x}{a}{b}{q},monic little $q$-Jacobi polynomial,little-q-Jacobi-polynomial-monic-p,P|F:P:PS,${\widehat p}_{n}$,http://drmf.wmflabs.org/wiki/Definition:moniclittleqJacobi, +\moniclittleqLaguerre{n}@@{x}{a}{q},monic little $q$-Laguerre/Wall polynomial,little-q-Laguerre-Wall-polynomial-monic-p,P|F:P:PS,${\widehat p}_{n}$,http://drmf.wmflabs.org/wiki/Definition:moniclittleqLaguerre, +\moniclittleqLegendre{n}@@{x}{q},monic little $q$-Legendre polynomial,little-q-Legendre-polynomial-monic-p,P|F:P:PS,${\widehat p}_{n}$,http://drmf.wmflabs.org/wiki/Definition:moniclittleqLegendre, +\monicpseudoJacobi{n}@@{x}{\nu}{N},monic pseudo-Jacobi polynomial,pseudo-Jacobi-polynomial-monic-p,P|F:P:PS,${\widehat P}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicpseudoJacobi, +\monicqBesselPoly{n}@@{x}{a}{q},monic $q$-Bessel polynomial,q-Bessel-polynomial-monic-p,P|F:P:PS,${\widehat y}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicqBessel, +\monicqCharlier{n}@@{q^{-x}}{a}{q},monic $q$-Charlier polynomial,q-Charlier-polynomial-monic-p,P|F:P:PS,${\widehat C}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicqCharlier, +\monicqHahn{n}@@{q^{-x}}{\alpha}{\beta}{N},monic $q$-Hahn polynomial,q-Hahn-polynomial-monic-p,P|F:P:PS,${\widehat Q}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicqHahn, +\monicqKrawtchouk{n}@@{q^{-x}}{p}{N}{q},monic $q$-Krawtchouk polynomial,q-Krawtchouk-polynomial-monic-p,P|F:P:PS,${\widehat K}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicqKrawtchouk, +\monicqLaguerre{n}@@{x}{q},monic $q$-Laguerre polynomial,q-Laguerre-polynomial-monic-p,P|F:P:PS,${\widehat L}^{(\alpha)}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicqLaguerre, +\monicqMeixner{n}@@{q^{-x}}{b}{c}{q},monic $q$-Meixner polynomial,q-Meixner-polynomial-monic-p,P|F:P:PS,${\widehat M}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicqMeixner, +\monicqMeixnerPollaczek{n}@@{x}{a}{q},monic $q$-Meixner-Pollaczek polynomial,q-Meixner-Pollaczek-polynomial-monic-p,P|F:P:PS,${\widehat P}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicqMeixnerPollaczek, +\monicqRacah{n}@@{\mu(x)}{\alpha}{\beta}{\gamma}{\delta}{q},monic $q$-Racah polynomial,q-Racah-polynomial-monic-R,P|F:P:PS,${\widehat R}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicqRacah, +\monicqinvAlSalamChihara{n}@{x}{a}{b}{q^{-1}},monic $q$-inverse Al-Salam-Chihara polynomial,q-inverse-AlSalam-Chihara-polynomial-monic-p,P|F:P,${\widehat Q}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicqinvAlSalamChihara, +\monicqtmqKrawtchouk{n}@@{q^{-x}}{p}{N}{q},monic quantum $q$-Krawtchouk polynomial,quantum-q-Krawtchouk-polynomial-monic-p,P|F:P:PS,${\widehat K^{\mathrm{qtm}}}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicqtmqKrawtchouk, +"\multinomial{n_1+n_2+\dots+n_k}{n_1,...,n_k}",multinomial coefficient,EMPTY,I|F,$\left(A;B\right)$,http://dlmf.nist.gov/26.4.E2, +\ninej@@{j_{11}}{j_{12}}{j_{13}}{j_{21}}{j_{22}}{j_{23}}{j_{31}}{j_{32}}{j_{33}},$\ninej$ symbol,EMPTY,F|F:P:nP,$$,http://dlmf.nist.gov/34.6#E1, +\normAskeyWilsonptilde{n}@@{x}{a}{b}{c}{d}{q},normalized Askey-Wilson polynomial ${\tilde p}$,Askey-Wilson-polynomial-normalized-p-tilde,P|F:P:PS,${\tilde p}_{n}$,http://drmf.wmflabs.org/wiki/Definition:normAskeyWilsonptilde, +\normJacobiR{\alpha}{\beta}{n}@{x},normalized Jacobi polynomial,Jacobi-polynomial-R,P|F:P,"$R^{(\alpha,\beta)}_{n}$",http://drmf.wmflabs.org/wiki/Definition:normJacobiR, +\normWilsonWtilde{n}@@{x^2}{a}{b}{c}{d},normalized Wilson polynomial ${\tilde W}$,Wilson-polynomial-normalized-W-tilde,P|F:P:PS,${\tilde W}_{n}$,http://drmf.wmflabs.org/wiki/Definition:normWilsonWtilde, +\normctsHahnptilde{n}@@{x}{a}{b}{c}{d},normalized continuous Hahn polynomial ${\tilde p}$,continuous-Hahn-polynomial-normalized-p-tilde,P|F:P:PS,${\tilde p}_{n}$,http://drmf.wmflabs.org/wiki/Definition:normctsHahnptilde, +\normctsdualHahnStilde{n}@@{x^2}{a}{b}{c}{d},normalized continuous dual Hahn polynomial ${\tilde S}$,continuous-dual-Hahn-polynomial-normalized-S-tilde,P|P:PS,${\tilde S}_{n}$,http://drmf.wmflabs.org/wiki/Definition:normctsdualHahnStilde, +\normctsdualqHahnptilde{n}@@{x}{a}{b}{c}{q},normalized continuous dual $q$-Hahn polynomial ${\tilde p}$,continuous-dual-q-Hahn-polynomial-normalized-p-tilde,P|F:P:PS,${\tilde p}_{n}$,http://drmf.wmflabs.org/wiki/Definition:normctsdualqHahnptilde, +\normctsqHahnptilde{n}@@{x}{a}{b}{c}{d}{q},normalized continuous $q$-Hahn polynomial ${\tilde p}$,continuous-q-Hahn-polynomial-normalized-p-tilde,P|F:P:PS,${\tilde p}_{n}$,http://drmf.wmflabs.org/wiki/Definition:normctsqHahnptilde, +\normctsqJacobiPtilde{n}@@{x}{q},normalized continuous $q$-Jacobi polynomial ${\tilde P}$,continuous-q-Jacobi-polynomial-P-tilde,P|F:P:PS,"${\tilde P}^{(\alpha,\beta)}_{n}$",http://drmf.wmflabs.org/wiki/Definition:normctsqJacobiPtilde, +\notin,not an element of,EMPTY,Q|F,$\notin$,http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r10, +\opminus,negative unity to an integer power,EMPTY,I|F,$(-1)$,http://dlmf.nist.gov/5.7.E7, +\pderiv{f}{x},partial derivative,EMPTY,O|F,$\frac{\partial f}{\partial x}$,http://dlmf.nist.gov/1.5#E3, +\pdiff{x},power of a partial differential,EMPTY,O|F,"$\partial^nx$",http://dlmf.nist.gov/1.5#E3, +\pgcd,greatest common divisor,EMPTY,I|F,"$\left(m,n\right)$",http://dlmf.nist.gov/27.1, +\ph@@{z},phase,phase,O|F:P:nP,$\mathrm{ph}$,http://dlmf.nist.gov/1.9#E7, +\pochhammer{a}{n},Pochhammer symbol,EMPTY,F|F,$(a)_n$,http://dlmf.nist.gov/5.2#iii, +\poly{p}{n}@{x},polynomial,EMPTY,SM|F:P,${p}_{n}$,http://drmf.wmflabs.org/wiki/Definition:poly, +\polygamma{n}@{z},polygamma functions,polygamma,F|F:P,$\psi^{(n)}$,http://dlmf.nist.gov/5.15, +\power,power function,EMPTY,F|F,$x^y$,http://dlmf.nist.gov/36.12.E9, +\prod,product,EMPTY,SM|F,$\Pi$,http://drmf.wmflabs.org/wiki/Definition:prod, +\pseudoJacobi{n}@{x}{\nu}{N},pseudo Jacobi polynomial,EMPTY,P|F:P,$P_{n}$,http://drmf.wmflabs.org/wiki/Definition:pseudoJacobi, +\psfactorial{a}{\kappa},partitional shifted factorial,EMPTY,F|F,${\left[a\right]_{\kappa}}$,http://dlmf.nist.gov/35.4#E1, +\pvint,Cauchy principal value,EMPTY,O|F,$\pvint_a^b$,http://dlmf.nist.gov/1.4#E24, +\qAppelli@{a}{b}{b'}{c}{q}{x}{y},first $q$-Appell function,q-Appell-Phi-1,F|F:P,$\Phi^{(1)}$,http://dlmf.nist.gov/17.4.E5, +\qAppellii@{a}{b}{b'}{c}{c'}{q}{x}{y},second $q$-Appell function,q-Appell-Phi-2,F|F:P,$\Phi^{(2)}$,http://dlmf.nist.gov/17.4.E6, +\qAppelliii@{a}{a'}{b}{b'}{c}{q}{x}{y},third $q$-Appell function,q-Appell-Phi-3,F|F:P,$\Phi^{(3)}$,http://dlmf.nist.gov/17.4.E7, +\qAppelliv@{a}{b}{c}{c'}{q}{x}{y},fourth $q$-Appell function,q-Appell-Phi-4,F|F:P,$\Phi^{(4)}$,http://dlmf.nist.gov/17.4.E8, +\qBernoulli{n}@{x}{q},$q$-Bernoulli polynomial,q-Bernoulli-polynomial,P|F:P,$\beta_{q}$,http://dlmf.nist.gov/17.3#E7, +\qBesselPoly{n}@{x}{b}{q},$q$-Bessel polynomial,EMPTY,P|F:P,$y_{n}$,http://drmf.wmflabs.org/wiki/Definition:qBessel, +\qBeta{q}@{a}{b},$q$-beta function,q-Beta,F|F:P,$\mathrm{B}_{q}$,http://dlmf.nist.gov/5.18#E11, +\qBinomial{n}{m}{q},$q$-binomial coefficient (or Gaussian polynomial),EMPTY,F|F,$\left[n \atop m\right]_{q}$,http://dlmf.nist.gov/17.2#E27,http://dlmf.nist.gov/26.9#SS2.p1 +\qCharlier{n}@{x}{c}{q},$q$-Charlier polynomial,EMPTY,P|F:P,$C_{n}$,http://drmf.wmflabs.org/wiki/Definition:qCharlier, +\qCos{q}@@{x},$q$-analogue of the $\cos$ function: ${\rm Cos}_q$,q-Cos,F|F:nP:P,$\mathrm{Cos}_{q}$,http://dlmf.nist.gov/17.3#E6, +\qCosKLS{q}@@{z},$q$-analogue of the $\cos$ function used in KLS: $\mathrm{Cos}_{q}$,q-Cos-KLS,P|F:nP:P,$\mathrm{Cos}_{q}$,http://drmf.wmflabs.org/wiki/Definition:qCosKLS, +\qDigamma{q}@{z},$q$-digamma function,q-digamma,F|F:P,$\psi_{q}$,http://drmf.wmflabs.org/wiki/Definition:qDigamma, +\qEuler{m}{s}@{q},$q$-Euler number,q-Euler-number,F|F:P,"$A_{m,n}$",http://dlmf.nist.gov/17.3#E8, +\qExp{q}@@{x},$q$-analogue of the $\exp$ function: $E_q$,q-Exp,F|F:nP:P,$E_{q}$,http://dlmf.nist.gov/17.3#E2, +\qExpKLS{q}@@{z},$q$-analogue of the $\exp$ function used in KLS: $\mathrm{E}_{q}$,q-Exp-KLS,F|F:nP:P,$\mathrm{E}_{q}$,http://drmf.wmflabs.org/wiki/Definition:qExpKLS, +\qFactorial{n}{q},$q$-factorial,q-factorial,F|F,$n!_q$,http://dlmf.nist.gov/5.18#E2, +\qGamma{q}@{z},$q$-gamma function,q-gamma,F|F:P,$\Gamma_{q}$,http://dlmf.nist.gov/5.18#E4, +\qHahn{n}@@{q^{-x}}{\alpha}{\beta}{N},$q$-Hahn polynomial,q-Hahn-polynomial-Q,P|P:PS,$Q_{n}$,http://drmf.wmflabs.org/wiki/Definition:qHahn, +\qHahnQ{n}@{x}{\alpha}{\beta}{N}{q},$q$-Hahn polynomial,q-Hahn-polynomial-Q,P|F:P,$Q_{n}$,http://dlmf.nist.gov/18.27#E3, +\qHermiteH{n}@{x}{q},continuous $q$-Hermite polynomial,continuous-q-Hermite-polynomial-H,P|F:P,$H_{n}$,http://dlmf.nist.gov/18.28#E16, +\qHermitehI{n}@{x}{q},discrete $q$-Hermite I polynomial,discrete-q-Hermite-polynomial-h-I,P|F:P,$h_{n}$,http://dlmf.nist.gov/18.27#E21, +\qHermitehII{n}@{x}{q},discrete $q$-Hermite II polynomial,discrete-q-Hermite-polynomial-h-II,P|F:P,$\tilde{h}_{n}$,http://dlmf.nist.gov/18.27#E23, +"\qHyperrWs{r}{r+1}@@@{a_1}{a_4,a_5,...,a_{r+1}}{q}{z}",very-well-poised basic hypergeometric (or $q$-hypergeometric) function,very-well-poised-q-hypergeometric-rWs,F|F:fo1:fo2:fo3,${{}_{r+1}W_{r}}$,http://drmf.wmflabs.org/wiki/Definition:qHyperrWs, +"\qHyperrphis{r}{s}@@@{a_1,...a_r}{b_1,...,b_s}{q}{z}",basic hypergeometric (or $q$-hypergeometric) function,q-hypergeometric-rphis,F|F:fo1:fo2:fo3,${{}_{r}\phi_{s}}$,http://dlmf.nist.gov/17.4#E1, +"\qHyperrpsis{r}{s}@@@{a_1,...,a_r}{b_1,...,b_s}{q}{z}",bilateral basic hypergeometric (or bilateral $q$-hypergeometric) function,q-hypergeometric-rpsis,F|F:fo1:fo2:fo3,${{}_{r}\psi_{s}}$,http://dlmf.nist.gov/17.4#E3, +\qJacobiP{n}@{x}{a}{b}{c}{q},"big $q$-Jacobi polynomial $P_n(x;a,b,c;q)$",q-Jacobi-polynomial-P,P|F:P,$P_{n}$,http://dlmf.nist.gov/18.27#E5, +\qJacobiPP{\alpha}{\beta}{n}@{x}{c}{d}{q},"big $q$-Jacobi polynomial $P_n^{(\alpha,\beta)}(x;c,d;q)$",big-q-Jacobi-polynomial-P-type-2,P|F:P,"$P^{(\alpha,\beta)}_{n}$",http://dlmf.nist.gov/18.27#E6, +\qJacobip{n}@{x}{a}{b}{q},little $q$-Jacobi polynomial,q-Jacobi-polynomial-p,P|F:P,$p_{n}$,http://dlmf.nist.gov/18.27#E13, +\qKrawtchouk{n}@@{q^{-x}}{p}{N}{q},$q$-Krawtchouk polynomial,q-Krawtchouk-polynomial-K,P|F:P:PS,$K_{n}$,http://drmf.wmflabs.org/wiki/Definition:qKrawtchouk, +\qLaguerre[\alpha]{n}@@{x}{q},$q$-Laguerre polynomial,q-Laguerre-polynomial-L,P|FnO:PnO:nPnO:F:P:PS,$L_n^{(\alpha)}$,http://drmf.wmflabs.org/wiki/Definition:qLaguerre, +\qLaguerreL[\alpha]{n}@{x}{q},$q$-Laguerre polynomial,q-Laguerre-polynomial-L,P|FnO:PnO:FO:PO,$L_n^{(\alpha)}$,http://dlmf.nist.gov/18.27#E15, +\qLegendre{n}@{x}{q},continuous $q$-Legendre polynomial,EMPTY,P|F:P,$\qLegendre{n}$,http://drmf.wmflabs.org/wiki/Definition:qLegendre, +\qMeixner{n}@{x}{b}{c}{q},$q$-Meixner polynomial,q-Meixner-polynomial-M,P|F:P,$M_{n}$,http://drmf.wmflabs.org/wiki/Definition:qMeixner, +\qMeixnerPollaczek{n}@{x}{a}{q},$q$-Meixner-Pollaczek polynomial,q-Meixner-Pollaczek-polynomial-M,P|F:P,$P_{n}$,http://drmf.wmflabs.org/wiki/Definition:qMeixnerPollaczek, +"\qMultiPochhammer{a_1,\ldots,a_n}{q}{n}",$q$-multi-Pochhammer symbol,q-multi-Pochhammer,F|F,"$\(a_1,\ldots,a_k;q)_n$",http://dlmf.nist.gov/17.2.E5, +"\qMultinomial{a_1+\ldots+a_n}{a_1,\ldots,a_n}{q}",$q$-multinomial,q-multinomial,F|F,"$\left[a_1+\ldots+a_n \atop a_1,\ldots,a_n\right]_{q}$",http://dlmf.nist.gov/26.16.E1, +\qPochhammer{a}{q}{n},$q$-Pochhammer symbol,EMPTY,F|F,$(a;q)_n$,http://dlmf.nist.gov/5.18#i,http://dlmf.nist.gov/17.2#SS1.p1 +\qPolygamma{n}{q}@{z},$q$-polygamma function,q-polygamma,F|F:P,$\psi_{q}^{(n)}$,http://drmf.wmflabs.org/wiki/Definition:qPolygamma, +\qRacah{n}@{x}{\alpha}{\beta}{\gamma}{\delta}{q},$q$-Racah polynomial,q-Racah-polynomial-R,P|F:P,$R_{n}$,http://dlmf.nist.gov/18.28#E19, +\qRacahR{n}@{x}{\alpha}{\beta}{\gamma}{\delta}{q},$q$-Racah polynomial,q-Racah-polynomial-R,P|F:P,$R_{n}$,http://dlmf.nist.gov/18.28#E19, +\qSin{q}@@{x},$q$-analogue of the $\sin$ function: ${\rm Sin}_q$,q-Sin,F|F:nP:P,$\mathrm{Sin}_{q}$,http://dlmf.nist.gov/17.3#E4, +\qSinKLS{q}@@{z},$q$-analogue of the $\sin$ function used in KLS: $\mathrm{Sin}_q$,q-Sin-KLS,F|F:nP:P,$\mathrm{Sin}_{q}$,http://drmf.wmflabs.org/wiki/Definition:qSinKLS, +\qStirling{m}{s}@{q},$q$-Stirling number,q-Stirling-number,F|F:P,"$a_{m,s}$",http://dlmf.nist.gov/17.3#E9, +\qUltraspherical{n}@{x}{\beta}{q},continuous $q$-ultraspherical polynomial,q-ultraspherical-polynomial,P|F:P,$C_{n}$,http://dlmf.nist.gov/18.28#E13, +\qcos{q}@@{x},$q$-analogue of the $\cos$ function: $\cos_q$,q-cos,F|F:nP:P,$\mathrm{cos}_{q}$,http://dlmf.nist.gov/17.3#E5, +\qcosKLS{q}@@{z},$q$-analogue of the $\cos$ function used in KLS: $\cos_q$,q-cos-KLS,P|F:nP:P,$\mathrm{cos}_{q}$,http://drmf.wmflabs.org/wiki/Definition:qcosKLS, +\qderiv[n]{q}@{z},$q$-derivative,q-derivative,O|FO:PO:FnO:PnO,$\mathcal{D}_q^n$,http://drmf.wmflabs.org/wiki/Definition:qderiv, +\qdiff{q}{x},$q$-differential,EMPTY,O|F,"${\mathrm d}^n_qx$",http://dlmf.nist.gov/17.2#SS5.p1, +\qexp{q}@@{x},$q$-analogue of the $\exp$ function: $e_q$,q-exp,F|F:nP:P,$e_{q}$,http://dlmf.nist.gov/17.3#E1, +\qexpKLS{q}@@{z},$q$-analogue of the $\exp$ function used in KLS: $\mathrm{e}_{q}$,q-exp-KLS,F|F:nP:P,$\mathrm{e}_{q}$,http://drmf.wmflabs.org/wiki/Definition:qexpKLS, +\qinvAlSalamChihara{n}@{x}{a}{b}{q^{-1}},$q$-inverse Al-Salam-Chihara polynomial,q-inverse-AlSalam-Chihara-polynomial-Q,P|F:P,$Q_{n}$,http://dlmf.nist.gov/23.1, +\qinvHermiteh{n}@{x}{q},continuous $q$-inverse Hermite polynomial,continuous-q-inverse-Hermite-polynomial-h,P|F:P,$h_{n}$,http://dlmf.nist.gov/23.1, +\qsin{q}@@{x},$q$-analogue of the $\sin$ function: $\sin_q$,q-sin,F|F:nP:P,$\mathrm{sin}_{q}$,http://dlmf.nist.gov/17.3#E3, +\qsinKLS{q}@@{z},$q$-analogue of the $\sin$ function used in KLS: $\sin_q$,q-sin-KLS,F|F:nP:P,$\mathrm{sin}_{q}$,http://drmf.wmflabs.org/wiki/Definition:qsinKLS, +\qtmqKrawtchouk{n}@@{q^{-x}}{p}{N}{q},quantum $q$-Krawtchouk polynomial,quantum-q-Krawtchouk-polynomial-K,P|F:P:PS,$K^{\mathrm{qtm}}_{n}$,http://drmf.wmflabs.org/wiki/Definition:qtmqKrawtchouk, +\realpart{z},real part,EMPTY,O|F,"$\Re {z}$",http://dlmf.nist.gov/1.9#E2, +\terminant{p}@{z},terminant function,terminant-function,F|F:P,$F_{p}$,http://dlmf.nist.gov/2.11#E11, +\rescaledTerminant{p}@{z},rescaled terminant functions,rescaled-terminant-function,F|F:P,$G_{p}$,http://dlmf.nist.gov/9.7#SS5.p1, +\rselection,right bracketed generalization of $\pm$,EMPTY,SM|F,$\left.\begin{matrix}x\end{matrix}\right\$,http://dlmf.nist.gov/28.4.E22, +\sec@@{z},secant function,secant,F|F:P:nP,$\mathrm{sec}$,http://dlmf.nist.gov/4.14#E6, +\sech@@{z},hyperbolic secant function,hyperbolic-secant,F|F:P:nP,$\mathrm{sech}$,http://dlmf.nist.gov/4.28#E6, +\selection,generalization of $\pm$,EMPTY,SM|F,$\;\begin{matrix}x\end{matrix}\;$,http://dlmf.nist.gov/10.23.E7, +\setminus,set subtraction,EMPTY,Q|F,$\setminus$,http://dlmf.nist.gov/front/introduction#Sx4.p2.t1.r19, +\setmod,$S_1 \setmod S_2$: set of all elements of $S_1$ modulo elements of $S_2$,EMPTY,SN|F,$/$,http://dlmf.nist.gov/21.1#p2.t1.r17, +\sign@@{x},sign function,sign,F|F:P:nP,$\mathrm{sign}$,http://dlmf.nist.gov/front/introduction#Sx4.p2.t1.r18, +\sim,asymptotic equality,EMPTY,Q|F,$\tilde$,http://dlmf.nist.gov/2.1#E1, +\sin@@{z},sine function,sin,F|F:P:nP,$\mathrm{sin}$,http://dlmf.nist.gov/4.14#E1, +\sinInt@{z},sine integral ${\rm si}$,shifted-sine-integral,F|F:P,$\mathrm{si}$,http://dlmf.nist.gov/6.2#E10, +\sinh@@{z},hyperbolic sine function,hyperbolic-sine,F|F:P:nP,$\mathrm{sinh}$,http://dlmf.nist.gov/4.28#E1, +\sinintg@{a}{z},generalized sine integral ${\rm si}$,generalized-sine-integral-si,F|F:P,$\mathrm{si}$,http://dlmf.nist.gov/8.21#E1, +\sixj@@{j_{1}}{j_{2}}{j_{3}}{l_{1}}{l_{2}}{l_{3}},$\sixj$ symbol,EMPTY,F|F:P:nP,$\left\{\begin{array}{ccc}j_1 & j_2 & j_3\\l_1 & l_2 & l_3\end{array}\right\}$,http://dlmf.nist.gov/23.1, +\subset,is contained in,EMPTY,Q|F,$\subset$,http://dlmf.nist.gov/front/introduction#Sx4.p2.t1.r2, +\subseteq,is in or is contained in,EMPTY,Q|F,$\subseteq$,http://dlmf.nist.gov/front/introduction#Sx4.p2.t1.r3, +\sum,sum,EMPTY,SM|F,$\Sigma$,http://drmf.wmflabs.org/wiki/Definition:sum, +\sup,least upper bound (supremum),EMPTY,Q|F,$\mathrm{sup}$,http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r25, +\tan@@{z},tangent function,tangent,F|F:P:nP,$\mathrm{tan}$,http://dlmf.nist.gov/4.14#E4, +\tanh@@{z},hyperbolic tangent function,hyperbolic-tangent,F|F:P:nP,$\mathrm{tanh}$,http://dlmf.nist.gov/4.28#E4, +\threej@@{j_1}{j_2}{j_3}{m_1}{m_2}{m_3},$\threej$ symbol,EMPTY,F|F:P:nP,$\left(\begin{array}{ccc}j_1 & j_2 & j_3\\m_1 & m_2 & m_3\end{array}\right)$,http://dlmf.nist.gov/34.2#E4, +\trace,trace of matrix,trace,L|F,$\mathrm{etr}$,http://dlmf.nist.gov/23.1, +\transpose,transpose of a matrix,EMPTY,L|F,$A^{\mathrm{T}}$,http://dlmf.nist.gov/1.3.E5, +\weight{w}@{x},weight function,EMPTY,SM|F:P,${w}$,http://drmf.wmflabs.org/wiki/Definition:weight, +\wigner{j_1}{m_1}{j_2}{m_2}{j_3}{-m_3},Clebsch-Gordan coefficient,EMPTY,F|F,"$\left(j_{1}\;m_{1}\;j_{2}\;m_{2} | j_{1}\;j_{2}\; j_{3}\,\,-m_{3}\right)$",http://dlmf.nist.gov/34.1#p4, +\BesselPolyIIparam{n}@{x}{a}{b},Bessel polynomial with two parameters $y_n$,Bessel-polynomial-two-parameters-y,P|F:P,$y_{n}$,http://drmf.wmflabs.org/wiki/Definition:BesselPolyIIparam, +\BesselPolyTheta{n}@{x}{a}{b},Bessel polynomial with two parameters $\theta_n$,Bessel-polynomial-two-parameters-theta,P|F:P,$\theta_{n}$,http://drmf.wmflabs.org/wiki/Definition:BesselPolyTheta, +\FourierTrans@{f}{s},Fourier transform,Fourier-transform,O|F:P,$\mathscr{F}$,http://dlmf.nist.gov/1.14#E1, +\Continuous@{A},space of continuous functions on a set $A$,set-of-continuous-functions,S|F,$C(A)$,http://dlmf.nist.gov/1.4#ii, +\SpaceTestFunTempered@{A},space of test functions for tempered distributions,space-of-test-functions-for-tempered-distributions-T,S|F,${\mathcal T}(\Real)$,http://dlmf.nist.gov/1.16#v, diff --git a/Pipeline/src/pipeline.py b/Pipeline/src/pipeline.py new file mode 100644 index 0000000..37c5cc7 --- /dev/null +++ b/Pipeline/src/pipeline.py @@ -0,0 +1,31 @@ +import glob +import sys + +sys.path.append('../../AlexDanoff/src') +from utilities import readin as readin +from utilities import writeout +from remove_excess import remove_excess as step1 +from replace_special import remove_special as step2 +from prepare_annotations import prepare_annotations as step3 + +sys.path.append('../../Azeem/src') +from tex2Wiki import readin as step4 +from tex2Wiki import writeout as step5 + + +# step1 = imp.load_source('remove_excess','../../AlexDanoff/src/remove_excess.py') + +def main(): + if len(sys.argv) != 2: + pattern = "../../data/[0-9]*ZE.tex" # [A-Z][A-Z] + else: + pattern = sys.argv[1] + for fname in glob.glob(pattern): + print "Processing " + fname + writeout(fname + "-processed.tex", step3(step2(step1(readin(fname))))) + step4(fname + "-processed.tex") + step5(fname + "-dump.xml") + + +if __name__ == "__main__": + main() From 213c341d3197b0aa711a4ee072f787690ceb7f0d Mon Sep 17 00:00:00 2001 From: physikerwelt Date: Wed, 2 Mar 2016 16:37:55 +0100 Subject: [PATCH 046/402] Use Formula namespace for formulae --- Azeem/src/tex2Wiki.py | 1 + 1 file changed, 1 insertion(+) diff --git a/Azeem/src/tex2Wiki.py b/Azeem/src/tex2Wiki.py index 9003bfe..5578db1 100644 --- a/Azeem/src/tex2Wiki.py +++ b/Azeem/src/tex2Wiki.py @@ -203,6 +203,7 @@ def modLabel(line): start_label = line.find("\\label{") + 7 end_label = line.find("}", start_label) label = line[start_label:end_label] + label = label.replace('eq:', 'Formula:') isNumer = False newlabel = "" num = "" From 553f46a73d69759e6ed6ea1bc0418ea3ea9d063a Mon Sep 17 00:00:00 2001 From: physikerwelt Date: Wed, 2 Mar 2016 17:57:41 +0100 Subject: [PATCH 047/402] Don't replace formula prefix * the formula prefix is needed by the wiki to create links to the formula homepages --- Azeem/src/tex2Wiki.py | 99 ++----------------------------------------- 1 file changed, 3 insertions(+), 96 deletions(-) diff --git a/Azeem/src/tex2Wiki.py b/Azeem/src/tex2Wiki.py index 5578db1..f4d2b46 100644 --- a/Azeem/src/tex2Wiki.py +++ b/Azeem/src/tex2Wiki.py @@ -87,7 +87,7 @@ def getEq(line): # Gets all data within constraints,substitutions elif count > 0 and per == 0: # not special character stringWrite += c # write the character - return (stringWrite.rstrip().lstrip()) + return stringWrite.rstrip().lstrip() def getEqP(line, Flag): # Gets all data within proofs @@ -133,7 +133,7 @@ def getEqP(line, Flag): # Gets all data within proofs elif count > 0 and per == 0: stringWrite += c - return (stringWrite.lstrip()) + return stringWrite.lstrip() def getSym(line): # Gets all symbols on a line for symbols list @@ -197,7 +197,6 @@ def secLabel(label): def modLabel(line): - # label.replace("Formula:KLS:","KLS;") start_label = line.find("\\formula{") + 9 if start_label == 8: start_label = line.find("\\label{") + 7 @@ -211,12 +210,10 @@ def modLabel(line): if isNumer and not isnumber(label[i]): if len(num) > 1: newlabel += num - isNumer = False num = "" else: newlabel += "0" + str(num) num = "" - isNumer = False if isnumber(label[i]): isNumer = True num += str(label[i]) @@ -279,13 +276,7 @@ def readin(ofname): mainText = main.read() mainPrepend = "" mainWrite = open("OrthogonalPolynomials.mmd.new", "w") - tester = open("testData.txt", 'w') - # glossary=open('Glossary', 'r') glossary = open('new.Glossary.csv', 'rb') - gCSV = csv.reader(glossary, delimiter=',', quotechar='\"') - # lLinks=open('BruceLabelLinks', 'r') - lGlos = glossary.readlines() - # lLink=lLinks.readlines() math = False constraint = False substitution = False @@ -298,7 +289,6 @@ def readin(ofname): refEqs = [] parse = False head = False - # chapRef=[("GA",open("GA.tex",'r')),("ZE",open("ZE.3.tex",'r'))] refLabels = [] '''for c in chapRef: refLines=refLines+(c[1].readlines()) @@ -321,7 +311,6 @@ def readin(ofname): if "\\begin{document}" in line: parse = True elif "\\end{document}" in line and parse: - # append_text("
\n") mainPrepend += "
\n" mainText = mainPrepend + mainText mainText = mainText.replace("\'\'\'Orthogonal Polynomials\'\'\'\n", "") @@ -339,8 +328,6 @@ def readin(ofname): stringWrite += getString(line) + "\'\'\'\n" labels.append("Orthogonal Polynomials") sections.append(["Orthogonal Polynomials", 0]) - # append_text(stringWrite) - # mainPrepend+=stringWrite chapter = getString(line) mainPrepend += ( "\n== Sections in " + chapter + " ==\n\n
\n") @@ -387,7 +374,6 @@ def readin(ofname): head = True elif "\\begin{equation}" in line and parse: - # symLine="" if head: append_text("\n") head = False @@ -395,8 +381,7 @@ def readin(ofname): eqCounter += 1 labels.append(label) eqs.append("") - # append_text("\n\n") - append_text("") + append_text("") math = True elif "\\begin{equation}" in line and not parse: label = modLabel(line) @@ -414,7 +399,6 @@ def readin(ofname): substitution = True math = False subLine = "" - # append_text("
Substitution(s): "+getEq(line)+"

\n") elif "\\proof" in line and parse: math = False elif "\\drmfn" in line and parse: @@ -474,16 +458,6 @@ def readin(ofname): eqCounter = 0 endNum = len(labels) - 1 - '''for n in range(1,len(labels)): - if n+1!=len(labels) and labels[n+1][0]=="*" and labels[n][0]!="*": - labels[n]=""+labels[n] - endNum=n - elif labels[n][0]=="*": - labels[n]=""+labels[n][1:] - else: - labels[n]=""+labels[n]''' - '''for n in range(0,len(refLabels)): - refLabels[n]=""+refLabels[n]''' parse = False constraint = False substitution = False @@ -657,8 +631,6 @@ def readin(ofname): symbol = symbolPar[0:symbolPar.find("{")] else: symbol = symbolPar - numArg = parCx - numPar = ArgCx gFlag = False checkFlag = False get = False @@ -723,31 +695,11 @@ def readin(ofname): ap = "" else: ap += Q[o] - '''websiteU=g[g.find("http://"):].strip("\n") - k=0 - websites=[] - for r in range(0,len(websiteU)): - if websiteU[r]==",": - if websiteU[k:r].find("http://")!=-1: - websites.append(websiteU[k:r+1].strip(" ")) - k=r - - - if websiteU[k:r].find("http://")!=-1: - websites.append(websiteU[k:r+1].strip(" ")) - websiteF="" - for d in websites: - websiteF=websiteF+" ["+d+" "+d+"]"''' - # websiteF=G[4].strip("\n") websiteF = "" web1 = G[5] for t in range(5, len(G)): if G[t] != "": websiteF = websiteF + " [" + G[t] + " " + G[t] + "]" - - # p1=Q - # if Q.find("@")!=-1: - # p1=Q[:Q.find("@")] p1 = G[4].strip("$") p1 = "" + p1 + "" # if checkFlag: @@ -755,25 +707,6 @@ def readin(ofname): new2 = "" pause = False mathF = True - '''for k in range(0,len(p1)): - if p1[k]=="$": - if mathF: - new1+="" - else: - new1+="" - mathF=not mathF - - elif p1[k]=="#" and p1[k+1].isdigit(): - pause=True - elif pause: - num=int(p1[k]) - #letter=chr(num+96) - letter=listArgs[num-1] - new1+=letter - pause=False - - else: - new1+=p1[k]''' p2 = G[1] for k in range(0, len(p2)): if p2[k] == "$": @@ -788,30 +721,10 @@ def readin(ofname): p2 = new2 finSym.append(web1 + " " + p1 + "] : " + p2 + " :" + websiteF) break - # gFlag=True - # if not symF: - # symF=True - # append_text("[") - # else: - # append_text("
\n") - # append_text("[") - # append_text(g[g.find("http://"):].strip("\n")) - # append_text(" ") - # append_text(getEq(g[g.find("{$"):]).strip("\n").replace("\\\\","\\")) - # append_text("] : ") - # append_text(g[g.find(" {")+2:g.find("} ",g.find(" {")+1)]) - # append_text(" : [") - # append_text(g[g.find("http://"):].strip("\n")) - # append_text(" ") - # append_text(g[g.find("http://"):].strip("\n")) - # append_text("] ")''' - - # preG=S if not gFlag: del newSym[s] gFlag = True - # finSym.reverse() if ampFlag: append_text("& : logical and") gFlag = False @@ -886,14 +799,12 @@ def readin(ofname): constraint = True math = False conLine = "" - # append_text("
"+getEq(line)+"

\n") elif "\\substitution" in line and parse: # symbols=symbols+getSym(line) symLine = line.strip("\n") if hSub: comToWrite = comToWrite + "\n== Substitution(s) ==\n\n" hSub = False - # append_text("
"+getEq(line)+"

\n") substitution = True math = False subLine = "" @@ -962,14 +873,11 @@ def readin(ofname): proofLine += (line[ind]) if "\\end{equation}" in lines[i + 1]: proof = False - # symLine+=line.strip("\n") append_text(comToWrite + getEqP(proofLine, False) + "
\n
\n") comToWrite = "" symbols = symbols + getSym(symLine) symLine = "" - # append_text(line) - elif proof: symLine += line.strip("\n") pauseP = False @@ -1011,7 +919,6 @@ def readin(ofname): proofLine += (line[ind]) if "\\end{equation}" in lines[i + 1]: proof = False - # symLine+=line.strip("\n") append_text(comToWrite + getEqP(proofLine, False).rstrip("\n") + "\n
\n") comToWrite = "" symbols = symbols + getSym(symLine) From 9632a53f5495c18deb500cb90cff1d930d482980 Mon Sep 17 00:00:00 2001 From: Edward Bian Date: Thu, 3 Mar 2016 17:27:58 -0500 Subject: [PATCH 048/402] fixed newcommands placement problems --- KLSadd_insertion/updateChapters.py | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/KLSadd_insertion/updateChapters.py b/KLSadd_insertion/updateChapters.py index a3d6ac1..09cab89 100644 --- a/KLSadd_insertion/updateChapters.py +++ b/KLSadd_insertion/updateChapters.py @@ -81,7 +81,7 @@ def insertCommands(kls, chap, cms): #find index of begin document in KLSadd index = 0 - for w in kls: + for word in kls: index+=1 if("begin{document}" in word): beginIndex += index From db24ced8b483f955777b8554058609cabddcd66e Mon Sep 17 00:00:00 2001 From: physikerwelt Date: Sat, 5 Mar 2016 19:02:56 +0100 Subject: [PATCH 049/402] Don't replace formula prefix * the formula prefix is needed by the wiki to create links to the formula homepages --- Azeem/src/tex2Wiki.py | 50 +++++++++++++++---------------------------- 1 file changed, 17 insertions(+), 33 deletions(-) diff --git a/Azeem/src/tex2Wiki.py b/Azeem/src/tex2Wiki.py index f4d2b46..27c941f 100644 --- a/Azeem/src/tex2Wiki.py +++ b/Azeem/src/tex2Wiki.py @@ -23,7 +23,7 @@ def getString(line): # Gets all data within curly braces on a line # ---------------------------- for c in line: if c == "{" or c == "}": # if there is a curly brace in the line - getStr = not (getStr) # toggle the getStr flag + getStr = not getStr # toggle the getStr flag if c == "}": # no more info needed break elif getStr: # if within curly braces @@ -33,7 +33,7 @@ def getString(line): # Gets all data within curly braces on a line else: # replace $ with '' stringWrite += "\'\'" # add double ' pW = c # change last character for finding double dashes - return (stringWrite.rstrip('\n').lstrip()) # return the data without newlines and without leading spaces + return stringWrite.rstrip('\n').lstrip() # return the data without newlines and without leading spaces def getG(line): # gets equation for symbols list @@ -48,7 +48,7 @@ def getG(line): # gets equation for symbols list start = True else: final += c - return (final) + return final def getEq(line): # Gets all data within constraints,substitutions @@ -72,7 +72,7 @@ def getEq(line): # Gets all data within constraints,substitutions if count > 0: stringWrite += c elif c == "$" and per == 0: # either begin or end equation - fEq = not (fEq) # toggle fEq flag to know if begin or end equation + fEq = not fEq # toggle fEq flag to know if begin or end equation if fEq: # if begin stringWrite += "" else: # if end @@ -92,9 +92,9 @@ def getEq(line): # Gets all data within constraints,substitutions def getEqP(line, Flag): # Gets all data within proofs if Flag: - a = 1 + pass else: - a = 0 + pass per = 1 stringWrite = "" fEq = False @@ -117,7 +117,7 @@ def getEqP(line, Flag): # Gets all data within proofs if count > 0: stringWrite += c elif c == "$" and per == 0: - fEq = not (fEq) + fEq = not fEq if fEq: stringWrite += " " else: @@ -183,17 +183,17 @@ def getSym(line): # Gets all symbols on a line for symbols list # if "MeixnerPollaczek{\\lambda}{n}@{x}{\\phi}=\frac{\\pochhammer{2\\lambda}{n}}" in line: symList.append(symbol) symList += getSym(symbol) - return (symList) + return symList def unmodLabel(label): label = label.replace(".0", ".") label = label.replace(":0", ":") - return (label) + return label def secLabel(label): - return (label.replace("\'\'", "")) + return label.replace("\'\'", "") def modLabel(line): @@ -276,7 +276,6 @@ def readin(ofname): mainText = main.read() mainPrepend = "" mainWrite = open("OrthogonalPolynomials.mmd.new", "w") - glossary = open('new.Glossary.csv', 'rb') math = False constraint = False substitution = False @@ -381,7 +380,7 @@ def readin(ofname): eqCounter += 1 labels.append(label) eqs.append("") - append_text("") + append_text("") math = True elif "\\begin{equation}" in line and not parse: label = modLabel(line) @@ -422,7 +421,7 @@ def readin(ofname): u] or "\\end{document}" in lines[u]: flagM = False flagM2 = True - if not (flagM2): + if not flagM2: append_text(line.rstrip("\n")) append_text("\n
\n") @@ -443,14 +442,14 @@ def readin(ofname): i + 1] or "\\drmfn" in lines[i + 1]: math = False if substitution and parse: - subLine = subLine + line.replace("&", "&
") + subLine += line.replace("&", "&
") if "\\end{equation}" in lines[i + 1] or "\\substitution" in lines[i + 1] or "\\constraint" in lines[ i + 1] or "\\drmfn" in lines[i + 1] or "\\proof" in lines[i + 1]: substitution = False append_text("
Substitution(s): " + getEq(subLine) + "

\n") if constraint and parse: - conLine = conLine + line.replace("&", "&
") + conLine += line.replace("&", "&
") if "\\end{equation}" in lines[i + 1] or "\\substitution" in lines[i + 1] or "\\constraint" in lines[ i + 1] or "\\drmfn" in lines[i + 1] or "\\proof" in lines[i + 1]: constraint = False @@ -597,7 +596,6 @@ def readin(ofname): if flagA: newSym.append(x) newSym.reverse() - symF = False ampFlag = False finSym = [] for s in range(len(newSym) - 1, -1, -1): @@ -644,9 +642,6 @@ def readin(ofname): parCx = 0 parFlag = False cC = 0 - ind = G[0].find("@") - if ind == -1: - ind = len(G[0]) - 1 for z in G[0]: if z == "@": parFlag = True @@ -673,11 +668,9 @@ def readin(ofname): get = True checkFlag = False - if (get): + if get: if get: G = preG - get = False - checkFlag = False if True: if symbolPar.find("@") != -1: Q = symbolPar[:symbolPar.find("@")] @@ -688,9 +681,8 @@ def readin(ofname): ap = "" for o in range(len(symbol), len(Q)): if Q[o] == "{" or z == "[": - argFlag = True + pass elif Q[o] == "}" or z == "]": - argFlag = False listArgs.append(ap) ap = "" else: @@ -703,7 +695,6 @@ def readin(ofname): p1 = G[4].strip("$") p1 = "" + p1 + "" # if checkFlag: - new1 = "" new2 = "" pause = False mathF = True @@ -717,7 +708,6 @@ def readin(ofname): mathF = not mathF else: new2 += p2[k] - # p1=new1 p2 = new2 finSym.append(web1 + " " + p1 + "] : " + p2 + " :" + websiteF) break @@ -744,10 +734,6 @@ def readin(ofname): q = r.find("KLS:") + 4 p = r.find(":", q) section = r[q:p] - equation = r[p + 1:] - if equation.find(":") != -1: - equation = equation[0:equation.find(":")] - append_text( "[http://homepage.tudelft.nl/11r49/askey/contents.html Equation in Section " + section + "] of [[Bibliography#KLS|'''KLS''']].\n\n") # Where should it link to? append_text("== URL links ==\n\nWe ask users to provide relevant URL links in this space.\n\n") @@ -784,7 +770,7 @@ def readin(ofname): append_text("
[[" + labels[0].replace(" ", "_") + "#" + labels[endNum][ 8:] + "|formula in " + labels[0] + "]]
\n") - append_text("
[[" + labels[(0) % endNum].replace(" ", "_") + "|" + labels[ + append_text("
[[" + labels[0 % endNum].replace(" ", "_") + "|" + labels[ 0 % endNum] + "]]
\n") append_text("
\n") @@ -835,7 +821,6 @@ def readin(ofname): if line[ind:ind + 7] == "\\eqref{": pause = True eqR = line[ind:line.find("}", ind) + 1] - rLab = getString(eqR) '''for l in lLink: if rLab==l[0:l.find("=")-1]: rlabel=l[l.find("=>")+3:l.find("\\n")] @@ -886,7 +871,6 @@ def readin(ofname): if line[ind:ind + 7] == "\\eqref{": pause = True eqR = line[ind:line.find("}", ind) + 1] - rLab = getString(eqR) '''for l in lLink: if rLab==l[0:l.find("=")-1]: rlabel=l[l.find("=>")+3:l.find("\\n")] From 414a61337a4483eecf39d001cd22ad2c0eedb5ec Mon Sep 17 00:00:00 2001 From: physikerwelt Date: Sat, 5 Mar 2016 19:03:52 +0100 Subject: [PATCH 050/402] Update with changes from chapter 15 --- AlexDanoff/src/remove_excess.py | 42 ++++++++++++++++++++++++++++++--- 1 file changed, 39 insertions(+), 3 deletions(-) diff --git a/AlexDanoff/src/remove_excess.py b/AlexDanoff/src/remove_excess.py index 183712c..f3b84e9 100644 --- a/AlexDanoff/src/remove_excess.py +++ b/AlexDanoff/src/remove_excess.py @@ -34,6 +34,15 @@ def main(): writeout(ofname, remove_excess(readin(fname))) +def remove_group(name, content, skip_lines): + """Removes the group bounded by the group name and then curly braces.""" + + str_pat = name + STD_REGEX + pattern = re.compile(str_pat, re.DOTALL) + + return pattern.sub(r'', content) + + def remove_section(start, end, content): """Removes the section or part bounded by start and end. :param start: @@ -59,6 +68,12 @@ def remove_begin_end(name, content): return pattern.sub(r'', content) +def remove_comments(content): + oldest = content + updated = _get_preamble() + old = content + + def remove_excess(content): """Removes the excess pieces from the given content and returns the updated version as a string. :param content: @@ -72,8 +87,23 @@ def remove_excess(content): # edit single lines content = re.sub(r'\\bibliography{\.\./bib/DLMF}', r'\\bibliographystyle{plain}' + "\n" + r'\\bibliography{/home/hcohl/DRMF/DLMF/DLMF.bib}', content) - content = re.sub(r'\\author\[.*?\]{(.*?)}', r'\\author{\1}', content) - content = re.sub(r'\{math\}', r'{equation}', content) + content = re.sub(r'\\acknowledgements{.*?}{.*?}', r' ', content) + content = re.sub(r'{math}', r'{equation}', content) + content = re.sub(r'\\galleryitem{.*?}{.*?}', r' ', content) + content = re.sub(r'\\origref{.*?}{.*?}', r' ', content) + content = re.sub(r'\\onlyelectronic{.*?{.*?}.*?}', r' ', content) + content = re.sub(r'\\onlyelectronic{.*?}', r' ', content) + content = re.sub(r'\\printonly{.*?}', r' ', content) + pattern = re.compile(r'\\onlyprint{.*?}', re.DOTALL) + content = re.sub(pattern, r' ', content) + pattern = re.compile(r'\\origref\[.*?\]{.*?}', re.DOTALL) + content = re.sub(pattern, r' ', content) + content = re.sub(r'\\begin{electroniconly}', r' ', content) + content = re.sub(r'\\end{electroniconly}', r' ', content) + content = re.sub(r'\\end{printonly}', r' ', content) + content = re.sub(r'\\begin{printonly}', r' ', content) + content = re.sub(r'\\DLMF\[.*?\]', r' ', content) + content = re.sub(r'\\author\[.*?\]{.*?}', r' ', content) # modify labels, etc content = re.sub(r'\\begin\{equation\}\[.*?\]+', r'\\begin{equation}', content, re.DOTALL) @@ -117,7 +147,12 @@ def remove_excess(content): content = remove_begin_end("comment", content) content = remove_begin_end("figure", content) content = remove_begin_end("errata", content) + content = remove_begin_end("table", content) + content = remove_begin_end("table", content) + + content = remove_begin_end("%", content) + content = remove_begin_end("sidebar", content) content = parentheses.remove(content, curly=True) to_remove = [ @@ -286,7 +321,8 @@ def remove_excess(content): # returns the preamble as a list of it's lines def _get_preamble(): - preamble = ['\\documentclass{article}', '\\usepackage{amsmath}', '\\usepackage{amsfonts}', '\\usepackage{breqn}', + preamble = ['\\documentclass{article}', '\\usepackage{amsmath}', '\\usepackage{amsfonts}', + '\\usepackage{DLMFbreqn}', '\\usepackage{DLMFmath}', '\\usepackage{DRMFfcns}', '', '\\oddsidemargin -0.7cm', '\\textwidth 18.3cm', '\\textheight 26.0cm', '\\topmargin -2.0cm', '', '% \constraint{', '% \substitution{', '% \drmfnote{', '% \drmfname{', ''] From c0db7a5a24f837ba5b2eb162557e138b54da7746 Mon Sep 17 00:00:00 2001 From: physikerwelt Date: Sat, 5 Mar 2016 19:51:24 +0100 Subject: [PATCH 051/402] Include changes from chapter 16 --- AlexDanoff/src/remove_excess.py | 15 ++++++++++----- AlexDanoff/src/replace_special.py | 5 ++++- 2 files changed, 14 insertions(+), 6 deletions(-) diff --git a/AlexDanoff/src/remove_excess.py b/AlexDanoff/src/remove_excess.py index f3b84e9..3d7b3a6 100644 --- a/AlexDanoff/src/remove_excess.py +++ b/AlexDanoff/src/remove_excess.py @@ -72,7 +72,9 @@ def remove_comments(content): oldest = content updated = _get_preamble() old = content - +def remove_macro(name, content): + pattern = re.compile(name+"{(.*?)}") + return pattern.sub(r'\1', content) def remove_excess(content): """Removes the excess pieces from the given content and returns the updated version as a string. @@ -83,11 +85,14 @@ def remove_excess(content): updated = _get_preamble() old = content - + content = remove_macro(r"\\lxID", content) # edit single lines - content = re.sub(r'\\bibliography{\.\./bib/DLMF}', - r'\\bibliographystyle{plain}' + "\n" + r'\\bibliography{/home/hcohl/DRMF/DLMF/DLMF.bib}', content) - content = re.sub(r'\\acknowledgements{.*?}{.*?}', r' ', content) + #removed unecessary information that caused an error with pdflatex + pattern = re.compile(r'\\acknowledgements{.*?}\n\n', re.DOTALL) + content = re.sub(pattern, r'', content) + content = re.sub(r'\\maketitle', r' ', content) + content = re.sub(r'\\bibliography{\.\./bib/DLMF}', r'\\bibliographystyle{plain}' + "\n" + r'\\bibliography{/home/hcohl/DRMF/DLMF/DLMF.bib}', content) + content = re.sub(r'\\acknowledgements{.*?}', r' ', content) content = re.sub(r'{math}', r'{equation}', content) content = re.sub(r'\\galleryitem{.*?}{.*?}', r' ', content) content = re.sub(r'\\origref{.*?}{.*?}', r' ', content) diff --git a/AlexDanoff/src/replace_special.py b/AlexDanoff/src/replace_special.py index 54ee6ae..f8a7e4f 100644 --- a/AlexDanoff/src/replace_special.py +++ b/AlexDanoff/src/replace_special.py @@ -86,7 +86,7 @@ def remove_special(content): # if this line is an index start storing it, or write it if we're done with the indexes if IND_START in line: in_ind = True - ind_str += line + "\n" + ind_str += line continue elif in_ind: @@ -95,6 +95,9 @@ def remove_special(content): # add a preceding newline if one is not already present if previous.strip() != "": + print ("first if: {0}".format(ind_str)) + if ("}" in ind_str): + ind_str+= "\n" ind_str = "\n" + ind_str fullsplit = ind_str.split("\n") From 57c45a255e5cbca3e4e5ccb5ec2b85a59ebd9845 Mon Sep 17 00:00:00 2001 From: physikerwelt Date: Sat, 5 Mar 2016 19:56:30 +0100 Subject: [PATCH 052/402] Include changes from chapter 17 --- AlexDanoff/src/replace_special.py | 5 +---- 1 file changed, 1 insertion(+), 4 deletions(-) diff --git a/AlexDanoff/src/replace_special.py b/AlexDanoff/src/replace_special.py index f8a7e4f..54ee6ae 100644 --- a/AlexDanoff/src/replace_special.py +++ b/AlexDanoff/src/replace_special.py @@ -86,7 +86,7 @@ def remove_special(content): # if this line is an index start storing it, or write it if we're done with the indexes if IND_START in line: in_ind = True - ind_str += line + ind_str += line + "\n" continue elif in_ind: @@ -95,9 +95,6 @@ def remove_special(content): # add a preceding newline if one is not already present if previous.strip() != "": - print ("first if: {0}".format(ind_str)) - if ("}" in ind_str): - ind_str+= "\n" ind_str = "\n" + ind_str fullsplit = ind_str.split("\n") From f20c9502e4d68412afe3f6777b4ffb666af7c1b1 Mon Sep 17 00:00:00 2001 From: physikerwelt Date: Sat, 5 Mar 2016 20:02:49 +0100 Subject: [PATCH 053/402] Fix IDE warnings --- AlexDanoff/src/remove_excess.py | 33 +++++++++++++++++---------------- 1 file changed, 17 insertions(+), 16 deletions(-) diff --git a/AlexDanoff/src/remove_excess.py b/AlexDanoff/src/remove_excess.py index 3d7b3a6..d4ef51a 100644 --- a/AlexDanoff/src/remove_excess.py +++ b/AlexDanoff/src/remove_excess.py @@ -34,8 +34,11 @@ def main(): writeout(ofname, remove_excess(readin(fname))) -def remove_group(name, content, skip_lines): - """Removes the group bounded by the group name and then curly braces.""" +def remove_group(name, content): + """Removes the group bounded by the group name and then curly braces. + :param content: + :param name: + """ str_pat = name + STD_REGEX pattern = re.compile(str_pat, re.DOTALL) @@ -68,13 +71,10 @@ def remove_begin_end(name, content): return pattern.sub(r'', content) -def remove_comments(content): - oldest = content - updated = _get_preamble() - old = content def remove_macro(name, content): - pattern = re.compile(name+"{(.*?)}") - return pattern.sub(r'\1', content) + pattern = re.compile(name + "{(.*?)}") + return pattern.sub(r'\1', content) + def remove_excess(content): """Removes the excess pieces from the given content and returns the updated version as a string. @@ -87,13 +87,14 @@ def remove_excess(content): old = content content = remove_macro(r"\\lxID", content) # edit single lines - #removed unecessary information that caused an error with pdflatex - pattern = re.compile(r'\\acknowledgements{.*?}\n\n', re.DOTALL) + # removed unecessary information that caused an error with pdflatex + pattern = re.compile(r'\\acknowledgements{.*?}\n\n', re.DOTALL) content = re.sub(pattern, r'', content) content = re.sub(r'\\maketitle', r' ', content) - content = re.sub(r'\\bibliography{\.\./bib/DLMF}', r'\\bibliographystyle{plain}' + "\n" + r'\\bibliography{/home/hcohl/DRMF/DLMF/DLMF.bib}', content) + content = re.sub(r'\\bibliography{\.\./bib/DLMF}', + r'\\bibliographystyle{plain}' + "\n" + r'\\bibliography{/home/hcohl/DRMF/DLMF/DLMF.bib}', content) content = re.sub(r'\\acknowledgements{.*?}', r' ', content) - content = re.sub(r'{math}', r'{equation}', content) + content = re.sub(r'\{math}', r'{equation}', content) content = re.sub(r'\\galleryitem{.*?}{.*?}', r' ', content) content = re.sub(r'\\origref{.*?}{.*?}', r' ', content) content = re.sub(r'\\onlyelectronic{.*?{.*?}.*?}', r' ', content) @@ -103,10 +104,10 @@ def remove_excess(content): content = re.sub(pattern, r' ', content) pattern = re.compile(r'\\origref\[.*?\]{.*?}', re.DOTALL) content = re.sub(pattern, r' ', content) - content = re.sub(r'\\begin{electroniconly}', r' ', content) - content = re.sub(r'\\end{electroniconly}', r' ', content) - content = re.sub(r'\\end{printonly}', r' ', content) - content = re.sub(r'\\begin{printonly}', r' ', content) + content = re.sub(r'\\begin\{electroniconly}', r' ', content) + content = re.sub(r'\\end\{electroniconly}', r' ', content) + content = re.sub(r'\\end\{printonly}', r' ', content) + content = re.sub(r'\\begin\{printonly}', r' ', content) content = re.sub(r'\\DLMF\[.*?\]', r' ', content) content = re.sub(r'\\author\[.*?\]{.*?}', r' ', content) From 0904f1ac3b0c9c64539a1aad1cfca941eefeba95 Mon Sep 17 00:00:00 2001 From: physikerwelt Date: Sun, 6 Mar 2016 13:38:19 +0100 Subject: [PATCH 054/402] Add hack to prevent infinit loop in i-replacements --- AlexDanoff/src/replace_special.py | 11 +++++++---- 1 file changed, 7 insertions(+), 4 deletions(-) diff --git a/AlexDanoff/src/replace_special.py b/AlexDanoff/src/replace_special.py index 54ee6ae..39631b4 100644 --- a/AlexDanoff/src/replace_special.py +++ b/AlexDanoff/src/replace_special.py @@ -250,11 +250,14 @@ def _replace_i(words): else: # neither of the characters surrounding the "i" are alphabetic, replace replacement = r'\iunit' - - # print("Surrounding: {0} - replacement made: {1}".format(surrounding, replacement != "i")) - words = words[:iloc] + replacement + words[iloc + 1:] - + # do not search in the replaced part + ilocOld=iloc iloc = words.find("i", iloc + len(replacement)) + # print("Surrounding: {0} - replacement made: {1}".format(surrounding, replacement != "i")) + wl=len(words) + words = words[:ilocOld] + replacement + words[ilocOld + 1:] + if ilocOld == -1: + break return words From ebb06845a7e38d9f43e900752ab33aa42c14de8e Mon Sep 17 00:00:00 2001 From: physikerwelt Date: Sun, 6 Mar 2016 13:59:06 +0100 Subject: [PATCH 055/402] Fix problem with i detection --- .gitignore | 1 + AlexDanoff/src/replace_special.py | 3 +-- AlexDanoff/test/test__replace_i.py | 9 +++++++++ AlexDanoff/test/test_remove_excess.py | 20 ++++++++++---------- 4 files changed, 21 insertions(+), 12 deletions(-) create mode 100644 AlexDanoff/test/test__replace_i.py diff --git a/.gitignore b/.gitignore index e9b040a..1376649 100644 --- a/.gitignore +++ b/.gitignore @@ -1,3 +1,4 @@ +/Azeem/src/BruceLabelLinks /data Macro-Replacement/sources/ KLSadd_insertion/chapterFilesIgnore diff --git a/AlexDanoff/src/replace_special.py b/AlexDanoff/src/replace_special.py index 39631b4..c81f5ab 100644 --- a/AlexDanoff/src/replace_special.py +++ b/AlexDanoff/src/replace_special.py @@ -254,10 +254,9 @@ def _replace_i(words): ilocOld=iloc iloc = words.find("i", iloc + len(replacement)) # print("Surrounding: {0} - replacement made: {1}".format(surrounding, replacement != "i")) - wl=len(words) - words = words[:ilocOld] + replacement + words[ilocOld + 1:] if ilocOld == -1: break + words = words[:ilocOld] + replacement + words[ilocOld + 1:] return words diff --git a/AlexDanoff/test/test__replace_i.py b/AlexDanoff/test/test__replace_i.py new file mode 100644 index 0000000..d4ad80a --- /dev/null +++ b/AlexDanoff/test/test__replace_i.py @@ -0,0 +1,9 @@ +from unittest import TestCase +from replace_special import _replace_i as replace_i + + +class Test_replace_i(TestCase): + def test__replace_i(self): + tex = 'z^a = \\underbrace{z \\cdot z \\cdots z}_{n \\text{ times}} = 1 / z^{-a}.' + result = replace_i(tex) + self.assertEqual(tex, result) diff --git a/AlexDanoff/test/test_remove_excess.py b/AlexDanoff/test/test_remove_excess.py index a070edd..cfefa95 100644 --- a/AlexDanoff/test/test_remove_excess.py +++ b/AlexDanoff/test/test_remove_excess.py @@ -1,10 +1,10 @@ -from unittest import TestCase -from remove_excess import remove_section - - -class TestRemoveExcess(TestCase): - def test_remove_section(self): - content = 'AA{Begin} This should be removed {End}DDBB' - result = remove_section(r'{Begin}', r'{End}', content) - self.assertEqual('AA{End}DDBB', result) - +from unittest import TestCase +from remove_excess import remove_section + + +class TestRemoveExcess(TestCase): + def test_remove_section(self): + content = 'AA{Begin} This should be removed {End}DDBB' + result = remove_section(r'{Begin}', r'{End}', content) + self.assertEqual('AA{End}DDBB', result) + From d1341164eb81ea7f490fd8181a7eb75d18cd0235 Mon Sep 17 00:00:00 2001 From: physikerwelt Date: Sun, 6 Mar 2016 17:47:50 +0100 Subject: [PATCH 056/402] Add test for formula numbering --- Azeem/src/tex2Wiki.py | 15 ++++++++++++--- Azeem/test/test_modlabel.py | 10 ++++++++++ 2 files changed, 22 insertions(+), 3 deletions(-) create mode 100644 Azeem/test/test_modlabel.py diff --git a/Azeem/src/tex2Wiki.py b/Azeem/src/tex2Wiki.py index 27c941f..4d5ec65 100644 --- a/Azeem/src/tex2Wiki.py +++ b/Azeem/src/tex2Wiki.py @@ -7,6 +7,7 @@ import datetime wiki = '' +next_formula_number = 0 ET.register_namespace('', 'http://www.mediawiki.org/xml/export-0.10/') root = ET.Element('{http://www.mediawiki.org/xml/export-0.10/}mediawiki') @@ -197,9 +198,17 @@ def secLabel(label): def modLabel(line): - start_label = line.find("\\formula{") + 9 - if start_label == 8: - start_label = line.find("\\label{") + 7 + start_label = line.find("\\formula{") + if start_label > 0: + start_label += len("\\formula{") + else: + start_label = line.find("\\label{") + if start_label > 0: + start_label += len("\\label{") + else: + global next_formula_number + next_formula_number += 1 + return 'auto-number-' + str(next_formula_number) end_label = line.find("}", start_label) label = line[start_label:end_label] label = label.replace('eq:', 'Formula:') diff --git a/Azeem/test/test_modlabel.py b/Azeem/test/test_modlabel.py new file mode 100644 index 0000000..3439e28 --- /dev/null +++ b/Azeem/test/test_modlabel.py @@ -0,0 +1,10 @@ +from unittest import TestCase +from tex2Wiki import modLabel + + +class TestModLabel(TestCase): + def test_modLabel(self): + self.assertEqual('Formula:EF.EX.TM', modLabel('\\begin{equation}\label{eq:EF.EX.TM}')) + self.assertEqual('Formula:EF.EX.TM', modLabel('\\begin{equation}\\formula{eq:EF.EX.TM}')) + self.assertEqual('auto-number-1', modLabel('\\begin{equation}')) + self.assertEqual('auto-number-2', modLabel('\\begin{equation}')) From bef4dba95b2efbd449549d2a7c731a4e8c1e97b9 Mon Sep 17 00:00:00 2001 From: physikerwelt Date: Sun, 6 Mar 2016 22:34:19 +0100 Subject: [PATCH 057/402] Add support for lookup tables for formula names --- Azeem/src/tex2Wiki.py | 24 +++++++++++++++++++----- Azeem/test/llinks | 1 + Azeem/test/test_modlabel.py | 6 ++++++ 3 files changed, 26 insertions(+), 5 deletions(-) create mode 100644 Azeem/test/llinks diff --git a/Azeem/src/tex2Wiki.py b/Azeem/src/tex2Wiki.py index 4d5ec65..02dae61 100644 --- a/Azeem/src/tex2Wiki.py +++ b/Azeem/src/tex2Wiki.py @@ -8,6 +8,7 @@ wiki = '' next_formula_number = 0 +lLink = '' ET.register_namespace('', 'http://www.mediawiki.org/xml/export-0.10/') root = ET.Element('{http://www.mediawiki.org/xml/export-0.10/}mediawiki') @@ -92,10 +93,6 @@ def getEq(line): # Gets all data within constraints,substitutions def getEqP(line, Flag): # Gets all data within proofs - if Flag: - pass - else: - pass per = 1 stringWrite = "" fEq = False @@ -198,6 +195,7 @@ def secLabel(label): def modLabel(line): + global lLink start_label = line.find("\\formula{") if start_label > 0: start_label += len("\\formula{") @@ -211,6 +209,14 @@ def modLabel(line): return 'auto-number-' + str(next_formula_number) end_label = line.find("}", start_label) label = line[start_label:end_label] + rlabel = label + for l in lLink: + if l.find(label) != -1 and l[len(label) + 1] == "=": + rlabel = l[l.find("=>") + 3:l.find("\\n")] + rlabel = rlabel.replace("/", "") + rlabel = rlabel.replace("#", ":") + break + label = rlabel label = label.replace('eq:', 'Formula:') isNumer = False newlabel = "" @@ -262,20 +268,28 @@ def writeout(ofname): def main(): - if len(sys.argv) != 3: + if len(sys.argv) != 4: fname = "../../data/ZE.3.tex" ofname = "../../data/ZE.4.xml" + lname = "../../data/BruceLabelLinks" else: fname = sys.argv[1] ofname = sys.argv[2] + lname = sys.argv[3] + setup_label_links(lname) readin(fname) writeout(ofname) +def setup_label_links(ofname): + global lLink + lLink = open(ofname, "r").readlines() + + def readin(ofname): # try: for jsahlfkjsd in range(0, 1): diff --git a/Azeem/test/llinks b/Azeem/test/llinks new file mode 100644 index 0000000..72ed132 --- /dev/null +++ b/Azeem/test/llinks @@ -0,0 +1 @@ +eq:EF.EX.TM => DLMF:/4.9#E4 \ No newline at end of file diff --git a/Azeem/test/test_modlabel.py b/Azeem/test/test_modlabel.py index 3439e28..d209ef5 100644 --- a/Azeem/test/test_modlabel.py +++ b/Azeem/test/test_modlabel.py @@ -1,5 +1,6 @@ from unittest import TestCase from tex2Wiki import modLabel +from tex2Wiki import setup_label_links class TestModLabel(TestCase): @@ -8,3 +9,8 @@ def test_modLabel(self): self.assertEqual('Formula:EF.EX.TM', modLabel('\\begin{equation}\\formula{eq:EF.EX.TM}')) self.assertEqual('auto-number-1', modLabel('\\begin{equation}')) self.assertEqual('auto-number-2', modLabel('\\begin{equation}')) + + def test_modLabelsWithList(self): + setup_label_links('llinks') + self.assertEqual('Formula:EF.EX.TM', modLabel('\\begin{equation}\label{eq:EF.EX.TM}')) + self.assertEqual('Formula:EF.EX.TM', modLabel('\\begin{equation}\\formula{eq:EF.EX.TM}')) From 389df984d7e3ed52da86fbd69a46e84d58ef8d97 Mon Sep 17 00:00:00 2001 From: physikerwelt Date: Mon, 7 Mar 2016 09:28:30 +0100 Subject: [PATCH 058/402] Fix tests --- Azeem/test/test_modlabel.py | 6 +++--- testdata/llinks | 1 + 2 files changed, 4 insertions(+), 3 deletions(-) create mode 100644 testdata/llinks diff --git a/Azeem/test/test_modlabel.py b/Azeem/test/test_modlabel.py index d209ef5..dfb21e4 100644 --- a/Azeem/test/test_modlabel.py +++ b/Azeem/test/test_modlabel.py @@ -11,6 +11,6 @@ def test_modLabel(self): self.assertEqual('auto-number-2', modLabel('\\begin{equation}')) def test_modLabelsWithList(self): - setup_label_links('llinks') - self.assertEqual('Formula:EF.EX.TM', modLabel('\\begin{equation}\label{eq:EF.EX.TM}')) - self.assertEqual('Formula:EF.EX.TM', modLabel('\\begin{equation}\\formula{eq:EF.EX.TM}')) + setup_label_links('testdata/llinks') + self.assertEqual('DLMF:04.09:E', modLabel('\\begin{equation}\label{eq:EF.EX.TM}')) + self.assertEqual('DLMF:04.09:E', modLabel('\\begin{equation}\\formula{eq:EF.EX.TM}')) diff --git a/testdata/llinks b/testdata/llinks new file mode 100644 index 0000000..72ed132 --- /dev/null +++ b/testdata/llinks @@ -0,0 +1 @@ +eq:EF.EX.TM => DLMF:/4.9#E4 \ No newline at end of file From 8f9ddcd86adb225783f747f9f89d3f224eb1d126 Mon Sep 17 00:00:00 2001 From: Moritz Schubotz Date: Fri, 25 Mar 2016 12:21:37 -0400 Subject: [PATCH 059/402] Create Documentation.md --- wikidata/Documentation.md | 2 ++ 1 file changed, 2 insertions(+) create mode 100644 wikidata/Documentation.md diff --git a/wikidata/Documentation.md b/wikidata/Documentation.md new file mode 100644 index 0000000..4baabd2 --- /dev/null +++ b/wikidata/Documentation.md @@ -0,0 +1,2 @@ +The relevant pages to manually create, and list semantic properties are availible from +http://wikidata-drmf2016.wmflabs.org/wiki/Special:SpecialPages#Wikibase From 887d3c518e468957d9715581e0c16bc00ae0197e Mon Sep 17 00:00:00 2001 From: Rahul Shah Date: Fri, 25 Mar 2016 14:53:09 -0400 Subject: [PATCH 060/402] Thoroughly explains updateChapters.py --- KLSadd_insertion/updateChaptersRoadMap.txt | 97 ++++++++++++++++++++++ 1 file changed, 97 insertions(+) create mode 100644 KLSadd_insertion/updateChaptersRoadMap.txt diff --git a/KLSadd_insertion/updateChaptersRoadMap.txt b/KLSadd_insertion/updateChaptersRoadMap.txt new file mode 100644 index 0000000..e89b864 --- /dev/null +++ b/KLSadd_insertion/updateChaptersRoadMap.txt @@ -0,0 +1,97 @@ +This is a roadmap of updateChapters.py. It will help explain every piece of code and exactly how everything works in relation to each other. + + +_______________________________________________________________________________ + + +First: the files + +>KLSadd.tex: This is the addendum we are working with. It has sections that correspond with sections in the chapter files, and it contains paragraphs that must be *inserted* into the chapter files. CAN BE FOUND ONINE IN PDF FORM: https://staff.fnwi.uva.nl/t.h.koornwinder/art/informal/KLSadd.pdf + +>chap09.tex and chap14.tex: these are chapter files. These are LATEX documents that are chapters in a book. This is a book written by smart math people. But they did some things wrong, so another math person wants to fix them. That math person wrote an addendum file called KLSadd.tex and our job is to pull paragraphs out of KLSadd and insert them at the end of the relevant section in the chapter files. Every chapter is made up of sections and every section has subsections + +>updateChapters.py: this is our code. This document will be going over the variables and methods in this program. This program should: take paragraphs from every section in KLSadd and insert them into the relevant chapter file and into the relevant section within that chapter. + +>linetest.py: this abomination of code is the original program. It did the job, and it did it well. However, I realized that it was very very unreadable. This program is the child I never wanted. This program does 100% work and its outputs can be used to check against updateChapters.py's output. + +>newtempchap09.tex and newtempchap14.tex these are output files of linetest.py and this is what the output to updateChapters.py should be like!!!!!!! +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +Next: the specifications + +Toward the end of every section in the chapter file, there is a subsection called "References". It basically just contains a bunch of references to sources and other papers. Every paragraph in every subsection in KLSadd.tex must be put *before* the "References" subsection in the corresponding section in the chapter file. For example: +**These aren't real sections, obviously, just an example** + +in chap09.tex: +>\section{Dogs}\index{Dog polynomials} +>\subsection*{Hypergeometric dogs} +>blah blah blah +>\subsection*{References} +>\cite{Dogs101} + +in KLSadd.tex: +>\subsection*{9.15 Dogs} +>\paragraph{Beagles} +>The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetric +>in $a,b,c,d$. + +So in the chapter 9 file, section 15 "Dogs", before the References subsection, we need to add in the Beagles paragraph. HOWEVER, there are some changes that need to be made before we insert. The easiest change is adding a comment that tells us something was inserted. We just append something like "%RS insertion begin" and "%RS insertion complete" before and after, respectively, the Beagles paragraph when we insert. Another change is adding \large\bf before the paragraph name. This changes the size and format so it looks prettier. So the heading in this example would be: + +>\paragraph{\large\bf Beagles} + +**It is important to note that these changes are currently (3/25/16) unimplemented in updateChapters.py and only exist in linetest.py, if you want to see how the output to updateChapters.py should look, look at newtempchap09.tex** + +So after updateChapters.py is called, the chapter 9 file should look like this: + +>\section{Dogs}\index{Dog polynomials} +>\subsection*{Hypergeometric dogs} +>blah blah blah +>%RS insertion begin +>\paragraph{\large\bf Beagles} +>The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetric +>in $a,b,c,d$. +>%RS insertion complete +>\subsection*{References} +>\cite{Dogs101} + + +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +The variables: + +These are the variables in the beginning + +-chapNums: chapNums is a list, denoted by the [], that holds the chapter number (in this case it's either a 9 or a 14) that corresponds to where a section in KLSadd should go + +-paras: holds all of the text in the paragraphs of KLSadd.tex. This is the list that gets read when its time to insert paragraphs into the chapter file. Every element is a loooong string + +-mathPeople: this holds the *name* of every section in KLSadd. It stores names like Wilson, Racah, Dual Hahn, etc. In our example with the Dogs above, it would hold "Dogs". This is useful to finding where to insert the correct paragraph + +-newCommands: this is a list that holds ints representing line numbers in KLSadd.tex that correspond to commands. There are a few special commands in the file that help turn LATEX files into PDF files and they need to be copied over into both chapter files + +-comms: holds the actual strings of the commands found from the line numbers stored in newCommands + +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +The methods: + +-prepareForPDF(chap): this method inserts some packages needed by LATEX files to be turned into pdf files. It takes a list called chap which is a String containing the contents of the chapter fie. It returns the chap String edited with the special packages in place. + +-getCommands(kls): takes the KLSadd.tex as a string as a parameter and stores the special commands in the comms variable mentioned earlier + +-insertCommands(kls, chap, cms): kls is KLSadd.tex, chap is a chapter file string, cms is a list of commands, it is the comms variable. This method returns the chap with the special commands inserted in place + +-findReferences(str): takes a string representation of a chapter file. Returns a list of the line numbers of the "References" subsections. This references variable is used as an index as to where the paragraphs in paras should be inserted + +-fixChapter(chap, references, p, kls): takes a chapter file string, a references list, the paras variable, and the KLSadd file string. This method basically just calls all of the methods above and adds all of the extra stuff like the commands and packages. +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +At this point, the code is like 90% comments. The big chunk at the bottom under the "with open..." can be explained through the comments. + +As a quick rundown: + +The chapters and KLSadd are turned into Strings and passed through the fixChapter methods. + +Then the strings are written into seperate updated chapter files. + +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +However, its broken. Inserted text comes in random places between paragraphs and sometimes stuff isn't added. This shouldn't be that hard to fix but it will require combing through the code a lot of times and changing little things here and there until everything works. As a last ditch effort, we can always resort to using linetext.py and adding stuff in from there. From 06d781cc0ed1c553e2b670ec043f6869e5813b3d Mon Sep 17 00:00:00 2001 From: Edward Bian Date: Fri, 25 Mar 2016 15:13:19 -0400 Subject: [PATCH 061/402] Commented out some lines and changed /myciteKLS to /mycite as Dr. cohl asked --- KLSadd_insertion/updateChapters.py | 28 ++++++++++++++++++++++++++-- 1 file changed, 26 insertions(+), 2 deletions(-) diff --git a/KLSadd_insertion/updateChapters.py b/KLSadd_insertion/updateChapters.py index 09cab89..520a6b9 100644 --- a/KLSadd_insertion/updateChapters.py +++ b/KLSadd_insertion/updateChapters.py @@ -126,6 +126,30 @@ def fixChapter(chap, references, p, kls): chap = prepareForPDF(chap) cms = getCommands(kls) chap = insertCommands(kls,chap, cms) + commentticker = 0 + for word in chap: + if ("\\newcommand\\half{\\frac12}" in word): + wordtoadd = "%" + word + chap[commentticker] = wordtoadd + elif ("\\newcommand{\\hyp}[5]{\\,\\mbox{}_{#1}F_{#2}\\!\\left(" in word): + wordtoadd = "%" + word + chap[commentticker] = wordtoadd + elif ("\\genfrac{}{}{0pt}{}{#3}{#4};#5\\right)}" in word): + wordtoadd = "%" + word + chap[commentticker] = wordtoadd + elif ("\\newcommand{\\qhyp}[5]{\\,\\mbox{}_{#1}\\phi_{#2}\\!\\left(" in word): + wordtoadd = "%" + word + chap[commentticker] = wordtoadd + elif ("\\genfrac{}{}{0pt}{}{#3}{#4};#5\\right)}" in word): + wordtoadd = "%" + word + chap[commentticker] = wordtoadd + commentticker += 1 + # Hopefully this works + ticker1 = 0 + while ticker1 < len(chap): + if ('\\myciteKLS' in chap[ticker1]): + chap[ticker1] = chap[ticker1].replace('\\myciteKLS', '\\cite') + ticker1 += 1 #probably won't work because I don't know how anything works return chap @@ -160,12 +184,12 @@ def fixChapter(chap, references, p, kls): #parse both files 9 and 14 as strings #chapter 9 - with open("tempchap9.tex", "r") as ch9: + with open("chap09.tex", "r") as ch9: entire9 = ch9.readlines() #reads in as a list of strings ch9.close() #chapter 14 - with open("tempchap14.tex", "r") as ch14: + with open("chap14.tex", "r") as ch14: entire14 = ch14.readlines() ch14.close() #call the findReferences method to find the index of the References paragraph in the two file strings From c2828d2939532129c7ba9b9a234b29ee4fbcef50 Mon Sep 17 00:00:00 2001 From: Moritz Schubotz Date: Tue, 29 Mar 2016 10:42:00 -0400 Subject: [PATCH 062/402] Add new code quality metric buttons to the readme --- README.md | 2 ++ 1 file changed, 2 insertions(+) diff --git a/README.md b/README.md index ef118d7..01cdcef 100644 --- a/README.md +++ b/README.md @@ -2,6 +2,8 @@ [![Build Status](https://travis-ci.org/DRMF/DRMF-Seeding-Project.svg?branch=master)](https://travis-ci.org/DRMF/DRMF-Seeding-Project) [![Coverage Status](https://coveralls.io/repos/github/DRMF/DRMF-Seeding-Project/badge.svg?branch=master)](https://coveralls.io/github/DRMF/DRMF-Seeding-Project?branch=master) +[![Code Issues](https://www.quantifiedcode.com/api/v1/project/b097e91550e147b2b02e345ffb1c5162/badge.svg)](https://www.quantifiedcode.com/app/project/b097e91550e147b2b02e345ffb1c5162) +[![Code Health](https://landscape.io/github/DRMF/DRMF-Seeding-Project/master/landscape.svg?style=flat)](https://landscape.io/github/DRMF/DRMF-Seeding-Project/master) This project is meant to convert different designated source formats to semantic LaTeX for the DRMF project. One major aspect of this conversion is the inclusion, insertion, and replacement From ca4911ba6f3ace8298f9bdca4f4ccbb7a1f66cec Mon Sep 17 00:00:00 2001 From: Rahul Shah Date: Tue, 29 Mar 2016 14:24:21 -0400 Subject: [PATCH 063/402] Attempt to fix linebreaks --- KLSadd_insertion/updateChapters.py | 12 ++++-------- 1 file changed, 4 insertions(+), 8 deletions(-) diff --git a/KLSadd_insertion/updateChapters.py b/KLSadd_insertion/updateChapters.py index 520a6b9..23ed89b 100644 --- a/KLSadd_insertion/updateChapters.py +++ b/KLSadd_insertion/updateChapters.py @@ -86,7 +86,6 @@ def insertCommands(kls, chap, cms): if("begin{document}" in word): beginIndex += index tempIndex = 0 - #D'OH FOUND THE PROBLEM newCommands stores just the integer value of the indexes of the commands. we need to get the actual commands. easy fix, do next time for i in cms: chap.insert(beginIndex+tempIndex,i) tempIndex +=1 @@ -114,15 +113,15 @@ def findReferences(str): def fixChapter(chap, references, p, kls): #chap is the file string(actually a list), references is the specific references for the file, #and p is the paras variable(not sure if needed) kls is the KLSadd.tex as a list - #TODO: OPTIMIZE(?) count = 0 #count is used to represent the values in count for i in references: - #add the paragraphs before the Reference paragraphs start - chap[i-3] += p[count] + #Place before References paragraph + chap[i-2] += "%KLS insert begin" + chap[i-2] += p[count] + chap[i-2] += "%KLS insert end" count+=1 chap[i-1] += p[count] - #I actually don't remember why I had to do this but I think you need it ^^^^^^^^^^^^^^^^^^^ chap = prepareForPDF(chap) cms = getCommands(kls) chap = insertCommands(kls,chap, cms) @@ -175,7 +174,6 @@ def fixChapter(chap, references, p, kls): for i in range(len(indexes)-1): str = ''.join(addendum[indexes[i]: indexes[i+1]-1]) paras.append(str) - #paras now holds the paragraphs that need to go into the chapter files, but they need to go in the appropriate #section(like Wilson, Racah, Hahn, etc.) so we use the mathPeople variable #we can use the section names to place the relevant paragraphs in the right place @@ -197,8 +195,6 @@ def fixChapter(chap, references, p, kls): references9 = findReferences(entire9) references14 = findReferences(entire14) - #ERROR! entire9 sometimes contains lists, must only contain strings. Should be fixed by next week. Check the - #getCommands method, I suspect that is the problem #call the fixChapter method to get a list with the addendum paragraphs added in str9 = ''.join(fixChapter(entire9, references9, paras, addendum)) str14 = ''.join(fixChapter(entire14, references14, paras, addendum)) From d025894dda42ec8b134dd145e071b8710e07f70c Mon Sep 17 00:00:00 2001 From: Rahul Shah Date: Tue, 29 Mar 2016 15:34:28 -0400 Subject: [PATCH 064/402] Fixed line breaks --- KLSadd_insertion/updateChapters.py | 9 ++++----- 1 file changed, 4 insertions(+), 5 deletions(-) diff --git a/KLSadd_insertion/updateChapters.py b/KLSadd_insertion/updateChapters.py index 23ed89b..773e97b 100644 --- a/KLSadd_insertion/updateChapters.py +++ b/KLSadd_insertion/updateChapters.py @@ -63,11 +63,7 @@ def getCommands(kls): newCommands.append(index-1) if("mybibitem[1]" in word): newCommands.append(index) - #add \large\bf KLSadd: to make KLSadd additions appear like the other paragraphs - if("paragraph{" in word): - #not sure if this works, needs testing! - temp = word[0:word.find("{")+ 1] + "\large\\bf KLSadd: " + word[word.find("{")+ 1: word.find("}")+1] - + #pretty sure I had to do something here but I forgot, so pass? #duh, obviously I need to store the commands somewhere! comms = kls[newCommands[0]:newCommands[1]] @@ -154,6 +150,7 @@ def fixChapter(chap, references, p, kls): #open the KLSadd file to do things with with open("KLSadd.tex", "r") as add: + #store the file as a string addendum = add.readlines() index = 0 @@ -199,6 +196,8 @@ def fixChapter(chap, references, p, kls): str9 = ''.join(fixChapter(entire9, references9, paras, addendum)) str14 = ''.join(fixChapter(entire14, references14, paras, addendum)) + print(len(references9) + len(references14)) + print(len(paras)) """ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ If you are writing something that will make a change to the chapter files, write it BEFORE this line, this part From adc7d77e1b013188675f1aae6a3050255c2d2516 Mon Sep 17 00:00:00 2001 From: Rahul Shah Date: Tue, 29 Mar 2016 15:43:03 -0400 Subject: [PATCH 065/402] fixed something i can't remember... --- KLSadd_insertion/updateChapters.py | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/KLSadd_insertion/updateChapters.py b/KLSadd_insertion/updateChapters.py index 773e97b..5df705d 100644 --- a/KLSadd_insertion/updateChapters.py +++ b/KLSadd_insertion/updateChapters.py @@ -99,7 +99,7 @@ def findReferences(str): w = word[word.find("{")+1: word.find("}")] if(w not in mathPeople): canAdd = True - if("\\subsection*{References}" in word and canAdd == True): + if(("\\subsection*{References}" in word or "subsubsection*{References" in word )and canAdd == True): references.append(index) canAdd = False return references From 9b77a4e84b75b4994190c749165c1c3ee8a2f019 Mon Sep 17 00:00:00 2001 From: edward-bian Date: Tue, 29 Mar 2016 15:44:41 -0400 Subject: [PATCH 066/402] Fixed Spacing, also added very ugly headings Pdf loads normally now and there are now very ugly headings to distinguish KLSadd sections from the original pdf. updateChapters.py still loads weirdly --- KLSadd_insertion/updateChapters.py | 8 ++------ 1 file changed, 2 insertions(+), 6 deletions(-) diff --git a/KLSadd_insertion/updateChapters.py b/KLSadd_insertion/updateChapters.py index 23ed89b..f3c709e 100644 --- a/KLSadd_insertion/updateChapters.py +++ b/KLSadd_insertion/updateChapters.py @@ -64,10 +64,6 @@ def getCommands(kls): if("mybibitem[1]" in word): newCommands.append(index) #add \large\bf KLSadd: to make KLSadd additions appear like the other paragraphs - if("paragraph{" in word): - #not sure if this works, needs testing! - temp = word[0:word.find("{")+ 1] + "\large\\bf KLSadd: " + word[word.find("{")+ 1: word.find("}")+1] - #pretty sure I had to do something here but I forgot, so pass? #duh, obviously I need to store the commands somewhere! comms = kls[newCommands[0]:newCommands[1]] @@ -116,9 +112,9 @@ def fixChapter(chap, references, p, kls): count = 0 #count is used to represent the values in count for i in references: #Place before References paragraph - chap[i-2] += "%KLS insert begin" + chap[i-2] += "\\large\\bf KLSadd additions: " chap[i-2] += p[count] - chap[i-2] += "%KLS insert end" + chap[i-2] += "\\large\\bf End of KLSadd additions" count+=1 chap[i-1] += p[count] From eb0c2e399a7e0eccb369250862146f569a1de15f Mon Sep 17 00:00:00 2001 From: Howard Cohl Date: Wed, 30 Mar 2016 11:22:50 -0400 Subject: [PATCH 067/402] Add .sty .cls files to .gitignore --- .gitignore | 2 ++ 1 file changed, 2 insertions(+) diff --git a/.gitignore b/.gitignore index 1376649..5c0f18c 100644 --- a/.gitignore +++ b/.gitignore @@ -116,6 +116,8 @@ crashlytics-build.properties ### TeX template ## Core latex/pdflatex auxiliary files: +*.sty +*.cls *.aux *.lof *.lot From 061a93a65617522b51823ad869b5c554814b8546 Mon Sep 17 00:00:00 2001 From: Edward Bian Date: Thu, 31 Mar 2016 15:26:38 -0400 Subject: [PATCH 068/402] Fixed formatting and made it look like linetest.py --- KLSadd_insertion/updateChapters.py | 300 ++++++++++++++--------------- 1 file changed, 149 insertions(+), 151 deletions(-) diff --git a/KLSadd_insertion/updateChapters.py b/KLSadd_insertion/updateChapters.py index f3c709e..ff57ebd 100644 --- a/KLSadd_insertion/updateChapters.py +++ b/KLSadd_insertion/updateChapters.py @@ -1,27 +1,21 @@ """ Rahul Shah -self-taught python don't kill me I know its bad -2/5/16 +self-taught python don't kill me I know its bad +2/5/16 Re-write of linetest.py so people other than me can read it - This project was/is being written to streamline the update process of the book used for the online repository, updated via the KLSadd addendum file which only affects chapters 9 and 14 - ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ NOTE! AS OF 2/26/16 THIS FILE IS NOT YET UPDATED TO THE FULL CAPACITY OF THE PREVIOUS FILE. IF FURTHER DEVELOPMENT IS NEEDED, REFER TO THE linetest.py FILE IF THIS FILE DOES NOT ADDRESS THE NEW/EXTRA GOALS. This means that XCITE PARSE etc HAS NOT BEEN ADDED - Additional goals (already addressed in linetest.py): -insert correct usepackages -insert correct commands found in KLSadd.tex to chapter files -insert editor's initials into sections (ie %RS: begin addition, %RS: end addition) - Additional goals (not already addressed in linetest.py): -rewrite for "smarter" edits (ex. add new limit relations straight to the limit relations section in the section itself, not at the end) - Current update status: incomplete - Goals:change the book chapter files to include paragraphs from the addendum, and after that insert some edits so they are intelligently integrated into the chapter """ @@ -35,165 +29,169 @@ chapNums = [] paras = [] mathPeople = [] -newCommands = [] #used to hold the indexes of the commands -comms = "" #holds the ACTUAL STRINGS of the commands +newCommands = [] #used to hold the indexes of the commands +comms = "" #holds the ACTUAL STRINGS of the commands #2/18/16 this method addresses the goal of hardcoding in the necessary packages to let the chapter files run as pdf's. #Currently only works with chapter 9, ask Dr. Cohl to help port your chapter 14 output file into a pdf def prepareForPDF(chap): - footmiscIndex = 0 - index = 0 - for word in chap: - index+=1 - if("footmisc" in word): - footmiscIndex+= index - #edits the chapter string sent to include hyperref, xparse, and cite packages - #str[footmiscIndex] += "\\usepackage[pdftex]{hyperref} \n\\usepackage {xparse} \n\\usepackage{cite} \n" - chap.insert(footmiscIndex, "\\usepackage[pdftex]{hyperref} \n\\usepackage {xparse} \n\\usepackage{cite} \n") + footmiscIndex = 0 + index = 0 + for word in chap: + index+=1 + if("footmisc" in word): + footmiscIndex+= index + #edits the chapter string sent to include hyperref, xparse, and cite packages + #str[footmiscIndex] += "\\usepackage[pdftex]{hyperref} \n\\usepackage {xparse} \n\\usepackage{cite} \n" + chap.insert(footmiscIndex, "\\usepackage[pdftex]{hyperref} \n\\usepackage {xparse} \n\\usepackage{cite} \n") - return chap + return chap -#2/18/16 this method reads in relevant commands that are in KLSadd.tex and returns them as a list and also adds +#2/18/16 this method reads in relevant commands that are in KLSadd.tex and returns them as a list and also adds def getCommands(kls): - index = 0 - for word in kls: - index+=1 - if("smallskipamount" in word): - newCommands.append(index-1) - if("mybibitem[1]" in word): - newCommands.append(index) - #add \large\bf KLSadd: to make KLSadd additions appear like the other paragraphs - #pretty sure I had to do something here but I forgot, so pass? - #duh, obviously I need to store the commands somewhere! - comms = kls[newCommands[0]:newCommands[1]] - return comms + index = 0 + for word in kls: + index+=1 + if("smallskipamount" in word): + newCommands.append(index-1) + if("mybibitem[1]" in word): + newCommands.append(index) + #add \large\bf KLSadd: to make KLSadd additions appear like the other paragraphs + #pretty sure I had to do something here but I forgot, so pass? + #duh, obviously I need to store the commands somewhere! + comms = kls[newCommands[0]:newCommands[1]] + return comms #2/18/16 this method addresses the goal of hardcoding in the necessary commands to let the chapter files run as pdf's. Currently only works with chapter 9 def insertCommands(kls, chap, cms): - #reads in the newCommands[] and puts them in chap - beginIndex = -1 #the index of the "begin document" keyphrase, this is where the new commands need to be inserted. - - #find index of begin document in KLSadd - index = 0 - for word in kls: - index+=1 - if("begin{document}" in word): - beginIndex += index - tempIndex = 0 - for i in cms: - chap.insert(beginIndex+tempIndex,i) - tempIndex +=1 - return chap + #reads in the newCommands[] and puts them in chap + beginIndex = -1 #the index of the "begin document" keyphrase, this is where the new commands need to be inserted. + + #find index of begin document in KLSadd + index = 0 + for word in kls: + index+=1 + if("begin{document}" in word): + beginIndex += index + tempIndex = 0 + for i in cms: + chap.insert(beginIndex+tempIndex,i) + tempIndex +=1 + return chap #method to find the indices of the reference paragraphs def findReferences(str): - references = [] - index = -1 - canAdd = False - for word in str: - index+=1 - #check sections and subsections - if("section{" in word or "subsection*{" in word): - w = word[word.find("{")+1: word.find("}")] - if(w not in mathPeople): - canAdd = True - if("\\subsection*{References}" in word and canAdd == True): - references.append(index) - canAdd = False - return references + references = [] + index = -1 + canAdd = False + for word in str: + index+=1 + #check sections and subsections + if("section{" in word or "subsection*{" in word): + w = word[word.find("{")+1: word.find("}")] + if(w not in mathPeople): + canAdd = True + if("\\subsection*{References}" in word and canAdd == True): + references.append(index) + canAdd = False + return references #method to change file string(actually a list right now), returns string to be written to file #If you write a method that changes something, it is preffered that you call the method in here def fixChapter(chap, references, p, kls): - #chap is the file string(actually a list), references is the specific references for the file, - #and p is the paras variable(not sure if needed) kls is the KLSadd.tex as a list - count = 0 #count is used to represent the values in count - for i in references: - #Place before References paragraph - chap[i-2] += "\\large\\bf KLSadd additions: " - chap[i-2] += p[count] - chap[i-2] += "\\large\\bf End of KLSadd additions" - count+=1 - chap[i-1] += p[count] - - chap = prepareForPDF(chap) - cms = getCommands(kls) - chap = insertCommands(kls,chap, cms) - commentticker = 0 - for word in chap: - if ("\\newcommand\\half{\\frac12}" in word): - wordtoadd = "%" + word - chap[commentticker] = wordtoadd - elif ("\\newcommand{\\hyp}[5]{\\,\\mbox{}_{#1}F_{#2}\\!\\left(" in word): - wordtoadd = "%" + word - chap[commentticker] = wordtoadd - elif ("\\genfrac{}{}{0pt}{}{#3}{#4};#5\\right)}" in word): - wordtoadd = "%" + word - chap[commentticker] = wordtoadd - elif ("\\newcommand{\\qhyp}[5]{\\,\\mbox{}_{#1}\\phi_{#2}\\!\\left(" in word): - wordtoadd = "%" + word - chap[commentticker] = wordtoadd - elif ("\\genfrac{}{}{0pt}{}{#3}{#4};#5\\right)}" in word): - wordtoadd = "%" + word - chap[commentticker] = wordtoadd - commentticker += 1 - # Hopefully this works - ticker1 = 0 - while ticker1 < len(chap): - if ('\\myciteKLS' in chap[ticker1]): - chap[ticker1] = chap[ticker1].replace('\\myciteKLS', '\\cite') - ticker1 += 1 - #probably won't work because I don't know how anything works - return chap - -#open the KLSadd file to do things with + #chap is the file string(actually a list), references is the specific references for the file, + #and p is the paras variable(not sure if needed) kls is the KLSadd.tex as a list + count = 0 #count is used to represent the values in count + for i in references: + #Place before References paragraph + chap[i-2] += "%KLSadd additions:" + chap[i-2] += p[count] + chap[i-2] += "%End of KLSadd additions" + count+=1 + chap[i-1] += p[count] + + chap = prepareForPDF(chap) + cms = getCommands(kls) + chap = insertCommands(kls,chap, cms) + commentticker = 0 + for word in chap: + if ("\\newcommand\\half{\\frac12}" in word): + wordtoadd = "%" + word + chap[commentticker] = wordtoadd + elif ("\\newcommand{\\hyp}[5]{\\,\\mbox{}_{#1}F_{#2}\\!\\left(" in word): + wordtoadd = "%" + word + chap[commentticker] = wordtoadd + elif ("\\genfrac{}{}{0pt}{}{#3}{#4};#5\\right)}" in word): + wordtoadd = "%" + word + chap[commentticker] = wordtoadd + elif ("\\newcommand{\\qhyp}[5]{\\,\\mbox{}_{#1}\\phi_{#2}\\!\\left(" in word): + wordtoadd = "%" + word + chap[commentticker] = wordtoadd + elif ("\\genfrac{}{}{0pt}{}{#3}{#4};#5\\right)}" in word): + wordtoadd = "%" + word + chap[commentticker] = wordtoadd + commentticker += 1 + # Hopefully this works + ticker1 = 0 + while ticker1 < len(chap): + if ('\\myciteKLS' in chap[ticker1]): + chap[ticker1] = chap[ticker1].replace('\\myciteKLS', '\\cite') + ticker1 += 1 + #probably won't work because I don't know how anything works + return chap + +#open the KLSadd file to do things with with open("KLSadd.tex", "r") as add: - #store the file as a string - addendum = add.readlines() - index = 0 - indexes = [] - for word in addendum: - index+=1 - if("." in word and "\\subsection*{" in word): - name = word[word.find(" "): word.find("}")] - mathPeople.append(name) - if("9." in word): - chapNums.append(9) - if("14." in word): - chapNums.append(9) - #get the index - indexes.append(index-1) - #now indexes holds all of the places there is a section - #using these indexes, get all of the words in between and add that to the paras[] - for i in range(len(indexes)-1): - str = ''.join(addendum[indexes[i]: indexes[i+1]-1]) - paras.append(str) - #paras now holds the paragraphs that need to go into the chapter files, but they need to go in the appropriate - #section(like Wilson, Racah, Hahn, etc.) so we use the mathPeople variable - #we can use the section names to place the relevant paragraphs in the right place - - #as of 2/8/16 the paragraphs will go before the References paragraph of the relevant section - #parse both files 9 and 14 as strings - - #chapter 9 - with open("chap09.tex", "r") as ch9: - entire9 = ch9.readlines() #reads in as a list of strings - ch9.close() - - #chapter 14 - with open("chap14.tex", "r") as ch14: - entire14 = ch14.readlines() - ch14.close() - #call the findReferences method to find the index of the References paragraph in the two file strings - #two variables for the references lists one for chapter 9 one for chapter 14 - references9 = findReferences(entire9) - references14 = findReferences(entire14) - - #call the fixChapter method to get a list with the addendum paragraphs added in - str9 = ''.join(fixChapter(entire9, references9, paras, addendum)) - str14 = ''.join(fixChapter(entire14, references14, paras, addendum)) + #store the file as a string + addendum = add.readlines() + for word in addendum: + if ("paragraph{" in word): + temp = word[0:word.find("{") + 1] + "\large\\bf KLSadd: " + word[word.find("{") + 1: word.find("}") + 1] + addendum[addendum.index(word): addendum.index(word) + 1] = temp + index = 0 + indexes = [] + for word in addendum: + index+=1 + if("." in word and "\\subsection*{" in word): + name = word[word.find(" "): word.find("}")] + mathPeople.append(name) + if("9." in word): + chapNums.append(9) + if("14." in word): + chapNums.append(9) + #get the index + indexes.append(index-1) + #now indexes holds all of the places there is a section + #using these indexes, get all of the words in between and add that to the paras[] + for i in range(len(indexes)-1): + str = ''.join(addendum[indexes[i]: indexes[i+1]-1]) + paras.append(str) + #paras now holds the paragraphs that need to go into the chapter files, but they need to go in the appropriate + #section(like Wilson, Racah, Hahn, etc.) so we use the mathPeople variable + #we can use the section names to place the relevant paragraphs in the right place + + #as of 2/8/16 the paragraphs will go before the References paragraph of the relevant section + #parse both files 9 and 14 as strings + + #chapter 9 + with open("chap09.tex", "r") as ch9: + entire9 = ch9.readlines() #reads in as a list of strings + ch9.close() + + #chapter 14 + with open("chap14.tex", "r") as ch14: + entire14 = ch14.readlines() + ch14.close() + #call the findReferences method to find the index of the References paragraph in the two file strings + #two variables for the references lists one for chapter 9 one for chapter 14 + references9 = findReferences(entire9) + references14 = findReferences(entire14) + + #call the fixChapter method to get a list with the addendum paragraphs added in + str9 = ''.join(fixChapter(entire9, references9, paras, addendum)) + str14 = ''.join(fixChapter(entire14, references14, paras, addendum)) """ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ @@ -203,10 +201,10 @@ def fixChapter(chap, references, p, kls): """ -#write to files +#write to files #new output files for safety with open("updated9.tex","w") as temp9: - temp9.write(str9) - + temp9.write(str9) + with open("updated14.tex", "w") as temp14: - temp14.write(str9) + temp14.write(str9) \ No newline at end of file From c80a03062b690637a6f9f2f9b5271c08ad579aea Mon Sep 17 00:00:00 2001 From: Edward Bian Date: Fri, 1 Apr 2016 13:03:48 -0400 Subject: [PATCH 069/402] Fixed very minor errors --- KLSadd_insertion/updateChapters.py | 11 +++++------ 1 file changed, 5 insertions(+), 6 deletions(-) diff --git a/KLSadd_insertion/updateChapters.py b/KLSadd_insertion/updateChapters.py index ff57ebd..7ca4275 100644 --- a/KLSadd_insertion/updateChapters.py +++ b/KLSadd_insertion/updateChapters.py @@ -52,10 +52,10 @@ def prepareForPDF(chap): def getCommands(kls): index = 0 for word in kls: - index+=1 - if("smallskipamount" in word): + index+=1 + if("smallskipamount" in word): newCommands.append(index-1) - if("mybibitem[1]" in word): + if("mybibitem[1]" in word): newCommands.append(index) #add \large\bf KLSadd: to make KLSadd additions appear like the other paragraphs #pretty sure I had to do something here but I forgot, so pass? @@ -110,7 +110,6 @@ def fixChapter(chap, references, p, kls): chap[i-2] += p[count] chap[i-2] += "%End of KLSadd additions" count+=1 - chap[i-1] += p[count] chap = prepareForPDF(chap) cms = getCommands(kls) @@ -166,8 +165,8 @@ def fixChapter(chap, references, p, kls): #now indexes holds all of the places there is a section #using these indexes, get all of the words in between and add that to the paras[] for i in range(len(indexes)-1): - str = ''.join(addendum[indexes[i]: indexes[i+1]-1]) - paras.append(str) + box = ''.join(addendum[indexes[i]: indexes[i+1]-1]) + paras.append(box) #paras now holds the paragraphs that need to go into the chapter files, but they need to go in the appropriate #section(like Wilson, Racah, Hahn, etc.) so we use the mathPeople variable #we can use the section names to place the relevant paragraphs in the right place From 801f37a52ebcb17a80e93ccc96dfe17df6f55536 Mon Sep 17 00:00:00 2001 From: Edward Bian Date: Fri, 1 Apr 2016 13:09:57 -0400 Subject: [PATCH 070/402] Fixed 1 more very minor error --- KLSadd_insertion/updateChapters.py | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/KLSadd_insertion/updateChapters.py b/KLSadd_insertion/updateChapters.py index 7ca4275..6350090 100644 --- a/KLSadd_insertion/updateChapters.py +++ b/KLSadd_insertion/updateChapters.py @@ -82,11 +82,11 @@ def insertCommands(kls, chap, cms): return chap #method to find the indices of the reference paragraphs -def findReferences(str): +def findReferences(chapter): references = [] index = -1 canAdd = False - for word in str: + for word in chapter: index+=1 #check sections and subsections if("section{" in word or "subsection*{" in word): From aaacd6dd86019d975b349924bce08b8bbcd33c5a Mon Sep 17 00:00:00 2001 From: Edward Bian Date: Fri, 1 Apr 2016 14:56:09 -0400 Subject: [PATCH 071/402] Fix reference errors --- KLSadd_insertion/updateChapters.py | 14 ++++++++------ 1 file changed, 8 insertions(+), 6 deletions(-) diff --git a/KLSadd_insertion/updateChapters.py b/KLSadd_insertion/updateChapters.py index 6350090..944ae1e 100644 --- a/KLSadd_insertion/updateChapters.py +++ b/KLSadd_insertion/updateChapters.py @@ -91,9 +91,11 @@ def findReferences(chapter): #check sections and subsections if("section{" in word or "subsection*{" in word): w = word[word.find("{")+1: word.find("}")] - if(w not in mathPeople): - canAdd = True - if("\\subsection*{References}" in word and canAdd == True): + for unit in mathPeople: + if (w in unit): + canAdd = True + + if("\\subsection*{References}" in word) and (canAdd == True): references.append(index) canAdd = False return references @@ -106,7 +108,7 @@ def fixChapter(chap, references, p, kls): count = 0 #count is used to represent the values in count for i in references: #Place before References paragraph - chap[i-2] += "%KLSadd additions:" + chap[i-2] += "%Begin KLSadd additions" chap[i-2] += p[count] chap[i-2] += "%End of KLSadd additions" count+=1 @@ -159,7 +161,7 @@ def fixChapter(chap, references, p, kls): if("9." in word): chapNums.append(9) if("14." in word): - chapNums.append(9) + chapNums.append(14) #get the index indexes.append(index-1) #now indexes holds all of the places there is a section @@ -206,4 +208,4 @@ def fixChapter(chap, references, p, kls): temp9.write(str9) with open("updated14.tex", "w") as temp14: - temp14.write(str9) \ No newline at end of file + temp14.write(str14) \ No newline at end of file From 0eda041c8df86bbb017d42db2725e4247b9c49f8 Mon Sep 17 00:00:00 2001 From: physikerwelt Date: Tue, 5 Apr 2016 12:02:03 -0400 Subject: [PATCH 072/402] Making tex2Wiki more readable * I went through the tex2Wiki code and tried to make it a little bit more readable --- Azeem/src/tex2Wiki.py | 170 ++++++++++++++++-------------------------- 1 file changed, 66 insertions(+), 104 deletions(-) diff --git a/Azeem/src/tex2Wiki.py b/Azeem/src/tex2Wiki.py index 02dae61..bcbe712 100644 --- a/Azeem/src/tex2Wiki.py +++ b/Azeem/src/tex2Wiki.py @@ -92,7 +92,7 @@ def getEq(line): # Gets all data within constraints,substitutions return stringWrite.rstrip().lstrip() -def getEqP(line, Flag): # Gets all data within proofs +def getEqP(line): # Gets all data within proofs per = 1 stringWrite = "" fEq = False @@ -295,8 +295,8 @@ def readin(ofname): for jsahlfkjsd in range(0, 1): global wiki tex = open(ofname, 'r') - main = open("OrthogonalPolynomials.mmd", "r") - mainText = main.read() + main_file = open("OrthogonalPolynomials.mmd", "r") + mainText = main_file.read() mainPrepend = "" mainWrite = open("OrthogonalPolynomials.mmd.new", "w") math = False @@ -312,9 +312,7 @@ def readin(ofname): parse = False head = False refLabels = [] - '''for c in chapRef: - refLines=refLines+(c[1].readlines()) - c[1].close()''' + chapter = '' for i in range(0, len(refLines)): line = refLines[i] if "\\begin{equation}" in line: @@ -342,7 +340,7 @@ def readin(ofname): mainText = "{{#set:Section=0}}\n" + mainText mainWrite.write(mainText) mainWrite.close() - main.close() + main_file.close() copyfile('OrthogonalPolynomials.mmd', 'OrthogonalPolynomials.mmd.new') parse = False elif "\\title" in line and parse: @@ -367,6 +365,8 @@ def readin(ofname): secCounter = 0 eqCounter = 0 + subLine = '' + conLine = '' for i in range(0, len(lines)): line = lines[i] if "\\section" in line: @@ -431,17 +431,16 @@ def readin(ofname): flagM = True eqs[len(eqs) - 1] += line - if "\\end{equation}" in lines[i + 1] and not "\\subsection" in lines[i + 3] and not "\\section" in \ - lines[ - i + 3] and not "\\part" in lines[i + 3]: + if "\\end{equation}" in lines[i + 1] and "\\subsection" not in lines[i + 3] and "\\section" not in \ + lines[i + 3] and "\\part" not in lines[i + 3]: u = i flagM2 = False while flagM: u += 1 if "\\begin{equation}" in lines[u] in lines[u]: flagM = False - if "\\section" in lines[u] or "\\subsection" in lines[i] or "\\part" in lines[ - u] or "\\end{document}" in lines[u]: + if "\\section" in lines[u] or "\\subsection" in lines[i] or "\\part" in lines[u] \ + or "\\end{document}" in lines[u]: flagM = False flagM2 = True if not flagM2: @@ -492,6 +491,11 @@ def readin(ofname): comToWrite = "" secCount = -1 newSec = False + symLine = '' + eqS = '' + noteLine = '' + proofLine = '' + pause = False for i in range(0, len(lines)): line = lines[i] @@ -521,17 +525,14 @@ def readin(ofname): if eqCounter < endNum: # FOR ANYTHING THAT IS NOT THE EXTRA EQUATIONS append_text("
\n") if newSec: - append_text("
<< [[" + secLabel(sections[secCount][0]).replace(" ", - "_") + "|" + secLabel( - sections[secCount][0]) + "]]
\n") + append_text("
<< [[" + secLabel(sections[secCount][0]).replace(" ", "_") + + "|" + secLabel(sections[secCount][0]) + "]]
\n") else: - append_text("
<< [[" + secLabel(labels[eqCounter - 1]).replace(" ", - "_") + "|" + secLabel( - labels[eqCounter - 1]) + "]]
\n") - append_text("
[[" + secLabel(sections[secCount + 1][0]).replace(" ", - "_") + "#" + secLabel( - labels[eqCounter][len("Formula:"):]) + "|formula in " + secLabel( - sections[secCount + 1][0]) + "]]
\n") + append_text("
<< [[" + secLabel(labels[eqCounter - 1]).replace(" ", "_") + + "|" + secLabel(labels[eqCounter - 1]) + "]]
\n") + append_text("
[[" + secLabel(sections[secCount + 1][0]).replace(" ", "_") + + "#" + secLabel(labels[eqCounter][len("Formula:"):]) + "|formula in " + + secLabel(sections[secCount + 1][0]) + "]]
\n") if eqS == sections[secCount][1]: append_text( "
[[" + secLabel( @@ -539,26 +540,22 @@ def readin(ofname): " ", "_") + "|" + secLabel( sections[(secCount + 1) % len(sections)][0]) + "]] >>
\n") else: - append_text( - "
[[" + secLabel(labels[(eqCounter + 1) % (endNum + 1)]).replace(" ", - "_") + "|" + secLabel( - labels[(eqCounter + 1) % (endNum + 1)]) + "]] >>
\n") + append_text("
[[" + + secLabel(labels[(eqCounter + 1) % (endNum + 1)]).replace(" ", "_") + + "|" + secLabel(labels[(eqCounter + 1) % (endNum + 1)]) + "]] >>
\n") append_text("
\n\n") elif eqCounter == endNum: append_text("
\n") if newSec: newSec = False - append_text("
<< [[" + secLabel(sections[secCount][0]).replace(" ", - "_") + "|" + secLabel( - sections[secCount][0]) + "]]
\n") + append_text("
<< [[" + secLabel(sections[secCount][0]).replace(" ", "_") + + "|" + secLabel(sections[secCount][0]) + "]]
\n") else: - append_text("
<< [[" + secLabel(labels[eqCounter - 1]).replace(" ", - "_") + "|" + secLabel( - labels[eqCounter - 1]) + "]]
\n") - append_text("
[[" + secLabel(sections[secCount + 1][0]).replace(" ", - "_") + "#" + secLabel( - labels[eqCounter][len("Formula:"):]) + "|formula in " + secLabel( - sections[secCount + 1][0]) + "]]
\n") + append_text("
<< [[" + secLabel(labels[eqCounter - 1]).replace(" ", "_") + + "|" + secLabel(labels[eqCounter - 1]) + "]]
\n") + append_text("
[[" + secLabel(sections[secCount + 1][0]).replace(" ", "_") + + "#" + secLabel(labels[eqCounter][len("Formula:"):]) + "|formula in " + + secLabel(sections[secCount + 1][0]) + "]]
\n") append_text("
[[" + secLabel( labels[(eqCounter + 1) % (endNum + 1)].replace(" ", "_")) + "|" + secLabel( labels[(eqCounter + 1) % (endNum + 1)]) + "]]
\n") @@ -571,9 +568,8 @@ def readin(ofname): parse = False math = False if hProof: - append_text( - "\n== Proof ==\n\nWe ask users to provide proof(s), reference(s) to proof(s), or further clarification on the proof(s) in this space.\n") - + append_text("\n== Proof ==\n\nWe ask users to provide proof(s), reference(s) to proof(s), or" + " further clarification on the proof(s) in this space.\n") append_text("\n== Symbols List ==\n\n") newSym = [] # if "09.07:04" in label: @@ -627,6 +623,7 @@ def readin(ofname): parCx = 0 parFlag = False cC = 0 + cN = 0 for z in symbolPar: if z == "@": parFlag = True @@ -680,17 +677,13 @@ def readin(ofname): parCx += 1 else: ArgCx += 1 - if G[0].find(symbol) == 0 and (len(G[0]) == len(symbol) or not G[0][ - len(symbol)].isalpha()): # and (numPar!=0 or numArg!=0): + if G[0].find(symbol) == 0 and (len(G[0]) == len(symbol) or not G[0][len(symbol)].isalpha()): checkFlag = True get = True preG = S - - elif checkFlag: get = True checkFlag = False - if get: if get: G = preG @@ -757,53 +750,45 @@ def readin(ofname): q = r.find("KLS:") + 4 p = r.find(":", q) section = r[q:p] - append_text( - "[http://homepage.tudelft.nl/11r49/askey/contents.html Equation in Section " + section + "] of [[Bibliography#KLS|'''KLS''']].\n\n") # Where should it link to? + # Where should it link to? + append_text("[http://homepage.tudelft.nl/11r49/askey/contents.html " + "Equation in Section " + section + "] of [[Bibliography#KLS|'''KLS''']].\n\n") append_text("== URL links ==\n\nWe ask users to provide relevant URL links in this space.\n\n") if eqCounter < endNum: append_text("
\n") if newSec: newSec = False - append_text("
<< [[" + secLabel(sections[secCount][0]).replace(" ", - "_") + "|" + secLabel( - sections[secCount][0]) + "]]
\n") + append_text("
<< [[" + secLabel(sections[secCount][0]).replace(" ", "_") + + "|" + secLabel(sections[secCount][0]) + "]]
\n") else: - append_text("
<< [[" + secLabel(labels[eqCounter - 1]).replace(" ", - "_") + "|" + secLabel( - labels[eqCounter - 1]) + "]]
\n") - append_text("
[[" + secLabel(sections[secCount + 1][0]).replace(" ", - "_") + "#" + secLabel( - labels[eqCounter][len("Formula:"):]) + "|formula in " + secLabel( - sections[secCount + 1][0]) + "]]
\n") + append_text("
<< [[" + secLabel(labels[eqCounter - 1]).replace(" ", "_") + + "|" + secLabel(labels[eqCounter - 1]) + "]]
\n") + append_text("
[[" + secLabel(sections[secCount + 1][0]).replace(" ", "_") + + "#" + secLabel(labels[eqCounter][len("Formula:"):]) + "|formula in " + + secLabel(sections[secCount + 1][0]) + "]]
\n") if eqS == sections[secCount][1]: - append_text( - "
[[" + sections[(secCount + 1) % len(sections)][0].replace(" ", - "_") + "|" + - sections[(secCount + 1) % len(sections)][0] + "]] >>
\n") + append_text("
[[" + + sections[(secCount + 1) % len(sections)][0].replace(" ", "_") + "|" + + sections[(secCount + 1) % len(sections)][0] + "]] >>
\n") else: - append_text( - "
[[" + secLabel(labels[(eqCounter + 1) % (endNum + 1)]).replace(" ", - "_") + "|" + secLabel( - labels[(eqCounter + 1) % (endNum + 1)]) + "]] >>
\n") + append_text("
[[" + + secLabel(labels[(eqCounter + 1) % (endNum + 1)]).replace(" ", "_") + "|" + + secLabel(labels[(eqCounter + 1) % (endNum + 1)]) + "]] >>
\n") append_text("
\n") else: # FOR EXTRA EQUATIONS append_text("
\n") append_text("
<< [[" + labels[endNum - 1].replace(" ", "_") + "|" + labels[ endNum - 1] + "]]
\n") - append_text("
[[" + labels[0].replace(" ", "_") + "#" + labels[endNum][ - 8:] + "|formula in " + - labels[0] + "]]
\n") + append_text("
[[" + labels[0].replace(" ", "_") + "#" + labels[endNum][8:] + + "|formula in " + labels[0] + "]]
\n") append_text("
[[" + labels[0 % endNum].replace(" ", "_") + "|" + labels[ 0 % endNum] + "]]
\n") append_text("
\n") - - - elif "\\constraint" in line and parse: # symbols=symbols+getSym(line) symLine = line.strip("\n") if hCon: - comToWrite = comToWrite + "\n== Constraint(s) ==\n\n" + comToWrite += "\n== Constraint(s) ==\n\n" hCon = False constraint = True math = False @@ -812,7 +797,7 @@ def readin(ofname): # symbols=symbols+getSym(line) symLine = line.strip("\n") if hSub: - comToWrite = comToWrite + "\n== Substitution(s) ==\n\n" + comToWrite += "\n== Substitution(s) ==\n\n" hSub = False substitution = True math = False @@ -824,7 +809,7 @@ def readin(ofname): elif "\\drmfnote" in line and parse: symbols = symbols + getSym(line) if hNote: - comToWrite = comToWrite + "\n== Note(s) ==\n\n" + comToWrite += "\n== Note(s) ==\n\n" hNote = False note = True math = False @@ -835,7 +820,9 @@ def readin(ofname): symLine = line.strip("\n") if hProof: hProof = False - comToWrite = comToWrite + "\n== Proof ==\n\nWe ask users to provide proof(s), reference(s) to proof(s), or further clarification on the proof(s) in this space. \n

\n
" + comToWrite += "\n== Proof ==\n\nWe ask users to provide proof(s), reference(s) to proof(s), or " \ + "further clarification on the proof(s) in this space. \n

\n" \ + "
" proof = True proofLine = "" pause = False @@ -843,15 +830,6 @@ def readin(ofname): for ind in range(0, len(line)): if line[ind:ind + 7] == "\\eqref{": pause = True - eqR = line[ind:line.find("}", ind) + 1] - '''for l in lLink: - if rLab==l[0:l.find("=")-1]: - rlabel=l[l.find("=>")+3:l.find("\\n")] - rlabel=rlabel.replace("/","") - rlabel=rlabel.replace("#",":") - rlabel=rlabel.replace("!",":") - break''' - eInd = refLabels.index("" + label) z = line[line.find("}", ind + 7) + 1] if z == "." or z == ",": @@ -867,8 +845,6 @@ def readin(ofname): proofLine += ( "
\n \n" + refEqs[ eInd] + "
\n") - - else: if pause: if line[ind] == "}": @@ -881,7 +857,7 @@ def readin(ofname): proofLine += (line[ind]) if "\\end{equation}" in lines[i + 1]: proof = False - append_text(comToWrite + getEqP(proofLine, False) + "
\n
\n") + append_text(comToWrite + getEqP(proofLine) + "
\n
\n") comToWrite = "" symbols = symbols + getSym(symLine) symLine = "" @@ -889,19 +865,9 @@ def readin(ofname): elif proof: symLine += line.strip("\n") pauseP = False - # symbols=symbols+getSym(line) for ind in range(0, len(line)): if line[ind:ind + 7] == "\\eqref{": pause = True - eqR = line[ind:line.find("}", ind) + 1] - '''for l in lLink: - if rLab==l[0:l.find("=")-1]: - rlabel=l[l.find("=>")+3:l.find("\\n")] - rlabel=rlabel.replace("/","") - rlabel=rlabel.replace("#",":") - rlabel=rlabel.replace("!",":") - break''' - eInd = refLabels.index("" + label) z = line[line.find("}", ind + 7) + 1] if z == "." or z == ",": @@ -926,7 +892,7 @@ def readin(ofname): proofLine += (line[ind]) if "\\end{equation}" in lines[i + 1]: proof = False - append_text(comToWrite + getEqP(proofLine, False).rstrip("\n") + "
\n
\n") + append_text(comToWrite + getEqP(proofLine).rstrip("\n") + "\n
\n") comToWrite = "" symbols = symbols + getSym(symLine) symLine = "" @@ -945,7 +911,7 @@ def readin(ofname): symLine += line.strip("\n") append_text(line) if note and parse: - noteLine = noteLine + line + noteLine += line symbols = symbols + getSym(line) if "\\end{equation}" in lines[i + 1] or "\\drmfn" in lines[i + 1] or "\\constraint" in lines[ i + 1] or "\\substitution" in lines[i + 1] or "\\proof" in lines[i + 1]: @@ -953,7 +919,7 @@ def readin(ofname): comToWrite = comToWrite + "
" + getEq(noteLine) + "

\n" if constraint and parse: - conLine = conLine + line.replace("&", "&
") + conLine += line.replace("&", "&
") symLine += line.strip("\n") # symbols=symbols+getSym(line) @@ -965,7 +931,7 @@ def readin(ofname): append_text(comToWrite + "
" + getEq(conLine) + "

\n") comToWrite = "" if substitution and parse: - subLine = subLine + line.replace("&", "&
") + subLine += line.replace("&", "&
") symLine += line.strip("\n") # symbols=symbols+getSym(line) @@ -976,10 +942,6 @@ def readin(ofname): symLine = "" append_text(comToWrite + "
" + getEq(subLine) + "

\n") comToWrite = "" - # except Exception as detail: #If exception occured - # print("Exception",detail) #print details of error - # except: #If anythin else occured... - # print ("ERROR",sys.exc_info()[0])#ERROR with basic info if __name__ == "__main__": From da682f19af1c27c51e470d581c0cee921a03a2f8 Mon Sep 17 00:00:00 2001 From: physikerwelt Date: Tue, 5 Apr 2016 12:56:25 -0400 Subject: [PATCH 073/402] Add tests for parentheses file Demonstrate how unit tests can help to test the functionality of a module. --- AlexDanoff/src/parentheses.py | 51 +++++++++++++++-------------- AlexDanoff/test/test_parentheses.py | 49 +++++++++++++++++++++++++++ 2 files changed, 75 insertions(+), 25 deletions(-) create mode 100644 AlexDanoff/test/test_parentheses.py diff --git a/AlexDanoff/src/parentheses.py b/AlexDanoff/src/parentheses.py index 6f995d1..63ad2bd 100644 --- a/AlexDanoff/src/parentheses.py +++ b/AlexDanoff/src/parentheses.py @@ -1,42 +1,42 @@ -"""Goes through input file, numbers every opening and corresponding closing parenthesis and writes result to output file.""" +"""Goes through input file, + numbers every opening and corresponding closing parenthesis and writes result to output file.""" -import sys import re _last_val = 0 -def remove(input, curly=False, cached=False): + +def remove(text, curly=False, cached=False): """Goes through input and replaces any opening and closing parentheses with numbered markers.""" - + global _last_val counter = 0 - #resume from previous count + # resume from previous count if cached: counter = _last_val updated = "" - open = "(" - close = ")" + open_delimiter = "(" + close_delimiter = ")" - #if client wants to replace curly braces, adjust accordingly + # if client wants to replace curly braces, adjust accordingly if curly: - open = "{" - close = "}" - - #go through input and replace parentheses with labels - for character in input: + open_delimiter = "{" + close_delimiter = "}" - #found an opening parenthesis, replace it with its label - if character == open: + # go through input and replace parentheses with labels + for character in text: + # found an opening parenthesis, replace it with its label + if character == open_delimiter: character = "###open_{0}###".format(counter) counter += 1 - #found a closing parenthesis, replace it with its label - if character == close: + # found a closing parenthesis, replace it with its label + if character == close_delimiter: counter -= 1 if counter < 0: @@ -46,17 +46,18 @@ def remove(input, curly=False, cached=False): updated += character - _last_val = counter #store our spot + _last_val = counter # store our spot return updated -def insert(input, curly=False): + +def insert(text, curly=False): """Goes through input and replaces any numbered markers with opening and closing parentheses.""" o_rep = "(" - c_rep = ")" - - #adjust for curly braces + c_rep = ")" + + # adjust for curly braces if curly: o_rep = "{" c_rep = "}" @@ -64,7 +65,7 @@ def insert(input, curly=False): open_pattern = r'###open_\d###' close_pattern = r'###close_\d###' - input = re.sub(open_pattern, o_rep, input) - input = re.sub(close_pattern, c_rep, input) + text = re.sub(open_pattern, o_rep, text) + text = re.sub(close_pattern, c_rep, text) - return input + return text diff --git a/AlexDanoff/test/test_parentheses.py b/AlexDanoff/test/test_parentheses.py new file mode 100644 index 0000000..d74fb21 --- /dev/null +++ b/AlexDanoff/test/test_parentheses.py @@ -0,0 +1,49 @@ +from unittest import TestCase +from parentheses import insert, remove + +testCases = [{ + 'normal': '\\sin(x)', + 'replaced': '\\sin###open_0###x###close_0###', +}, { + 'normal': '\\sin{x}', + 'replaced': '\\sin{x}', +}, + { + 'normal': '\\sin{x}', + 'replaced': '\\sin###open_0###x###close_0###', + 'curly': True + } +] + +roundTripTests = [{ + 'normal': '\\sin(\\arcsin(x))', +}, { + 'normal': '\\sin{\\arcsin(x)}', +}, { + 'normal': '\\sin{\\arcsin(x)}', + 'curly': True +}, { + 'normal': '\\sin{\\arcsin{x}}', + 'curly': True +}, +] + + +class TestInsert(TestCase): + def test_insert(self): + for case in testCases: + self.assertEqual(case['normal'], insert(case['replaced'], case.get('curly', False))) + + def test_remove(self): + for case in testCases: + self.assertEqual(case['replaced'], remove(case['normal'], case.get('curly', False))) + + def test_round_trip(self): + for case in testCases: + output = insert(remove(case['normal'], case.get('curly', False)), case.get('curly', False)) + self.assertEqual(case['normal'], output) + + def test_extended_round_trip(self): + for case in roundTripTests: + output = insert(remove(case['normal'], case.get('curly', False)), case.get('curly', False)) + self.assertEqual(case['normal'], output) From ed0ca6ca831653964cd6e9880b02d26bf7e32f3c Mon Sep 17 00:00:00 2001 From: physikerwelt Date: Tue, 5 Apr 2016 13:15:15 -0400 Subject: [PATCH 074/402] Fix warnings from QuantifiedCode --- AlexDanoff/test/test_parentheses.py | 12 ++++++------ 1 file changed, 6 insertions(+), 6 deletions(-) diff --git a/AlexDanoff/test/test_parentheses.py b/AlexDanoff/test/test_parentheses.py index d74fb21..a502ab3 100644 --- a/AlexDanoff/test/test_parentheses.py +++ b/AlexDanoff/test/test_parentheses.py @@ -1,7 +1,7 @@ from unittest import TestCase from parentheses import insert, remove -testCases = [{ +test_cases = [{ 'normal': '\\sin(x)', 'replaced': '\\sin###open_0###x###close_0###', }, { @@ -15,7 +15,7 @@ } ] -roundTripTests = [{ +round_trip_tests = [{ 'normal': '\\sin(\\arcsin(x))', }, { 'normal': '\\sin{\\arcsin(x)}', @@ -31,19 +31,19 @@ class TestInsert(TestCase): def test_insert(self): - for case in testCases: + for case in test_cases: self.assertEqual(case['normal'], insert(case['replaced'], case.get('curly', False))) def test_remove(self): - for case in testCases: + for case in test_cases: self.assertEqual(case['replaced'], remove(case['normal'], case.get('curly', False))) def test_round_trip(self): - for case in testCases: + for case in test_cases: output = insert(remove(case['normal'], case.get('curly', False)), case.get('curly', False)) self.assertEqual(case['normal'], output) def test_extended_round_trip(self): - for case in roundTripTests: + for case in round_trip_tests: output = insert(remove(case['normal'], case.get('curly', False)), case.get('curly', False)) self.assertEqual(case['normal'], output) From 9b3664d1b05d83ca2842c5a0d5d3a80c67c442d2 Mon Sep 17 00:00:00 2001 From: Philip Wang Date: Wed, 6 Apr 2016 17:16:43 -0400 Subject: [PATCH 075/402] Update tex2Wiki.py --- Azeem/src/OrthogonalPolynomials.mmd | 81 --- Azeem/src/new.Glossary.csv | 691 ------------------------- Azeem/src/tex2Wiki.py | 19 +- Pipeline/src/OrthogonalPolynomials.mmd | 81 --- Pipeline/src/new.Glossary.csv | 691 ------------------------- 5 files changed, 10 insertions(+), 1553 deletions(-) delete mode 100644 Azeem/src/OrthogonalPolynomials.mmd delete mode 100644 Azeem/src/new.Glossary.csv delete mode 100644 Pipeline/src/OrthogonalPolynomials.mmd delete mode 100644 Pipeline/src/new.Glossary.csv diff --git a/Azeem/src/OrthogonalPolynomials.mmd b/Azeem/src/OrthogonalPolynomials.mmd deleted file mode 100644 index c841fb5..0000000 --- a/Azeem/src/OrthogonalPolynomials.mmd +++ /dev/null @@ -1,81 +0,0 @@ -drmf_bof -'''Orthogonal Polynomials''' -{{#set:Section=0}} -== Sections in KLS Chapter 1 == - -
-* [[Orthogonal polynomials|Orthogonal polynomials]] -* [[The gamma and beta function|The gamma and beta function]] -* [[The shifted factorial and binomial coefficients|The shifted factorial and binomial coefficients]] -* [[Hypergeometric functions|Hypergeometric functions]] -* [[The binomial theorem and other summation formulas|The binomial theorem and other summation formulas]] -* [[Some integrals|Some integrals]] -* [[Transformation formulas|Transformation formulas]] -* [[The q-shifted factorial|The ''q''-shifted factorial]] -* [[The q-gamma function and q-binomial coefficients|The ''q''-gamma function and ''q''-binomial coefficients]] -* [[Basic hypergeometric functions|Basic hypergeometric functions]] -* [[The q-binomial theorem and other summation formulas|The ''q''-binomial theorem and other summation formulas]] -* [[More integrals|More integrals]] -* [[Transformation formulas|Transformation formulas]] -* [[Some q-analogues of special functions|Some ''q''-analogues of special functions]] -* [[The q-derivative and q-integral|The ''q''-derivative and ''q''-integral]] -* [[Shift operators and Rodrigues-type formulas|Shift operators and Rodrigues-type formulas]] -
-== Sections in KLS Chapter 9 == - -
-* [[Wilson|Wilson]] -* [[Racah|Racah]] -* [[Continuous dual Hahn|Continuous dual Hahn]] -* [[Continuous Hahn|Continuous Hahn]] -* [[Hahn|Hahn]] -* [[Dual Hahn|Dual Hahn]] -* [[Meixner-Pollaczek|Meixner-Pollaczek]] -* [[Jacobi|Jacobi]] -* [[Jacobi: Special cases|Jacobi: Special cases]] -* [[Pseudo Jacobi|Pseudo Jacobi]] -* [[Meixner|Meixner]] -* [[Krawtchouk|Krawtchouk]] -* [[Laguerre|Laguerre]] -* [[Bessel|Bessel]] -* [[Charlier|Charlier]] -* [[Hermite|Hermite]] -
-== Sections in KLS Chapter 14 == - -
-* [[Askey-Wilson|Askey-Wilson]] -* [[q-Racah|''q''-Racah]] -* [[Continuous dual q-Hahn|Continuous dual ''q''-Hahn]] -* [[Continuous q-Hahn|Continuous ''q''-Hahn]] -* [[Big q-Jacobi|Big ''q''-Jacobi]] -* [[Big q-Jacobi: Special case|Big ''q''-Jacobi: Special case]] -* [[q-Hahn|''q''-Hahn]] -* [[Dual q-Hahn|Dual ''q''-Hahn]] -* [[Al-Salam-Chihara|Al-Salam-Chihara]] -* [[q-Meixner-Pollaczek|''q''-Meixner-Pollaczek]] -* [[Continuous q-Jacobi|Continuous ''q''-Jacobi]] -* [[Continuous q-Jacobi: Special cases|Continuous ''q''-Jacobi: Special cases]] -* [[Big q-Laguerre|Big ''q''-Laguerre]] -* [[Little q-Jacobi|Little ''q''-Jacobi]] -* [[Little q-Jacobi: Special case|Little ''q''-Jacobi: Special case]] -* [[q-Meixner|''q''-Meixner]] -* [[Quantum q-Krawtchouk|Quantum ''q''-Krawtchouk]] -* [[q-Krawtchouk|''q''-Krawtchouk]] -* [[Affine q-Krawtchouk|Affine ''q''-Krawtchouk]] -* [[Dual q-Krawtchouk|Dual ''q''-Krawtchouk]] -* [[Continuous big q-Hermite|Continuous big ''q''-Hermite]] -* [[Continuous q-Laguerre|Continuous ''q''-Laguerre]] -* [[Little q-Laguerre / Wall|Little ''q''-Laguerre / Wall]] -* [[q-Laguerre|''q''-Laguerre]] -* [[q-Bessel|''q''-Bessel]] -* [[q-Charlier|''q''-Charlier]] -* [[Al-Salam-Carlitz I|Al-Salam-Carlitz I]] -* [[Al-Salam-Carlitz II|Al-Salam-Carlitz II]] -* [[Continuous q-Hermite|Continuous ''q''-Hermite]] -* [[Stieltjes-Wigert|Stieltjes-Wigert]] -* [[Discrete q-Hermite I|Discrete ''q''-Hermite I]] -* [[Discrete q-Hermite II|Discrete ''q''-Hermite II]] -
- -drmf_eof diff --git a/Azeem/src/new.Glossary.csv b/Azeem/src/new.Glossary.csv deleted file mode 100644 index 8875e6b..0000000 --- a/Azeem/src/new.Glossary.csv +++ /dev/null @@ -1,691 +0,0 @@ -\AGM@{a}{g},arithmetic-geometric mean,arithmetic-geometric-mean,F|F:P,$M$,http://dlmf.nist.gov/19.8#SS1.p1, -\AThetaFunction@{x},Jacobi theta function $\vartheta$,a-theta-function,F|F:P,$\vartheta$,http://dlmf.nist.gov/23.1, -\AbstractJacobi{p}{q},generic Jacobian elliptic function,generic-Jacobian-elliptic-function,F|F,$\mathrm{pq}$,http://dlmf.nist.gov/22.2#E10, -\Acos@@{z},general inverse cosine function,multivalued-inverse-cosine,F|F:P:nP,$\mathrm{Arccos}$,http://dlmf.nist.gov/4.23#E2, -\Acosh@@{z},general inverse hyperbolic cosine function,multivalued-hyperbolic-inverse-cosine,F|F:P:nP,$\mathrm{Arccosh}$,http://dlmf.nist.gov/4.37#E2, -\Acot@@{z},general inverse cotangent function,multivalued-inverse-cotangent,F|F:P:nP,$\mathrm{Arccot}$,http://dlmf.nist.gov/4.23#E6, -\Acoth@@{z},general inverse hyperbolic cotangent function,multivalued-hyperbolic-inverse-cotangent,F|F:P:nP,$\mathrm{Arccoth}$,http://dlmf.nist.gov/4.37#E6, -\Acsc@@{z},general inverse cosecant function,multivalued-inverse-cosecant,F|F:P:nP,$\mathrm{Arccsc}$,http://dlmf.nist.gov/4.23#E4, -\Acsch@@{z},general inverse hyperbolic cosecant function,multivalued-hyperbolic-inverse-cosecant,F|F:P:nP,$\mathrm{Arccsch}$,http://dlmf.nist.gov/4.37#E4, -\AffqKrawtchouk{n}@@{q^{-x}}{p}{N}{q},affine $q$-Krawtchouk polynomial,affine-q-Krawtchouk-polynomial-K,P|F:P:PS,$K^{\mathrm{Aff}}_{n}$,http://drmf.wmflabs.org/wiki/Definition:AffqKrawtchouk, -\AiryAi@{z},Airy function ${\rm Ai}$,Airy-Ai,F|F:P,$\mathrm{Ai}$,http://dlmf.nist.gov/9.2#i, -\AiryBi@{z},Airy function ${\rm Bi}$,Airy-Bi,F|F:P,$\mathrm{Bi}$,http://dlmf.nist.gov/9.2#i, -\AiryModulusM@{z},Airy modulus function $M$,modulus-Airy-M,F|F:P,$M$,http://dlmf.nist.gov/9.8#i, -\AiryModulusN@{z},Airy modulus function $N$,modulus-Airy-N,F|F:P,$N$,http://dlmf.nist.gov/9.8#i, -\AiryPhasePhi@{z},Airy phase function $\phi$,phase-Airy-Phi,F|F:P,$\phi$,http://dlmf.nist.gov/9.8#i, -\AiryPhaseTheta@{z},Airy phase function $\theta$,phase-Airy-Theta,F|F:P,$\theta$,http://dlmf.nist.gov/9.8#i, -\AlSalamCarlitzI{a}{n}@{x}{q},Al-Salam-Carlitz I polynomial,EMPTY,P|F:P,$U^{(n)}_{\alpha}$,http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzI, -\AlSalamCarlitzII{a}{n}@{x}{q},Al-Salam-Carlitz II polynomial,EMPTY,P|F:P,$V^{(n)}_{\alpha}$,http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzII, -\AlSalamChihara{n}@@{x}{a}{b}{q},Al-Salam-Chihara polynomial,Al-Salam-Chihara-polynomial-Q,P|F:P:PS,$Q_{n}$,http://drmf.wmflabs.org/wiki/Definition:AlSalamChihara, -\AlSalamChiharaQ{n}@{x}{a}{b}{q},Al-Salam-Chihara polynomial,AlSalam-Chihara-polynomial-Q,P|F:P,$Q_{n}$,http://dlmf.nist.gov/18.28#E7, -\AlSalamIsmail{n}@{x}{s}{q},Al-Salam Ismail polynomial,Al-Salam-Ismail-q-polynomial-U,P|P:F,$U_{n}$,http://drmf.wmflabs.org/wiki/Definition:AlSalamIsmail, -\AngerA{\nu}@{z},Anger-Weber function,associated-Anger-Weber-A,F|F:P,$\mathbf{A}_{\nu}$,http://dlmf.nist.gov/11.10#E4, -\AngerJ{\nu}@{z},Anger function,Anger-J,F|F:P,$\mathbf{J}_{\nu}$,http://dlmf.nist.gov/11.10#E1, -\AntiDer@{x}{f(x)},semantic antiderivative,antiderivative,SM|O,"$\int f(x){\mathrm d}\!x$",http://drmf.wmflabs.org/wiki/Definition:AntiDer, -\AppellFi@{\alpha}{\beta}{\beta'}{\gamma}{x}{y},Appell function $F_1$,Appell-F-1,F|F:P,${F_{1}}$,http://dlmf.nist.gov/16.13#E1, -\AppellFii@{\alpha}{\beta}{\beta'}{\gamma}{\gamma'}{x}{y},Appell function $F_2$,Appell-F-2,F|F:P,${F_{2}}$,http://dlmf.nist.gov/16.13#E2, -\AppellFiii@{\alpha}{\alpha'}{\beta}{\beta'}{\gamma}{x}{y},Appell function $F_3$,Appell-F-3,F|F:P,${F_{3}}$,http://dlmf.nist.gov/16.13#E3, -\AppellFiv@{\alpha}{\beta}{\gamma}{\gamma'}{x}{y},Appell function $F_4$,Appell-F-4,F|F:P,${F_{4}}$,http://dlmf.nist.gov/16.13#E4, -\Asec@@{z},general inverse secant function,multivalued-inverse-secant,F|F:P:nP,$\mathrm{Arcsec}$,http://dlmf.nist.gov/4.23#E5, -\Asech@@{z},general inverse hyperbolic secant function,multivalued-hyperbolic-inverse-secant,F|F:P:nP,$\mathrm{Arcsech}$,http://dlmf.nist.gov/4.37#E5, -\Asin@@{z},general inverse sine function,multivalued-inverse-sine,F|F:P:nP,$\mathrm{Arcsin}$,http://dlmf.nist.gov/4.23#E1, -\Asinh@@{z},general inverse hyperbolic sine function,multivalued-hyperbolic-inverse-sine,F|F:P:nP,$\mathrm{Arcsinh}$,http://dlmf.nist.gov/4.37#E1, -\AskeyWilson{n}@{x}{a}{b}{c}{d}{q},Askey-Wilson polynomial,Askey-Wilson-polynomial-p,P|F:P,$p_{n}$,http://dlmf.nist.gov/18.28#E1, -\AskeyWilsonp{n}@{x}{a}{b}{c}{d}{q},Askey-Wilson polynomial,Askey-Wilson-polynomial-p,P|F:P,$p_{n}$,http://dlmf.nist.gov/18.28#E1, -\AssJacobiP{\alpha}{\beta}{n}@{x}{c},associated Jacobi polynomial,associated-Jacobi-polynomial-P,P|F:P,"$P^{(\alpha,\alpha)}_{n}$",http://dlmf.nist.gov/18.30#E4, -\AssLegendrePoly{n}@{x}{c},associated Legendre polynomial,Legendre-spherical-polynomial-P,F|F:P,$P_{n}$,http://dlmf.nist.gov/18.30#E6, -\Atan@@{z},general inverse tangent function,multivalued-inverse-tangent,F|F:P:nP,$\mathrm{Arctan}$,http://dlmf.nist.gov/4.23#E3, -\Atanh@@{z},general inverse hyperbolic tangent function,multivalued-hyperbolic-inverse-tangent,F|F:P:nP,$\mathrm{Arctanh}$,http://dlmf.nist.gov/4.37#E3, -\BarnesGamma@{z},Barnes $G$-function (or double gamma function),Barnes-Gamma,F|F:P,$G$,http://dlmf.nist.gov/5.17#E1, -\BellNumber@{n},Bell number,Bell-number,I|F:P,$B$,http://dlmf.nist.gov/26.7#i, -\BernoulliB{n}@{x},Bernoulli polynomial,Bernoulli-polynomial-B,P|F:P,$B_{n}$,http://dlmf.nist.gov/24.2#i, -\BesselA{\nu}@{\Matrix{T}},Bessel function of matrix argument of the first kind,Bessel-of-matrix-A,F|F:P,$A_{\nu}$,http://dlmf.nist.gov/35.5#i, -\BesselB{\nu}@{\Matrix{T}},Bessel function of matrix argument of the second kind,Bessel-of-matrix-B,F|F:P,$B_{\nu}$,http://dlmf.nist.gov/35.5#E3, -\BesselI{\nu}@{z},modified Bessel function of the first kind,modified-Bessel-first-kind,F|F:P,$I_{\nu}$,http://dlmf.nist.gov/23.1, -\BesselItilde{\nu}@{x},modified Bessel function of the first kind of imaginary order,modified-Bessel-first-kind-imaginary-order,F|F:P,$\widetilde{I}_{\nu}$,http://dlmf.nist.gov/23.1, -\BesselJ{\nu}@{z},Bessel function of the first kind,Bessel-J,F|F:P,$J_{\nu}$,http://dlmf.nist.gov/10.2#E2, -\BesselJtilde{\nu}@{x},Bessel function of the first kind of imaginary order,Bessel-J-imaginary-order,F|F:P,$\widetilde{J}_{\nu}$,http://dlmf.nist.gov/10.24#Ex1, -\BesselK{\nu}@{z},modified Bessel function of the second kind,modified-Bessel-second-kind,F|F:P,$K_{\nu}$,http://dlmf.nist.gov/23.1, -\BesselKtilde{\nu}@{x},modified Bessel function of the second kind of imaginary order,modified-Bessel-second-kind-imaginary-order,F|F:P,$\widetilde{K}_{\nu}$,http://dlmf.nist.gov/10.45#E2, -\BesselModulusM{\nu}@{x},modulus of Bessel functions,modulus-Bessel-M,F|F:P,$M_{\nu}$,http://dlmf.nist.gov/10.18#E1, -\BesselModulusN{\nu}@{x},modulus of derivatives of Bessel functions,modulus-Bessel-N,F|F:P,$N_{\nu}$,http://dlmf.nist.gov/10.18#E2, -\BesselPhasePhi{\nu}@{x},phase of derivatives of Bessel functions,phase-Bessel-Phi,F|F:P,$\phi_{\nu}$,http://dlmf.nist.gov/10.18#E3, -\BesselPhaseTheta{\nu}@{x},phase of Bessel functions,phase-Bessel-Theta,F|F:P,$\theta_{\nu}$,http://dlmf.nist.gov/10.18#E3, -\BesselPoly{n}@{x}{a},Bessel polynomial,Bessel-polynomial-y,P|F:P,$y_{n}$,http://dlmf.nist.gov/18.34#E1, -\BesselPolyy{n}@{x}{a},Bessel polynomial,Bessel-polynomial-y,P|F:P,$y_{n}$,http://dlmf.nist.gov/18.34#E1, -\BesselY{\nu}@{z},Bessel function of the second kind,Bessel-Y-Weber,F|F:P,$Y_{\nu}$,http://dlmf.nist.gov/10.2#E3, -\BesselYtilde{\nu}@{x},Bessel function of the second kind of imaginary order,Bessel-Y-Weber-imaginary-order,F|F:P,$\widetilde{Y}_{\nu}$,http://dlmf.nist.gov/10.24#Ex1, -\BickleyKi{\alpha}@{x},Bickley function,Bickley-Ki,F|F:P,$\mathrm{Ki}_{\alpha}$,http://dlmf.nist.gov/10.43#E11, -\BigO@{x},order not exceeding,Big-O,Q|F:P,$O$,http://dlmf.nist.gov/2.1#E3, -\CanonicInt{K}@{\Vector{x}},canonical integral $\Psi_K$,canonical-integral,F|F:P,$\Psi_{K}$,http://dlmf.nist.gov/36.2#E4, -\CanonicIntU@{\Vector{x}},umbilic canonical integral $\Psi^{({\rm U})}$,canonical-umbilic-integral,F|F:P,$\Psi^{(\mathrm{U})}$,http://dlmf.nist.gov/36.2#E5, -\CanonicIntE@{\Vector{x}},elliptic umbilic canonical integral $\Psi^{({\rm E})}$,elliptic-umbilic-canonical-integral,F|F:P,$\Psi^{(\mathrm{E})}$,http://dlmf.nist.gov/36.2#E6, -\CanonicIntH@{\Vector{x}},hyperbolic umbilic canonical integral $\Psi^{({\rm H})}$,hyperbolic-umbilic-canonical-integral,F|F:P,$\Psi^{(\mathrm{H})}$,http://dlmf.nist.gov/36.2#E8, -\CatalanNumber@{n},Catalan number,Catalan-number,I|F:P,$C$,http://dlmf.nist.gov/26.5#E1, -\CatalansConstant,Catalan's constant,Catalan's-constant,C|F,$G$,http://dlmf.nist.gov/25.11.E40, -\Charlier{n}@{x}{a},Charlier polynomial,Charlier-polynomial-C,P|F:P,$C_{n}$,http://dlmf.nist.gov/18.19#T1.t1.r11, -\CharlierC{n}@{x}{a},Charlier polynomial,Charlier-polynomial-C,P|F:P,$C_{n}$,http://dlmf.nist.gov/18.19#T1.t1.r11, -\ChebyC{n}@{x},dilated Chebyshev polynomial of the first kind,dilated-Chebyshev-polynomial-second-kind-C,P|F:P,$C_{n}$,http://dlmf.nist.gov/18.1#E3, -\ChebyS{n}@{x},dilated Chebyshev polynomial of the second kind,dilated-Chebyshev-polynomial-first-kind-S,P|F:P,$S_{n}$,http://dlmf.nist.gov/18.1#E3, -\ChebyT{n}@{x},Chebyshev polynomial of the first kind,Chebyshev-polynomial-first-kind-T,P|F:P,$T_{n}$,http://dlmf.nist.gov/18.3#T1.t1.r8, -\ChebyTs{n}@{x},shifted Chebyshev polynomial of the first kind,shifted-Chebyshev-polynomial-first-kind-T-star,P|F:P,$T^{*}_{n}$,http://dlmf.nist.gov/18.3#T1.t1.r20, -\ChebyU{n}@{x},Chebyshev polynomial of the second kind,Chebyshev-polynomial-second-kind-U,P|F:P,$U_{n}$,http://dlmf.nist.gov/18.3#T1.t1.r11, -\ChebyUs{n}@{x},shifted Chebyshev polynomial of the second kind,shifted-Chebyshev-polynomial-second-kind-U-star,P|F:P,$U^{*}_{n}$,http://dlmf.nist.gov/18.3#T1.t1.r23, -\ChebyV{n}@{x},Chebyshev polynomial of the third kind,Chebyshev-polynomial-third-kind-V,P|F:P,$V_{n}$,http://dlmf.nist.gov/18.3#T1.t1.r14, -\ChebyW{n}@{x},Chebyshev polynomial of the fourth kind,Chebyshev-polynomial-fourth-kind-W,P|F:P,$W_{n}$,http://dlmf.nist.gov/18.3#T1.t1.r17, -\ChebyshevPsi@{x},Chebyshev $\ChebyshevPsi$-function,Chebyshev-psi,F|F:P,$\psi$,http://dlmf.nist.gov/25.16#E1, -\CiglerqChebyT{n}@{x}{s}{q},Cigler $q$-Chebyshev polynomial $T$,Cigler-q-Chebyshev-polynomial-T,P|P:F,$T_{n}$,http://drmf.wmflabs.org/wiki/Definition:CiglerqChebyT, -\CiglerqChebyU{n}@{x}{s}{q},Cigler $q$-Chebyshev polynomial $U$,Cigler-q-Chebyshev-polynomial-U,P|P:F,$U_{n}$,http://drmf.wmflabs.org/wiki/Definition:CiglerqChebyU, -\CompEllIntCE@@{k},Legendre's complementary complete elliptic integral of the second kind,complementary-complete-elliptic-integral-second-kind-E,F|F:P:nP,${E'}$,http://dlmf.nist.gov/19.2#E9, -\CompEllIntCK@@{k},Legendre's complementary complete elliptic integral of the first kind,complementary-complete-elliptic-integral-first-kind-K,F|F:P:nP,${K'}$,http://dlmf.nist.gov/19.2#E9, -\CompEllIntD@@{k},complete elliptic integral of Legendre's type,complete-elliptic-integral-D,F|F:P:nP,$D$,http://dlmf.nist.gov/19.2#E8, -\CompEllIntE@@{k},Legendre's complete elliptic integral of the second kind,complete-elliptic-integral-second-kind-E,F|F:P:nP,$E$,http://dlmf.nist.gov/19.2#E8, -\CompEllIntK@@{k},Legendre's complete elliptic integral of the first kind,complete-elliptic-integral-first-kind-K,F|F:P:nP,$K$,http://dlmf.nist.gov/19.2#E8, -\CompEllIntPi@{\alpha^2}{k},Legendre's complete elliptic integral of the third kind,complete-elliptic-integral-third-kind-Pi,F|F:P,$\Pi$,http://dlmf.nist.gov/19.2#E8, -\Complex,set of complex numbers,EMPTY,SN|F,$\mathbb{C}$,http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r1, -\ComplexZeroAiryBi{k},$k$\textsuperscript{th} complex zero of Airy $\AiryBi$,complex-zero-Airy-Bi,F|F,$\beta_{k}$,http://dlmf.nist.gov/9.9#SS1.p2, -\ComplexZeroAiryBiPrime{k},$k$\textsuperscript{th} complex zero of Airy $\AiryBi'$,complex-zero-derivative-Airy-Bi,F|F,$\beta'_{k}$,http://dlmf.nist.gov/9.9#SS1.p2, -\CompositionsC[m]@{n},number of compositions,number-of-compositions-C,I|F:F:P:P,$c_{m}$,http://dlmf.nist.gov/26.11#p1, -\CosInt@{z},cosine integral ${\rm Ci}$,cosine-integral,F|F:P,$\mathrm{Ci}$,http://dlmf.nist.gov/6.2#E11, -\CosIntCin@{z},cosine integral ${\rm Cin}$,cosine-integral-Cin,F|F:P,$\mathrm{Cin}$,http://dlmf.nist.gov/6.2#E12, -\CosIntg@{a}{z},generalized cosine integral ${\rm Ci}$,generalized-cosine-integral-Ci,F|F:P,$\mathrm{Ci}$,http://dlmf.nist.gov/8.21#E2, -\CoshInt@{z},hyperbolic cosine integral,hyperbolic-cosine-integral,F|F:P,$\mathrm{Chi}$,http://dlmf.nist.gov/6.2#E16, -\CoulombC{\ell}@{\eta},normalizing constant for Coulomb radial functions,Coulomb-C,F|F:P,$C_{\ell}$,http://dlmf.nist.gov/33.2#E5, -\CoulombF{\ell}@{\eta}{\rho},regular Coulomb radial function,regular-Coulomb-F,F|F:P,$F_{\ell}$,http://dlmf.nist.gov/33.2#E3, -\CoulombG{\ell}@{\eta}{\rho},irregular Coulomb radial function,irregular-Coulomb-G,F|F:P,$G_{\ell}$,http://dlmf.nist.gov/33.2#E11, -\CoulombH{\pm}{\ell}@{\eta}{\rho},irregular Coulomb radial functions,irregular-Coulomb-H,F|F:P,${H^{\pm}_{\ell}}$,http://dlmf.nist.gov/33.2#E7, -\CoulombM{\ell}@{\eta}{\rho},envelope of Coulomb functions,envelope-Coulomb-M,F|F:P,$M_{\ell}$,http://dlmf.nist.gov/33.3#E1, -\CoulombSigma{\ell}@{\eta},Coulomb phase shift,Coulomb-phase-shift,F|F:P,${\sigma_{\ell}}$,http://dlmf.nist.gov/33.2#E10, -\CoulombTheta{\ell}@{\eta}{\rho},phase of Coulomb functions,Coulomb-phase,F|F:P,${\theta_{\ell}}$,http://dlmf.nist.gov/33.2#E9, -\Coulombc@{\epsilon}{\ell}{r},irregular Coulomb function $c$,Coulomb-c,F|F:P,$c$,http://dlmf.nist.gov/23.1, -\Coulombf@{\epsilon}{\ell}{r},regular Coulomb function $f$,Coulomb-f,F|F:P,$f$,http://dlmf.nist.gov/33.14#E4, -\Coulombh@{\epsilon}{\ell}{r},irregular Coulomb function $h$,Coulomb-h,F|F:P,$h$,http://dlmf.nist.gov/33.14#E7, -\Coulombrhotp@{\eta}{\ell},outer turning point for Coulomb radial functions,Coulomb-radial-outer-turning-point,F|F:P,$\rho_{\mathrm{tp}}$,http://dlmf.nist.gov/33.2#E2, -\Coulombrtp@{\epsilon}{\ell},outer turning point for Coulomb functions,Coulomb-outer-turning-point,F|F:P,$r_{\mathrm{tp}}$,http://dlmf.nist.gov/33.14#E3, -\Coulombs@{\epsilon}{\ell}{r},regular Coulomb function $s$,Coulomb-s,F|F:P,$s$,http://dlmf.nist.gov/33.14#E9, -\CuspCat{K}@{t}{\Vector{x}},cuspoid catastrophe,cuspoid-catastrophe,F|F:P,$\Phi_{K}$,http://dlmf.nist.gov/36.2#E1, -\Cylinder{\nu}@{z},cylinder function,cylinder-function,F|F:P,$\mathscr{C}_{\nu}$,http://dlmf.nist.gov/10.2#P5.p1, -\DawsonsInt@{z},Dawson's integral,Dawsons-integral,F|F:P,$F$,http://dlmf.nist.gov/7.2#E5, -\DedekindModularEta@{\tau},Dedekind's eta function (or Dedekind modular function),Dedekind-modular-Eta,F|F:P,$\eta$,http://dlmf.nist.gov/23.15#E9,http://dlmf.nist.gov/27.14#E12 -\DiffCat{K}@{\Vector{x}}{k},diffraction catastrophe,diffraction-catastrophe,F|F:P,$\Psi_{K}$,http://dlmf.nist.gov/36.2#E10, -\DiffCatE@{\Vector{x}}{k},elliptic umbilic canonical integral function,diffraction-elliptic-umbilic-catastrophe,F|F:P,$\Psi^{(\mathrm{E})}$,http://dlmf.nist.gov/23.1, -\DiffCatH@{\Vector{x}}{k},hyperbolic umbilic canonical integral function,diffraction-hyperbolic-umbilic-catastrophe,F|F:P,$\Psi^{(\mathrm{H})}$,http://dlmf.nist.gov/36.2#E11, -\DiffCatU@{\Vector{x}}{k},umbilic canonical integral function,diffraction-umbilic-catastrophe,F|F:P,$\Psi^{(\mathrm{U})}$,http://dlmf.nist.gov/36.2#E11, -\Dilogarithm@{z},dilogarithm,dilogarithm,F|F:P,$\mathrm{Li}_2$,http://dlmf.nist.gov/25.12#E1, -\Diracdelta@{x-a},Dirac delta (or Dirac delta function),Dirac-delta,D|F:P,$\Diracdelta@{x-a}$,http://dlmf.nist.gov/1.17#SS1.p1, -\DirichletCharacter@@{n}{k},Dirichlet character,Dirichlet-character,I|F:P:nP,$\chi$,http://dlmf.nist.gov/27.8, -\DirichletL@{s}{\chi},Dirichlet $\DirichletL$-function,Dirichlet-L,F|F:P,$L$,http://dlmf.nist.gov/25.15#E1, -\DiscriminantDelta@{\tau},discriminant function,discriminant-delta,F|F:P,$\Delta$,http://dlmf.nist.gov/27.14#E16, -\DiskOP{\alpha}{m}{n}@{z},disk polynomial,disk-orthogonal-polynomial-R,P|F:P,"$R^{(\alpha)}_{m,n}$",http://dlmf.nist.gov/18.37#E1, -\Distribution{f}{g},distribution,EMPTY,D|F,"$\left\langle f,g\right\rangle$",http://dlmf.nist.gov/1.16#SS1.p5, -\DivisorFunctionD[k]@{n},divisor function,divisor-function-D,F|F:P,$d_{k}$,http://dlmf.nist.gov/27.2#E9, -\DivisorSigma{\alpha}@{n},sum of powers of divisors,divisor-sigma,I|F:P,$\sigma_{\alpha}$,http://dlmf.nist.gov/27.2#E10, -\EllIntD@{\phi}{k},incomplete elliptic integral of Legendre's type,elliptic-integral-third-kind-D,F|F:P,$D$,http://dlmf.nist.gov/19.2#E6, -\EllIntE@{\phi}{k},Legendre's incomplete elliptic integral of the second kind,elliptic-integral-second-kind-E,F|F:P,$E$,http://dlmf.nist.gov/19.2#E5, -\EllIntF@{\phi}{k},Legendre's incomplete elliptic integral of the first kind,elliptic-integral-first-kind-F,F|F:P,$F$,http://dlmf.nist.gov/19.2#E4, -\EllIntPi@{\phi}{\alpha^2}{k},Legendre's incomplete elliptic integral of the third kind,elliptic-integral-third-kind-Pi,F|F:P,$\Pi$,http://dlmf.nist.gov/19.2#E7, -\EllIntR{-a}@{\Vector{b}}{\Vector{z}},Carlson integral $R_{-a}$,Carlson-integral-R,F|F:P,$R_{-a}$,http://dlmf.nist.gov/19.16#E9, -\EllIntRC@@{x}{y},Carlson's elliptic integral $R_C$,Carlson-integral-RC,F|F:P:nP,$R_C$,http://dlmf.nist.gov/19.2#E17, -\EllIntRD@@{x}{y}{z},Carlson's elliptic integral $R_D$,Carlson-integral-RD,F|F:P:nP,$R_D$,http://dlmf.nist.gov/19.16#E5, -\EllIntRF@@{x}{y}{z},Carlson's symmetric elliptic integral of first kind,Carlson-integral-RF,F|F:P:nP,$R_F$,http://dlmf.nist.gov/19.16#E1, -\EllIntRG@@{x}{y}{z},Carlson's symmetric elliptic integral of second kind,Carlson-integral-RG,F|F:P:nP,$R_G$,http://dlmf.nist.gov/19.16#E3, -\EllIntRJ@@{x}{y}{z}{p},Carlson's symmetric elliptic integral of third kind,Carlson-integral-RJ,F|F:P:nP,$R_J$,http://dlmf.nist.gov/19.16#E2, -\EllIntcel@{k_c}{p}{a}{b},Bulirsch's complete elliptic integral,Bulirsch-integral-cel,F|F:P,$\mathrm{cel}$,http://dlmf.nist.gov/19.2#E11, -\EllIntelone@{x}{k_c},Bulirsch's incomplete elliptic integral of the first kind,Bulirsch-integral-el-1,F|F:P,$\mathrm{el1}$,http://dlmf.nist.gov/19.2#E15, -\EllIntelthree@{x}{k_c}{p},Bulirsch's incomplete elliptic integral of the third kind,Bulirsch-integral-el-3,F|F:P,$\mathrm{el3}$,http://dlmf.nist.gov/19.2#E16, -\EllInteltwo@{x}{k_c}{a}{b},Bulirsch's incomplete elliptic integral of the second kind,Bulirsch-integral-el-2,F|F:P,$\mathrm{el2}$,http://dlmf.nist.gov/19.2#E12, -\EulerBeta@{a}{b},beta function,Euler-Beta,F|F:P,$\mathrm{B}$,http://dlmf.nist.gov/5.12#E1, -\EulerConstant,Euler's (Euler-Mascheroni) constant,EMPTY,C|F,$\gamma$,http://dlmf.nist.gov/5.2#E3, -\EulerE{n}@{x},Euler polynomial,Euler-polynomial-E,P|F:P,$E_{n}$,http://dlmf.nist.gov/24.2#ii, -\EulerGamma@{z},Euler's gamma function,Euler-Gamma,F|F:P,$\Gamma$,http://dlmf.nist.gov/5.2#E1, -\EulerPhi@{x},Euler's reciprocal function,Euler-phi,F|F:P,$\mathit{f}$,http://dlmf.nist.gov/27.14#E2, -\EulerSumH@{s},Euler sums,Euler-sum-H,F|F:P,$H$,http://dlmf.nist.gov/25.16#SS2.p1, -\EulerTotientPhi[k]@{n},Euler's totient $\phi_k(n)$,Euler-totient-phi,I|F:P,$\phi_{k}$,http://dlmf.nist.gov/27.2#E6, -\EulerianNumber{n}{k},Eulerian number,Eulerian-number,I|F,$\left\langle n\atop k\right\rangle$,http://dlmf.nist.gov/26.14#SS1.p3, -\ExpInt@{z},exponential integral $E_1$,exponential-integral,F|F:P,$E_1$,http://dlmf.nist.gov/6.2#E1, -\ExpIntEin@{z},complementary exponential integral ${\rm Ein}$,complementary-exponential-integral,F|F:P,$\mathrm{Ein}$,http://dlmf.nist.gov/6.2#E3, -\ExpInti@{x},exponential integral ${\rm Ei}$,exponential-integral-Ei,F|F:P,$\mathrm{Ei}$,http://dlmf.nist.gov/6.2#E1, -\ExpIntn{p}@{z},generalized exponential integral,exponential-integral-En,F|F:P,$E_{p}$,http://dlmf.nist.gov/8.19#E1, -\FerrersHatQ[-\mu]{-\frac{1}{2}+i\tau}@{x},conical function $\widehat{\mathsf{Q}}_\nu^\mu$,Ferrers-conical-legendre-Q-hat,F|FO:PO:FnO:PnO,$\widehat{\mathsf{Q}}_\nu^\mu$,http://dlmf.nist.gov/14.20#E2, -\FerrersP[\mu]{\nu}@{x},Ferrers function of the first kind,Ferrers-Legendre-P-first-kind,F|FO:PO:FnO:PnO,$\mathsf{P}_\nu^\mu$,http://dlmf.nist.gov/14.3#E1, -\FerrersQ[\mu]{\nu}@{x},Ferrers function of the second kind,Ferrers-Legendre-Q-first-kind,F|FO:PO:FnO:PnO,$\mathsf{Q}_\nu^\mu$,http://dlmf.nist.gov/14.3#E2, -\FishersHh{n}@{z},probability function,Fishers-Hh,F|F:P,$\mathit{Hh}_{n}$,http://dlmf.nist.gov/7.18#E12, -\FresnelCos@{z},Fresnel integral $C$,Fresnel-cosine-integral,F|F:P,$C$,http://dlmf.nist.gov/7.2#E7, -\FresnelF@{z},Fresnel integral ${\mathcal F}$,Fresnel-integral,F|F:P,$\mathcal{F}$,http://dlmf.nist.gov/7.2#E6, -\FresnelSin@{z},Fresnel integral $S$,Fresnel-sine-integral,F|F:P,$S$,http://dlmf.nist.gov/7.2#E8, -\Fresnelf@{z},auxiliary function for Fresnel integrals ${\rm f}$,Fresnel-auxilliary-function-f,F|F:P,$\mathrm{f}$,http://dlmf.nist.gov/7.2#E10, -\Fresnelg@{z},auxiliary function for Fresnel integrals ${\rm g}$,Fresnel-auxilliary-function-g,F|F:P,$\mathrm{g}$,http://dlmf.nist.gov/7.2#E11, -\GammaP@{a}{z},normalized incomplete gamma function $P$,incomplete-gamma-P,F|F:P,$P$,http://dlmf.nist.gov/8.2#E4, -\GammaQ@{a}{z},normalized incomplete gamma function $Q$,incomplete-gamma-Q,F|F:P,$Q$,http://dlmf.nist.gov/8.2#E4, -\GaussSum@{n}{\DirichletCharacter},Gauss sum,Gauss-sum,F|F:P,$G$,http://dlmf.nist.gov/27.10#E9, -\GenBernoulliB{\ell}{n}@{x},generalized Bernoulli polynomial,generalized-Bernoulli-polynomial-B,P|F:P,$B^{(\ell)}_{n}$,http://dlmf.nist.gov/24.16#P1.p1, -\GenBesselFun@{\rho}{\beta}{z},generalized Bessel function,generalized-Bessel-function,F|F:P,$\phi$,http://dlmf.nist.gov/10.46#E1, -\GenEulerE{\ell}{n}@{x},generalized Euler polynomial,generalized-Euler-polynomial-E,P|F:P,$E^{(\ell)}_{n}$,http://dlmf.nist.gov/24.16#P1.p1, -\GenEulerSumH@{s}{z},generalized Euler sums,generalized-Euler-sum-H,F|F:P,$H$,http://dlmf.nist.gov/25.16#SS2.p7, -\GenGegenbauer{\alpha}{\beta}{n}@{x},Generalized Gegenbauer polynomial,generalized-Gegenbauer-polynomial-S,P|F:P,"$S^{(\alpha,\beta)}_{n}$",http://drmf.wmflabs.org/wiki/Definition:GenGegenbauer, -\GenHermite[\alpha]{n}@{x},generalized Hermite polynomial,generalized-Hermite-polynomial-H,P|FO:PO:FnO:PnO,$H_n^{\mu}$,http://drmf.wmflabs.org/wiki/Definition:GenHermite, -\GoodStat@{z},Goodwin-Staton integral,Goodwin-Staton-integral,F|F:P,$G$,http://dlmf.nist.gov/7.2#E12, -\GottliebLaguerre{n}@{x},Gottlieb-Laguerre polynomial,Gottlieb-Laguerre-polynomial-l,P|F:P,$l_{n}$,http://drmf.wmflabs.org/wiki/Definition:GottliebLaguerre, -\Gudermannian@@{x},Gudermannian function,inverse-Gudermannian,F|F:P:nP,$\mathrm{gd}$,http://dlmf.nist.gov/4.23#E39, -\Hahn{n}@{x}{\alpha}{\beta}{N},Hahn polynomial,Hahn-polynomial-Q,P|F:P,$Q_{n}$,http://dlmf.nist.gov/18.19#T1.t1.r3, -\HahnQ{n}@{x}{\alpha}{\beta}{N},Hahn polynomial,Hahn-polynomial-Q,P|F:P,$Q_{n}$,http://dlmf.nist.gov/18.19#T1.t1.r3, -\HahnR{n}@{x}{\gamma}{\delta}{N},dual Hahn polynomial,dual-Hahn-R,P|F:P,$R_{n}$,http://dlmf.nist.gov/18.25#T1.t1.r5, -\HahnS{n}@{x}{a}{b}{c},continuous dual Hahn polynomial,continuous-dual-Hahn-S,P|F:P,$S_{n}$,http://dlmf.nist.gov/18.25#T1.t1.r3, -\Hahnp{n}@{x}{a}{b}{\conj{a}}{\conj{b}},continuous Hahn polynomial,continuous-Hahn-polynomial-p,P|F:P,$p_{n}$,http://dlmf.nist.gov/18.19#P2.p1, -\HankelHi{\nu}@{z},Hankel function of the first kind,Hankel-H-1-Bessel-third-kind,F|F:P,${H^{(1)}_{\nu}}$,http://dlmf.nist.gov/10.2#E5, -\HankelHii{\nu}@{z},Hankel function of the second kind,Hankel-H-2-Bessel-third-kind,F|F:P,${H^{(2)}_{\nu}}$,http://dlmf.nist.gov/10.2#E6, -\HarmonicNumber{n},Harmonic number,Harmonic-number,I|F,$H_{n}$,http://dlmf.nist.gov/25.11#E33, -\HeavisideH@{x},Heaviside step function,Heaviside-H,F|F:P,$H$,http://dlmf.nist.gov/1.16#E13, -\Hermite{n}@{x},Hermite polynomial $H_n$,Hermite-polynomial-H,P|F:P,$H_{n}$,http://dlmf.nist.gov/18.3#T1.t1.r28, -\HermiteH{n}@{x},Hermite polynomial $H_n$,Hermite-polynomial-H,P|F:P,$H_{n}$,http://dlmf.nist.gov/18.3#T1.t1.r28, -\HermiteHe{n}@{x},Hermite polynomial $He_n$,Hermite-polynomial-He,P|F:P,$\mathit{He}_{n}$,http://dlmf.nist.gov/18.3#T1.t1.r29, -\HeunFunction{m}{a}{b}@@{a}{q}{\alpha}{\beta}{\gamma}{\delta}{z},Heun functions $(a;b)Hf_m^\nu$,Heun-function,F|FO:PO:nPO:FnO:PnO:nPnO,"$(s_1,s_2)\mathit{Hf}_{m}^\nu$",http://dlmf.nist.gov/31.4#p1, -\HeunLocal@@{a}{q}{\alpha}{\beta}{\gamma}{\delta}{z},Heun functions $Hl$,Heun-local,F|F:P:nP,$\mathit{H\!\ell}$,http://dlmf.nist.gov/31.3#E1, -\HeunPolynom{n}{m}@@{a}{q}{-n}{\beta}{\gamma}{\delta}{z},Heun polynomial,Heun-polynomial-m,P|F:P:nP,"$\mathit{Hp}_{n,m}$",http://dlmf.nist.gov/31.5#E2, -\HilbertTrans@{f}{x},Hilbert transform,Hilbert-transform,O|F:P,$\mathcal{H}$,http://dlmf.nist.gov/1.14#SS5.p1, -\HurwitzZeta@{s}{a},Hurwitz zeta function,Hurwitz-zeta,F|F:P,$\zeta$,http://dlmf.nist.gov/25.11#E1, -\HyperPsi@{a}{b}{\Matrix{T}},confluent hypergeometric function of matrix argument of the second kind,confluent-hypergeometric-of-matrix-Psi,F|F:P,$\Psi$,http://dlmf.nist.gov/35.6#E2, -\HyperboldpFq{p}{q}@@@{\Vector{a}}{\Vector{b}}{z},scaled (or Olver's) generalized hypergeometric function,hypergeometric-bold-pFq,F|F:fo1:fo2:fo3,${{}_{p}{\mathbf{F}}_{q}}$,http://dlmf.nist.gov/16.2#E5, -\HypergeoF@@@{a}{b}{c}{z},hypergeometric function,Gauss-hypergeometric-F,F|F:fo1:fo2:fo3,$F$,http://dlmf.nist.gov/15.2#E1, -\HypergeoboldF@@@{a}{b}{c}{z},Olver's hypergeometric function,scaled-hypergeometric-bold-F,F|F:fo1:fo2:fo3,$\mathbf{F}$,http://dlmf.nist.gov/15.2#E2, -"\HyperpFq{p}{q}@@@{a_1,...,a_p}{b_1,...,b_q}{z}",generalized hypergeometric function,Gauss-hypergeometric-pFq,F|F:fo1:fo2:fo3,${{}_{p}F_{q}}$,http://dlmf.nist.gov/16.2#E1, -"\HyperpHq{p}{q}@@@{{a_1,...,a_p}{b_1,...,b_q}{z}",bilateral hypergeometric function,hypergeometric-pHq,F|F:fo1:fo2:fo3,${{}_{p}H_{q}}$,http://dlmf.nist.gov/16.4#E16, -\IncBeta{x}@{a}{b},incomplete beta function,incomplete-Beta,F|F:P,$\mathrm{B}_{x}$,http://dlmf.nist.gov/8.17#E1, -\IncGamma@{a}{z},incomplete gamma function $\Gamma$,incomplete-Gamma,F|F:P,$\Gamma$,http://dlmf.nist.gov/8.2#E2, -\IncI{x}@{a}{b},regularized incomplete beta function,IncI,F|F:P,$I_{x}$,http://dlmf.nist.gov/8.17#E2, -\Int{a}{b}@{x}{f(x)},semantic definite integral,semantic-definite-integral,SM|O,"$\int_{a}^{b}f(x){\mathrm d}\!x$",http://drmf.wmflabs.org/wiki/Definition:Int, -\Integer,set of integers,EMPTY,SN|F,$\mathbb{Z}$,http://dlmf.nist.gov/front/introduction#Sx4.p2.t1.r20, -\IntgenAiryA{k}@{z}{p},generalized Airy function from integral representation $A_k$,Integral-generalized-Airy-A,F|F:P,$A_{k}$,http://dlmf.nist.gov/9.13#SS2.p1, -\IntgenAiryB{k}@{z}{p},generalized Airy function from integral representation $B_k$,Integral-generalized-Airy-B,F|F:P,$B_{k}$,http://dlmf.nist.gov/9.13#SS2.p1, -\JacksonqBesselI{\nu}@{z}{q},Jackson $q$-Bessel function 1,EMPTY,F|F:P,$J^{(1)}_{q}$,http://drmf.wmflabs.org/wiki/Definition:JacksonqBesselI, -\JacksonqBesselII{\nu}@{z}{q},Jackson $q$-Bessel function 2,EMPTY,F|F:P,$J^{(2)}_{q}$,http://drmf.wmflabs.org/wiki/Definition:JacksonqBesselII, -\JacksonqBesselIII{\nu}@{z}{q},Jackson/Hahn-Exton $q$-Bessel function 3,EMPTY,F|F:P,$J^{(3)}_{q}$,http://drmf.wmflabs.org/wiki/Definition:JacksonqBesselIII, -\Jacobi{\alpha}{\beta}{n}@{x},Jacobi polynomial,Jacobi-polynomial-P,P|F:P,"$P^{(\alpha,\beta)}_{n}$",http://dlmf.nist.gov/18.3#T1.t1.r3, -\JacobiEpsilon@{x}{k},Jacobi's epsilon function,Jacobi-Epsilon,F|F:P,$\mathcal{E}$,http://dlmf.nist.gov/22.16#E14, -\JacobiG{n}@{p}{q}{x},shifted Jacobi polynomial,shifted-Jacobi-polynomial-G,P|F:P,$G_{n}$,http://dlmf.nist.gov/18.1#E2, -\JacobiP{\alpha}{\beta}{n}@{x},Jacobi polynomial,Jacobi-polynomial-P,P|F:P,"$P^{(\alpha,\beta)}_{n}$",http://dlmf.nist.gov/18.3#T1.t1.r3, -\JacobiSymbol{n}{P},Jacobi symbol,Jacobi-symbol,I|F,$(n|P)$,http://dlmf.nist.gov/27.9#p3, -\JacobiTheta{j}@{z}{q},Jacobi theta function,Jacobi-theta,F|F:P,$\theta_{p}$,http://dlmf.nist.gov/20.2#i, -\JacobiThetaCombined{n}{m}@{z}{q},combined theta function,Jacobi-theta-combined,F|F:P,"$\varphi_{m,n}$",http://dlmf.nist.gov/20.11#SS5.p2, -\JacobiThetaTau{j}@{z}{\tau},Jacobi theta $\tau$ function $\theta_j$,Jacobi-theta-tau,F|F:P,$\theta_{p}$,http://dlmf.nist.gov/20.2#i, -\JacobiZeta@{x}{k},Jacobi's zeta function,Jacobi-Zeta,F|F:P,$\mathrm{Z}$,http://dlmf.nist.gov/22.16#E32, -\Jacobiam@@{x}{k},Jacobi's amplitude function ${\rm am}$,Jacobi-elliptic-amplitude,F|F:P:nP,$\mathrm{am}$,http://dlmf.nist.gov/23.1, -\Jacobicd@@{z}{k},Jacobian elliptic function ${\rm cd}$,Jacobi-elliptic-cd,F|F:P:nP,$\mathrm{cd}$,http://dlmf.nist.gov/22.2#E8, -\Jacobicn@@{z}{k},Jacobian elliptic function ${\rm cn}$,Jacobi-elliptic-cn,F|F:P:nP,$\mathrm{cn}$,http://dlmf.nist.gov/22.2#E5, -\Jacobics@@{z}{k},Jacobian elliptic function ${\rm cs}$,Jacobi-elliptic-cs,F|F:P:nP,$\mathrm{cs}$,http://dlmf.nist.gov/23.1, -\Jacobidc@@{z}{k},Jacobian elliptic function ${\rm dc}$,Jacobi-elliptic-dc,F|F:P:nP,$\mathrm{dc}$,http://dlmf.nist.gov/23.1, -\Jacobidn@@{z}{k},Jacobian elliptic function ${\rm dn}$,Jacobi-elliptic-dn,F|F:P:nP,$\mathrm{dn}$,http://dlmf.nist.gov/22.2#E6, -\Jacobids@@{z}{k},Jacobian elliptic function ${\rm ds}$,Jacobi-elliptic-ds,F|F:P:nP,$\mathrm{ds}$,http://dlmf.nist.gov/23.1, -\Jacobinc@@{z}{k},Jacobian elliptic function ${\rm nc}$,Jacobi-elliptic-nc,F|F:P:nP,$\mathrm{nc}$,http://dlmf.nist.gov/23.1, -\Jacobind@@{z}{k},Jacobian elliptic function ${\rm nd}$,Jacobi-elliptic-nd,F|F:P:nP,$\mathrm{nd}$,http://dlmf.nist.gov/23.1, -\Jacobins@@{z}{k},Jacobian elliptic function ${\rm ns}$,Jacobi-elliptic-ns,F|F:P:nP,$\mathrm{ns}$,http://dlmf.nist.gov/23.1, -\Jacobiphi{\alpha}{\beta}{\lambda}@{t},Jacobi function,Jacobi-hypergeometric-phi,F|F:P,"$\phi^{(\alpha,\beta)}_{\lambda}$",http://dlmf.nist.gov/15.9#E11, -\Jacobisc@@{z}{k},Jacobian elliptic function ${\rm sc}$,Jacobi-elliptic-sc,F|F:P:nP,$\mathrm{sc}$,http://dlmf.nist.gov/22.2#E9, -\Jacobisd@@{z}{k},Jacobian elliptic function ${\rm sd}$,Jacobi-elliptic-sd,F|F:P:nP,$\mathrm{sd}$,http://dlmf.nist.gov/22.2#E7, -\Jacobisn@@{z}{k},Jacobian elliptic function ${\rm sn}$,Jacobi-elliptic-sn,F|F:P:nP,$\mathrm{sn}$,http://dlmf.nist.gov/22.2#E4, -\JonquierePhi@{z}{s}=\Polylogarithm{s}@{z},Jonquiére's function,Jonquiere-phi,F|F:P,$\phi$,http://dlmf.nist.gov/23.1, -\JordanJ{k}@{n},Jordan's function,Jordan-J,F|F:P,$J_{k}$,http://dlmf.nist.gov/27.2#E11, -\Kelvinbei{\nu}@@{x},Kelvin function ${\rm bei}_\nu$,Kelvin-bei,F|F:P:nP,$\mathrm{bei}_{\nu}$,http://dlmf.nist.gov/10.61#E1, -\Kelvinber{\nu}@@{x},Kelvin function ${\rm ber}_\nu$,Kelvin-ber,F|F:P:nP,$\mathrm{ber}_{\nu}$,http://dlmf.nist.gov/10.61#E1, -\Kelvinkei{\nu}@@{x},Kelvin function ${\rm kei}_\nu$,Kelvin-kei,F|F:P:nP,$\mathrm{kei}_{\nu}$,http://dlmf.nist.gov/10.61#E2, -\Kelvinker{\nu}@@{x},Kelvin function ${\rm ker}_\nu$,Kelvin-ker,F|F:P:nP,$\mathrm{ker}_{\nu}$,http://dlmf.nist.gov/10.61#E2, -\Krawtchouk{n}@{x}{p}{N},Krawtchouk polynomial,Krawtchouk-polynomial-K,P|F:P,$K_{n}$,http://dlmf.nist.gov/18.19#T1.t1.r6, -\KrawtchoukK{n}@{x}{p}{N},Krawtchouk polynomial,Krawtchouk-polynomial-K,P|F:P,$K_{n}$,http://dlmf.nist.gov/18.19#T1.t1.r6, -\Kronecker{j}{k},Kronecker delta,EMPTY,I|F,"$\delta_{m,n}$",http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r4, -\KummerM@{a}{b}{z},Kummer confluent hypergeometric function $M$,Kummer-confluent-hypergeometric-M,F|F:P,$M$,http://dlmf.nist.gov/13.2#E2, -\KummerU@{a}{b}{z},Kummer confluent hypergeometric function $U$,Kummer-confluent-hypergeometric-U,F|F:P,$U$,http://dlmf.nist.gov/13.2#E6, -\KummerboldM@{a}{b}{z},Olver's confluent hypergeometric function,Kummer-confluent-hypergeometric-bold-M,F|F:P,${\mathbf{M}}$,http://dlmf.nist.gov/13.2#E3, -\Laguerre[\alpha]{n}@{x},Laguerre (or generalized Laguerre) polynomial,generalized-Laguerre-polynomial-L,P|FnO:PnO:FO:PO,$L_n^{(\alpha)}$,http://dlmf.nist.gov/18.3#T1.t1.r27, -\LaguerreL[\alpha]{n}@{x},Laguerre (or generalized Laguerre) polynomial,generalized-Laguerre-polynomial-L,P|FnO:PnO:FO:PO,$L_n^{(\alpha)}$,http://dlmf.nist.gov/18.3#T1.t1.r27, -\LambertW@{x},Lambert $\LambertW$-function,Lambert-W,F|F:P,$W$,http://dlmf.nist.gov/23.1, -\LambertWm@{x},non-principal branch of Lambert $\LambertW$-function,Lambert-Wm,F|F:P,$\mathrm{Wm}$,http://dlmf.nist.gov/4.13#p2, -\LambertWp@{x},principal branch of Lambert $\LambertW$-function,Lambert-Wp,F|F:P,$\mathrm{Wp}$,http://dlmf.nist.gov/23.1, -\LameEc{m}{\nu}@{z}{k^2},Lamé function $Ec_\nu^m$,Lame-Ec,F|F:P,$\mathit{Ec}^{m}_{\nu}$,http://dlmf.nist.gov/23.1, -\LameEs{m}{\nu}@{z}{k^2},Lamé function $Es_\nu^m$,Lame-Es,F|F:P,$\mathit{Es}^{m}_{\nu}$,http://dlmf.nist.gov/23.1, -\Lamea{n}{\nu}@{k^2},eigenvalues of Lamé's equation $a_\nu^m$,Lame-eigenvalue-a,F|F:P,$a^{n}_{\nu}$,http://dlmf.nist.gov/29.3#SS1.p1, -\Lameb{n}{\nu}@{k^2},eigenvalues of Lamé's equation $b_\nu^m$,Lame-eigenvalue-b,F|F:P,$b^{n}_{\nu}$,http://dlmf.nist.gov/29.3#SS1.p1, -\LamecE{m}{2n+1}@{z}{k^2},Lamé polynomial $cE_{2n+1}^m$,Lame-polynomial-cE,P|F:P,$\mathit{cE}^{m}_{\nu}$,http://dlmf.nist.gov/23.1, -\LamecdE{m}{2n+2}@{z}{k^2},Lamé polynomial $cdE_{2n+2}^m$,Lame-polynomial-cdE,P|F:P,$\mathit{cdE}^{m}_{\nu}$,http://dlmf.nist.gov/23.1, -\LamedE{m}{2n+1}@{z}{k^2},Lamé polynomial $dE_{2n+1}^m$,Lame-polynomial-dE,P|F:P,$\mathit{dE}^{m}_{\nu}$,http://dlmf.nist.gov/29.12#E4, -\LamesE{m}{2n+1}@{z}{k^2},Lamé polynomial $sE_{2n+1}^m$,Lame-polynomial-sE,P|F:P,$\mathit{sE}^{m}_{\nu}$,http://dlmf.nist.gov/29.12#E2, -\LamescE{m}{2n+2}@{z}{k^2},Lamé polynomial $scE_{2n+2}^m$,Lame-polynomial-scE,P|F:P,$\mathit{scE}^{m}_{\nu}$,http://dlmf.nist.gov/29.12#E5, -\LamescdE{m}{2n+3}@{z}{k^2},Lamé polynomial $scdE_{2n+3}^m$,Lame-polynomial-scdE,P|F:P,$\mathit{scdE}^{m}_{\nu}$,http://dlmf.nist.gov/29.12#E8, -\LamesdE{m}{2n+2}@{z}{k^2},Lamé polynomial $sdE_{2n+2}^m$,Lame-polynomial-sdE,P|F:P,$\mathit{sdE}^{m}_{\nu}$,http://dlmf.nist.gov/29.12#E6, -\LameuE{m}{2n}@{z}{k^2},Lamé polynomial $uE_{2n}^m$,Lame-polynomial-uE,P|F:P,$\mathit{uE}^{m}_{\nu}$,http://dlmf.nist.gov/29.12#E1, -\LaplaceTrans@{f}{s},Laplace transform,Laplace-transform,O|F:P,$\mathscr{L}$,http://dlmf.nist.gov/1.14#E17, -\Lattice{L},lattice in $\Complex$,EMPTY,SN|F,$\mathbb{L}$,http://dlmf.nist.gov/23.2#i, -"\LauricellaFD@{a}{b_1,...,b_n}{c}{z_1,...,z_n}",Lauricella's multivariate hypergeometric function,Lauricella-FD,F|F:P,$F_D$,http://dlmf.nist.gov/19.15#p1, -\LegendreBlackQ[\mu]{\nu}@{z},Olver's associated Legendre function,associated-Legendre-black-Q,F|FO:PO:FnO:PnO,$\boldsymbol{Q}_\nu^\mu$,http://dlmf.nist.gov/14.3#E10, -\LegendreP[\mu]{\nu}@{z},associated Legendre function of the first kind,Legendre-P-first-kind,F|FO:PO:FnO:PnO,$P_\nu^\mu$,http://dlmf.nist.gov/14.3#E6, -\LegendrePoly{n}@{x},Legendre polynomial,Legendre-spherical-polynomial,P|F:P,$P_{n}$,http://dlmf.nist.gov/18.3#T1.t1.r25, -\LegendrePolys{n}@{x},shifted Legendre polynomial,shifted-spherical-Legendre-polynomial-s,P|F:P,$P^{*}_{n}$,http://dlmf.nist.gov/18.3#T1.t1.r26, -\LegendreQ[\mu]{\nu}@{z},associated Legendre function of the second kind,Legendre-Q-second-kind,F|FO:PO:FnO:PnO,$Q_\nu^\mu$,http://dlmf.nist.gov/14.3#E7, -\LegendreSymbol{n}{p},Legendre symbol,Legendre-symbol,I|F,$(n|p)$,http://dlmf.nist.gov/27.9, -\LerchPhi@{z}{s}{a},Lerch's transcendent,Lerch-Phi,F|F:P,$\Phi$,http://dlmf.nist.gov/25.14#E1, -\LeviCivita{j}{k}{\ell},Levi-Civita symbol,EMPTY,I|F,$\epsilon_{jkl}$,http://dlmf.nist.gov/1.6#E14, -\LinBrF@{a}{u},line-broadening function,line-broadening-function,F|F:P,$H$,http://dlmf.nist.gov/7.19#E4, -\LiouvilleLambda@{n},Liouville's function,Liouville-lambda,F|F:P,$\lambda$,http://dlmf.nist.gov/27.2#E13, -\Ln@@{z},general logarithm function,multivalued-natural-logarithm,F|F:P:nP,$\mathrm{Ln}$,http://dlmf.nist.gov/4.2#E1, -\LogInt@{x},logarithmic integral,logarithmic-integral,F|F:P,$\mathrm{li}$,http://dlmf.nist.gov/6.2#E8, -\LommelS{\mu}{\nu}@{z},"Lommel function $S$",Lommel-S,F|F:P,"$S_{{\mu},{\nu}}$",http://dlmf.nist.gov/11.9#E5, -\Lommels{\mu}{\nu}@{z},"Lommel function $s$",Lommel-s,F|F:P,"$s_{{\mu},{\nu}}$",http://dlmf.nist.gov/11.9#E3, -\Longleftrightarrow,is equivalent to,equivalent-to,SM|F,$\Longleftrightarrow $,http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r13, -\Longrightarrow,implies,EMPTY,SM|F,$\Longrightarrow$,http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r12, -\MangoldtLambda@{n},Mangoldt's function,Mangoldt-Lambda,F|F:P,$\Lambda$,http://dlmf.nist.gov/27.2#E14, -\MathieuCe{\nu}@@{z}{q},modified Mathieu function ${\rm Ce}_\nu$,modified-Mathieu-Ce,F|F:P:nP,$\mathrm{Ce}_{n}$,http://dlmf.nist.gov/28.20#E3, -\MathieuD{j}@{\nu}{\mu}{z},cross-products of modified Mathieu functions and their derivatives ${\rm D}_{j}$,Mathieu-D,F|F:P,$\mathrm{D}_{j}$,http://dlmf.nist.gov/28.28#E24, -\MathieuDc{j}@{n}{m}{z},cross-products of radial Mathieu functions and their derivatives ${\rm Dc}_j$,Mathieu-Dc,F|F:P,$\mathrm{Dc}_{j}$,http://dlmf.nist.gov/28.28#E39, -\MathieuDs{j}@{n}{m}{z},cross-products of radial Mathieu functions and their derivatives ${\rm Ds}_j$,Mathieu-Ds,F|F:P,$\mathrm{Ds}_{j}$,http://dlmf.nist.gov/28.28#E35, -\MathieuDsc{j}@{n}{m}{z},cross-products of radial Mathieu functions and their derivatives ${\rm Dsc}_j$,Mathieu-Dsc,F|F:P,$\mathrm{Dsc}_{j}$,http://dlmf.nist.gov/28.28#E40, -\MathieuEigenvaluea{n}@@{q},eigenvalues of Mathieu equation $a_n(q)$,Mathieu-eigenvalue-a,F|F:P:nP,$a_{n}$,http://dlmf.nist.gov/28.2#SS5.p1, -\MathieuEigenvalueb{n}@@{q},eigenvalues of Mathieu equation $b_n(q)$,Mathieu-eigenvalue-b,F|F:P:nP,$b_{n}$,http://dlmf.nist.gov/28.2#SS5.p1, -\MathieuEigenvaluelambda{\nu+2n}@@{q},eigenvalues of Mathieu equation $\lambda_\nu(q)$,Mathieu-eigenvalue-lambda,F|F:P:nP,$\lambda_{\nu}$,http://dlmf.nist.gov/28.12#SS1.p2, -\MathieuFc{m}@{z}{h},Mathieu function ${\rm Fc}_m$,Mathieu-Fc,F|F:P,$\mathrm{Fc}_{m}$,http://dlmf.nist.gov/23.1, -\MathieuFe{n}@@{z}{q},modified Mathieu function ${\rm Fe}_n$,modified-Mathieu-Fe,F|F:P:nP,$\mathrm{Fe}_{n}$,http://dlmf.nist.gov/28.20#E6, -\MathieuFs{m}@{z}{h},Mathieu function ${\rm Fs}_m$,Mathieu-Fs,F|F:P,$\mathrm{Fs}_{m}$,http://dlmf.nist.gov/23.1, -\MathieuGc{m}@{z}{h},Mathieu function ${\rm Gc}_m$,Mathieu-Gc,F|F:P,$\mathrm{Gc}_{m}$,http://dlmf.nist.gov/23.1, -\MathieuGe{n}@@{z}{q},modified Mathieu function ${\rm Ge}_n$,modified-Mathieu-Ge,F|F:P:nP,$\mathrm{Ge}_{n}$,http://dlmf.nist.gov/28.20#E7, -\MathieuGs{m}@{z}{h},Mathieu function ${\rm Gs}_m$,Mathieu-Gs,F|F:P,$\mathrm{Gs}_{m}$,http://dlmf.nist.gov/23.1, -\MathieuIe{n}@@{z}{h},modified Mathieu function ${\rm Ie}_n$,modified-Mathieu-Ie,F|F:P:nP,$\mathrm{Ie}_{n}$,http://dlmf.nist.gov/28.20#E17, -\MathieuIo{n}@@{z}{h},modified Mathieu function ${\rm Io}_n$,modified-Mathieu-Io,F|F:P:nP,$\mathrm{Io}_{n}$,http://dlmf.nist.gov/28.20#E18, -\MathieuKe{n}@@{z}{h},modified Mathieu function ${\rm Ke}_n$,modified-Mathieu-Ke,F|F:P:nP,$\mathrm{Ke}_{n}$,http://dlmf.nist.gov/28.20#E19, -\MathieuKo{n}@@{z}{h},modified Mathieu function ${\rm Ko}_n$,modified-Mathieu-Ko,F|F:P:nP,$\mathrm{Ko}_{n}$,http://dlmf.nist.gov/28.20#E20, -\MathieuM{j}{\nu}@@{z}{h},modified Mathieu function ${\rm M}_\nu^{(j)}$,modified-Mathieu-M,F|F:P:nP,${\mathrm{M}^{(j)}_{\nu}}$,http://dlmf.nist.gov/23.1, -\MathieuMc{j}{n}@@{z}{h},radial Mathieu function ${\rm Mc}_n^{(j)}$,modified-Mathieu-Mc,F|F:P:nP,${\mathrm{Mc}^{(j)}_{\nu}}$,http://dlmf.nist.gov/28.20#E15, -\MathieuMe{\nu}@@{z}{q},modified Mathieu function ${\rm Me}_\nu$,modified-Mathieu-Me,F|F:P:nP,$\mathrm{Me}_{n}$,http://dlmf.nist.gov/28.20#E5, -\MathieuMs{j}{n}@@{z}{h},radial Mathieu function ${\rm Ms}_n^{(j)}$,modified-Mathieu-Ms,F|F:P:nP,${\mathrm{Ms}^{(j)}_{\nu}}$,http://dlmf.nist.gov/28.20#E16, -\MathieuSe{\nu}@@{z}{q},modified Mathieu function ${\rm Se}_\nu$,modified-Mathieu-Se,F|F:P:nP,$\mathrm{Se}_{n}$,http://dlmf.nist.gov/28.20#E4, -\Mathieuce{n}@@{z}{q},Mathieu function ${\rm ce}_\nu$,Mathieu-ce,F|F:P:nP,$\mathrm{ce}_{n}$,http://dlmf.nist.gov/23.1, -\Mathieufe{n}@@{z}{q},second solution of Mathieu's equation ${\rm fe}_n$,Mathieu-fe,F|F:P:nP,$\mathrm{fe}_{n}$,http://dlmf.nist.gov/28.5#E1, -\Mathieuge{n}@@{z}{q},second solution of Mathieu's equation ${\rm ge}_n$,Mathieu-ge,F|F:P:nP,$\mathrm{ge}_{n}$,http://dlmf.nist.gov/28.5#E2, -\Mathieume{n}@@{z}{q},Mathieu function ${\rm me}_\nu$,Mathieu-me,F|F:P:nP,$\mathrm{me}_{\nu}$,http://dlmf.nist.gov/23.1, -\Mathieuse{n}@@{z}{q},Mathieu function ${\rm se}_\nu$,Mathieu-se,F|F:P:nP,$\mathrm{se}_{\nu}$,http://dlmf.nist.gov/23.1, -\Matrix{I},Bessel function of matrix argument (first kind),EMPTY,F|F,$\Matrix{T}$,http://dlmf.nist.gov/front/introduction#Sx4.p2.t1.r9, -\MeijerG{m}{n}{p}{q}@@@{z}{\Vector{a}}{\Vector{b}},Meijer $G$-function,Meijer-G,F|F:fo1:fo2:fo3,"${G^{m,n}_{p,q}}$",http://dlmf.nist.gov/16.17#E1, -\Meixner{n}@{x}{\beta}{c},Meixner polynomial,Meixner-polynomial-M,P|F:P,$M_{n}$,http://dlmf.nist.gov/18.19#T1.t1.r9, -\MeixnerM{n}@{x}{\beta}{c},Meixner polynomial,Meixner-polynomial-M,P|F:P,$M_{n}$,http://dlmf.nist.gov/18.19#T1.t1.r9, -\MeixnerPollaczek{\lambda}{n}@{x}{\phi},Meixner-Pollaczek polynomial,Meixner-Pollaczek-polynomial-P,P|F:P,$P^{(\alpha)}_{n}$,http://dlmf.nist.gov/18.19#P3.p1, -\MeixnerPollaczekP{\lambda}{n}@{x}{\phi},Meixner-Pollaczek polynomial,Meixner-Pollaczek-polynomial-P,P|F:P,$P^{(\alpha)}_{n}$,http://dlmf.nist.gov/18.19#P3.p1, -\MellinTrans@{f}{s},Mellin transform,Mellin-transform,O|F:P,$\mathscr{M}$,http://dlmf.nist.gov/1.14#E32, -\Mills@{x},Mills ratio ${\sf M}$,Mills-ratio,F|F:P,$\mathsf{M}$,http://dlmf.nist.gov/7.8#E1, -\MittagLeffler{a}{b}@{z},Mittag-Leffler function,Mittag-Leffler-function,F|F:P,"$E_{a,b}$",http://dlmf.nist.gov/10.46#E3, -\ModCylinder{\nu}@{z},modified cylinder function,modified-cylinder-function,F|F:P,$\mathscr{Z}_{\nu}$,http://dlmf.nist.gov/10.25#P2.p1, -\ModularJ@{\tau},Klein's complete invariant,Kleins-invariant-modular-J,F|F:P,$J$,http://dlmf.nist.gov/23.15#E7, -\ModularLambda@{\tau},elliptic modular function,modular-Lambda,F|F:P,$\lambda$,http://dlmf.nist.gov/23.15#E6, -\MoebiusMu@{n},Möbius function,Moebius-mu,F|F:P,$\mu$,http://dlmf.nist.gov/27.2#E12, -\NatNumber,set of positive integers,EMPTY,SN|F,$\mathbb{N}$,http://dlmf.nist.gov/front/introduction#Sx4.p2.t1.r11, -\NeumannFactor{n},Neumann factor,Neumann-factor,I|F,$\epsilon_{m}$,http://drmf.wmflabs.org/wiki/Definition:NeumannFactor, -\NeumannPoly{n}@{x},Neumann's polynomial,Neumann-polynomial,P|F:P,$O_{k}$,http://dlmf.nist.gov/10.23#E12, -\NumPrimeDivNu@{n},$\nu(n)$ : number of distinct primes dividing $n$,number-of-primes-dividing-nu,I|F:P,$\nu$,http://dlmf.nist.gov/27.2#SS1.p1, -\NumPrimesLessPi@{x},number of primes not exceeding $x$,number-of-primes-less-pi,I|F:P,$\pi$,http://dlmf.nist.gov/27.2#E2, -\NumSquaresR{k}@{n},number of squares,number-of-squares-R,I|F:P,$r_{k}$,http://dlmf.nist.gov/27.13#SS4.p1, -\ODEgenAiryA{n}@{z},generalized Airy function from differential equation $A_n$,ODE-generalized-Airy-A,F|F:P,$A_{n}$,http://dlmf.nist.gov/9.13#SS1.p2, -\ODEgenAiryB{n}@{z},generalized Airy function from differential equation $B_n$,ODE-generalized-Airy-B,F|F:P,$B_{n}$,http://dlmf.nist.gov/9.13#SS1.p2, -\Pade{p}{q}{f},Padé approximant,Pade-approximant,O|F,${[p/q]_{f}}$,http://dlmf.nist.gov/3.11#SS4.p1, -\ParabolicU@{a}{z},parabolic cylinder function $U$,parabolic-U,F|F:P,$U$,http://dlmf.nist.gov/12.4.E1, -\ParabolicUbar@{a}{x},parabolic cylinder function $\overline{U}$,parabolic-U-bar,F|F:P,$\overline{U}$,http://dlmf.nist.gov/12.2.21, -\ParabolicV@{a}{z},parabolic cylinder function $V$,parabolic-V,F|F:P,$V$,http://dlmf.nist.gov/12.4.E2, -\ParabolicW@{a}{x},parabolic cylinder function $W$,parabolic-W,F|F:P,$W$,http://dlmf.nist.gov/12.14.i, -\PartitionsP[k]@{n},total number of partitions,partition-function,I|F:F:P:P,$p_{k}$,http://dlmf.nist.gov/26.2#P4.p2, -\PeriodicBernoulliB{n}@{x},periodic Bernoulli functions,periodic-Bernoulli-polynomial-B,F|F:P,$\widetilde{B}_{n}$,http://dlmf.nist.gov/24.2#iii, -\PeriodicEulerE{n}@{x},periodic Euler functions,periodic-Euler-polynomial-E,F|F:P,$\widetilde{E}_{n}$,http://dlmf.nist.gov/24.2#iii, -\PeriodicZeta@{x}{s},periodic zeta function,periodic-zeta,F|F:P,$F$,http://dlmf.nist.gov/25.13#E1, -\Permutations{n},"set of permutations of ${\bf n}$",set-of-permutations,SN|F,$\mathfrak{S}_{n}$,http://dlmf.nist.gov/23.1, -\PlanePartitionsPP@{n},number of plane partitions of $n$,plane-partitions-PP,I|F:P,$\mathit{pp}$,http://dlmf.nist.gov/26.12#SS1.p4, -\PollaczekP{\lambda}{n}@{x}{a}{b},Pollaczek polynomial,Pollaczek-polynomial-P,P|F:P,$P^{(\lambda)}_{n}$,http://dlmf.nist.gov/18.35#E4, -\Polylogarithm{s}@{z},polylogarithm,polylogarithm,F|F:P,$\mathrm{Li}_{s}$,http://dlmf.nist.gov/25.12#E10, -\Prod{n}{k1}{k2}@{f(n)},semantic product,EMPTY,SM|O:O,$\prod_{n=\alpha}^{\beta}$,http://drmf.wmflabs.org/wiki/Definition:Prod, -\Racah{n}@{x}{\alpha}{\beta}{\gamma}{\delta},Racah polynomial,Racah-polynomial-R,P|F:P,$R_{n}$,http://dlmf.nist.gov/18.25#T1.t1.r4, -\RacahR{n}@{x}{\alpha}{\beta}{\gamma}{\delta},Racah polynomial,Racah-polynomial-R,P|F:P,$R_{n}$,http://dlmf.nist.gov/18.25#T1.t1.r4, -\RamanujanSum{k}@{n},Ramanujan's sum,Ramanujan-sum,F|F:P,$c_{k}$,http://dlmf.nist.gov/27.10#E4, -\RamanujanTau@{n},Ramanujan's tau function,Ramanujan-tau,F|F:P,$\tau$,http://dlmf.nist.gov/27.14#E18, -\Rational,set of rational numbers,EMPTY,SN|F,$\mathbb{Q}$,http://dlmf.nist.gov/front/introduction#Sx4.p2.t1.r13, -\RayleighFun{n}@{\nu},Rayleigh function,Rayleigh-function,F|F:P,$\sigma_{n}$,http://dlmf.nist.gov/10.21#E55, -\Real,set of real numbers,EMPTY,SN|F,$\mathbb{R}$,http://dlmf.nist.gov/front/introduction#Sx4.p2.t1.r14, -\RepInterfc{n}@{z},repeated integrals of the complementary error function,repeated-integral-complementary-error-function,F|F:P,$\mathrm{i}^{n}\mathrm{erfc}$,http://dlmf.nist.gov/7.18#E2, -\Residue,residue,EMPTY,O|F,$\mathrm{res}$,http://dlmf.nist.gov/1.10#SS3.p5, -\RestrictedCompositionsC@{\mathrm{condition}}{n},restricted number of compositions,restricted-number-of-compositions-C,I|F:P,$c_{m}$,http://dlmf.nist.gov/26.11#p1, -\RestrictedPartitionsP[k]@{\mathrm{condition}}{n},restricted number of partitions,restricted-partitions-P,I|F:F:P:P,$p_{k}$,http://dlmf.nist.gov/26.10#i, -\RiemannP@{\begin{Bmatrix} \alpha & \beta &\gamma & \\ a_1 & b_1 & c_1 & z \\ a_2 & b_2 & c_2 & \end{Bmatrix}},Riemann-Papperitz $\RiemannP$-symbol for solutions of the generalized hypergeometric differential equation,Riemann-P-symbol,Q|F:P,$P$,http://dlmf.nist.gov/15.11#E3, -\RiemannTheta@{\Vector{z}}{\Matrix{\Omega}},Riemann theta function,Riemann-theta,F|F:P,$\theta$,http://dlmf.nist.gov/21.2#E1, -\RiemannThetaChar{\Vector{\alpha}}{\Vector{\beta}}@{\Vector{z}}{\Matrix{\Omega}},Riemann theta function with characteristics,Riemann-theta-characterstics,F|F:P,$\theta\left[\Vector{\alpha} \atop \Vector{\beta} \right]$,http://dlmf.nist.gov/21.2#E5, -\RiemannThetaHat@{\Vector{z}}{\Matrix{\Omega}},scaled Riemann theta function,Riemann-theta-hat,F|F:P,$\hat{\theta}$,http://dlmf.nist.gov/21.2#E2, -\RiemannXi@{s},Riemann's $\RiemannXi$-function,Riemann-xi,F|F:P,$\xi$,http://dlmf.nist.gov/25.4#E4, -\RiemannZeta@{s},Riemann zeta function,Riemann-zeta,F|F:P,$\zeta$,http://dlmf.nist.gov/25.2#E1, -\Schwarzian{z}{\zeta},Schwarzian derivative,EMPTY,O|F,"$\left\{f,g\right\}$",http://dlmf.nist.gov/1.13#E20, -\ScorerGi@{z},Scorer function (inhomogeneous Airy function) ${\rm Gi}$,Scorer-Gi,F|F:P,$\mathrm{Gi}$,http://dlmf.nist.gov/9.12#i, -\ScorerHi@{z},Scorer function (inhomogeneous Airy function) ${\rm Hi}$,Scorer-Hi,F|F:P,$\mathrm{Hi}$,http://dlmf.nist.gov/9.12#i, -\SinCosIntf@{z},auxiliary function for sine and cosine integrals ${\rm f}$,sine-cosine-integral-f,F|F:P,$\mathrm{f}$,http://dlmf.nist.gov/6.2#E17, -\SinCosIntg@{z},auxiliary function for sine and cosine integrals ${\rm g}$,sine-cosine-integral-g,F|F:P,$\mathrm{g}$,http://dlmf.nist.gov/6.2#E18, -\SinInt@{z},sine integral ${\rm Si}$,sine-integral,F|F:P,$\mathrm{Si}$,http://dlmf.nist.gov/6.2#E9, -\SinIntg@{a}{z},generalized sine integral ${\rm Si}$,generalized-sine-integral-Si,F|F:P,$\mathrm{Si}$,http://dlmf.nist.gov/8.21#E2, -\SinhInt@{z},hyperbolic sine integral,hyperbolic-sine-integral,F|F:P,$\mathrm{Shi}$,http://dlmf.nist.gov/6.2#E15, -\SphBesselIi{n}@{z},modified spherical Bessel function of the first kind ${\sf i}_n^{(1)}$,spherical-Bessel-I-1,F|F:P,${\mathsf{i}^{(1)}_{n}}$,http://dlmf.nist.gov/10.47#E7, -\SphBesselIii{n}@{z},modified spherical Bessel function of the first kind ${\sf i}_n^{(2)}$,spherical-Bessel-I-2,F|F:P,${\mathsf{i}^{(2)}_{n}}$,http://dlmf.nist.gov/10.47#E8, -\SphBesselJ{n}@{z},spherical Bessel function of the first kind ${\sf j}_n$,spherical-Bessel-J,F|F:P,$\mathsf{j}_{n}$,http://dlmf.nist.gov/10.47#E3, -\SphBesselK{n}@{z},modified spherical Bessel function of the second kind ${\sf k}_n$,spherical-Bessel-K,F|F:P,$\mathsf{k}_{n}$,http://dlmf.nist.gov/10.47#E9, -\SphBesselY{n}@{z},spherical Bessel function of the second kind ${\sf y}_n$,spherical-Bessel-Y,F|F:P,$\mathsf{y}_{n}$,http://dlmf.nist.gov/10.47#E4, -\SphHankelHi{n}@{z},spherical Hankel function of the first kind ${\sf h}_n^{(1)}$,spherical-Hankel-H-1-Bessel-third-kind,F|F:P,${\mathsf{h}^{(1)}_{n}}$,http://dlmf.nist.gov/10.47#E5, -\SphHankelHii{n}@{z},spherical Hankel function of the second kind ${\sf h}_n^{(2)}$,spherical-Hankel-H-2-Bessel-third-kind,F|F:P,${\mathsf{h}^{(2)}_{n}}$,http://dlmf.nist.gov/10.47#E6, -\SphericalHarmonicY{l}{m}@{\theta}{\phi},spherical harmonic,spherical-harmonic-Y,F|F:P,"$Y_{{\ell},{m}}$",http://dlmf.nist.gov/14.30#E1, -\SpheroidalEigenvalueLambda{m}{n}@{\gamma^2},eigenvalues of the spheroidal differential equation,spheroidal-eigenvalue-lambda,O|F:P,$\lambda^{m}_{n}$,http://dlmf.nist.gov/30.3#SS1.p1, -\SpheroidalOnCutPs{m}{n}@{x}{\gamma^2},spheroidal wave function of the first kind ${\sf Ps}_n^m$,spheroidal-wave-on-cut-Ps,F|F:P,$\mathsf{Ps}^{m}_{n}$,http://dlmf.nist.gov/30.4#i, -\SpheroidalOnCutQs{m}{n}@{x}{\gamma^2},spheroidal wave function of the second kind ${\sf Qs}_n^m$,spheroidal-wave-on-cut-Qs,F|F:P,$\mathsf{Qs}^{m}_{n}$,http://dlmf.nist.gov/30.5, -\SpheroidalPs{m}{n}@{z}{\gamma^2},spheroidal wave function of complex argument $Ps_n^m$,spheroidal-wave-Ps,F|F:P,$\mathit{Ps}^{m}_{n}$,http://dlmf.nist.gov/30.6#p1, -\SpheroidalQs{m}{n}@{z}{\gamma^2},spheroidal wave function of complex argument $Qs_n^m$,spheroidal-wave-Qs,F|F:P,$\mathit{Qs}^{m}_{n}$,http://dlmf.nist.gov/30.6#p1, -\SpheroidalRadialS{m}{j}{n}@{z}{\gamma},radial spheroidal wave function,radial-spheroidal-wave-S,F|F:P,$S^{m(j)}_{n}$,http://dlmf.nist.gov/30.11#E3, -\StieltjesConstants{n},Stieltjes constants,Stieltjes-constants,C|F,$\gamma_{n}$,http://drmf.wmflabs.org/wiki/Definition:StieltjesConstants, -\StieltjesTrans@{f}{s},Stieltjes transform,Stieltjes-transform,O|F:P,$\mathcal{S}$,http://dlmf.nist.gov/1.14#E47, -\StieltjesWigert{n}@@{x}{q},Stieltjes-Wigert polynomial,Stieltjes-Wigert-polynomial-S,P|F:P:PS,$S_{n}$,http://drmf.wmflabs.org/wiki/Definition:StieltjesWigert, -\StieltjesWigertS{n}@{x}{q},Stieltjes-Wigert polynomial,Stieltjes-Wigert-polynomial-S,P|F:P,$S_{n}$,http://dlmf.nist.gov/18.27#E18, -\StirlingS@{n}{k},Stirling number of the first kind,Stirling-number-first-kind-S,I|F:P,$s$,http://dlmf.nist.gov/26.8#SS1.p1, -\StirlingSS@{n}{k},Stirling number of the second kind,Stirling-number-second-kind-S,I|F:P,$S$,http://dlmf.nist.gov/26.8#SS1.p3, -\StruveH{\nu}@{z},Struve function ${\bf H}_{\nu}$,Struve-H,F|F:P,$\mathbf{H}_{\nu}$,http://dlmf.nist.gov/11.2#E1, -\StruveK{\nu}@{z},Struve function ${\bf K}_{\nu}$,associated-Struve-K,F|F:P,$\mathbf{K}_{\nu}$,http://dlmf.nist.gov/11.2#E5, -\StruveL{\nu}@{z},modified Struve function ${\bf L}_{\nu}$,modified-Struve-L,F|F:P,$\mathbf{L}_{\nu}$,http://dlmf.nist.gov/11.2#E2, -\StruveM{\nu}@{z},modified Struve function ${\bf M}_{\nu}$,associated-Struve-M,F|F:P,$\mathbf{M}_{\nu}$,http://dlmf.nist.gov/11.2#E6, -\Sum{n}{0}{k}@{f(n)},semantic sum,EMPTY,SM|F:P,$\sum_{n=k}^{N}$,http://drmf.wmflabs.org/wiki/Definition:Sum, -\SurfaceHarmonicY{l}{m}@{\theta}{\phi},surface harmonic of the first kind,surface-harmonic-Y,F|F:P,$Y_{l}^{m}$,http://dlmf.nist.gov/14.30#E2, -\TriangleOP{\alpha}{\beta}{\gamma}{m}{n}@{x}{y},triangle polynomial,triangle-orthogonal-polynomial-P,P|F:P,"$P^{\alpha,\beta,\gamma}_{m,n}$",http://dlmf.nist.gov/18.37#E7, -\Ultra{\lambda}{n}@{x},ultraspherical/Gegenbauer polynomial,ultraspherical-Gegenbauer-polynomial,P|F:P,$C^{\mu}_{n}$,http://dlmf.nist.gov/18.3#T1.t1.r5, -\Ultraspherical{\lambda}{n}@{x},ultraspherical (or Gegenbauer) polynomial,ultraspherical-Gegenbauer-polynomial,P|F:P,$C^{(\mu)}_{n}$,http://dlmf.nist.gov/18.3#T1.t1.r5, -\UmbilicCatE@{s}{t}{\Vector{x}},elliptic umbilic catastrophe,elliptic-umbilic-catastrophe,F|F:P,$\Phi^{(\mathrm{E})}$,http://dlmf.nist.gov/36.2#E2, -\UmbilicCatH@{s}{t}{\Vector{x}},hyperbolic umbilic catastrophe,hyperbolic-umbilic-catastrophe,F|F:P,$\Phi^{(\mathrm{H})}$,http://dlmf.nist.gov/36.2#E3, -\VariationalOp[a;b]@{f},total variation,variational-operator,F|FO:PO:FnO:PnO,"$\mathcal{V}_{a,b}$",http://dlmf.nist.gov/1.4#E33, -\VoigtU@{x}{t},Voigt function ${\sf U}$,Voigt-U,F|F:P,$\mathsf{U}$,http://dlmf.nist.gov/7.19#E1, -\VoigtV@{x}{t},Voigt function ${\sf V}$,Voigt-V,F|F:P,$\mathsf{V}$,http://dlmf.nist.gov/7.19#E2, -\WaringG@{k},Waring's function $G$,Waring-G,F|F:P,$G$,http://dlmf.nist.gov/27.13#SS3.p1, -\Waringg@{k},Waring's function $g$,Waring-g,F|F:P,$g$,http://dlmf.nist.gov/27.13#SS3.p1, -\WeberE{\nu}@{z},Weber function,Weber-E,F|F:P,$\mathbf{E}_{\nu}$,http://dlmf.nist.gov/11.10#E2, -\WeierPInv@@{z}{g_2}{g_3},Weierstrass $\WeierPInv$-function on invariants $\WeierPInv@{z}{g_2}{g_3}$,Weierstrass-P-on-invariants,F|F:P:nP,$\wp$,http://dlmf.nist.gov/23.1, -\WeierPLat@@{z}{\Lattice{L}},Weierstrass $\WeierPLat$-function on lattice $\WeierPLat@{z}{\Lattice{L}}$,Weierstrass-P-on-lattice,F|F:P:nP,$\wp$,http://dlmf.nist.gov/23.1, -\WeiersigmaInv@@{z}{g_2}{g_3},Weierstrass sigma function on invariants,Weierstrass-sigma-on-invariants,F|F:P:nP,$\sigma$,http://dlmf.nist.gov/23.2#E6, -\WeiersigmaLat@@{z}{\Lattice{L}},Weierstrass sigma function on lattice,Weierstrass-sigma-on-lattice,F|F:P:nP,$\sigma$,http://dlmf.nist.gov/23.2#E6, -\WeierzetaInv@@{z}{g_2}{g_3},Weierstrass zeta function on invariants,Weierstrass-zeta-on-invariants,F|F:P:nP,$\zeta$,http://dlmf.nist.gov/23.2#E5, -\WeierzetaLat@@{z}{\Lattice{L}},Weierstrass zeta function on lattice,Weierstrass-zeta-on-lattice,F|F:P:nP,$\zeta$,http://dlmf.nist.gov/23.2#E5, -\WhitD{\nu}@{z},parabolic cylinder function,Whittaker-D,F|F:P,$D_{n}$,http://dlmf.nist.gov/23.1, -\WhitM{\kappa}{\mu}@{z},Whittaker function $M_{\kappa;\mu}$,Whittaker-confluent-hypergeometric-M,F|F:P,"$M_{\kappa,\mu}$",http://dlmf.nist.gov/13.14#E2, -\WhitW{\kappa}{\mu}@{z},Whittaker function $W_{\kappa;\mu}$,Whittaker-confluent-hypergeometric-W,F|F:P,"$W_{\kappa,\mu}$",http://dlmf.nist.gov/13.14#E3, -\Wilson{n}@{x}{a}{b}{c}{d},Wilson polynomial,Wilson-polynomial-W,P|F:P,$W_{n}$,http://dlmf.nist.gov/18.25#T1.t1.r2, -\WilsonW{n}@{x}{a}{b}{c}{d},Wilson polynomial,Wilson-polynomial-W,P|F:P,$W_{n}$,http://dlmf.nist.gov/18.25#T1.t1.r2, -\Wronskian@{w_1(z);w_2(z)},Wronskian,Wronskian,F|F:P,$\mathscr{W}$,http://dlmf.nist.gov/1.13#E4, -\ZeroAiryAi{k},$k$\textsuperscript{th} zero of Airy $\AiryAi$,zero-Airy-Ai,F|F,$a_{k}$,http://dlmf.nist.gov/23.1, -\ZeroAiryAiPrime{k},$k$\textsuperscript{th} zero of Airy $\AiryAi'$,zero-derivative-Airy-Ai,F|F,$a'_{k}$,http://dlmf.nist.gov/23.1, -\ZeroAiryBi{k},$k$\textsuperscript{th} zero of Airy $\AiryBi$,zero-Airy-Bi,F|F,$b_{k}$,http://dlmf.nist.gov/23.1, -\ZeroAiryBiPrime{k},$k$\textsuperscript{th} zero of Airy $\AiryBi'$,zero-derivative-Airy-Bi,F|F,$b'_{k}$,http://dlmf.nist.gov/23.1, -\ZeroBesselJ{\nu}{m},zeros of the Bessel function $\BesselJ{\nu}@{x}$,zero-Bessel-J,F|F,"$j_{\nu,m}$",http://dlmf.nist.gov/23.1, -\ZeroBesselJPrime{\nu}{m},zeros of the Bessel function derivative $\BesselJ{\nu}'@{x}$,zero-derivative-Bessel-J,F|F,"${j'_{\nu,m}}$",http://dlmf.nist.gov/23.1, -\ZeroBesselY{\nu}{m},zeros of the Bessel function $\BesselY{\nu}@{x}$,zero-Bessel-Y-Weber,F|F,"$y_{\nu,m}$",http://dlmf.nist.gov/23.1, -\ZeroBesselYPrime{\nu}{m},zeros of the Bessel function derivative $\BesselY{\nu}'@{x}$,zero-derivative-Bessel-Y-Weber,F|F,"${y'_{\nu,m}}$",http://dlmf.nist.gov/23.1, -\ZonalPoly{\kappa}@{\Matrix{T}},zonal polynomial,zonal-polynomial,P|F:P,$Z_{\kappa}$,http://dlmf.nist.gov/35.4#SS1.p3, -\acos@@{z},inverse cosine function,inverse-cosine,F|F:P:nP,$\mathrm{arccos}$,http://dlmf.nist.gov/4.23#SS2.p1, -\acosh@@{z},inverse hyperbolic cosine function,hyperbolic-inverse-cosine,F|F:P:nP,$\mathrm{arccosh}$,http://dlmf.nist.gov/4.37#SS2.p1, -\acot@@{z},inverse cotangent function,inverse-cotangent,F|F:P:nP,$\mathrm{arccot}$,http://dlmf.nist.gov/4.23#E9, -\acoth@@{z},inverse hyperbolic cotangent function,hyperbolic-inverse-cotangent,F|F:P:nP,$\mathrm{arccoth}$,http://dlmf.nist.gov/4.37#E9, -\acsc@@{z},inverse cosecant function,inverse-cosecant,F|F:P:nP,$\mathrm{arccsc}$,http://dlmf.nist.gov/4.23#E7, -\acsch@@{z},inverse hyperbolic cosecant function,hyperbolic-inverse-cosecant,F|F:P:nP,$\mathrm{arccsch}$,http://dlmf.nist.gov/4.37#E7, -\arcGudermannian@@{x},inverse Gudermannian function,inverse-Gudermannian,F|F:P:nP,${\mathrm{gd}^{-1}}$,http://dlmf.nist.gov/4.23#E41, -\arcJacobicd@{x}{k},inverse Jacobian elliptic function ${\rm arccd}$,inverse-Jacobi-elliptic-cd,F|F:P,$\mathrm{arccd}$,http://dlmf.nist.gov/22.15#SS1.p1, -\arcJacobicn@{x}{k},inverse Jacobian elliptic function ${\rm arccn}$,inverse-Jacobi-elliptic-cn,F|F:P,$\mathrm{arccn}$,http://dlmf.nist.gov/22.15#SS1.p1, -\arcJacobics@{x}{k},inverse Jacobian elliptic function ${\rm arccs}$,inverse-Jacobi-elliptic-cs,F|F:P,$\mathrm{arccs}$,http://dlmf.nist.gov/22.15#SS1.p1, -\arcJacobidc@{x}{k},inverse Jacobian elliptic function ${\rm arcdc}$,inverse-Jacobi-elliptic-dc,F|F:P,$\mathrm{arcdc}$,http://dlmf.nist.gov/22.15#SS1.p1, -\arcJacobidn@{x}{k},inverse Jacobian elliptic function ${\rm arcdn}$,inverse-Jacobi-elliptic-dn,F|F:P,$\mathrm{arcdn}$,http://dlmf.nist.gov/22.15#SS1.p1, -\arcJacobids@{x}{k},inverse Jacobian elliptic function ${\rm arcds}$,inverse-Jacobi-elliptic-ds,F|F:P,$\mathrm{arcds}$,http://dlmf.nist.gov/22.15#SS1.p1, -\arcJacobinc@{x}{k},inverse Jacobian elliptic function ${\rm arcnc}$,inverse-Jacobi-elliptic-nc,F|F:P,$\mathrm{arcnc}$,http://dlmf.nist.gov/22.15#SS1.p1, -\arcJacobind@{x}{k},inverse Jacobian elliptic function ${\rm arcnd}$,inverse-Jacobi-elliptic-nd,F|F:P,$\mathrm{arcnd}$,http://dlmf.nist.gov/22.15#SS1.p1, -\arcJacobins@{x}{k},inverse Jacobian elliptic function ${\rm arcns}$,inverse-Jacobi-elliptic-ns,F|F:P,$\mathrm{arcns}$,http://dlmf.nist.gov/22.15#SS1.p1, -\arcJacobisc@{x}{k},inverse Jacobian elliptic function ${\rm arcsc}$,inverse-Jacobi-elliptic-sc,F|F:P,$\mathrm{arcsc}$,http://dlmf.nist.gov/22.15#SS1.p1, -\arcJacobisd@{x}{k},inverse Jacobian elliptic function ${\rm arcsd}$,inverse-Jacobi-elliptic-sd,F|F:P,$\mathrm{arcsd}$,http://dlmf.nist.gov/22.15#SS1.p1, -\arcJacobisn@{x}{k},inverse Jacobian elliptic function ${\rm arcsn}$,inverse-Jacobi-elliptic-sn,F|F:P,$\mathrm{arcsn}$,http://dlmf.nist.gov/22.15#SS1.p1, -\asec@@{z},inverse secant function,inverse-secant,F|F:P:nP,$\mathrm{arcsec}$,http://dlmf.nist.gov/4.23#E8, -\asech@@{z},inverse hyperbolic secant function,hyperbolic-inverse-secant,F|F:P:nP,$\mathrm{arcsech}$,http://dlmf.nist.gov/4.37#E8, -\asin@@{z},inverse sine function,inverse-sine,F|F:P:nP,$\mathrm{arcsin}$,http://dlmf.nist.gov/4.23#SS2.p1, -\asinh@@{z},inverse hyperbolic sine function,hyperbolic-inverse-sine,F|F:P:nP,$\mathrm{arcsinh}$,http://dlmf.nist.gov/4.37#SS2.p1, -\atan@@{z},inverse tangent function,inverse-tangent,F|F:P:nP,$\mathrm{arctan}$,http://dlmf.nist.gov/4.23#SS2.p1, -\atanh@@{z},inverse hyperbolic tangent function,hyperbolic-inverse-tangent,F|F:P:nP,$\mathrm{arctanh}$,http://dlmf.nist.gov/4.37#SS2.p1, -\bDiff[x],backward difference,EMPTY,O|F,"$\,\nabla_{x}$",http://dlmf.nist.gov/18.1#EGx2, -\bigqJacobi{n}@@{x}{a}{b}{c}{q},big $q$-Jacobi polynomial,big-q-Jacobi-polynomial-P,P|F:P:PS,$P_{n}$,http://drmf.wmflabs.org/wiki/Definition:bigqJacobi, -\bigqJacobiIVparam{n}@@{x}{a}{b}{c}{d}{q},big $q$-Jacobi polynomial with four parameters,big-q-Jacobi-polynomial-four-parameters-P,P|F:P,$P_{n}$,http://drmf.wmflabs.org/wiki/Definition:bigqJacobiIVparam, -\bigqLaguerre{n}@{x}{a}{b}{q},big $q$-Laguerre polynomial,EMPTY,P|F:P,$P_{n}$,http://drmf.wmflabs.org/wiki/Definition:bigqLaguerre, -\bigqLegendre{n}@{x}{c}{q},big $q$-Legendre polynomial,EMPTY,P|F:P,$P_{n}$,http://drmf.wmflabs.org/wiki/Definition:bigqLegendre, -\binom{n}{k},binomial coefficient,EMPTY,I|F,$\binom{n}{k}$,http://dlmf.nist.gov/1.2#E1,http://dlmf.nist.gov/26.3#SS1.p1 -\binomial{n}{k},binomial coefficient,EMPTY,I|F,$\binom{n}{k}$,http://dlmf.nist.gov/1.2#E1,http://dlmf.nist.gov/26.3#SS1.p1 -\cDiff[x],central difference,EMPTY,O|F,"$\,\delta_{x}$",http://dlmf.nist.gov/18.1#EGx3, -\card,number of elements of a finite set,EMPTY,SN|F,$\left|A\right|$,http://dlmf.nist.gov/26.1, -\cartprod,$G\cartprod H$: Cartesian product of groups $G$ and $H$,EMPTY,O|F,$\times$,http://dlmf.nist.gov/23.1, -\ceiling{x},ceiling,EMPTY,I|F,$\left\lceil a\right\rceil$,http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r17, -\conj{z},complex conjugate,EMPTY,O|F,$\overline{a}$,http://dlmf.nist.gov/1.9#E11, -\cos@@{z},cosine function,cos,F|F:P:nP,$\mathrm{cos}$,http://dlmf.nist.gov/4.14#E2, -\cosh@@{z},hyperbolic cosine function,hyperbolic-cosine,F|F:P:nP,$\mathrm{cosh}$,http://dlmf.nist.gov/4.28#E2, -\cosintg@{a}{z},generalized cosine integral ${\rm ci}$,generalized-cosine-integral-ci,F|F:P,$\mathrm{ci}$,http://dlmf.nist.gov/8.21#E1, -\cot@@{z},cotangent function,cotangent,F|F:P:nP,$\mathrm{cot}$,http://dlmf.nist.gov/4.14#E7, -\coth@@{z},hyperbolic cotangent function,hyperbolic-cotangent,F|F:P:nP,$\mathrm{coth}$,http://dlmf.nist.gov/4.28#E7, -\cpi,ratio of a circle's circumference to its diameter,ratio-of-a-circle's-circumference-to-its-diameter,C|F,$\pi$,http://dlmf.nist.gov/5.19.E4, -\csc@@{z},cosecant function,cosecant,F|F:P:nP,$\mathrm{csc}$,http://dlmf.nist.gov/4.14#E5, -\csch@@{z},hyperbolic cosecant function,hyperbolic-cosecant,F|F:P:nP,$\mathrm{csch}$,http://dlmf.nist.gov/4.28#E5, -\ctsHahn{n}@@{x}{a}{b}{c}{d},continuous Hahn polynomial,continuous-Hahn-polynomial,P|F:P:PS,$p_{n}$,http://dlmf.nist.gov/18.19#P2.p1, -\ctsbigqHermite{n}@{x}{a}{q},continuous big $q$-Hermite polynomial,EMPTY,P|F:P,$H_{n}$,http://drmf.wmflabs.org/wiki/Definition:ctsbigqHermite, -\ctsdualHahn{n}@@{x^2}{a}{b}{c},continuous dual Hahn polynomial,continuous-dual-Hahn-normalized-S,P|F:P:PS,$S_{n}$,http://dlmf.nist.gov/18.25#T1.t1.r3, -\ctsdualqHahn{n}@{x}{a}{b}{c}{q},continuous dual $q$-Hahn polynomial,EMPTY,P|F:P,$p_{n}$,http://drmf.wmflabs.org/wiki/Definition:ctsdualqHahn, -\ctsqHahn{n}@{x}{a}{b}{c}{d}{q},continuous $q$-Hahn polynomial,EMPTY,P|F:P,$p_{n}$,http://drmf.wmflabs.org/wiki/Definition:ctsqHahn, -\ctsqHermite{n}@@{x}{q},continuous $q$-Hermite polynomial,continuous-q-Hermite-polynomial-H,P|F:P:PS,$H_{n}$,http://drmf.wmflabs.org/wiki/Definition:ctsqHermite, -\ctsqJacobi{\alpha}{\beta}{m}@{x}{q},continuous $q$-Jacobi polynomial,EMPTY,P|F:P,"$P^{(\alpha,\beta)}_{n}$",http://drmf.wmflabs.org/wiki/Definition:ctsqJacobi, -\ctsqJacobiRahman{n}@@{x}{q},continuous $q$-Jacobi-Rahman-polynomial,continuous-q-Jacobi-Rahman-polynomial-P,P|F:P:PS,"$P^{(\alpha,\beta)}_{n}$",http://drmf.wmflabs.org/wiki/Definition:ctsqJacobiRahman, -\ctsqLaguerre{\alpha}{n}@{x}{q},continuous $q$-Laguerre polynomial,EMPTY,P|F:P,$P^{(n)}_{\alpha}$,http://drmf.wmflabs.org/wiki/Definition:ctsqLaguerre, -\ctsqLegendre{n}@@{x}{q},continuous $q$-Legendre polynomial,continuous-q-Legendre-polynomial-P,P|F:P:PS,$P_{n}$,http://drmf.wmflabs.org/wiki/Definition:ctsqLegendre, -\ctsqLegendreRahman{n}@@{x}{q},continuous $q$-Legendre-Rahman polynomial,continuous-q-Legendre-Rahman-polynomial-P,P|F:P:PS,$P_{n}$,http://drmf.wmflabs.org/wiki/Definition:ctsqLegendreRahman, -\ctsqUltra{n}@{x}{\beta}{q},continuous $q$-ultraspherical/Rogers polynomial,continuous-q-ultraspherical-Rogers-polynomial,P|F:P,$C_{n}$,http://dlmf.nist.gov/18.28#E13, -\ctsqUltrae{n}@{e^{i\theta}}{\beta}{q},continuous $q$-ultraspherical/Rogers polynomial exponential argument,continuous-q-ultraspherical-Rogers-polynomial-exponential-argument,P|F:P,$C_{n}$,http://drmf.wmflabs.org/wiki/Definition:ctsqUltrae, -\curl,curl of vector-valued function,EMPTY,F|F,$\nabla\times$,http://dlmf.nist.gov/1.6#E22, -\deltaDistribution,Dirac delta distribution,Dirac delta distribution,D|F,$\delta_x$,http://dlmf.nist.gov/1.17.i, -\deriv{f}{x},derivative,EMPTY,O|F,$\deriv{}{x}}$,http://dlmf.nist.gov/1.4#E4, -\det,determinant,EMPTY,L|F,$\det$,http://dlmf.nist.gov/1.3#i, -\diag,diagonal part of a matrix,EMPTY,L|F,$\diag[\Matrix{a}]$,http://dlmf.nist.gov/21.5.E7, -\diff{x},differential,EMPTY,O|F,$\mathrm{d}^nx$,http://dlmf.nist.gov/1.4#iv, -\digamma@{z},psi (or digamma) function,digamma,F|F:P,$\psi$,http://dlmf.nist.gov/5.2#E2, -\discrqHermiteI{n}@@{x}{q},discrete $q$-Hermite I polynomial,discrete-q-Hermite-polynomial-h-I,P|F:P:PS,$h_{n}$,http://drmf.wmflabs.org/wiki/Definition:discrqHermiteI, -\discrqHermiteII{n}@@{x}{q},discrete $q$-Hermite II polynomial,discrete-q-Hermite-polynomial-II-h-tilde,P|F:P:PS,$\tilde{h}_{n}$,http://drmf.wmflabs.org/wiki/Definition:discrqHermiteII, -\divergence,divergence of vector-valued function,EMPTY,F|F,$\mathrm{div}$,http://dlmf.nist.gov/1.6#E21, -\divides,divides,EMPTY,O|F,$\nabla\cdot$,http://dlmf.nist.gov/24.1, -\dualHahn{n}@@{\lambda(x)}{\gamma}{\delta}{N},dual Hahn polynomial,dual-Hahn-polynomial-R,P|F:P:PS,$R_{n}$,http://dlmf.nist.gov/18.25#T1.t1.r5, -\dualqHahn{n}@@{\mu(x)}{\gamma}{\delta}{N}{q},dual $q$-Hahn polynomial,dual-q-Hahn-R,P|F:P:PS,$R_{n}$,http://drmf.wmflabs.org/wiki/Definition:dualqHahn, -\dualqKrawtchouk{n}@@{\lambda(x)}{c}{N}{q},dual $q$-Krawtchouk polynomial,dual-q-Krawtchouk-polynomial-K,P|F:P:PS,$K_{n}$,http://drmf.wmflabs.org/wiki/Definition:dualqKrawtchouk, -\erf@@{z},error function,error-function,F|F:P:nP,$\mathrm{erf}$,http://dlmf.nist.gov/7.2#E1, -\erfc@@{z},complementary error function,repeated-integral-complementary-error-function,F|F:P:nP,$\mathrm{erfc}$,http://dlmf.nist.gov/7.2#E2, -\erfw@@{z},complementary error function $w$,error-function-w,F|F:P:nP,$w$,http://dlmf.nist.gov/7.2#E3, -\exp@@{z},exponential function,exponential,F|F:P:nP,$\mathrm{exp}$,http://dlmf.nist.gov/4.2#E19, -\expe,the base of the natural logarithm,EMPTY,F|F,$\mathrm{e}$,http://dlmf.nist.gov/4.2.E11, -\exptrace@{\Matrix{X}},exponential of trace of a matrix,exponential-trace,L|F:P,$\mathrm{etr}$,http://dlmf.nist.gov/35.1#p3.t1.r10, -\f{f}@{x},function,EMPTY,SM|F:P,${f}$,http://drmf.wmflabs.org/wiki/Definition:f, -\fDiff,forward difference operator,EMPTY,O|F,"$\,\Delta_{y}$",http://dlmf.nist.gov/3.6#SS1.p1, -\floor{x},floor,EMPTY,I|F,$\left\lfloor a\right\rfloor$,http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r16, -\forall,for every,EMPTY,Q|F,$\forall$,http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r11, -\gradient,gradient of differentiable scalar function,EMPTY,F|F,$\nabla$,http://dlmf.nist.gov/1.6#E20, -"\idem@{\chi_1}{\chi_1,...,\chi_n}",$\idem$ function,idem,F|F:P,$\mathrm{idem}$,http://dlmf.nist.gov/17.1#p3, -\ifrac,semantic slash fraction,EMPTY,SM|F,$(a)/(b)$,http://dlmf.nist.gov/4.2.E8, -\imagpart{z},imaginary part,EMPTY,O|F,"$\Im {z}$",http://dlmf.nist.gov/1.9#E2, -\in,element of,EMPTY,Q|F,$\in$,http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r9, -\incgamma@{a}{z},incomplete gamma function $\gamma$,incomplete-gamma,F|F:P,$\gamma$,http://dlmf.nist.gov/8.2#E1, -\incgammastar@{a}{z},incomplete gamma function $\gamma^{*}$,incomplete-gamma-star,F|F:P,$\gamma^{*}$,http://dlmf.nist.gov/8.2#E6, -\inf,greatest lower bound (infimum),EMPTY,Q|F,$\mathrm{inf}$,http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r24, -\int,integral,EMPTY,F|F,$\int$,http://dlmf.nist.gov/1.4#iv, -\inverf@@{x},inverse error function,inverse-error-function,F|F:P:nP,$\mathrm{inverf}$,http://dlmf.nist.gov/7.17#E1, -\inverfc@@{x},inverse complementary error function,inverse-complementary-error-function,F|F:P:nP,$\mathrm{inverfc}$,http://dlmf.nist.gov/7.17#E1, -\iunit,imaginary unit,EMPTY,C|F,$\mathrm{i}$,http://dlmf.nist.gov/1.9.i, -\liminf,least limit point,EMPTY,Q|F,$\mathrm{liminf}$,http://dlmf.nist.gov/front/introduction#Sx4.p2.t1.r4, -\littleo@{x},order less than,little-o,Q|F:P,$o$,http://dlmf.nist.gov/2.1#E2, -\littleqJacobi{n}@@{x}{a}{b}{q},little $q$-Jacobi polynomial,little-q-Jacobi-polynomial-p,P|F:P:PS,$p_{n}$,http://drmf.wmflabs.org/wiki/Definition:littleqJacobi, -\littleqLaguerre{n}@{x}{a}{q},little $q$-Laguerre / Wall polynomial,EMPTY,P|F:P,$p_{n}$,http://drmf.wmflabs.org/wiki/Definition:littleqLaguerre, -\littleqLegendre{m}@{x}{q},little $q$-Legendre polynomial,EMPTY,P|F:P,$p_{n}$,http://drmf.wmflabs.org/wiki/Definition:littleqLegendre, -\ln@@{z},principal branch of logarithm function,natural-logarithm,F|F:P:nP,$\mathrm{ln}$,http://dlmf.nist.gov/4.2#E2, -\log@@{z},principle branch of natural logarithm,logarithm,F|F:P:nP,$\mathrm{log}$,http://dlmf.nist.gov/4.2#E2, -\logb{a}@@{z},logarithm to general base,logarithm,F|F:P:nP,$\mathrm{log}_{b}$,http://dlmf.nist.gov/4.2#EGx1, -\lrselection,bracketed generalization of $\pm$,EMPTY,SM|F,$\left\{\begin{matrix}x\end{matrix}\right\}$,http://drmf.wmflabs.org/wiki/Definition:Lrselection, -\lselection,left bracketed generalization of $\pm$,EMPTY,SM|F,$\left\{\begin{matrix}x\end{matrix}\right.$,http://dlmf.nist.gov/28.4.E22, -\mEulerBeta{m}@{a}{b},multivariate beta function,multivariate-Euler-Beta,F|F:P,$\mathrm{B}_{m}$,http://dlmf.nist.gov/35.3#E3, -\mEulerGamma{m}@{a},multivariate gamma function,multivariate-Euler-Gamma,F|F:P,$\Gamma_{m}$,http://dlmf.nist.gov/35.3#i, -\mod,modulo,EMPTY,Q|F,$\mod$,http://dlmf.nist.gov/front/introduction#Sx4.p2.t1.r10, -\monicAffqKrawtchouk{n}@@{q^{-x}}{p}{N}{q},monic affine $q$-Krawtchouk polynomial,affine-q-Krawtchouk-polynomial-monic-p,P|F:P:PS,$\widehat{K}^{\mathrm{Aff}}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicAffqKrawtchouk, -\monicAlSalamCarlitzI{a}{n}@@{x}{q},monic Al-Salam-Carlitz I polynomial,q-Al-Salam-Carlitz-I-polynomial-monic-p,P|F:P:PS,${\widehat U}^{(\alpha)}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicAlSalamCarlitzI, -\monicAlSalamCarlitzII{a}{n}@@{x}{q},monic Al-Salam-Carlitz II polynomial,q-Al-Salam-Carlitz-II-polynomial-monic-p,P|F:P:PS,${\widehat V}^{(n)}_{\alpha}$,http://drmf.wmflabs.org/wiki/Definition:monicAlSalamCarlitzII, -\monicAlSalamChihara{n}@@{x}{a}{b}{q},monic Al-Salam-Chihara polynomial,Al-Salam-Chihara-polynomial-monic-p,P|F:P:PS,${\widehat Q}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicAlSalamChihara, -\monicAskeyWilson{n}@@{x}{a}{b}{c}{d}{q},monic Askey-Wilson polynomial,Askey-Wilson-polynomial-monic-p,P|F:P:PS,${\widehat p}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicAskeyWilson, -\monicBesselPoly{n}@@{x}{a},monic Bessel polynomial,Bessel-polynomial-monic-p,P|F:P:PS,${\widehat y}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicBesselPoly, -\monicCharlier{n}@@{x}{a},monic Charlier polynomial,Charlier-polynomial-monic-p,P|F:P:PS,${\widehat C}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicCharlier, -\monicChebyT{n}@@{x},monic Chebyshev polynomial of the first kind,Chebyshev-polynomial-first-kind-monic-p,P|F:P:PS,${\widehat T}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicChebyT, -\monicChebyU{n}@@{x},monic Chebyshev polynomial of the second kind,Chebyshev-polynomial-second-kind-monic-p,P|F:P:PS,${\widehat U}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicChebyU, -\monicHahn{n}@@{x}{\alpha}{\beta}{N},monic Hahn polynomial,Hahn-polynomial-monic-p,P|F:P:PS,${\widehat Q}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicHahn, -\monicHermite{n}@@{x},monic Hermite polynomial,Hermite-polynomial-monic,P|F:P:PS,${\widehat H}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicHermite, -\monicJacobi{n}@@{x},monic Jacobi polynomial,Jacobi-polynomial-monic-p,P|F:P:PS,"${\widehat P}^{(\alpha,\beta)}_{n}$",http://drmf.wmflabs.org/wiki/Definition:monicJacobi, -\monicKrawtchouk{n}@@{q^{-x}}{p}{N}{q},monic Krawtchouk polynomial,Krawtchouk-polynomial-monic-p,P|F:P:PS,${\widehat K}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicKrawtchouk, -\monicLaguerre{n}@@{x},monic Laguerre polynomial,generalized-Laguerre-polynomial-monic-p,P|F:P:PS,${\widehat L}^{\alpha}_n$,http://drmf.wmflabs.org/wiki/Definition:monicLaguerre, -\monicLegendrePoly{n}@@{x},monic Legendre polynomial,Legendre-spherical-polynomial-monic-p,P|F:P:PS,${\widehat P}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicLegendrePoly, -\monicMeixner{n}@@{x}{\beta}{c},monic Meixner polynomial,Meixner-polynomial-monic-p,P|F:P:PS,${\widehat M}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicMeixner, -\monicMeixnerPollaczek{\lambda}{n}@@{x}{\phi},monic Meixner-Pollaczek polynomial,Meixner-Pollaczek-polynomial-monic-p,P|F:P:PS,${\widehat P}^{(\alpha)}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicMeixnerPollaczek, -\monicRacah{n}@@{\lambda(x)}{\alpha}{\beta}{\gamma}{\delta},monic Racah polynomial,Racah-polynomial-monic-p,P|F:P:PS,${\widehat R}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicRacah, -\monicStieltjesWigert{n}@@{x}{q},monic Stieltjes-Wigert polynomial,Stieltjes-Wigert-polynomial-monic-p,P|F:P:PS,${\widehat S}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicStieltjesWigert, -\monicUltra{n}@@{x},monic ultraspherical/Gegenbauer polynomial,ultraspherical-Gegenbauer-polynomial-monic-p,P|F:P:PS,${\widehat C}^{\mu}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicUltra, -\monicWilson{n}@@{x^2}{a}{b}{c}{d},monic Wilson polynomial,Wilson-polynomial-monic-p,P|F:P:PS,${\widehat W}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicWilson, -\monicbigqJacobi{n}@@{x}{a}{b}{c}{q},monic big $q$-Jacobi polynomial,big-q-Jacobi-polynomial-monic-p,P|F:P:PS,${\widehat P}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicbigqJacobi, -\monicbigqLaguerre{n}@@{x}{a}{b}{q},monic big $q$-Laguerre polynomial,big-q-Laguerre-polynomial-monic-p,P|F:P:PS,${\widehat P}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicbigqLaguerre, -\monicbigqLegendre{n}@@{x}{c}{q},monic big $q$-Legendre polynomial,big-q-Legendre-polynomial-monic-p,P|F:P:PS,${\widehat P}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicbigqLegendre, -\monicctsHahn{n}@@{x}{a}{b}{c}{d},monic continuous Hahn polynomial,continuous-Hahn-polynomial-monic-p,P|F:P:PS,${\widehat p}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicctsHahn, -\monicctsbigqHermite{n}@@{x}{a}{q},monic continuous big $q$-Hermite polynomial,continuous-big-q-Hermite-polynomial-monic-p,P|F:P:PS,${\widehat H}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicctsbigqHermite, -\monicctsdualHahn{n}@@{x^2}{a}{b}{c}{d},monic continuous dual Hahn polynomial,continuous-dual-Hahn-polynomial-monic-p,P|F:P:PS,${\widehat S}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicctsdualHahn, -\monicctsdualqHahn{n}@@{x^2}{a}{b}{c},monic continuous dual $q$-Hahn polynomial,continuous-dual-q-Hahn-polynomial-monic-p,P|F:P:PS,${\widehat p}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicctsdualqHahn, -\monicctsqHahn{n}@@{x}{a}{b}{c}{d}{q},monic continuous $q$-Hahn polynomial,continuous-q-Hahn-polynomial-monic-p,P|F:P:PS,${\widehat p}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicctsqHahn, -\monicctsqHermite{n}@@{x}{q},monic continuous $q$-Hermite polynomial,continuous-q-Hermite-polynomial-monic-p,P|F:P:PS,${\widehat H}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicctsqHermite, -\monicctsqJacobi{n}@@{x}{q},monic continuous $q$-Jacobi polynomial,continuous-q-Jacobi-polynomial-monic-p,P|F:P:PS,"${\widehat P}^{(\alpha,\beta)}_{n}$",http://drmf.wmflabs.org/wiki/Definition:monicctsqJacobi, -\monicctsqLaguerre{n}@@{x}{q},monic continuous $q$-Laguerre polynomial,continuous-q-Laguerre-polynomial-monic-p,P|F:P:PS,${\widehat P}^{(\alpha)}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicctsqLaguerre, -\monicctsqLegendre{n}@@{x}{q},monic continuous $q$-Legendre polynomial,continuous-q-Legendre-polynomial-monic-p,P|F:P:PS,${\widehat P}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicctsqLegendre, -\monicctsqUltra{n}@@{x}{\beta}{q},monic continuous $q$-ultraspherical/Rogers polynomial,continuous-q-ultraspherical-Rogers-polynomial-monic-p,P|F:P:PS,${\widehat C}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicctsqUltra, -\monicdiscrqHermiteI{n}@@{x}{q},monic discrete $q$-Hermite I polynomial,discrete-q-Hermite-polynomial-I-monic-p,P|F:P:PS,${\widehat h}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicdiscrqHermiteI, -\monicdiscrqHermiteII{n}@@{x}{q},monic discrete $q$-Hermite II polynomial,discrete-q-Hermite-polynomial-II-monic-p,P|F:P:PS,${\widehat \tilde{p}_{n}}$,http://drmf.wmflabs.org/wiki/Definition:monicdiscrqHermiteII, -\monicdualHahn{n}@@{\lambda(x)}{\gamma}{\delta}{N},monic dual Hahn polynomial,dual-Hahn-polynomial-monic-p,P|F:P:PS,${\widehat R}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicdualHahn, -\monicdualqHahn{n}@@{\mu(x)}{\gamma}{\delta}{N}{q},monic dual $q$-Hahn polynomial,dual-q-Hahn-monic-p,P|F:P:PS,${\widehat R}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicdualqHahn, -\monicdualqKrawtchouk{n}@@{\lambda(x)}{c}{N}{q},monic dual $q$-Krawtchouk polynomial,dual-q-Krawtchouk-polynomial-monic-p,P|F:P:PS,${\widehat K}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicdualqKrawtchouk, -\moniclittleqJacobi{n}@@{x}{a}{b}{q},monic little $q$-Jacobi polynomial,little-q-Jacobi-polynomial-monic-p,P|F:P:PS,${\widehat p}_{n}$,http://drmf.wmflabs.org/wiki/Definition:moniclittleqJacobi, -\moniclittleqLaguerre{n}@@{x}{a}{q},monic little $q$-Laguerre/Wall polynomial,little-q-Laguerre-Wall-polynomial-monic-p,P|F:P:PS,${\widehat p}_{n}$,http://drmf.wmflabs.org/wiki/Definition:moniclittleqLaguerre, -\moniclittleqLegendre{n}@@{x}{q},monic little $q$-Legendre polynomial,little-q-Legendre-polynomial-monic-p,P|F:P:PS,${\widehat p}_{n}$,http://drmf.wmflabs.org/wiki/Definition:moniclittleqLegendre, -\monicpseudoJacobi{n}@@{x}{\nu}{N},monic pseudo-Jacobi polynomial,pseudo-Jacobi-polynomial-monic-p,P|F:P:PS,${\widehat P}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicpseudoJacobi, -\monicqBesselPoly{n}@@{x}{a}{q},monic $q$-Bessel polynomial,q-Bessel-polynomial-monic-p,P|F:P:PS,${\widehat y}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicqBessel, -\monicqCharlier{n}@@{q^{-x}}{a}{q},monic $q$-Charlier polynomial,q-Charlier-polynomial-monic-p,P|F:P:PS,${\widehat C}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicqCharlier, -\monicqHahn{n}@@{q^{-x}}{\alpha}{\beta}{N},monic $q$-Hahn polynomial,q-Hahn-polynomial-monic-p,P|F:P:PS,${\widehat Q}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicqHahn, -\monicqKrawtchouk{n}@@{q^{-x}}{p}{N}{q},monic $q$-Krawtchouk polynomial,q-Krawtchouk-polynomial-monic-p,P|F:P:PS,${\widehat K}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicqKrawtchouk, -\monicqLaguerre{n}@@{x}{q},monic $q$-Laguerre polynomial,q-Laguerre-polynomial-monic-p,P|F:P:PS,${\widehat L}^{(\alpha)}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicqLaguerre, -\monicqMeixner{n}@@{q^{-x}}{b}{c}{q},monic $q$-Meixner polynomial,q-Meixner-polynomial-monic-p,P|F:P:PS,${\widehat M}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicqMeixner, -\monicqMeixnerPollaczek{n}@@{x}{a}{q},monic $q$-Meixner-Pollaczek polynomial,q-Meixner-Pollaczek-polynomial-monic-p,P|F:P:PS,${\widehat P}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicqMeixnerPollaczek, -\monicqRacah{n}@@{\mu(x)}{\alpha}{\beta}{\gamma}{\delta}{q},monic $q$-Racah polynomial,q-Racah-polynomial-monic-R,P|F:P:PS,${\widehat R}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicqRacah, -\monicqinvAlSalamChihara{n}@{x}{a}{b}{q^{-1}},monic $q$-inverse Al-Salam-Chihara polynomial,q-inverse-AlSalam-Chihara-polynomial-monic-p,P|F:P,${\widehat Q}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicqinvAlSalamChihara, -\monicqtmqKrawtchouk{n}@@{q^{-x}}{p}{N}{q},monic quantum $q$-Krawtchouk polynomial,quantum-q-Krawtchouk-polynomial-monic-p,P|F:P:PS,${\widehat K^{\mathrm{qtm}}}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicqtmqKrawtchouk, -"\multinomial{n_1+n_2+\dots+n_k}{n_1,...,n_k}",multinomial coefficient,EMPTY,I|F,$\left(A;B\right)$,http://dlmf.nist.gov/26.4.E2, -\ninej@@{j_{11}}{j_{12}}{j_{13}}{j_{21}}{j_{22}}{j_{23}}{j_{31}}{j_{32}}{j_{33}},$\ninej$ symbol,EMPTY,F|F:P:nP,$$,http://dlmf.nist.gov/34.6#E1, -\normAskeyWilsonptilde{n}@@{x}{a}{b}{c}{d}{q},normalized Askey-Wilson polynomial ${\tilde p}$,Askey-Wilson-polynomial-normalized-p-tilde,P|F:P:PS,${\tilde p}_{n}$,http://drmf.wmflabs.org/wiki/Definition:normAskeyWilsonptilde, -\normJacobiR{\alpha}{\beta}{n}@{x},normalized Jacobi polynomial,Jacobi-polynomial-R,P|F:P,"$R^{(\alpha,\beta)}_{n}$",http://drmf.wmflabs.org/wiki/Definition:normJacobiR, -\normWilsonWtilde{n}@@{x^2}{a}{b}{c}{d},normalized Wilson polynomial ${\tilde W}$,Wilson-polynomial-normalized-W-tilde,P|F:P:PS,${\tilde W}_{n}$,http://drmf.wmflabs.org/wiki/Definition:normWilsonWtilde, -\normctsHahnptilde{n}@@{x}{a}{b}{c}{d},normalized continuous Hahn polynomial ${\tilde p}$,continuous-Hahn-polynomial-normalized-p-tilde,P|F:P:PS,${\tilde p}_{n}$,http://drmf.wmflabs.org/wiki/Definition:normctsHahnptilde, -\normctsdualHahnStilde{n}@@{x^2}{a}{b}{c}{d},normalized continuous dual Hahn polynomial ${\tilde S}$,continuous-dual-Hahn-polynomial-normalized-S-tilde,P|P:PS,${\tilde S}_{n}$,http://drmf.wmflabs.org/wiki/Definition:normctsdualHahnStilde, -\normctsdualqHahnptilde{n}@@{x}{a}{b}{c}{q},normalized continuous dual $q$-Hahn polynomial ${\tilde p}$,continuous-dual-q-Hahn-polynomial-normalized-p-tilde,P|F:P:PS,${\tilde p}_{n}$,http://drmf.wmflabs.org/wiki/Definition:normctsdualqHahnptilde, -\normctsqHahnptilde{n}@@{x}{a}{b}{c}{d}{q},normalized continuous $q$-Hahn polynomial ${\tilde p}$,continuous-q-Hahn-polynomial-normalized-p-tilde,P|F:P:PS,${\tilde p}_{n}$,http://drmf.wmflabs.org/wiki/Definition:normctsqHahnptilde, -\normctsqJacobiPtilde{n}@@{x}{q},normalized continuous $q$-Jacobi polynomial ${\tilde P}$,continuous-q-Jacobi-polynomial-P-tilde,P|F:P:PS,"${\tilde P}^{(\alpha,\beta)}_{n}$",http://drmf.wmflabs.org/wiki/Definition:normctsqJacobiPtilde, -\notin,not an element of,EMPTY,Q|F,$\notin$,http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r10, -\opminus,negative unity to an integer power,EMPTY,I|F,$(-1)$,http://dlmf.nist.gov/5.7.E7, -\pderiv{f}{x},partial derivative,EMPTY,O|F,$\frac{\partial f}{\partial x}$,http://dlmf.nist.gov/1.5#E3, -\pdiff{x},power of a partial differential,EMPTY,O|F,"$\partial^nx$",http://dlmf.nist.gov/1.5#E3, -\pgcd,greatest common divisor,EMPTY,I|F,"$\left(m,n\right)$",http://dlmf.nist.gov/27.1, -\ph@@{z},phase,phase,O|F:P:nP,$\mathrm{ph}$,http://dlmf.nist.gov/1.9#E7, -\pochhammer{a}{n},Pochhammer symbol,EMPTY,F|F,$(a)_n$,http://dlmf.nist.gov/5.2#iii, -\poly{p}{n}@{x},polynomial,EMPTY,SM|F:P,${p}_{n}$,http://drmf.wmflabs.org/wiki/Definition:poly, -\polygamma{n}@{z},polygamma functions,polygamma,F|F:P,$\psi^{(n)}$,http://dlmf.nist.gov/5.15, -\power,power function,EMPTY,F|F,$x^y$,http://dlmf.nist.gov/36.12.E9, -\prod,product,EMPTY,SM|F,$\Pi$,http://drmf.wmflabs.org/wiki/Definition:prod, -\pseudoJacobi{n}@{x}{\nu}{N},pseudo Jacobi polynomial,EMPTY,P|F:P,$P_{n}$,http://drmf.wmflabs.org/wiki/Definition:pseudoJacobi, -\psfactorial{a}{\kappa},partitional shifted factorial,EMPTY,F|F,${\left[a\right]_{\kappa}}$,http://dlmf.nist.gov/35.4#E1, -\pvint,Cauchy principal value,EMPTY,O|F,$\pvint_a^b$,http://dlmf.nist.gov/1.4#E24, -\qAppelli@{a}{b}{b'}{c}{q}{x}{y},first $q$-Appell function,q-Appell-Phi-1,F|F:P,$\Phi^{(1)}$,http://dlmf.nist.gov/17.4.E5, -\qAppellii@{a}{b}{b'}{c}{c'}{q}{x}{y},second $q$-Appell function,q-Appell-Phi-2,F|F:P,$\Phi^{(2)}$,http://dlmf.nist.gov/17.4.E6, -\qAppelliii@{a}{a'}{b}{b'}{c}{q}{x}{y},third $q$-Appell function,q-Appell-Phi-3,F|F:P,$\Phi^{(3)}$,http://dlmf.nist.gov/17.4.E7, -\qAppelliv@{a}{b}{c}{c'}{q}{x}{y},fourth $q$-Appell function,q-Appell-Phi-4,F|F:P,$\Phi^{(4)}$,http://dlmf.nist.gov/17.4.E8, -\qBernoulli{n}@{x}{q},$q$-Bernoulli polynomial,q-Bernoulli-polynomial,P|F:P,$\beta_{q}$,http://dlmf.nist.gov/17.3#E7, -\qBesselPoly{n}@{x}{b}{q},$q$-Bessel polynomial,EMPTY,P|F:P,$y_{n}$,http://drmf.wmflabs.org/wiki/Definition:qBessel, -\qBeta{q}@{a}{b},$q$-beta function,q-Beta,F|F:P,$\mathrm{B}_{q}$,http://dlmf.nist.gov/5.18#E11, -\qBinomial{n}{m}{q},$q$-binomial coefficient (or Gaussian polynomial),EMPTY,F|F,$\left[n \atop m\right]_{q}$,http://dlmf.nist.gov/17.2#E27,http://dlmf.nist.gov/26.9#SS2.p1 -\qCharlier{n}@{x}{c}{q},$q$-Charlier polynomial,EMPTY,P|F:P,$C_{n}$,http://drmf.wmflabs.org/wiki/Definition:qCharlier, -\qCos{q}@@{x},$q$-analogue of the $\cos$ function: ${\rm Cos}_q$,q-Cos,F|F:nP:P,$\mathrm{Cos}_{q}$,http://dlmf.nist.gov/17.3#E6, -\qCosKLS{q}@@{z},$q$-analogue of the $\cos$ function used in KLS: $\mathrm{Cos}_{q}$,q-Cos-KLS,P|F:nP:P,$\mathrm{Cos}_{q}$,http://drmf.wmflabs.org/wiki/Definition:qCosKLS, -\qDigamma{q}@{z},$q$-digamma function,q-digamma,F|F:P,$\psi_{q}$,http://drmf.wmflabs.org/wiki/Definition:qDigamma, -\qEuler{m}{s}@{q},$q$-Euler number,q-Euler-number,F|F:P,"$A_{m,n}$",http://dlmf.nist.gov/17.3#E8, -\qExp{q}@@{x},$q$-analogue of the $\exp$ function: $E_q$,q-Exp,F|F:nP:P,$E_{q}$,http://dlmf.nist.gov/17.3#E2, -\qExpKLS{q}@@{z},$q$-analogue of the $\exp$ function used in KLS: $\mathrm{E}_{q}$,q-Exp-KLS,F|F:nP:P,$\mathrm{E}_{q}$,http://drmf.wmflabs.org/wiki/Definition:qExpKLS, -\qFactorial{n}{q},$q$-factorial,q-factorial,F|F,$n!_q$,http://dlmf.nist.gov/5.18#E2, -\qGamma{q}@{z},$q$-gamma function,q-gamma,F|F:P,$\Gamma_{q}$,http://dlmf.nist.gov/5.18#E4, -\qHahn{n}@@{q^{-x}}{\alpha}{\beta}{N},$q$-Hahn polynomial,q-Hahn-polynomial-Q,P|P:PS,$Q_{n}$,http://drmf.wmflabs.org/wiki/Definition:qHahn, -\qHahnQ{n}@{x}{\alpha}{\beta}{N}{q},$q$-Hahn polynomial,q-Hahn-polynomial-Q,P|F:P,$Q_{n}$,http://dlmf.nist.gov/18.27#E3, -\qHermiteH{n}@{x}{q},continuous $q$-Hermite polynomial,continuous-q-Hermite-polynomial-H,P|F:P,$H_{n}$,http://dlmf.nist.gov/18.28#E16, -\qHermitehI{n}@{x}{q},discrete $q$-Hermite I polynomial,discrete-q-Hermite-polynomial-h-I,P|F:P,$h_{n}$,http://dlmf.nist.gov/18.27#E21, -\qHermitehII{n}@{x}{q},discrete $q$-Hermite II polynomial,discrete-q-Hermite-polynomial-h-II,P|F:P,$\tilde{h}_{n}$,http://dlmf.nist.gov/18.27#E23, -"\qHyperrWs{r}{r+1}@@@{a_1}{a_4,a_5,...,a_{r+1}}{q}{z}",very-well-poised basic hypergeometric (or $q$-hypergeometric) function,very-well-poised-q-hypergeometric-rWs,F|F:fo1:fo2:fo3,${{}_{r+1}W_{r}}$,http://drmf.wmflabs.org/wiki/Definition:qHyperrWs, -"\qHyperrphis{r}{s}@@@{a_1,...a_r}{b_1,...,b_s}{q}{z}",basic hypergeometric (or $q$-hypergeometric) function,q-hypergeometric-rphis,F|F:fo1:fo2:fo3,${{}_{r}\phi_{s}}$,http://dlmf.nist.gov/17.4#E1, -"\qHyperrpsis{r}{s}@@@{a_1,...,a_r}{b_1,...,b_s}{q}{z}",bilateral basic hypergeometric (or bilateral $q$-hypergeometric) function,q-hypergeometric-rpsis,F|F:fo1:fo2:fo3,${{}_{r}\psi_{s}}$,http://dlmf.nist.gov/17.4#E3, -\qJacobiP{n}@{x}{a}{b}{c}{q},"big $q$-Jacobi polynomial $P_n(x;a,b,c;q)$",q-Jacobi-polynomial-P,P|F:P,$P_{n}$,http://dlmf.nist.gov/18.27#E5, -\qJacobiPP{\alpha}{\beta}{n}@{x}{c}{d}{q},"big $q$-Jacobi polynomial $P_n^{(\alpha,\beta)}(x;c,d;q)$",big-q-Jacobi-polynomial-P-type-2,P|F:P,"$P^{(\alpha,\beta)}_{n}$",http://dlmf.nist.gov/18.27#E6, -\qJacobip{n}@{x}{a}{b}{q},little $q$-Jacobi polynomial,q-Jacobi-polynomial-p,P|F:P,$p_{n}$,http://dlmf.nist.gov/18.27#E13, -\qKrawtchouk{n}@@{q^{-x}}{p}{N}{q},$q$-Krawtchouk polynomial,q-Krawtchouk-polynomial-K,P|F:P:PS,$K_{n}$,http://drmf.wmflabs.org/wiki/Definition:qKrawtchouk, -\qLaguerre[\alpha]{n}@@{x}{q},$q$-Laguerre polynomial,q-Laguerre-polynomial-L,P|FnO:PnO:nPnO:F:P:PS,$L_n^{(\alpha)}$,http://drmf.wmflabs.org/wiki/Definition:qLaguerre, -\qLaguerreL[\alpha]{n}@{x}{q},$q$-Laguerre polynomial,q-Laguerre-polynomial-L,P|FnO:PnO:FO:PO,$L_n^{(\alpha)}$,http://dlmf.nist.gov/18.27#E15, -\qLegendre{n}@{x}{q},continuous $q$-Legendre polynomial,EMPTY,P|F:P,$\qLegendre{n}$,http://drmf.wmflabs.org/wiki/Definition:qLegendre, -\qMeixner{n}@{x}{b}{c}{q},$q$-Meixner polynomial,q-Meixner-polynomial-M,P|F:P,$M_{n}$,http://drmf.wmflabs.org/wiki/Definition:qMeixner, -\qMeixnerPollaczek{n}@{x}{a}{q},$q$-Meixner-Pollaczek polynomial,q-Meixner-Pollaczek-polynomial-M,P|F:P,$P_{n}$,http://drmf.wmflabs.org/wiki/Definition:qMeixnerPollaczek, -"\qMultiPochhammer{a_1,\ldots,a_n}{q}{n}",$q$-multi-Pochhammer symbol,q-multi-Pochhammer,F|F,"$\(a_1,\ldots,a_k;q)_n$",http://dlmf.nist.gov/17.2.E5, -"\qMultinomial{a_1+\ldots+a_n}{a_1,\ldots,a_n}{q}",$q$-multinomial,q-multinomial,F|F,"$\left[a_1+\ldots+a_n \atop a_1,\ldots,a_n\right]_{q}$",http://dlmf.nist.gov/26.16.E1, -\qPochhammer{a}{q}{n},$q$-Pochhammer symbol,EMPTY,F|F,$(a;q)_n$,http://dlmf.nist.gov/5.18#i,http://dlmf.nist.gov/17.2#SS1.p1 -\qPolygamma{n}{q}@{z},$q$-polygamma function,q-polygamma,F|F:P,$\psi_{q}^{(n)}$,http://drmf.wmflabs.org/wiki/Definition:qPolygamma, -\qRacah{n}@{x}{\alpha}{\beta}{\gamma}{\delta}{q},$q$-Racah polynomial,q-Racah-polynomial-R,P|F:P,$R_{n}$,http://dlmf.nist.gov/18.28#E19, -\qRacahR{n}@{x}{\alpha}{\beta}{\gamma}{\delta}{q},$q$-Racah polynomial,q-Racah-polynomial-R,P|F:P,$R_{n}$,http://dlmf.nist.gov/18.28#E19, -\qSin{q}@@{x},$q$-analogue of the $\sin$ function: ${\rm Sin}_q$,q-Sin,F|F:nP:P,$\mathrm{Sin}_{q}$,http://dlmf.nist.gov/17.3#E4, -\qSinKLS{q}@@{z},$q$-analogue of the $\sin$ function used in KLS: $\mathrm{Sin}_q$,q-Sin-KLS,F|F:nP:P,$\mathrm{Sin}_{q}$,http://drmf.wmflabs.org/wiki/Definition:qSinKLS, -\qStirling{m}{s}@{q},$q$-Stirling number,q-Stirling-number,F|F:P,"$a_{m,s}$",http://dlmf.nist.gov/17.3#E9, -\qUltraspherical{n}@{x}{\beta}{q},continuous $q$-ultraspherical polynomial,q-ultraspherical-polynomial,P|F:P,$C_{n}$,http://dlmf.nist.gov/18.28#E13, -\qcos{q}@@{x},$q$-analogue of the $\cos$ function: $\cos_q$,q-cos,F|F:nP:P,$\mathrm{cos}_{q}$,http://dlmf.nist.gov/17.3#E5, -\qcosKLS{q}@@{z},$q$-analogue of the $\cos$ function used in KLS: $\cos_q$,q-cos-KLS,P|F:nP:P,$\mathrm{cos}_{q}$,http://drmf.wmflabs.org/wiki/Definition:qcosKLS, -\qderiv[n]{q}@{z},$q$-derivative,q-derivative,O|FO:PO:FnO:PnO,$\mathcal{D}_q^n$,http://drmf.wmflabs.org/wiki/Definition:qderiv, -\qdiff{q}{x},$q$-differential,EMPTY,O|F,"${\mathrm d}^n_qx$",http://dlmf.nist.gov/17.2#SS5.p1, -\qexp{q}@@{x},$q$-analogue of the $\exp$ function: $e_q$,q-exp,F|F:nP:P,$e_{q}$,http://dlmf.nist.gov/17.3#E1, -\qexpKLS{q}@@{z},$q$-analogue of the $\exp$ function used in KLS: $\mathrm{e}_{q}$,q-exp-KLS,F|F:nP:P,$\mathrm{e}_{q}$,http://drmf.wmflabs.org/wiki/Definition:qexpKLS, -\qinvAlSalamChihara{n}@{x}{a}{b}{q^{-1}},$q$-inverse Al-Salam-Chihara polynomial,q-inverse-AlSalam-Chihara-polynomial-Q,P|F:P,$Q_{n}$,http://dlmf.nist.gov/23.1, -\qinvHermiteh{n}@{x}{q},continuous $q$-inverse Hermite polynomial,continuous-q-inverse-Hermite-polynomial-h,P|F:P,$h_{n}$,http://dlmf.nist.gov/23.1, -\qsin{q}@@{x},$q$-analogue of the $\sin$ function: $\sin_q$,q-sin,F|F:nP:P,$\mathrm{sin}_{q}$,http://dlmf.nist.gov/17.3#E3, -\qsinKLS{q}@@{z},$q$-analogue of the $\sin$ function used in KLS: $\sin_q$,q-sin-KLS,F|F:nP:P,$\mathrm{sin}_{q}$,http://drmf.wmflabs.org/wiki/Definition:qsinKLS, -\qtmqKrawtchouk{n}@@{q^{-x}}{p}{N}{q},quantum $q$-Krawtchouk polynomial,quantum-q-Krawtchouk-polynomial-K,P|F:P:PS,$K^{\mathrm{qtm}}_{n}$,http://drmf.wmflabs.org/wiki/Definition:qtmqKrawtchouk, -\realpart{z},real part,EMPTY,O|F,"$\Re {z}$",http://dlmf.nist.gov/1.9#E2, -\terminant{p}@{z},terminant function,terminant-function,F|F:P,$F_{p}$,http://dlmf.nist.gov/2.11#E11, -\rescaledTerminant{p}@{z},rescaled terminant functions,rescaled-terminant-function,F|F:P,$G_{p}$,http://dlmf.nist.gov/9.7#SS5.p1, -\rselection,right bracketed generalization of $\pm$,EMPTY,SM|F,$\left.\begin{matrix}x\end{matrix}\right\$,http://dlmf.nist.gov/28.4.E22, -\sec@@{z},secant function,secant,F|F:P:nP,$\mathrm{sec}$,http://dlmf.nist.gov/4.14#E6, -\sech@@{z},hyperbolic secant function,hyperbolic-secant,F|F:P:nP,$\mathrm{sech}$,http://dlmf.nist.gov/4.28#E6, -\selection,generalization of $\pm$,EMPTY,SM|F,$\;\begin{matrix}x\end{matrix}\;$,http://dlmf.nist.gov/10.23.E7, -\setminus,set subtraction,EMPTY,Q|F,$\setminus$,http://dlmf.nist.gov/front/introduction#Sx4.p2.t1.r19, -\setmod,$S_1 \setmod S_2$: set of all elements of $S_1$ modulo elements of $S_2$,EMPTY,SN|F,$/$,http://dlmf.nist.gov/21.1#p2.t1.r17, -\sign@@{x},sign function,sign,F|F:P:nP,$\mathrm{sign}$,http://dlmf.nist.gov/front/introduction#Sx4.p2.t1.r18, -\sim,asymptotic equality,EMPTY,Q|F,$\tilde$,http://dlmf.nist.gov/2.1#E1, -\sin@@{z},sine function,sin,F|F:P:nP,$\mathrm{sin}$,http://dlmf.nist.gov/4.14#E1, -\sinInt@{z},sine integral ${\rm si}$,shifted-sine-integral,F|F:P,$\mathrm{si}$,http://dlmf.nist.gov/6.2#E10, -\sinh@@{z},hyperbolic sine function,hyperbolic-sine,F|F:P:nP,$\mathrm{sinh}$,http://dlmf.nist.gov/4.28#E1, -\sinintg@{a}{z},generalized sine integral ${\rm si}$,generalized-sine-integral-si,F|F:P,$\mathrm{si}$,http://dlmf.nist.gov/8.21#E1, -\sixj@@{j_{1}}{j_{2}}{j_{3}}{l_{1}}{l_{2}}{l_{3}},$\sixj$ symbol,EMPTY,F|F:P:nP,$\left\{\begin{array}{ccc}j_1 & j_2 & j_3\\l_1 & l_2 & l_3\end{array}\right\}$,http://dlmf.nist.gov/23.1, -\subset,is contained in,EMPTY,Q|F,$\subset$,http://dlmf.nist.gov/front/introduction#Sx4.p2.t1.r2, -\subseteq,is in or is contained in,EMPTY,Q|F,$\subseteq$,http://dlmf.nist.gov/front/introduction#Sx4.p2.t1.r3, -\sum,sum,EMPTY,SM|F,$\Sigma$,http://drmf.wmflabs.org/wiki/Definition:sum, -\sup,least upper bound (supremum),EMPTY,Q|F,$\mathrm{sup}$,http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r25, -\tan@@{z},tangent function,tangent,F|F:P:nP,$\mathrm{tan}$,http://dlmf.nist.gov/4.14#E4, -\tanh@@{z},hyperbolic tangent function,hyperbolic-tangent,F|F:P:nP,$\mathrm{tanh}$,http://dlmf.nist.gov/4.28#E4, -\threej@@{j_1}{j_2}{j_3}{m_1}{m_2}{m_3},$\threej$ symbol,EMPTY,F|F:P:nP,$\left(\begin{array}{ccc}j_1 & j_2 & j_3\\m_1 & m_2 & m_3\end{array}\right)$,http://dlmf.nist.gov/34.2#E4, -\trace,trace of matrix,trace,L|F,$\mathrm{etr}$,http://dlmf.nist.gov/23.1, -\transpose,transpose of a matrix,EMPTY,L|F,$A^{\mathrm{T}}$,http://dlmf.nist.gov/1.3.E5, -\weight{w}@{x},weight function,EMPTY,SM|F:P,${w}$,http://drmf.wmflabs.org/wiki/Definition:weight, -\wigner{j_1}{m_1}{j_2}{m_2}{j_3}{-m_3},Clebsch-Gordan coefficient,EMPTY,F|F,"$\left(j_{1}\;m_{1}\;j_{2}\;m_{2} | j_{1}\;j_{2}\; j_{3}\,\,-m_{3}\right)$",http://dlmf.nist.gov/34.1#p4, -\BesselPolyIIparam{n}@{x}{a}{b},Bessel polynomial with two parameters $y_n$,Bessel-polynomial-two-parameters-y,P|F:P,$y_{n}$,http://drmf.wmflabs.org/wiki/Definition:BesselPolyIIparam, -\BesselPolyTheta{n}@{x}{a}{b},Bessel polynomial with two parameters $\theta_n$,Bessel-polynomial-two-parameters-theta,P|F:P,$\theta_{n}$,http://drmf.wmflabs.org/wiki/Definition:BesselPolyTheta, -\FourierTrans@{f}{s},Fourier transform,Fourier-transform,O|F:P,$\mathscr{F}$,http://dlmf.nist.gov/1.14#E1, -\Continuous@{A},space of continuous functions on a set $A$,set-of-continuous-functions,S|F,$C(A)$,http://dlmf.nist.gov/1.4#ii, -\SpaceTestFunTempered@{A},space of test functions for tempered distributions,space-of-test-functions-for-tempered-distributions-T,S|F,${\mathcal T}(\Real)$,http://dlmf.nist.gov/1.16#v, diff --git a/Azeem/src/tex2Wiki.py b/Azeem/src/tex2Wiki.py index bcbe712..a89ab01 100644 --- a/Azeem/src/tex2Wiki.py +++ b/Azeem/src/tex2Wiki.py @@ -268,20 +268,21 @@ def writeout(ofname): def main(): - if len(sys.argv) != 4: - + if len(sys.argv) != 6: fname = "../../data/ZE.3.tex" ofname = "../../data/ZE.4.xml" lname = "../../data/BruceLabelLinks" + glossary = "../../data/new.Glossary.csv" + mmd = "../../data/OrthogonalPolynomials.mmd" else: - fname = sys.argv[1] ofname = sys.argv[2] lname = sys.argv[3] - + glossary = sys.argv[4] + mmd = sys.argv[5] setup_label_links(lname) - readin(fname) + readin(fname,glossary,mmd) writeout(ofname) @@ -290,12 +291,12 @@ def setup_label_links(ofname): lLink = open(ofname, "r").readlines() -def readin(ofname): +def readin(ofname,glossary,mmd): # try: for jsahlfkjsd in range(0, 1): global wiki tex = open(ofname, 'r') - main_file = open("OrthogonalPolynomials.mmd", "r") + main_file = open(mmd, "r") mainText = main_file.read() mainPrepend = "" mainWrite = open("OrthogonalPolynomials.mmd.new", "w") @@ -341,7 +342,7 @@ def readin(ofname): mainWrite.write(mainText) mainWrite.close() main_file.close() - copyfile('OrthogonalPolynomials.mmd', 'OrthogonalPolynomials.mmd.new') + copyfile(mmd, 'OrthogonalPolynomials.mmd.new') parse = False elif "\\title" in line and parse: stringWrite = "\'\'\'" @@ -652,7 +653,7 @@ def readin(ofname): gFlag = False checkFlag = False get = False - gCSV = csv.reader(open('new.Glossary.csv', 'rb'), delimiter=',', quotechar='\"') + gCSV = csv.reader(open(glossary, 'rb'), delimiter=',', quotechar='\"') preG = "" if symbol == "\\&": ampFlag = True diff --git a/Pipeline/src/OrthogonalPolynomials.mmd b/Pipeline/src/OrthogonalPolynomials.mmd deleted file mode 100644 index c841fb5..0000000 --- a/Pipeline/src/OrthogonalPolynomials.mmd +++ /dev/null @@ -1,81 +0,0 @@ -drmf_bof -'''Orthogonal Polynomials''' -{{#set:Section=0}} -== Sections in KLS Chapter 1 == - -
-* [[Orthogonal polynomials|Orthogonal polynomials]] -* [[The gamma and beta function|The gamma and beta function]] -* [[The shifted factorial and binomial coefficients|The shifted factorial and binomial coefficients]] -* [[Hypergeometric functions|Hypergeometric functions]] -* [[The binomial theorem and other summation formulas|The binomial theorem and other summation formulas]] -* [[Some integrals|Some integrals]] -* [[Transformation formulas|Transformation formulas]] -* [[The q-shifted factorial|The ''q''-shifted factorial]] -* [[The q-gamma function and q-binomial coefficients|The ''q''-gamma function and ''q''-binomial coefficients]] -* [[Basic hypergeometric functions|Basic hypergeometric functions]] -* [[The q-binomial theorem and other summation formulas|The ''q''-binomial theorem and other summation formulas]] -* [[More integrals|More integrals]] -* [[Transformation formulas|Transformation formulas]] -* [[Some q-analogues of special functions|Some ''q''-analogues of special functions]] -* [[The q-derivative and q-integral|The ''q''-derivative and ''q''-integral]] -* [[Shift operators and Rodrigues-type formulas|Shift operators and Rodrigues-type formulas]] -
-== Sections in KLS Chapter 9 == - -
-* [[Wilson|Wilson]] -* [[Racah|Racah]] -* [[Continuous dual Hahn|Continuous dual Hahn]] -* [[Continuous Hahn|Continuous Hahn]] -* [[Hahn|Hahn]] -* [[Dual Hahn|Dual Hahn]] -* [[Meixner-Pollaczek|Meixner-Pollaczek]] -* [[Jacobi|Jacobi]] -* [[Jacobi: Special cases|Jacobi: Special cases]] -* [[Pseudo Jacobi|Pseudo Jacobi]] -* [[Meixner|Meixner]] -* [[Krawtchouk|Krawtchouk]] -* [[Laguerre|Laguerre]] -* [[Bessel|Bessel]] -* [[Charlier|Charlier]] -* [[Hermite|Hermite]] -
-== Sections in KLS Chapter 14 == - -
-* [[Askey-Wilson|Askey-Wilson]] -* [[q-Racah|''q''-Racah]] -* [[Continuous dual q-Hahn|Continuous dual ''q''-Hahn]] -* [[Continuous q-Hahn|Continuous ''q''-Hahn]] -* [[Big q-Jacobi|Big ''q''-Jacobi]] -* [[Big q-Jacobi: Special case|Big ''q''-Jacobi: Special case]] -* [[q-Hahn|''q''-Hahn]] -* [[Dual q-Hahn|Dual ''q''-Hahn]] -* [[Al-Salam-Chihara|Al-Salam-Chihara]] -* [[q-Meixner-Pollaczek|''q''-Meixner-Pollaczek]] -* [[Continuous q-Jacobi|Continuous ''q''-Jacobi]] -* [[Continuous q-Jacobi: Special cases|Continuous ''q''-Jacobi: Special cases]] -* [[Big q-Laguerre|Big ''q''-Laguerre]] -* [[Little q-Jacobi|Little ''q''-Jacobi]] -* [[Little q-Jacobi: Special case|Little ''q''-Jacobi: Special case]] -* [[q-Meixner|''q''-Meixner]] -* [[Quantum q-Krawtchouk|Quantum ''q''-Krawtchouk]] -* [[q-Krawtchouk|''q''-Krawtchouk]] -* [[Affine q-Krawtchouk|Affine ''q''-Krawtchouk]] -* [[Dual q-Krawtchouk|Dual ''q''-Krawtchouk]] -* [[Continuous big q-Hermite|Continuous big ''q''-Hermite]] -* [[Continuous q-Laguerre|Continuous ''q''-Laguerre]] -* [[Little q-Laguerre / Wall|Little ''q''-Laguerre / Wall]] -* [[q-Laguerre|''q''-Laguerre]] -* [[q-Bessel|''q''-Bessel]] -* [[q-Charlier|''q''-Charlier]] -* [[Al-Salam-Carlitz I|Al-Salam-Carlitz I]] -* [[Al-Salam-Carlitz II|Al-Salam-Carlitz II]] -* [[Continuous q-Hermite|Continuous ''q''-Hermite]] -* [[Stieltjes-Wigert|Stieltjes-Wigert]] -* [[Discrete q-Hermite I|Discrete ''q''-Hermite I]] -* [[Discrete q-Hermite II|Discrete ''q''-Hermite II]] -
- -drmf_eof diff --git a/Pipeline/src/new.Glossary.csv b/Pipeline/src/new.Glossary.csv deleted file mode 100644 index 8875e6b..0000000 --- a/Pipeline/src/new.Glossary.csv +++ /dev/null @@ -1,691 +0,0 @@ -\AGM@{a}{g},arithmetic-geometric mean,arithmetic-geometric-mean,F|F:P,$M$,http://dlmf.nist.gov/19.8#SS1.p1, -\AThetaFunction@{x},Jacobi theta function $\vartheta$,a-theta-function,F|F:P,$\vartheta$,http://dlmf.nist.gov/23.1, -\AbstractJacobi{p}{q},generic Jacobian elliptic function,generic-Jacobian-elliptic-function,F|F,$\mathrm{pq}$,http://dlmf.nist.gov/22.2#E10, -\Acos@@{z},general inverse cosine function,multivalued-inverse-cosine,F|F:P:nP,$\mathrm{Arccos}$,http://dlmf.nist.gov/4.23#E2, -\Acosh@@{z},general inverse hyperbolic cosine function,multivalued-hyperbolic-inverse-cosine,F|F:P:nP,$\mathrm{Arccosh}$,http://dlmf.nist.gov/4.37#E2, -\Acot@@{z},general inverse cotangent function,multivalued-inverse-cotangent,F|F:P:nP,$\mathrm{Arccot}$,http://dlmf.nist.gov/4.23#E6, -\Acoth@@{z},general inverse hyperbolic cotangent function,multivalued-hyperbolic-inverse-cotangent,F|F:P:nP,$\mathrm{Arccoth}$,http://dlmf.nist.gov/4.37#E6, -\Acsc@@{z},general inverse cosecant function,multivalued-inverse-cosecant,F|F:P:nP,$\mathrm{Arccsc}$,http://dlmf.nist.gov/4.23#E4, -\Acsch@@{z},general inverse hyperbolic cosecant function,multivalued-hyperbolic-inverse-cosecant,F|F:P:nP,$\mathrm{Arccsch}$,http://dlmf.nist.gov/4.37#E4, -\AffqKrawtchouk{n}@@{q^{-x}}{p}{N}{q},affine $q$-Krawtchouk polynomial,affine-q-Krawtchouk-polynomial-K,P|F:P:PS,$K^{\mathrm{Aff}}_{n}$,http://drmf.wmflabs.org/wiki/Definition:AffqKrawtchouk, -\AiryAi@{z},Airy function ${\rm Ai}$,Airy-Ai,F|F:P,$\mathrm{Ai}$,http://dlmf.nist.gov/9.2#i, -\AiryBi@{z},Airy function ${\rm Bi}$,Airy-Bi,F|F:P,$\mathrm{Bi}$,http://dlmf.nist.gov/9.2#i, -\AiryModulusM@{z},Airy modulus function $M$,modulus-Airy-M,F|F:P,$M$,http://dlmf.nist.gov/9.8#i, -\AiryModulusN@{z},Airy modulus function $N$,modulus-Airy-N,F|F:P,$N$,http://dlmf.nist.gov/9.8#i, -\AiryPhasePhi@{z},Airy phase function $\phi$,phase-Airy-Phi,F|F:P,$\phi$,http://dlmf.nist.gov/9.8#i, -\AiryPhaseTheta@{z},Airy phase function $\theta$,phase-Airy-Theta,F|F:P,$\theta$,http://dlmf.nist.gov/9.8#i, -\AlSalamCarlitzI{a}{n}@{x}{q},Al-Salam-Carlitz I polynomial,EMPTY,P|F:P,$U^{(n)}_{\alpha}$,http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzI, -\AlSalamCarlitzII{a}{n}@{x}{q},Al-Salam-Carlitz II polynomial,EMPTY,P|F:P,$V^{(n)}_{\alpha}$,http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzII, -\AlSalamChihara{n}@@{x}{a}{b}{q},Al-Salam-Chihara polynomial,Al-Salam-Chihara-polynomial-Q,P|F:P:PS,$Q_{n}$,http://drmf.wmflabs.org/wiki/Definition:AlSalamChihara, -\AlSalamChiharaQ{n}@{x}{a}{b}{q},Al-Salam-Chihara polynomial,AlSalam-Chihara-polynomial-Q,P|F:P,$Q_{n}$,http://dlmf.nist.gov/18.28#E7, -\AlSalamIsmail{n}@{x}{s}{q},Al-Salam Ismail polynomial,Al-Salam-Ismail-q-polynomial-U,P|P:F,$U_{n}$,http://drmf.wmflabs.org/wiki/Definition:AlSalamIsmail, -\AngerA{\nu}@{z},Anger-Weber function,associated-Anger-Weber-A,F|F:P,$\mathbf{A}_{\nu}$,http://dlmf.nist.gov/11.10#E4, -\AngerJ{\nu}@{z},Anger function,Anger-J,F|F:P,$\mathbf{J}_{\nu}$,http://dlmf.nist.gov/11.10#E1, -\AntiDer@{x}{f(x)},semantic antiderivative,antiderivative,SM|O,"$\int f(x){\mathrm d}\!x$",http://drmf.wmflabs.org/wiki/Definition:AntiDer, -\AppellFi@{\alpha}{\beta}{\beta'}{\gamma}{x}{y},Appell function $F_1$,Appell-F-1,F|F:P,${F_{1}}$,http://dlmf.nist.gov/16.13#E1, -\AppellFii@{\alpha}{\beta}{\beta'}{\gamma}{\gamma'}{x}{y},Appell function $F_2$,Appell-F-2,F|F:P,${F_{2}}$,http://dlmf.nist.gov/16.13#E2, -\AppellFiii@{\alpha}{\alpha'}{\beta}{\beta'}{\gamma}{x}{y},Appell function $F_3$,Appell-F-3,F|F:P,${F_{3}}$,http://dlmf.nist.gov/16.13#E3, -\AppellFiv@{\alpha}{\beta}{\gamma}{\gamma'}{x}{y},Appell function $F_4$,Appell-F-4,F|F:P,${F_{4}}$,http://dlmf.nist.gov/16.13#E4, -\Asec@@{z},general inverse secant function,multivalued-inverse-secant,F|F:P:nP,$\mathrm{Arcsec}$,http://dlmf.nist.gov/4.23#E5, -\Asech@@{z},general inverse hyperbolic secant function,multivalued-hyperbolic-inverse-secant,F|F:P:nP,$\mathrm{Arcsech}$,http://dlmf.nist.gov/4.37#E5, -\Asin@@{z},general inverse sine function,multivalued-inverse-sine,F|F:P:nP,$\mathrm{Arcsin}$,http://dlmf.nist.gov/4.23#E1, -\Asinh@@{z},general inverse hyperbolic sine function,multivalued-hyperbolic-inverse-sine,F|F:P:nP,$\mathrm{Arcsinh}$,http://dlmf.nist.gov/4.37#E1, -\AskeyWilson{n}@{x}{a}{b}{c}{d}{q},Askey-Wilson polynomial,Askey-Wilson-polynomial-p,P|F:P,$p_{n}$,http://dlmf.nist.gov/18.28#E1, -\AskeyWilsonp{n}@{x}{a}{b}{c}{d}{q},Askey-Wilson polynomial,Askey-Wilson-polynomial-p,P|F:P,$p_{n}$,http://dlmf.nist.gov/18.28#E1, -\AssJacobiP{\alpha}{\beta}{n}@{x}{c},associated Jacobi polynomial,associated-Jacobi-polynomial-P,P|F:P,"$P^{(\alpha,\alpha)}_{n}$",http://dlmf.nist.gov/18.30#E4, -\AssLegendrePoly{n}@{x}{c},associated Legendre polynomial,Legendre-spherical-polynomial-P,F|F:P,$P_{n}$,http://dlmf.nist.gov/18.30#E6, -\Atan@@{z},general inverse tangent function,multivalued-inverse-tangent,F|F:P:nP,$\mathrm{Arctan}$,http://dlmf.nist.gov/4.23#E3, -\Atanh@@{z},general inverse hyperbolic tangent function,multivalued-hyperbolic-inverse-tangent,F|F:P:nP,$\mathrm{Arctanh}$,http://dlmf.nist.gov/4.37#E3, -\BarnesGamma@{z},Barnes $G$-function (or double gamma function),Barnes-Gamma,F|F:P,$G$,http://dlmf.nist.gov/5.17#E1, -\BellNumber@{n},Bell number,Bell-number,I|F:P,$B$,http://dlmf.nist.gov/26.7#i, -\BernoulliB{n}@{x},Bernoulli polynomial,Bernoulli-polynomial-B,P|F:P,$B_{n}$,http://dlmf.nist.gov/24.2#i, -\BesselA{\nu}@{\Matrix{T}},Bessel function of matrix argument of the first kind,Bessel-of-matrix-A,F|F:P,$A_{\nu}$,http://dlmf.nist.gov/35.5#i, -\BesselB{\nu}@{\Matrix{T}},Bessel function of matrix argument of the second kind,Bessel-of-matrix-B,F|F:P,$B_{\nu}$,http://dlmf.nist.gov/35.5#E3, -\BesselI{\nu}@{z},modified Bessel function of the first kind,modified-Bessel-first-kind,F|F:P,$I_{\nu}$,http://dlmf.nist.gov/23.1, -\BesselItilde{\nu}@{x},modified Bessel function of the first kind of imaginary order,modified-Bessel-first-kind-imaginary-order,F|F:P,$\widetilde{I}_{\nu}$,http://dlmf.nist.gov/23.1, -\BesselJ{\nu}@{z},Bessel function of the first kind,Bessel-J,F|F:P,$J_{\nu}$,http://dlmf.nist.gov/10.2#E2, -\BesselJtilde{\nu}@{x},Bessel function of the first kind of imaginary order,Bessel-J-imaginary-order,F|F:P,$\widetilde{J}_{\nu}$,http://dlmf.nist.gov/10.24#Ex1, -\BesselK{\nu}@{z},modified Bessel function of the second kind,modified-Bessel-second-kind,F|F:P,$K_{\nu}$,http://dlmf.nist.gov/23.1, -\BesselKtilde{\nu}@{x},modified Bessel function of the second kind of imaginary order,modified-Bessel-second-kind-imaginary-order,F|F:P,$\widetilde{K}_{\nu}$,http://dlmf.nist.gov/10.45#E2, -\BesselModulusM{\nu}@{x},modulus of Bessel functions,modulus-Bessel-M,F|F:P,$M_{\nu}$,http://dlmf.nist.gov/10.18#E1, -\BesselModulusN{\nu}@{x},modulus of derivatives of Bessel functions,modulus-Bessel-N,F|F:P,$N_{\nu}$,http://dlmf.nist.gov/10.18#E2, -\BesselPhasePhi{\nu}@{x},phase of derivatives of Bessel functions,phase-Bessel-Phi,F|F:P,$\phi_{\nu}$,http://dlmf.nist.gov/10.18#E3, -\BesselPhaseTheta{\nu}@{x},phase of Bessel functions,phase-Bessel-Theta,F|F:P,$\theta_{\nu}$,http://dlmf.nist.gov/10.18#E3, -\BesselPoly{n}@{x}{a},Bessel polynomial,Bessel-polynomial-y,P|F:P,$y_{n}$,http://dlmf.nist.gov/18.34#E1, -\BesselPolyy{n}@{x}{a},Bessel polynomial,Bessel-polynomial-y,P|F:P,$y_{n}$,http://dlmf.nist.gov/18.34#E1, -\BesselY{\nu}@{z},Bessel function of the second kind,Bessel-Y-Weber,F|F:P,$Y_{\nu}$,http://dlmf.nist.gov/10.2#E3, -\BesselYtilde{\nu}@{x},Bessel function of the second kind of imaginary order,Bessel-Y-Weber-imaginary-order,F|F:P,$\widetilde{Y}_{\nu}$,http://dlmf.nist.gov/10.24#Ex1, -\BickleyKi{\alpha}@{x},Bickley function,Bickley-Ki,F|F:P,$\mathrm{Ki}_{\alpha}$,http://dlmf.nist.gov/10.43#E11, -\BigO@{x},order not exceeding,Big-O,Q|F:P,$O$,http://dlmf.nist.gov/2.1#E3, -\CanonicInt{K}@{\Vector{x}},canonical integral $\Psi_K$,canonical-integral,F|F:P,$\Psi_{K}$,http://dlmf.nist.gov/36.2#E4, -\CanonicIntU@{\Vector{x}},umbilic canonical integral $\Psi^{({\rm U})}$,canonical-umbilic-integral,F|F:P,$\Psi^{(\mathrm{U})}$,http://dlmf.nist.gov/36.2#E5, -\CanonicIntE@{\Vector{x}},elliptic umbilic canonical integral $\Psi^{({\rm E})}$,elliptic-umbilic-canonical-integral,F|F:P,$\Psi^{(\mathrm{E})}$,http://dlmf.nist.gov/36.2#E6, -\CanonicIntH@{\Vector{x}},hyperbolic umbilic canonical integral $\Psi^{({\rm H})}$,hyperbolic-umbilic-canonical-integral,F|F:P,$\Psi^{(\mathrm{H})}$,http://dlmf.nist.gov/36.2#E8, -\CatalanNumber@{n},Catalan number,Catalan-number,I|F:P,$C$,http://dlmf.nist.gov/26.5#E1, -\CatalansConstant,Catalan's constant,Catalan's-constant,C|F,$G$,http://dlmf.nist.gov/25.11.E40, -\Charlier{n}@{x}{a},Charlier polynomial,Charlier-polynomial-C,P|F:P,$C_{n}$,http://dlmf.nist.gov/18.19#T1.t1.r11, -\CharlierC{n}@{x}{a},Charlier polynomial,Charlier-polynomial-C,P|F:P,$C_{n}$,http://dlmf.nist.gov/18.19#T1.t1.r11, -\ChebyC{n}@{x},dilated Chebyshev polynomial of the first kind,dilated-Chebyshev-polynomial-second-kind-C,P|F:P,$C_{n}$,http://dlmf.nist.gov/18.1#E3, -\ChebyS{n}@{x},dilated Chebyshev polynomial of the second kind,dilated-Chebyshev-polynomial-first-kind-S,P|F:P,$S_{n}$,http://dlmf.nist.gov/18.1#E3, -\ChebyT{n}@{x},Chebyshev polynomial of the first kind,Chebyshev-polynomial-first-kind-T,P|F:P,$T_{n}$,http://dlmf.nist.gov/18.3#T1.t1.r8, -\ChebyTs{n}@{x},shifted Chebyshev polynomial of the first kind,shifted-Chebyshev-polynomial-first-kind-T-star,P|F:P,$T^{*}_{n}$,http://dlmf.nist.gov/18.3#T1.t1.r20, -\ChebyU{n}@{x},Chebyshev polynomial of the second kind,Chebyshev-polynomial-second-kind-U,P|F:P,$U_{n}$,http://dlmf.nist.gov/18.3#T1.t1.r11, -\ChebyUs{n}@{x},shifted Chebyshev polynomial of the second kind,shifted-Chebyshev-polynomial-second-kind-U-star,P|F:P,$U^{*}_{n}$,http://dlmf.nist.gov/18.3#T1.t1.r23, -\ChebyV{n}@{x},Chebyshev polynomial of the third kind,Chebyshev-polynomial-third-kind-V,P|F:P,$V_{n}$,http://dlmf.nist.gov/18.3#T1.t1.r14, -\ChebyW{n}@{x},Chebyshev polynomial of the fourth kind,Chebyshev-polynomial-fourth-kind-W,P|F:P,$W_{n}$,http://dlmf.nist.gov/18.3#T1.t1.r17, -\ChebyshevPsi@{x},Chebyshev $\ChebyshevPsi$-function,Chebyshev-psi,F|F:P,$\psi$,http://dlmf.nist.gov/25.16#E1, -\CiglerqChebyT{n}@{x}{s}{q},Cigler $q$-Chebyshev polynomial $T$,Cigler-q-Chebyshev-polynomial-T,P|P:F,$T_{n}$,http://drmf.wmflabs.org/wiki/Definition:CiglerqChebyT, -\CiglerqChebyU{n}@{x}{s}{q},Cigler $q$-Chebyshev polynomial $U$,Cigler-q-Chebyshev-polynomial-U,P|P:F,$U_{n}$,http://drmf.wmflabs.org/wiki/Definition:CiglerqChebyU, -\CompEllIntCE@@{k},Legendre's complementary complete elliptic integral of the second kind,complementary-complete-elliptic-integral-second-kind-E,F|F:P:nP,${E'}$,http://dlmf.nist.gov/19.2#E9, -\CompEllIntCK@@{k},Legendre's complementary complete elliptic integral of the first kind,complementary-complete-elliptic-integral-first-kind-K,F|F:P:nP,${K'}$,http://dlmf.nist.gov/19.2#E9, -\CompEllIntD@@{k},complete elliptic integral of Legendre's type,complete-elliptic-integral-D,F|F:P:nP,$D$,http://dlmf.nist.gov/19.2#E8, -\CompEllIntE@@{k},Legendre's complete elliptic integral of the second kind,complete-elliptic-integral-second-kind-E,F|F:P:nP,$E$,http://dlmf.nist.gov/19.2#E8, -\CompEllIntK@@{k},Legendre's complete elliptic integral of the first kind,complete-elliptic-integral-first-kind-K,F|F:P:nP,$K$,http://dlmf.nist.gov/19.2#E8, -\CompEllIntPi@{\alpha^2}{k},Legendre's complete elliptic integral of the third kind,complete-elliptic-integral-third-kind-Pi,F|F:P,$\Pi$,http://dlmf.nist.gov/19.2#E8, -\Complex,set of complex numbers,EMPTY,SN|F,$\mathbb{C}$,http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r1, -\ComplexZeroAiryBi{k},$k$\textsuperscript{th} complex zero of Airy $\AiryBi$,complex-zero-Airy-Bi,F|F,$\beta_{k}$,http://dlmf.nist.gov/9.9#SS1.p2, -\ComplexZeroAiryBiPrime{k},$k$\textsuperscript{th} complex zero of Airy $\AiryBi'$,complex-zero-derivative-Airy-Bi,F|F,$\beta'_{k}$,http://dlmf.nist.gov/9.9#SS1.p2, -\CompositionsC[m]@{n},number of compositions,number-of-compositions-C,I|F:F:P:P,$c_{m}$,http://dlmf.nist.gov/26.11#p1, -\CosInt@{z},cosine integral ${\rm Ci}$,cosine-integral,F|F:P,$\mathrm{Ci}$,http://dlmf.nist.gov/6.2#E11, -\CosIntCin@{z},cosine integral ${\rm Cin}$,cosine-integral-Cin,F|F:P,$\mathrm{Cin}$,http://dlmf.nist.gov/6.2#E12, -\CosIntg@{a}{z},generalized cosine integral ${\rm Ci}$,generalized-cosine-integral-Ci,F|F:P,$\mathrm{Ci}$,http://dlmf.nist.gov/8.21#E2, -\CoshInt@{z},hyperbolic cosine integral,hyperbolic-cosine-integral,F|F:P,$\mathrm{Chi}$,http://dlmf.nist.gov/6.2#E16, -\CoulombC{\ell}@{\eta},normalizing constant for Coulomb radial functions,Coulomb-C,F|F:P,$C_{\ell}$,http://dlmf.nist.gov/33.2#E5, -\CoulombF{\ell}@{\eta}{\rho},regular Coulomb radial function,regular-Coulomb-F,F|F:P,$F_{\ell}$,http://dlmf.nist.gov/33.2#E3, -\CoulombG{\ell}@{\eta}{\rho},irregular Coulomb radial function,irregular-Coulomb-G,F|F:P,$G_{\ell}$,http://dlmf.nist.gov/33.2#E11, -\CoulombH{\pm}{\ell}@{\eta}{\rho},irregular Coulomb radial functions,irregular-Coulomb-H,F|F:P,${H^{\pm}_{\ell}}$,http://dlmf.nist.gov/33.2#E7, -\CoulombM{\ell}@{\eta}{\rho},envelope of Coulomb functions,envelope-Coulomb-M,F|F:P,$M_{\ell}$,http://dlmf.nist.gov/33.3#E1, -\CoulombSigma{\ell}@{\eta},Coulomb phase shift,Coulomb-phase-shift,F|F:P,${\sigma_{\ell}}$,http://dlmf.nist.gov/33.2#E10, -\CoulombTheta{\ell}@{\eta}{\rho},phase of Coulomb functions,Coulomb-phase,F|F:P,${\theta_{\ell}}$,http://dlmf.nist.gov/33.2#E9, -\Coulombc@{\epsilon}{\ell}{r},irregular Coulomb function $c$,Coulomb-c,F|F:P,$c$,http://dlmf.nist.gov/23.1, -\Coulombf@{\epsilon}{\ell}{r},regular Coulomb function $f$,Coulomb-f,F|F:P,$f$,http://dlmf.nist.gov/33.14#E4, -\Coulombh@{\epsilon}{\ell}{r},irregular Coulomb function $h$,Coulomb-h,F|F:P,$h$,http://dlmf.nist.gov/33.14#E7, -\Coulombrhotp@{\eta}{\ell},outer turning point for Coulomb radial functions,Coulomb-radial-outer-turning-point,F|F:P,$\rho_{\mathrm{tp}}$,http://dlmf.nist.gov/33.2#E2, -\Coulombrtp@{\epsilon}{\ell},outer turning point for Coulomb functions,Coulomb-outer-turning-point,F|F:P,$r_{\mathrm{tp}}$,http://dlmf.nist.gov/33.14#E3, -\Coulombs@{\epsilon}{\ell}{r},regular Coulomb function $s$,Coulomb-s,F|F:P,$s$,http://dlmf.nist.gov/33.14#E9, -\CuspCat{K}@{t}{\Vector{x}},cuspoid catastrophe,cuspoid-catastrophe,F|F:P,$\Phi_{K}$,http://dlmf.nist.gov/36.2#E1, -\Cylinder{\nu}@{z},cylinder function,cylinder-function,F|F:P,$\mathscr{C}_{\nu}$,http://dlmf.nist.gov/10.2#P5.p1, -\DawsonsInt@{z},Dawson's integral,Dawsons-integral,F|F:P,$F$,http://dlmf.nist.gov/7.2#E5, -\DedekindModularEta@{\tau},Dedekind's eta function (or Dedekind modular function),Dedekind-modular-Eta,F|F:P,$\eta$,http://dlmf.nist.gov/23.15#E9,http://dlmf.nist.gov/27.14#E12 -\DiffCat{K}@{\Vector{x}}{k},diffraction catastrophe,diffraction-catastrophe,F|F:P,$\Psi_{K}$,http://dlmf.nist.gov/36.2#E10, -\DiffCatE@{\Vector{x}}{k},elliptic umbilic canonical integral function,diffraction-elliptic-umbilic-catastrophe,F|F:P,$\Psi^{(\mathrm{E})}$,http://dlmf.nist.gov/23.1, -\DiffCatH@{\Vector{x}}{k},hyperbolic umbilic canonical integral function,diffraction-hyperbolic-umbilic-catastrophe,F|F:P,$\Psi^{(\mathrm{H})}$,http://dlmf.nist.gov/36.2#E11, -\DiffCatU@{\Vector{x}}{k},umbilic canonical integral function,diffraction-umbilic-catastrophe,F|F:P,$\Psi^{(\mathrm{U})}$,http://dlmf.nist.gov/36.2#E11, -\Dilogarithm@{z},dilogarithm,dilogarithm,F|F:P,$\mathrm{Li}_2$,http://dlmf.nist.gov/25.12#E1, -\Diracdelta@{x-a},Dirac delta (or Dirac delta function),Dirac-delta,D|F:P,$\Diracdelta@{x-a}$,http://dlmf.nist.gov/1.17#SS1.p1, -\DirichletCharacter@@{n}{k},Dirichlet character,Dirichlet-character,I|F:P:nP,$\chi$,http://dlmf.nist.gov/27.8, -\DirichletL@{s}{\chi},Dirichlet $\DirichletL$-function,Dirichlet-L,F|F:P,$L$,http://dlmf.nist.gov/25.15#E1, -\DiscriminantDelta@{\tau},discriminant function,discriminant-delta,F|F:P,$\Delta$,http://dlmf.nist.gov/27.14#E16, -\DiskOP{\alpha}{m}{n}@{z},disk polynomial,disk-orthogonal-polynomial-R,P|F:P,"$R^{(\alpha)}_{m,n}$",http://dlmf.nist.gov/18.37#E1, -\Distribution{f}{g},distribution,EMPTY,D|F,"$\left\langle f,g\right\rangle$",http://dlmf.nist.gov/1.16#SS1.p5, -\DivisorFunctionD[k]@{n},divisor function,divisor-function-D,F|F:P,$d_{k}$,http://dlmf.nist.gov/27.2#E9, -\DivisorSigma{\alpha}@{n},sum of powers of divisors,divisor-sigma,I|F:P,$\sigma_{\alpha}$,http://dlmf.nist.gov/27.2#E10, -\EllIntD@{\phi}{k},incomplete elliptic integral of Legendre's type,elliptic-integral-third-kind-D,F|F:P,$D$,http://dlmf.nist.gov/19.2#E6, -\EllIntE@{\phi}{k},Legendre's incomplete elliptic integral of the second kind,elliptic-integral-second-kind-E,F|F:P,$E$,http://dlmf.nist.gov/19.2#E5, -\EllIntF@{\phi}{k},Legendre's incomplete elliptic integral of the first kind,elliptic-integral-first-kind-F,F|F:P,$F$,http://dlmf.nist.gov/19.2#E4, -\EllIntPi@{\phi}{\alpha^2}{k},Legendre's incomplete elliptic integral of the third kind,elliptic-integral-third-kind-Pi,F|F:P,$\Pi$,http://dlmf.nist.gov/19.2#E7, -\EllIntR{-a}@{\Vector{b}}{\Vector{z}},Carlson integral $R_{-a}$,Carlson-integral-R,F|F:P,$R_{-a}$,http://dlmf.nist.gov/19.16#E9, -\EllIntRC@@{x}{y},Carlson's elliptic integral $R_C$,Carlson-integral-RC,F|F:P:nP,$R_C$,http://dlmf.nist.gov/19.2#E17, -\EllIntRD@@{x}{y}{z},Carlson's elliptic integral $R_D$,Carlson-integral-RD,F|F:P:nP,$R_D$,http://dlmf.nist.gov/19.16#E5, -\EllIntRF@@{x}{y}{z},Carlson's symmetric elliptic integral of first kind,Carlson-integral-RF,F|F:P:nP,$R_F$,http://dlmf.nist.gov/19.16#E1, -\EllIntRG@@{x}{y}{z},Carlson's symmetric elliptic integral of second kind,Carlson-integral-RG,F|F:P:nP,$R_G$,http://dlmf.nist.gov/19.16#E3, -\EllIntRJ@@{x}{y}{z}{p},Carlson's symmetric elliptic integral of third kind,Carlson-integral-RJ,F|F:P:nP,$R_J$,http://dlmf.nist.gov/19.16#E2, -\EllIntcel@{k_c}{p}{a}{b},Bulirsch's complete elliptic integral,Bulirsch-integral-cel,F|F:P,$\mathrm{cel}$,http://dlmf.nist.gov/19.2#E11, -\EllIntelone@{x}{k_c},Bulirsch's incomplete elliptic integral of the first kind,Bulirsch-integral-el-1,F|F:P,$\mathrm{el1}$,http://dlmf.nist.gov/19.2#E15, -\EllIntelthree@{x}{k_c}{p},Bulirsch's incomplete elliptic integral of the third kind,Bulirsch-integral-el-3,F|F:P,$\mathrm{el3}$,http://dlmf.nist.gov/19.2#E16, -\EllInteltwo@{x}{k_c}{a}{b},Bulirsch's incomplete elliptic integral of the second kind,Bulirsch-integral-el-2,F|F:P,$\mathrm{el2}$,http://dlmf.nist.gov/19.2#E12, -\EulerBeta@{a}{b},beta function,Euler-Beta,F|F:P,$\mathrm{B}$,http://dlmf.nist.gov/5.12#E1, -\EulerConstant,Euler's (Euler-Mascheroni) constant,EMPTY,C|F,$\gamma$,http://dlmf.nist.gov/5.2#E3, -\EulerE{n}@{x},Euler polynomial,Euler-polynomial-E,P|F:P,$E_{n}$,http://dlmf.nist.gov/24.2#ii, -\EulerGamma@{z},Euler's gamma function,Euler-Gamma,F|F:P,$\Gamma$,http://dlmf.nist.gov/5.2#E1, -\EulerPhi@{x},Euler's reciprocal function,Euler-phi,F|F:P,$\mathit{f}$,http://dlmf.nist.gov/27.14#E2, -\EulerSumH@{s},Euler sums,Euler-sum-H,F|F:P,$H$,http://dlmf.nist.gov/25.16#SS2.p1, -\EulerTotientPhi[k]@{n},Euler's totient $\phi_k(n)$,Euler-totient-phi,I|F:P,$\phi_{k}$,http://dlmf.nist.gov/27.2#E6, -\EulerianNumber{n}{k},Eulerian number,Eulerian-number,I|F,$\left\langle n\atop k\right\rangle$,http://dlmf.nist.gov/26.14#SS1.p3, -\ExpInt@{z},exponential integral $E_1$,exponential-integral,F|F:P,$E_1$,http://dlmf.nist.gov/6.2#E1, -\ExpIntEin@{z},complementary exponential integral ${\rm Ein}$,complementary-exponential-integral,F|F:P,$\mathrm{Ein}$,http://dlmf.nist.gov/6.2#E3, -\ExpInti@{x},exponential integral ${\rm Ei}$,exponential-integral-Ei,F|F:P,$\mathrm{Ei}$,http://dlmf.nist.gov/6.2#E1, -\ExpIntn{p}@{z},generalized exponential integral,exponential-integral-En,F|F:P,$E_{p}$,http://dlmf.nist.gov/8.19#E1, -\FerrersHatQ[-\mu]{-\frac{1}{2}+i\tau}@{x},conical function $\widehat{\mathsf{Q}}_\nu^\mu$,Ferrers-conical-legendre-Q-hat,F|FO:PO:FnO:PnO,$\widehat{\mathsf{Q}}_\nu^\mu$,http://dlmf.nist.gov/14.20#E2, -\FerrersP[\mu]{\nu}@{x},Ferrers function of the first kind,Ferrers-Legendre-P-first-kind,F|FO:PO:FnO:PnO,$\mathsf{P}_\nu^\mu$,http://dlmf.nist.gov/14.3#E1, -\FerrersQ[\mu]{\nu}@{x},Ferrers function of the second kind,Ferrers-Legendre-Q-first-kind,F|FO:PO:FnO:PnO,$\mathsf{Q}_\nu^\mu$,http://dlmf.nist.gov/14.3#E2, -\FishersHh{n}@{z},probability function,Fishers-Hh,F|F:P,$\mathit{Hh}_{n}$,http://dlmf.nist.gov/7.18#E12, -\FresnelCos@{z},Fresnel integral $C$,Fresnel-cosine-integral,F|F:P,$C$,http://dlmf.nist.gov/7.2#E7, -\FresnelF@{z},Fresnel integral ${\mathcal F}$,Fresnel-integral,F|F:P,$\mathcal{F}$,http://dlmf.nist.gov/7.2#E6, -\FresnelSin@{z},Fresnel integral $S$,Fresnel-sine-integral,F|F:P,$S$,http://dlmf.nist.gov/7.2#E8, -\Fresnelf@{z},auxiliary function for Fresnel integrals ${\rm f}$,Fresnel-auxilliary-function-f,F|F:P,$\mathrm{f}$,http://dlmf.nist.gov/7.2#E10, -\Fresnelg@{z},auxiliary function for Fresnel integrals ${\rm g}$,Fresnel-auxilliary-function-g,F|F:P,$\mathrm{g}$,http://dlmf.nist.gov/7.2#E11, -\GammaP@{a}{z},normalized incomplete gamma function $P$,incomplete-gamma-P,F|F:P,$P$,http://dlmf.nist.gov/8.2#E4, -\GammaQ@{a}{z},normalized incomplete gamma function $Q$,incomplete-gamma-Q,F|F:P,$Q$,http://dlmf.nist.gov/8.2#E4, -\GaussSum@{n}{\DirichletCharacter},Gauss sum,Gauss-sum,F|F:P,$G$,http://dlmf.nist.gov/27.10#E9, -\GenBernoulliB{\ell}{n}@{x},generalized Bernoulli polynomial,generalized-Bernoulli-polynomial-B,P|F:P,$B^{(\ell)}_{n}$,http://dlmf.nist.gov/24.16#P1.p1, -\GenBesselFun@{\rho}{\beta}{z},generalized Bessel function,generalized-Bessel-function,F|F:P,$\phi$,http://dlmf.nist.gov/10.46#E1, -\GenEulerE{\ell}{n}@{x},generalized Euler polynomial,generalized-Euler-polynomial-E,P|F:P,$E^{(\ell)}_{n}$,http://dlmf.nist.gov/24.16#P1.p1, -\GenEulerSumH@{s}{z},generalized Euler sums,generalized-Euler-sum-H,F|F:P,$H$,http://dlmf.nist.gov/25.16#SS2.p7, -\GenGegenbauer{\alpha}{\beta}{n}@{x},Generalized Gegenbauer polynomial,generalized-Gegenbauer-polynomial-S,P|F:P,"$S^{(\alpha,\beta)}_{n}$",http://drmf.wmflabs.org/wiki/Definition:GenGegenbauer, -\GenHermite[\alpha]{n}@{x},generalized Hermite polynomial,generalized-Hermite-polynomial-H,P|FO:PO:FnO:PnO,$H_n^{\mu}$,http://drmf.wmflabs.org/wiki/Definition:GenHermite, -\GoodStat@{z},Goodwin-Staton integral,Goodwin-Staton-integral,F|F:P,$G$,http://dlmf.nist.gov/7.2#E12, -\GottliebLaguerre{n}@{x},Gottlieb-Laguerre polynomial,Gottlieb-Laguerre-polynomial-l,P|F:P,$l_{n}$,http://drmf.wmflabs.org/wiki/Definition:GottliebLaguerre, -\Gudermannian@@{x},Gudermannian function,inverse-Gudermannian,F|F:P:nP,$\mathrm{gd}$,http://dlmf.nist.gov/4.23#E39, -\Hahn{n}@{x}{\alpha}{\beta}{N},Hahn polynomial,Hahn-polynomial-Q,P|F:P,$Q_{n}$,http://dlmf.nist.gov/18.19#T1.t1.r3, -\HahnQ{n}@{x}{\alpha}{\beta}{N},Hahn polynomial,Hahn-polynomial-Q,P|F:P,$Q_{n}$,http://dlmf.nist.gov/18.19#T1.t1.r3, -\HahnR{n}@{x}{\gamma}{\delta}{N},dual Hahn polynomial,dual-Hahn-R,P|F:P,$R_{n}$,http://dlmf.nist.gov/18.25#T1.t1.r5, -\HahnS{n}@{x}{a}{b}{c},continuous dual Hahn polynomial,continuous-dual-Hahn-S,P|F:P,$S_{n}$,http://dlmf.nist.gov/18.25#T1.t1.r3, -\Hahnp{n}@{x}{a}{b}{\conj{a}}{\conj{b}},continuous Hahn polynomial,continuous-Hahn-polynomial-p,P|F:P,$p_{n}$,http://dlmf.nist.gov/18.19#P2.p1, -\HankelHi{\nu}@{z},Hankel function of the first kind,Hankel-H-1-Bessel-third-kind,F|F:P,${H^{(1)}_{\nu}}$,http://dlmf.nist.gov/10.2#E5, -\HankelHii{\nu}@{z},Hankel function of the second kind,Hankel-H-2-Bessel-third-kind,F|F:P,${H^{(2)}_{\nu}}$,http://dlmf.nist.gov/10.2#E6, -\HarmonicNumber{n},Harmonic number,Harmonic-number,I|F,$H_{n}$,http://dlmf.nist.gov/25.11#E33, -\HeavisideH@{x},Heaviside step function,Heaviside-H,F|F:P,$H$,http://dlmf.nist.gov/1.16#E13, -\Hermite{n}@{x},Hermite polynomial $H_n$,Hermite-polynomial-H,P|F:P,$H_{n}$,http://dlmf.nist.gov/18.3#T1.t1.r28, -\HermiteH{n}@{x},Hermite polynomial $H_n$,Hermite-polynomial-H,P|F:P,$H_{n}$,http://dlmf.nist.gov/18.3#T1.t1.r28, -\HermiteHe{n}@{x},Hermite polynomial $He_n$,Hermite-polynomial-He,P|F:P,$\mathit{He}_{n}$,http://dlmf.nist.gov/18.3#T1.t1.r29, -\HeunFunction{m}{a}{b}@@{a}{q}{\alpha}{\beta}{\gamma}{\delta}{z},Heun functions $(a;b)Hf_m^\nu$,Heun-function,F|FO:PO:nPO:FnO:PnO:nPnO,"$(s_1,s_2)\mathit{Hf}_{m}^\nu$",http://dlmf.nist.gov/31.4#p1, -\HeunLocal@@{a}{q}{\alpha}{\beta}{\gamma}{\delta}{z},Heun functions $Hl$,Heun-local,F|F:P:nP,$\mathit{H\!\ell}$,http://dlmf.nist.gov/31.3#E1, -\HeunPolynom{n}{m}@@{a}{q}{-n}{\beta}{\gamma}{\delta}{z},Heun polynomial,Heun-polynomial-m,P|F:P:nP,"$\mathit{Hp}_{n,m}$",http://dlmf.nist.gov/31.5#E2, -\HilbertTrans@{f}{x},Hilbert transform,Hilbert-transform,O|F:P,$\mathcal{H}$,http://dlmf.nist.gov/1.14#SS5.p1, -\HurwitzZeta@{s}{a},Hurwitz zeta function,Hurwitz-zeta,F|F:P,$\zeta$,http://dlmf.nist.gov/25.11#E1, -\HyperPsi@{a}{b}{\Matrix{T}},confluent hypergeometric function of matrix argument of the second kind,confluent-hypergeometric-of-matrix-Psi,F|F:P,$\Psi$,http://dlmf.nist.gov/35.6#E2, -\HyperboldpFq{p}{q}@@@{\Vector{a}}{\Vector{b}}{z},scaled (or Olver's) generalized hypergeometric function,hypergeometric-bold-pFq,F|F:fo1:fo2:fo3,${{}_{p}{\mathbf{F}}_{q}}$,http://dlmf.nist.gov/16.2#E5, -\HypergeoF@@@{a}{b}{c}{z},hypergeometric function,Gauss-hypergeometric-F,F|F:fo1:fo2:fo3,$F$,http://dlmf.nist.gov/15.2#E1, -\HypergeoboldF@@@{a}{b}{c}{z},Olver's hypergeometric function,scaled-hypergeometric-bold-F,F|F:fo1:fo2:fo3,$\mathbf{F}$,http://dlmf.nist.gov/15.2#E2, -"\HyperpFq{p}{q}@@@{a_1,...,a_p}{b_1,...,b_q}{z}",generalized hypergeometric function,Gauss-hypergeometric-pFq,F|F:fo1:fo2:fo3,${{}_{p}F_{q}}$,http://dlmf.nist.gov/16.2#E1, -"\HyperpHq{p}{q}@@@{{a_1,...,a_p}{b_1,...,b_q}{z}",bilateral hypergeometric function,hypergeometric-pHq,F|F:fo1:fo2:fo3,${{}_{p}H_{q}}$,http://dlmf.nist.gov/16.4#E16, -\IncBeta{x}@{a}{b},incomplete beta function,incomplete-Beta,F|F:P,$\mathrm{B}_{x}$,http://dlmf.nist.gov/8.17#E1, -\IncGamma@{a}{z},incomplete gamma function $\Gamma$,incomplete-Gamma,F|F:P,$\Gamma$,http://dlmf.nist.gov/8.2#E2, -\IncI{x}@{a}{b},regularized incomplete beta function,IncI,F|F:P,$I_{x}$,http://dlmf.nist.gov/8.17#E2, -\Int{a}{b}@{x}{f(x)},semantic definite integral,semantic-definite-integral,SM|O,"$\int_{a}^{b}f(x){\mathrm d}\!x$",http://drmf.wmflabs.org/wiki/Definition:Int, -\Integer,set of integers,EMPTY,SN|F,$\mathbb{Z}$,http://dlmf.nist.gov/front/introduction#Sx4.p2.t1.r20, -\IntgenAiryA{k}@{z}{p},generalized Airy function from integral representation $A_k$,Integral-generalized-Airy-A,F|F:P,$A_{k}$,http://dlmf.nist.gov/9.13#SS2.p1, -\IntgenAiryB{k}@{z}{p},generalized Airy function from integral representation $B_k$,Integral-generalized-Airy-B,F|F:P,$B_{k}$,http://dlmf.nist.gov/9.13#SS2.p1, -\JacksonqBesselI{\nu}@{z}{q},Jackson $q$-Bessel function 1,EMPTY,F|F:P,$J^{(1)}_{q}$,http://drmf.wmflabs.org/wiki/Definition:JacksonqBesselI, -\JacksonqBesselII{\nu}@{z}{q},Jackson $q$-Bessel function 2,EMPTY,F|F:P,$J^{(2)}_{q}$,http://drmf.wmflabs.org/wiki/Definition:JacksonqBesselII, -\JacksonqBesselIII{\nu}@{z}{q},Jackson/Hahn-Exton $q$-Bessel function 3,EMPTY,F|F:P,$J^{(3)}_{q}$,http://drmf.wmflabs.org/wiki/Definition:JacksonqBesselIII, -\Jacobi{\alpha}{\beta}{n}@{x},Jacobi polynomial,Jacobi-polynomial-P,P|F:P,"$P^{(\alpha,\beta)}_{n}$",http://dlmf.nist.gov/18.3#T1.t1.r3, -\JacobiEpsilon@{x}{k},Jacobi's epsilon function,Jacobi-Epsilon,F|F:P,$\mathcal{E}$,http://dlmf.nist.gov/22.16#E14, -\JacobiG{n}@{p}{q}{x},shifted Jacobi polynomial,shifted-Jacobi-polynomial-G,P|F:P,$G_{n}$,http://dlmf.nist.gov/18.1#E2, -\JacobiP{\alpha}{\beta}{n}@{x},Jacobi polynomial,Jacobi-polynomial-P,P|F:P,"$P^{(\alpha,\beta)}_{n}$",http://dlmf.nist.gov/18.3#T1.t1.r3, -\JacobiSymbol{n}{P},Jacobi symbol,Jacobi-symbol,I|F,$(n|P)$,http://dlmf.nist.gov/27.9#p3, -\JacobiTheta{j}@{z}{q},Jacobi theta function,Jacobi-theta,F|F:P,$\theta_{p}$,http://dlmf.nist.gov/20.2#i, -\JacobiThetaCombined{n}{m}@{z}{q},combined theta function,Jacobi-theta-combined,F|F:P,"$\varphi_{m,n}$",http://dlmf.nist.gov/20.11#SS5.p2, -\JacobiThetaTau{j}@{z}{\tau},Jacobi theta $\tau$ function $\theta_j$,Jacobi-theta-tau,F|F:P,$\theta_{p}$,http://dlmf.nist.gov/20.2#i, -\JacobiZeta@{x}{k},Jacobi's zeta function,Jacobi-Zeta,F|F:P,$\mathrm{Z}$,http://dlmf.nist.gov/22.16#E32, -\Jacobiam@@{x}{k},Jacobi's amplitude function ${\rm am}$,Jacobi-elliptic-amplitude,F|F:P:nP,$\mathrm{am}$,http://dlmf.nist.gov/23.1, -\Jacobicd@@{z}{k},Jacobian elliptic function ${\rm cd}$,Jacobi-elliptic-cd,F|F:P:nP,$\mathrm{cd}$,http://dlmf.nist.gov/22.2#E8, -\Jacobicn@@{z}{k},Jacobian elliptic function ${\rm cn}$,Jacobi-elliptic-cn,F|F:P:nP,$\mathrm{cn}$,http://dlmf.nist.gov/22.2#E5, -\Jacobics@@{z}{k},Jacobian elliptic function ${\rm cs}$,Jacobi-elliptic-cs,F|F:P:nP,$\mathrm{cs}$,http://dlmf.nist.gov/23.1, -\Jacobidc@@{z}{k},Jacobian elliptic function ${\rm dc}$,Jacobi-elliptic-dc,F|F:P:nP,$\mathrm{dc}$,http://dlmf.nist.gov/23.1, -\Jacobidn@@{z}{k},Jacobian elliptic function ${\rm dn}$,Jacobi-elliptic-dn,F|F:P:nP,$\mathrm{dn}$,http://dlmf.nist.gov/22.2#E6, -\Jacobids@@{z}{k},Jacobian elliptic function ${\rm ds}$,Jacobi-elliptic-ds,F|F:P:nP,$\mathrm{ds}$,http://dlmf.nist.gov/23.1, -\Jacobinc@@{z}{k},Jacobian elliptic function ${\rm nc}$,Jacobi-elliptic-nc,F|F:P:nP,$\mathrm{nc}$,http://dlmf.nist.gov/23.1, -\Jacobind@@{z}{k},Jacobian elliptic function ${\rm nd}$,Jacobi-elliptic-nd,F|F:P:nP,$\mathrm{nd}$,http://dlmf.nist.gov/23.1, -\Jacobins@@{z}{k},Jacobian elliptic function ${\rm ns}$,Jacobi-elliptic-ns,F|F:P:nP,$\mathrm{ns}$,http://dlmf.nist.gov/23.1, -\Jacobiphi{\alpha}{\beta}{\lambda}@{t},Jacobi function,Jacobi-hypergeometric-phi,F|F:P,"$\phi^{(\alpha,\beta)}_{\lambda}$",http://dlmf.nist.gov/15.9#E11, -\Jacobisc@@{z}{k},Jacobian elliptic function ${\rm sc}$,Jacobi-elliptic-sc,F|F:P:nP,$\mathrm{sc}$,http://dlmf.nist.gov/22.2#E9, -\Jacobisd@@{z}{k},Jacobian elliptic function ${\rm sd}$,Jacobi-elliptic-sd,F|F:P:nP,$\mathrm{sd}$,http://dlmf.nist.gov/22.2#E7, -\Jacobisn@@{z}{k},Jacobian elliptic function ${\rm sn}$,Jacobi-elliptic-sn,F|F:P:nP,$\mathrm{sn}$,http://dlmf.nist.gov/22.2#E4, -\JonquierePhi@{z}{s}=\Polylogarithm{s}@{z},Jonquiére's function,Jonquiere-phi,F|F:P,$\phi$,http://dlmf.nist.gov/23.1, -\JordanJ{k}@{n},Jordan's function,Jordan-J,F|F:P,$J_{k}$,http://dlmf.nist.gov/27.2#E11, -\Kelvinbei{\nu}@@{x},Kelvin function ${\rm bei}_\nu$,Kelvin-bei,F|F:P:nP,$\mathrm{bei}_{\nu}$,http://dlmf.nist.gov/10.61#E1, -\Kelvinber{\nu}@@{x},Kelvin function ${\rm ber}_\nu$,Kelvin-ber,F|F:P:nP,$\mathrm{ber}_{\nu}$,http://dlmf.nist.gov/10.61#E1, -\Kelvinkei{\nu}@@{x},Kelvin function ${\rm kei}_\nu$,Kelvin-kei,F|F:P:nP,$\mathrm{kei}_{\nu}$,http://dlmf.nist.gov/10.61#E2, -\Kelvinker{\nu}@@{x},Kelvin function ${\rm ker}_\nu$,Kelvin-ker,F|F:P:nP,$\mathrm{ker}_{\nu}$,http://dlmf.nist.gov/10.61#E2, -\Krawtchouk{n}@{x}{p}{N},Krawtchouk polynomial,Krawtchouk-polynomial-K,P|F:P,$K_{n}$,http://dlmf.nist.gov/18.19#T1.t1.r6, -\KrawtchoukK{n}@{x}{p}{N},Krawtchouk polynomial,Krawtchouk-polynomial-K,P|F:P,$K_{n}$,http://dlmf.nist.gov/18.19#T1.t1.r6, -\Kronecker{j}{k},Kronecker delta,EMPTY,I|F,"$\delta_{m,n}$",http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r4, -\KummerM@{a}{b}{z},Kummer confluent hypergeometric function $M$,Kummer-confluent-hypergeometric-M,F|F:P,$M$,http://dlmf.nist.gov/13.2#E2, -\KummerU@{a}{b}{z},Kummer confluent hypergeometric function $U$,Kummer-confluent-hypergeometric-U,F|F:P,$U$,http://dlmf.nist.gov/13.2#E6, -\KummerboldM@{a}{b}{z},Olver's confluent hypergeometric function,Kummer-confluent-hypergeometric-bold-M,F|F:P,${\mathbf{M}}$,http://dlmf.nist.gov/13.2#E3, -\Laguerre[\alpha]{n}@{x},Laguerre (or generalized Laguerre) polynomial,generalized-Laguerre-polynomial-L,P|FnO:PnO:FO:PO,$L_n^{(\alpha)}$,http://dlmf.nist.gov/18.3#T1.t1.r27, -\LaguerreL[\alpha]{n}@{x},Laguerre (or generalized Laguerre) polynomial,generalized-Laguerre-polynomial-L,P|FnO:PnO:FO:PO,$L_n^{(\alpha)}$,http://dlmf.nist.gov/18.3#T1.t1.r27, -\LambertW@{x},Lambert $\LambertW$-function,Lambert-W,F|F:P,$W$,http://dlmf.nist.gov/23.1, -\LambertWm@{x},non-principal branch of Lambert $\LambertW$-function,Lambert-Wm,F|F:P,$\mathrm{Wm}$,http://dlmf.nist.gov/4.13#p2, -\LambertWp@{x},principal branch of Lambert $\LambertW$-function,Lambert-Wp,F|F:P,$\mathrm{Wp}$,http://dlmf.nist.gov/23.1, -\LameEc{m}{\nu}@{z}{k^2},Lamé function $Ec_\nu^m$,Lame-Ec,F|F:P,$\mathit{Ec}^{m}_{\nu}$,http://dlmf.nist.gov/23.1, -\LameEs{m}{\nu}@{z}{k^2},Lamé function $Es_\nu^m$,Lame-Es,F|F:P,$\mathit{Es}^{m}_{\nu}$,http://dlmf.nist.gov/23.1, -\Lamea{n}{\nu}@{k^2},eigenvalues of Lamé's equation $a_\nu^m$,Lame-eigenvalue-a,F|F:P,$a^{n}_{\nu}$,http://dlmf.nist.gov/29.3#SS1.p1, -\Lameb{n}{\nu}@{k^2},eigenvalues of Lamé's equation $b_\nu^m$,Lame-eigenvalue-b,F|F:P,$b^{n}_{\nu}$,http://dlmf.nist.gov/29.3#SS1.p1, -\LamecE{m}{2n+1}@{z}{k^2},Lamé polynomial $cE_{2n+1}^m$,Lame-polynomial-cE,P|F:P,$\mathit{cE}^{m}_{\nu}$,http://dlmf.nist.gov/23.1, -\LamecdE{m}{2n+2}@{z}{k^2},Lamé polynomial $cdE_{2n+2}^m$,Lame-polynomial-cdE,P|F:P,$\mathit{cdE}^{m}_{\nu}$,http://dlmf.nist.gov/23.1, -\LamedE{m}{2n+1}@{z}{k^2},Lamé polynomial $dE_{2n+1}^m$,Lame-polynomial-dE,P|F:P,$\mathit{dE}^{m}_{\nu}$,http://dlmf.nist.gov/29.12#E4, -\LamesE{m}{2n+1}@{z}{k^2},Lamé polynomial $sE_{2n+1}^m$,Lame-polynomial-sE,P|F:P,$\mathit{sE}^{m}_{\nu}$,http://dlmf.nist.gov/29.12#E2, -\LamescE{m}{2n+2}@{z}{k^2},Lamé polynomial $scE_{2n+2}^m$,Lame-polynomial-scE,P|F:P,$\mathit{scE}^{m}_{\nu}$,http://dlmf.nist.gov/29.12#E5, -\LamescdE{m}{2n+3}@{z}{k^2},Lamé polynomial $scdE_{2n+3}^m$,Lame-polynomial-scdE,P|F:P,$\mathit{scdE}^{m}_{\nu}$,http://dlmf.nist.gov/29.12#E8, -\LamesdE{m}{2n+2}@{z}{k^2},Lamé polynomial $sdE_{2n+2}^m$,Lame-polynomial-sdE,P|F:P,$\mathit{sdE}^{m}_{\nu}$,http://dlmf.nist.gov/29.12#E6, -\LameuE{m}{2n}@{z}{k^2},Lamé polynomial $uE_{2n}^m$,Lame-polynomial-uE,P|F:P,$\mathit{uE}^{m}_{\nu}$,http://dlmf.nist.gov/29.12#E1, -\LaplaceTrans@{f}{s},Laplace transform,Laplace-transform,O|F:P,$\mathscr{L}$,http://dlmf.nist.gov/1.14#E17, -\Lattice{L},lattice in $\Complex$,EMPTY,SN|F,$\mathbb{L}$,http://dlmf.nist.gov/23.2#i, -"\LauricellaFD@{a}{b_1,...,b_n}{c}{z_1,...,z_n}",Lauricella's multivariate hypergeometric function,Lauricella-FD,F|F:P,$F_D$,http://dlmf.nist.gov/19.15#p1, -\LegendreBlackQ[\mu]{\nu}@{z},Olver's associated Legendre function,associated-Legendre-black-Q,F|FO:PO:FnO:PnO,$\boldsymbol{Q}_\nu^\mu$,http://dlmf.nist.gov/14.3#E10, -\LegendreP[\mu]{\nu}@{z},associated Legendre function of the first kind,Legendre-P-first-kind,F|FO:PO:FnO:PnO,$P_\nu^\mu$,http://dlmf.nist.gov/14.3#E6, -\LegendrePoly{n}@{x},Legendre polynomial,Legendre-spherical-polynomial,P|F:P,$P_{n}$,http://dlmf.nist.gov/18.3#T1.t1.r25, -\LegendrePolys{n}@{x},shifted Legendre polynomial,shifted-spherical-Legendre-polynomial-s,P|F:P,$P^{*}_{n}$,http://dlmf.nist.gov/18.3#T1.t1.r26, -\LegendreQ[\mu]{\nu}@{z},associated Legendre function of the second kind,Legendre-Q-second-kind,F|FO:PO:FnO:PnO,$Q_\nu^\mu$,http://dlmf.nist.gov/14.3#E7, -\LegendreSymbol{n}{p},Legendre symbol,Legendre-symbol,I|F,$(n|p)$,http://dlmf.nist.gov/27.9, -\LerchPhi@{z}{s}{a},Lerch's transcendent,Lerch-Phi,F|F:P,$\Phi$,http://dlmf.nist.gov/25.14#E1, -\LeviCivita{j}{k}{\ell},Levi-Civita symbol,EMPTY,I|F,$\epsilon_{jkl}$,http://dlmf.nist.gov/1.6#E14, -\LinBrF@{a}{u},line-broadening function,line-broadening-function,F|F:P,$H$,http://dlmf.nist.gov/7.19#E4, -\LiouvilleLambda@{n},Liouville's function,Liouville-lambda,F|F:P,$\lambda$,http://dlmf.nist.gov/27.2#E13, -\Ln@@{z},general logarithm function,multivalued-natural-logarithm,F|F:P:nP,$\mathrm{Ln}$,http://dlmf.nist.gov/4.2#E1, -\LogInt@{x},logarithmic integral,logarithmic-integral,F|F:P,$\mathrm{li}$,http://dlmf.nist.gov/6.2#E8, -\LommelS{\mu}{\nu}@{z},"Lommel function $S$",Lommel-S,F|F:P,"$S_{{\mu},{\nu}}$",http://dlmf.nist.gov/11.9#E5, -\Lommels{\mu}{\nu}@{z},"Lommel function $s$",Lommel-s,F|F:P,"$s_{{\mu},{\nu}}$",http://dlmf.nist.gov/11.9#E3, -\Longleftrightarrow,is equivalent to,equivalent-to,SM|F,$\Longleftrightarrow $,http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r13, -\Longrightarrow,implies,EMPTY,SM|F,$\Longrightarrow$,http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r12, -\MangoldtLambda@{n},Mangoldt's function,Mangoldt-Lambda,F|F:P,$\Lambda$,http://dlmf.nist.gov/27.2#E14, -\MathieuCe{\nu}@@{z}{q},modified Mathieu function ${\rm Ce}_\nu$,modified-Mathieu-Ce,F|F:P:nP,$\mathrm{Ce}_{n}$,http://dlmf.nist.gov/28.20#E3, -\MathieuD{j}@{\nu}{\mu}{z},cross-products of modified Mathieu functions and their derivatives ${\rm D}_{j}$,Mathieu-D,F|F:P,$\mathrm{D}_{j}$,http://dlmf.nist.gov/28.28#E24, -\MathieuDc{j}@{n}{m}{z},cross-products of radial Mathieu functions and their derivatives ${\rm Dc}_j$,Mathieu-Dc,F|F:P,$\mathrm{Dc}_{j}$,http://dlmf.nist.gov/28.28#E39, -\MathieuDs{j}@{n}{m}{z},cross-products of radial Mathieu functions and their derivatives ${\rm Ds}_j$,Mathieu-Ds,F|F:P,$\mathrm{Ds}_{j}$,http://dlmf.nist.gov/28.28#E35, -\MathieuDsc{j}@{n}{m}{z},cross-products of radial Mathieu functions and their derivatives ${\rm Dsc}_j$,Mathieu-Dsc,F|F:P,$\mathrm{Dsc}_{j}$,http://dlmf.nist.gov/28.28#E40, -\MathieuEigenvaluea{n}@@{q},eigenvalues of Mathieu equation $a_n(q)$,Mathieu-eigenvalue-a,F|F:P:nP,$a_{n}$,http://dlmf.nist.gov/28.2#SS5.p1, -\MathieuEigenvalueb{n}@@{q},eigenvalues of Mathieu equation $b_n(q)$,Mathieu-eigenvalue-b,F|F:P:nP,$b_{n}$,http://dlmf.nist.gov/28.2#SS5.p1, -\MathieuEigenvaluelambda{\nu+2n}@@{q},eigenvalues of Mathieu equation $\lambda_\nu(q)$,Mathieu-eigenvalue-lambda,F|F:P:nP,$\lambda_{\nu}$,http://dlmf.nist.gov/28.12#SS1.p2, -\MathieuFc{m}@{z}{h},Mathieu function ${\rm Fc}_m$,Mathieu-Fc,F|F:P,$\mathrm{Fc}_{m}$,http://dlmf.nist.gov/23.1, -\MathieuFe{n}@@{z}{q},modified Mathieu function ${\rm Fe}_n$,modified-Mathieu-Fe,F|F:P:nP,$\mathrm{Fe}_{n}$,http://dlmf.nist.gov/28.20#E6, -\MathieuFs{m}@{z}{h},Mathieu function ${\rm Fs}_m$,Mathieu-Fs,F|F:P,$\mathrm{Fs}_{m}$,http://dlmf.nist.gov/23.1, -\MathieuGc{m}@{z}{h},Mathieu function ${\rm Gc}_m$,Mathieu-Gc,F|F:P,$\mathrm{Gc}_{m}$,http://dlmf.nist.gov/23.1, -\MathieuGe{n}@@{z}{q},modified Mathieu function ${\rm Ge}_n$,modified-Mathieu-Ge,F|F:P:nP,$\mathrm{Ge}_{n}$,http://dlmf.nist.gov/28.20#E7, -\MathieuGs{m}@{z}{h},Mathieu function ${\rm Gs}_m$,Mathieu-Gs,F|F:P,$\mathrm{Gs}_{m}$,http://dlmf.nist.gov/23.1, -\MathieuIe{n}@@{z}{h},modified Mathieu function ${\rm Ie}_n$,modified-Mathieu-Ie,F|F:P:nP,$\mathrm{Ie}_{n}$,http://dlmf.nist.gov/28.20#E17, -\MathieuIo{n}@@{z}{h},modified Mathieu function ${\rm Io}_n$,modified-Mathieu-Io,F|F:P:nP,$\mathrm{Io}_{n}$,http://dlmf.nist.gov/28.20#E18, -\MathieuKe{n}@@{z}{h},modified Mathieu function ${\rm Ke}_n$,modified-Mathieu-Ke,F|F:P:nP,$\mathrm{Ke}_{n}$,http://dlmf.nist.gov/28.20#E19, -\MathieuKo{n}@@{z}{h},modified Mathieu function ${\rm Ko}_n$,modified-Mathieu-Ko,F|F:P:nP,$\mathrm{Ko}_{n}$,http://dlmf.nist.gov/28.20#E20, -\MathieuM{j}{\nu}@@{z}{h},modified Mathieu function ${\rm M}_\nu^{(j)}$,modified-Mathieu-M,F|F:P:nP,${\mathrm{M}^{(j)}_{\nu}}$,http://dlmf.nist.gov/23.1, -\MathieuMc{j}{n}@@{z}{h},radial Mathieu function ${\rm Mc}_n^{(j)}$,modified-Mathieu-Mc,F|F:P:nP,${\mathrm{Mc}^{(j)}_{\nu}}$,http://dlmf.nist.gov/28.20#E15, -\MathieuMe{\nu}@@{z}{q},modified Mathieu function ${\rm Me}_\nu$,modified-Mathieu-Me,F|F:P:nP,$\mathrm{Me}_{n}$,http://dlmf.nist.gov/28.20#E5, -\MathieuMs{j}{n}@@{z}{h},radial Mathieu function ${\rm Ms}_n^{(j)}$,modified-Mathieu-Ms,F|F:P:nP,${\mathrm{Ms}^{(j)}_{\nu}}$,http://dlmf.nist.gov/28.20#E16, -\MathieuSe{\nu}@@{z}{q},modified Mathieu function ${\rm Se}_\nu$,modified-Mathieu-Se,F|F:P:nP,$\mathrm{Se}_{n}$,http://dlmf.nist.gov/28.20#E4, -\Mathieuce{n}@@{z}{q},Mathieu function ${\rm ce}_\nu$,Mathieu-ce,F|F:P:nP,$\mathrm{ce}_{n}$,http://dlmf.nist.gov/23.1, -\Mathieufe{n}@@{z}{q},second solution of Mathieu's equation ${\rm fe}_n$,Mathieu-fe,F|F:P:nP,$\mathrm{fe}_{n}$,http://dlmf.nist.gov/28.5#E1, -\Mathieuge{n}@@{z}{q},second solution of Mathieu's equation ${\rm ge}_n$,Mathieu-ge,F|F:P:nP,$\mathrm{ge}_{n}$,http://dlmf.nist.gov/28.5#E2, -\Mathieume{n}@@{z}{q},Mathieu function ${\rm me}_\nu$,Mathieu-me,F|F:P:nP,$\mathrm{me}_{\nu}$,http://dlmf.nist.gov/23.1, -\Mathieuse{n}@@{z}{q},Mathieu function ${\rm se}_\nu$,Mathieu-se,F|F:P:nP,$\mathrm{se}_{\nu}$,http://dlmf.nist.gov/23.1, -\Matrix{I},Bessel function of matrix argument (first kind),EMPTY,F|F,$\Matrix{T}$,http://dlmf.nist.gov/front/introduction#Sx4.p2.t1.r9, -\MeijerG{m}{n}{p}{q}@@@{z}{\Vector{a}}{\Vector{b}},Meijer $G$-function,Meijer-G,F|F:fo1:fo2:fo3,"${G^{m,n}_{p,q}}$",http://dlmf.nist.gov/16.17#E1, -\Meixner{n}@{x}{\beta}{c},Meixner polynomial,Meixner-polynomial-M,P|F:P,$M_{n}$,http://dlmf.nist.gov/18.19#T1.t1.r9, -\MeixnerM{n}@{x}{\beta}{c},Meixner polynomial,Meixner-polynomial-M,P|F:P,$M_{n}$,http://dlmf.nist.gov/18.19#T1.t1.r9, -\MeixnerPollaczek{\lambda}{n}@{x}{\phi},Meixner-Pollaczek polynomial,Meixner-Pollaczek-polynomial-P,P|F:P,$P^{(\alpha)}_{n}$,http://dlmf.nist.gov/18.19#P3.p1, -\MeixnerPollaczekP{\lambda}{n}@{x}{\phi},Meixner-Pollaczek polynomial,Meixner-Pollaczek-polynomial-P,P|F:P,$P^{(\alpha)}_{n}$,http://dlmf.nist.gov/18.19#P3.p1, -\MellinTrans@{f}{s},Mellin transform,Mellin-transform,O|F:P,$\mathscr{M}$,http://dlmf.nist.gov/1.14#E32, -\Mills@{x},Mills ratio ${\sf M}$,Mills-ratio,F|F:P,$\mathsf{M}$,http://dlmf.nist.gov/7.8#E1, -\MittagLeffler{a}{b}@{z},Mittag-Leffler function,Mittag-Leffler-function,F|F:P,"$E_{a,b}$",http://dlmf.nist.gov/10.46#E3, -\ModCylinder{\nu}@{z},modified cylinder function,modified-cylinder-function,F|F:P,$\mathscr{Z}_{\nu}$,http://dlmf.nist.gov/10.25#P2.p1, -\ModularJ@{\tau},Klein's complete invariant,Kleins-invariant-modular-J,F|F:P,$J$,http://dlmf.nist.gov/23.15#E7, -\ModularLambda@{\tau},elliptic modular function,modular-Lambda,F|F:P,$\lambda$,http://dlmf.nist.gov/23.15#E6, -\MoebiusMu@{n},Möbius function,Moebius-mu,F|F:P,$\mu$,http://dlmf.nist.gov/27.2#E12, -\NatNumber,set of positive integers,EMPTY,SN|F,$\mathbb{N}$,http://dlmf.nist.gov/front/introduction#Sx4.p2.t1.r11, -\NeumannFactor{n},Neumann factor,Neumann-factor,I|F,$\epsilon_{m}$,http://drmf.wmflabs.org/wiki/Definition:NeumannFactor, -\NeumannPoly{n}@{x},Neumann's polynomial,Neumann-polynomial,P|F:P,$O_{k}$,http://dlmf.nist.gov/10.23#E12, -\NumPrimeDivNu@{n},$\nu(n)$ : number of distinct primes dividing $n$,number-of-primes-dividing-nu,I|F:P,$\nu$,http://dlmf.nist.gov/27.2#SS1.p1, -\NumPrimesLessPi@{x},number of primes not exceeding $x$,number-of-primes-less-pi,I|F:P,$\pi$,http://dlmf.nist.gov/27.2#E2, -\NumSquaresR{k}@{n},number of squares,number-of-squares-R,I|F:P,$r_{k}$,http://dlmf.nist.gov/27.13#SS4.p1, -\ODEgenAiryA{n}@{z},generalized Airy function from differential equation $A_n$,ODE-generalized-Airy-A,F|F:P,$A_{n}$,http://dlmf.nist.gov/9.13#SS1.p2, -\ODEgenAiryB{n}@{z},generalized Airy function from differential equation $B_n$,ODE-generalized-Airy-B,F|F:P,$B_{n}$,http://dlmf.nist.gov/9.13#SS1.p2, -\Pade{p}{q}{f},Padé approximant,Pade-approximant,O|F,${[p/q]_{f}}$,http://dlmf.nist.gov/3.11#SS4.p1, -\ParabolicU@{a}{z},parabolic cylinder function $U$,parabolic-U,F|F:P,$U$,http://dlmf.nist.gov/12.4.E1, -\ParabolicUbar@{a}{x},parabolic cylinder function $\overline{U}$,parabolic-U-bar,F|F:P,$\overline{U}$,http://dlmf.nist.gov/12.2.21, -\ParabolicV@{a}{z},parabolic cylinder function $V$,parabolic-V,F|F:P,$V$,http://dlmf.nist.gov/12.4.E2, -\ParabolicW@{a}{x},parabolic cylinder function $W$,parabolic-W,F|F:P,$W$,http://dlmf.nist.gov/12.14.i, -\PartitionsP[k]@{n},total number of partitions,partition-function,I|F:F:P:P,$p_{k}$,http://dlmf.nist.gov/26.2#P4.p2, -\PeriodicBernoulliB{n}@{x},periodic Bernoulli functions,periodic-Bernoulli-polynomial-B,F|F:P,$\widetilde{B}_{n}$,http://dlmf.nist.gov/24.2#iii, -\PeriodicEulerE{n}@{x},periodic Euler functions,periodic-Euler-polynomial-E,F|F:P,$\widetilde{E}_{n}$,http://dlmf.nist.gov/24.2#iii, -\PeriodicZeta@{x}{s},periodic zeta function,periodic-zeta,F|F:P,$F$,http://dlmf.nist.gov/25.13#E1, -\Permutations{n},"set of permutations of ${\bf n}$",set-of-permutations,SN|F,$\mathfrak{S}_{n}$,http://dlmf.nist.gov/23.1, -\PlanePartitionsPP@{n},number of plane partitions of $n$,plane-partitions-PP,I|F:P,$\mathit{pp}$,http://dlmf.nist.gov/26.12#SS1.p4, -\PollaczekP{\lambda}{n}@{x}{a}{b},Pollaczek polynomial,Pollaczek-polynomial-P,P|F:P,$P^{(\lambda)}_{n}$,http://dlmf.nist.gov/18.35#E4, -\Polylogarithm{s}@{z},polylogarithm,polylogarithm,F|F:P,$\mathrm{Li}_{s}$,http://dlmf.nist.gov/25.12#E10, -\Prod{n}{k1}{k2}@{f(n)},semantic product,EMPTY,SM|O:O,$\prod_{n=\alpha}^{\beta}$,http://drmf.wmflabs.org/wiki/Definition:Prod, -\Racah{n}@{x}{\alpha}{\beta}{\gamma}{\delta},Racah polynomial,Racah-polynomial-R,P|F:P,$R_{n}$,http://dlmf.nist.gov/18.25#T1.t1.r4, -\RacahR{n}@{x}{\alpha}{\beta}{\gamma}{\delta},Racah polynomial,Racah-polynomial-R,P|F:P,$R_{n}$,http://dlmf.nist.gov/18.25#T1.t1.r4, -\RamanujanSum{k}@{n},Ramanujan's sum,Ramanujan-sum,F|F:P,$c_{k}$,http://dlmf.nist.gov/27.10#E4, -\RamanujanTau@{n},Ramanujan's tau function,Ramanujan-tau,F|F:P,$\tau$,http://dlmf.nist.gov/27.14#E18, -\Rational,set of rational numbers,EMPTY,SN|F,$\mathbb{Q}$,http://dlmf.nist.gov/front/introduction#Sx4.p2.t1.r13, -\RayleighFun{n}@{\nu},Rayleigh function,Rayleigh-function,F|F:P,$\sigma_{n}$,http://dlmf.nist.gov/10.21#E55, -\Real,set of real numbers,EMPTY,SN|F,$\mathbb{R}$,http://dlmf.nist.gov/front/introduction#Sx4.p2.t1.r14, -\RepInterfc{n}@{z},repeated integrals of the complementary error function,repeated-integral-complementary-error-function,F|F:P,$\mathrm{i}^{n}\mathrm{erfc}$,http://dlmf.nist.gov/7.18#E2, -\Residue,residue,EMPTY,O|F,$\mathrm{res}$,http://dlmf.nist.gov/1.10#SS3.p5, -\RestrictedCompositionsC@{\mathrm{condition}}{n},restricted number of compositions,restricted-number-of-compositions-C,I|F:P,$c_{m}$,http://dlmf.nist.gov/26.11#p1, -\RestrictedPartitionsP[k]@{\mathrm{condition}}{n},restricted number of partitions,restricted-partitions-P,I|F:F:P:P,$p_{k}$,http://dlmf.nist.gov/26.10#i, -\RiemannP@{\begin{Bmatrix} \alpha & \beta &\gamma & \\ a_1 & b_1 & c_1 & z \\ a_2 & b_2 & c_2 & \end{Bmatrix}},Riemann-Papperitz $\RiemannP$-symbol for solutions of the generalized hypergeometric differential equation,Riemann-P-symbol,Q|F:P,$P$,http://dlmf.nist.gov/15.11#E3, -\RiemannTheta@{\Vector{z}}{\Matrix{\Omega}},Riemann theta function,Riemann-theta,F|F:P,$\theta$,http://dlmf.nist.gov/21.2#E1, -\RiemannThetaChar{\Vector{\alpha}}{\Vector{\beta}}@{\Vector{z}}{\Matrix{\Omega}},Riemann theta function with characteristics,Riemann-theta-characterstics,F|F:P,$\theta\left[\Vector{\alpha} \atop \Vector{\beta} \right]$,http://dlmf.nist.gov/21.2#E5, -\RiemannThetaHat@{\Vector{z}}{\Matrix{\Omega}},scaled Riemann theta function,Riemann-theta-hat,F|F:P,$\hat{\theta}$,http://dlmf.nist.gov/21.2#E2, -\RiemannXi@{s},Riemann's $\RiemannXi$-function,Riemann-xi,F|F:P,$\xi$,http://dlmf.nist.gov/25.4#E4, -\RiemannZeta@{s},Riemann zeta function,Riemann-zeta,F|F:P,$\zeta$,http://dlmf.nist.gov/25.2#E1, -\Schwarzian{z}{\zeta},Schwarzian derivative,EMPTY,O|F,"$\left\{f,g\right\}$",http://dlmf.nist.gov/1.13#E20, -\ScorerGi@{z},Scorer function (inhomogeneous Airy function) ${\rm Gi}$,Scorer-Gi,F|F:P,$\mathrm{Gi}$,http://dlmf.nist.gov/9.12#i, -\ScorerHi@{z},Scorer function (inhomogeneous Airy function) ${\rm Hi}$,Scorer-Hi,F|F:P,$\mathrm{Hi}$,http://dlmf.nist.gov/9.12#i, -\SinCosIntf@{z},auxiliary function for sine and cosine integrals ${\rm f}$,sine-cosine-integral-f,F|F:P,$\mathrm{f}$,http://dlmf.nist.gov/6.2#E17, -\SinCosIntg@{z},auxiliary function for sine and cosine integrals ${\rm g}$,sine-cosine-integral-g,F|F:P,$\mathrm{g}$,http://dlmf.nist.gov/6.2#E18, -\SinInt@{z},sine integral ${\rm Si}$,sine-integral,F|F:P,$\mathrm{Si}$,http://dlmf.nist.gov/6.2#E9, -\SinIntg@{a}{z},generalized sine integral ${\rm Si}$,generalized-sine-integral-Si,F|F:P,$\mathrm{Si}$,http://dlmf.nist.gov/8.21#E2, -\SinhInt@{z},hyperbolic sine integral,hyperbolic-sine-integral,F|F:P,$\mathrm{Shi}$,http://dlmf.nist.gov/6.2#E15, -\SphBesselIi{n}@{z},modified spherical Bessel function of the first kind ${\sf i}_n^{(1)}$,spherical-Bessel-I-1,F|F:P,${\mathsf{i}^{(1)}_{n}}$,http://dlmf.nist.gov/10.47#E7, -\SphBesselIii{n}@{z},modified spherical Bessel function of the first kind ${\sf i}_n^{(2)}$,spherical-Bessel-I-2,F|F:P,${\mathsf{i}^{(2)}_{n}}$,http://dlmf.nist.gov/10.47#E8, -\SphBesselJ{n}@{z},spherical Bessel function of the first kind ${\sf j}_n$,spherical-Bessel-J,F|F:P,$\mathsf{j}_{n}$,http://dlmf.nist.gov/10.47#E3, -\SphBesselK{n}@{z},modified spherical Bessel function of the second kind ${\sf k}_n$,spherical-Bessel-K,F|F:P,$\mathsf{k}_{n}$,http://dlmf.nist.gov/10.47#E9, -\SphBesselY{n}@{z},spherical Bessel function of the second kind ${\sf y}_n$,spherical-Bessel-Y,F|F:P,$\mathsf{y}_{n}$,http://dlmf.nist.gov/10.47#E4, -\SphHankelHi{n}@{z},spherical Hankel function of the first kind ${\sf h}_n^{(1)}$,spherical-Hankel-H-1-Bessel-third-kind,F|F:P,${\mathsf{h}^{(1)}_{n}}$,http://dlmf.nist.gov/10.47#E5, -\SphHankelHii{n}@{z},spherical Hankel function of the second kind ${\sf h}_n^{(2)}$,spherical-Hankel-H-2-Bessel-third-kind,F|F:P,${\mathsf{h}^{(2)}_{n}}$,http://dlmf.nist.gov/10.47#E6, -\SphericalHarmonicY{l}{m}@{\theta}{\phi},spherical harmonic,spherical-harmonic-Y,F|F:P,"$Y_{{\ell},{m}}$",http://dlmf.nist.gov/14.30#E1, -\SpheroidalEigenvalueLambda{m}{n}@{\gamma^2},eigenvalues of the spheroidal differential equation,spheroidal-eigenvalue-lambda,O|F:P,$\lambda^{m}_{n}$,http://dlmf.nist.gov/30.3#SS1.p1, -\SpheroidalOnCutPs{m}{n}@{x}{\gamma^2},spheroidal wave function of the first kind ${\sf Ps}_n^m$,spheroidal-wave-on-cut-Ps,F|F:P,$\mathsf{Ps}^{m}_{n}$,http://dlmf.nist.gov/30.4#i, -\SpheroidalOnCutQs{m}{n}@{x}{\gamma^2},spheroidal wave function of the second kind ${\sf Qs}_n^m$,spheroidal-wave-on-cut-Qs,F|F:P,$\mathsf{Qs}^{m}_{n}$,http://dlmf.nist.gov/30.5, -\SpheroidalPs{m}{n}@{z}{\gamma^2},spheroidal wave function of complex argument $Ps_n^m$,spheroidal-wave-Ps,F|F:P,$\mathit{Ps}^{m}_{n}$,http://dlmf.nist.gov/30.6#p1, -\SpheroidalQs{m}{n}@{z}{\gamma^2},spheroidal wave function of complex argument $Qs_n^m$,spheroidal-wave-Qs,F|F:P,$\mathit{Qs}^{m}_{n}$,http://dlmf.nist.gov/30.6#p1, -\SpheroidalRadialS{m}{j}{n}@{z}{\gamma},radial spheroidal wave function,radial-spheroidal-wave-S,F|F:P,$S^{m(j)}_{n}$,http://dlmf.nist.gov/30.11#E3, -\StieltjesConstants{n},Stieltjes constants,Stieltjes-constants,C|F,$\gamma_{n}$,http://drmf.wmflabs.org/wiki/Definition:StieltjesConstants, -\StieltjesTrans@{f}{s},Stieltjes transform,Stieltjes-transform,O|F:P,$\mathcal{S}$,http://dlmf.nist.gov/1.14#E47, -\StieltjesWigert{n}@@{x}{q},Stieltjes-Wigert polynomial,Stieltjes-Wigert-polynomial-S,P|F:P:PS,$S_{n}$,http://drmf.wmflabs.org/wiki/Definition:StieltjesWigert, -\StieltjesWigertS{n}@{x}{q},Stieltjes-Wigert polynomial,Stieltjes-Wigert-polynomial-S,P|F:P,$S_{n}$,http://dlmf.nist.gov/18.27#E18, -\StirlingS@{n}{k},Stirling number of the first kind,Stirling-number-first-kind-S,I|F:P,$s$,http://dlmf.nist.gov/26.8#SS1.p1, -\StirlingSS@{n}{k},Stirling number of the second kind,Stirling-number-second-kind-S,I|F:P,$S$,http://dlmf.nist.gov/26.8#SS1.p3, -\StruveH{\nu}@{z},Struve function ${\bf H}_{\nu}$,Struve-H,F|F:P,$\mathbf{H}_{\nu}$,http://dlmf.nist.gov/11.2#E1, -\StruveK{\nu}@{z},Struve function ${\bf K}_{\nu}$,associated-Struve-K,F|F:P,$\mathbf{K}_{\nu}$,http://dlmf.nist.gov/11.2#E5, -\StruveL{\nu}@{z},modified Struve function ${\bf L}_{\nu}$,modified-Struve-L,F|F:P,$\mathbf{L}_{\nu}$,http://dlmf.nist.gov/11.2#E2, -\StruveM{\nu}@{z},modified Struve function ${\bf M}_{\nu}$,associated-Struve-M,F|F:P,$\mathbf{M}_{\nu}$,http://dlmf.nist.gov/11.2#E6, -\Sum{n}{0}{k}@{f(n)},semantic sum,EMPTY,SM|F:P,$\sum_{n=k}^{N}$,http://drmf.wmflabs.org/wiki/Definition:Sum, -\SurfaceHarmonicY{l}{m}@{\theta}{\phi},surface harmonic of the first kind,surface-harmonic-Y,F|F:P,$Y_{l}^{m}$,http://dlmf.nist.gov/14.30#E2, -\TriangleOP{\alpha}{\beta}{\gamma}{m}{n}@{x}{y},triangle polynomial,triangle-orthogonal-polynomial-P,P|F:P,"$P^{\alpha,\beta,\gamma}_{m,n}$",http://dlmf.nist.gov/18.37#E7, -\Ultra{\lambda}{n}@{x},ultraspherical/Gegenbauer polynomial,ultraspherical-Gegenbauer-polynomial,P|F:P,$C^{\mu}_{n}$,http://dlmf.nist.gov/18.3#T1.t1.r5, -\Ultraspherical{\lambda}{n}@{x},ultraspherical (or Gegenbauer) polynomial,ultraspherical-Gegenbauer-polynomial,P|F:P,$C^{(\mu)}_{n}$,http://dlmf.nist.gov/18.3#T1.t1.r5, -\UmbilicCatE@{s}{t}{\Vector{x}},elliptic umbilic catastrophe,elliptic-umbilic-catastrophe,F|F:P,$\Phi^{(\mathrm{E})}$,http://dlmf.nist.gov/36.2#E2, -\UmbilicCatH@{s}{t}{\Vector{x}},hyperbolic umbilic catastrophe,hyperbolic-umbilic-catastrophe,F|F:P,$\Phi^{(\mathrm{H})}$,http://dlmf.nist.gov/36.2#E3, -\VariationalOp[a;b]@{f},total variation,variational-operator,F|FO:PO:FnO:PnO,"$\mathcal{V}_{a,b}$",http://dlmf.nist.gov/1.4#E33, -\VoigtU@{x}{t},Voigt function ${\sf U}$,Voigt-U,F|F:P,$\mathsf{U}$,http://dlmf.nist.gov/7.19#E1, -\VoigtV@{x}{t},Voigt function ${\sf V}$,Voigt-V,F|F:P,$\mathsf{V}$,http://dlmf.nist.gov/7.19#E2, -\WaringG@{k},Waring's function $G$,Waring-G,F|F:P,$G$,http://dlmf.nist.gov/27.13#SS3.p1, -\Waringg@{k},Waring's function $g$,Waring-g,F|F:P,$g$,http://dlmf.nist.gov/27.13#SS3.p1, -\WeberE{\nu}@{z},Weber function,Weber-E,F|F:P,$\mathbf{E}_{\nu}$,http://dlmf.nist.gov/11.10#E2, -\WeierPInv@@{z}{g_2}{g_3},Weierstrass $\WeierPInv$-function on invariants $\WeierPInv@{z}{g_2}{g_3}$,Weierstrass-P-on-invariants,F|F:P:nP,$\wp$,http://dlmf.nist.gov/23.1, -\WeierPLat@@{z}{\Lattice{L}},Weierstrass $\WeierPLat$-function on lattice $\WeierPLat@{z}{\Lattice{L}}$,Weierstrass-P-on-lattice,F|F:P:nP,$\wp$,http://dlmf.nist.gov/23.1, -\WeiersigmaInv@@{z}{g_2}{g_3},Weierstrass sigma function on invariants,Weierstrass-sigma-on-invariants,F|F:P:nP,$\sigma$,http://dlmf.nist.gov/23.2#E6, -\WeiersigmaLat@@{z}{\Lattice{L}},Weierstrass sigma function on lattice,Weierstrass-sigma-on-lattice,F|F:P:nP,$\sigma$,http://dlmf.nist.gov/23.2#E6, -\WeierzetaInv@@{z}{g_2}{g_3},Weierstrass zeta function on invariants,Weierstrass-zeta-on-invariants,F|F:P:nP,$\zeta$,http://dlmf.nist.gov/23.2#E5, -\WeierzetaLat@@{z}{\Lattice{L}},Weierstrass zeta function on lattice,Weierstrass-zeta-on-lattice,F|F:P:nP,$\zeta$,http://dlmf.nist.gov/23.2#E5, -\WhitD{\nu}@{z},parabolic cylinder function,Whittaker-D,F|F:P,$D_{n}$,http://dlmf.nist.gov/23.1, -\WhitM{\kappa}{\mu}@{z},Whittaker function $M_{\kappa;\mu}$,Whittaker-confluent-hypergeometric-M,F|F:P,"$M_{\kappa,\mu}$",http://dlmf.nist.gov/13.14#E2, -\WhitW{\kappa}{\mu}@{z},Whittaker function $W_{\kappa;\mu}$,Whittaker-confluent-hypergeometric-W,F|F:P,"$W_{\kappa,\mu}$",http://dlmf.nist.gov/13.14#E3, -\Wilson{n}@{x}{a}{b}{c}{d},Wilson polynomial,Wilson-polynomial-W,P|F:P,$W_{n}$,http://dlmf.nist.gov/18.25#T1.t1.r2, -\WilsonW{n}@{x}{a}{b}{c}{d},Wilson polynomial,Wilson-polynomial-W,P|F:P,$W_{n}$,http://dlmf.nist.gov/18.25#T1.t1.r2, -\Wronskian@{w_1(z);w_2(z)},Wronskian,Wronskian,F|F:P,$\mathscr{W}$,http://dlmf.nist.gov/1.13#E4, -\ZeroAiryAi{k},$k$\textsuperscript{th} zero of Airy $\AiryAi$,zero-Airy-Ai,F|F,$a_{k}$,http://dlmf.nist.gov/23.1, -\ZeroAiryAiPrime{k},$k$\textsuperscript{th} zero of Airy $\AiryAi'$,zero-derivative-Airy-Ai,F|F,$a'_{k}$,http://dlmf.nist.gov/23.1, -\ZeroAiryBi{k},$k$\textsuperscript{th} zero of Airy $\AiryBi$,zero-Airy-Bi,F|F,$b_{k}$,http://dlmf.nist.gov/23.1, -\ZeroAiryBiPrime{k},$k$\textsuperscript{th} zero of Airy $\AiryBi'$,zero-derivative-Airy-Bi,F|F,$b'_{k}$,http://dlmf.nist.gov/23.1, -\ZeroBesselJ{\nu}{m},zeros of the Bessel function $\BesselJ{\nu}@{x}$,zero-Bessel-J,F|F,"$j_{\nu,m}$",http://dlmf.nist.gov/23.1, -\ZeroBesselJPrime{\nu}{m},zeros of the Bessel function derivative $\BesselJ{\nu}'@{x}$,zero-derivative-Bessel-J,F|F,"${j'_{\nu,m}}$",http://dlmf.nist.gov/23.1, -\ZeroBesselY{\nu}{m},zeros of the Bessel function $\BesselY{\nu}@{x}$,zero-Bessel-Y-Weber,F|F,"$y_{\nu,m}$",http://dlmf.nist.gov/23.1, -\ZeroBesselYPrime{\nu}{m},zeros of the Bessel function derivative $\BesselY{\nu}'@{x}$,zero-derivative-Bessel-Y-Weber,F|F,"${y'_{\nu,m}}$",http://dlmf.nist.gov/23.1, -\ZonalPoly{\kappa}@{\Matrix{T}},zonal polynomial,zonal-polynomial,P|F:P,$Z_{\kappa}$,http://dlmf.nist.gov/35.4#SS1.p3, -\acos@@{z},inverse cosine function,inverse-cosine,F|F:P:nP,$\mathrm{arccos}$,http://dlmf.nist.gov/4.23#SS2.p1, -\acosh@@{z},inverse hyperbolic cosine function,hyperbolic-inverse-cosine,F|F:P:nP,$\mathrm{arccosh}$,http://dlmf.nist.gov/4.37#SS2.p1, -\acot@@{z},inverse cotangent function,inverse-cotangent,F|F:P:nP,$\mathrm{arccot}$,http://dlmf.nist.gov/4.23#E9, -\acoth@@{z},inverse hyperbolic cotangent function,hyperbolic-inverse-cotangent,F|F:P:nP,$\mathrm{arccoth}$,http://dlmf.nist.gov/4.37#E9, -\acsc@@{z},inverse cosecant function,inverse-cosecant,F|F:P:nP,$\mathrm{arccsc}$,http://dlmf.nist.gov/4.23#E7, -\acsch@@{z},inverse hyperbolic cosecant function,hyperbolic-inverse-cosecant,F|F:P:nP,$\mathrm{arccsch}$,http://dlmf.nist.gov/4.37#E7, -\arcGudermannian@@{x},inverse Gudermannian function,inverse-Gudermannian,F|F:P:nP,${\mathrm{gd}^{-1}}$,http://dlmf.nist.gov/4.23#E41, -\arcJacobicd@{x}{k},inverse Jacobian elliptic function ${\rm arccd}$,inverse-Jacobi-elliptic-cd,F|F:P,$\mathrm{arccd}$,http://dlmf.nist.gov/22.15#SS1.p1, -\arcJacobicn@{x}{k},inverse Jacobian elliptic function ${\rm arccn}$,inverse-Jacobi-elliptic-cn,F|F:P,$\mathrm{arccn}$,http://dlmf.nist.gov/22.15#SS1.p1, -\arcJacobics@{x}{k},inverse Jacobian elliptic function ${\rm arccs}$,inverse-Jacobi-elliptic-cs,F|F:P,$\mathrm{arccs}$,http://dlmf.nist.gov/22.15#SS1.p1, -\arcJacobidc@{x}{k},inverse Jacobian elliptic function ${\rm arcdc}$,inverse-Jacobi-elliptic-dc,F|F:P,$\mathrm{arcdc}$,http://dlmf.nist.gov/22.15#SS1.p1, -\arcJacobidn@{x}{k},inverse Jacobian elliptic function ${\rm arcdn}$,inverse-Jacobi-elliptic-dn,F|F:P,$\mathrm{arcdn}$,http://dlmf.nist.gov/22.15#SS1.p1, -\arcJacobids@{x}{k},inverse Jacobian elliptic function ${\rm arcds}$,inverse-Jacobi-elliptic-ds,F|F:P,$\mathrm{arcds}$,http://dlmf.nist.gov/22.15#SS1.p1, -\arcJacobinc@{x}{k},inverse Jacobian elliptic function ${\rm arcnc}$,inverse-Jacobi-elliptic-nc,F|F:P,$\mathrm{arcnc}$,http://dlmf.nist.gov/22.15#SS1.p1, -\arcJacobind@{x}{k},inverse Jacobian elliptic function ${\rm arcnd}$,inverse-Jacobi-elliptic-nd,F|F:P,$\mathrm{arcnd}$,http://dlmf.nist.gov/22.15#SS1.p1, -\arcJacobins@{x}{k},inverse Jacobian elliptic function ${\rm arcns}$,inverse-Jacobi-elliptic-ns,F|F:P,$\mathrm{arcns}$,http://dlmf.nist.gov/22.15#SS1.p1, -\arcJacobisc@{x}{k},inverse Jacobian elliptic function ${\rm arcsc}$,inverse-Jacobi-elliptic-sc,F|F:P,$\mathrm{arcsc}$,http://dlmf.nist.gov/22.15#SS1.p1, -\arcJacobisd@{x}{k},inverse Jacobian elliptic function ${\rm arcsd}$,inverse-Jacobi-elliptic-sd,F|F:P,$\mathrm{arcsd}$,http://dlmf.nist.gov/22.15#SS1.p1, -\arcJacobisn@{x}{k},inverse Jacobian elliptic function ${\rm arcsn}$,inverse-Jacobi-elliptic-sn,F|F:P,$\mathrm{arcsn}$,http://dlmf.nist.gov/22.15#SS1.p1, -\asec@@{z},inverse secant function,inverse-secant,F|F:P:nP,$\mathrm{arcsec}$,http://dlmf.nist.gov/4.23#E8, -\asech@@{z},inverse hyperbolic secant function,hyperbolic-inverse-secant,F|F:P:nP,$\mathrm{arcsech}$,http://dlmf.nist.gov/4.37#E8, -\asin@@{z},inverse sine function,inverse-sine,F|F:P:nP,$\mathrm{arcsin}$,http://dlmf.nist.gov/4.23#SS2.p1, -\asinh@@{z},inverse hyperbolic sine function,hyperbolic-inverse-sine,F|F:P:nP,$\mathrm{arcsinh}$,http://dlmf.nist.gov/4.37#SS2.p1, -\atan@@{z},inverse tangent function,inverse-tangent,F|F:P:nP,$\mathrm{arctan}$,http://dlmf.nist.gov/4.23#SS2.p1, -\atanh@@{z},inverse hyperbolic tangent function,hyperbolic-inverse-tangent,F|F:P:nP,$\mathrm{arctanh}$,http://dlmf.nist.gov/4.37#SS2.p1, -\bDiff[x],backward difference,EMPTY,O|F,"$\,\nabla_{x}$",http://dlmf.nist.gov/18.1#EGx2, -\bigqJacobi{n}@@{x}{a}{b}{c}{q},big $q$-Jacobi polynomial,big-q-Jacobi-polynomial-P,P|F:P:PS,$P_{n}$,http://drmf.wmflabs.org/wiki/Definition:bigqJacobi, -\bigqJacobiIVparam{n}@@{x}{a}{b}{c}{d}{q},big $q$-Jacobi polynomial with four parameters,big-q-Jacobi-polynomial-four-parameters-P,P|F:P,$P_{n}$,http://drmf.wmflabs.org/wiki/Definition:bigqJacobiIVparam, -\bigqLaguerre{n}@{x}{a}{b}{q},big $q$-Laguerre polynomial,EMPTY,P|F:P,$P_{n}$,http://drmf.wmflabs.org/wiki/Definition:bigqLaguerre, -\bigqLegendre{n}@{x}{c}{q},big $q$-Legendre polynomial,EMPTY,P|F:P,$P_{n}$,http://drmf.wmflabs.org/wiki/Definition:bigqLegendre, -\binom{n}{k},binomial coefficient,EMPTY,I|F,$\binom{n}{k}$,http://dlmf.nist.gov/1.2#E1,http://dlmf.nist.gov/26.3#SS1.p1 -\binomial{n}{k},binomial coefficient,EMPTY,I|F,$\binom{n}{k}$,http://dlmf.nist.gov/1.2#E1,http://dlmf.nist.gov/26.3#SS1.p1 -\cDiff[x],central difference,EMPTY,O|F,"$\,\delta_{x}$",http://dlmf.nist.gov/18.1#EGx3, -\card,number of elements of a finite set,EMPTY,SN|F,$\left|A\right|$,http://dlmf.nist.gov/26.1, -\cartprod,$G\cartprod H$: Cartesian product of groups $G$ and $H$,EMPTY,O|F,$\times$,http://dlmf.nist.gov/23.1, -\ceiling{x},ceiling,EMPTY,I|F,$\left\lceil a\right\rceil$,http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r17, -\conj{z},complex conjugate,EMPTY,O|F,$\overline{a}$,http://dlmf.nist.gov/1.9#E11, -\cos@@{z},cosine function,cos,F|F:P:nP,$\mathrm{cos}$,http://dlmf.nist.gov/4.14#E2, -\cosh@@{z},hyperbolic cosine function,hyperbolic-cosine,F|F:P:nP,$\mathrm{cosh}$,http://dlmf.nist.gov/4.28#E2, -\cosintg@{a}{z},generalized cosine integral ${\rm ci}$,generalized-cosine-integral-ci,F|F:P,$\mathrm{ci}$,http://dlmf.nist.gov/8.21#E1, -\cot@@{z},cotangent function,cotangent,F|F:P:nP,$\mathrm{cot}$,http://dlmf.nist.gov/4.14#E7, -\coth@@{z},hyperbolic cotangent function,hyperbolic-cotangent,F|F:P:nP,$\mathrm{coth}$,http://dlmf.nist.gov/4.28#E7, -\cpi,ratio of a circle's circumference to its diameter,ratio-of-a-circle's-circumference-to-its-diameter,C|F,$\pi$,http://dlmf.nist.gov/5.19.E4, -\csc@@{z},cosecant function,cosecant,F|F:P:nP,$\mathrm{csc}$,http://dlmf.nist.gov/4.14#E5, -\csch@@{z},hyperbolic cosecant function,hyperbolic-cosecant,F|F:P:nP,$\mathrm{csch}$,http://dlmf.nist.gov/4.28#E5, -\ctsHahn{n}@@{x}{a}{b}{c}{d},continuous Hahn polynomial,continuous-Hahn-polynomial,P|F:P:PS,$p_{n}$,http://dlmf.nist.gov/18.19#P2.p1, -\ctsbigqHermite{n}@{x}{a}{q},continuous big $q$-Hermite polynomial,EMPTY,P|F:P,$H_{n}$,http://drmf.wmflabs.org/wiki/Definition:ctsbigqHermite, -\ctsdualHahn{n}@@{x^2}{a}{b}{c},continuous dual Hahn polynomial,continuous-dual-Hahn-normalized-S,P|F:P:PS,$S_{n}$,http://dlmf.nist.gov/18.25#T1.t1.r3, -\ctsdualqHahn{n}@{x}{a}{b}{c}{q},continuous dual $q$-Hahn polynomial,EMPTY,P|F:P,$p_{n}$,http://drmf.wmflabs.org/wiki/Definition:ctsdualqHahn, -\ctsqHahn{n}@{x}{a}{b}{c}{d}{q},continuous $q$-Hahn polynomial,EMPTY,P|F:P,$p_{n}$,http://drmf.wmflabs.org/wiki/Definition:ctsqHahn, -\ctsqHermite{n}@@{x}{q},continuous $q$-Hermite polynomial,continuous-q-Hermite-polynomial-H,P|F:P:PS,$H_{n}$,http://drmf.wmflabs.org/wiki/Definition:ctsqHermite, -\ctsqJacobi{\alpha}{\beta}{m}@{x}{q},continuous $q$-Jacobi polynomial,EMPTY,P|F:P,"$P^{(\alpha,\beta)}_{n}$",http://drmf.wmflabs.org/wiki/Definition:ctsqJacobi, -\ctsqJacobiRahman{n}@@{x}{q},continuous $q$-Jacobi-Rahman-polynomial,continuous-q-Jacobi-Rahman-polynomial-P,P|F:P:PS,"$P^{(\alpha,\beta)}_{n}$",http://drmf.wmflabs.org/wiki/Definition:ctsqJacobiRahman, -\ctsqLaguerre{\alpha}{n}@{x}{q},continuous $q$-Laguerre polynomial,EMPTY,P|F:P,$P^{(n)}_{\alpha}$,http://drmf.wmflabs.org/wiki/Definition:ctsqLaguerre, -\ctsqLegendre{n}@@{x}{q},continuous $q$-Legendre polynomial,continuous-q-Legendre-polynomial-P,P|F:P:PS,$P_{n}$,http://drmf.wmflabs.org/wiki/Definition:ctsqLegendre, -\ctsqLegendreRahman{n}@@{x}{q},continuous $q$-Legendre-Rahman polynomial,continuous-q-Legendre-Rahman-polynomial-P,P|F:P:PS,$P_{n}$,http://drmf.wmflabs.org/wiki/Definition:ctsqLegendreRahman, -\ctsqUltra{n}@{x}{\beta}{q},continuous $q$-ultraspherical/Rogers polynomial,continuous-q-ultraspherical-Rogers-polynomial,P|F:P,$C_{n}$,http://dlmf.nist.gov/18.28#E13, -\ctsqUltrae{n}@{e^{i\theta}}{\beta}{q},continuous $q$-ultraspherical/Rogers polynomial exponential argument,continuous-q-ultraspherical-Rogers-polynomial-exponential-argument,P|F:P,$C_{n}$,http://drmf.wmflabs.org/wiki/Definition:ctsqUltrae, -\curl,curl of vector-valued function,EMPTY,F|F,$\nabla\times$,http://dlmf.nist.gov/1.6#E22, -\deltaDistribution,Dirac delta distribution,Dirac delta distribution,D|F,$\delta_x$,http://dlmf.nist.gov/1.17.i, -\deriv{f}{x},derivative,EMPTY,O|F,$\deriv{}{x}}$,http://dlmf.nist.gov/1.4#E4, -\det,determinant,EMPTY,L|F,$\det$,http://dlmf.nist.gov/1.3#i, -\diag,diagonal part of a matrix,EMPTY,L|F,$\diag[\Matrix{a}]$,http://dlmf.nist.gov/21.5.E7, -\diff{x},differential,EMPTY,O|F,$\mathrm{d}^nx$,http://dlmf.nist.gov/1.4#iv, -\digamma@{z},psi (or digamma) function,digamma,F|F:P,$\psi$,http://dlmf.nist.gov/5.2#E2, -\discrqHermiteI{n}@@{x}{q},discrete $q$-Hermite I polynomial,discrete-q-Hermite-polynomial-h-I,P|F:P:PS,$h_{n}$,http://drmf.wmflabs.org/wiki/Definition:discrqHermiteI, -\discrqHermiteII{n}@@{x}{q},discrete $q$-Hermite II polynomial,discrete-q-Hermite-polynomial-II-h-tilde,P|F:P:PS,$\tilde{h}_{n}$,http://drmf.wmflabs.org/wiki/Definition:discrqHermiteII, -\divergence,divergence of vector-valued function,EMPTY,F|F,$\mathrm{div}$,http://dlmf.nist.gov/1.6#E21, -\divides,divides,EMPTY,O|F,$\nabla\cdot$,http://dlmf.nist.gov/24.1, -\dualHahn{n}@@{\lambda(x)}{\gamma}{\delta}{N},dual Hahn polynomial,dual-Hahn-polynomial-R,P|F:P:PS,$R_{n}$,http://dlmf.nist.gov/18.25#T1.t1.r5, -\dualqHahn{n}@@{\mu(x)}{\gamma}{\delta}{N}{q},dual $q$-Hahn polynomial,dual-q-Hahn-R,P|F:P:PS,$R_{n}$,http://drmf.wmflabs.org/wiki/Definition:dualqHahn, -\dualqKrawtchouk{n}@@{\lambda(x)}{c}{N}{q},dual $q$-Krawtchouk polynomial,dual-q-Krawtchouk-polynomial-K,P|F:P:PS,$K_{n}$,http://drmf.wmflabs.org/wiki/Definition:dualqKrawtchouk, -\erf@@{z},error function,error-function,F|F:P:nP,$\mathrm{erf}$,http://dlmf.nist.gov/7.2#E1, -\erfc@@{z},complementary error function,repeated-integral-complementary-error-function,F|F:P:nP,$\mathrm{erfc}$,http://dlmf.nist.gov/7.2#E2, -\erfw@@{z},complementary error function $w$,error-function-w,F|F:P:nP,$w$,http://dlmf.nist.gov/7.2#E3, -\exp@@{z},exponential function,exponential,F|F:P:nP,$\mathrm{exp}$,http://dlmf.nist.gov/4.2#E19, -\expe,the base of the natural logarithm,EMPTY,F|F,$\mathrm{e}$,http://dlmf.nist.gov/4.2.E11, -\exptrace@{\Matrix{X}},exponential of trace of a matrix,exponential-trace,L|F:P,$\mathrm{etr}$,http://dlmf.nist.gov/35.1#p3.t1.r10, -\f{f}@{x},function,EMPTY,SM|F:P,${f}$,http://drmf.wmflabs.org/wiki/Definition:f, -\fDiff,forward difference operator,EMPTY,O|F,"$\,\Delta_{y}$",http://dlmf.nist.gov/3.6#SS1.p1, -\floor{x},floor,EMPTY,I|F,$\left\lfloor a\right\rfloor$,http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r16, -\forall,for every,EMPTY,Q|F,$\forall$,http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r11, -\gradient,gradient of differentiable scalar function,EMPTY,F|F,$\nabla$,http://dlmf.nist.gov/1.6#E20, -"\idem@{\chi_1}{\chi_1,...,\chi_n}",$\idem$ function,idem,F|F:P,$\mathrm{idem}$,http://dlmf.nist.gov/17.1#p3, -\ifrac,semantic slash fraction,EMPTY,SM|F,$(a)/(b)$,http://dlmf.nist.gov/4.2.E8, -\imagpart{z},imaginary part,EMPTY,O|F,"$\Im {z}$",http://dlmf.nist.gov/1.9#E2, -\in,element of,EMPTY,Q|F,$\in$,http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r9, -\incgamma@{a}{z},incomplete gamma function $\gamma$,incomplete-gamma,F|F:P,$\gamma$,http://dlmf.nist.gov/8.2#E1, -\incgammastar@{a}{z},incomplete gamma function $\gamma^{*}$,incomplete-gamma-star,F|F:P,$\gamma^{*}$,http://dlmf.nist.gov/8.2#E6, -\inf,greatest lower bound (infimum),EMPTY,Q|F,$\mathrm{inf}$,http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r24, -\int,integral,EMPTY,F|F,$\int$,http://dlmf.nist.gov/1.4#iv, -\inverf@@{x},inverse error function,inverse-error-function,F|F:P:nP,$\mathrm{inverf}$,http://dlmf.nist.gov/7.17#E1, -\inverfc@@{x},inverse complementary error function,inverse-complementary-error-function,F|F:P:nP,$\mathrm{inverfc}$,http://dlmf.nist.gov/7.17#E1, -\iunit,imaginary unit,EMPTY,C|F,$\mathrm{i}$,http://dlmf.nist.gov/1.9.i, -\liminf,least limit point,EMPTY,Q|F,$\mathrm{liminf}$,http://dlmf.nist.gov/front/introduction#Sx4.p2.t1.r4, -\littleo@{x},order less than,little-o,Q|F:P,$o$,http://dlmf.nist.gov/2.1#E2, -\littleqJacobi{n}@@{x}{a}{b}{q},little $q$-Jacobi polynomial,little-q-Jacobi-polynomial-p,P|F:P:PS,$p_{n}$,http://drmf.wmflabs.org/wiki/Definition:littleqJacobi, -\littleqLaguerre{n}@{x}{a}{q},little $q$-Laguerre / Wall polynomial,EMPTY,P|F:P,$p_{n}$,http://drmf.wmflabs.org/wiki/Definition:littleqLaguerre, -\littleqLegendre{m}@{x}{q},little $q$-Legendre polynomial,EMPTY,P|F:P,$p_{n}$,http://drmf.wmflabs.org/wiki/Definition:littleqLegendre, -\ln@@{z},principal branch of logarithm function,natural-logarithm,F|F:P:nP,$\mathrm{ln}$,http://dlmf.nist.gov/4.2#E2, -\log@@{z},principle branch of natural logarithm,logarithm,F|F:P:nP,$\mathrm{log}$,http://dlmf.nist.gov/4.2#E2, -\logb{a}@@{z},logarithm to general base,logarithm,F|F:P:nP,$\mathrm{log}_{b}$,http://dlmf.nist.gov/4.2#EGx1, -\lrselection,bracketed generalization of $\pm$,EMPTY,SM|F,$\left\{\begin{matrix}x\end{matrix}\right\}$,http://drmf.wmflabs.org/wiki/Definition:Lrselection, -\lselection,left bracketed generalization of $\pm$,EMPTY,SM|F,$\left\{\begin{matrix}x\end{matrix}\right.$,http://dlmf.nist.gov/28.4.E22, -\mEulerBeta{m}@{a}{b},multivariate beta function,multivariate-Euler-Beta,F|F:P,$\mathrm{B}_{m}$,http://dlmf.nist.gov/35.3#E3, -\mEulerGamma{m}@{a},multivariate gamma function,multivariate-Euler-Gamma,F|F:P,$\Gamma_{m}$,http://dlmf.nist.gov/35.3#i, -\mod,modulo,EMPTY,Q|F,$\mod$,http://dlmf.nist.gov/front/introduction#Sx4.p2.t1.r10, -\monicAffqKrawtchouk{n}@@{q^{-x}}{p}{N}{q},monic affine $q$-Krawtchouk polynomial,affine-q-Krawtchouk-polynomial-monic-p,P|F:P:PS,$\widehat{K}^{\mathrm{Aff}}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicAffqKrawtchouk, -\monicAlSalamCarlitzI{a}{n}@@{x}{q},monic Al-Salam-Carlitz I polynomial,q-Al-Salam-Carlitz-I-polynomial-monic-p,P|F:P:PS,${\widehat U}^{(\alpha)}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicAlSalamCarlitzI, -\monicAlSalamCarlitzII{a}{n}@@{x}{q},monic Al-Salam-Carlitz II polynomial,q-Al-Salam-Carlitz-II-polynomial-monic-p,P|F:P:PS,${\widehat V}^{(n)}_{\alpha}$,http://drmf.wmflabs.org/wiki/Definition:monicAlSalamCarlitzII, -\monicAlSalamChihara{n}@@{x}{a}{b}{q},monic Al-Salam-Chihara polynomial,Al-Salam-Chihara-polynomial-monic-p,P|F:P:PS,${\widehat Q}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicAlSalamChihara, -\monicAskeyWilson{n}@@{x}{a}{b}{c}{d}{q},monic Askey-Wilson polynomial,Askey-Wilson-polynomial-monic-p,P|F:P:PS,${\widehat p}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicAskeyWilson, -\monicBesselPoly{n}@@{x}{a},monic Bessel polynomial,Bessel-polynomial-monic-p,P|F:P:PS,${\widehat y}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicBesselPoly, -\monicCharlier{n}@@{x}{a},monic Charlier polynomial,Charlier-polynomial-monic-p,P|F:P:PS,${\widehat C}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicCharlier, -\monicChebyT{n}@@{x},monic Chebyshev polynomial of the first kind,Chebyshev-polynomial-first-kind-monic-p,P|F:P:PS,${\widehat T}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicChebyT, -\monicChebyU{n}@@{x},monic Chebyshev polynomial of the second kind,Chebyshev-polynomial-second-kind-monic-p,P|F:P:PS,${\widehat U}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicChebyU, -\monicHahn{n}@@{x}{\alpha}{\beta}{N},monic Hahn polynomial,Hahn-polynomial-monic-p,P|F:P:PS,${\widehat Q}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicHahn, -\monicHermite{n}@@{x},monic Hermite polynomial,Hermite-polynomial-monic,P|F:P:PS,${\widehat H}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicHermite, -\monicJacobi{n}@@{x},monic Jacobi polynomial,Jacobi-polynomial-monic-p,P|F:P:PS,"${\widehat P}^{(\alpha,\beta)}_{n}$",http://drmf.wmflabs.org/wiki/Definition:monicJacobi, -\monicKrawtchouk{n}@@{q^{-x}}{p}{N}{q},monic Krawtchouk polynomial,Krawtchouk-polynomial-monic-p,P|F:P:PS,${\widehat K}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicKrawtchouk, -\monicLaguerre{n}@@{x},monic Laguerre polynomial,generalized-Laguerre-polynomial-monic-p,P|F:P:PS,${\widehat L}^{\alpha}_n$,http://drmf.wmflabs.org/wiki/Definition:monicLaguerre, -\monicLegendrePoly{n}@@{x},monic Legendre polynomial,Legendre-spherical-polynomial-monic-p,P|F:P:PS,${\widehat P}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicLegendrePoly, -\monicMeixner{n}@@{x}{\beta}{c},monic Meixner polynomial,Meixner-polynomial-monic-p,P|F:P:PS,${\widehat M}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicMeixner, -\monicMeixnerPollaczek{\lambda}{n}@@{x}{\phi},monic Meixner-Pollaczek polynomial,Meixner-Pollaczek-polynomial-monic-p,P|F:P:PS,${\widehat P}^{(\alpha)}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicMeixnerPollaczek, -\monicRacah{n}@@{\lambda(x)}{\alpha}{\beta}{\gamma}{\delta},monic Racah polynomial,Racah-polynomial-monic-p,P|F:P:PS,${\widehat R}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicRacah, -\monicStieltjesWigert{n}@@{x}{q},monic Stieltjes-Wigert polynomial,Stieltjes-Wigert-polynomial-monic-p,P|F:P:PS,${\widehat S}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicStieltjesWigert, -\monicUltra{n}@@{x},monic ultraspherical/Gegenbauer polynomial,ultraspherical-Gegenbauer-polynomial-monic-p,P|F:P:PS,${\widehat C}^{\mu}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicUltra, -\monicWilson{n}@@{x^2}{a}{b}{c}{d},monic Wilson polynomial,Wilson-polynomial-monic-p,P|F:P:PS,${\widehat W}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicWilson, -\monicbigqJacobi{n}@@{x}{a}{b}{c}{q},monic big $q$-Jacobi polynomial,big-q-Jacobi-polynomial-monic-p,P|F:P:PS,${\widehat P}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicbigqJacobi, -\monicbigqLaguerre{n}@@{x}{a}{b}{q},monic big $q$-Laguerre polynomial,big-q-Laguerre-polynomial-monic-p,P|F:P:PS,${\widehat P}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicbigqLaguerre, -\monicbigqLegendre{n}@@{x}{c}{q},monic big $q$-Legendre polynomial,big-q-Legendre-polynomial-monic-p,P|F:P:PS,${\widehat P}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicbigqLegendre, -\monicctsHahn{n}@@{x}{a}{b}{c}{d},monic continuous Hahn polynomial,continuous-Hahn-polynomial-monic-p,P|F:P:PS,${\widehat p}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicctsHahn, -\monicctsbigqHermite{n}@@{x}{a}{q},monic continuous big $q$-Hermite polynomial,continuous-big-q-Hermite-polynomial-monic-p,P|F:P:PS,${\widehat H}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicctsbigqHermite, -\monicctsdualHahn{n}@@{x^2}{a}{b}{c}{d},monic continuous dual Hahn polynomial,continuous-dual-Hahn-polynomial-monic-p,P|F:P:PS,${\widehat S}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicctsdualHahn, -\monicctsdualqHahn{n}@@{x^2}{a}{b}{c},monic continuous dual $q$-Hahn polynomial,continuous-dual-q-Hahn-polynomial-monic-p,P|F:P:PS,${\widehat p}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicctsdualqHahn, -\monicctsqHahn{n}@@{x}{a}{b}{c}{d}{q},monic continuous $q$-Hahn polynomial,continuous-q-Hahn-polynomial-monic-p,P|F:P:PS,${\widehat p}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicctsqHahn, -\monicctsqHermite{n}@@{x}{q},monic continuous $q$-Hermite polynomial,continuous-q-Hermite-polynomial-monic-p,P|F:P:PS,${\widehat H}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicctsqHermite, -\monicctsqJacobi{n}@@{x}{q},monic continuous $q$-Jacobi polynomial,continuous-q-Jacobi-polynomial-monic-p,P|F:P:PS,"${\widehat P}^{(\alpha,\beta)}_{n}$",http://drmf.wmflabs.org/wiki/Definition:monicctsqJacobi, -\monicctsqLaguerre{n}@@{x}{q},monic continuous $q$-Laguerre polynomial,continuous-q-Laguerre-polynomial-monic-p,P|F:P:PS,${\widehat P}^{(\alpha)}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicctsqLaguerre, -\monicctsqLegendre{n}@@{x}{q},monic continuous $q$-Legendre polynomial,continuous-q-Legendre-polynomial-monic-p,P|F:P:PS,${\widehat P}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicctsqLegendre, -\monicctsqUltra{n}@@{x}{\beta}{q},monic continuous $q$-ultraspherical/Rogers polynomial,continuous-q-ultraspherical-Rogers-polynomial-monic-p,P|F:P:PS,${\widehat C}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicctsqUltra, -\monicdiscrqHermiteI{n}@@{x}{q},monic discrete $q$-Hermite I polynomial,discrete-q-Hermite-polynomial-I-monic-p,P|F:P:PS,${\widehat h}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicdiscrqHermiteI, -\monicdiscrqHermiteII{n}@@{x}{q},monic discrete $q$-Hermite II polynomial,discrete-q-Hermite-polynomial-II-monic-p,P|F:P:PS,${\widehat \tilde{p}_{n}}$,http://drmf.wmflabs.org/wiki/Definition:monicdiscrqHermiteII, -\monicdualHahn{n}@@{\lambda(x)}{\gamma}{\delta}{N},monic dual Hahn polynomial,dual-Hahn-polynomial-monic-p,P|F:P:PS,${\widehat R}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicdualHahn, -\monicdualqHahn{n}@@{\mu(x)}{\gamma}{\delta}{N}{q},monic dual $q$-Hahn polynomial,dual-q-Hahn-monic-p,P|F:P:PS,${\widehat R}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicdualqHahn, -\monicdualqKrawtchouk{n}@@{\lambda(x)}{c}{N}{q},monic dual $q$-Krawtchouk polynomial,dual-q-Krawtchouk-polynomial-monic-p,P|F:P:PS,${\widehat K}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicdualqKrawtchouk, -\moniclittleqJacobi{n}@@{x}{a}{b}{q},monic little $q$-Jacobi polynomial,little-q-Jacobi-polynomial-monic-p,P|F:P:PS,${\widehat p}_{n}$,http://drmf.wmflabs.org/wiki/Definition:moniclittleqJacobi, -\moniclittleqLaguerre{n}@@{x}{a}{q},monic little $q$-Laguerre/Wall polynomial,little-q-Laguerre-Wall-polynomial-monic-p,P|F:P:PS,${\widehat p}_{n}$,http://drmf.wmflabs.org/wiki/Definition:moniclittleqLaguerre, -\moniclittleqLegendre{n}@@{x}{q},monic little $q$-Legendre polynomial,little-q-Legendre-polynomial-monic-p,P|F:P:PS,${\widehat p}_{n}$,http://drmf.wmflabs.org/wiki/Definition:moniclittleqLegendre, -\monicpseudoJacobi{n}@@{x}{\nu}{N},monic pseudo-Jacobi polynomial,pseudo-Jacobi-polynomial-monic-p,P|F:P:PS,${\widehat P}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicpseudoJacobi, -\monicqBesselPoly{n}@@{x}{a}{q},monic $q$-Bessel polynomial,q-Bessel-polynomial-monic-p,P|F:P:PS,${\widehat y}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicqBessel, -\monicqCharlier{n}@@{q^{-x}}{a}{q},monic $q$-Charlier polynomial,q-Charlier-polynomial-monic-p,P|F:P:PS,${\widehat C}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicqCharlier, -\monicqHahn{n}@@{q^{-x}}{\alpha}{\beta}{N},monic $q$-Hahn polynomial,q-Hahn-polynomial-monic-p,P|F:P:PS,${\widehat Q}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicqHahn, -\monicqKrawtchouk{n}@@{q^{-x}}{p}{N}{q},monic $q$-Krawtchouk polynomial,q-Krawtchouk-polynomial-monic-p,P|F:P:PS,${\widehat K}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicqKrawtchouk, -\monicqLaguerre{n}@@{x}{q},monic $q$-Laguerre polynomial,q-Laguerre-polynomial-monic-p,P|F:P:PS,${\widehat L}^{(\alpha)}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicqLaguerre, -\monicqMeixner{n}@@{q^{-x}}{b}{c}{q},monic $q$-Meixner polynomial,q-Meixner-polynomial-monic-p,P|F:P:PS,${\widehat M}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicqMeixner, -\monicqMeixnerPollaczek{n}@@{x}{a}{q},monic $q$-Meixner-Pollaczek polynomial,q-Meixner-Pollaczek-polynomial-monic-p,P|F:P:PS,${\widehat P}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicqMeixnerPollaczek, -\monicqRacah{n}@@{\mu(x)}{\alpha}{\beta}{\gamma}{\delta}{q},monic $q$-Racah polynomial,q-Racah-polynomial-monic-R,P|F:P:PS,${\widehat R}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicqRacah, -\monicqinvAlSalamChihara{n}@{x}{a}{b}{q^{-1}},monic $q$-inverse Al-Salam-Chihara polynomial,q-inverse-AlSalam-Chihara-polynomial-monic-p,P|F:P,${\widehat Q}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicqinvAlSalamChihara, -\monicqtmqKrawtchouk{n}@@{q^{-x}}{p}{N}{q},monic quantum $q$-Krawtchouk polynomial,quantum-q-Krawtchouk-polynomial-monic-p,P|F:P:PS,${\widehat K^{\mathrm{qtm}}}_{n}$,http://drmf.wmflabs.org/wiki/Definition:monicqtmqKrawtchouk, -"\multinomial{n_1+n_2+\dots+n_k}{n_1,...,n_k}",multinomial coefficient,EMPTY,I|F,$\left(A;B\right)$,http://dlmf.nist.gov/26.4.E2, -\ninej@@{j_{11}}{j_{12}}{j_{13}}{j_{21}}{j_{22}}{j_{23}}{j_{31}}{j_{32}}{j_{33}},$\ninej$ symbol,EMPTY,F|F:P:nP,$$,http://dlmf.nist.gov/34.6#E1, -\normAskeyWilsonptilde{n}@@{x}{a}{b}{c}{d}{q},normalized Askey-Wilson polynomial ${\tilde p}$,Askey-Wilson-polynomial-normalized-p-tilde,P|F:P:PS,${\tilde p}_{n}$,http://drmf.wmflabs.org/wiki/Definition:normAskeyWilsonptilde, -\normJacobiR{\alpha}{\beta}{n}@{x},normalized Jacobi polynomial,Jacobi-polynomial-R,P|F:P,"$R^{(\alpha,\beta)}_{n}$",http://drmf.wmflabs.org/wiki/Definition:normJacobiR, -\normWilsonWtilde{n}@@{x^2}{a}{b}{c}{d},normalized Wilson polynomial ${\tilde W}$,Wilson-polynomial-normalized-W-tilde,P|F:P:PS,${\tilde W}_{n}$,http://drmf.wmflabs.org/wiki/Definition:normWilsonWtilde, -\normctsHahnptilde{n}@@{x}{a}{b}{c}{d},normalized continuous Hahn polynomial ${\tilde p}$,continuous-Hahn-polynomial-normalized-p-tilde,P|F:P:PS,${\tilde p}_{n}$,http://drmf.wmflabs.org/wiki/Definition:normctsHahnptilde, -\normctsdualHahnStilde{n}@@{x^2}{a}{b}{c}{d},normalized continuous dual Hahn polynomial ${\tilde S}$,continuous-dual-Hahn-polynomial-normalized-S-tilde,P|P:PS,${\tilde S}_{n}$,http://drmf.wmflabs.org/wiki/Definition:normctsdualHahnStilde, -\normctsdualqHahnptilde{n}@@{x}{a}{b}{c}{q},normalized continuous dual $q$-Hahn polynomial ${\tilde p}$,continuous-dual-q-Hahn-polynomial-normalized-p-tilde,P|F:P:PS,${\tilde p}_{n}$,http://drmf.wmflabs.org/wiki/Definition:normctsdualqHahnptilde, -\normctsqHahnptilde{n}@@{x}{a}{b}{c}{d}{q},normalized continuous $q$-Hahn polynomial ${\tilde p}$,continuous-q-Hahn-polynomial-normalized-p-tilde,P|F:P:PS,${\tilde p}_{n}$,http://drmf.wmflabs.org/wiki/Definition:normctsqHahnptilde, -\normctsqJacobiPtilde{n}@@{x}{q},normalized continuous $q$-Jacobi polynomial ${\tilde P}$,continuous-q-Jacobi-polynomial-P-tilde,P|F:P:PS,"${\tilde P}^{(\alpha,\beta)}_{n}$",http://drmf.wmflabs.org/wiki/Definition:normctsqJacobiPtilde, -\notin,not an element of,EMPTY,Q|F,$\notin$,http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r10, -\opminus,negative unity to an integer power,EMPTY,I|F,$(-1)$,http://dlmf.nist.gov/5.7.E7, -\pderiv{f}{x},partial derivative,EMPTY,O|F,$\frac{\partial f}{\partial x}$,http://dlmf.nist.gov/1.5#E3, -\pdiff{x},power of a partial differential,EMPTY,O|F,"$\partial^nx$",http://dlmf.nist.gov/1.5#E3, -\pgcd,greatest common divisor,EMPTY,I|F,"$\left(m,n\right)$",http://dlmf.nist.gov/27.1, -\ph@@{z},phase,phase,O|F:P:nP,$\mathrm{ph}$,http://dlmf.nist.gov/1.9#E7, -\pochhammer{a}{n},Pochhammer symbol,EMPTY,F|F,$(a)_n$,http://dlmf.nist.gov/5.2#iii, -\poly{p}{n}@{x},polynomial,EMPTY,SM|F:P,${p}_{n}$,http://drmf.wmflabs.org/wiki/Definition:poly, -\polygamma{n}@{z},polygamma functions,polygamma,F|F:P,$\psi^{(n)}$,http://dlmf.nist.gov/5.15, -\power,power function,EMPTY,F|F,$x^y$,http://dlmf.nist.gov/36.12.E9, -\prod,product,EMPTY,SM|F,$\Pi$,http://drmf.wmflabs.org/wiki/Definition:prod, -\pseudoJacobi{n}@{x}{\nu}{N},pseudo Jacobi polynomial,EMPTY,P|F:P,$P_{n}$,http://drmf.wmflabs.org/wiki/Definition:pseudoJacobi, -\psfactorial{a}{\kappa},partitional shifted factorial,EMPTY,F|F,${\left[a\right]_{\kappa}}$,http://dlmf.nist.gov/35.4#E1, -\pvint,Cauchy principal value,EMPTY,O|F,$\pvint_a^b$,http://dlmf.nist.gov/1.4#E24, -\qAppelli@{a}{b}{b'}{c}{q}{x}{y},first $q$-Appell function,q-Appell-Phi-1,F|F:P,$\Phi^{(1)}$,http://dlmf.nist.gov/17.4.E5, -\qAppellii@{a}{b}{b'}{c}{c'}{q}{x}{y},second $q$-Appell function,q-Appell-Phi-2,F|F:P,$\Phi^{(2)}$,http://dlmf.nist.gov/17.4.E6, -\qAppelliii@{a}{a'}{b}{b'}{c}{q}{x}{y},third $q$-Appell function,q-Appell-Phi-3,F|F:P,$\Phi^{(3)}$,http://dlmf.nist.gov/17.4.E7, -\qAppelliv@{a}{b}{c}{c'}{q}{x}{y},fourth $q$-Appell function,q-Appell-Phi-4,F|F:P,$\Phi^{(4)}$,http://dlmf.nist.gov/17.4.E8, -\qBernoulli{n}@{x}{q},$q$-Bernoulli polynomial,q-Bernoulli-polynomial,P|F:P,$\beta_{q}$,http://dlmf.nist.gov/17.3#E7, -\qBesselPoly{n}@{x}{b}{q},$q$-Bessel polynomial,EMPTY,P|F:P,$y_{n}$,http://drmf.wmflabs.org/wiki/Definition:qBessel, -\qBeta{q}@{a}{b},$q$-beta function,q-Beta,F|F:P,$\mathrm{B}_{q}$,http://dlmf.nist.gov/5.18#E11, -\qBinomial{n}{m}{q},$q$-binomial coefficient (or Gaussian polynomial),EMPTY,F|F,$\left[n \atop m\right]_{q}$,http://dlmf.nist.gov/17.2#E27,http://dlmf.nist.gov/26.9#SS2.p1 -\qCharlier{n}@{x}{c}{q},$q$-Charlier polynomial,EMPTY,P|F:P,$C_{n}$,http://drmf.wmflabs.org/wiki/Definition:qCharlier, -\qCos{q}@@{x},$q$-analogue of the $\cos$ function: ${\rm Cos}_q$,q-Cos,F|F:nP:P,$\mathrm{Cos}_{q}$,http://dlmf.nist.gov/17.3#E6, -\qCosKLS{q}@@{z},$q$-analogue of the $\cos$ function used in KLS: $\mathrm{Cos}_{q}$,q-Cos-KLS,P|F:nP:P,$\mathrm{Cos}_{q}$,http://drmf.wmflabs.org/wiki/Definition:qCosKLS, -\qDigamma{q}@{z},$q$-digamma function,q-digamma,F|F:P,$\psi_{q}$,http://drmf.wmflabs.org/wiki/Definition:qDigamma, -\qEuler{m}{s}@{q},$q$-Euler number,q-Euler-number,F|F:P,"$A_{m,n}$",http://dlmf.nist.gov/17.3#E8, -\qExp{q}@@{x},$q$-analogue of the $\exp$ function: $E_q$,q-Exp,F|F:nP:P,$E_{q}$,http://dlmf.nist.gov/17.3#E2, -\qExpKLS{q}@@{z},$q$-analogue of the $\exp$ function used in KLS: $\mathrm{E}_{q}$,q-Exp-KLS,F|F:nP:P,$\mathrm{E}_{q}$,http://drmf.wmflabs.org/wiki/Definition:qExpKLS, -\qFactorial{n}{q},$q$-factorial,q-factorial,F|F,$n!_q$,http://dlmf.nist.gov/5.18#E2, -\qGamma{q}@{z},$q$-gamma function,q-gamma,F|F:P,$\Gamma_{q}$,http://dlmf.nist.gov/5.18#E4, -\qHahn{n}@@{q^{-x}}{\alpha}{\beta}{N},$q$-Hahn polynomial,q-Hahn-polynomial-Q,P|P:PS,$Q_{n}$,http://drmf.wmflabs.org/wiki/Definition:qHahn, -\qHahnQ{n}@{x}{\alpha}{\beta}{N}{q},$q$-Hahn polynomial,q-Hahn-polynomial-Q,P|F:P,$Q_{n}$,http://dlmf.nist.gov/18.27#E3, -\qHermiteH{n}@{x}{q},continuous $q$-Hermite polynomial,continuous-q-Hermite-polynomial-H,P|F:P,$H_{n}$,http://dlmf.nist.gov/18.28#E16, -\qHermitehI{n}@{x}{q},discrete $q$-Hermite I polynomial,discrete-q-Hermite-polynomial-h-I,P|F:P,$h_{n}$,http://dlmf.nist.gov/18.27#E21, -\qHermitehII{n}@{x}{q},discrete $q$-Hermite II polynomial,discrete-q-Hermite-polynomial-h-II,P|F:P,$\tilde{h}_{n}$,http://dlmf.nist.gov/18.27#E23, -"\qHyperrWs{r}{r+1}@@@{a_1}{a_4,a_5,...,a_{r+1}}{q}{z}",very-well-poised basic hypergeometric (or $q$-hypergeometric) function,very-well-poised-q-hypergeometric-rWs,F|F:fo1:fo2:fo3,${{}_{r+1}W_{r}}$,http://drmf.wmflabs.org/wiki/Definition:qHyperrWs, -"\qHyperrphis{r}{s}@@@{a_1,...a_r}{b_1,...,b_s}{q}{z}",basic hypergeometric (or $q$-hypergeometric) function,q-hypergeometric-rphis,F|F:fo1:fo2:fo3,${{}_{r}\phi_{s}}$,http://dlmf.nist.gov/17.4#E1, -"\qHyperrpsis{r}{s}@@@{a_1,...,a_r}{b_1,...,b_s}{q}{z}",bilateral basic hypergeometric (or bilateral $q$-hypergeometric) function,q-hypergeometric-rpsis,F|F:fo1:fo2:fo3,${{}_{r}\psi_{s}}$,http://dlmf.nist.gov/17.4#E3, -\qJacobiP{n}@{x}{a}{b}{c}{q},"big $q$-Jacobi polynomial $P_n(x;a,b,c;q)$",q-Jacobi-polynomial-P,P|F:P,$P_{n}$,http://dlmf.nist.gov/18.27#E5, -\qJacobiPP{\alpha}{\beta}{n}@{x}{c}{d}{q},"big $q$-Jacobi polynomial $P_n^{(\alpha,\beta)}(x;c,d;q)$",big-q-Jacobi-polynomial-P-type-2,P|F:P,"$P^{(\alpha,\beta)}_{n}$",http://dlmf.nist.gov/18.27#E6, -\qJacobip{n}@{x}{a}{b}{q},little $q$-Jacobi polynomial,q-Jacobi-polynomial-p,P|F:P,$p_{n}$,http://dlmf.nist.gov/18.27#E13, -\qKrawtchouk{n}@@{q^{-x}}{p}{N}{q},$q$-Krawtchouk polynomial,q-Krawtchouk-polynomial-K,P|F:P:PS,$K_{n}$,http://drmf.wmflabs.org/wiki/Definition:qKrawtchouk, -\qLaguerre[\alpha]{n}@@{x}{q},$q$-Laguerre polynomial,q-Laguerre-polynomial-L,P|FnO:PnO:nPnO:F:P:PS,$L_n^{(\alpha)}$,http://drmf.wmflabs.org/wiki/Definition:qLaguerre, -\qLaguerreL[\alpha]{n}@{x}{q},$q$-Laguerre polynomial,q-Laguerre-polynomial-L,P|FnO:PnO:FO:PO,$L_n^{(\alpha)}$,http://dlmf.nist.gov/18.27#E15, -\qLegendre{n}@{x}{q},continuous $q$-Legendre polynomial,EMPTY,P|F:P,$\qLegendre{n}$,http://drmf.wmflabs.org/wiki/Definition:qLegendre, -\qMeixner{n}@{x}{b}{c}{q},$q$-Meixner polynomial,q-Meixner-polynomial-M,P|F:P,$M_{n}$,http://drmf.wmflabs.org/wiki/Definition:qMeixner, -\qMeixnerPollaczek{n}@{x}{a}{q},$q$-Meixner-Pollaczek polynomial,q-Meixner-Pollaczek-polynomial-M,P|F:P,$P_{n}$,http://drmf.wmflabs.org/wiki/Definition:qMeixnerPollaczek, -"\qMultiPochhammer{a_1,\ldots,a_n}{q}{n}",$q$-multi-Pochhammer symbol,q-multi-Pochhammer,F|F,"$\(a_1,\ldots,a_k;q)_n$",http://dlmf.nist.gov/17.2.E5, -"\qMultinomial{a_1+\ldots+a_n}{a_1,\ldots,a_n}{q}",$q$-multinomial,q-multinomial,F|F,"$\left[a_1+\ldots+a_n \atop a_1,\ldots,a_n\right]_{q}$",http://dlmf.nist.gov/26.16.E1, -\qPochhammer{a}{q}{n},$q$-Pochhammer symbol,EMPTY,F|F,$(a;q)_n$,http://dlmf.nist.gov/5.18#i,http://dlmf.nist.gov/17.2#SS1.p1 -\qPolygamma{n}{q}@{z},$q$-polygamma function,q-polygamma,F|F:P,$\psi_{q}^{(n)}$,http://drmf.wmflabs.org/wiki/Definition:qPolygamma, -\qRacah{n}@{x}{\alpha}{\beta}{\gamma}{\delta}{q},$q$-Racah polynomial,q-Racah-polynomial-R,P|F:P,$R_{n}$,http://dlmf.nist.gov/18.28#E19, -\qRacahR{n}@{x}{\alpha}{\beta}{\gamma}{\delta}{q},$q$-Racah polynomial,q-Racah-polynomial-R,P|F:P,$R_{n}$,http://dlmf.nist.gov/18.28#E19, -\qSin{q}@@{x},$q$-analogue of the $\sin$ function: ${\rm Sin}_q$,q-Sin,F|F:nP:P,$\mathrm{Sin}_{q}$,http://dlmf.nist.gov/17.3#E4, -\qSinKLS{q}@@{z},$q$-analogue of the $\sin$ function used in KLS: $\mathrm{Sin}_q$,q-Sin-KLS,F|F:nP:P,$\mathrm{Sin}_{q}$,http://drmf.wmflabs.org/wiki/Definition:qSinKLS, -\qStirling{m}{s}@{q},$q$-Stirling number,q-Stirling-number,F|F:P,"$a_{m,s}$",http://dlmf.nist.gov/17.3#E9, -\qUltraspherical{n}@{x}{\beta}{q},continuous $q$-ultraspherical polynomial,q-ultraspherical-polynomial,P|F:P,$C_{n}$,http://dlmf.nist.gov/18.28#E13, -\qcos{q}@@{x},$q$-analogue of the $\cos$ function: $\cos_q$,q-cos,F|F:nP:P,$\mathrm{cos}_{q}$,http://dlmf.nist.gov/17.3#E5, -\qcosKLS{q}@@{z},$q$-analogue of the $\cos$ function used in KLS: $\cos_q$,q-cos-KLS,P|F:nP:P,$\mathrm{cos}_{q}$,http://drmf.wmflabs.org/wiki/Definition:qcosKLS, -\qderiv[n]{q}@{z},$q$-derivative,q-derivative,O|FO:PO:FnO:PnO,$\mathcal{D}_q^n$,http://drmf.wmflabs.org/wiki/Definition:qderiv, -\qdiff{q}{x},$q$-differential,EMPTY,O|F,"${\mathrm d}^n_qx$",http://dlmf.nist.gov/17.2#SS5.p1, -\qexp{q}@@{x},$q$-analogue of the $\exp$ function: $e_q$,q-exp,F|F:nP:P,$e_{q}$,http://dlmf.nist.gov/17.3#E1, -\qexpKLS{q}@@{z},$q$-analogue of the $\exp$ function used in KLS: $\mathrm{e}_{q}$,q-exp-KLS,F|F:nP:P,$\mathrm{e}_{q}$,http://drmf.wmflabs.org/wiki/Definition:qexpKLS, -\qinvAlSalamChihara{n}@{x}{a}{b}{q^{-1}},$q$-inverse Al-Salam-Chihara polynomial,q-inverse-AlSalam-Chihara-polynomial-Q,P|F:P,$Q_{n}$,http://dlmf.nist.gov/23.1, -\qinvHermiteh{n}@{x}{q},continuous $q$-inverse Hermite polynomial,continuous-q-inverse-Hermite-polynomial-h,P|F:P,$h_{n}$,http://dlmf.nist.gov/23.1, -\qsin{q}@@{x},$q$-analogue of the $\sin$ function: $\sin_q$,q-sin,F|F:nP:P,$\mathrm{sin}_{q}$,http://dlmf.nist.gov/17.3#E3, -\qsinKLS{q}@@{z},$q$-analogue of the $\sin$ function used in KLS: $\sin_q$,q-sin-KLS,F|F:nP:P,$\mathrm{sin}_{q}$,http://drmf.wmflabs.org/wiki/Definition:qsinKLS, -\qtmqKrawtchouk{n}@@{q^{-x}}{p}{N}{q},quantum $q$-Krawtchouk polynomial,quantum-q-Krawtchouk-polynomial-K,P|F:P:PS,$K^{\mathrm{qtm}}_{n}$,http://drmf.wmflabs.org/wiki/Definition:qtmqKrawtchouk, -\realpart{z},real part,EMPTY,O|F,"$\Re {z}$",http://dlmf.nist.gov/1.9#E2, -\terminant{p}@{z},terminant function,terminant-function,F|F:P,$F_{p}$,http://dlmf.nist.gov/2.11#E11, -\rescaledTerminant{p}@{z},rescaled terminant functions,rescaled-terminant-function,F|F:P,$G_{p}$,http://dlmf.nist.gov/9.7#SS5.p1, -\rselection,right bracketed generalization of $\pm$,EMPTY,SM|F,$\left.\begin{matrix}x\end{matrix}\right\$,http://dlmf.nist.gov/28.4.E22, -\sec@@{z},secant function,secant,F|F:P:nP,$\mathrm{sec}$,http://dlmf.nist.gov/4.14#E6, -\sech@@{z},hyperbolic secant function,hyperbolic-secant,F|F:P:nP,$\mathrm{sech}$,http://dlmf.nist.gov/4.28#E6, -\selection,generalization of $\pm$,EMPTY,SM|F,$\;\begin{matrix}x\end{matrix}\;$,http://dlmf.nist.gov/10.23.E7, -\setminus,set subtraction,EMPTY,Q|F,$\setminus$,http://dlmf.nist.gov/front/introduction#Sx4.p2.t1.r19, -\setmod,$S_1 \setmod S_2$: set of all elements of $S_1$ modulo elements of $S_2$,EMPTY,SN|F,$/$,http://dlmf.nist.gov/21.1#p2.t1.r17, -\sign@@{x},sign function,sign,F|F:P:nP,$\mathrm{sign}$,http://dlmf.nist.gov/front/introduction#Sx4.p2.t1.r18, -\sim,asymptotic equality,EMPTY,Q|F,$\tilde$,http://dlmf.nist.gov/2.1#E1, -\sin@@{z},sine function,sin,F|F:P:nP,$\mathrm{sin}$,http://dlmf.nist.gov/4.14#E1, -\sinInt@{z},sine integral ${\rm si}$,shifted-sine-integral,F|F:P,$\mathrm{si}$,http://dlmf.nist.gov/6.2#E10, -\sinh@@{z},hyperbolic sine function,hyperbolic-sine,F|F:P:nP,$\mathrm{sinh}$,http://dlmf.nist.gov/4.28#E1, -\sinintg@{a}{z},generalized sine integral ${\rm si}$,generalized-sine-integral-si,F|F:P,$\mathrm{si}$,http://dlmf.nist.gov/8.21#E1, -\sixj@@{j_{1}}{j_{2}}{j_{3}}{l_{1}}{l_{2}}{l_{3}},$\sixj$ symbol,EMPTY,F|F:P:nP,$\left\{\begin{array}{ccc}j_1 & j_2 & j_3\\l_1 & l_2 & l_3\end{array}\right\}$,http://dlmf.nist.gov/23.1, -\subset,is contained in,EMPTY,Q|F,$\subset$,http://dlmf.nist.gov/front/introduction#Sx4.p2.t1.r2, -\subseteq,is in or is contained in,EMPTY,Q|F,$\subseteq$,http://dlmf.nist.gov/front/introduction#Sx4.p2.t1.r3, -\sum,sum,EMPTY,SM|F,$\Sigma$,http://drmf.wmflabs.org/wiki/Definition:sum, -\sup,least upper bound (supremum),EMPTY,Q|F,$\mathrm{sup}$,http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r25, -\tan@@{z},tangent function,tangent,F|F:P:nP,$\mathrm{tan}$,http://dlmf.nist.gov/4.14#E4, -\tanh@@{z},hyperbolic tangent function,hyperbolic-tangent,F|F:P:nP,$\mathrm{tanh}$,http://dlmf.nist.gov/4.28#E4, -\threej@@{j_1}{j_2}{j_3}{m_1}{m_2}{m_3},$\threej$ symbol,EMPTY,F|F:P:nP,$\left(\begin{array}{ccc}j_1 & j_2 & j_3\\m_1 & m_2 & m_3\end{array}\right)$,http://dlmf.nist.gov/34.2#E4, -\trace,trace of matrix,trace,L|F,$\mathrm{etr}$,http://dlmf.nist.gov/23.1, -\transpose,transpose of a matrix,EMPTY,L|F,$A^{\mathrm{T}}$,http://dlmf.nist.gov/1.3.E5, -\weight{w}@{x},weight function,EMPTY,SM|F:P,${w}$,http://drmf.wmflabs.org/wiki/Definition:weight, -\wigner{j_1}{m_1}{j_2}{m_2}{j_3}{-m_3},Clebsch-Gordan coefficient,EMPTY,F|F,"$\left(j_{1}\;m_{1}\;j_{2}\;m_{2} | j_{1}\;j_{2}\; j_{3}\,\,-m_{3}\right)$",http://dlmf.nist.gov/34.1#p4, -\BesselPolyIIparam{n}@{x}{a}{b},Bessel polynomial with two parameters $y_n$,Bessel-polynomial-two-parameters-y,P|F:P,$y_{n}$,http://drmf.wmflabs.org/wiki/Definition:BesselPolyIIparam, -\BesselPolyTheta{n}@{x}{a}{b},Bessel polynomial with two parameters $\theta_n$,Bessel-polynomial-two-parameters-theta,P|F:P,$\theta_{n}$,http://drmf.wmflabs.org/wiki/Definition:BesselPolyTheta, -\FourierTrans@{f}{s},Fourier transform,Fourier-transform,O|F:P,$\mathscr{F}$,http://dlmf.nist.gov/1.14#E1, -\Continuous@{A},space of continuous functions on a set $A$,set-of-continuous-functions,S|F,$C(A)$,http://dlmf.nist.gov/1.4#ii, -\SpaceTestFunTempered@{A},space of test functions for tempered distributions,space-of-test-functions-for-tempered-distributions-T,S|F,${\mathcal T}(\Real)$,http://dlmf.nist.gov/1.16#v, From 3c1b2dad9b25954f3960da64f7b0bfe144b769d6 Mon Sep 17 00:00:00 2001 From: Edward Bian Date: Thu, 14 Apr 2016 17:53:42 -0400 Subject: [PATCH 076/402] Fixed up paragraph placement and reading --- KLSadd_insertion/updateChapters.py | 95 +++++++++++++++++++++--------- 1 file changed, 66 insertions(+), 29 deletions(-) diff --git a/KLSadd_insertion/updateChapters.py b/KLSadd_insertion/updateChapters.py index 944ae1e..0a5b2ee 100644 --- a/KLSadd_insertion/updateChapters.py +++ b/KLSadd_insertion/updateChapters.py @@ -85,19 +85,26 @@ def insertCommands(kls, chap, cms): def findReferences(chapter): references = [] index = -1 + chaptercheck = 0 + if chapticker == 0: + chaptercheck = str(9) + elif chapticker == 1: + chaptercheck = str(14) canAdd = False for word in chapter: index+=1 #check sections and subsections - if("section{" in word or "subsection*{" in word): + if("section{" in word or "subsection*{" in word) and ("subsubsection*{" not in word): w = word[word.find("{")+1: word.find("}")] + ws = word[word.find("{")+1: word.find("~")] for unit in mathPeople: - if (w in unit): + subunit = unit[unit.find(" ")+1: unit.find("#")] + if ((w in subunit) or (ws in subunit)) and (chaptercheck in unit) and (len(w) == len(subunit)) or (("Pseudo Jacobi" in w) and ("Pseudo Jacobi (or Routh-Romanovski)" in subunit)): canAdd = True - if("\\subsection*{References}" in word) and (canAdd == True): references.append(index) canAdd = False + return references #method to change file string(actually a list right now), returns string to be written to file @@ -106,33 +113,53 @@ def fixChapter(chap, references, p, kls): #chap is the file string(actually a list), references is the specific references for the file, #and p is the paras variable(not sure if needed) kls is the KLSadd.tex as a list count = 0 #count is used to represent the values in count + designator = 0 #Tells which chapter it's on + if chapticker2 == 0: + designator = "9." + elif chapticker2 == 1: + designator = "14." + designator = str(designator) for i in references: #Place before References paragraph - chap[i-2] += "%Begin KLSadd additions" - chap[i-2] += p[count] - chap[i-2] += "%End of KLSadd additions" - count+=1 - + if count > 34: + word1 = "14. Banana" + else: + word1 = str(p[count]) + if (designator in word1[word1.find("\\subsection*{") + 1: word1.find("}") ]): + chap[i-2] += "%Begin KLSadd additions" + chap[i-2] += p[count] + chap[i-2] += "%End of KLSadd additions" + count += 1 + else: + while (designator not in word1[word1.find("\\subsection*{") + 1: word1.find("}") ]): + word1 = str(p[count]) + if (designator in word1[word1.find("\\subsection*{") + 1: word1.find("}") ]): + chap[i - 2] += "%Begin KLSadd additions" + chap[i - 2] += p[count] + chap[i - 2] += "%End of KLSadd additions" + count += 1 + else: + + count+=1 chap = prepareForPDF(chap) cms = getCommands(kls) chap = insertCommands(kls,chap, cms) commentticker = 0 for word in chap: - if ("\\newcommand\\half{\\frac12}" in word): - wordtoadd = "%" + word - chap[commentticker] = wordtoadd - elif ("\\newcommand{\\hyp}[5]{\\,\\mbox{}_{#1}F_{#2}\\!\\left(" in word): - wordtoadd = "%" + word - chap[commentticker] = wordtoadd - elif ("\\genfrac{}{}{0pt}{}{#3}{#4};#5\\right)}" in word): - wordtoadd = "%" + word - chap[commentticker] = wordtoadd - elif ("\\newcommand{\\qhyp}[5]{\\,\\mbox{}_{#1}\\phi_{#2}\\!\\left(" in word): - wordtoadd = "%" + word - chap[commentticker] = wordtoadd - elif ("\\genfrac{}{}{0pt}{}{#3}{#4};#5\\right)}" in word): - wordtoadd = "%" + word - chap[commentticker] = wordtoadd + word2 = chap[chap.index(word)-1] + if ("\\newcommand{\qhypK}[5]{\,\mbox{}_{#1}\phi_{#2}\!\left(" not in word2): + if ("\\newcommand\\half{\\frac12}" in word): + wordtoadd = "%" + word + chap[commentticker] = wordtoadd + elif ("\\newcommand{\\hyp}[5]{\\,\\mbox{}_{#1}F_{#2}\\!\\left(" in word): + wordtoadd = "%" + word + chap[commentticker] = wordtoadd + elif ("\\genfrac{}{}{0pt}{}{#3}{#4};#5\\right)}" in word): + wordtoadd = "%" + word + chap[commentticker] = wordtoadd + elif ("\\newcommand{\\qhyp}[5]{\\,\\mbox{}_{#1}\\phi_{#2}\\!\\left(" in word): + wordtoadd = "%" + word + chap[commentticker] = wordtoadd commentticker += 1 # Hopefully this works ticker1 = 0 @@ -149,19 +176,25 @@ def fixChapter(chap, references, p, kls): addendum = add.readlines() for word in addendum: if ("paragraph{" in word): - temp = word[0:word.find("{") + 1] + "\large\\bf KLSadd: " + word[word.find("{") + 1: word.find("}") + 1] - addendum[addendum.index(word): addendum.index(word) + 1] = temp + lenword = len(word) - 1 + temp = word[0:word.find("{") + 1] + "\large\\bf KLSadd: " + word[word.find("{") + 1: lenword] + addendum[addendum.index(word)] = temp + if ("subsubsection*{" in word): + lenword = len(word) - 1 + addendum[addendum.index(word)] = word[0:word.find("{") + 1] + "\large\\bf KLSadd: " + word[word.find("{") + 1: lenword] index = 0 indexes = [] for word in addendum: index+=1 if("." in word and "\\subsection*{" in word): - name = word[word.find(" "): word.find("}")] - mathPeople.append(name) - if("9." in word): + if ("9." in word): chapNums.append(9) + name = word[word.find("{") + 1: word.find("}") ] + mathPeople.append(name + "#") if("14." in word): chapNums.append(14) + name = word[word.find("{") + 1: word.find("}") ] + mathPeople.append(name + "#") #get the index indexes.append(index-1) #now indexes holds all of the places there is a section @@ -169,6 +202,7 @@ def fixChapter(chap, references, p, kls): for i in range(len(indexes)-1): box = ''.join(addendum[indexes[i]: indexes[i+1]-1]) paras.append(box) + paras.append("% This is a test") #paras now holds the paragraphs that need to go into the chapter files, but they need to go in the appropriate #section(like Wilson, Racah, Hahn, etc.) so we use the mathPeople variable #we can use the section names to place the relevant paragraphs in the right place @@ -187,11 +221,14 @@ def fixChapter(chap, references, p, kls): ch14.close() #call the findReferences method to find the index of the References paragraph in the two file strings #two variables for the references lists one for chapter 9 one for chapter 14 + chapticker = 0 references9 = findReferences(entire9) + chapticker += 1 references14 = findReferences(entire14) - #call the fixChapter method to get a list with the addendum paragraphs added in + chapticker2 = 0 str9 = ''.join(fixChapter(entire9, references9, paras, addendum)) + chapticker2 += 1 str14 = ''.join(fixChapter(entire14, references14, paras, addendum)) """ From c4915f9e448f0fab181c4155647046af6cd40255 Mon Sep 17 00:00:00 2001 From: Moritz Schubotz Date: Fri, 15 Apr 2016 17:07:40 -0400 Subject: [PATCH 077/402] Create CONTRIBUTING #Closes #8 --- CONTRIBUTING | 2 ++ 1 file changed, 2 insertions(+) create mode 100644 CONTRIBUTING diff --git a/CONTRIBUTING b/CONTRIBUTING new file mode 100644 index 0000000..93c00da --- /dev/null +++ b/CONTRIBUTING @@ -0,0 +1,2 @@ +Plese use pull request to contribute to the project. +Check out our guide at http://drmf.wmflabs.org/wiki/GitHub how to use git. From e979e36d60328c12d782757cdf6564a7787c2b93 Mon Sep 17 00:00:00 2001 From: Moritz Schubotz Date: Fri, 15 Apr 2016 17:09:38 -0400 Subject: [PATCH 078/402] Update CONTRIBUTING provide an example for a good pull request --- CONTRIBUTING | 11 +++++++++++ 1 file changed, 11 insertions(+) diff --git a/CONTRIBUTING b/CONTRIBUTING index 93c00da..d834a55 100644 --- a/CONTRIBUTING +++ b/CONTRIBUTING @@ -1,2 +1,13 @@ Plese use pull request to contribute to the project. Check out our guide at http://drmf.wmflabs.org/wiki/GitHub how to use git. + +## Example +See the following pull request: + +https://github.com/DRMF/DRMF-Seeding-Project/pull/19 + +This pull request satisfies the following properties: +1. It is small (incremental) +2. It improves the readability +3. It adds more tests +4. It demonstrates how to add issues for aspects that can be fixed later From 11a777973978bd0fb004d5898968c9790acf8013 Mon Sep 17 00:00:00 2001 From: Moritz Schubotz Date: Fri, 15 Apr 2016 17:23:26 -0400 Subject: [PATCH 079/402] Rename CONTRIBUTING to CONTRIBUTING.md Enable the use markdown --- CONTRIBUTING => CONTRIBUTING.md | 0 1 file changed, 0 insertions(+), 0 deletions(-) rename CONTRIBUTING => CONTRIBUTING.md (100%) diff --git a/CONTRIBUTING b/CONTRIBUTING.md similarity index 100% rename from CONTRIBUTING rename to CONTRIBUTING.md From 09650b53ede32e061b064c6d42a08aed900951e1 Mon Sep 17 00:00:00 2001 From: pzw Date: Wed, 20 Apr 2016 17:58:42 -0400 Subject: [PATCH 080/402] Fix spelling and code convention issues This fixes the issues identified by the code analysis as described in #17. --- AlexDanoff/src/conversion_ecf.py | 24 ++--- Azeem/src/tex2Wiki.py | 1 + KLSadd_insertion/FileChangerLatex.py | 128 +++++++++++++-------------- 3 files changed, 77 insertions(+), 76 deletions(-) diff --git a/AlexDanoff/src/conversion_ecf.py b/AlexDanoff/src/conversion_ecf.py index 4ea201a..801ef49 100644 --- a/AlexDanoff/src/conversion_ecf.py +++ b/AlexDanoff/src/conversion_ecf.py @@ -327,7 +327,7 @@ def markup(file): if location==-1: location=len(equation) - newFile=newFile+"\\begin{equation} \n" + equation[1:location] + "% \\constraint{\n" + newFile=newFile+"\\begin{equation} \n" + equation[1:location] + "% \\constraints{\n" #find constraints c=findArgs(equation,"Element") @@ -340,7 +340,7 @@ def markup(file): z=element[1:location].strip() elif element.count("|") >0: for index in range(element.count("|")+1): - #if it is the first constraint then it is from beginning to the first location of the |, the loc is increased each time to get through all of the constraints + #if it is the first constraints then it is from beginning to the first location of the |, the loc is increased each time to get through all of the constraints if index==0: loc=element.find("|") constraints.append(element[1:loc]) @@ -353,23 +353,23 @@ def markup(file): print(var) for index in range(len(constraints)): - print(constraint) + print(constraints) if (var=="Complexes"): - newFile="% "+ newFile+ consraint+ "\in \Complex" + newFile="% "+ newFile+ constraints+ "\in \Complex" elif (var=="Wholes"): - newFile="% "+ newFile+ consraint+ "\in \\NonNegInteger" + newFile="% "+ newFile+ constraints+ "\in \\NonNegInteger" elif (var=="Naturals"): - newFile="% "+ newFile+ consraint+ "\in \\NatNumber" + newFile="% "+ newFile+ constraints+ "\in \\NatNumber" elif (var=="Integers"): - newFile="% "+ newFile+ consraint+ "\in \Integer" + newFile="% "+ newFile+ constraints+ "\in \Integer" elif (var=="Irrationals"): - newFile="% "+ newFile+ consraint+ "\in \Irrationals" + newFile="% "+ newFile+ constraints+ "\in \Irrationals" elif (var=="Reals"): - newFile="% "+ newFile+ consraint+ "\in \Real" + newFile="% "+ newFile+ constraints+ "\in \Real" elif (var=="Rational"): - newFile="% "+ newFile+ consraint+ "\in \Rational" + newFile="% "+ newFile+ constraints+ "\in \Rational" elif (var=="Primes"): - newFile="% "+ newFile+ consraint+ "\in \Prime" + newFile="% "+ newFile+ constraints+ "\in \Prime" return newFile def equationSetUp(s): @@ -383,7 +383,7 @@ def equationSetUp(s): constraints=[] if (s.find("Element["))>0: elements=s[s.find("Element[",end)+ len("Element["): len(s)-1] - newFile=newFile+"% \constraint{\n" + newFile=newFile+"% \constraints{\n" numCon=elements.count("|")+1 location=0 for index in range(numCon): diff --git a/Azeem/src/tex2Wiki.py b/Azeem/src/tex2Wiki.py index a89ab01..1180a38 100644 --- a/Azeem/src/tex2Wiki.py +++ b/Azeem/src/tex2Wiki.py @@ -830,6 +830,7 @@ def readin(ofname,glossary,mmd): pauseP = False for ind in range(0, len(line)): if line[ind:ind + 7] == "\\eqref{": + rLab = getString(eqR) pause = True eInd = refLabels.index("" + label) z = line[line.find("}", ind + 7) + 1] diff --git a/KLSadd_insertion/FileChangerLatex.py b/KLSadd_insertion/FileChangerLatex.py index 1ef75ff..3ff483e 100644 --- a/KLSadd_insertion/FileChangerLatex.py +++ b/KLSadd_insertion/FileChangerLatex.py @@ -1,64 +1,64 @@ -""" -Rahul Shah -11/20/15 -FileChangerLatex.py -Reads in from KLSadd.tex, finds places to be changed and adds -to end of section -""" - -#Open the document to be read -#Figure out which chapter to read - -subsection = "" -a = -1 -chapter ="" -list = [] -with open("KLSadd.tex", "rb") as file: - for line in file: - #finds all mentions of subsections in KLSadd - #used to determine chapter number - #tests every subsection to find if it has a number - #eliminates Introduction, etc - chapter = "" - subsection = "" - toCopy = False - if("\\subsection*{" in line): - #finds the subsection and subsubsections - for word in line: - try: - chapter+= str(int(word)) - except: - a = -1 - - if(word =="."): - chapter += "." - - if(chapter is not ""): - list.append(chapter) - #END OF LOOP why can't I have nice brackets like java :( - """ - list contains all of the subsections. Begin chapter file searching - """ - words = "" - for line2 in file: - words += line2 - - print(words) - numFile = "" - fileList = [] - for s in list: - numFile = "" - if("." in s[0:2]): - numFile = s[0] - else: - numFile = s[0:2] - fileList.append(int(numFile)) - - #numFile now contains the right number of the file to access - #stored in fileList - - - - - file.close() -#end of FileChangerLatex.py +""" +Rahul Shah +11/20/15 +FileChangerLatex.py +Reads in from KLSadd.tex, finds places to be changed and adds +to end of section +""" + +#Open the document to be read +#Figure out which chapter to read + +subsection = "" +a = -1 +chapter ="" +list = [] +with open("KLSadd.tex", "rb") as file: + for line in file: + #finds all mentions of subsections in KLSadd + #used to determine chapter number + #tests every subsection to find if it has a number + #eliminates Introduction, etc + chapter = "" + subsection = "" + toCopy = False + if("\\subsection*{" in line): + #finds the subsection and subsubsections + for word in line: + try: + chapter+= str(int(word)) + except: + a = -1 + + if(word =="."): + chapter += "." + + if(chapter is not ""): + list.append(chapter) + #END OF LOOP why can't I have nice brackets like java :( + """ + list contains all of the subsections. Begin chapter file searching + """ + words = "" + for line2 in file: + words += line2 + + print(words) + numFile = "" + fileList = [] + for s in list: + numFile = "" + if("." in s[0:2]): + numFile = s[0] + else: + numFile = s[0:2] + fileList.append(int(numFile)) + + #numFile now contains the right number of the file to access + #stored in fileList + + + + + file.close() +#end of FileChangerLatex.py From 69f6727f638ad42ce40d9f2ef7b174ac607d0616 Mon Sep 17 00:00:00 2001 From: physikerwelt Date: Fri, 22 Apr 2016 17:19:23 -0400 Subject: [PATCH 081/402] Add exception handler as reported in https://www.quantifiedcode.com/app/project/gh:DRMF:DRMF-Seeding-Project/diff/265d1e606c7e3e4eafba11ca6a08ea6b40a053b2/d3b969ab2dc03c4e83fe3c074a83c3a400db3995/issues?groups=code_patterns%3A3JwOg9ad%3Af0,code_patterns%3A4m6i5Dpz --- KLSadd_insertion/FileChangerLatex.py | 90 ++++++++++++++-------------- 1 file changed, 45 insertions(+), 45 deletions(-) diff --git a/KLSadd_insertion/FileChangerLatex.py b/KLSadd_insertion/FileChangerLatex.py index 3ff483e..9bbbb2d 100644 --- a/KLSadd_insertion/FileChangerLatex.py +++ b/KLSadd_insertion/FileChangerLatex.py @@ -6,59 +6,59 @@ to end of section """ -#Open the document to be read -#Figure out which chapter to read +# Open the document to be read +# Figure out which chapter to read subsection = "" a = -1 -chapter ="" +chapter = "" list = [] with open("KLSadd.tex", "rb") as file: - for line in file: - #finds all mentions of subsections in KLSadd - #used to determine chapter number - #tests every subsection to find if it has a number - #eliminates Introduction, etc - chapter = "" - subsection = "" - toCopy = False - if("\\subsection*{" in line): - #finds the subsection and subsubsections - for word in line: - try: - chapter+= str(int(word)) - except: - a = -1 - - if(word =="."): - chapter += "." + for line in file: + # finds all mentions of subsections in KLSadd + # used to determine chapter number + # tests every subsection to find if it has a number + # eliminates Introduction, etc + chapter = "" + subsection = "" + toCopy = False + if "\\subsection*{" in line: + # finds the subsection and subsubsections + for word in line: + try: + chapter += str(int(word)) + except ValueError: + a = -1 - if(chapter is not ""): - list.append(chapter) - #END OF LOOP why can't I have nice brackets like java :( - """ - list contains all of the subsections. Begin chapter file searching - """ - words = "" - for line2 in file: - words += line2 + if word == ".": + chapter += "." - print(words) + if chapter is not "": + list.append(chapter) + # END OF LOOP why can't I have nice brackets like java :( + """ + list contains all of the subsections. Begin chapter file searching + """ + words = "" + for line2 in file: + words += line2 + + print(words) + numFile = "" + fileList = [] + for s in list: numFile = "" - fileList = [] - for s in list: - numFile = "" - if("." in s[0:2]): - numFile = s[0] - else: - numFile = s[0:2] - fileList.append(int(numFile)) + if ("." in s[0:2]): + numFile = s[0] + else: + numFile = s[0:2] + fileList.append(int(numFile)) + + # numFile now contains the right number of the file to access + # stored in fileList + - #numFile now contains the right number of the file to access - #stored in fileList - - - file.close() -#end of FileChangerLatex.py + file.close() +# end of FileChangerLatex.py From 19104ff91de0aba6aa586f478d4d4ca5598e50f0 Mon Sep 17 00:00:00 2001 From: physikerwelt Date: Fri, 22 Apr 2016 17:45:05 -0400 Subject: [PATCH 082/402] Compare repository tex2Wiki and local tex2Wiki Add new code into tex2Wiki (local version) Fix unnecessary parentheses Get rid of unnecessary comments --- Azeem/src/tex2Wiki.py | 62 +++++++++++++++++++++++++++++++++---------- 1 file changed, 48 insertions(+), 14 deletions(-) diff --git a/Azeem/src/tex2Wiki.py b/Azeem/src/tex2Wiki.py index 1180a38..aa6600d 100644 --- a/Azeem/src/tex2Wiki.py +++ b/Azeem/src/tex2Wiki.py @@ -1,4 +1,4 @@ -# Version 14: Templates +# Version 15: Minor Fixes # Convert tex to wikiText import csv # imported for using csv format import sys # imported for getting args @@ -13,7 +13,7 @@ root = ET.Element('{http://www.mediawiki.org/xml/export-0.10/}mediawiki') -def isnumber(char): +def isnumber(char): # Function to check if char is a number (assuming 1 character) return char[0] in "0123456789" @@ -165,7 +165,7 @@ def getSym(line): # Gets all symbols on a line for symbols list p = "" else: p = line[i + 1] - if cC == 0 and p != "{" and p != "[" and p != "@": + if cC <= 0 and p != "{" and p != "[" and p != "@": symFlag = False # if "monicAskeyWilson{n+1}@{x}{a}{b}{c}{d}{q}+\\frac{1}{2}" in line: symList.append(symbol) @@ -175,7 +175,7 @@ def getSym(line): # Gets all symbols on a line for symbols list elif c == "\\": symFlag = True - elif c == "&": + elif c == "&" and not (i > line.find("\\begin{array}") and i < line.find("\\end{array}")): symList.append("&") # if "MeixnerPollaczek{\\lambda}{n}@{x}{\\phi}=\frac{\\pochhammer{2\\lambda}{n}}" in line: @@ -294,12 +294,18 @@ def setup_label_links(ofname): def readin(ofname,glossary,mmd): # try: for jsahlfkjsd in range(0, 1): - global wiki tex = open(ofname, 'r') main_file = open(mmd, "r") mainText = main_file.read() mainPrepend = "" mainWrite = open("OrthogonalPolynomials.mmd.new", "w") + tester = open("testData.txt", 'w') + # glossary=open('Glossary', 'r') + glossary = open('new.Glossary.csv', 'rb') + gCSV = csv.reader(glossary, delimiter=',', quotechar='\"') + # lLinks=open('BruceLabelLinks', 'r') + lGlos = glossary.readlines() + # lLink=lLinks.readlines() math = False constraint = False substitution = False @@ -351,7 +357,7 @@ def readin(ofname,glossary,mmd): sections.append(["Orthogonal Polynomials", 0]) chapter = getString(line) mainPrepend += ( - "\n== Sections in " + chapter + " ==\n\n
\n") + "\n== Sections in " + chapter + " ==\n\n
\n") elif "\\part" in line: if getString(line) == "BOF": parse = False @@ -468,6 +474,13 @@ def readin(ofname,glossary,mmd): subLine += line.replace("&", "&
") if "\\end{equation}" in lines[i + 1] or "\\substitution" in lines[i + 1] or "\\constraint" in lines[ i + 1] or "\\drmfn" in lines[i + 1] or "\\proof" in lines[i + 1]: + lineR = "" + for i in range(0, len(subLine)): + if subLine[i] == "&" and not ( + i > subLine.find("\\begin{array}") and i < subLine.find("\\end{array}")): + lineR += "&
" + else: + lineR += subLine[i] substitution = False append_text("
Substitution(s): " + getEq(subLine) + "

\n") @@ -663,6 +676,9 @@ def readin(ofname,glossary,mmd): parCx = 0 parFlag = False cC = 0 + ind = G[0].find("@") + if ind == -1: + ind = len(G[0]) - 1 for z in G[0]: if z == "@": parFlag = True @@ -697,9 +713,7 @@ def readin(ofname,glossary,mmd): if len(Q) > len(symbol) and (Q[len(symbol)] == "{" or Q[len(symbol)] == "["): ap = "" for o in range(len(symbol), len(Q)): - if Q[o] == "{" or z == "[": - pass - elif Q[o] == "}" or z == "]": + if Q[o] == "}" or z == "]": listArgs.append(ap) ap = "" else: @@ -751,7 +765,9 @@ def readin(ofname,glossary,mmd): q = r.find("KLS:") + 4 p = r.find(":", q) section = r[q:p] - # Where should it link to? + equation = r[p + 1:] + if equation.find(":") != -1: + equation = equation[0:equation.find(":")] append_text("[http://homepage.tudelft.nl/11r49/askey/contents.html " "Equation in Section " + section + "] of [[Bibliography#KLS|'''KLS''']].\n\n") append_text("== URL links ==\n\nWe ask users to provide relevant URL links in this space.\n\n") @@ -828,11 +844,13 @@ def readin(ofname,glossary,mmd): proofLine = "" pause = False pauseP = False + eqR = '' for ind in range(0, len(line)): if line[ind:ind + 7] == "\\eqref{": - rLab = getString(eqR) + # TODO: figure out how eqR is defined + # rLab = getString(eqR) pause = True - eInd = refLabels.index("" + label) + eInd = refLabels.index("" + label) # This should be rLab z = line[line.find("}", ind + 7) + 1] if z == "." or z == ",": pauseP = True @@ -870,6 +888,8 @@ def readin(ofname,glossary,mmd): for ind in range(0, len(line)): if line[ind:ind + 7] == "\\eqref{": pause = True + eqR = line[ind:line.find("}", ind) + 1] + rLab = getString(eqR) eInd = refLabels.index("" + label) z = line[line.find("}", ind + 7) + 1] if z == "." or z == ",": @@ -918,6 +938,14 @@ def readin(ofname,glossary,mmd): if "\\end{equation}" in lines[i + 1] or "\\drmfn" in lines[i + 1] or "\\constraint" in lines[ i + 1] or "\\substitution" in lines[i + 1] or "\\proof" in lines[i + 1]: note = False + if "\\emph" in noteLine: + noteLine = noteLine[0:noteLine.find("\\emph{")] + "\'\'" + noteLine[noteLine.find("\\emph{") + len( + "\\emph{"):noteLine.find("}", noteLine.find("\\emph{") + len("\\emph{"))] + "\'\'" + noteLine[ + noteLine.find( + "}", + noteLine.find( + "\\emph{") + len( + "\\emph{")) + 1:] comToWrite = comToWrite + "
" + getEq(noteLine) + "

\n" if constraint and parse: @@ -933,15 +961,21 @@ def readin(ofname,glossary,mmd): append_text(comToWrite + "
" + getEq(conLine) + "

\n") comToWrite = "" if substitution and parse: - subLine += line.replace("&", "&
") + subLine += line.replace("&", "&
") #TODO: Figure out if .replace is needed symLine += line.strip("\n") - # symbols=symbols+getSym(line) if "\\end{equation}" in lines[i + 1] or "\\drmfn" in lines[i + 1] or "\\substitution" in lines[ i + 1] or "\\constraint" in lines[i + 1] or "\\proof" in lines[i + 1]: substitution = False symbols = symbols + getSym(symLine) symLine = "" + lineR = "" + for i in range(0, len(subLine)): + if subLine[i] == "&" and not ( + i > subLine.find("\\begin{array}") and i < subLine.find("\\end{array}")): + lineR += "&
" + else: + lineR += subLine[i] append_text(comToWrite + "
" + getEq(subLine) + "

\n") comToWrite = "" From 0f3f89af13d41351aba2721cc2b5c26bb0233372 Mon Sep 17 00:00:00 2001 From: physikerwelt Date: Fri, 22 Apr 2016 17:51:21 -0400 Subject: [PATCH 083/402] Fix warnings from Landscape https://landscape.io/diff/332573 --- Azeem/src/tex2Wiki.py | 12 +++--------- 1 file changed, 3 insertions(+), 9 deletions(-) diff --git a/Azeem/src/tex2Wiki.py b/Azeem/src/tex2Wiki.py index aa6600d..3c5200d 100644 --- a/Azeem/src/tex2Wiki.py +++ b/Azeem/src/tex2Wiki.py @@ -299,13 +299,7 @@ def readin(ofname,glossary,mmd): mainText = main_file.read() mainPrepend = "" mainWrite = open("OrthogonalPolynomials.mmd.new", "w") - tester = open("testData.txt", 'w') - # glossary=open('Glossary', 'r') glossary = open('new.Glossary.csv', 'rb') - gCSV = csv.reader(glossary, delimiter=',', quotechar='\"') - # lLinks=open('BruceLabelLinks', 'r') - lGlos = glossary.readlines() - # lLink=lLinks.readlines() math = False constraint = False substitution = False @@ -844,7 +838,6 @@ def readin(ofname,glossary,mmd): proofLine = "" pause = False pauseP = False - eqR = '' for ind in range(0, len(line)): if line[ind:ind + 7] == "\\eqref{": # TODO: figure out how eqR is defined @@ -888,8 +881,9 @@ def readin(ofname,glossary,mmd): for ind in range(0, len(line)): if line[ind:ind + 7] == "\\eqref{": pause = True - eqR = line[ind:line.find("}", ind) + 1] - rLab = getString(eqR) + # TODO: Figure out how this is used + # eqR = line[ind:line.find("}", ind) + 1] + # rLab = getString(eqR) eInd = refLabels.index("" + label) z = line[line.find("}", ind + 7) + 1] if z == "." or z == ",": From 7e09a6cec3e5e1777fa01a3fe29eaba3f2427c6c Mon Sep 17 00:00:00 2001 From: svellala2001 Date: Tue, 26 Apr 2016 12:17:20 -0400 Subject: [PATCH 084/402] Correct Problem with \iunit (#4) * Correct Problem with \iunit * Create new python file to replace units using Jagan's math mode program * Don't call main when program is included * Change replace_i with replace_special * Fix code to work with unit test * Fix Quantified Code issues in replace_special.py and test__replace_i.py * Find and fix problem with reading files * Find and fix problem with math mode not recognizing \label{ * Polish code and make sure code follows Quantifed Code Standards * Found and fix error with \iunit replacement concerning surrounding characters --- AlexDanoff/src/math_mode.py | 159 +++++++++++++++++ AlexDanoff/src/replace_special.py | 275 ++++++++--------------------- AlexDanoff/test/test__replace_i.py | 8 +- 3 files changed, 235 insertions(+), 207 deletions(-) create mode 100644 AlexDanoff/src/math_mode.py diff --git a/AlexDanoff/src/math_mode.py b/AlexDanoff/src/math_mode.py new file mode 100644 index 0000000..16954f8 --- /dev/null +++ b/AlexDanoff/src/math_mode.py @@ -0,0 +1,159 @@ +math_start = {"\\[": "\\]", + "\\(": "\\)", + "$$": "$$", + "$": "$", + "\\begin{equation}": "\\end{equation}", + "\\begin{equation*}": "\\end{equation*}", + "\\begin{align}": "\\end{align}", + "\\begin{align*}": "\\end{align*}", + "\\begin{multline}": "\\end{multline}", + "\\begin{multline*}": "\\end{multline*}"} + +math_end = ["\\hbox{", + "\\mbox{", + "\\text{"] + + +def find_first(string, delim, start=0): + # type: (str, str, int) -> int + """ + Finds the first occurrence of a delimiter within a string. + :param string: The string to search within. + :param delim: The delimiter to search for. + :param start: The starting index in the string. + :return: The index of the first occurrence. + """ + i = string.find(delim, start) + if delim in ["$", "\\[", "\\("]: + if i == -1: + return -1 + elif i != 0 and string[i - 1] == "\\": + return find_first(string, delim, i + len(delim)) + return i + + +def first_delim(string, enter=True): + # type: (str, bool) -> str + """ + Finds the first math or text mode delimiter within a string. + :param string: The string to search within. + :param enter: Whether to find a math mode or a text mode delimiter. + :return: The delimiter that first appears in the string. + """ + minm = ["\\hbox{", "\\]"][enter] + for delim in [math_end, math_start][enter]: + i = find_first(string, delim) + if i != -1 and i < find_first(string, minm) or minm not in string: + minm = delim + if minm == "$" and string.find("$$") == string.find("$"): + minm = "$$" + return minm + + +def does_exit(string): + # type: (str) -> bool + """ + Returns whether a string starts with a text mode delimiter. + :param string: The string to check. + :return: Whether the string starts with a text mode delimiter. + """ + return any(string.startswith(delim) for delim in math_end) + + +def does_enter(string): + # type: (str) -> bool + """ + Returns whether a string starts with a math mode delimiter. + :param string: The string to check. + :return: Whether the string starts with a math mode delimiter. + """ + return any(string.startswith(delim) for delim in math_start) + + +def parse_math(string, start, ranges): + # type: (str) -> str, int + """ + Finds the ranges of the current math mode segment. + :param string: The string to parse. + :param start: The starting index in the original string. + :param ranges: The ranges of all math mode segments. + :return i: The length of the math mode segment. + """ + delim = first_delim(string) + i = len(delim) + begin = start + i + while i < len(string): + if string[i:].startswith("\\$"): + i += 1 + elif string[i:].startswith("\\\\]"): + i += 2 + elif string[i:].startswith("\\\\)"): + i += 2 + else: + if does_exit(string[i:]): + if begin != start + i: + ranges.append((begin, start + i)) + i += parse_non_math(string[i:], start + i, ranges) + begin = start + i + if string[i:].startswith(math_start[delim]): + if begin != start + i: + ranges.append((begin, start + i)) + return i + len(math_start[delim]) - 1 + i += 1 + return i + + +def parse_non_math(string, start, ranges): + # type: (str) -> str, int + """ + Finds occurrences of math mode within a segment of text mode. + :param string: The string to parse. + :param start: The starting index in the original string. + :param ranges: The ranges of all math mode segments. + :return i: The length of the text mode segment. + """ + delim = first_delim(string, False) + if not string.startswith(delim): + delim = "" + level = 0 + i = len(delim) + while i < len(string): + if string[i:].startswith("\\$"): + i += 1 + elif string[i:].startswith("\\\\["): + i += 2 + elif string[i:].startswith("\\\\("): + i += 2 + elif does_enter(string[i:]): + i += parse_math(string[i:], start + i, ranges) + elif string[i] == "{": + level += 1 + elif string[i] == "}": + if level == 0 and delim != "": + i += 1 + return i + else: + level -= 1 + i += 1 + return i + + +def find_math_ranges(string,drmf): + # type: (str) -> list + """ + Returns a list of tuples, each tuple denoting a range of math mode. + :param string: The string to search within. + :return: A list of tuples denoting math mode ranges. + """ + global math_end + if drmf == True: + string = string.replace("\\drmfnote","\\drmfname") + math_end.extend(["\\constraint{", + "\\substitution" + "\\drmfnote{", + "\\drmfname{", + "\\proof{", + "\\label{"]) + ranges = [] + parse_non_math(string, 0, ranges) + return ranges[:] \ No newline at end of file diff --git a/AlexDanoff/src/replace_special.py b/AlexDanoff/src/replace_special.py index c81f5ab..22f4799 100644 --- a/AlexDanoff/src/replace_special.py +++ b/AlexDanoff/src/replace_special.py @@ -1,265 +1,134 @@ """Replace i, e, and \pi with \iunit, \expe, and \cpi respectively.""" +import math_mode import re import sys + # for compatibility with Python 3 try: from itertools import izip except ImportError: izip = zip -import parentheses from utilities import writeout from utilities import readin -EQ_START = r'\begin{equation}' -EQ_END = r'\end{equation}' - -IND_START = r'\index{' - -NAME = 0 -SEEN = 1 - -CASES_START = r'\begin{cases}' -CASES_END = r'\end{cases}' - -EQMIX_START = r'\begin{equationmix}' -EQMIX_END = r'\end{equationmix}' - -STD_REGEX = r'.*?###open_(\d+)###.*?###close_\1###' - - def main(): if len(sys.argv) != 3: - fname = "../../data/ZE.1.tex" - ofname = "../../data/ZE.2.tex" - + fname = "ZE.1.tex" + ofname = "ZE.2.tex" else: fname = sys.argv[1] ofname = sys.argv[2] - writeout(ofname, remove_special(readin(fname))) - - -def remove_special(content): - """Removes the excess pieces from the given content and returns the updated version as a string.""" - - lines = content.split("\n") - - # various flags that will help us keep track of what elements we are inside of currently - inside = { - "constraint": [r'\constraint{', False], - "substitution": [r'\substitution{', False], - "drmfnote": [r'\drmfnote{', False], - "drmfname": [r'\drmfname{', False], - "proof": [r'\proof{', False] - } - - should_replace = False - in_ind = False - in_eq = False - - pi_pat = re.compile(r'(\s*)\\pi(\s*\b|[aeiou])') - expe_pat = re.compile(r'\b([^\\]?\W*)\s*e\s*\^') - - spaces_pat = re.compile(r' {2,}') - paren_pat = re.compile(r'\(\s*(.*?)\s*\)') - - dollar_pat = re.compile(r'(? Date: Wed, 4 May 2016 16:13:08 -0400 Subject: [PATCH 085/402] Added files via upload New version of replace_special, takes string of math mode as input, and only runs that string through the remove_speciall function. --- AlexDanoff/src/math_function.py | 72 ++++++++ AlexDanoff/src/replace_special.py | 291 ++++++++++++++++++++++-------- 2 files changed, 290 insertions(+), 73 deletions(-) create mode 100644 AlexDanoff/src/math_function.py diff --git a/AlexDanoff/src/math_function.py b/AlexDanoff/src/math_function.py new file mode 100644 index 0000000..9536ffc --- /dev/null +++ b/AlexDanoff/src/math_function.py @@ -0,0 +1,72 @@ +import math_mode +import string +from utilities import readin +from utilities import writeout +import re + +def math_string(file): + # Takes input file and returns list of strings when in math mode + string = open(file).read() + output = [] + ranges = math_mode.find_math_ranges(string) + for i in ranges: + new = string[i[0]:i[1]] + output.append(new) + return output + + +def change_original(o_file, changed_math_string): + # Places changed string from math mode back into place in the original function + o_string = open(o_file).read() + ranges = math_mode.find_math_ranges(o_string) + num = 0 + edited = o_string + for i in ranges: + edited = string.replace(edited, o_string[i[0]:i[1]], changed_math_string[num]) + num += 1 + return edited + + +def index_spacing(file, out_file): + used = open(file).read() + + updated = [] + IND_START = r'\index{' + ind_str = "" + in_ind = False + previous = "" + lines = used.split("\n") + + for line in lines: + # if this line is an index start storing it, or write it if we're done with the indexes + if IND_START in line: + in_ind = True + ind_str += line + "\n" + continue + + elif in_ind: + in_ind = False + + # add a preceding newline if one is not already present + if previous.strip() != "": + ind_str = "\n" + ind_str + + fullsplit = ind_str.split("\n") + updated.extend(fullsplit) + ind_str = "" + previous = line + updated.append(line) + + wrote = "\n".join(updated) + + # remove consecutive blank lines and blank lines between \index groups + spaces_pat = re.compile(r'\n{2,}[ ]?\n+') + wrote = spaces_pat.sub('\n\n', wrote) + wrote = re.sub(r'\\index{(.*?)}\n\n\\index{(.*?)}', r'\\index{\1}\n\\index{\2}', wrote) + + out = open(out_file, 'w') + out.write(wrote) + out.close() + return out +# 4/26, look at chapter 16 and prevent converting the \ApellFiii into \ApellFii\iunit + diff --git a/AlexDanoff/src/replace_special.py b/AlexDanoff/src/replace_special.py index 22f4799..eca08a9 100644 --- a/AlexDanoff/src/replace_special.py +++ b/AlexDanoff/src/replace_special.py @@ -1,134 +1,279 @@ -"""Replace i, e, and \pi with \iunit, \expe, and \cpi respectively.""" +"""Replace i, e, and \pi with \iunit, \expe, and \cpi respectively. Creates proper spacing for indexes and removes +unnecessary characters.""" -import math_mode import re import sys - # for compatibility with Python 3 try: from itertools import izip except ImportError: izip = zip +import math_function +import parentheses from utilities import writeout from utilities import readin + +EQ_START = r'\begin{equation}' +EQ_END = r'\end{equation}' + +IND_START = r'\index{' + +NAME = 0 +SEEN = 1 + +CASES_START = r'\begin{cases}' +CASES_END = r'\end{cases}' + +EQMIX_START = r'\begin{equationmix}' +EQMIX_END = r'\end{equationmix}' + +STD_REGEX = r'.*?###open_(\d+)###.*?###close_\1###' + + def main(): if len(sys.argv) != 3: - fname = "ZE.1.tex" - ofname = "ZE.2.tex" + fname = "../../data/ZE.1.tex" + ofname = "../../data/ZE.2.tex" + else: fname = sys.argv[1] ofname = sys.argv[2] - string = open(fname).read() + # Below: index_str writes to output file, math_string takes output file as input, and change_original + # writes to the output based on the previous output file + index_str = math_function.index_spacing(fname, ofname) + my_string = math_function.math_string(ofname) - writeout(ofname, replace_special(string)) + math_string = remove_special(my_string) -def string_to_list(_line, _list): + writeout(ofname, math_function.change_original(ofname, math_string)) - _output = [] - if _list != []: +def remove_special(content): + """Removes the excess pieces from the given content and returns the updated version as a string.""" + print(content) + counter = 0 + for function in content: - for i in range(0, len(_list)): - if i == 0: + lines = function.split('\n') - _output.append(_line[0:_list[i][0]]) + # various flags that will help us keep track of what elements we are inside of currently + inside = { + "constraint": [r'\constraint{', False], + "substitution": [r'\substitution{', False], + "drmfnote": [r'\drmfnote{', False], + "drmfname": [r'\drmfname{', False], + "proof": [r'\proof{', False] + } - else: + should_replace = False + in_ind = False + in_eq = True - _output.append(_line[_list[i-1][1]:_list[i][0]]) + pi_pat = re.compile(r'(\s*)\\pi(\s*\b|[aeiou])') + expe_pat = re.compile(r'\b([^\\]?\W*)\s*e\s*\^') - _output.append(_line[_list[i][0]:_list[i][1]]) + spaces_pat = re.compile(r' {2,}') + paren_pat = re.compile(r'\(\s*(.*?)\s*\)') - if len(_line) != _list[i][1] + 1: + dollar_pat = re.compile(r'(? Date: Thu, 5 May 2016 07:15:11 -0400 Subject: [PATCH 086/402] Hardcoded in a fix for the parargraph placement (#30) * Hardcoded in a fix for the parargraph placement * New indexes --- KLSadd_insertion/updateChapters.py | 121 +++++++++++++++++++++-------- 1 file changed, 87 insertions(+), 34 deletions(-) diff --git a/KLSadd_insertion/updateChapters.py b/KLSadd_insertion/updateChapters.py index 0a5b2ee..6867c00 100644 --- a/KLSadd_insertion/updateChapters.py +++ b/KLSadd_insertion/updateChapters.py @@ -17,6 +17,9 @@ -rewrite for "smarter" edits (ex. add new limit relations straight to the limit relations section in the section itself, not at the end) Current update status: incomplete Goals:change the book chapter files to include paragraphs from the addendum, and after that insert some edits so they are intelligently integrated into the chapter + +Edward Bian +Currently under heavy modification; sections may not work and/or look inefficient/confusing """ #start out by reading KLSadd.tex to get all of the paragraphs that must be added to the chapter files @@ -25,11 +28,17 @@ #variables: chapNums for the chapter number each section belongs to paras will hold the sections that will be copied over #chapNums is needed to know which file to open (9 or 14) #mathPeople is just the name for the sections that are used, like Wilson, Racah, etc. I know some are not people, its just a var name -#yes I like lists chapNums = [] paras = [] +klsparas = [] mathPeople = [] newCommands = [] #used to hold the indexes of the commands +ref9II = [] #Hold section search indexes +ref14II = [] +ref9III = [] #Holds all indexes +ref14III = [] +specref9 = [] +specref14 = [] comms = "" #holds the ACTUAL STRINGS of the commands #2/18/16 this method addresses the goal of hardcoding in the necessary packages to let the chapter files run as pdf's. #Currently only works with chapter 9, ask Dr. Cohl to help port your chapter 14 output file into a pdf @@ -44,10 +53,9 @@ def prepareForPDF(chap): #edits the chapter string sent to include hyperref, xparse, and cite packages #str[footmiscIndex] += "\\usepackage[pdftex]{hyperref} \n\\usepackage {xparse} \n\\usepackage{cite} \n" chap.insert(footmiscIndex, "\\usepackage[pdftex]{hyperref} \n\\usepackage {xparse} \n\\usepackage{cite} \n") - return chap -#2/18/16 this method reads in relevant commands that are in KLSadd.tex and returns them as a list and also adds +#2/18/16 this method reads in relevant commands that are in KLSadd.tex and returns them as a list def getCommands(kls): index = 0 @@ -57,9 +65,6 @@ def getCommands(kls): newCommands.append(index-1) if("mybibitem[1]" in word): newCommands.append(index) - #add \large\bf KLSadd: to make KLSadd additions appear like the other paragraphs - #pretty sure I had to do something here but I forgot, so pass? - #duh, obviously I need to store the commands somewhere! comms = kls[newCommands[0]:newCommands[1]] return comms @@ -68,7 +73,6 @@ def getCommands(kls): def insertCommands(kls, chap, cms): #reads in the newCommands[] and puts them in chap beginIndex = -1 #the index of the "begin document" keyphrase, this is where the new commands need to be inserted. - #find index of begin document in KLSadd index = 0 for word in kls: @@ -85,11 +89,13 @@ def insertCommands(kls, chap, cms): def findReferences(chapter): references = [] index = -1 + #chaptercheck designates which chapter is being searched for references chaptercheck = 0 if chapticker == 0: chaptercheck = str(9) elif chapticker == 1: chaptercheck = str(14) + #canAdd tells the program whether the next section is a reference canAdd = False for word in chapter: index+=1 @@ -99,52 +105,88 @@ def findReferences(chapter): ws = word[word.find("{")+1: word.find("~")] for unit in mathPeople: subunit = unit[unit.find(" ")+1: unit.find("#")] + # System of checks that verifies if section is in chapter if ((w in subunit) or (ws in subunit)) and (chaptercheck in unit) and (len(w) == len(subunit)) or (("Pseudo Jacobi" in w) and ("Pseudo Jacobi (or Routh-Romanovski)" in subunit)): canAdd = True + if chapticker == 0: + ref9II.append(index) + ref9III.append(index) + elif chapticker == 1: + ref14II.append(index) + ref14III.append(index) if("\\subsection*{References}" in word) and (canAdd == True): + # Appends valid locations references.append(index) + if chapticker == 0: + ref9II.append(index) + ref9III.append(index) + elif chapticker == 1: + ref14II.append(index) + ref14III.append(index) canAdd = False - + if ("subsection*{" in word and "References" not in word): + if chapticker == 0: + ref9III.append(index) + elif chapticker == 1: + ref14III.append(index) + print(ref9II) + print (ref14II) + print(ref9III) + print(ref14III) return references -#method to change file string(actually a list right now), returns string to be written to file -#If you write a method that changes something, it is preffered that you call the method in here -def fixChapter(chap, references, p, kls): - #chap is the file string(actually a list), references is the specific references for the file, - #and p is the paras variable(not sure if needed) kls is the KLSadd.tex as a list - count = 0 #count is used to represent the values in count - designator = 0 #Tells which chapter it's on +def referencePlacer(chap, references, p, kls,refII,refIII,specref): + # count is used to represent the values in count + count = 0 + # Tells which chapter it's on + designator = 0 if chapticker2 == 0: designator = "9." elif chapticker2 == 1: designator = "14." - designator = str(designator) for i in references: - #Place before References paragraph - if count > 34: - word1 = "14. Banana" - else: - word1 = str(p[count]) - if (designator in word1[word1.find("\\subsection*{") + 1: word1.find("}") ]): - chap[i-2] += "%Begin KLSadd additions" - chap[i-2] += p[count] - chap[i-2] += "%End of KLSadd additions" + # Place before References paragraph + word1 = str(p[count]) + if (designator in word1[word1.find("\\subsection*{") + 1: word1.find("}")]): + chap[i - 2] += "%Begin KLSadd additions" + chap[i - 2] += p[count] + chap[i - 2] += "%End of KLSadd additions" count += 1 else: - while (designator not in word1[word1.find("\\subsection*{") + 1: word1.find("}") ]): + while (designator not in word1[word1.find("\\subsection*{") + 1: word1.find("}")]): word1 = str(p[count]) - if (designator in word1[word1.find("\\subsection*{") + 1: word1.find("}") ]): + if (designator in word1[word1.find("\\subsection*{") + 1: word1.find("}")]): chap[i - 2] += "%Begin KLSadd additions" chap[i - 2] += p[count] chap[i - 2] += "%End of KLSadd additions" count += 1 else: + count += 1 - count+=1 +def referencePlacerII(chap, references, p, kls,refII,refIII,specref): + if chapticker2 == 0: + designator = "9." + elif chapticker2 == 1: + designator = "14." + count = 0 + sectionnuma = specref[0] + sectionnumb = specref[1] + for i in specref: + pass + + + +#method to change file string(actually a list right now), returns string to be written to file +#If you write a method that changes something, it is preffered that you call the method in here +def fixChapter(chap, references, p, kls,refII,refIII,specref): + #chap is the file string(actually a list), references is the specific references for the file, + #and p is the paras variable(not sure if needed) kls is the KLSadd.tex as a list + referencePlacer(chap, references, p, kls, refII,refIII,specref) chap = prepareForPDF(chap) cms = getCommands(kls) chap = insertCommands(kls,chap, cms) commentticker = 0 + # Hard coded command remover for word in chap: word2 = chap[chap.index(word)-1] if ("\\newcommand{\qhypK}[5]{\,\mbox{}_{#1}\phi_{#2}\!\left(" not in word2): @@ -161,19 +203,19 @@ def fixChapter(chap, references, p, kls): wordtoadd = "%" + word chap[commentticker] = wordtoadd commentticker += 1 - # Hopefully this works ticker1 = 0 + # Formatting to make the Latex file run while ticker1 < len(chap): if ('\\myciteKLS' in chap[ticker1]): chap[ticker1] = chap[ticker1].replace('\\myciteKLS', '\\cite') ticker1 += 1 - #probably won't work because I don't know how anything works return chap #open the KLSadd file to do things with with open("KLSadd.tex", "r") as add: #store the file as a string addendum = add.readlines() + #Makes sections look like other sections for word in addendum: if ("paragraph{" in word): lenword = len(word) - 1 @@ -184,6 +226,8 @@ def fixChapter(chap, references, p, kls): addendum[addendum.index(word)] = word[0:word.find("{") + 1] + "\large\\bf KLSadd: " + word[word.find("{") + 1: lenword] index = 0 indexes = [] + # Designates sections that need stuff added + # get the index for word in addendum: index+=1 if("." in word and "\\subsection*{" in word): @@ -191,18 +235,27 @@ def fixChapter(chap, references, p, kls): chapNums.append(9) name = word[word.find("{") + 1: word.find("}") ] mathPeople.append(name + "#") + specref9.append(index-1) if("14." in word): chapNums.append(14) name = word[word.find("{") + 1: word.find("}") ] mathPeople.append(name + "#") - #get the index + specref14.append(index - 1) indexes.append(index-1) + if ("paragraph{" in word) and (index > 313): + klsparas.append(index-1) + print(indexes) + print(specref9) + print(specref14) + print(klsparas) + print(mathPeople) #now indexes holds all of the places there is a section #using these indexes, get all of the words in between and add that to the paras[] for i in range(len(indexes)-1): box = ''.join(addendum[indexes[i]: indexes[i+1]-1]) paras.append(box) - paras.append("% This is a test") + box2 = ''.join(addendum[indexes[35]: 2245]) + paras.append(box2) #paras now holds the paragraphs that need to go into the chapter files, but they need to go in the appropriate #section(like Wilson, Racah, Hahn, etc.) so we use the mathPeople variable #we can use the section names to place the relevant paragraphs in the right place @@ -227,9 +280,9 @@ def fixChapter(chap, references, p, kls): references14 = findReferences(entire14) #call the fixChapter method to get a list with the addendum paragraphs added in chapticker2 = 0 - str9 = ''.join(fixChapter(entire9, references9, paras, addendum)) + str9 = ''.join(fixChapter(entire9, references9, paras, addendum,ref9II,ref9III,specref9)) chapticker2 += 1 - str14 = ''.join(fixChapter(entire14, references14, paras, addendum)) + str14 = ''.join(fixChapter(entire14, references14, paras, addendum,ref14II,ref14III,specref14)) """ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ From 0f8533de36ba6b30ad522a90647f3aaa05c8b1e0 Mon Sep 17 00:00:00 2001 From: joonbang Date: Thu, 5 May 2016 09:01:00 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\n") + mainPrepend+="
\n" + mainText="" + mainText=mainPrepend+mainText + mainText=mainText.replace("drmf_bof\n","") + mainText=mainText.replace("drmf_eof\n","") + mainText=mainText.replace("\'\'\'Zeta and Related Functions\'\'\'\n","") + mainText=mainText.replace("{{#set:Section=0}}\n","") + mainText="drmf_bof\n\'\'\'Zeta and Related Functions\'\'\'\n{{#set:Section=0}}"+mainText+"\ndrmf_eof\n" + mainWrite.write(mainText) + mainWrite.close() + main.close() + os.system("cp -f ZetaFunctions.mmd.new ZetaFunctions.mmd") + #wiki.write("\ndrmf_eof\n") + parse=False + elif "\\title" in line and parse: + labels.append("Zeta and Related Functions") + sections.append(["Zeta and Related Functions",0]) + elif "\\part" in line: + if getString(line)=="BOF": + parse=False + elif getString(line)=="EOF": + parse=True + elif parse: + stringWrite="\'\'\'" + stringWrite+=getString(line)+"\'\'\'\n" + chapter=getString(line) + if startFlag: + mainPrepend+=("\n== Sections in "+chapter+" ==\n\n
\n") + startFlag=False + else: + mainPrepend+=("
\n\n== Sections in "+chapter+" ==\n\n
\n") + head=True + elif "\\section" in line: + mainPrepend+=("* [["+secLabel(getString(line))+"|"+getString(line)+"]]\n") + sections.append([getString(line)]) + + secCounter=0 + eqCounter=0 + for i in range(0,len(lines)): + line=lines[i] + if "\\section" in line: + parse=True + secCounter+=1 + wiki.write("drmf_bof\n") + wiki.write("\'\'\'"+secLabel(getString(line))+"\'\'\'\n") + wiki.write("{{DISPLAYTITLE:"+(sections[secCounter][0])+"}}\n") + wiki.write("
\n") + wiki.write("
<< [["+secLabel(sections[secCounter-1][0])+"|"+secLabel(sections[secCounter-1][0])+"]]
\n") + wiki.write("
[[Zeta_and_Related_Functions#"+ + "Sections_in_"+chapter.replace(" ","_")+"|"+secLabel(sections[secCounter][0])+"]]
\n") + wiki.write("
[["+secLabel(sections[(secCounter+1)%len(sections)][0])+ + "|"+secLabel(sections[(secCounter+1)%len(sections)][0])+"]] >>
\n
\n\n") + head=True + wiki.write("== "+getString(line)+" ==\n") + elif ("\\section" in lines[(i+1)%len(lines)] or "\\end{document}" in lines[(i+1)%len(lines)]) and parse: + wiki.write("
\n") + wiki.write("
<< [["+secLabel(sections[secCounter-1][0])+"|"+secLabel(sections[secCounter-1][0])+"]]
\n") + wiki.write("
[[Zeta_and_Related_Functions#"+"Sections_in_" + ""+chapter.replace(" ","_")+"|"+secLabel(sections[secCounter][0])+"]]
\n") + wiki.write("
[["+secLabel(sections[(secCounter+1)%len(sections)][0])+ + "|"+secLabel(sections[(secCounter+1)%len(sections)][0])+"]] >>
\n
\n\n") + wiki.write("drmf_eof\n") + sections[secCounter].append(eqCounter) + eqCounter=0 + + elif "\\subsection" in line and parse: + wiki.write("\n== "+getString(line)+" ==\n") + head=True + elif "\\paragraph" in line and parse: + wiki.write("\n=== "+getString(line)+" ===\n") + head=True + elif "\\subsubsection" in line and parse: + wiki.write("\n=== "+getString(line)+" ===\n") + head=True + + elif "\\begin{equation}" in line and parse: + # symLine="" + if head: + wiki.write("\n") + head=False + sLabel=line.find("\\label{")+7 + eLabel=line.find("}",sLabel) + label=(line[sLabel:eLabel]) + eqCounter+=1 + for l in lLink: + if label==l[0:l.find("=")-1]: + rlabel=l[l.find("=>")+3:l.find("\\n")] + rlabel=rlabel.replace("/","") + rlabel=rlabel.replace("#",":") + rlabel=rlabel.replace("!",":") + break + label=modLabel(rlabel) + labels.append("Formula:"+rlabel) + eqs.append("") + #wiki.write("\n\n") + wiki.write("{\displaystyle \n") + math=True + elif "\\begin{equation}" in line and not parse: + sLabel=line.find("\\label{")+7 + eLabel=line.find("}",sLabel) + label=modLabel(line[sLabel:eLabel]) + labels.append("*"+label) #special marker + eqs.append("") + math=True + elif "\\end{equation}" in line: + + math=False + elif "\\constraint" in line and parse: + constraint=True + math=False + conLine="" + elif "\\substitution" in line and parse: + substitution=True + math=False + subLine="" + #wiki.write("
Substitution(s): "+getEq(line)+"

\n") + elif "\\proof" in line and parse: + math=False + elif "\\drmfn" in line and parse: + math=False + if "\\drmfname" in line and parse: + wiki.write("
This formula has the name: "+getString(line)+"

\n") + elif math and parse: + flagM=True + eqs[len(eqs)-1]+=line + + if "\\end{equation}" in lines[i+1] and not "\\subsection" in lines[i+3] and not "\\section" in lines[i+3] and not "\\part" in lines[i+3]: + u=i + flagM2=False + while flagM: + u+=1 + if "\\begin{equation}"in lines[u] in lines[u]: + flagM=False + if "\\section" in lines[u] or "\\subsection" in lines[i] or "\\part" in lines[u] or "\\end{document}" in lines[u]: + flagM=False + flagM2=True + if not(flagM2): + + wiki.write(line.rstrip("\n")) + wiki.write("\n}
\n") + else: + wiki.write(line.rstrip("\n")) + wiki.write("\n}\n") + elif "\\end{equation}" in lines[i+1]: + wiki.write(line.rstrip("\n")) + wiki.write("\n}\n") + elif "\\constraint" in lines[i+1] or "\\substitution" in lines[i+1] or "\\drmfn" in lines[i+1]: + wiki.write(line.rstrip("\n")) + wiki.write("\n}\n") + else: + wiki.write(line) + elif math and not parse: + eqs[len(eqs)-1]+=line + if "\\end{equation}" in lines[i+1] or "\\constraint" in lines[i+1] or "\\substitution" in lines[i+1] or "\\drmfn" in lines[i+1]: + math=False + if substitution and parse: + subLine=subLine+line.replace("&","&
") + if "\\end{equation}" in lines[i+1] or "\\substitution" in lines[i+1]\ + or "\\constraint" in lines[i+1] or "\\drmfn" in lines[i+1] or "\\proof" in lines[i+1]: + substitution=False + wiki.write("
Substitution(s): "+getEq(subLine)+"

\n") + + if constraint and parse: + conLine=conLine+line.replace("&","&
") + if "\\end{equation}" in lines[i+1] or "\\substitution" \ + in lines[i+1] or "\\constraint" in lines[i+1] or "\\drmfn" in lines[i+1] or "\\proof" in lines[i+1]: + constraint=False + wiki.write("
Constraint(s): "+getEq(conLine)+"

\n") + + eqCounter=n + endNum=len(labels)-1 + parse=False + constraint=False + substitution=False + note=False + hCon=True + hSub=True + hNote=True + hProof=True + proof=False + comToWrite="" + secCount=-1 + newSec=False + for i in range(0,len(lines)): + line=lines[i] + + if "\\section" in line: + secCount+=1 + newSec=True + eqS=0 + + if "\\begin{equation}" in line: + symLine=line.strip("\n") + eqS+=1 + constraint=False + substitution=False + note=False + comToWrite="" + hCon=True + hSub=True + hNote=True + hProof=True + proof=False + parse=True + symbols=[] + eqCounter+=1 + wiki.write("drmf_bof\n") + label=labels[eqCounter] + #if not "DLMF" in label:eqCounter+=1 + label=labels[eqCounter] + wiki.write("\'\'\'"+secLabel(label)+"\'\'\'\n") + wiki.write("{{DISPLAYTITLE:"+(labels[eqCounter])+"}}\n") + if eqCounter==len(labels)-1: + break + if eqCounter\n") + if newSec: + wiki.write("
" + "<< [["+secLabel(sections[secCount][0]).replace(" ","_")+"|"+secLabel(sections[secCount][0])+"]]
\n") + else: + wiki.write("
" + "<< [["+secLabel(labels[eqCounter-1]).replace(" ","_")+"|"+secLabel(labels[eqCounter-1])+"]]
\n") + wiki.write("
[["+secLabel(sections[secCount+1][0]).replace(" ","_")+"#"+ + secLabel(labels[eqCounter][len("Formula:"):])+"|formula in "+secLabel(sections[secCount+1][0])+"]]
\n") + #if eqS==sections[secCount][1]: + if True: + wiki.write("
[["+secLabel(labels[(eqCounter+1)%(endNum+1)]).replace(" ","_")+ + "|"+secLabel(labels[(eqCounter+1)%(endNum+1)])+"]] >>
\n") + wiki.write("
\n\n") + elif eqCounter==endNum: + wiki.write("
\n") + if newSec: + newSec=False + wiki.write("
<< [["+secLabel(sections[secCount][0]).replace(" ","_")+"|"+ + secLabel(sections[secCount][0])+"]]
\n") + else: + wiki.write("
" + "<< [["+secLabel(labels[eqCounter-1]).replace(" ","_")+"|"+secLabel(labels[eqCounter-1])+"]]
\n") + wiki.write("
[["+ + secLabel(sections[secCount+1][0]).replace(" ","_")+ + "#"+secLabel(labels[eqCounter][len("Formula:"):])+ + "|formula in "+secLabel(sections[secCount+1][0])+ + "]]
\n") + wiki.write("
[["+secLabel(labels[(eqCounter+1)%(endNum+1)].replace(" ","_"))+ + "|"+secLabel(labels[(eqCounter+1)%(endNum+1)])+"]]
\n") + wiki.write("
\n\n") + + wiki.write("
{\displaystyle \n") + math=True + elif "\\end{equation}" in line: + wiki.write(comToWrite) + parse=False + math=False + if hProof: + wiki.write("\n== Proof ==\n\nWe ask users to provide proof(s), \ + reference(s) to proof(s), or further clarification on the proof(s) in this space.\n") + + wiki.write("\n== Symbols List ==\n\n") + newSym=[] + #if "09.07:04" in label: + for x in symbols: + flagA=True + #if x not in newSym: + cN=0 + cC=0 + flag=True + ArgCx=0 + for z in x: + if z.isalpha() and flag or z=="&": + cN+=1 + else: + flag=False + if z=="{" or z=="[": + cC+=1 + if z=="}" or z=="]": + cC-=1 + if cC==0: + ArgCx+=1 + noA=x[:cN] + for y in newSym: + cN=0 + cC=0 + ArgC=0 + flag=True + for z in y: + if z.isalpha() and flag or z=="&": + cN+=1 + else: + flag=False + if z=="{" or z=="[": + cC+=1 + if z=="}" or z=="]": + cC-=1 + if cC==0: + ArgC+=1 + + + if y[:cN]==noA: #and ArgC==ArgCx: + flagA=False + break + if flagA: + newSym.append(x) + newSym.reverse() + ampFlag=False + finSym=[] + for s in range(len(newSym)-1,-1,-1): + symbolPar="\\"+newSym[s] + ArgCx=0 + parCx=0 + parFlag=False + cC=0 + for z in symbolPar: + if z=="@": + parFlag=True + elif z.isalpha() or z=="&": + cN+=1 + else: + if z=="{" or z=="[": + cC+=1 + if z=="}" or z=="]": + cC-=1 + if cC==0: + if parFlag: + parCx+=1 + else: + ArgCx+=1 + + if symbolPar.find("{")!=-1 or symbolPar.find("[")!=-1: + if symbolPar.find("[")==-1: + symbol=symbolPar[0:symbolPar.find("{")] + elif symbolPar.find("{")==-1 or symbolPar.find("[")len(symbol) and (Q[len(symbol)]=="{" or Q[len(symbol)]=="["): + ap="" + for o in range(len(symbol),len(Q)): + if Q[o]=="{" or z=="[": + pass + elif Q[o]=="}" or z=="]": + listArgs.append(ap) + ap="" + else: + ap+=Q[o] + #websiteF=G[4].strip("\n") + websiteF="" + web1=G[5] + for t in range(5,len(G)): + if G[t]!="": + websiteF=websiteF+" ["+G[t]+" "+G[t]+"]" + + #p1=Q + #if Q.find("@")!=-1: + #p1=Q[:Q.find("@")] + p1=G[4].strip("$") + p1="{\\displaystyle "+p1+"}" + #if checkFlag: + new2="" + pause=False + mathF=True + p2=G[1] + for k in range(0,len(p2)): + if p2[k]=="$": + if mathF: + new2+="{\\displaystyle " + else: + new2+="}" + mathF=not mathF + else: + new2+=p2[k] + p2=new2 + finSym.append(web1+" "+p1+"] : "+p2+" :"+websiteF) + break + + #preG=S + if not gFlag: + del newSym[s] + + gFlag=True + #finSym.reverse() + if ampFlag: + wiki.write("& : logical and") + gFlag=False + for y in finSym: + if y=="& : logical and": + pass + elif gFlag: + gFlag=False + wiki.write("["+y) + else: + wiki.write("
\n["+y) + + + wiki.write("\n
\n") + + wiki.write("\n== Bibliography==\n\n")#should there be a space between bibliography and ==? + r=unmodLabel(labels[eqCounter]) + q=r.find("DLMF:")+5 + p=r.find(":",q) + section=r[q:p] + equation=r[p+1:] + #print("Section", section, r, p, q,) + if equation.find(":")!=-1: + equation=equation[0:equation.find(":")] + if is_number(section) == False: + return eqCounter + wiki.write("[HTTP://DLMF.NIST.GOV/"+ + section+"#"+equation+" Equation ("+equation[1:]+"), " + "Section "+section+"] of [[Bibliography#DLMF|'''DLMF''']].\n\n") + wiki.write("== URL links ==\n\nWe ask users to provide relevant URL links in this space.\n\n") + if eqCounter
\n") + if newSec: + newSec=False + wiki.write("
<< [["+secLabel(sections[secCount][0]).replace(" ","_")+ + "|"+secLabel(sections[secCount][0])+"]]
\n") + else: + wiki.write("
<< [["+secLabel(labels[eqCounter-1]).replace(" ","_")+"|"+ + secLabel(labels[eqCounter-1])+"]]
\n") + wiki.write("
[["+secLabel(sections[secCount+1][0]).replace(" ","_")+"#"+ + secLabel(labels[eqCounter][len("Formula:"):])+"|formula in "+secLabel(sections[secCount+1][0])+"]]
\n") + #if eqS==sections[secCount][1]: + #else: + if True: + wiki.write("
[["+secLabel(labels[(eqCounter+1)% + (endNum+1)]).replace(" ","_")+"|"+secLabel(labels[(eqCounter+1)%(endNum+1)])+"]] >>
\n") + wiki.write("
\n\ndrmf_eof\n") + else: #FOR EXTRA EQUATIONS + wiki.write("
\n") + wiki.write("
<< [["+labels[endNum-1].replace(" ","_")+"|"+labels[endNum-1]+"]]
\n") + wiki.write("
[["+labels[0].replace(" ","_")+"#"+labels[endNum][8:]+"|formula in "+labels[0]+"]]
\n") + wiki.write("
[["+labels[(0)%endNum].replace(" ","_")+"|"+labels[0%endNum]+"]]
\n") + wiki.write("
\n\ndrmf_eof\n") + + + + elif "\\constraint" in line and parse: + #symbols=symbols+getSym(line) + symLine=line.strip("\n") + if hCon: + comToWrite=comToWrite+"\n== Constraint(s) ==\n\n" + hCon=False + constraint=True + math=False + conLine="" + #wiki.write("
"+getEq(line)+"

\n") + elif "\\substitution" in line and parse: + #symbols=symbols+getSym(line) + symLine=line.strip("\n") + if hSub: + comToWrite=comToWrite+"\n== Substitution(s) ==\n\n" + hSub=False + #wiki.write("
"+getEq(line)+"

\n") + substitution=True + math=False + subLine="" + elif "\\drmfname" in line and parse: + math=False + comToWrite="\n== Name ==\n\n
"+getString(line)+"

\n"+comToWrite + + elif "\\drmfnote" in line and parse: + symbols=symbols+getSym(line) + if hNote: + comToWrite=comToWrite+"\n== Note(s) ==\n\n" + hNote=False + note=True + math=False + noteLine="" + + elif "\\proof" in line and parse: + #symbols=symbols+getSym(line) + symLine=line.strip("\n") + if hProof: + hProof=False + comToWrite=comToWrite+"\n== Proof ==\n\nWe ask users to provide proof(s), reference(s) to proof(s), " \ + "or further clarification on the proof(s) in this space. \n

\n
" + proof=True + proofLine="" + pause=False + pauseP=False + for ind in range(0,len(line)): + if line[ind:ind+7]=="\\eqref{": + pause=True + eqR=line[ind:line.find("}",ind)+1] + rLab=getString(eqR) + for l in lLink: + if rLab==l[0:l.find("=")-1]: + rlabel=l[l.find("=>")+3:l.find("\\n")] + rlabel=rlabel.replace("/","") + rlabel=rlabel.replace("#",":") + rlabel=rlabel.replace("!",":") + break + + eInd=refLabels.index(""+rlabel) + z=line[line.find("}",ind+7)+1] + if z=="." or z==",": + pauseP=True + proofLine+=("
\n{\displaystyle \n"+refEqs[eInd]+"}"+z+"
\n") + else: + if z=="}": + proofLine+=("
\n{\displaystyle \n"+refEqs[eInd]+"}
") + else: + proofLine+=("
\n{\displaystyle \n"+refEqs[eInd]+"}
\n") + + + else: + if pause: + if line[ind]=="}": + pause=False + elif pauseP: + pauseP=False + elif line[ind]=="\n" and "\\end{equation}" in lines[i+1]: + pass + else: + proofLine+=(line[ind]) + if "\\end{equation}" in lines[i+1]: + proof=False + #symLine+=line.strip("\n") + wiki.write(comToWrite+getEqP(proofLine)+"
\n
\n") + comToWrite="" + symbols=symbols+getSym(symLine) + symLine="" + + #wiki.write(line) + + elif proof: + symLine+=line.strip("\n") + pauseP=False + #symbols=symbols+getSym(line) + for ind in range(0,len(line)): + if line[ind:ind+7]=="\\eqref{": + pause=True + eqR=line[ind:line.find("}",ind)+1] + rLab=getString(eqR) + for l in lLink: + if rLab==l[0:l.find("=")-1]: + rlabel=l[l.find("=>")+3:l.find("\\n")] + rlabel=rlabel.replace("/","") + rlabel=rlabel.replace("#",":") + rlabel=rlabel.replace("!",":") + break + + eInd=refLabels.index(""+rlabel.lstrip("Formula:")) + z=line[line.find("}",ind+7)+1] + if z=="." or z==",": + pauseP=True + proofLine+=("
\n{\displaystyle \n"+refEqs[eInd]+"}"+z+"
\n") + else: + proofLine+=("
\n{\displaystyle \n"+refEqs[eInd]+"}
\n") + + else: + if pause: + if line[ind]=="}": + pause=False + elif pauseP: + pauseP=False + elif line[ind]=="\n" and "\\end{equation}" in lines[i+1]: + pass + + else: + proofLine+=(line[ind]) + if "\\end{equation}" in lines[i+1]: + proof=False + #symLine+=line.strip("\n") + wiki.write(comToWrite+getEqP(proofLine).rstrip("\n")+"
\n
\n") + comToWrite="" + symbols=symbols+getSym(symLine) + symLine="" + + elif math: + if "\\end{equation}" in lines[i+1] or "\\constraint" in lines[i+1] \ + or "\\substitution" in lines[i+1] or "\\proof" in lines[i+1] or "\\drmfnote" in lines[i+1] or "\\drmfname" in lines[i+1]: + wiki.write(line.rstrip("\n")) + symLine+=line.strip("\n") + symbols=symbols+getSym(symLine) + symLine="" + wiki.write("\n}\n") + else: + symLine+=line.strip("\n") + wiki.write(line) + if note and parse: + noteLine=noteLine+line + symbols=symbols+getSym(line) + if "\\end{equation}" in lines[i+1] or "\\drmfn" in lines[i+1] \ + or "\\constraint" in lines[i+1] \ + or "\\substitution" in lines[i+1] \ + or "\\proof" in lines[i+1]: + note=False + if "\\emph" in noteLine: + noteLine=noteLine[0:noteLine.find("\\emph{")]+"\'\'"+noteLine[noteLine.find("\\emph{")+len("\\emph{"): + noteLine.find("}",noteLine.find("\\emph{")+len("\\emph{"))]+"\'\'"+\ + noteLine[noteLine.find("}",noteLine.find("\\emph{")+len("\\emph{"))+1:] + comToWrite=comToWrite+"
"+getEq(noteLine)+"

\n" + + if constraint and parse: + conLine=conLine+line.replace("&","&
") + + symLine+=line.strip("\n") + #symbols=symbols+getSym(line) + if "\\end{equation}" in lines[i+1] or "\\drmfn" in lines[i+1] \ + or "\\constraint" in lines[i+1]\ + or "\\substitution" in lines[i+1] \ + or "\\proof" in lines[i+1]: + constraint=False + symbols=symbols+getSym(symLine) + symLine="" + wiki.write(comToWrite+"
"+getEq(conLine)+"

\n") + comToWrite="" + if substitution and parse: + subLine=subLine+line.replace("&","&
") + + symLine+=line.strip("\n") + #symbols=symbols+getSym(line) + if "\\end{equation}" in lines[i+1] or "\\drmfn" in lines[i+1] or \ + "\\substitution" in lines[i+1] or "\\constraint" in lines[i+1] or "\\proof" in lines[i+1]: + substitution=False + symbols=symbols+getSym(symLine) + symLine="" + wiki.write(comToWrite+"
"+getEq(subLine)+"

\n") + comToWrite="" \ No newline at end of file From 0af70f22b894ad7f2dd961f7f85c80c7317a134f Mon Sep 17 00:00:00 2001 From: Moritz Schubotz Date: Thu, 19 May 2016 23:50:19 +0200 Subject: [PATCH 091/402] Improve DLMFtex2Wiki (#44) * Remove duplicate functions in DLMFtex2Wiki.py, import tex2Wiki from DLMFtex2Wiki * Change iterations from jsahlfkjsd in tex2Wiki * Add back modlabel to DLMFtex2Wiki * Add back modlabel to DLMFtex2Wiki.py, fix syntax error * Add back modlabel * Add back modlabel to DLMFtex2Wiki.py * Remove >>head and >>parent messages --- Azeem/src/DLMFtex2Wiki.py | 317 ++++++++++---------------------------- Azeem/src/tex2Wiki.py | 2 +- 2 files changed, 82 insertions(+), 237 deletions(-) diff --git a/Azeem/src/DLMFtex2Wiki.py b/Azeem/src/DLMFtex2Wiki.py index 26bac46..06b4d3e 100644 --- a/Azeem/src/DLMFtex2Wiki.py +++ b/Azeem/src/DLMFtex2Wiki.py @@ -3,181 +3,22 @@ import csv #imported for using csv format import sys #imported for getting args import os #imported for copying file +import tex2Wiki as KLS #Change Accordingly based on project directory. -def isnumber(char): #Function to check if char is a number (assuming 1 character) - return char[0] in "0123456789" def is_number(char): try: "".join(["0"+str(float(char)) if float(char) < 10 else float(char)]) return True except ValueError: return False -def getString(line): #Gets all data within curly braces on a line - #-------Initialization------- - stringWrite="" - getStr=False - pW="" - #---------------------------- - for c in line: - if c=="{" or c=="}": #if there is a curly brace in the line - getStr=not(getStr) #toggle the getStr flag - if c=="}": #no more info needed - break - elif getStr: #if within curly braces - if not(pW==c and c=="-"): #if there is no double dash (makes single dash) - if c!="$": #if not a $ sign - stringWrite+=c #this character is part of the data - else: #replace $ with '' - stringWrite+="\'\'" #add double ' - pW=c #change last character for finding double dashes - return (stringWrite.rstrip('\n').lstrip()) #return the data without newlines and without leading spaces -def getG(line): #gets equation for symbols list - start=True - final="" - for c in line: - if c=="$" and start: - final+="{\\displaystyle " - start=False - elif c=="$": - final+="}" - start=True - else: - final+=c - return(final) -def getEq(line): #Gets all data within constraints,substitutions - #-------Initialization------- - per=1 - stringWrite="" - fEq=False - count=0 - #---------------------------- - for c in line: #read each character - if count>=0 and per!=0: - per+=1 - if c!=" " and c!="%": - per=0 - if c=="{" and per==0: - if count>0: - stringWrite+=c - count+=1 - elif c=="}" and per==0: - count-=1 - if count>0: - stringWrite+=c - elif c=="$" and per==0: #either begin or end equation - fEq=not(fEq)#toggle fEq flag to know if begin or end equation - if fEq:#if begin - stringWrite+="{\\displaystyle " - else:#if end - stringWrite+="}" - elif c=="\n" and per==0:#if newline - stringWrite=stringWrite.strip()#remove all leading and trailing whitespace - - #should above be rstrip?<-------------------------------------------------------------------CHECK THIS - - stringWrite+=c#add the newline character - per+=1 #watch for % signs - elif count>0 and per==0: #not special character - stringWrite+=c #write the character - - return (stringWrite.rstrip().lstrip()) -def getEqP(line): #Gets all data within proofs - per=1 - stringWrite="" - fEq=False - count=0 - length=0 - for c in line: - if fEq and c!=" " and c!="$" and c!="{" and c!="}" and not c.isalpha(): - length+=1 - if count>=0 and per!=0: - per+=1 - if c!=" " and c!="%": - #if c!="%": - per=0 - if c=="{" and per==0: - if count>0: - stringWrite+=c - count+=1 - elif c=="}" and per==0: - count-=1 - if count>0: - stringWrite+=c - elif c=="$" and per==0: - fEq=not(fEq) - if fEq: - stringWrite+="{\\displaystyle " - else: - if length<10: - stringWrite+="}" - else: - stringWrite+="}"+"
" - length=0 - elif c=="\n" and per==0: - stringWrite=stringWrite.strip() - stringWrite+=c - per+=1 - elif count>0 and per==0: - stringWrite+=c - - return (stringWrite.lstrip()) -def getSym(line): #Gets all symbols on a line for symbols list - symList=[] - if line=="": - return symList - symbol="" - symFlag=False - cC=0 - for i in range(0,len(line)): - #if "MeixnerPollaczek{\\lambda}{n}@{x}{\\phi}=\frac{\\pochhammer{2\\lambda}{n}}" in line: - c=line[i] - if symFlag: - if c=="{" or c=="[": - cC+=1 - if c!="}" and c!="]": - if c.isalpha(): - symbol+=c - else: - symFlag=False - symList.append(symbol) - symList+=(getSym(symbol)) - symbol="" - else: - cC-=1 - symbol+=c - if i+1==len(line): - p="" - else: - p=line[i+1] - if cC<=0 and p!="{" and p!="[" and p!="@": - symFlag=False - #if "monicAskeyWilson{n+1}@{x}{a}{b}{c}{d}{q}+\\frac{1}{2}" in line: - symList.append(symbol) - symList+=(getSym(symbol)) - symbol="" - - elif c=="\\": - symFlag=True - elif c=="&" and not (i>line.find("\\begin{array}") and i1: newlabel+=num isNumer=False @@ -186,7 +27,7 @@ def modLabel(label): newlabel+="0"+str(num) num="" isNumer=False - if isnumber(label[i]): + if is_number(label[i]): isNumer=True num+=str(label[i]) else: @@ -197,6 +38,7 @@ def modLabel(label): elif len(num)==1: newlabel+="0"+num return(newlabel) + def DLMF(n): for iterations in range(0,1): try: @@ -288,14 +130,14 @@ def DLMF(n): labels.append("Zeta and Related Functions") sections.append(["Zeta and Related Functions",0]) elif "\\part" in line: - if getString(line)=="BOF": + if KLS.getString(line)=="BOF": parse=False - elif getString(line)=="EOF": + elif KLS.getString(line)=="EOF": parse=True elif parse: stringWrite="\'\'\'" - stringWrite+=getString(line)+"\'\'\'\n" - chapter=getString(line) + stringWrite+=KLS.getString(line)+"\'\'\'\n" + chapter=KLS.getString(line) if startFlag: mainPrepend+=("\n== Sections in "+chapter+" ==\n\n
\n") startFlag=False @@ -304,8 +146,8 @@ def DLMF(n): "column-count:2;-webkit-column-count:2\">\n") head=True elif "\\section" in line: - mainPrepend+=("* [["+secLabel(getString(line))+"|"+getString(line)+"]]\n") - sections.append([getString(line)]) + mainPrepend+=("* [["+KLS.secLabel(KLS.getString(line))+"|"+KLS.getString(line)+"]]\n") + sections.append([KLS.getString(line)]) secCounter=0 eqCounter=0 @@ -315,35 +157,35 @@ def DLMF(n): parse=True secCounter+=1 wiki.write("drmf_bof\n") - wiki.write("\'\'\'"+secLabel(getString(line))+"\'\'\'\n") + wiki.write("\'\'\'"+KLS.secLabel(KLS.getString(line))+"\'\'\'\n") wiki.write("{{DISPLAYTITLE:"+(sections[secCounter][0])+"}}\n") wiki.write("
\n") - wiki.write("
<< [["+secLabel(sections[secCounter-1][0])+"|"+secLabel(sections[secCounter-1][0])+"]]
\n") + wiki.write("
<< [["+KLS.secLabel(sections[secCounter-1][0])+"|"+KLS.secLabel(sections[secCounter-1][0])+"]]
\n") wiki.write("
[[Zeta_and_Related_Functions#"+ - "Sections_in_"+chapter.replace(" ","_")+"|"+secLabel(sections[secCounter][0])+"]]
\n") - wiki.write("
[["+secLabel(sections[(secCounter+1)%len(sections)][0])+ - "|"+secLabel(sections[(secCounter+1)%len(sections)][0])+"]] >>
\n
\n\n") + "Sections_in_"+chapter.replace(" ","_")+"|"+KLS.secLabel(sections[secCounter][0])+"]]
\n") + wiki.write("
[["+KLS.secLabel(sections[(secCounter+1)%len(sections)][0])+ + "|"+KLS.secLabel(sections[(secCounter+1)%len(sections)][0])+"]] >>
\n\n\n") head=True - wiki.write("== "+getString(line)+" ==\n") + wiki.write("== "+KLS.getString(line)+" ==\n") elif ("\\section" in lines[(i+1)%len(lines)] or "\\end{document}" in lines[(i+1)%len(lines)]) and parse: wiki.write("
\n") - wiki.write("
<< [["+secLabel(sections[secCounter-1][0])+"|"+secLabel(sections[secCounter-1][0])+"]]
\n") + wiki.write("
<< [["+KLS.secLabel(sections[secCounter-1][0])+"|"+KLS.secLabel(sections[secCounter-1][0])+"]]
\n") wiki.write("
[[Zeta_and_Related_Functions#"+"Sections_in_" - ""+chapter.replace(" ","_")+"|"+secLabel(sections[secCounter][0])+"]]
\n") - wiki.write("
[["+secLabel(sections[(secCounter+1)%len(sections)][0])+ - "|"+secLabel(sections[(secCounter+1)%len(sections)][0])+"]] >>
\n
\n\n") + ""+chapter.replace(" ","_")+"|"+KLS.secLabel(sections[secCounter][0])+"]] \n") + wiki.write("
[["+KLS.secLabel(sections[(secCounter+1)%len(sections)][0])+ + "|"+KLS.secLabel(sections[(secCounter+1)%len(sections)][0])+"]] >>
\n\n\n") wiki.write("drmf_eof\n") sections[secCounter].append(eqCounter) eqCounter=0 elif "\\subsection" in line and parse: - wiki.write("\n== "+getString(line)+" ==\n") + wiki.write("\n== "+KLS.getString(line)+" ==\n") head=True elif "\\paragraph" in line and parse: - wiki.write("\n=== "+getString(line)+" ===\n") + wiki.write("\n=== "+KLS.getString(line)+" ===\n") head=True elif "\\subsubsection" in line and parse: - wiki.write("\n=== "+getString(line)+" ===\n") + wiki.write("\n=== "+KLS.getString(line)+" ===\n") head=True elif "\\begin{equation}" in line and parse: @@ -362,7 +204,7 @@ def DLMF(n): rlabel=rlabel.replace("#",":") rlabel=rlabel.replace("!",":") break - label=modLabel(rlabel) + label=KLS.modLabel(rlabel) labels.append("Formula:"+rlabel) eqs.append("") #wiki.write("\n\n") @@ -371,7 +213,7 @@ def DLMF(n): elif "\\begin{equation}" in line and not parse: sLabel=line.find("\\label{")+7 eLabel=line.find("}",sLabel) - label=modLabel(line[sLabel:eLabel]) + label=KLS.modLabel(line[sLabel:eLabel]) labels.append("*"+label) #special marker eqs.append("") math=True @@ -386,13 +228,13 @@ def DLMF(n): substitution=True math=False subLine="" - #wiki.write("
Substitution(s): "+getEq(line)+"

\n") + #wiki.write("
Substitution(s): "+KLS.getEq(line)+"

\n") elif "\\proof" in line and parse: math=False elif "\\drmfn" in line and parse: math=False if "\\drmfname" in line and parse: - wiki.write("
This formula has the name: "+getString(line)+"

\n") + wiki.write("
This formula has the name: "+KLS.getString(line)+"

\n") elif math and parse: flagM=True eqs[len(eqs)-1]+=line @@ -431,14 +273,14 @@ def DLMF(n): if "\\end{equation}" in lines[i+1] or "\\substitution" in lines[i+1]\ or "\\constraint" in lines[i+1] or "\\drmfn" in lines[i+1] or "\\proof" in lines[i+1]: substitution=False - wiki.write("
Substitution(s): "+getEq(subLine)+"

\n") + wiki.write("
Substitution(s): "+KLS.getEq(subLine)+"

\n") if constraint and parse: conLine=conLine+line.replace("&","&
") if "\\end{equation}" in lines[i+1] or "\\substitution" \ in lines[i+1] or "\\constraint" in lines[i+1] or "\\drmfn" in lines[i+1] or "\\proof" in lines[i+1]: constraint=False - wiki.write("
Constraint(s): "+getEq(conLine)+"

\n") + wiki.write("
Constraint(s): "+KLS.getEq(conLine)+"

\n") eqCounter=n endNum=len(labels)-1 @@ -481,7 +323,7 @@ def DLMF(n): label=labels[eqCounter] #if not "DLMF" in label:eqCounter+=1 label=labels[eqCounter] - wiki.write("\'\'\'"+secLabel(label)+"\'\'\'\n") + wiki.write("\'\'\'"+KLS.secLabel(label)+"\'\'\'\n") wiki.write("{{DISPLAYTITLE:"+(labels[eqCounter])+"}}\n") if eqCounter==len(labels)-1: break @@ -489,33 +331,33 @@ def DLMF(n): wiki.write("
\n") if newSec: wiki.write("
" - "<< [["+secLabel(sections[secCount][0]).replace(" ","_")+"|"+secLabel(sections[secCount][0])+"]]
\n") + "<< [["+KLS.secLabel(sections[secCount][0]).replace(" ","_")+"|"+KLS.secLabel(sections[secCount][0])+"]]
\n") else: wiki.write("
" - "<< [["+secLabel(labels[eqCounter-1]).replace(" ","_")+"|"+secLabel(labels[eqCounter-1])+"]]
\n") - wiki.write("
[["+secLabel(sections[secCount+1][0]).replace(" ","_")+"#"+ - secLabel(labels[eqCounter][len("Formula:"):])+"|formula in "+secLabel(sections[secCount+1][0])+"]]
\n") + "<< [["+KLS.secLabel(labels[eqCounter-1]).replace(" ","_")+"|"+KLS.secLabel(labels[eqCounter-1])+"]] \n") + wiki.write("
[["+KLS.secLabel(sections[secCount+1][0]).replace(" ","_")+"#"+ + KLS.secLabel(labels[eqCounter][len("Formula:"):])+"|formula in "+KLS.secLabel(sections[secCount+1][0])+"]]
\n") #if eqS==sections[secCount][1]: if True: - wiki.write("
[["+secLabel(labels[(eqCounter+1)%(endNum+1)]).replace(" ","_")+ - "|"+secLabel(labels[(eqCounter+1)%(endNum+1)])+"]] >>
\n") + wiki.write("
[["+KLS.secLabel(labels[(eqCounter+1)%(endNum+1)]).replace(" ","_")+ + "|"+KLS.secLabel(labels[(eqCounter+1)%(endNum+1)])+"]] >>
\n") wiki.write("\n\n") elif eqCounter==endNum: wiki.write("
\n") if newSec: newSec=False - wiki.write("
<< [["+secLabel(sections[secCount][0]).replace(" ","_")+"|"+ - secLabel(sections[secCount][0])+"]]
\n") + wiki.write("
<< [["+KLS.secLabel(sections[secCount][0]).replace(" ","_")+"|"+ + KLS.secLabel(sections[secCount][0])+"]]
\n") else: wiki.write("
" - "<< [["+secLabel(labels[eqCounter-1]).replace(" ","_")+"|"+secLabel(labels[eqCounter-1])+"]]
\n") + "<< [["+KLS.secLabel(labels[eqCounter-1]).replace(" ","_")+"|"+KLS.secLabel(labels[eqCounter-1])+"]]
\n") wiki.write("
[["+ - secLabel(sections[secCount+1][0]).replace(" ","_")+ - "#"+secLabel(labels[eqCounter][len("Formula:"):])+ - "|formula in "+secLabel(sections[secCount+1][0])+ + KLS.secLabel(sections[secCount+1][0]).replace(" ","_")+ + "#"+KLS.secLabel(labels[eqCounter][len("Formula:"):])+ + "|formula in "+KLS.secLabel(sections[secCount+1][0])+ "]]
\n") - wiki.write("
[["+secLabel(labels[(eqCounter+1)%(endNum+1)].replace(" ","_"))+ - "|"+secLabel(labels[(eqCounter+1)%(endNum+1)])+"]]
\n") + wiki.write("
[["+KLS.secLabel(labels[(eqCounter+1)%(endNum+1)].replace(" ","_"))+ + "|"+KLS.secLabel(labels[(eqCounter+1)%(endNum+1)])+"]]
\n") wiki.write("\n\n") wiki.write("
{\displaystyle \n") @@ -723,7 +565,7 @@ def DLMF(n): wiki.write("\n
\n") wiki.write("\n== Bibliography==\n\n")#should there be a space between bibliography and ==? - r=unmodLabel(labels[eqCounter]) + r=KLS.unmodlabel(labels[eqCounter]) q=r.find("DLMF:")+5 p=r.find(":",q) section=r[q:p] @@ -741,18 +583,18 @@ def DLMF(n): wiki.write("
\n") if newSec: newSec=False - wiki.write("
<< [["+secLabel(sections[secCount][0]).replace(" ","_")+ - "|"+secLabel(sections[secCount][0])+"]]
\n") + wiki.write("
<< [["+KLS.secLabel(sections[secCount][0]).replace(" ","_")+ + "|"+KLS.secLabel(sections[secCount][0])+"]]
\n") else: - wiki.write("
<< [["+secLabel(labels[eqCounter-1]).replace(" ","_")+"|"+ - secLabel(labels[eqCounter-1])+"]]
\n") - wiki.write("
[["+secLabel(sections[secCount+1][0]).replace(" ","_")+"#"+ - secLabel(labels[eqCounter][len("Formula:"):])+"|formula in "+secLabel(sections[secCount+1][0])+"]]
\n") + wiki.write("
<< [["+KLS.secLabel(labels[eqCounter-1]).replace(" ","_")+"|"+ + KLS.secLabel(labels[eqCounter-1])+"]]
\n") + wiki.write("
[["+KLS.secLabel(sections[secCount+1][0]).replace(" ","_")+"#"+ + KLS.secLabel(labels[eqCounter][len("Formula:"):])+"|formula in "+KLS.secLabel(sections[secCount+1][0])+"]]
\n") #if eqS==sections[secCount][1]: #else: if True: - wiki.write("
[["+secLabel(labels[(eqCounter+1)% - (endNum+1)]).replace(" ","_")+"|"+secLabel(labels[(eqCounter+1)%(endNum+1)])+"]] >>
\n") + wiki.write("
[["+KLS.secLabel(labels[(eqCounter+1)% + (endNum+1)]).replace(" ","_")+"|"+KLS.secLabel(labels[(eqCounter+1)%(endNum+1)])+"]] >>
\n") wiki.write("
\n\ndrmf_eof\n") else: #FOR EXTRA EQUATIONS wiki.write("
\n") @@ -764,7 +606,7 @@ def DLMF(n): elif "\\constraint" in line and parse: - #symbols=symbols+getSym(line) + #symbols=symbols+KLS.getSym(line) symLine=line.strip("\n") if hCon: comToWrite=comToWrite+"\n== Constraint(s) ==\n\n" @@ -772,23 +614,23 @@ def DLMF(n): constraint=True math=False conLine="" - #wiki.write("
"+getEq(line)+"

\n") + #wiki.write("
"+KLS.getEq(line)+"

\n") elif "\\substitution" in line and parse: - #symbols=symbols+getSym(line) + #symbols=symbols+KLS.getSym(line) symLine=line.strip("\n") if hSub: comToWrite=comToWrite+"\n== Substitution(s) ==\n\n" hSub=False - #wiki.write("
"+getEq(line)+"

\n") + #wiki.write("
"+KLS.getEq(line)+"

\n") substitution=True math=False subLine="" elif "\\drmfname" in line and parse: math=False - comToWrite="\n== Name ==\n\n
"+getString(line)+"

\n"+comToWrite + comToWrite="\n== Name ==\n\n
"+KLS.getString(line)+"

\n"+comToWrite elif "\\drmfnote" in line and parse: - symbols=symbols+getSym(line) + symbols=symbols+KLS.getSym(line) if hNote: comToWrite=comToWrite+"\n== Note(s) ==\n\n" hNote=False @@ -797,7 +639,7 @@ def DLMF(n): noteLine="" elif "\\proof" in line and parse: - #symbols=symbols+getSym(line) + #symbols=symbols+KLS.getSym(line) symLine=line.strip("\n") if hProof: hProof=False @@ -811,7 +653,7 @@ def DLMF(n): if line[ind:ind+7]=="\\eqref{": pause=True eqR=line[ind:line.find("}",ind)+1] - rLab=getString(eqR) + rLab=KLS.getString(eqR) for l in lLink: if rLab==l[0:l.find("=")-1]: rlabel=l[l.find("=>")+3:l.find("\\n")] @@ -845,9 +687,9 @@ def DLMF(n): if "\\end{equation}" in lines[i+1]: proof=False #symLine+=line.strip("\n") - wiki.write(comToWrite+getEqP(proofLine)+"
\n
\n") + wiki.write(comToWrite+KLS.getEqP(proofLine)+"
\n
\n") comToWrite="" - symbols=symbols+getSym(symLine) + symbols=symbols+KLS.getSym(symLine) symLine="" #wiki.write(line) @@ -855,12 +697,12 @@ def DLMF(n): elif proof: symLine+=line.strip("\n") pauseP=False - #symbols=symbols+getSym(line) + #symbols=symbols+KLS.getSym(line) for ind in range(0,len(line)): if line[ind:ind+7]=="\\eqref{": pause=True eqR=line[ind:line.find("}",ind)+1] - rLab=getString(eqR) + rLab=KLS.getString(eqR) for l in lLink: if rLab==l[0:l.find("=")-1]: rlabel=l[l.find("=>")+3:l.find("\\n")] @@ -891,9 +733,9 @@ def DLMF(n): if "\\end{equation}" in lines[i+1]: proof=False #symLine+=line.strip("\n") - wiki.write(comToWrite+getEqP(proofLine).rstrip("\n")+"\n
\n") + wiki.write(comToWrite+KLS.getEqP(proofLine).rstrip("\n")+"\n
\n") comToWrite="" - symbols=symbols+getSym(symLine) + symbols=symbols+KLS.getSym(symLine) symLine="" elif math: @@ -901,7 +743,7 @@ def DLMF(n): or "\\substitution" in lines[i+1] or "\\proof" in lines[i+1] or "\\drmfnote" in lines[i+1] or "\\drmfname" in lines[i+1]: wiki.write(line.rstrip("\n")) symLine+=line.strip("\n") - symbols=symbols+getSym(symLine) + symbols=symbols+KLS.getSym(symLine) symLine="" wiki.write("\n}\n") else: @@ -909,7 +751,7 @@ def DLMF(n): wiki.write(line) if note and parse: noteLine=noteLine+line - symbols=symbols+getSym(line) + symbols=symbols+KLS.getSym(line) if "\\end{equation}" in lines[i+1] or "\\drmfn" in lines[i+1] \ or "\\constraint" in lines[i+1] \ or "\\substitution" in lines[i+1] \ @@ -919,31 +761,34 @@ def DLMF(n): noteLine=noteLine[0:noteLine.find("\\emph{")]+"\'\'"+noteLine[noteLine.find("\\emph{")+len("\\emph{"): noteLine.find("}",noteLine.find("\\emph{")+len("\\emph{"))]+"\'\'"+\ noteLine[noteLine.find("}",noteLine.find("\\emph{")+len("\\emph{"))+1:] - comToWrite=comToWrite+"
"+getEq(noteLine)+"

\n" + comToWrite=comToWrite+"
"+KLS.getEq(noteLine)+"

\n" if constraint and parse: conLine=conLine+line.replace("&","&
") symLine+=line.strip("\n") - #symbols=symbols+getSym(line) + #symbols=symbols+KLS.getSym(line) if "\\end{equation}" in lines[i+1] or "\\drmfn" in lines[i+1] \ or "\\constraint" in lines[i+1]\ or "\\substitution" in lines[i+1] \ or "\\proof" in lines[i+1]: constraint=False - symbols=symbols+getSym(symLine) + symbols=symbols+KLS.getSym(symLine) symLine="" - wiki.write(comToWrite+"
"+getEq(conLine)+"

\n") + wiki.write(comToWrite+"
"+KLS.getEq(conLine)+"

\n") comToWrite="" if substitution and parse: subLine=subLine+line.replace("&","&
") symLine+=line.strip("\n") - #symbols=symbols+getSym(line) + #symbols=symbols+KLS.getSym(line) if "\\end{equation}" in lines[i+1] or "\\drmfn" in lines[i+1] or \ "\\substitution" in lines[i+1] or "\\constraint" in lines[i+1] or "\\proof" in lines[i+1]: substitution=False - symbols=symbols+getSym(symLine) + symbols=symbols+KLS.getSym(symLine) symLine="" - wiki.write(comToWrite+"
"+getEq(subLine)+"

\n") - comToWrite="" \ No newline at end of file + wiki.write(comToWrite+"
"+KLS.getEq(subLine)+"

\n") + comToWrite="" + +if __name__ == "__main__": + DLMF(0); \ No newline at end of file diff --git a/Azeem/src/tex2Wiki.py b/Azeem/src/tex2Wiki.py index 3c5200d..20f5cd2 100644 --- a/Azeem/src/tex2Wiki.py +++ b/Azeem/src/tex2Wiki.py @@ -293,7 +293,7 @@ def setup_label_links(ofname): def readin(ofname,glossary,mmd): # try: - for jsahlfkjsd in range(0, 1): + for iterations in range(0, 1): tex = open(ofname, 'r') main_file = open(mmd, "r") mainText = main_file.read() From 633742482dfbb441398b8c04744ebc6d0b176c64 Mon Sep 17 00:00:00 2001 From: Philip Wang Date: Thu, 19 May 2016 18:24:23 -0400 Subject: [PATCH 092/402] change wiki.write() to append_text() in DLMFtex2Wiki.py (#45) --- Azeem/src/DLMFtex2Wiki.py | 165 +++++++++++++++++++------------------- 1 file changed, 83 insertions(+), 82 deletions(-) diff --git a/Azeem/src/DLMFtex2Wiki.py b/Azeem/src/DLMFtex2Wiki.py index 06b4d3e..f78f42f 100644 --- a/Azeem/src/DLMFtex2Wiki.py +++ b/Azeem/src/DLMFtex2Wiki.py @@ -39,6 +39,7 @@ def modLabel(label): newlabel+="0"+num return(newlabel) + def DLMF(n): for iterations in range(0,1): try: @@ -108,10 +109,10 @@ def DLMF(n): for i in range(0,len(lines)): line=lines[i] if "\\begin{document}" in line: - #wiki.write("drmf_bof\n") + #KLS.append_text("drmf_bof\n") parse=True elif "\\end{document}" in line: - #wiki.write("\n") + #KLS.append_text("\n") mainPrepend+="\n" mainText="" mainText=mainPrepend+mainText @@ -124,7 +125,7 @@ def DLMF(n): mainWrite.close() main.close() os.system("cp -f ZetaFunctions.mmd.new ZetaFunctions.mmd") - #wiki.write("\ndrmf_eof\n") + #KLS.append_text("\ndrmf_eof\n") parse=False elif "\\title" in line and parse: labels.append("Zeta and Related Functions") @@ -156,42 +157,42 @@ def DLMF(n): if "\\section" in line: parse=True secCounter+=1 - wiki.write("drmf_bof\n") - wiki.write("\'\'\'"+KLS.secLabel(KLS.getString(line))+"\'\'\'\n") - wiki.write("{{DISPLAYTITLE:"+(sections[secCounter][0])+"}}\n") - wiki.write("") + len(">> \n"):] + if "
" in h: + h = h[0:h.find("
")] + prev = pages[(x - 1) % len(pages)] + # print h + next = pages[(x + 1) % len(pages)] + cur = "Main Page" + header = "\n
\n" + header += "
<< [[" + prev.replace(" ", "_") + "|" + prev + "]]
\n" + header += "
[[" + cur.replace(" ", "_") + "|" + cur + "]]
\n" + header += "
[[" + next.replace(" ", "_") + "|" + next + "]] >>
\n" + header += "
" + + footer = "
\n" + footer += "
<< [[" + prev.replace(" ", "_") + "|" + prev + "]]
\n" + footer += "
[[" + cur.replace(" ", "_") + "|" + cur + "]]
\n" + footer += "
[[" + next.replace(" ", "_") + "|" + next + "]] >>
\n" + footer += "
" + main.write("drmf_bof\n" + title + header + h + footer + "\ndrmf_eof\n") + x += 1 diff --git a/KLS-main-page/list.py b/KLS-main-page/list.py new file mode 100644 index 0000000..1e33e9a --- /dev/null +++ b/KLS-main-page/list.py @@ -0,0 +1,86 @@ +import os +import time + +localtime = time.asctime(time.localtime(time.time())).replace(" ", "_") +os.system("cp main_page.mmd main_page.mmd" + str(localtime)) + + +def compString(a, b): + x = a[0:a.find("|")] + y = b[0:b.find("|")] + if len(x) > len(y): + max = len(y) + ret = False + else: + max = len(x) + ret = True + if True: + for i in range(0, max): + if ord(x[i].lower()) < ord(y[i].lower()): + return True + elif ord(x[i].lower()) > ord(y[i].lower()): + return False + return ret + + +def sumOrd(x): + sum = 0 + for s in x: + sum += ord(s) + return (sum) + + +def sort(x): + ret = [] + ret.append(x[0]) + for i in range(1, len(x)): + if compString(x[i], ret[0]): + ret = [x[i]] + ret + elif compString(ret[len(ret) - 1], x[i]): + ret.append(x[i]) + else: + for p in range(1, len(ret)): + if compString(x[i], ret[p]): + ret = ret[0:p] + [x[i]] + ret[p:] + break + return (ret) + + +'''listFile=open("list.txt","w") +files_in_dir=os.listdir(".") +toWrite=[] +for g in files_in_dir: + if ".new" in g and "Definition:" in g: + g=g[:g.find(".new")] + element="* [["+g+"|"+g[len("Definition:"):]+"]]" + toWrite.append(element) + toWrite=sort(toWrite) +text="" +for g in toWrite: + text+=(g+"\n") +listFile.close() +listFile=open("list.txt","r") +text=listFile.read()''' +main = open("main_page.mmd", "r") +mainText = main.read() +mainLines = mainText.split("\n") +main.close() +toWrite = [] +for z in mainLines: + if "\'\'\'Definition:" in z: + g = z[z.find("\'\'\'Definition:") + len("\'\'\'"):z.find("\'\'\'", z.find("\'\'\'Definition:") + 4)] + element = "* [[" + g + "|" + g[len("Definition:"):] + "]]" + toWrite.append(element) +main = open("main_page.mmd", "w") +s = mainText.find("== Definition Pages ==") +e = mainText.find("= Copyright =") +text = "" +for g in toWrite: + text += (g + "\n") +# print text +# print mainText[:s] +newText = mainText[ + :s] + "== Definition Pages ==\n\n
\n" + text + "
\n\n" + mainText[ + e:] +main.write(newText) +main.close() diff --git a/KLS-main-page/modMain.py b/KLS-main-page/modMain.py new file mode 100644 index 0000000..3ec6d8f --- /dev/null +++ b/KLS-main-page/modMain.py @@ -0,0 +1,117 @@ +file = open("main_page.mmd", "r") +import csv + +lines = file.read() +lines = lines.replace("The LaTeX DLMF macro \'\'\'\\", "The LaTeX DLMF and DRMF macro \'\'\'\\") +file.close() +file = open("main_page.mmd", "w") +flag = True +gCSV = csv.reader(open('new.Glossary.csv', 'rb'), delimiter=',', quotechar='\"') +toWrite = "" +i = 0 + + +def atSort(l): + return (sorted(l, key=lambda x: x.count("@"))) + + +def remOpt(p): + parse = True + ret = "" + for x in p: + if parse: + if x == "[": + parse = False + else: + ret += x + else: + if x == "]": + parse = True + return ret + + +def findAllPos(g): + key = g[3] + orig = g[0] + r = key[:key.find("|")] + key = key[key.find("|") + 1:] + keys = key.split(":") + ret = [] + for x in keys: + if x == "F" or x == "FO" or x == "O" or x == "O1": + if orig.find("@") != -1: + ret.append(orig[0:orig.find("@")]) + if x == "O": + ret.append(orig) + else: + ret.append(orig) + if x == "FnO": # remove optional parameters + if orig.find("@") != -1: + ret.append(remOpt(orig[0:orig.find("@")])) + else: + ret.append(remOpt(orig)) + if x == "P" or x == "PO": + ret.append(orig.replace("@@", "@")) + if x == "PnO": # remove optional parameters + ret.append(remOpt(orig.replace("@@", "@"))) + if x == "nP" or x == "nPO" or x == "PS" or x == "O2": + ret.append(orig) + if x == "nPnO": # remove optional paramters + ret.append(remOpt(orig)) + if "fo" in x: + ret.append(orig.replace("@@@", "@" * int(x[2]))) + return atSort(ret), r + + +categories = open("Categories.txt", "r") +cats = categories.readlines() +categories.close() +for x in range(0, len(cats)): + cats[x] = cats[x].strip() +while flag: + n = lines.find("\'\'\'Definition:", i) + if n == -1: + flag = False + # elif lines.find("This macro can be called in the following way")1: plural =1; if count=1: plural=0 + if count == 1: plural = 0 + for t in cats: + if s + " -" in t: + category = "This macro is in the category of" + t[t.find("-") + 2:] + break + new = category + "\n\nIn math mode, this macro can be called in the following way" + "s" * plural + ":\n\n" + for q in range(0, len(listCalls)): + c = listCalls[q] + new += ":\'\'\'" + c + "\'\'\'" + " produces {\\displaystyle " + c + "}
\n" + # Now add the multiple ways \macroname{n}@... produces \macroname{n}@... + toWrite += new + "\n" + i = lines.find("These are defined by", p) +file.write(toWrite + lines[i:]) diff --git a/KLS-main-page/symbolsList2.py b/KLS-main-page/symbolsList2.py new file mode 100644 index 0000000..a79c590 --- /dev/null +++ b/KLS-main-page/symbolsList2.py @@ -0,0 +1,373 @@ +def getSym(line): # Gets all symbols on a line for symbols list + line = line.replace("\n", " ") + symList = [] + if line == "": + return symList + symbol = "" + symFlag = False + argFlag = False + cC = 0 + for i in range(0, len(line)): + # if "MeixnerPollaczek{\\lambda}{n}@{x}{\\phi}=\frac{\\pochhammer{2\\lambda}{n}}" in line: + # print symbol + c = line[i] + if symFlag: + if c == "{" or c == "[": + cC += 1 + argFlag = True + if c != "}" and c != "]": + if argFlag or c.isalpha(): + symbol += c + else: + symFlag = False + argFlag = False + symList.append(symbol) + symList += (getSym(symbol)) + symbol = "" + else: + cC -= 1 + symbol += c + if i + 1 == len(line): + p = "" + else: + p = line[i + 1] + if cC == 0 and p != "{" and p != "[" and p != "@": + symFlag = False + # if "monicAskeyWilson{n+1}@{x}{a}{b}{c}{d}{q}+\\frac{1}{2}" in line: + # print(symbol,p) + symList.append(symbol) + # if symbol=="binom{n}{k}":print line + symList += (getSym(symbol)) + argFlag = False + symbol = "" + + elif c == "\\": + symFlag = True + + # if "MeixnerPollaczek{\\lambda}{n}@{x}{\\phi}=\frac{\\pochhammer{2\\lambda}{n}}" in line: + # print symbol + symList.append(symbol) + # if "ctsbigqHermite" in line: + # print(symList) + symList += getSym(symbol) + return (symList) + + +import csv + +# files_in_dir=os.listdir(".") +mainPage = open("main_page.mmd", "r") +mainLines = mainPage.read() +mainLines = mainLines[mainLines.find("drmf_bof"):mainLines.rfind("drmf_eof") + len("drmf_eof\n")] +mainPage.close() +main = open("main_page.mmd", "w") +mainPages = mainLines.split("drmf_eof") +if mainPages[-1].strip() == "": + mainPages = mainPages[0:len(mainPages) - 1] +print len(mainPages) +glossary = open('new.Glossary.csv', 'rb') +gCSV = csv.reader(glossary, delimiter=',', quotechar='\"') +lGlos = glossary.readlines() +wiki = "" +for g in mainPages: + if not "\'\'\'Definition:" in g: + h = g.replace("drmf_eof", "") + h = g.replace("drmf_bof", "") + toWrite = "drmf_bof" + h + "drmf_eof\n" + main.write(toWrite) + else: + h = g.replace("drmf_eof", "") + h = g.replace("drmf_bof", "") + h = h.strip("\n") + sflag1 = False + sflag2 = False + if h.find("== Symbols List ==") != -1: + toWrite = h[0:h.find("== Symbols List ==")] + h = toWrite + else: + sflag1 = True + if h.find("drmf_foot") == -1: + toWrite = h + sflag2 = True + else: + toWrite = h[0:h.find("
"):]+t[0:t.find("
")] + t = t.replace("\n", " ") + # if "Symbol" in t:print t + symbols = getSym(t) + # print symbols + # if "qsin" in g:print t,symbols + # if "Neumann" in g:print t,"\n",symbols + if True: + toWrite += ("== Symbols List ==\n\n") + newSym = [] + # if "09.07:04" in label: + # print(symbols) + for x in symbols: + flagA = True + # if x not in newSym: + cN = 0 + cC = 0 + flag = True + ArgCx = 0 + for z in x: + if z.isalpha() and flag: + cN += 1 + else: + flag = False + if z == "{" or z == "[": + cC += 1 + if z == "}" or z == "]": + cC -= 1 + if cC == 0: + ArgCx += 1 + noA = x[:cN] + for y in newSym: + cN = 0 + cC = 0 + ArgC = 0 + flag = True + for z in y: + if z.isalpha() and flag: + cN += 1 + else: + flag = False + if z == "{" or z == "[": + cC += 1 + if z == "}" or z == "]": + cC -= 1 + if cC == 0: + ArgC += 1 + + if y[:cN].strip() == noA.strip(): # and ArgC==ArgCx: + flagA = False + # print(x,y) + break + if flagA: + newSym.append(x) + newSym.reverse() + # print newSym + # if "14.27:02" in label: + # print newSym + symF = False + finSym = [] + for s in range(len(newSym) - 1, -1, -1): + # print(newSym[s]) + symbolPar = "\\" + newSym[s] + ArgCx = 0 + parCx = 0 + parFlag = False + cC = 0 + for z in symbolPar: + if z == "@": + parFlag = True + elif z.isalpha(): + cN += 1 + else: + if z == "{" or z == "[": + cC += 1 + if z == "}" or z == "]": + cC -= 1 + if cC == 0: + if parFlag: + parCx += 1 + else: + ArgCx += 1 + + if symbolPar.find("{") != -1 or symbolPar.find("[") != -1: + if symbolPar.find("[") == -1: + symbol = symbolPar[0:symbolPar.find("{")] + elif symbolPar.find("{") == -1 or symbolPar.find("[") < symbolPar.find("{"): + symbol = symbolPar[0:symbolPar.find("[")] + else: + symbol = symbolPar[0:symbolPar.find("{")] + else: + symbol = symbolPar + # if "14.27:02" in label:print symbolPar,symbol + numArg = parCx + numPar = ArgCx + gFlag = False + checkFlag = False + get = False + gCSV = csv.reader(open('new.Glossary.csv', 'rb'), delimiter=',', quotechar='\"') + preG = "" + symbol = symbol.strip("}").strip("]") + for S in gCSV: + G = S + ArgCx = 0 + parCx = 0 + parFlag = False + cC = 0 + ind = G[0].find("@") + if ind == -1: + ind = len(G[0]) - 1 + for z in G[0]: + if z == "@": + parFlag = True + elif z.isalpha(): + cN += 1 + else: + if z == "{" or z == "[": + cC += 1 + if z == "}" or z == "]": + cC -= 1 + if cC == 0: + if parFlag: + parCx += 1 + else: + ArgCx += 1 + if G[0].find(symbol) == 0 and (len(G[0]) == len(symbol) or not G[0][ + len(symbol)].isalpha()): # and (numPar!=0 or numArg!=0): + checkFlag = True + get = True + preG = S + # if "14.27:02" in label:print "ok" + symbol + + # if "9.07:04" in label:print symbol,numPar,numArg,parCx,ArgCx + + elif checkFlag: + get = True + checkFlag = False + if (get): + if get: + G = preG + # print preG + get = False + checkFlag = False + # if "14.27:02" in label:print "yes"+symbol + if True: + if symbolPar.find("@") != -1: + Q = symbolPar[:symbolPar.find("@")] + else: + Q = symbolPar + listArgs = [] + if len(Q) > len(symbol) and (Q[len(symbol)] == "{" or Q[len(symbol)] == "["): + ap = "" + for o in range(len(symbol), len(Q)): + if Q[o] == "{" or z == "[": + argFlag = True + elif Q[o] == "}" or z == "]": + argFlag = False + listArgs.append(ap) + ap = "" + else: + ap += Q[o] + '''websiteU=g[g.find("http://"):].strip("\n") + k=0 + websites=[] + for r in range(0,len(websiteU)): + if websiteU[r]==",": + if websiteU[k:r].find("http://")!=-1: + websites.append(websiteU[k:r+1].strip(" ")) + k=r + + + if websiteU[k:r].find("http://")!=-1: + websites.append(websiteU[k:r+1].strip(" ")) + websiteF="" + for d in websites: + websiteF=websiteF+" ["+d+" "+d+"]"''' + # websiteF=G[4].strip("\n") + websiteF = "" + # print("(",G,")") + web1 = G[5] + for t in range(5, len(G)): + if G[t] != "": + websiteF = websiteF + " [" + G[t] + " " + G[t] + "]" + + # p1=Q + # if Q.find("@")!=-1: + # p1=Q[:Q.find("@")] + p1 = G[4].strip("$") + p1 = "{\\displaystyle " + p1 + "}" + # if checkFlag: + # print(p1) + new1 = "" + new2 = "" + pause = False + mathF = True + '''for k in range(0,len(p1)): + if p1[k]=="$": + if mathF: + new1+="{\\displaystyle " + else: + new1+="}" + mathF=not mathF + + elif p1[k]=="#" and p1[k+1].isdigit(): + pause=True + elif pause: + num=int(p1[k]) + #letter=chr(num+96) + letter=listArgs[num-1] + new1+=letter + pause=False + + else: + new1+=p1[k]''' + p2 = G[1] + for k in range(0, len(p2)): + if p2[k] == "$": + if mathF: + new2 += "{\\displaystyle " + else: + new2 += "}" + mathF = not mathF + else: + new2 += p2[k] + # p1=new1 + p2 = new2 + finSym.append(web1 + " " + p1 + "] : " + p2 + " :" + websiteF) + break + # gFlag=True + # if not symF: + # symF=True + # wiki.write("[") + # else: + # wiki.write("
\n") + # wiki.write("[") + # wiki.write(g[g.find("http://"):].strip("\n")) + # wiki.write(" ") + # wiki.write(getEq(g[g.find("{$"):]).strip("\n").replace("\\\\","\\")) + # wiki.write("] : ") + # wiki.write(g[g.find(" {")+2:g.find("} ",g.find(" {")+1)]) + # wiki.write(" : [") + # wiki.write(g[g.find("http://"):].strip("\n")) + # wiki.write(" ") + # wiki.write(g[g.find("http://"):].strip("\n")) + # wiki.write("] ")''' + + # preG=S + if not gFlag: + del newSym[s] + + gFlag = True + # finSym.reverse() + for y in finSym: + # print(finSym) + if gFlag: + gFlag = False + toWrite += ("[" + y) + else: + toWrite += ("
\n[" + y) + + toWrite += ("\n
\ndrmf_eof\n\n") + if sflag2: + toWrite += "\n" + main.write(toWrite) From 2bdb8455a232fbf6dbf5c23a6046697409f3f7ae Mon Sep 17 00:00:00 2001 From: Joon Bang Date: Wed, 20 Jul 2016 14:58:36 -0400 Subject: [PATCH 184/402] Fix issues with Azeem's code --- main_page/main_page.mmd | 4943 +++++++++++++++++++++++++++++++++ main_page/src/KLS.py | 19 + main_page/src/__init__.py | 0 main_page/src/headers.py | 64 + main_page/src/list.py | 59 + main_page/src/main.py | 8 + main_page/src/mod_main.py | 125 + main_page/src/symbols_list.py | 285 ++ main_page/symbolsList.py~ | 278 ++ main_page/tex2Wiki.py~ | 966 +++++++ 10 files changed, 6747 insertions(+) create mode 100644 main_page/main_page.mmd create mode 100644 main_page/src/KLS.py create mode 100644 main_page/src/__init__.py create mode 100644 main_page/src/headers.py create mode 100644 main_page/src/list.py create mode 100644 main_page/src/main.py create mode 100644 main_page/src/mod_main.py create mode 100644 main_page/src/symbols_list.py create mode 100644 main_page/symbolsList.py~ create mode 100644 main_page/tex2Wiki.py~ diff --git a/main_page/main_page.mmd b/main_page/main_page.mmd new file mode 100644 index 0000000..ff6a93c --- /dev/null +++ b/main_page/main_page.mmd @@ -0,0 +1,4943 @@ +drmf_bof +'''Main Page''' +The '''NIST Digital Repository of Mathematical Formulae''' is designed for +a mathematically literate audience and should + +# facilitate interaction among a [[community]] of mathematicians and scientists interested in [[wikipedia:compendia|compendia]] formulae data for '''orthogonal polynomials and special functions'''; +# be [[expandable]], allowing the input of new formulae from the literature; +# represent the context-free full [[semantic]] information concerning individual formulas; +# have a user friendly, consistent, and hyperlinkable viewpoint and authoring [[perspective]]; +# contain easily [[Special:MathSearch|searchable]] mathematics; and +# take advantage of modern [[MathML]] tools for easy to read, scalably rendered content driven mathematics. + +For more information see [http://link.springer.com/chapter/10.1007%2F978-3-319-08434-3_30 Digital Repository of Mathematical Formulae] or [http://arxiv.org/abs/1404.6519 arXiv:1404.6519]. + +== Sample Seeding Project Implementations == + +DLMF: [[Zeta and Related Functions]] + +KLS and KLSadd: [[Orthogonal Polynomials]] + +== Useful Pages == + +[[GitHub|How to Upload Your Project to GitHub]] + +[[Wikilabs|How to get access to Wikilabs/Wikitech]] + +[https://github.com/DRMF/DRMF/blob/master/doc/upload_and_connect.MD Upload and Connect] + +== Sample formula home pages with DLMF proofs given == + +
+* [[Formula:DLMF:25.2:E9]] +* [[Formula:DLMF:25.5:E2]] +* [[Formula:DLMF:25.5:E4]] +* [[Formula:DLMF:25.5:E6]] +* [[Formula:DLMF:25.8:E7]] +* [[Formula:DLMF:25.8:E8]] +* [[Formula:DLMF:25.11:E7]] +* [[Formula:DLMF:25.11:E10]] +* [[Formula:DLMF:25.11:E11]] +* [[Formula:DLMF:25.11:E15]] +* [[Formula:DLMF:25.11:E17]] +* [[Formula:DLMF:25.11:E24]] +* [[Formula:DLMF:25.11:E27]] +* [[Formula:DLMF:25.11:E28]] +* [[Formula:DLMF:25.11:E30]] +* [[Formula:DLMF:25.11:E31]] +* [[Formula:DLMF:25.11:E35]] +
+ += Digital Repository of Mathematical Formulae = +
+# [[Algebraic and Analytic Methods]] +# [[Asymptotic Approximations]] +# [[Numerical Methods]] +# [[Elementary Functions]] +# [[Gamma Function]] +# [[Exponential, Logarithmic, Sine, and Cosine Integrals]] +# [[Error Functions, Dawson’s and Fresnel Integrals]] +# [[Incomplete Gamma and Related Functions]] +# [[Airy and Related Functions]] +# [[Bessel Functions]] +# [[Struve and Related Functions]] +# [[Parabolic Cylinder Functions]] +# [[Confluent Hypergeometric Functions]] +# [[Legendre and Related Functions]] +# [[Hypergeometric Function]] +# [[Generalized Hypergeometric Functions and Meijer G-Function|Generalized Hypergeometric Functions and Meijer ''G''-Function]] +# [[q-Hypergeometric and Related Functions|''q''-Hypergeometric and Related Functions]] +# [[Orthogonal Polynomials]] +# [[Elliptic Integrals]] +# [[Theta Functions]] +# [[Multidimensional Theta Functions]] +# [[Jacobian Elliptic Functions]] +# [[Weierstrass Elliptic and Modular Functions]] +# [[Bernoulli and Euler Polynomials]] +# [[Zeta and Related Functions]] +# [[Combinatorial Analysis]] +# [[Functions of Number Theory]] +# [[Mathieu Functions and Hill’s Equation]] +# [[Lamé Functions]] +# [[Spheroidal Wave Functions]] +# [[Heun Functions]] +# [[Painlevé Transcendents]] +# [[Coulomb Functions]] +# [[3j,6j,9j Symbols|''3j,6j,9j'' Symbols]] +# [[Functions of Matrix Argument]] +# [[Integrals with Coalescing Saddles]] +
+ +== Definition Pages == + +
+* [[Definition:AffqKrawtchouk|AffqKrawtchouk]] +* [[Definition:AlSalamCarlitzI|AlSalamCarlitzI]] +* [[Definition:AlSalamCarlitzII|AlSalamCarlitzII]] +* [[Definition:AlSalamIsmail|AlSalamIsmail]] +* [[Definition:AntiDer|AntiDer]] +* [[Definition:BesselPolyIIparam|BesselPolyIIparam]] +* [[Definition:BesselPolyTheta|BesselPolyTheta]] +* [[Definition:bigqJacobiIVparam|bigqJacobiIVparam]] +* [[Definition:bigqLaguerre|bigqLaguerre]] +* [[Definition:bigqLegendre|bigqLegendre]] +* [[Definition:CiglerqChebyT|CiglerqChebyT]] +* [[Definition:CiglerqChebyU|CiglerqChebyU]] +* [[Definition:ctsbigqHermite|ctsbigqHermite]] +* [[Definition:ctsdualqHahn|ctsdualqHahn]] +* [[Definition:ctsqHahn|ctsqHahn]] +* [[Definition:ctsqJacobi|ctsqJacobi]] +* [[Definition:ctsqLaguerre|ctsqLaguerre]] +* [[Definition:ctsqLegendre|ctsqLegendre]] +* [[Definition:f|f]] +* [[Definition:GenHermite|GenHermite]] +* [[Definition:GottliebLaguerre|GottliebLaguerre]] +* [[Definition:Int|Int]] +* [[Definition:JacksonqBesselII|JacksonqBesselII]] +* [[Definition:JacksonqBesselIII|JacksonqBesselIII]] +* [[Definition:littleqLegendre|littleqLegendre]] +* [[Definition:lrselection|lrselection]] +* [[Definition:monicAlSalamCarlitzI|monicAlSalamCarlitzI]] +* [[Definition:monicAlSalamCarlitzII|monicAlSalamCarlitzII]] +* [[Definition:monicAlSalamChihara|monicAlSalamChihara]] +* [[Definition:monicAskeyWilson|monicAskeyWilson]] +* [[Definition:monicBesselPoly|monicBesselPoly]] +* [[Definition:monicbigqJacobi|monicbigqJacobi]] +* [[Definition:monicbigqLaguerre|monicbigqLaguerre]] +* [[Definition:monicbigqLegendre|monicbigqLegendre]] +* [[Definition:monicCharlier|monicCharlier]] +* [[Definition:monicChebyT|monicChebyT]] +* [[Definition:monicChebyU|monicChebyU]] +* [[Definition:monicctsbigqHermite|monicctsbigqHermite]] +* [[Definition:monicctsdualHahn|monicctsdualHahn]] +* [[Definition:monicctsdualqHahn|monicctsdualqHahn]] +* [[Definition:monicctsHahn|monicctsHahn]] +* [[Definition:monicctsqHahn|monicctsqHahn]] +* [[Definition:monicctsqHermite|monicctsqHermite]] +* [[Definition:monicctsqJacobi|monicctsqJacobi]] +* [[Definition:monicctsqLaguerre|monicctsqLaguerre]] +* [[Definition:monicctsqLegendre|monicctsqLegendre]] +* [[Definition:monicctsqUltra|monicctsqUltra]] +* [[Definition:monicdiscrqHermiteI|monicdiscrqHermiteI]] +* [[Definition:monicdiscrqHermiteII|monicdiscrqHermiteII]] +* [[Definition:monicdualHahn|monicdualHahn]] +* [[Definition:monicdualqHahn|monicdualqHahn]] +* [[Definition:monicdualqKrawtchouk|monicdualqKrawtchouk]] +* [[Definition:monicHahn|monicHahn]] +* [[Definition:monicHermite|monicHermite]] +* [[Definition:monicJacobi|monicJacobi]] +* [[Definition:monicKrawtchouk|monicKrawtchouk]] +* [[Definition:monicLaguerre|monicLaguerre]] +* [[Definition:monicLegendrePoly|monicLegendrePoly]] +* [[Definition:moniclittleqJacobi|moniclittleqJacobi]] +* [[Definition:moniclittleqLaguerre|moniclittleqLaguerre]] +* [[Definition:moniclittleqLegendre|moniclittleqLegendre]] +* [[Definition:monicMeixner|monicMeixner]] +* [[Definition:monicMeixnerPollaczek|monicMeixnerPollaczek]] +* [[Definition:monicpseudoJacobi|monicpseudoJacobi]] +* [[Definition:monicqBesselPoly|monicqBesselPoly]] +* [[Definition:monicqCharlier|monicqCharlier]] +* [[Definition:monicqKrawtchouk|monicqKrawtchouk]] +* [[Definition:monicqLaguerre|monicqLaguerre]] +* [[Definition:monicqMeixner|monicqMeixner]] +* [[Definition:monicqMeixnerPollaczek|monicqMeixnerPollaczek]] +* [[Definition:monicqRacah|monicqRacah]] +* [[Definition:monicqtmqKrawtchouk|monicqtmqKrawtchouk]] +* [[Definition:monicRacah|monicRacah]] +* [[Definition:monicStieltjesWigert|monicStieltjesWigert]] +* [[Definition:monicUltra|monicUltra]] +* [[Definition:monicWilson|monicWilson]] +* [[Definition:NeumannFactor|NeumannFactor]] +* [[Definition:normctsdualHahnStilde|normctsdualHahnStilde]] +* [[Definition:normctsHahnptilde|normctsHahnptilde]] +* [[Definition:normJacobiR|normJacobiR]] +* [[Definition:normWilsonWtilde|normWilsonWtilde]] +* [[Definition:poly|poly]] +* [[Definition:qBesselPoly|qBesselPoly]] +* [[Definition:qCharlier|qCharlier]] +* [[Definition:qDigamma|qDigamma]] +* [[Definition:qHyperrWs|qHyperrWs]] +* [[Definition:qExpKLS|qExpKLS]] +* [[Definition:qexpKLS|qexpKLS]] +* [[Definition:qcosKLS|qcosKLS]] +* [[Definition:qMeixner|qMeixner]] +* [[Definition:qKrawtchouk|qKrawtchouk]] +* [[Definition:monicqHahn|monicqHahn]] +* [[Definition:dualqHahn|dualqHahn]] +* [[Definition:dualqKrawtchouk|dualqKrawtchouk]] +* [[Definition:littleqLaguerre|littleqLaguerre]] +* [[Definition:qsinKLS|qsinKLS]] +* [[Definition:qSinKLS|qSinKLS]] +* [[Definition:qCosKLS|qCosKLS]] +* [[Definition:Wilson|Wilson]] +* [[Definition:Racah|Racah]] +* [[Definition:ctsdualHahn|ctsdualHahn]] +* [[Definition:ctsHahn|ctsHahn]] +* [[Definition:Hahn|Hahn]] +* [[Definition:dualHahn|dualHahn]] +* [[Definition:qRacah|qRacah]] +* [[Definition:normctsdualqHahnptilde|normctsdualqHahnptilde]] +* [[Definition:normctsqHahnptilde|normctsqHahnptilde]] +* [[Definition:qHahn|qHahn]] +* [[Definition:AlSalamChihara|AlSalamChihara]] +* [[Definition:qinvAlSalamChihara|qinvAlSalamChihara]] +* [[Definition:monicqinvAlSalamChihara|monicqinvAlSalamChihara]] +* [[Definition:monicAffqKrawtchouk|monicAffqKrawtchouk]] +* [[Definition:qMeixnerPollaczek|qMeixnerPollaczek]] +* [[Definition:qtmqKrawtchouk|qtmqKrawtchouk]] +* [[Definition:qLaguerre|qLaguerre]] +* [[Definition:ctsqHermite|ctsqHermite]] +* [[Definition:StieltjesWigert|StieltjesWigert]] +* [[Definition:discrqHermiteI|discrqHermiteI]] +* [[Definition:discrqHermiteII|discrqHermiteII]] +* [[Definition:StieltjesConstants|StieltjesConstants]] +* [[Definition:PolylogarithmS|PolylogarithmS]] +* [[Definition:HarmonicNumber|HarmonicNumber]] +* [[Definition:LucasL|LucasL]] +* [[Definition:HurwitzLerchPhi|HurwitzLerchPhi]] +* [[Definition:GoldenRatio|GoldenRatio]] +* [[Definition:WhitPsi|WhitPsi]] +* [[Definition:GompertzConstant|GompertzConstant]] +* [[Definition:rabbitConstant|rabbitConstant]] +* [[Definition:FibonacciNumber|FibonacciNumber]] +* [[Definition:Fibonacci|Fibonacci]] +
+ += Copyright = + +Pursuant to U.S. Code, Title 17, Chapter 1, Section 105: +
+Copyright protection under this title is not available for any +work of the United States Government, but the United States Government is not +precluded from receiving and holding copyrights transferred to it by assignment, +bequest, or otherwise. +
+the National Institute of Standards and Technology (NIST), United States Department +of Commerce, this website, a work of the United States Government, is in the public +domain. + += Disclaimer = + +While NIST has made every effort to ensure the accuracy and reliability of the +information in the DLMF, the DLMF is expressly provided “AS-IS.” NIST makes NO WARRANTY +OF ANY TYPE, including no warranties of merchantability or fitness for a particular +purpose. NIST makes no warranties or representations as to the correctness, accuracy, +or reliability of the DLMF. As a condition of using the DLMF, you explicitly release +NIST from any and all liabilities for any damage of any type that may result from +errors or omissions in the DLMF. + +Certain products, commercial and otherwise, are mentioned in the DLMF. These mentions +are for informational purposes only, and do not imply recommendation or endorsement by NIST. + += Privacy and Security Notice = + +The information we receive depends upon what you do when visiting our site. We collect +no personally identifying information about you when you visit our site, unless you +choose to provide that information to us. + +''If you visit our site to read or download information, we collect and store'' '''only''' +''the following information:'' + +* The name of the internet domain from which you accessed our site and the Internet Protocol (IP) address or host name which may or may not identify a specific computer. +* The date and time you access our site. +* The pages you visit. +* The Internet address of the website from which you linked directly to our site. + +This information may be preserved in web access logs for an indefinite period of time. +The data are used to prevent security breaches, insure the integrity of data on our +servers, and summarize usage of our systems. We do not collect information through use +of cookies. Other than the information described above which is collected from all +visitors, we do not collect any information online from children. + +''If you identify yourself by sending an E-mail containing personal information:'' + +* You also may decide to send us personally identifying information (for example, your mailing address) in an electronic mail message or web-based form requesting that information or products be mailed to you. Information collected in this manner is used solely for responding to your request unless noted in the specific web page where your request is made. + +'''Note:''' NIST is ''not'' responsible for the privacy practices employed by non-NIST sites that may link to or from the NIST website. +drmf_eof +drmf_bof +'''Definition:AffqKrawtchouk''' +
+
<< [[Definition:Fibonacci|Definition:Fibonacci]]
+
[[Main_Page|Main Page]]
+
[[Definition:AlSalamCarlitzI|Definition:AlSalamCarlitzI]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\AffqKrawtchouk''' represents the Affine q-Krawtchouk polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\AffqKrawtchouk{n}''' produces {\displaystyle \AffqKrawtchouk{n}}
+:'''\AffqKrawtchouk{n}@{q^{-x}}{p}{N}{q}''' produces {\displaystyle \AffqKrawtchouk{n}@{q^{-x}}{p}{N}{q}}
+:'''\AffqKrawtchouk{n}@@{q^{-x}}{p}{N}{q}''' produces {\displaystyle \AffqKrawtchouk{n}@@{q^{-x}}{p}{N}{q}}
+ +These are defined by + +{\displaystyle +\AffqKrawtchouk{n}@{q^{-x}}{p}{N}{q}:=\qHyperrphis{3}{2}@@{q^{-n},0,q^{-x}}{pq,q^{-N}}{q}{q} +} +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:AffqKrawtchouk {\displaystyle K^{\mathrm{Aff}}_{n}}] : affine {\displaystyle q}-Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:AffqKrawtchouk http://drmf.wmflabs.org/wiki/Definition:AffqKrawtchouk]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1] +
+
<< [[Definition:Fibonacci|Definition:Fibonacci]]
+
[[Main_Page|Main Page]]
+
[[Definition:AlSalamCarlitzI|Definition:AlSalamCarlitzI]] >>
+
+drmf_eof +drmf_bof +'''Definition:AlSalamCarlitzI''' +
+
<< [[Definition:AffqKrawtchouk|Definition:AffqKrawtchouk]]
+
[[Main_Page|Main Page]]
+
[[Definition:AlSalamCarlitzII|Definition:AlSalamCarlitzII]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\AlSalamCarlitzI''' represents the Al-Salam-Carlitz I polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\AlSalamCarlitzI{a}{n}''' produces {\displaystyle \AlSalamCarlitzI{a}{n}}
+:'''\AlSalamCarlitzI{a}{n}@{x}{q}''' produces {\displaystyle \AlSalamCarlitzI{a}{n}@{x}{q}}
+ +These are defined by +{\displaystyle +\AlSalamCarlitzI{a}{n}@{x}{q}:=(-a)^nq^{\binom{n}{2}}\,\qHyperrphis{2}{1}@@{q^{-n},x^{-1}}{0}{q}{\frac{qx}{a}}. +} +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzI {\displaystyle U^{(n)}_{\alpha}}] : Al-Salam-Carlitz I polynomial : [http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzI http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzI]
+[http://dlmf.nist.gov/1.2#E1 {\displaystyle \binom{n}{k}}] : binomial coefficient : [http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/1.2#E1] [http://dlmf.nist.gov/26.3#SS1.p1 http://dlmf.nist.gov/26.3#SS1.p1]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1] +
+
<< [[Definition:AffqKrawtchouk|Definition:AffqKrawtchouk]]
+
[[Main_Page|Main Page]]
+
[[Definition:AlSalamCarlitzII|Definition:AlSalamCarlitzII]] >>
+
+drmf_eof +drmf_bof +'''Definition:AlSalamCarlitzII''' +
+
<< [[Definition:AlSalamCarlitzI|Definition:AlSalamCarlitzI]]
+
[[Main_Page|Main Page]]
+
[[Definition:AlSalamIsmail|Definition:AlSalamIsmail]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\AlSalamCarlitzII''' represents the Al-Salam-Carlitz II polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\AlSalamCarlitzII{a}{n}''' produces {\displaystyle \AlSalamCarlitzII{a}{n}}
+:'''\AlSalamCarlitzII{a}{n}@{x}{q}''' produces {\displaystyle \AlSalamCarlitzII{a}{n}@{x}{q}}
+ +These are defined by +{\displaystyle +\AlSalamCarlitzII{a}{n}@{x}{q}:= +(-a)^nq^{-\binom{n}{2}}\,\qHyperrphis{2}{0}@@{q^{-n},x}{-}{q}{\frac{q^n}{a}}. +} +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzII {\displaystyle V^{(n)}_{\alpha}}] : Al-Salam-Carlitz II polynomial : [http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzII http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzII]
+[http://dlmf.nist.gov/1.2#E1 {\displaystyle \binom{n}{k}}] : binomial coefficient : [http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/1.2#E1] [http://dlmf.nist.gov/26.3#SS1.p1 http://dlmf.nist.gov/26.3#SS1.p1]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1] +
+
<< [[Definition:AlSalamCarlitzI|Definition:AlSalamCarlitzI]]
+
[[Main_Page|Main Page]]
+
[[Definition:AlSalamIsmail|Definition:AlSalamIsmail]] >>
+
+drmf_eof +drmf_bof +'''Definition:AlSalamIsmail''' +
+
<< [[Definition:AlSalamCarlitzII|Definition:AlSalamCarlitzII]]
+
[[Main_Page|Main Page]]
+
[[Definition:AntiDer|Definition:AntiDer]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\AlSalamIsmail''' represents the Al-Salam Ismail q polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\AlSalamIsmail{n}''' produces {\displaystyle \AlSalamIsmail{n}}
+:'''\AlSalamIsmail{n}@{x}{s}{q}''' produces {\displaystyle \AlSalamIsmail{n}@{x}{s}{q}}
+ +These are defined by + +\AlSalamIsmail{2n}@{x^{1/2}}{a}{b}:=q^{n^2-n}(-b)^n +\qHyperrphis{4}{3}@@{q^{-n},-q^{1-n}/a,-aq^n,q^{n+1}} +{-q,q^{1/2},-q^{1/2}} +{q}{\frac{xaq}{b}} +
+ +and + + +\AlSalamIsmail{2n+1}@{x^{1/2}}{a}{b}=q^{n^2-n}\frac{(-b)^n(1+aq^n)(1-q^{n+1})x^{1/2}}{(1-q)} +\qHyperrphis{4}{3}@@{q^{-n},-q^{1-n}/a,-aq^{n+1},q^{n+2}} +{-q,q^{3/2},-q^{3/2}} +{q}{\frac{xaq}{b}}. + +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:AlSalamIsmail {\displaystyle U_{n}}] : Al-Salam Ismail polynomial : [http://drmf.wmflabs.org/wiki/Definition:AlSalamIsmail http://drmf.wmflabs.org/wiki/Definition:AlSalamIsmail]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1] +
+
<< [[Definition:AlSalamCarlitzII|Definition:AlSalamCarlitzII]]
+
[[Main_Page|Main Page]]
+
[[Definition:AntiDer|Definition:AntiDer]] >>
+
+drmf_eof +drmf_bof +'''Definition:AntiDer''' +
+
<< [[Definition:AlSalamIsmail|Definition:AlSalamIsmail]]
+
[[Main_Page|Main Page]]
+
[[Definition:BesselPolyIIparam|Definition:BesselPolyIIparam]] >>
+
+ +In the DRMF, the LaTeX semantic macro '''\AntiDer''' represents the anti-derivative notation, +which is defined as follows. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\AntiDer''' produces {\displaystyle \AntiDer}
+:'''\AntiDer@{x}{f(x)}''' produces {\displaystyle \AntiDer@{x}{f(x)}}
+ +These are defined by +{\displaystyle +\AntiDer@{f(x)}{x}. +} +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:AntiDer {\displaystyle \int f(x){\mathrm d}\!x}] : semantic antiderivative : [http://drmf.wmflabs.org/wiki/Definition:AntiDer http://drmf.wmflabs.org/wiki/Definition:AntiDer] +
+
<< [[Definition:AlSalamIsmail|Definition:AlSalamIsmail]]
+
[[Main_Page|Main Page]]
+
[[Definition:BesselPolyIIparam|Definition:BesselPolyIIparam]] >>
+
+drmf_eof +drmf_bof +'''Definition:BesselPolyIIparam''' +
+
<< [[Definition:AntiDer|Definition:AntiDer]]
+
[[Main_Page|Main Page]]
+
[[Definition:BesselPolyTheta|Definition:BesselPolyTheta]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\BesselPolyIIparam''' represents the Bessel polynomial with two-parameters. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\BesselPolyIIparam{n}''' produces {\displaystyle \BesselPolyIIparam{n}}
+:'''\BesselPolyIIparam{n}@{x}{a}{b}''' produces {\displaystyle \BesselPolyIIparam{n}@{x}{a}{b}}
+ +These are defined by +{\displaystyle +\BesselPolyIIparam{n}@{x}{a}{b}:=\BesselPoly{n}@{2b^{-1}x}{a} +} +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:BesselPolyIIparam {\displaystyle y_{n}}] : Bessel polynomial with two parameters {\displaystyle y_n} : [http://drmf.wmflabs.org/wiki/Definition:BesselPolyIIparam http://drmf.wmflabs.org/wiki/Definition:BesselPolyIIparam]
+[http://dlmf.nist.gov/18.34#E1 {\displaystyle y_{n}}] : Bessel polynomial : [http://dlmf.nist.gov/18.34#E1 http://dlmf.nist.gov/18.34#E1] +
+
<< [[Definition:AntiDer|Definition:AntiDer]]
+
[[Main_Page|Main Page]]
+
[[Definition:BesselPolyTheta|Definition:BesselPolyTheta]] >>
+
+drmf_eof +drmf_bof +'''Definition:BesselPolyTheta''' +
+
<< [[Definition:BesselPolyIIparam|Definition:BesselPolyIIparam]]
+
[[Main_Page|Main Page]]
+
[[Definition:bigqJacobiIVparam|Definition:bigqJacobiIVparam]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\BesselPolyTheta''' represents the Bessel polynomial \theta_nwith two-parameters. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\BesselPolyTheta{n}''' produces {\displaystyle \BesselPolyTheta{n}}
+:'''\BesselPolyTheta{n}@{x}{a}{b}''' produces {\displaystyle \BesselPolyTheta{n}@{x}{a}{b}}
+ +These are defined by +{\displaystyle +\BesselPolyTheta{n}@{x}{a}{b}:=x^n\BesselPolyIIparam{n}@{x^{-1}}{a}{b} +} +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:BesselPolyTheta {\displaystyle \theta_{n}}] : Bessel polynomial with two parameters {\displaystyle \theta_n} : [http://drmf.wmflabs.org/wiki/Definition:BesselPolyTheta http://drmf.wmflabs.org/wiki/Definition:BesselPolyTheta]
+[http://drmf.wmflabs.org/wiki/Definition:BesselPolyIIparam {\displaystyle y_{n}}] : Bessel polynomial with two parameters {\displaystyle y_n} : [http://drmf.wmflabs.org/wiki/Definition:BesselPolyIIparam http://drmf.wmflabs.org/wiki/Definition:BesselPolyIIparam] +
+
<< [[Definition:BesselPolyIIparam|Definition:BesselPolyIIparam]]
+
[[Main_Page|Main Page]]
+
[[Definition:bigqJacobiIVparam|Definition:bigqJacobiIVparam]] >>
+
+drmf_eof +drmf_bof +'''Definition:bigqJacobiIVparam''' +
+
<< [[Definition:BesselPolyTheta|Definition:BesselPolyTheta]]
+
[[Main_Page|Main Page]]
+
[[Definition:bigqLaguerre|Definition:bigqLaguerre]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\bigqJacobiIVparam''' represents the Big q-Jacobi polynomial with four-parameters. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\bigqJacobiIVparam{n}''' produces {\displaystyle \bigqJacobiIVparam{n}}
+:'''\bigqJacobiIVparam{n}@{x}{a}{b}{c}{d}{q}''' produces {\displaystyle \bigqJacobiIVparam{n}@{x}{a}{b}{c}{d}{q}}
+ +These are defined by +{\displaystyle +\bigqJacobiIVparam{n}@{x}{a}{b}{c}{d}{q}:=\qHyperrphis{3}{2}@@{q^{-n},abq^{n+1},ac^{-1}qx}{aq,-ac^{-1}dq}{q}{q} +} +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:bigqJacobiIVparam {\displaystyle P_{n}}] : big {\displaystyle q}-Jacobi polynomial with four parameters : [http://drmf.wmflabs.org/wiki/Definition:bigqJacobiIVparam http://drmf.wmflabs.org/wiki/Definition:bigqJacobiIVparam]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1] +
+
<< [[Definition:BesselPolyTheta|Definition:BesselPolyTheta]]
+
[[Main_Page|Main Page]]
+
[[Definition:bigqLaguerre|Definition:bigqLaguerre]] >>
+
+drmf_eof +drmf_bof +'''Definition:bigqLaguerre''' +
+
<< [[Definition:bigqJacobiIVparam|Definition:bigqJacobiIVparam]]
+
[[Main_Page|Main Page]]
+
[[Definition:bigqLegendre|Definition:bigqLegendre]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\bigqLaguerre''' represents big q-Laguerre polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\bigqLaguerre{n}''' produces {\displaystyle \bigqLaguerre{n}}
+:'''\bigqLaguerre{n}@{x}{a}{b}{q}''' produces {\displaystyle \bigqLaguerre{n}@{x}{a}{b}{q}}
+ +These are defined by +{\displaystyle +\bigqLaguerre{n}@{x}{a}{b}{q}:=\qHyperrphis{3}{2}@@{q^{-n},0,x}{aq,bq}{q}{q}. +} +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:bigqLaguerre {\displaystyle P_{n}}] : big {\displaystyle q}-Laguerre polynomial : [http://drmf.wmflabs.org/wiki/Definition:bigqLaguerre http://drmf.wmflabs.org/wiki/Definition:bigqLaguerre]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1] +
+
<< [[Definition:bigqJacobiIVparam|Definition:bigqJacobiIVparam]]
+
[[Main_Page|Main Page]]
+
[[Definition:bigqLegendre|Definition:bigqLegendre]] >>
+
+drmf_eof +drmf_bof +'''Definition:bigqLegendre''' +
+
<< [[Definition:bigqLaguerre|Definition:bigqLaguerre]]
+
[[Main_Page|Main Page]]
+
[[Definition:CiglerqChebyT|Definition:CiglerqChebyT]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\bigqLegendre''' represents the big q-Legendre polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\bigqLegendre{n}''' produces {\displaystyle \bigqLegendre{n}}
+:'''\bigqLegendre{n}@{x}{c}{q}''' produces {\displaystyle \bigqLegendre{n}@{x}{c}{q}}
+ +These are defined by +{\displaystyle +\bigqLegendre{n}@{x}{c}{q}:=\qHyperrphis{3}{2}@@{q^{-n},q^{n+1},x}{q,cq}{q}{q}. +} +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:bigqLegendre {\displaystyle P_{n}}] : big {\displaystyle q}-Legendre polynomial : [http://drmf.wmflabs.org/wiki/Definition:bigqLegendre http://drmf.wmflabs.org/wiki/Definition:bigqLegendre]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1] +
+
<< [[Definition:bigqLaguerre|Definition:bigqLaguerre]]
+
[[Main_Page|Main Page]]
+
[[Definition:CiglerqChebyT|Definition:CiglerqChebyT]] >>
+
+drmf_eof +drmf_bof +'''Definition:CiglerqChebyT''' +
+
<< [[Definition:bigqLegendre|Definition:bigqLegendre]]
+
[[Main_Page|Main Page]]
+
[[Definition:CiglerqChebyU|Definition:CiglerqChebyU]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\CiglerqChebyT''' represents the Cigler q-Chebyshev polynomial of the first kind. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\CiglerqChebyT{n}''' produces {\displaystyle \CiglerqChebyT{n}}
+:'''\CiglerqChebyT{n}@{x}{s}{q}''' produces {\displaystyle \CiglerqChebyT{n}@{x}{s}{q}}
+ +These are defined by + +{\displaystyle +\CiglerqChebyT{n}@{x}{s}{q}:=(-s)^{\frac12 n} \bigqJacobiIVparam{n}@{(-qs)^{-\frac12} x}{q^{-\frac12}}{q^{-\frac12}}{1}{1}{q}. +} + +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:CiglerqChebyT {\displaystyle T_{n}}] : Cigler {\displaystyle q}-Chebyshev polynomial {\displaystyle T} : [http://drmf.wmflabs.org/wiki/Definition:CiglerqChebyT http://drmf.wmflabs.org/wiki/Definition:CiglerqChebyT]
+[http://drmf.wmflabs.org/wiki/Definition:bigqJacobiIVparam {\displaystyle P_{n}}] : big {\displaystyle q}-Jacobi polynomial with four parameters : [http://drmf.wmflabs.org/wiki/Definition:bigqJacobiIVparam http://drmf.wmflabs.org/wiki/Definition:bigqJacobiIVparam] +
+
<< [[Definition:bigqLegendre|Definition:bigqLegendre]]
+
[[Main_Page|Main Page]]
+
[[Definition:CiglerqChebyU|Definition:CiglerqChebyU]] >>
+
+drmf_eof +drmf_bof +'''Definition:CiglerqChebyU''' +
+
<< [[Definition:CiglerqChebyT|Definition:CiglerqChebyT]]
+
[[Main_Page|Main Page]]
+
[[Definition:ctsbigqHermite|Definition:ctsbigqHermite]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\CiglerqChebyU''' represents the Cigler q-Chebyshev polynomial of the second kind. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\CiglerqChebyU{n}''' produces {\displaystyle \CiglerqChebyU{n}}
+:'''\CiglerqChebyU{n}@{x}{s}{q}''' produces {\displaystyle \CiglerqChebyU{n}@{x}{s}{q}}
+ +These are defined by + +{\displaystyle +\CiglerqChebyU{n}@{x}{s}{q}:=(-q^{-2}s)^{\frac12 n} \frac{1-q^{n+1}}{1-q}. +} +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:CiglerqChebyU {\displaystyle U_{n}}] : Cigler {\displaystyle q}-Chebyshev polynomial {\displaystyle U} : [http://drmf.wmflabs.org/wiki/Definition:CiglerqChebyU http://drmf.wmflabs.org/wiki/Definition:CiglerqChebyU] +
+
<< [[Definition:CiglerqChebyT|Definition:CiglerqChebyT]]
+
[[Main_Page|Main Page]]
+
[[Definition:ctsbigqHermite|Definition:ctsbigqHermite]] >>
+
+drmf_eof +drmf_bof +'''Definition:ctsbigqHermite''' +
+
<< [[Definition:CiglerqChebyU|Definition:CiglerqChebyU]]
+
[[Main_Page|Main Page]]
+
[[Definition:ctsdualqHahn|Definition:ctsdualqHahn]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\ctsbigqHermite''' represents continuous big q-Hermite polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\ctsbigqHermite{n}''' produces {\displaystyle \ctsbigqHermite{n}}
+:'''\ctsbigqHermite{n}@{x}{a}{q}''' produces {\displaystyle \ctsbigqHermite{n}@{x}{a}{q}}
+ +These are defined by +{\displaystyle +\ctsbigqHermite{n}@{x}{a}{q}:=a^{-n}\,\qHyperrphis{3}{2}@@{q^{-n},a\expe^{i\theta},a\expe^{-i\theta}}{0,0}{q}{q}. + +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:ctsbigqHermite {\displaystyle H_{n}}] : continuous big {\displaystyle q}-Hermite polynomial : [http://drmf.wmflabs.org/wiki/Definition:ctsbigqHermite http://drmf.wmflabs.org/wiki/Definition:ctsbigqHermite]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
+[http://dlmf.nist.gov/4.2.E11 {\displaystyle \mathrm{e}}] : the base of the natural logarithm : [http://dlmf.nist.gov/4.2.E11 http://dlmf.nist.gov/4.2.E11] +
+
<< [[Definition:CiglerqChebyU|Definition:CiglerqChebyU]]
+
[[Main_Page|Main Page]]
+
[[Definition:ctsdualqHahn|Definition:ctsdualqHahn]] >>
+
+drmf_eof +drmf_bof +'''Definition:ctsdualqHahn''' +
+
<< [[Definition:ctsbigqHermite|Definition:ctsbigqHermite]]
+
[[Main_Page|Main Page]]
+
[[Definition:ctsqHahn|Definition:ctsqHahn]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\ctsdualqHahn''' represents the continuous dual q-Hahn polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\ctsdualqHahn{n}''' produces {\displaystyle \ctsdualqHahn{n}}
+:'''\ctsdualqHahn{n}@{x}{a}{b}{c}{q}''' produces {\displaystyle \ctsdualqHahn{n}@{x}{a}{b}{c}{q}}
+ +These are defined by + +\frac{a^n\ctsdualqHahn{n}@{x}{a}{b}{c}{q}}{\qPochhammer{ab,ac}{q}{n}} +:=\qHyperrphis{3}{2}@@{q^{-n},a\expe^{i\theta},a\expe^{-i\theta}}{ab,ac}{q}{q}. +} +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:ctsdualqHahn {\displaystyle p_{n}}] : continuous dual {\displaystyle q}-Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:ctsdualqHahn http://drmf.wmflabs.org/wiki/Definition:ctsdualqHahn]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
+[http://dlmf.nist.gov/4.2.E11 {\displaystyle \mathrm{e}}] : the base of the natural logarithm : [http://dlmf.nist.gov/4.2.E11 http://dlmf.nist.gov/4.2.E11] +
+
<< [[Definition:ctsbigqHermite|Definition:ctsbigqHermite]]
+
[[Main_Page|Main Page]]
+
[[Definition:ctsqHahn|Definition:ctsqHahn]] >>
+
+drmf_eof +drmf_bof +'''Definition:ctsqHahn''' +
+
<< [[Definition:ctsdualqHahn|Definition:ctsdualqHahn]]
+
[[Main_Page|Main Page]]
+
[[Definition:ctsqJacobi|Definition:ctsqJacobi]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\ctsqHahn''' represents the continuous q-Hahn polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\ctsqHahn{n}''' produces {\displaystyle \ctsqHahn{n}}
+:'''\ctsqHahn{n}@{x}{a}{b}{c}{d}{q}''' produces {\displaystyle \ctsqHahn{n}@{x}{a}{b}{c}{d}{q}}
+ +These are defined by +{\displaystyle +\frac{(a\expe^{i\phi})^n\ctsqHahn{n}@{x}{a}{b}{c}{d}{q}}{\qPochhammer{ab\expe^{2i\phi},ac,ad}{q}{n}} +{}:=\qHyperrphis{4}{3}@@{q^{-n},abcdq^{n-1},a\expe^{i(\theta+2\phi)},a\expe^{-i\theta}}{ab\expe^{2i\phi},ac,ad}{q}{q}. +} +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:ctsqHahn {\displaystyle p_{n}}] : continuous {\displaystyle q}-Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:ctsqHahn http://drmf.wmflabs.org/wiki/Definition:ctsqHahn]
+[http://dlmf.nist.gov/4.2.E11 {\displaystyle \mathrm{e}}] : the base of the natural logarithm : [http://dlmf.nist.gov/4.2.E11 http://dlmf.nist.gov/4.2.E11]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1] +
+
<< [[Definition:ctsdualqHahn|Definition:ctsdualqHahn]]
+
[[Main_Page|Main Page]]
+
[[Definition:ctsqJacobi|Definition:ctsqJacobi]] >>
+
+drmf_eof +drmf_bof +'''Definition:ctsqJacobi''' +
+
<< [[Definition:ctsqHahn|Definition:ctsqHahn]]
+
[[Main_Page|Main Page]]
+
[[Definition:ctsqLaguerre|Definition:ctsqLaguerre]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\ctsqJacobi''' represents the continuous q-Jacobi polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\ctsqJacobi{\alpha}{\beta}{m}''' produces {\displaystyle \ctsqJacobi{\alpha}{\beta}{m}}
+:'''\ctsqJacobi{\alpha}{\beta}{m}@{x}{q}''' produces {\displaystyle \ctsqJacobi{\alpha}{\beta}{m}@{x}{q}}
+ +These are defined by
+ +\ctsqJacobi{\alpha}{\beta}{n}@{x}{q} +:=\frac{\qPochhammer{q^{\alpha+1}}{q}{n}}{\qPochhammer{q}{q}{n}} +\qHyperrphis{4}{3}@@{q^{-n},q^{n+\alpha+\beta+1},q^{\frac{1}{2}\alpha+\frac{1}{4}}\expe^{i\theta},q^{\frac{1}{2}\alpha+\frac{1}{4}}\expe^{-i\theta}} +{q^{\alpha+1},-q^{\frac{1}{2}(\alpha+\beta+1)},-q^{\frac{1}{2}(\alpha+\beta+2)}}{q}{q} + + +with x=\cos@@{\theta}. +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:ctsqJacobi {\displaystyle P^{(\alpha,\beta)}_{n}}] : continuous {\displaystyle q}-Jacobi polynomial : [http://drmf.wmflabs.org/wiki/Definition:ctsqJacobi http://drmf.wmflabs.org/wiki/Definition:ctsqJacobi]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
+[http://dlmf.nist.gov/4.2.E11 {\displaystyle \mathrm{e}}] : the base of the natural logarithm : [http://dlmf.nist.gov/4.2.E11 http://dlmf.nist.gov/4.2.E11]
+[http://dlmf.nist.gov/4.14#E2 {\displaystyle \mathrm{cos}}] : cosine function : [http://dlmf.nist.gov/4.14#E2 http://dlmf.nist.gov/4.14#E2] +
+
<< [[Definition:ctsqHahn|Definition:ctsqHahn]]
+
[[Main_Page|Main Page]]
+
[[Definition:ctsqLaguerre|Definition:ctsqLaguerre]] >>
+
+drmf_eof +drmf_bof +'''Definition:ctsqLaguerre''' +
+
<< [[Definition:ctsqJacobi|Definition:ctsqJacobi]]
+
[[Main_Page|Main Page]]
+
[[Definition:ctsqLegendre|Definition:ctsqLegendre]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\ctsqLaguerre''' represents the continuous q-Laguerre polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\ctsqLaguerre{\alpha}{n}''' produces {\displaystyle \ctsqLaguerre{\alpha}{n}}
+:'''\ctsqLaguerre{\alpha}{n}@{x}{q}''' produces {\displaystyle \ctsqLaguerre{\alpha}{n}@{x}{q}}
+ +These are defined by +{\displaystyle +\ctsqLaguerre{\alpha}{n}@{x}{q}:=\frac{\qPochhammer{q^{\alpha+1}}{q}{n}}{\qPochhammer{q}{q}{n}}\,\qHyperrphis{3}{2}@@{q^{-n},q^{\frac{1}{2}\alpha+\frac{1}{4}}\expe^{i\theta},q^{\frac{1}{2}\alpha+\frac{1}{4}}\expe^{-i\theta}}{q^{\alpha+1},0}{q}{q}. + +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:ctsqLaguerre {\displaystyle P^{(n)}_{\alpha}}] : continuous {\displaystyle q}-Laguerre polynomial : [http://drmf.wmflabs.org/wiki/Definition:ctsqLaguerre http://drmf.wmflabs.org/wiki/Definition:ctsqLaguerre]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
+[http://dlmf.nist.gov/4.2.E11 {\displaystyle \mathrm{e}}] : the base of the natural logarithm : [http://dlmf.nist.gov/4.2.E11 http://dlmf.nist.gov/4.2.E11] +
+
<< [[Definition:ctsqJacobi|Definition:ctsqJacobi]]
+
[[Main_Page|Main Page]]
+
[[Definition:ctsqLegendre|Definition:ctsqLegendre]] >>
+
+drmf_eof +drmf_bof +'''Definition:ctsqLegendre''' +
+
<< [[Definition:ctsqLaguerre|Definition:ctsqLaguerre]]
+
[[Main_Page|Main Page]]
+
[[Definition:f|Definition:f]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\ctsqLegendre''' represents the continuous q-Legendre polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\ctsqLegendre{n}''' produces {\displaystyle \ctsqLegendre{n}}
+:'''\ctsqLegendre{n}@{x}{q}''' produces {\displaystyle \ctsqLegendre{n}@{x}{q}}
+:'''\ctsqLegendre{n}@@{x}{q}''' produces {\displaystyle \ctsqLegendre{n}@@{x}{q}}
+ +These are defined by + +{\displaystyle +\dualqKrawtchouk{n}@{\lambda(x)}{c}{N}{q}:=\qHyperrphis{3}{2}@@{q^{-n},q^{-x},cq^{x-N}}{q^{-N},0}{q}{q} +}. +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:ctsqLegendre {\displaystyle P_{n}}] : continuous {\displaystyle q}-Legendre polynomial : [http://drmf.wmflabs.org/wiki/Definition:ctsqLegendre http://drmf.wmflabs.org/wiki/Definition:ctsqLegendre]
+[http://drmf.wmflabs.org/wiki/Definition:dualqKrawtchouk {\displaystyle K_{n}}] : dual {\displaystyle q}-Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:dualqKrawtchouk http://drmf.wmflabs.org/wiki/Definition:dualqKrawtchouk]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1] +
+
<< [[Definition:ctsqLaguerre|Definition:ctsqLaguerre]]
+
[[Main_Page|Main Page]]
+
[[Definition:f|Definition:f]] >>
+
+drmf_eof +drmf_bof +'''Definition:f''' +
+
<< [[Definition:ctsqLegendre|Definition:ctsqLegendre]]
+
[[Main_Page|Main Page]]
+
[[Definition:GenHermite|Definition:GenHermite]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\f''' represents Function. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\f{f}''' produces {\displaystyle \f{f}}
+:'''\f{f}@{x}''' produces {\displaystyle \f{f}@{x}}
+ +These are defined by + + +\GenGegenbauer{\alpha}{\beta}{2m}@{x}:={\rm const}\times \Jacobi{\alpha}{\beta}{m}@{2x^2-1}, +
+ + +\GenGegenbauer{\alpha}{\beta}{2m+1}@{x}:={\rm const}\times x\,\Jacobi{\alpha}{\beta+1}{m}@{2x^2-1}. +
+ +Then for \alpha,\beta>-1, we have the orthogonality relation
+ + +\int_{-1}^1 \GenGegenbauer{\alpha}{\beta}{m}@{x}\,\GenGegenbauer{\alpha}{\beta}{n}@{x}\,|x|^{2\beta+1} +(1-x^2)^\alpha\,dx=0, + + +for m\ne n. + +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:f {\displaystyle {f}}] : function : [http://drmf.wmflabs.org/wiki/Definition:f http://drmf.wmflabs.org/wiki/Definition:f]
+[http://drmf.wmflabs.org/wiki/Definition:GenGegenbauer {\displaystyle S^{(\alpha,\beta)}_{n}}] : Generalized Gegenbauer polynomial : [http://drmf.wmflabs.org/wiki/Definition:GenGegenbauer http://drmf.wmflabs.org/wiki/Definition:GenGegenbauer]
+[http://dlmf.nist.gov/18.3#T1.t1.r3 {\displaystyle P^{(\alpha,\beta)}_{n}}] : Jacobi polynomial : [http://dlmf.nist.gov/18.3#T1.t1.r3 http://dlmf.nist.gov/18.3#T1.t1.r3]
+[http://dlmf.nist.gov/1.4#iv {\displaystyle \int}] : integral : [http://dlmf.nist.gov/1.4#iv http://dlmf.nist.gov/1.4#iv] +
+
<< [[Definition:ctsqLegendre|Definition:ctsqLegendre]]
+
[[Main_Page|Main Page]]
+
[[Definition:GenHermite|Definition:GenHermite]] >>
+
+drmf_eof +drmf_bof +'''Definition:GenHermite''' +
+
<< [[Definition:f|Definition:f]]
+
[[Main_Page|Main Page]]
+
[[Definition:GottliebLaguerre|Definition:GottliebLaguerre]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\GenHermite''' represents the generalized Hermite polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\GenHermite[\alpha]{n}''' produces {\displaystyle \GenHermite[\alpha]{n}}
+:'''\GenHermite{n}''' produces {\displaystyle \GenHermite{n}}
+:'''\GenHermite[\alpha]{n}@{x}''' produces {\displaystyle \GenHermite[\alpha]{n}@{x}}
+:'''\GenHermite{n}@{x}''' produces {\displaystyle \GenHermite{n}@{x}}
+ +These are defined by + + +\GenHermite[\mu]{2m}@{x}:={\rm const}\times \Laguerre[\mu-\frac12]{m}@{x^2},\qquad +
+ +\GenHermite[\mu]{2m+1}@{x}:={\rm \const}\times x \Laguerre[\mu+\frac12]{m}@{x^2}. + +
+Then for \mu>-\tfrac12 we have orthogonality relation + +\int_{-\infty}^{\infty} \GenHermite[\mu]{m}@{x} \GenHermite[\mu]{n}@{x} |x|^{2\mu}e^{-x^2} dx, +=0 +
+for m\ne n. + +== Bibliography == +
+Page 156 of [[Bibliography#CHI|'''CHI''']]. +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:GenHermite {\displaystyle H_n^{\mu}}] : generalized Hermite polynomial : [http://drmf.wmflabs.org/wiki/Definition:GenHermite http://drmf.wmflabs.org/wiki/Definition:GenHermite]
+[http://dlmf.nist.gov/18.3#T1.t1.r27 {\displaystyle L_n^{(\alpha)}}] : Laguerre (or generalized Laguerre) polynomial : [http://dlmf.nist.gov/18.3#T1.t1.r27 http://dlmf.nist.gov/18.3#T1.t1.r27]
+[http://dlmf.nist.gov/1.4#iv {\displaystyle \int}] : integral : [http://dlmf.nist.gov/1.4#iv http://dlmf.nist.gov/1.4#iv] +
+
<< [[Definition:f|Definition:f]]
+
[[Main_Page|Main Page]]
+
[[Definition:GottliebLaguerre|Definition:GottliebLaguerre]] >>
+
+drmf_eof +drmf_bof +'''Definition:GottliebLaguerre''' +
+
<< [[Definition:GenHermite|Definition:GenHermite]]
+
[[Main_Page|Main Page]]
+
[[Definition:Int|Definition:Int]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\GottliebLaguerre''' represents the Gottlieb-Laguerre polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\GottliebLaguerre{n}''' produces {\displaystyle \GottliebLaguerre{n}}
+:'''\GottliebLaguerre{n}@{x}''' produces {\displaystyle \GottliebLaguerre{n}@{x}}
+ +These are defined by + +{\displaystyle +\GottliebLaguerre{n}@{x}:=e^{-n\la} \Meixner{n}@{x}{1}{e^{-\la}}. +} + +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:GottliebLaguerre {\displaystyle l_{n}}] : Gottlieb-Laguerre polynomial : [http://drmf.wmflabs.org/wiki/Definition:GottliebLaguerre http://drmf.wmflabs.org/wiki/Definition:GottliebLaguerre]
+[http://dlmf.nist.gov/18.19#T1.t1.r9 {\displaystyle M_{n}}] : Meixner polynomial : [http://dlmf.nist.gov/18.19#T1.t1.r9 http://dlmf.nist.gov/18.19#T1.t1.r9] +
+
<< [[Definition:GenHermite|Definition:GenHermite]]
+
[[Main_Page|Main Page]]
+
[[Definition:Int|Definition:Int]] >>
+
+drmf_eof +drmf_bof +'''Definition:Int''' +
+
<< [[Definition:GottliebLaguerre|Definition:GottliebLaguerre]]
+
[[Main_Page|Main Page]]
+
[[Definition:JacksonqBesselII|Definition:JacksonqBesselII]] >>
+
+ +In the DRMF, the LaTeX semantic macro '''\Int''' represents the definite integral notation, +which is defined as follows. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\Int{a}{b}''' produces {\displaystyle \Int{a}{b}}
+:'''\Int{a}{b}@{x}{f(x)}''' produces {\displaystyle \Int{a}{b}@{x}{f(x)}}
+ +These are defined by +{\displaystyle +\JacksonqBesselI{\nu}@{z}{q}:=\frac{\qPochhammer{q^{\nu+1}}{q}{\infty}}{\qPochhammer{q}{q}{\infty}} +\left(\frac{z}{2}\right)^{\nu}\,\qHyperrphis{2}{1}@@{0,0}{q^{\nu+1}}{q}{-\frac{z^2}{4}}. +} +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:Int {\displaystyle \int_{a}^{b}f(x){\mathrm d}\!x}] : semantic definite integral : [http://drmf.wmflabs.org/wiki/Definition:Int http://drmf.wmflabs.org/wiki/Definition:Int]
+[http://drmf.wmflabs.org/wiki/Definition:JacksonqBesselI {\displaystyle J^{(1)}_{q}}] : Jackson {\displaystyle q}-Bessel function 1 : [http://drmf.wmflabs.org/wiki/Definition:JacksonqBesselI http://drmf.wmflabs.org/wiki/Definition:JacksonqBesselI]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1] +
+
<< [[Definition:GottliebLaguerre|Definition:GottliebLaguerre]]
+
[[Main_Page|Main Page]]
+
[[Definition:JacksonqBesselII|Definition:JacksonqBesselII]] >>
+
+drmf_eof +drmf_bof +'''Definition:JacksonqBesselII''' +
+
<< [[Definition:Int|Definition:Int]]
+
[[Main_Page|Main Page]]
+
[[Definition:JacksonqBesselIII|Definition:JacksonqBesselIII]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\JacksonqBesselII''' represents the Jackson q-Bessel function 2. + +This macro is in the category of real or complex valued functions. + +In math mode, this macro can be called in the following ways: + +:'''\JacksonqBesselII{\nu}''' produces {\displaystyle \JacksonqBesselII{\nu}}
+:'''\JacksonqBesselII{\nu}@{z}{q}''' produces {\displaystyle \JacksonqBesselII{\nu}@{z}{q}}
+ +These are defined by + +{\displaystyle +\JacksonqBesselII{\nu}@{z}{q}:=\frac{\qPochhammer{q^{\nu+1}}{q}{\infty}}{\qPochhammer{q}{q}{\infty}} +\left(\frac{z}{2}\right)^{\nu}\,\qHyperrphis{0}{1}@@{-}{q^{\nu+1}}{q}{-\frac{q^{\nu+1}z^2}{4}}. +} +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:JacksonqBesselII {\displaystyle J^{(2)}_{q}}] : Jackson {\displaystyle q}-Bessel function 2 : [http://drmf.wmflabs.org/wiki/Definition:JacksonqBesselII http://drmf.wmflabs.org/wiki/Definition:JacksonqBesselII]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1] +
+
<< [[Definition:Int|Definition:Int]]
+
[[Main_Page|Main Page]]
+
[[Definition:JacksonqBesselIII|Definition:JacksonqBesselIII]] >>
+
+drmf_eof +drmf_bof +'''Definition:JacksonqBesselIII''' +
+
<< [[Definition:JacksonqBesselII|Definition:JacksonqBesselII]]
+
[[Main_Page|Main Page]]
+
[[Definition:littleqLegendre|Definition:littleqLegendre]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\JacksonqBesselIII''' represents the Jackson q-Bessel function 3. + +This macro is in the category of real or complex valued functions. + +In math mode, this macro can be called in the following ways: + +:'''\JacksonqBesselIII{\nu}''' produces {\displaystyle \JacksonqBesselIII{\nu}}
+:'''\JacksonqBesselIII{\nu}@{z}{q}''' produces {\displaystyle \JacksonqBesselIII{\nu}@{z}{q}}
+ +These are defined by + +{\displaystyle +\littleqLaguerre{n}@{x}{a}{q}:=\qHyperrphis{2}{1}@@{q^{-n},0}{aq}{q}{qx}. +} +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:JacksonqBesselIII {\displaystyle J^{(3)}_{q}}] : Jackson/Hahn-Exton {\displaystyle q}-Bessel function 3 : [http://drmf.wmflabs.org/wiki/Definition:JacksonqBesselIII http://drmf.wmflabs.org/wiki/Definition:JacksonqBesselIII]
+[http://drmf.wmflabs.org/wiki/Definition:littleqLaguerre {\displaystyle p_{n}}] : little {\displaystyle q}-Laguerre / Wall polynomial : [http://drmf.wmflabs.org/wiki/Definition:littleqLaguerre http://drmf.wmflabs.org/wiki/Definition:littleqLaguerre]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1] +
+
<< [[Definition:JacksonqBesselII|Definition:JacksonqBesselII]]
+
[[Main_Page|Main Page]]
+
[[Definition:littleqLegendre|Definition:littleqLegendre]] >>
+
+drmf_eof +drmf_bof +'''Definition:littleqLegendre''' +
+
<< [[Definition:JacksonqBesselIII|Definition:JacksonqBesselIII]]
+
[[Main_Page|Main Page]]
+
[[Definition:lrselection|Definition:lrselection]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\littleqLegendre''' represents the little q-Legendre polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\littleqLegendre{m}''' produces {\displaystyle \littleqLegendre{m}}
+:'''\littleqLegendre{m}@{x}{q}''' produces {\displaystyle \littleqLegendre{m}@{x}{q}}
+ +These are defined by +{\displaystyle +\littleqLegendre{n}@{x}{q}:=\qHyperrphis{2}{1}@@{q^{-n},q^{n+1}}{q}{q}{qx}. +} +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:littleqLegendre {\displaystyle p_{n}}] : little {\displaystyle q}-Legendre polynomial : [http://drmf.wmflabs.org/wiki/Definition:littleqLegendre http://drmf.wmflabs.org/wiki/Definition:littleqLegendre]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1] +
+
<< [[Definition:JacksonqBesselIII|Definition:JacksonqBesselIII]]
+
[[Main_Page|Main Page]]
+
[[Definition:lrselection|Definition:lrselection]] >>
+
+drmf_eof +drmf_bof +'''Definition:lrselection''' +
+
<< [[Definition:littleqLegendre|Definition:littleqLegendre]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicAlSalamCarlitzI|Definition:monicAlSalamCarlitzI]] >>
+
+ +For the sake of compactness, we use the abbreviated '''\lrselection''' notation in a number of formulas. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following way: + +:'''\lrselection''' produces {\displaystyle \lrselection}
+ +These are defined by + +{\displaystyle +\AffqKrawtchouk{n}@{q^{-x}}{p}{N}{q}=\frac{1}{\qPochhammer{pq,q^{-N}}{q}{n}}\monicAffqKrawtchouk{n}@{q^{-x}}{p}{N}{q}. +} + +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:Lrselection {\displaystyle \left\{\begin{matrix}x\end{matrix}\right\}}] : bracketed generalization of {\displaystyle \pm} : [http://drmf.wmflabs.org/wiki/Definition:Lrselection http://drmf.wmflabs.org/wiki/Definition:Lrselection]
+[http://drmf.wmflabs.org/wiki/Definition:AffqKrawtchouk {\displaystyle K^{\mathrm{Aff}}_{n}}] : affine {\displaystyle q}-Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:AffqKrawtchouk http://drmf.wmflabs.org/wiki/Definition:AffqKrawtchouk]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1]
+[http://drmf.wmflabs.org/wiki/Definition:monicAffqKrawtchouk {\displaystyle \widehat{K}^{\mathrm{Aff}}_{n}}] : monic affine {\displaystyle q}-Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicAffqKrawtchouk http://drmf.wmflabs.org/wiki/Definition:monicAffqKrawtchouk] +
+
<< [[Definition:littleqLegendre|Definition:littleqLegendre]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicAlSalamCarlitzI|Definition:monicAlSalamCarlitzI]] >>
+
+drmf_eof +drmf_bof +'''Definition:monicAlSalamCarlitzI''' +
+
<< [[Definition:lrselection|Definition:lrselection]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicAlSalamCarlitzII|Definition:monicAlSalamCarlitzII]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\monicAlSalamCarlitzI''' represents the monic q Al-Salam Carlitz I polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\monicAlSalamCarlitzI{a}{n}''' produces {\displaystyle \monicAlSalamCarlitzI{a}{n}}
+:'''\monicAlSalamCarlitzI{a}{n}@{x}{q}''' produces {\displaystyle \monicAlSalamCarlitzI{a}{n}@{x}{q}}
+:'''\monicAlSalamCarlitzI{a}{n}@@{x}{q}''' produces {\displaystyle \monicAlSalamCarlitzI{a}{n}@@{x}{q}}
+ +These are defined by + +{\displaystyle +\AlSalamCarlitzI{a}{n}@{x}{q}:=\monicAlSalamCarlitzI{a}{n}@{x}{q}. +} +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:monicAlSalamCarlitzI {\displaystyle {\widehat U}^{(\alpha)}_{n}}] : monic Al-Salam-Carlitz I polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicAlSalamCarlitzI http://drmf.wmflabs.org/wiki/Definition:monicAlSalamCarlitzI]
+[http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzI {\displaystyle U^{(n)}_{\alpha}}] : Al-Salam-Carlitz I polynomial : [http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzI http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzI] +
+
<< [[Definition:lrselection|Definition:lrselection]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicAlSalamCarlitzII|Definition:monicAlSalamCarlitzII]] >>
+
+drmf_eof +drmf_bof +'''Definition:monicAlSalamCarlitzII''' +
+
<< [[Definition:monicAlSalamCarlitzI|Definition:monicAlSalamCarlitzI]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicAlSalamChihara|Definition:monicAlSalamChihara]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\monicAlSalamCarlitzII''' represents the monic q Al-Salam Carlitz II polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\monicAlSalamCarlitzII{a}{n}''' produces {\displaystyle \monicAlSalamCarlitzII{a}{n}}
+:'''\monicAlSalamCarlitzII{a}{n}@{x}{q}''' produces {\displaystyle \monicAlSalamCarlitzII{a}{n}@{x}{q}}
+:'''\monicAlSalamCarlitzII{a}{n}@@{x}{q}''' produces {\displaystyle \monicAlSalamCarlitzII{a}{n}@@{x}{q}}
+ +These are defined by + +{\displaystyle +\AlSalamCarlitzII{a}{n}@{x}{q}=:\monicAlSalamCarlitzII{a}{n}@{x}{q}. +} +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:monicAlSalamCarlitzII {\displaystyle {\widehat V}^{(n)}_{\alpha}}] : monic Al-Salam-Carlitz II polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicAlSalamCarlitzII http://drmf.wmflabs.org/wiki/Definition:monicAlSalamCarlitzII]
+[http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzII {\displaystyle V^{(n)}_{\alpha}}] : Al-Salam-Carlitz II polynomial : [http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzII http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzII] +
+
<< [[Definition:monicAlSalamCarlitzI|Definition:monicAlSalamCarlitzI]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicAlSalamChihara|Definition:monicAlSalamChihara]] >>
+
+drmf_eof +drmf_bof +'''Definition:monicAlSalamChihara''' +
+
<< [[Definition:monicAlSalamCarlitzII|Definition:monicAlSalamCarlitzII]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicAskeyWilson|Definition:monicAskeyWilson]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\monicAlSalamChihara''' represents the monic Al-Salam Chihara q polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\monicAlSalamChihara{n}''' produces {\displaystyle \monicAlSalamChihara{n}}
+:'''\monicAlSalamChihara{n}@{x}{a}{b}{q}''' produces {\displaystyle \monicAlSalamChihara{n}@{x}{a}{b}{q}}
+:'''\monicAlSalamChihara{n}@@{x}{a}{b}{q}''' produces {\displaystyle \monicAlSalamChihara{n}@@{x}{a}{b}{q}}
+ +These are defined by + +{\displaystyle +\AlSalamChihara{n}@{x}{a}{b}{q}=:2^n\monicAlSalamChihara{n}@{x}{a}{b}{q}. +} +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:monicAlSalamChihara {\displaystyle {\widehat Q}_{n}}] : monic Al-Salam-Chihara polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicAlSalamChihara http://drmf.wmflabs.org/wiki/Definition:monicAlSalamChihara]
+[http://drmf.wmflabs.org/wiki/Definition:AlSalamChihara {\displaystyle Q_{n}}] : Al-Salam-Chihara polynomial : [http://drmf.wmflabs.org/wiki/Definition:AlSalamChihara http://drmf.wmflabs.org/wiki/Definition:AlSalamChihara] +
+
<< [[Definition:monicAlSalamCarlitzII|Definition:monicAlSalamCarlitzII]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicAskeyWilson|Definition:monicAskeyWilson]] >>
+
+drmf_eof +drmf_bof +'''Definition:monicAskeyWilson''' +
+
<< [[Definition:monicAlSalamChihara|Definition:monicAlSalamChihara]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicBesselPoly|Definition:monicBesselPoly]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\monicAskeyWilson''' represents the monic Askey Wilson polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\monicAskeyWilson{n}''' produces {\displaystyle \monicAskeyWilson{n}}
+:'''\monicAskeyWilson{n}@{x}{a}{b}{c}{d}{q}''' produces {\displaystyle \monicAskeyWilson{n}@{x}{a}{b}{c}{d}{q}}
+:'''\monicAskeyWilson{n}@@{x}{a}{b}{c}{d}{q}''' produces {\displaystyle \monicAskeyWilson{n}@@{x}{a}{b}{c}{d}{q}}
+ +These are defined by + +{\displaystyle +\AskeyWilson{n}@{x}{a}{b}{c}{d}{q}=:2^n\qPochhammer{abcdq^{n-1}}{q}{n}\monicAskeyWilson{n}@{x}{a}{b}{c}{d}{q}. +} +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:monicAskeyWilson {\displaystyle {\widehat p}_{n}}] : monic Askey-Wilson polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicAskeyWilson http://drmf.wmflabs.org/wiki/Definition:monicAskeyWilson]
+[http://dlmf.nist.gov/18.28#E1 {\displaystyle p_{n}}] : Askey-Wilson polynomial : [http://dlmf.nist.gov/18.28#E1 http://dlmf.nist.gov/18.28#E1]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1] +
+
<< [[Definition:monicAlSalamChihara|Definition:monicAlSalamChihara]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicBesselPoly|Definition:monicBesselPoly]] >>
+
+drmf_eof +drmf_bof +'''Definition:monicBesselPoly''' +
+
<< [[Definition:monicAskeyWilson|Definition:monicAskeyWilson]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicbigqJacobi|Definition:monicbigqJacobi]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\monicBesselPoly''' represents the monic Bessel polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\monicBesselPoly{n}''' produces {\displaystyle \monicBesselPoly{n}}
+:'''\monicBesselPoly{n}@{x}{a}''' produces {\displaystyle \monicBesselPoly{n}@{x}{a}}
+:'''\monicBesselPoly{n}@@{x}{a}''' produces {\displaystyle \monicBesselPoly{n}@@{x}{a}}
+ +These are defined by +{\displaystyle +\BesselPoly{n}@{x}{a}=:\frac{\pochhammer{n+a+1}{n}}{2^n}\monicBesselPoly{n}@{x}{a} +} +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:monicBesselPoly {\displaystyle {\widehat y}_{n}}] : monic Bessel polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicBesselPoly http://drmf.wmflabs.org/wiki/Definition:monicBesselPoly]
+[http://dlmf.nist.gov/18.34#E1 {\displaystyle y_{n}}] : Bessel polynomial : [http://dlmf.nist.gov/18.34#E1 http://dlmf.nist.gov/18.34#E1]
+[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii] +
+
<< [[Definition:monicAskeyWilson|Definition:monicAskeyWilson]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicbigqJacobi|Definition:monicbigqJacobi]] >>
+
+drmf_eof +drmf_bof +'''Definition:monicbigqJacobi''' +
+
<< [[Definition:monicBesselPoly|Definition:monicBesselPoly]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicbigqLaguerre|Definition:monicbigqLaguerre]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\monicbigqJacobi''' represents the monic big q Jacobi polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\monicbigqJacobi{n}''' produces {\displaystyle \monicbigqJacobi{n}}
+:'''\monicbigqJacobi{n}@{x}{a}{b}{c}{q}''' produces {\displaystyle \monicbigqJacobi{n}@{x}{a}{b}{c}{q}}
+:'''\monicbigqJacobi{n}@@{x}{a}{b}{c}{q}''' produces {\displaystyle \monicbigqJacobi{n}@@{x}{a}{b}{c}{q}}
+ +These are defined by + +{\displaystyle +\bigqJacobi{n}@{x}{a}{b}{c}{q}=:\frac{\qPochhammer{abq^{n+1}}{q}{n}}{\qPochhammer{aq,cq}{q}{n}}\monicbigqJacobi{n}@{x}{a}{b}{c}{q}. +} +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:monicbigqJacobi {\displaystyle {\widehat P}_{n}}] : monic big {\displaystyle q}-Jacobi polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicbigqJacobi http://drmf.wmflabs.org/wiki/Definition:monicbigqJacobi]
+[http://drmf.wmflabs.org/wiki/Definition:bigqJacobi {\displaystyle P_{n}}] : big {\displaystyle q}-Jacobi polynomial : [http://drmf.wmflabs.org/wiki/Definition:bigqJacobi http://drmf.wmflabs.org/wiki/Definition:bigqJacobi]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1] +
+
<< [[Definition:monicBesselPoly|Definition:monicBesselPoly]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicbigqLaguerre|Definition:monicbigqLaguerre]] >>
+
+drmf_eof +drmf_bof +'''Definition:monicbigqLaguerre''' +
+
<< [[Definition:monicbigqJacobi|Definition:monicbigqJacobi]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicbigqLegendre|Definition:monicbigqLegendre]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\monicbigqLaguerre''' represents the monic big q Laguerre polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\monicbigqLaguerre{n}''' produces {\displaystyle \monicbigqLaguerre{n}}
+:'''\monicbigqLaguerre{n}@{x}{a}{b}{q}''' produces {\displaystyle \monicbigqLaguerre{n}@{x}{a}{b}{q}}
+:'''\monicbigqLaguerre{n}@@{x}{a}{b}{q}''' produces {\displaystyle \monicbigqLaguerre{n}@@{x}{a}{b}{q}}
+ +These are defined by + +{\displaystyle +\bigqLaguerre{n}@{x}{a}{b}{q}=:\frac{1}{\qPochhammer{aq,bq}{q}{n}}\monicbigqLaguerre{n}@{x}{a}{b}{q}. +} +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:monicbigqLaguerre {\displaystyle {\widehat P}_{n}}] : monic big {\displaystyle q}-Laguerre polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicbigqLaguerre http://drmf.wmflabs.org/wiki/Definition:monicbigqLaguerre]
+[http://drmf.wmflabs.org/wiki/Definition:bigqLaguerre {\displaystyle P_{n}}] : big {\displaystyle q}-Laguerre polynomial : [http://drmf.wmflabs.org/wiki/Definition:bigqLaguerre http://drmf.wmflabs.org/wiki/Definition:bigqLaguerre]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1] +
+
<< [[Definition:monicbigqJacobi|Definition:monicbigqJacobi]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicbigqLegendre|Definition:monicbigqLegendre]] >>
+
+drmf_eof +drmf_bof +'''Definition:monicbigqLegendre''' +
+
<< [[Definition:monicbigqLaguerre|Definition:monicbigqLaguerre]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicCharlier|Definition:monicCharlier]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\monicbigqLegendre''' represents the monic big q polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\monicbigqLegendre{n}''' produces {\displaystyle \monicbigqLegendre{n}}
+:'''\monicbigqLegendre{n}@{x}{c}{q}''' produces {\displaystyle \monicbigqLegendre{n}@{x}{c}{q}}
+:'''\monicbigqLegendre{n}@@{x}{c}{q}''' produces {\displaystyle \monicbigqLegendre{n}@@{x}{c}{q}}
+ +These are defined by + +{\displaystyle +\bigqLegendre{n}@{x}{c}{q}=:\frac{\qPochhammer{q^{n+1}}{q}{n}}{\qPochhammer{q,cq}{q}{n}}\monicbigqLegendre{n}@{x}{c}{q}. +} +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:monicbigqLegendre {\displaystyle {\widehat P}_{n}}] : monic big {\displaystyle q}-Legendre polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicbigqLegendre http://drmf.wmflabs.org/wiki/Definition:monicbigqLegendre]
+[http://drmf.wmflabs.org/wiki/Definition:bigqLegendre {\displaystyle P_{n}}] : big {\displaystyle q}-Legendre polynomial : [http://drmf.wmflabs.org/wiki/Definition:bigqLegendre http://drmf.wmflabs.org/wiki/Definition:bigqLegendre]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1] +
+
<< [[Definition:monicbigqLaguerre|Definition:monicbigqLaguerre]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicCharlier|Definition:monicCharlier]] >>
+
+drmf_eof +drmf_bof +'''Definition:monicCharlier''' +
+
<< [[Definition:monicbigqLegendre|Definition:monicbigqLegendre]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicChebyT|Definition:monicChebyT]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\monicCharlier''' represents the monic Charlier polynomial monic p. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\monicCharlier{n}''' produces {\displaystyle \monicCharlier{n}}
+:'''\monicCharlier{n}@{x}{a}''' produces {\displaystyle \monicCharlier{n}@{x}{a}}
+:'''\monicCharlier{n}@@{x}{a}''' produces {\displaystyle \monicCharlier{n}@@{x}{a}}
+ +These are defined by + +{\displaystyle +\Charlier{n}@{x}{a}=:\left(-\frac{1}{a}\right)^n\monicCharlier{n}@{x}{a} +} + +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:monicCharlier {\displaystyle {\widehat C}_{n}}] : monic Charlier polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicCharlier http://drmf.wmflabs.org/wiki/Definition:monicCharlier]
+[http://dlmf.nist.gov/18.19#T1.t1.r11 {\displaystyle C_{n}}] : Charlier polynomial : [http://dlmf.nist.gov/18.19#T1.t1.r11 http://dlmf.nist.gov/18.19#T1.t1.r11] +
+
<< [[Definition:monicbigqLegendre|Definition:monicbigqLegendre]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicChebyT|Definition:monicChebyT]] >>
+
+drmf_eof +drmf_bof +'''Definition:monicChebyT''' +
+
<< [[Definition:monicCharlier|Definition:monicCharlier]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicChebyU|Definition:monicChebyU]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\monicChebyT''' represents the monic monic Chebyshev polynomial first kind. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\monicChebyT{n}''' produces {\displaystyle \monicChebyT{n}}
+:'''\monicChebyT{n}@{x}''' produces {\displaystyle \monicChebyT{n}@{x}}
+:'''\monicChebyT{n}@@{x}''' produces {\displaystyle \monicChebyT{n}@@{x}}
+ +These are defined by + +{\displaystyle +\ChebyT{n}@{x}:=2^n\monicChebyT{n}@{x} +} +for n\ne 1 and \ChebyT{1}@{x}=\monicChebyT{1}@{x}=x. +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:monicChebyT {\displaystyle {\widehat T}_{n}}] : monic Chebyshev polynomial of the first kind : [http://drmf.wmflabs.org/wiki/Definition:monicChebyT http://drmf.wmflabs.org/wiki/Definition:monicChebyT]
+[http://dlmf.nist.gov/18.3#T1.t1.r8 {\displaystyle T_{n}}] : Chebyshev polynomial of the first kind : [http://dlmf.nist.gov/18.3#T1.t1.r8 http://dlmf.nist.gov/18.3#T1.t1.r8] +
+
<< [[Definition:monicCharlier|Definition:monicCharlier]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicChebyU|Definition:monicChebyU]] >>
+
+drmf_eof +drmf_bof +'''Definition:monicChebyU''' +
+
<< [[Definition:monicChebyT|Definition:monicChebyT]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicctsbigqHermite|Definition:monicctsbigqHermite]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\monicChebyU''' represents the monic Chebyshev polynomial second kind. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\monicChebyU{n}''' produces {\displaystyle \monicChebyU{n}}
+:'''\monicChebyU{n}@{x}''' produces {\displaystyle \monicChebyU{n}@{x}}
+:'''\monicChebyU{n}@@{x}''' produces {\displaystyle \monicChebyU{n}@@{x}}
+ +These are defined by + +{\displaystyle +\ChebyU{n}@{x}:=2^n\monicChebyU{n}@{x} +}. +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:monicChebyU {\displaystyle {\widehat U}_{n}}] : monic Chebyshev polynomial of the second kind : [http://drmf.wmflabs.org/wiki/Definition:monicChebyU http://drmf.wmflabs.org/wiki/Definition:monicChebyU]
+[http://dlmf.nist.gov/18.3#T1.t1.r11 {\displaystyle U_{n}}] : Chebyshev polynomial of the second kind : [http://dlmf.nist.gov/18.3#T1.t1.r11 http://dlmf.nist.gov/18.3#T1.t1.r11] +
+
<< [[Definition:monicChebyT|Definition:monicChebyT]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicctsbigqHermite|Definition:monicctsbigqHermite]] >>
+
+drmf_eof +drmf_bof +'''Definition:monicctsbigqHermite''' +
+
<< [[Definition:monicChebyU|Definition:monicChebyU]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicctsdualHahn|Definition:monicctsdualHahn]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\monicctsbigqHermite''' represents the monic continuous big q polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\monicctsbigqHermite{n}''' produces {\displaystyle \monicctsbigqHermite{n}}
+:'''\monicctsbigqHermite{n}@{x}{a}{q}''' produces {\displaystyle \monicctsbigqHermite{n}@{x}{a}{q}}
+:'''\monicctsbigqHermite{n}@@{x}{a}{q}''' produces {\displaystyle \monicctsbigqHermite{n}@@{x}{a}{q}}
+ +These are defined by + +{\displaystyle +\ctsbigqHermite{n}@{x}{a}{q}=:2^n\monicctsbigqHermite{n}@{x}{a}{q} +} +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:monicctsbigqHermite {\displaystyle {\widehat H}_{n}}] : monic continuous big {\displaystyle q}-Hermite polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicctsbigqHermite http://drmf.wmflabs.org/wiki/Definition:monicctsbigqHermite]
+[http://drmf.wmflabs.org/wiki/Definition:ctsbigqHermite {\displaystyle H_{n}}] : continuous big {\displaystyle q}-Hermite polynomial : [http://drmf.wmflabs.org/wiki/Definition:ctsbigqHermite http://drmf.wmflabs.org/wiki/Definition:ctsbigqHermite] +
+
<< [[Definition:monicChebyU|Definition:monicChebyU]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicctsdualHahn|Definition:monicctsdualHahn]] >>
+
+drmf_eof +drmf_bof +'''Definition:monicctsdualHahn''' +
+
<< [[Definition:monicctsbigqHermite|Definition:monicctsbigqHermite]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicctsdualqHahn|Definition:monicctsdualqHahn]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\monicctsdualHahn''' represents the monic continuous dual Hahn polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\monicctsdualHahn{n}''' produces {\displaystyle \monicctsdualHahn{n}}
+:'''\monicctsdualHahn{n}@{x^2}{a}{b}{c}{d}''' produces {\displaystyle \monicctsdualHahn{n}@{x^2}{a}{b}{c}{d}}
+:'''\monicctsdualHahn{n}@@{x^2}{a}{b}{c}{d}''' produces {\displaystyle \monicctsdualHahn{n}@@{x^2}{a}{b}{c}{d}}
+ +These are defined by + +{\displaystyle +\ctsdualHahn{n}@{x^2}{a}{b}{c}=:(-1)^n\monicctsdualHahn{n}@{x^2}{a}{b}{c} +} +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:monicctsdualHahn {\displaystyle {\widehat S}_{n}}] : monic continuous dual Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicctsdualHahn http://drmf.wmflabs.org/wiki/Definition:monicctsdualHahn]
+[http://dlmf.nist.gov/18.25#T1.t1.r3 {\displaystyle S_{n}}] : continuous dual Hahn polynomial : [http://dlmf.nist.gov/18.25#T1.t1.r3 http://dlmf.nist.gov/18.25#T1.t1.r3] +
+
<< [[Definition:monicctsbigqHermite|Definition:monicctsbigqHermite]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicctsdualqHahn|Definition:monicctsdualqHahn]] >>
+
+drmf_eof +drmf_bof +'''Definition:monicctsdualqHahn''' +
+
<< [[Definition:monicctsdualHahn|Definition:monicctsdualHahn]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicctsHahn|Definition:monicctsHahn]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\monicctsdualqHahn''' represents the monic continuous dual q Hahn polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\monicctsdualqHahn{n}''' produces {\displaystyle \monicctsdualqHahn{n}}
+:'''\monicctsdualqHahn{n}@{x^2}{a}{b}{c}''' produces {\displaystyle \monicctsdualqHahn{n}@{x^2}{a}{b}{c}}
+:'''\monicctsdualqHahn{n}@@{x^2}{a}{b}{c}''' produces {\displaystyle \monicctsdualqHahn{n}@@{x^2}{a}{b}{c}}
+ +These are defined by + +{\displaystyle +\ctsdualqHahn{n}@{x}{a}{b}{c}{q}=:2^n\monicctsdualqHahn{n}@{x}{a}{b}{c}{q} +} +
+
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:monicctsdualqHahn {\displaystyle {\widehat p}_{n}}] : monic continuous dual {\displaystyle q}-Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicctsdualqHahn http://drmf.wmflabs.org/wiki/Definition:monicctsdualqHahn]
+[http://drmf.wmflabs.org/wiki/Definition:ctsdualqHahn {\displaystyle p_{n}}] : continuous dual {\displaystyle q}-Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:ctsdualqHahn http://drmf.wmflabs.org/wiki/Definition:ctsdualqHahn] +
+
<< [[Definition:monicctsdualHahn|Definition:monicctsdualHahn]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicctsHahn|Definition:monicctsHahn]] >>
+
+drmf_eof +drmf_bof +'''Definition:monicctsHahn''' +
+
<< [[Definition:monicctsdualqHahn|Definition:monicctsdualqHahn]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicctsqHahn|Definition:monicctsqHahn]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\monicctsHahn''' represents the monic continuous Hahn polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\monicctsHahn{n}''' produces {\displaystyle \monicctsHahn{n}}
+:'''\monicctsHahn{n}@{x}{a}{b}{c}{d}''' produces {\displaystyle \monicctsHahn{n}@{x}{a}{b}{c}{d}}
+:'''\monicctsHahn{n}@@{x}{a}{b}{c}{d}''' produces {\displaystyle \monicctsHahn{n}@@{x}{a}{b}{c}{d}}
+ +These are defined by + +{\displaystyle +\ctsHahn{n}@{x}{a}{b}{c}{d}=:\frac{\pochhammer{n+a+b+c+d-1}{n}}{n!}\monicctsHahn{n}@{x}{a}{b}{c}{d} +} +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:monicctsHahn {\displaystyle {\widehat p}_{n}}] : monic continuous Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicctsHahn http://drmf.wmflabs.org/wiki/Definition:monicctsHahn]
+[http://dlmf.nist.gov/18.19#P2.p1 {\displaystyle p_{n}}] : continuous Hahn polynomial : [http://dlmf.nist.gov/18.19#P2.p1 http://dlmf.nist.gov/18.19#P2.p1]
+[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii] +
+
<< [[Definition:monicctsdualqHahn|Definition:monicctsdualqHahn]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicctsqHahn|Definition:monicctsqHahn]] >>
+
+drmf_eof +drmf_bof +'''Definition:monicctsqHahn''' +
+
<< [[Definition:monicctsHahn|Definition:monicctsHahn]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicctsqHermite|Definition:monicctsqHermite]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\monicctsqHahn''' represents the monic continuous q Hahn polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\monicctsqHahn{n}''' produces {\displaystyle \monicctsqHahn{n}}
+:'''\monicctsqHahn{n}@{x}{a}{b}{c}{d}{q}''' produces {\displaystyle \monicctsqHahn{n}@{x}{a}{b}{c}{d}{q}}
+:'''\monicctsqHahn{n}@@{x}{a}{b}{c}{d}{q}''' produces {\displaystyle \monicctsqHahn{n}@@{x}{a}{b}{c}{d}{q}}
+ +These are defined by + +{\displaystyle +\ctsqHahn{n}@{x}{a}{b}{c}{d}{q}=:2^n\qPochhammer{abcdq^{n-1}}{q}{n}\monicctsqHahn{n}@{x}{a}{b}{c}{d}{q} +} +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:monicctsqHahn {\displaystyle {\widehat p}_{n}}] : monic continuous {\displaystyle q}-Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicctsqHahn http://drmf.wmflabs.org/wiki/Definition:monicctsqHahn]
+[http://drmf.wmflabs.org/wiki/Definition:ctsqHahn {\displaystyle p_{n}}] : continuous {\displaystyle q}-Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:ctsqHahn http://drmf.wmflabs.org/wiki/Definition:ctsqHahn]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1] +
+
<< [[Definition:monicctsHahn|Definition:monicctsHahn]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicctsqHermite|Definition:monicctsqHermite]] >>
+
+drmf_eof +drmf_bof +'''Definition:monicctsqHermite''' +
+
<< [[Definition:monicctsqHahn|Definition:monicctsqHahn]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicctsqJacobi|Definition:monicctsqJacobi]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\monicctsqHermite''' represents the monic continuous q Hermite polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\monicctsqHermite{n}''' produces {\displaystyle \monicctsqHermite{n}}
+:'''\monicctsqHermite{n}@{x}{q}''' produces {\displaystyle \monicctsqHermite{n}@{x}{q}}
+:'''\monicctsqHermite{n}@@{x}{q}''' produces {\displaystyle \monicctsqHermite{n}@@{x}{q}}
+ +These are defined by + +{\displaystyle +\ctsqHermite{n}@{x}{q}=:2^n\monicctsqHermite{n}@{x}{q} +} +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:monicctsqHermite {\displaystyle {\widehat H}_{n}}] : monic continuous {\displaystyle q}-Hermite polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicctsqHermite http://drmf.wmflabs.org/wiki/Definition:monicctsqHermite]
+[http://drmf.wmflabs.org/wiki/Definition:ctsqHermite {\displaystyle H_{n}}] : continuous {\displaystyle q}-Hermite polynomial : [http://drmf.wmflabs.org/wiki/Definition:ctsqHermite http://drmf.wmflabs.org/wiki/Definition:ctsqHermite] +
+
<< [[Definition:monicctsqHahn|Definition:monicctsqHahn]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicctsqJacobi|Definition:monicctsqJacobi]] >>
+
+drmf_eof +drmf_bof +'''Definition:monicctsqJacobi''' +
+
<< [[Definition:monicctsqHermite|Definition:monicctsqHermite]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicctsqLaguerre|Definition:monicctsqLaguerre]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\monicctsqJacobi''' represents the monic continous q Jacobi polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\monicctsqJacobi{n}''' produces {\displaystyle \monicctsqJacobi{n}}
+:'''\monicctsqJacobi{n}@{x}{q}''' produces {\displaystyle \monicctsqJacobi{n}@{x}{q}}
+:'''\monicctsqJacobi{n}@@{x}{q}''' produces {\displaystyle \monicctsqJacobi{n}@@{x}{q}}
+ +These are defined by +{\displaystyle +\ctsqJacobi{\alpha}{\beta}{n}@{x}{q}=:\frac{2^nq^{(\frac{1}{2}\alpha+\frac{1}{4})n}\qPochhammer{q^{n+\alpha+\beta+1}}{q}{n}} +{\qPochhammer{q,-q^{\frac{1}{2}(\alpha+\beta+1)},-q^{\frac{1}{2}(\alpha+\beta+2)}}{q}{n}}\monicctsqJacobi{\alpha}{\beta}{n}@@{x}{q}. + +
+
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:monicctsqJacobi {\displaystyle {\widehat P}^{(\alpha,\beta)}_{n}}] : monic continuous {\displaystyle q}-Jacobi polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicctsqJacobi http://drmf.wmflabs.org/wiki/Definition:monicctsqJacobi]
+[http://drmf.wmflabs.org/wiki/Definition:ctsqJacobi {\displaystyle P^{(\alpha,\beta)}_{n}}] : continuous {\displaystyle q}-Jacobi polynomial : [http://drmf.wmflabs.org/wiki/Definition:ctsqJacobi http://drmf.wmflabs.org/wiki/Definition:ctsqJacobi]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1] +
+
<< [[Definition:monicctsqHermite|Definition:monicctsqHermite]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicctsqLaguerre|Definition:monicctsqLaguerre]] >>
+
+drmf_eof +drmf_bof +'''Definition:monicctsqLaguerre''' +
+
<< [[Definition:monicctsqJacobi|Definition:monicctsqJacobi]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicctsqLegendre|Definition:monicctsqLegendre]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\monicctsqLaguerre''' represents the monic continuous q Laguerre polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\monicctsqLaguerre{n}''' produces {\displaystyle \monicctsqLaguerre{n}}
+:'''\monicctsqLaguerre{n}@{x}{q}''' produces {\displaystyle \monicctsqLaguerre{n}@{x}{q}}
+:'''\monicctsqLaguerre{n}@@{x}{q}''' produces {\displaystyle \monicctsqLaguerre{n}@@{x}{q}}
+ +These are defined by + +{\displaystyle +\ctsqLaguerre{\alpha}{n}@{x}{q}=:\frac{2^nq^{(\frac{1}{2}\alpha+\frac{1}{4})n}}{\qPochhammer{q}{q}{n}}\monicctsqLaguerre{\alpha}{n}@@{x}{q}. +} +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:monicctsqLaguerre {\displaystyle {\widehat P}^{(\alpha)}_{n}}] : monic continuous {\displaystyle q}-Laguerre polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicctsqLaguerre http://drmf.wmflabs.org/wiki/Definition:monicctsqLaguerre]
+[http://drmf.wmflabs.org/wiki/Definition:ctsqLaguerre {\displaystyle P^{(n)}_{\alpha}}] : continuous {\displaystyle q}-Laguerre polynomial : [http://drmf.wmflabs.org/wiki/Definition:ctsqLaguerre http://drmf.wmflabs.org/wiki/Definition:ctsqLaguerre]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1] +
+
<< [[Definition:monicctsqJacobi|Definition:monicctsqJacobi]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicctsqLegendre|Definition:monicctsqLegendre]] >>
+
+drmf_eof +drmf_bof +'''Definition:monicctsqLegendre''' +
+
<< [[Definition:monicctsqLaguerre|Definition:monicctsqLaguerre]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicctsqUltra|Definition:monicctsqUltra]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\monicctsqLegendre''' represents the monic continuous q Legendre polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\monicctsqLegendre{n}''' produces {\displaystyle \monicctsqLegendre{n}}
+:'''\monicctsqLegendre{n}@{x}{q}''' produces {\displaystyle \monicctsqLegendre{n}@{x}{q}}
+:'''\monicctsqLegendre{n}@@{x}{q}''' produces {\displaystyle \monicctsqLegendre{n}@@{x}{q}}
+ +These are defined by + +{\displaystyle +\ctsqLegendre{n}@{x}{q}=:\frac{2^nq^{\frac{1}{4}n}\qPochhammer{q^{\frac{1}{2}}}{q}{n}}{\qPochhammer{q}{q}{n}}\monicctsqLegendre{n}@@{x}{q}. +} +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:monicctsqLegendre {\displaystyle {\widehat P}_{n}}] : monic continuous {\displaystyle q}-Legendre polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicctsqLegendre http://drmf.wmflabs.org/wiki/Definition:monicctsqLegendre]
+[http://drmf.wmflabs.org/wiki/Definition:ctsqLegendre {\displaystyle P_{n}}] : continuous {\displaystyle q}-Legendre polynomial : [http://drmf.wmflabs.org/wiki/Definition:ctsqLegendre http://drmf.wmflabs.org/wiki/Definition:ctsqLegendre]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1] +
+
<< [[Definition:monicctsqLaguerre|Definition:monicctsqLaguerre]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicctsqUltra|Definition:monicctsqUltra]] >>
+
+drmf_eof +drmf_bof +'''Definition:monicctsqUltra''' +
+
<< [[Definition:monicctsqLegendre|Definition:monicctsqLegendre]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicdiscrqHermiteI|Definition:monicdiscrqHermiteI]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\monicctsqUltra''' represents the monic continuous q ultraspherical Rogers polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\monicctsqUltra{n}''' produces {\displaystyle \monicctsqUltra{n}}
+:'''\monicctsqUltra{n}@{x}{\beta}{q}''' produces {\displaystyle \monicctsqUltra{n}@{x}{\beta}{q}}
+:'''\monicctsqUltra{n}@@{x}{\beta}{q}''' produces {\displaystyle \monicctsqUltra{n}@@{x}{\beta}{q}}
+ +These are defined by + +{\displaystyle +\ctsqUltra{n}@{x}{\beta}{q}=:\frac{2^n\qPochhammer{\beta}{q}{n}}{\qPochhammer{q}{q}{n}}\monicctsqUltra{n}@@{x}{\beta}{q}. +} +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:monicctsqUltra {\displaystyle {\widehat C}_{n}}] : monic continuous {\displaystyle q}-ultraspherical/Rogers polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicctsqUltra http://drmf.wmflabs.org/wiki/Definition:monicctsqUltra]
+[http://dlmf.nist.gov/18.28#E13 {\displaystyle C_{n}}] : continuous {\displaystyle q}-ultraspherical/Rogers polynomial : [http://dlmf.nist.gov/18.28#E13 http://dlmf.nist.gov/18.28#E13]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1] +
+
<< [[Definition:monicctsqLegendre|Definition:monicctsqLegendre]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicdiscrqHermiteI|Definition:monicdiscrqHermiteI]] >>
+
+drmf_eof +drmf_bof +'''Definition:monicdiscrqHermiteI''' +
+
<< [[Definition:monicctsqUltra|Definition:monicctsqUltra]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicdiscrqHermiteII|Definition:monicdiscrqHermiteII]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\monicdiscrqHermiteI''' represents the monic discrete q Hermite polynomial I. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\monicdiscrqHermiteI{n}''' produces {\displaystyle \monicdiscrqHermiteI{n}}
+:'''\monicdiscrqHermiteI{n}@{x}{q}''' produces {\displaystyle \monicdiscrqHermiteI{n}@{x}{q}}
+:'''\monicdiscrqHermiteI{n}@@{x}{q}''' produces {\displaystyle \monicdiscrqHermiteI{n}@@{x}{q}}
+ +These are defined by + +{\displaystyle +\discrqHermiteI{n}@{x}{q}=:\monicdiscrqHermiteI{n}@{x}{q}. +} +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:monicdiscrqHermiteI {\displaystyle {\widehat h}_{n}}] : monic discrete {\displaystyle q}-Hermite I polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicdiscrqHermiteI http://drmf.wmflabs.org/wiki/Definition:monicdiscrqHermiteI]
+[http://drmf.wmflabs.org/wiki/Definition:discrqHermiteI {\displaystyle h_{n}}] : discrete {\displaystyle q}-Hermite I polynomial : [http://drmf.wmflabs.org/wiki/Definition:discrqHermiteI http://drmf.wmflabs.org/wiki/Definition:discrqHermiteI] +
+
<< [[Definition:monicctsqUltra|Definition:monicctsqUltra]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicdiscrqHermiteII|Definition:monicdiscrqHermiteII]] >>
+
+drmf_eof +drmf_bof +'''Definition:monicdiscrqHermiteII''' +
+
<< [[Definition:monicdiscrqHermiteI|Definition:monicdiscrqHermiteI]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicdualHahn|Definition:monicdualHahn]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\monicdiscrqHermiteII''' represents the monic discrete q Hermite polynomial II. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\monicdiscrqHermiteII{n}''' produces {\displaystyle \monicdiscrqHermiteII{n}}
+:'''\monicdiscrqHermiteII{n}@{x}{q}''' produces {\displaystyle \monicdiscrqHermiteII{n}@{x}{q}}
+:'''\monicdiscrqHermiteII{n}@@{x}{q}''' produces {\displaystyle \monicdiscrqHermiteII{n}@@{x}{q}}
+ +These are defined by + +{\displaystyle +\discrqHermiteII{n}@{x}{q}=:\monicdiscrqHermiteII{n}@@{x}{q}. +} +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:monicdiscrqHermiteII {\displaystyle {\widehat \tilde{p}_{n}}}] : monic discrete {\displaystyle q}-Hermite II polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicdiscrqHermiteII http://drmf.wmflabs.org/wiki/Definition:monicdiscrqHermiteII]
+[http://drmf.wmflabs.org/wiki/Definition:discrqHermiteII {\displaystyle \tilde{h}_{n}}] : discrete {\displaystyle q}-Hermite II polynomial : [http://drmf.wmflabs.org/wiki/Definition:discrqHermiteII http://drmf.wmflabs.org/wiki/Definition:discrqHermiteII] +
+
<< [[Definition:monicdiscrqHermiteI|Definition:monicdiscrqHermiteI]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicdualHahn|Definition:monicdualHahn]] >>
+
+drmf_eof +drmf_bof +'''Definition:monicdualHahn''' +
+
<< [[Definition:monicdiscrqHermiteII|Definition:monicdiscrqHermiteII]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicdualqHahn|Definition:monicdualqHahn]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\monicdualHahn''' represents the monic dual Hahn. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\monicdualHahn{n}''' produces {\displaystyle \monicdualHahn{n}}
+:'''\monicdualHahn{n}@{\lambda(x)}{\gamma}{\delta}{N}''' produces {\displaystyle \monicdualHahn{n}@{\lambda(x)}{\gamma}{\delta}{N}}
+:'''\monicdualHahn{n}@@{\lambda(x)}{\gamma}{\delta}{N}''' produces {\displaystyle \monicdualHahn{n}@@{\lambda(x)}{\gamma}{\delta}{N}}
+ +These are defined by + +{\displaystyle +\dualHahn{n}@{\lambda(x)}{\gamma}{\delta}{N}=:\frac{1}{\pochhammer{\gamma+1}{n}\pochhammer{-N}{n}}\monicdualHahn{n}@{\lambda(x}{\gamma}{\delta}{N}) +} + +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:monicdualHahn {\displaystyle {\widehat R}_{n}}] : monic dual Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicdualHahn http://drmf.wmflabs.org/wiki/Definition:monicdualHahn]
+[http://dlmf.nist.gov/18.25#T1.t1.r5 {\displaystyle R_{n}}] : dual Hahn polynomial : [http://dlmf.nist.gov/18.25#T1.t1.r5 http://dlmf.nist.gov/18.25#T1.t1.r5]
+[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii] +
+
<< [[Definition:monicdiscrqHermiteII|Definition:monicdiscrqHermiteII]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicdualqHahn|Definition:monicdualqHahn]] >>
+
+drmf_eof +drmf_bof +'''Definition:monicdualqHahn''' +
+
<< [[Definition:monicdualHahn|Definition:monicdualHahn]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicdualqKrawtchouk|Definition:monicdualqKrawtchouk]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\monicdualqHahn''' represents the monic dual q Hahn. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\monicdualqHahn{n}''' produces {\displaystyle \monicdualqHahn{n}}
+:'''\monicdualqHahn{n}@{\mu(x)}{\gamma}{\delta}{N}{q}''' produces {\displaystyle \monicdualqHahn{n}@{\mu(x)}{\gamma}{\delta}{N}{q}}
+:'''\monicdualqHahn{n}@@{\mu(x)}{\gamma}{\delta}{N}{q}''' produces {\displaystyle \monicdualqHahn{n}@@{\mu(x)}{\gamma}{\delta}{N}{q}}
+ +These are defined by + +{\displaystyle +\dualqHahn{n}@{\mu(x)}{\gamma}{\delta}{N}{q}=:\frac{1}{\qPochhammer{\gamma q,q^{-N}}{q}{n}}\monicdualqHahn{n}@@{\mu(x}{\gamma}{\delta}{N}). +} +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:monicdualqHahn {\displaystyle {\widehat R}_{n}}] : monic dual {\displaystyle q}-Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicdualqHahn http://drmf.wmflabs.org/wiki/Definition:monicdualqHahn]
+[http://drmf.wmflabs.org/wiki/Definition:dualqHahn {\displaystyle R_{n}}] : dual {\displaystyle q}-Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:dualqHahn http://drmf.wmflabs.org/wiki/Definition:dualqHahn]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1] +
+
<< [[Definition:monicdualHahn|Definition:monicdualHahn]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicdualqKrawtchouk|Definition:monicdualqKrawtchouk]] >>
+
+drmf_eof +drmf_bof +'''Definition:monicdualqKrawtchouk''' +
+
<< [[Definition:monicdualqHahn|Definition:monicdualqHahn]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicHahn|Definition:monicHahn]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\monicdualqKrawtchouk''' represents the monic dual q Krawtchouk polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\monicdualqKrawtchouk{n}''' produces {\displaystyle \monicdualqKrawtchouk{n}}
+:'''\monicdualqKrawtchouk{n}@{\lambda(x)}{c}{N}{q}''' produces {\displaystyle \monicdualqKrawtchouk{n}@{\lambda(x)}{c}{N}{q}}
+:'''\monicdualqKrawtchouk{n}@@{\lambda(x)}{c}{N}{q}''' produces {\displaystyle \monicdualqKrawtchouk{n}@@{\lambda(x)}{c}{N}{q}}
+ +These are defined by + +{\displaystyle +\dualqKrawtchouk{n}@{\lambda(x)}{c}{N}{q}=:\frac{1}{\qPochhammer{q^{-N}}{q}{n}}\monicdualqKrawtchouk{n}@@{\lambda(x}{c}{N}{q}). +} +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:monicdualqKrawtchouk {\displaystyle {\widehat K}_{n}}] : monic dual {\displaystyle q}-Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicdualqKrawtchouk http://drmf.wmflabs.org/wiki/Definition:monicdualqKrawtchouk]
+[http://drmf.wmflabs.org/wiki/Definition:dualqKrawtchouk {\displaystyle K_{n}}] : dual {\displaystyle q}-Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:dualqKrawtchouk http://drmf.wmflabs.org/wiki/Definition:dualqKrawtchouk]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1] +
+
<< [[Definition:monicdualqHahn|Definition:monicdualqHahn]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicHahn|Definition:monicHahn]] >>
+
+drmf_eof +drmf_bof +'''Definition:monicHahn''' +
+
<< [[Definition:monicdualqKrawtchouk|Definition:monicdualqKrawtchouk]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicHermite|Definition:monicHermite]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\monicHahn''' represents the monic Hahn polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\monicHahn{n}''' produces {\displaystyle \monicHahn{n}}
+:'''\monicHahn{n}@{x}{\alpha}{\beta}{N}''' produces {\displaystyle \monicHahn{n}@{x}{\alpha}{\beta}{N}}
+:'''\monicHahn{n}@@{x}{\alpha}{\beta}{N}''' produces {\displaystyle \monicHahn{n}@@{x}{\alpha}{\beta}{N}}
+ +These are defined by + +{\displaystyle +\Hahn{n}@{x}{\alpha}{\beta}{N}=:\frac{\pochhammer{n+\alpha+\beta+1}{n}}{\pochhammer{\alpha+1}{n}\pochhammer{-N}{n}}\monicHahn{n}@{x}{\alpha}{\beta}{N} +} +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:monicHahn {\displaystyle {\widehat Q}_{n}}] : monic Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicHahn http://drmf.wmflabs.org/wiki/Definition:monicHahn]
+[http://dlmf.nist.gov/18.19#T1.t1.r3 {\displaystyle Q_{n}}] : Hahn polynomial : [http://dlmf.nist.gov/18.19#T1.t1.r3 http://dlmf.nist.gov/18.19#T1.t1.r3]
+[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii] +
+
<< [[Definition:monicdualqKrawtchouk|Definition:monicdualqKrawtchouk]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicHermite|Definition:monicHermite]] >>
+
+drmf_eof +drmf_bof +'''Definition:monicHermite''' +
+
<< [[Definition:monicHahn|Definition:monicHahn]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicJacobi|Definition:monicJacobi]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\monicHermite''' represents the monic Hermite polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\monicHermite{n}''' produces {\displaystyle \monicHermite{n}}
+:'''\monicHermite{n}@{x}''' produces {\displaystyle \monicHermite{n}@{x}}
+:'''\monicHermite{n}@@{x}''' produces {\displaystyle \monicHermite{n}@@{x}}
+ +These are defined by + +{\displaystyle +\Hermite{n}@{x}=:2^n\monicHermite{n}@{x}. +} +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:monicHermite {\displaystyle {\widehat H}_{n}}] : monic Hermite polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicHermite http://drmf.wmflabs.org/wiki/Definition:monicHermite]
+[http://dlmf.nist.gov/18.3#T1.t1.r28 {\displaystyle H_{n}}] : Hermite polynomial {\displaystyle H_n} : [http://dlmf.nist.gov/18.3#T1.t1.r28 http://dlmf.nist.gov/18.3#T1.t1.r28] +
+
<< [[Definition:monicHahn|Definition:monicHahn]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicJacobi|Definition:monicJacobi]] >>
+
+drmf_eof +drmf_bof +'''Definition:monicJacobi''' +
+
<< [[Definition:monicHermite|Definition:monicHermite]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicKrawtchouk|Definition:monicKrawtchouk]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\monicJacobi''' represents the monic Jacobi polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\monicJacobi{n}''' produces {\displaystyle \monicJacobi{n}}
+:'''\monicJacobi{n}@{x}''' produces {\displaystyle \monicJacobi{n}@{x}}
+:'''\monicJacobi{n}@@{x}''' produces {\displaystyle \monicJacobi{n}@@{x}}
+ +These are defined by + +{\displaystyle +\Jacobi{\alpha}{\beta}{n}@{x}=:\frac{\pochhammer{n+\alpha+\beta+1}{n}}{2^nn!}\monicJacobi{\alpha}{\beta}{n}@{x} +} +
+
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:monicJacobi {\displaystyle {\widehat P}^{(\alpha,\beta)}_{n}}] : monic Jacobi polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicJacobi http://drmf.wmflabs.org/wiki/Definition:monicJacobi]
+[http://dlmf.nist.gov/18.3#T1.t1.r3 {\displaystyle P^{(\alpha,\beta)}_{n}}] : Jacobi polynomial : [http://dlmf.nist.gov/18.3#T1.t1.r3 http://dlmf.nist.gov/18.3#T1.t1.r3]
+[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii] +
+
<< [[Definition:monicHermite|Definition:monicHermite]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicKrawtchouk|Definition:monicKrawtchouk]] >>
+
+drmf_eof +drmf_bof +'''Definition:monicKrawtchouk''' +
+
<< [[Definition:monicJacobi|Definition:monicJacobi]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicLaguerre|Definition:monicLaguerre]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\monicKrawtchouk''' represents the monic Krawtchouk polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\monicKrawtchouk{n}''' produces {\displaystyle \monicKrawtchouk{n}}
+:'''\monicKrawtchouk{n}@{q^{-x}}{p}{N}{q}''' produces {\displaystyle \monicKrawtchouk{n}@{q^{-x}}{p}{N}{q}}
+:'''\monicKrawtchouk{n}@@{q^{-x}}{p}{N}{q}''' produces {\displaystyle \monicKrawtchouk{n}@@{q^{-x}}{p}{N}{q}}
+ +These are defined by + +{\displaystyle +\Krawtchouk{n}@{x}{p}{N}=:\frac{1}{\pochhammer{-N}{n}p^n}\monicKrawtchouk{n}@{x}{p}{N}. +} +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:monicKrawtchouk {\displaystyle {\widehat K}_{n}}] : monic Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicKrawtchouk http://drmf.wmflabs.org/wiki/Definition:monicKrawtchouk]
+[http://dlmf.nist.gov/18.19#T1.t1.r6 {\displaystyle K_{n}}] : Krawtchouk polynomial : [http://dlmf.nist.gov/18.19#T1.t1.r6 http://dlmf.nist.gov/18.19#T1.t1.r6]
+[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii] +
+
<< [[Definition:monicJacobi|Definition:monicJacobi]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicLaguerre|Definition:monicLaguerre]] >>
+
+drmf_eof +drmf_bof +'''Definition:monicLaguerre''' +
+
<< [[Definition:monicKrawtchouk|Definition:monicKrawtchouk]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicLegendrePoly|Definition:monicLegendrePoly]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\monicLaguerre''' represents the monic generalized Laguerre polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\monicLaguerre{n}''' produces {\displaystyle \monicLaguerre{n}}
+:'''\monicLaguerre{n}@{x}''' produces {\displaystyle \monicLaguerre{n}@{x}}
+:'''\monicLaguerre{n}@@{x}''' produces {\displaystyle \monicLaguerre{n}@@{x}}
+ +These are defined by + +{\displaystyle +\Laguerre[\alpha]{n}@{x}=:\frac{(-1)^n}{n!}\monicLaguerre[\alpha]{n}@{x}. +} +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:monicLaguerre {\displaystyle {\widehat L}^{\alpha}_n}] : monic Laguerre polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicLaguerre http://drmf.wmflabs.org/wiki/Definition:monicLaguerre]
+[http://dlmf.nist.gov/18.3#T1.t1.r27 {\displaystyle L_n^{(\alpha)}}] : Laguerre (or generalized Laguerre) polynomial : [http://dlmf.nist.gov/18.3#T1.t1.r27 http://dlmf.nist.gov/18.3#T1.t1.r27] +
+
<< [[Definition:monicKrawtchouk|Definition:monicKrawtchouk]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicLegendrePoly|Definition:monicLegendrePoly]] >>
+
+drmf_eof +drmf_bof +'''Definition:monicLegendrePoly''' +
+
<< [[Definition:monicLaguerre|Definition:monicLaguerre]]
+
[[Main_Page|Main Page]]
+
[[Definition:moniclittleqJacobi|Definition:moniclittleqJacobi]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\monicLegendrePoly''' represents the monic Legendre spherical polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\monicLegendrePoly{n}''' produces {\displaystyle \monicLegendrePoly{n}}
+:'''\monicLegendrePoly{n}@{x}''' produces {\displaystyle \monicLegendrePoly{n}@{x}}
+:'''\monicLegendrePoly{n}@@{x}''' produces {\displaystyle \monicLegendrePoly{n}@@{x}}
+ +These are defined by + +{\displaystyle +\LegendrePoly{n}@{x}=:\binomial{2n}{n}\frac{1}{2^n}\monicLegendrePoly{n}@{x}. +} + +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:monicLegendrePoly {\displaystyle {\widehat P}_{n}}] : monic Legendre polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicLegendrePoly http://drmf.wmflabs.org/wiki/Definition:monicLegendrePoly]
+[http://dlmf.nist.gov/18.3#T1.t1.r25 {\displaystyle P_{n}}] : Legendre polynomial : [http://dlmf.nist.gov/18.3#T1.t1.r25 http://dlmf.nist.gov/18.3#T1.t1.r25]
+[http://dlmf.nist.gov/1.2#E1 {\displaystyle \binom{n}{k}}] : binomial coefficient : [http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/1.2#E1] [http://dlmf.nist.gov/26.3#SS1.p1 http://dlmf.nist.gov/26.3#SS1.p1] +
+
<< [[Definition:monicLaguerre|Definition:monicLaguerre]]
+
[[Main_Page|Main Page]]
+
[[Definition:moniclittleqJacobi|Definition:moniclittleqJacobi]] >>
+
+drmf_eof +drmf_bof +'''Definition:moniclittleqJacobi''' +
+
<< [[Definition:monicLegendrePoly|Definition:monicLegendrePoly]]
+
[[Main_Page|Main Page]]
+
[[Definition:moniclittleqLaguerre|Definition:moniclittleqLaguerre]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\moniclittleqJacobi''' represents the monic little q Jacobi polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\moniclittleqJacobi{n}''' produces {\displaystyle \moniclittleqJacobi{n}}
+:'''\moniclittleqJacobi{n}@{x}{a}{b}{q}''' produces {\displaystyle \moniclittleqJacobi{n}@{x}{a}{b}{q}}
+:'''\moniclittleqJacobi{n}@@{x}{a}{b}{q}''' produces {\displaystyle \moniclittleqJacobi{n}@@{x}{a}{b}{q}}
+ +These are defined by + +{\displaystyle +\littleqJacobi{n}@{x}{a}{b}{q}=:\frac{(-1)^nq^{-\binomial{n}{2}}\qPochhammer{abq^{n+1}}{q}{n}}{\qPochhammer{aq}{q}{n}}\moniclittleqJacobi{n}@@{x}{a}{b}{q}. +} +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:moniclittleqJacobi {\displaystyle {\widehat p}_{n}}] : monic little {\displaystyle q}-Jacobi polynomial : [http://drmf.wmflabs.org/wiki/Definition:moniclittleqJacobi http://drmf.wmflabs.org/wiki/Definition:moniclittleqJacobi]
+[http://drmf.wmflabs.org/wiki/Definition:littleqJacobi {\displaystyle p_{n}}] : little {\displaystyle q}-Jacobi polynomial : [http://drmf.wmflabs.org/wiki/Definition:littleqJacobi http://drmf.wmflabs.org/wiki/Definition:littleqJacobi]
+[http://dlmf.nist.gov/1.2#E1 {\displaystyle \binom{n}{k}}] : binomial coefficient : [http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/1.2#E1] [http://dlmf.nist.gov/26.3#SS1.p1 http://dlmf.nist.gov/26.3#SS1.p1]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1] +
+
<< [[Definition:monicLegendrePoly|Definition:monicLegendrePoly]]
+
[[Main_Page|Main Page]]
+
[[Definition:moniclittleqLaguerre|Definition:moniclittleqLaguerre]] >>
+
+drmf_eof +drmf_bof +'''Definition:moniclittleqLaguerre''' +
+
<< [[Definition:moniclittleqJacobi|Definition:moniclittleqJacobi]]
+
[[Main_Page|Main Page]]
+
[[Definition:moniclittleqLegendre|Definition:moniclittleqLegendre]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\moniclittleqLaguerre''' represents the monic little q Laguerre Wall polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\moniclittleqLaguerre{n}''' produces {\displaystyle \moniclittleqLaguerre{n}}
+:'''\moniclittleqLaguerre{n}@{x}{a}{q}''' produces {\displaystyle \moniclittleqLaguerre{n}@{x}{a}{q}}
+:'''\moniclittleqLaguerre{n}@@{x}{a}{q}''' produces {\displaystyle \moniclittleqLaguerre{n}@@{x}{a}{q}}
+ +These are defined by +
+ +\littleqLaguerre{n}@{x}{a}{q}=:\frac{(-1)^nq^{-\binomial{n}{2}}}{\qPochhammer{aq}{q}{n}}\moniclittleqLaguerre{n}@@{x}{a}{q}. + + +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:moniclittleqLaguerre {\displaystyle {\widehat p}_{n}}] : monic little {\displaystyle q}-Laguerre/Wall polynomial : [http://drmf.wmflabs.org/wiki/Definition:moniclittleqLaguerre http://drmf.wmflabs.org/wiki/Definition:moniclittleqLaguerre]
+[http://drmf.wmflabs.org/wiki/Definition:littleqLaguerre {\displaystyle p_{n}}] : little {\displaystyle q}-Laguerre / Wall polynomial : [http://drmf.wmflabs.org/wiki/Definition:littleqLaguerre http://drmf.wmflabs.org/wiki/Definition:littleqLaguerre]
+[http://dlmf.nist.gov/1.2#E1 {\displaystyle \binom{n}{k}}] : binomial coefficient : [http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/1.2#E1] [http://dlmf.nist.gov/26.3#SS1.p1 http://dlmf.nist.gov/26.3#SS1.p1]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1] +
+
<< [[Definition:moniclittleqJacobi|Definition:moniclittleqJacobi]]
+
[[Main_Page|Main Page]]
+
[[Definition:moniclittleqLegendre|Definition:moniclittleqLegendre]] >>
+
+drmf_eof +drmf_bof +'''Definition:moniclittleqLegendre''' +
+
<< [[Definition:moniclittleqLaguerre|Definition:moniclittleqLaguerre]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicMeixner|Definition:monicMeixner]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\moniclittleqLegendre''' represents the monic little q Legendre polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\moniclittleqLegendre{n}''' produces {\displaystyle \moniclittleqLegendre{n}}
+:'''\moniclittleqLegendre{n}@{x}{q}''' produces {\displaystyle \moniclittleqLegendre{n}@{x}{q}}
+:'''\moniclittleqLegendre{n}@@{x}{q}''' produces {\displaystyle \moniclittleqLegendre{n}@@{x}{q}}
+ +These are defined by + +{\displaystyle +\littleqLegendre{n}@{x}{q}=:\frac{(-1)^nq^{-\binomial{n}{2}}\qPochhammer{q^{n+1}}{q}{n}}{\qPochhammer{q}{q}{n}}\moniclittleqLegendre{n}@@{x}{q}. +} +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:moniclittleqLegendre {\displaystyle {\widehat p}_{n}}] : monic little {\displaystyle q}-Legendre polynomial : [http://drmf.wmflabs.org/wiki/Definition:moniclittleqLegendre http://drmf.wmflabs.org/wiki/Definition:moniclittleqLegendre]
+[http://drmf.wmflabs.org/wiki/Definition:littleqLegendre {\displaystyle p_{n}}] : little {\displaystyle q}-Legendre polynomial : [http://drmf.wmflabs.org/wiki/Definition:littleqLegendre http://drmf.wmflabs.org/wiki/Definition:littleqLegendre]
+[http://dlmf.nist.gov/1.2#E1 {\displaystyle \binom{n}{k}}] : binomial coefficient : [http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/1.2#E1] [http://dlmf.nist.gov/26.3#SS1.p1 http://dlmf.nist.gov/26.3#SS1.p1]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1] +
+
<< [[Definition:moniclittleqLaguerre|Definition:moniclittleqLaguerre]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicMeixner|Definition:monicMeixner]] >>
+
+drmf_eof +drmf_bof +'''Definition:monicMeixner''' +
+
<< [[Definition:moniclittleqLegendre|Definition:moniclittleqLegendre]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicMeixnerPollaczek|Definition:monicMeixnerPollaczek]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\monicMeixner''' represents the monic Meixner polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\monicMeixner{n}''' produces {\displaystyle \monicMeixner{n}}
+:'''\monicMeixner{n}@{x}{\beta}{c}''' produces {\displaystyle \monicMeixner{n}@{x}{\beta}{c}}
+:'''\monicMeixner{n}@@{x}{\beta}{c}''' produces {\displaystyle \monicMeixner{n}@@{x}{\beta}{c}}
+ +These are defined by + +{\displaystyle +\Meixner{n}@{x}{\beta}{c}=:\frac{1}{\pochhammer{\beta}{n}}\left(\frac{c-1}{c}\right)^n\monicMeixner{n}@@{x}{\beta}{c}. +} +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:monicMeixner {\displaystyle {\widehat M}_{n}}] : monic Meixner polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicMeixner http://drmf.wmflabs.org/wiki/Definition:monicMeixner]
+[http://dlmf.nist.gov/18.19#T1.t1.r9 {\displaystyle M_{n}}] : Meixner polynomial : [http://dlmf.nist.gov/18.19#T1.t1.r9 http://dlmf.nist.gov/18.19#T1.t1.r9]
+[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii] +
+
<< [[Definition:moniclittleqLegendre|Definition:moniclittleqLegendre]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicMeixnerPollaczek|Definition:monicMeixnerPollaczek]] >>
+
+drmf_eof +drmf_bof +'''Definition:monicMeixnerPollaczek''' +
+
<< [[Definition:monicMeixner|Definition:monicMeixner]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicpseudoJacobi|Definition:monicpseudoJacobi]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\monicMeixnerPollaczek''' represents the monic Meixner Pollaczek polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\monicMeixnerPollaczek{\lambda}{n}''' produces {\displaystyle \monicMeixnerPollaczek{\lambda}{n}}
+:'''\monicMeixnerPollaczek{\lambda}{n}@{x}{\phi}''' produces {\displaystyle \monicMeixnerPollaczek{\lambda}{n}@{x}{\phi}}
+:'''\monicMeixnerPollaczek{\lambda}{n}@@{x}{\phi}''' produces {\displaystyle \monicMeixnerPollaczek{\lambda}{n}@@{x}{\phi}}
+ +These are defined by + +{\displaystyle +\MeixnerPollaczek{\lambda}{n}@{x}{\phi}=:\frac{(2\sin@@{\phi})^n}{n!}\monicMeixnerPollaczek{\lambda}{n}@@{x}{\phi}. +} +
+
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:monicMeixnerPollaczek {\displaystyle {\widehat P}^{(\alpha)}_{n}}] : monic Meixner-Pollaczek polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicMeixnerPollaczek http://drmf.wmflabs.org/wiki/Definition:monicMeixnerPollaczek]
+[http://dlmf.nist.gov/18.19#P3.p1 {\displaystyle P^{(\alpha)}_{n}}] : Meixner-Pollaczek polynomial : [http://dlmf.nist.gov/18.19#P3.p1 http://dlmf.nist.gov/18.19#P3.p1]
+[http://dlmf.nist.gov/4.14#E1 {\displaystyle \mathrm{sin}}] : sine function : [http://dlmf.nist.gov/4.14#E1 http://dlmf.nist.gov/4.14#E1] +
+
<< [[Definition:monicMeixner|Definition:monicMeixner]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicpseudoJacobi|Definition:monicpseudoJacobi]] >>
+
+drmf_eof +drmf_bof +'''Definition:monicpseudoJacobi''' +
+
<< [[Definition:monicMeixnerPollaczek|Definition:monicMeixnerPollaczek]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicqBesselPoly|Definition:monicqBesselPoly]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\monicpseudoJacobi''' represents the monic pseudo Jacobi polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\monicpseudoJacobi{n}''' produces {\displaystyle \monicpseudoJacobi{n}}
+:'''\monicpseudoJacobi{n}@{x}{\nu}{N}''' produces {\displaystyle \monicpseudoJacobi{n}@{x}{\nu}{N}}
+:'''\monicpseudoJacobi{n}@@{x}{\nu}{N}''' produces {\displaystyle \monicpseudoJacobi{n}@@{x}{\nu}{N}}
+ +These are defined by + +{\displaystyle +\pseudoJacobi{n}@{x}{\nu}{N}=:\monicpseudoJacobi{n}@@{x}{\nu}{N}. +} +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:monicpseudoJacobi {\displaystyle {\widehat P}_{n}}] : monic pseudo-Jacobi polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicpseudoJacobi http://drmf.wmflabs.org/wiki/Definition:monicpseudoJacobi]
+[http://drmf.wmflabs.org/wiki/Definition:pseudoJacobi {\displaystyle P_{n}}] : pseudo Jacobi polynomial : [http://drmf.wmflabs.org/wiki/Definition:pseudoJacobi http://drmf.wmflabs.org/wiki/Definition:pseudoJacobi] +
+
<< [[Definition:monicMeixnerPollaczek|Definition:monicMeixnerPollaczek]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicqBesselPoly|Definition:monicqBesselPoly]] >>
+
+drmf_eof +drmf_bof +'''Definition:monicqBesselPoly''' +
+
<< [[Definition:monicpseudoJacobi|Definition:monicpseudoJacobi]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicqCharlier|Definition:monicqCharlier]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\monicqBesselPoly''' represents the monic q Bessel polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\monicqBesselPoly{n}''' produces {\displaystyle \monicqBesselPoly{n}}
+:'''\monicqBesselPoly{n}@{x}{a}{q}''' produces {\displaystyle \monicqBesselPoly{n}@{x}{a}{q}}
+:'''\monicqBesselPoly{n}@@{x}{a}{q}''' produces {\displaystyle \monicqBesselPoly{n}@@{x}{a}{q}}
+ +These are defined by + +{\displaystyle +\qBesselPoly{n}@{x}{a}{q}=:(-1)^nq^{-\binomial{n}{2}}\qPochhammer{-aq^n}{q}{n}\monicqBesselPoly{n}@@{x}{a}{q}. +} +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:monicqBessel {\displaystyle {\widehat y}_{n}}] : monic {\displaystyle q}-Bessel polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicqBessel http://drmf.wmflabs.org/wiki/Definition:monicqBessel]
+[http://drmf.wmflabs.org/wiki/Definition:qBessel {\displaystyle y_{n}}] : {\displaystyle q}-Bessel polynomial : [http://drmf.wmflabs.org/wiki/Definition:qBessel http://drmf.wmflabs.org/wiki/Definition:qBessel]
+[http://dlmf.nist.gov/1.2#E1 {\displaystyle \binom{n}{k}}] : binomial coefficient : [http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/1.2#E1] [http://dlmf.nist.gov/26.3#SS1.p1 http://dlmf.nist.gov/26.3#SS1.p1]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1] +
+
<< [[Definition:monicpseudoJacobi|Definition:monicpseudoJacobi]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicqCharlier|Definition:monicqCharlier]] >>
+
+drmf_eof +drmf_bof +'''Definition:monicqCharlier''' +
+
<< [[Definition:monicqBesselPoly|Definition:monicqBesselPoly]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicqKrawtchouk|Definition:monicqKrawtchouk]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\monicqCharlier''' represents the monic q Charlier polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\monicqCharlier{n}''' produces {\displaystyle \monicqCharlier{n}}
+:'''\monicqCharlier{n}@{q^{-x}}{a}{q}''' produces {\displaystyle \monicqCharlier{n}@{q^{-x}}{a}{q}}
+:'''\monicqCharlier{n}@@{q^{-x}}{a}{q}''' produces {\displaystyle \monicqCharlier{n}@@{q^{-x}}{a}{q}}
+ +These are defined by + +{\displaystyle +\qHahn{n}@{q^{-x}}{\alpha}{\beta}{N}{q}=: +\frac{\qPochhammer{\alpha\beta q^{n+1}}{q}{n}}{\qPochhammer{\alpha q,q^{-N}}{q}{n}}\monicqHahn{n}@@{q^{-x}}{\alpha}{\beta}{N}. +} +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:monicqCharlier {\displaystyle {\widehat C}_{n}}] : monic {\displaystyle q}-Charlier polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicqCharlier http://drmf.wmflabs.org/wiki/Definition:monicqCharlier]
+[http://drmf.wmflabs.org/wiki/Definition:qHahn {\displaystyle Q_{n}}] : {\displaystyle q}-Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:qHahn http://drmf.wmflabs.org/wiki/Definition:qHahn]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1]
+[http://drmf.wmflabs.org/wiki/Definition:monicqHahn {\displaystyle {\widehat Q}_{n}}] : monic {\displaystyle q}-Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicqHahn http://drmf.wmflabs.org/wiki/Definition:monicqHahn] +
+
<< [[Definition:monicqBesselPoly|Definition:monicqBesselPoly]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicqKrawtchouk|Definition:monicqKrawtchouk]] >>
+
+drmf_eof +drmf_bof +'''Definition:monicqKrawtchouk''' +
+
<< [[Definition:monicqCharlier|Definition:monicqCharlier]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicqLaguerre|Definition:monicqLaguerre]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\monicqKrawtchouk''' represents the monic q Krawtchouk polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\monicqKrawtchouk{n}''' produces {\displaystyle \monicqKrawtchouk{n}}
+:'''\monicqKrawtchouk{n}@{q^{-x}}{p}{N}{q}''' produces {\displaystyle \monicqKrawtchouk{n}@{q^{-x}}{p}{N}{q}}
+:'''\monicqKrawtchouk{n}@@{q^{-x}}{p}{N}{q}''' produces {\displaystyle \monicqKrawtchouk{n}@@{q^{-x}}{p}{N}{q}}
+ +These are defined by + +{\displaystyle +\qKrawtchouk{n}@{q^{-x}}{p}{N}{q}=:\frac{\qPochhammer{-pq^n}{q}{n}}{\qPochhammer{q^{-N}}{q}{n}}\monicqKrawtchouk{n}@@{q^{-x}}{p}{N}{q}. +} +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:monicqKrawtchouk {\displaystyle {\widehat K}_{n}}] : monic {\displaystyle q}-Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicqKrawtchouk http://drmf.wmflabs.org/wiki/Definition:monicqKrawtchouk]
+[http://drmf.wmflabs.org/wiki/Definition:qKrawtchouk {\displaystyle K_{n}}] : {\displaystyle q}-Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:qKrawtchouk http://drmf.wmflabs.org/wiki/Definition:qKrawtchouk]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1] +
+
<< [[Definition:monicqCharlier|Definition:monicqCharlier]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicqLaguerre|Definition:monicqLaguerre]] >>
+
+drmf_eof +drmf_bof +'''Definition:monicqLaguerre''' +
+
<< [[Definition:monicqKrawtchouk|Definition:monicqKrawtchouk]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicqMeixner|Definition:monicqMeixner]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\monicqLaguerre''' represents the monic q Laguerre polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\monicqLaguerre{n}''' produces {\displaystyle \monicqLaguerre{n}}
+:'''\monicqLaguerre{n}@{x}{q}''' produces {\displaystyle \monicqLaguerre{n}@{x}{q}}
+:'''\monicqLaguerre{n}@@{x}{q}''' produces {\displaystyle \monicqLaguerre{n}@@{x}{q}}
+ +These are defined by + +{\displaystyle +\qLaguerre[\alpha]{n}@{x}{q}=:\frac{(-1)^nq^{n(n+\alpha)}}{\qPochhammer{q}{q}{n}}\monicqLaguerre[\alpha]{n}@@{x}{q}. +} +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:monicqLaguerre {\displaystyle {\widehat L}^{(\alpha)}_{n}}] : monic {\displaystyle q}-Laguerre polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicqLaguerre http://drmf.wmflabs.org/wiki/Definition:monicqLaguerre]
+[http://drmf.wmflabs.org/wiki/Definition:qLaguerre {\displaystyle L_n^{(\alpha)}}] : {\displaystyle q}-Laguerre polynomial : [http://drmf.wmflabs.org/wiki/Definition:qLaguerre http://drmf.wmflabs.org/wiki/Definition:qLaguerre]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1] +
+
<< [[Definition:monicqKrawtchouk|Definition:monicqKrawtchouk]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicqMeixner|Definition:monicqMeixner]] >>
+
+drmf_eof +drmf_bof +'''Definition:monicqMeixner''' +
+
<< [[Definition:monicqLaguerre|Definition:monicqLaguerre]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicqMeixnerPollaczek|Definition:monicqMeixnerPollaczek]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\monicqMeixner''' represents the monic q Meixner polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\monicqMeixner{n}''' produces {\displaystyle \monicqMeixner{n}}
+:'''\monicqMeixner{n}@{q^{-x}}{b}{c}{q}''' produces {\displaystyle \monicqMeixner{n}@{q^{-x}}{b}{c}{q}}
+:'''\monicqMeixner{n}@@{q^{-x}}{b}{c}{q}''' produces {\displaystyle \monicqMeixner{n}@@{q^{-x}}{b}{c}{q}}
+ +These are defined by +{\displaystyle +\qMeixner{n}@{q^{-x}}{b}{c}{q}=:\frac{(-1)^nq^{n^2}}{\qPochhammer{bq}{q}{n}c^n}\monicqMeixner{n}@@{q^{-x}}{b}{c}{q}. +} +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:monicqMeixner {\displaystyle {\widehat M}_{n}}] : monic {\displaystyle q}-Meixner polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicqMeixner http://drmf.wmflabs.org/wiki/Definition:monicqMeixner]
+[http://drmf.wmflabs.org/wiki/Definition:qMeixner {\displaystyle M_{n}}] : {\displaystyle q}-Meixner polynomial : [http://drmf.wmflabs.org/wiki/Definition:qMeixner http://drmf.wmflabs.org/wiki/Definition:qMeixner]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1] +
+
<< [[Definition:monicqLaguerre|Definition:monicqLaguerre]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicqMeixnerPollaczek|Definition:monicqMeixnerPollaczek]] >>
+
+drmf_eof +drmf_bof +'''Definition:monicqMeixnerPollaczek''' +
+
<< [[Definition:monicqMeixner|Definition:monicqMeixner]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicqRacah|Definition:monicqRacah]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\monicqMeixnerPollaczek''' represents the monic q Meixner Pollaczek polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\monicqMeixnerPollaczek{n}''' produces {\displaystyle \monicqMeixnerPollaczek{n}}
+:'''\monicqMeixnerPollaczek{n}@{x}{a}{q}''' produces {\displaystyle \monicqMeixnerPollaczek{n}@{x}{a}{q}}
+:'''\monicqMeixnerPollaczek{n}@@{x}{a}{q}''' produces {\displaystyle \monicqMeixnerPollaczek{n}@@{x}{a}{q}}
+ +These are defined by + +{\displaystyle +\qMeixnerPollaczek{n}@{x}{a}{q}=:\frac{2^n}{\qPochhammer{q}{q}{n}}\monicqMeixnerPollaczek{n}@@{x}{a}{q}. +} +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:monicqMeixnerPollaczek {\displaystyle {\widehat P}_{n}}] : monic {\displaystyle q}-Meixner-Pollaczek polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicqMeixnerPollaczek http://drmf.wmflabs.org/wiki/Definition:monicqMeixnerPollaczek]
+[http://drmf.wmflabs.org/wiki/Definition:qMeixnerPollaczek {\displaystyle P_{n}}] : {\displaystyle q}-Meixner-Pollaczek polynomial : [http://drmf.wmflabs.org/wiki/Definition:qMeixnerPollaczek http://drmf.wmflabs.org/wiki/Definition:qMeixnerPollaczek]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1] +
+
<< [[Definition:monicqMeixner|Definition:monicqMeixner]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicqRacah|Definition:monicqRacah]] >>
+
+drmf_eof +drmf_bof +'''Definition:monicqRacah''' +
+
<< [[Definition:monicqMeixnerPollaczek|Definition:monicqMeixnerPollaczek]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicqtmqKrawtchouk|Definition:monicqtmqKrawtchouk]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\monicqRacah''' represents the monic q Racah polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\monicqRacah{n}''' produces {\displaystyle \monicqRacah{n}}
+:'''\monicqRacah{n}@{\mu(x)}{\alpha}{\beta}{\gamma}{\delta}{q}''' produces {\displaystyle \monicqRacah{n}@{\mu(x)}{\alpha}{\beta}{\gamma}{\delta}{q}}
+:'''\monicqRacah{n}@@{\mu(x)}{\alpha}{\beta}{\gamma}{\delta}{q}''' produces {\displaystyle \monicqRacah{n}@@{\mu(x)}{\alpha}{\beta}{\gamma}{\delta}{q}}
+ +These are defined by + +{\displaystyle +\qRacah{n}@{\mu(x)}{\alpha}{\beta}{\gamma}{\delta}{q}=: +\frac{\qPochhammer{\alpha\beta q^{n+1}}{q}{n}}{\qPochhammer{\alpha q,\beta\delta q,\gamma q}{q}{n}}\monicqRacah{n}@@{\mu(x)}{\alpha}{\beta}{\gamma}{\delta}{q}. +} +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:monicqRacah {\displaystyle {\widehat R}_{n}}] : monic {\displaystyle q}-Racah polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicqRacah http://drmf.wmflabs.org/wiki/Definition:monicqRacah]
+[http://dlmf.nist.gov/18.28#E19 {\displaystyle R_{n}}] : {\displaystyle q}-Racah polynomial : [http://dlmf.nist.gov/18.28#E19 http://dlmf.nist.gov/18.28#E19]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1] +
+
<< [[Definition:monicqMeixnerPollaczek|Definition:monicqMeixnerPollaczek]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicqtmqKrawtchouk|Definition:monicqtmqKrawtchouk]] >>
+
+drmf_eof +drmf_bof +'''Definition:monicqtmqKrawtchouk''' +
+
<< [[Definition:monicqRacah|Definition:monicqRacah]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicRacah|Definition:monicRacah]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\monicqtmqKrawtchouk''' represents the monic quantum q Krawtchouk polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\monicqtmqKrawtchouk{n}''' produces {\displaystyle \monicqtmqKrawtchouk{n}}
+:'''\monicqtmqKrawtchouk{n}@{q^{-x}}{p}{N}{q}''' produces {\displaystyle \monicqtmqKrawtchouk{n}@{q^{-x}}{p}{N}{q}}
+:'''\monicqtmqKrawtchouk{n}@@{q^{-x}}{p}{N}{q}''' produces {\displaystyle \monicqtmqKrawtchouk{n}@@{q^{-x}}{p}{N}{q}}
+ +These are defined by + +{\displaystyle +\qtmqKrawtchouk{n}@{q^{-x}}{p}{N}{q}=:\frac{p^nq^{n^2}}{\qPochhammer{q^{-N}}{q}{n}}\monicqtmqKrawtchouk{n}@@{q^{-x}}{p}{N}{q}. +} +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:monicqtmqKrawtchouk {\displaystyle {\widehat K^{\mathrm{qtm}}}_{n}}] : monic quantum {\displaystyle q}-Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicqtmqKrawtchouk http://drmf.wmflabs.org/wiki/Definition:monicqtmqKrawtchouk]
+[http://drmf.wmflabs.org/wiki/Definition:qtmqKrawtchouk {\displaystyle K^{\mathrm{qtm}}_{n}}] : quantum {\displaystyle q}-Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:qtmqKrawtchouk http://drmf.wmflabs.org/wiki/Definition:qtmqKrawtchouk]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1] +
+
<< [[Definition:monicqRacah|Definition:monicqRacah]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicRacah|Definition:monicRacah]] >>
+
+drmf_eof +drmf_bof +'''Definition:monicRacah''' +
+
<< [[Definition:monicqtmqKrawtchouk|Definition:monicqtmqKrawtchouk]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicStieltjesWigert|Definition:monicStieltjesWigert]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\monicRacah''' represents the monic Racah polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\monicRacah{n}''' produces {\displaystyle \monicRacah{n}}
+:'''\monicRacah{n}@{\lambda(x)}{\alpha}{\beta}{\gamma}{\delta}''' produces {\displaystyle \monicRacah{n}@{\lambda(x)}{\alpha}{\beta}{\gamma}{\delta}}
+:'''\monicRacah{n}@@{\lambda(x)}{\alpha}{\beta}{\gamma}{\delta}''' produces {\displaystyle \monicRacah{n}@@{\lambda(x)}{\alpha}{\beta}{\gamma}{\delta}}
+ +These are defined by + +{\displaystyle +\Racah{n}@{\lambda(x)}{\alpha}{\beta}{\gamma}{\delta}=: +\frac{\pochhammer{n+\alpha+\beta+1}{n}}{\pochhammer{\alpha+1}{n}\pochhammer{\beta+\delta+1}{n}\pochhammer{\gamma+1}{n}}\monicRacah{n}@@{\lambda(x)}{\alpha}{\beta}{\gamma}{\delta}. +} +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:monicRacah {\displaystyle {\widehat R}_{n}}] : monic Racah polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicRacah http://drmf.wmflabs.org/wiki/Definition:monicRacah]
+[http://dlmf.nist.gov/18.25#T1.t1.r4 {\displaystyle R_{n}}] : Racah polynomial : [http://dlmf.nist.gov/18.25#T1.t1.r4 http://dlmf.nist.gov/18.25#T1.t1.r4]
+[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii] +
+
<< [[Definition:monicqtmqKrawtchouk|Definition:monicqtmqKrawtchouk]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicStieltjesWigert|Definition:monicStieltjesWigert]] >>
+
+drmf_eof +drmf_bof +'''Definition:monicStieltjesWigert''' +
+
<< [[Definition:monicRacah|Definition:monicRacah]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicUltra|Definition:monicUltra]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\monicStieltjesWigert''' represents the monic Stieltjes Wigert polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\monicStieltjesWigert{n}''' produces {\displaystyle \monicStieltjesWigert{n}}
+:'''\monicStieltjesWigert{n}@{x}{q}''' produces {\displaystyle \monicStieltjesWigert{n}@{x}{q}}
+:'''\monicStieltjesWigert{n}@@{x}{q}''' produces {\displaystyle \monicStieltjesWigert{n}@@{x}{q}}
+ +These are defined by + +{\displaystyle +\StieltjesWigert{n}@{x}{q}=:\frac{(-1)^nq^{n^2}}{\qPochhammer{q}{q}{n}}\monicStieltjesWigert{n}@@{x}{q}. +} +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:monicStieltjesWigert {\displaystyle {\widehat S}_{n}}] : monic Stieltjes-Wigert polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicStieltjesWigert http://drmf.wmflabs.org/wiki/Definition:monicStieltjesWigert]
+[http://drmf.wmflabs.org/wiki/Definition:StieltjesWigert {\displaystyle S_{n}}] : Stieltjes-Wigert polynomial : [http://drmf.wmflabs.org/wiki/Definition:StieltjesWigert http://drmf.wmflabs.org/wiki/Definition:StieltjesWigert]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1] +
+
<< [[Definition:monicRacah|Definition:monicRacah]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicUltra|Definition:monicUltra]] >>
+
+drmf_eof +drmf_bof +'''Definition:monicUltra''' +
+
<< [[Definition:monicStieltjesWigert|Definition:monicStieltjesWigert]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicWilson|Definition:monicWilson]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\monicUltra''' represents the monic ultraspherical Gegenbauer polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\monicUltra{n}''' produces {\displaystyle \monicUltra{n}}
+:'''\monicUltra{n}@{x}''' produces {\displaystyle \monicUltra{n}@{x}}
+:'''\monicUltra{n}@@{x}''' produces {\displaystyle \monicUltra{n}@@{x}}
+ +These are defined by + +{\displaystyle +\Ultra{\lambda}{n}@{x}=:\frac{2^n\pochhammer{\lambda}{n}}{n!}\monicUltra{\lambda}{n}@@{x}. +} +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:monicUltra {\displaystyle {\widehat C}^{\mu}_{n}}] : monic ultraspherical/Gegenbauer polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicUltra http://drmf.wmflabs.org/wiki/Definition:monicUltra]
+[http://dlmf.nist.gov/18.3#T1.t1.r5 {\displaystyle C^{\mu}_{n}}] : ultraspherical/Gegenbauer polynomial : [http://dlmf.nist.gov/18.3#T1.t1.r5 http://dlmf.nist.gov/18.3#T1.t1.r5]
+[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii] +
+
<< [[Definition:monicStieltjesWigert|Definition:monicStieltjesWigert]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicWilson|Definition:monicWilson]] >>
+
+drmf_eof +drmf_bof +'''Definition:monicWilson''' +
+
<< [[Definition:monicUltra|Definition:monicUltra]]
+
[[Main_Page|Main Page]]
+
[[Definition:NeumannFactor|Definition:NeumannFactor]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\monicWilson''' represents the monic Wilson polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\monicWilson{n}''' produces {\displaystyle \monicWilson{n}}
+:'''\monicWilson{n}@{x^2}{a}{b}{c}{d}''' produces {\displaystyle \monicWilson{n}@{x^2}{a}{b}{c}{d}}
+:'''\monicWilson{n}@@{x^2}{a}{b}{c}{d}''' produces {\displaystyle \monicWilson{n}@@{x^2}{a}{b}{c}{d}}
+ +These are defined by + +{\displaystyle +\Wilson{n}@{x^2}{a}{b}{c}{d}=:(-1)^n\pochhammer{n+a+b+c+d-1}{n}\monicWilson{n}@@{x^2}{a}{b}{c}{d}. +} +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:monicWilson {\displaystyle {\widehat W}_{n}}] : monic Wilson polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicWilson http://drmf.wmflabs.org/wiki/Definition:monicWilson]
+[http://dlmf.nist.gov/18.25#T1.t1.r2 {\displaystyle W_{n}}] : Wilson polynomial : [http://dlmf.nist.gov/18.25#T1.t1.r2 http://dlmf.nist.gov/18.25#T1.t1.r2]
+[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii] +
+
<< [[Definition:monicUltra|Definition:monicUltra]]
+
[[Main_Page|Main Page]]
+
[[Definition:NeumannFactor|Definition:NeumannFactor]] >>
+
+drmf_eof +drmf_bof +'''Definition:NeumannFactor''' +
+
<< [[Definition:monicWilson|Definition:monicWilson]]
+
[[Main_Page|Main Page]]
+
[[Definition:normctsdualHahnStilde|Definition:normctsdualHahnStilde]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\NeumannFactor''' represents the Neumann factor. + +This macro is in the category of integer valued functions. + +In math mode, this macro can be called in the following way: + +:'''\NeumannFactor{n}''' produces {\displaystyle \NeumannFactor{n}}
+ +These are defined by + +\NeumannFactor{n}:=2-\Kronecker{n}{0}. + + +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:NeumannFactor {\displaystyle \epsilon_{m}}] : Neumann factor : [http://drmf.wmflabs.org/wiki/Definition:NeumannFactor http://drmf.wmflabs.org/wiki/Definition:NeumannFactor]
+[http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r4 {\displaystyle \delta_{m,n}}] : Kronecker delta : [http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r4 http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r4] +
+
<< [[Definition:monicWilson|Definition:monicWilson]]
+
[[Main_Page|Main Page]]
+
[[Definition:normctsdualHahnStilde|Definition:normctsdualHahnStilde]] >>
+
+drmf_eof +drmf_bof +'''Definition:normctsdualHahnStilde''' +
+
<< [[Definition:NeumannFactor|Definition:NeumannFactor]]
+
[[Main_Page|Main Page]]
+
[[Definition:normctsHahnptilde|Definition:normctsHahnptilde]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\normctsdualHahnStilde''' represents the normalized continuous dual Hahn polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\normctsdualHahnStilde{n}@{x^2}{a}{b}{c}{d}''' produces {\displaystyle \normctsdualHahnStilde{n}@{x^2}{a}{b}{c}{d}}
+:'''\normctsdualHahnStilde{n}@@{x^2}{a}{b}{c}{d}''' produces {\displaystyle \normctsdualHahnStilde{n}@@{x^2}{a}{b}{c}{d}}
+ +These are defined by + +\normctsdualHahnStilde{n}@@{x^2}{a}{b}{c}:=\normctsdualHahnStilde{n}@{x^2}{a}{b}{c}=\frac{\ctsdualHahn{n}@{x^2}{a}{b}{c}}{\pochhammer{a+b}{n}\pochhammer{a+c}{n}}. + + +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:normctsdualHahnStilde {\displaystyle {\tilde S}_{n}}] : normalized continuous dual Hahn polynomial {\displaystyle {\tilde S}} : [http://drmf.wmflabs.org/wiki/Definition:normctsdualHahnStilde http://drmf.wmflabs.org/wiki/Definition:normctsdualHahnStilde]
+[http://dlmf.nist.gov/18.25#T1.t1.r3 {\displaystyle S_{n}}] : continuous dual Hahn polynomial : [http://dlmf.nist.gov/18.25#T1.t1.r3 http://dlmf.nist.gov/18.25#T1.t1.r3]
+[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii] +
+
<< [[Definition:NeumannFactor|Definition:NeumannFactor]]
+
[[Main_Page|Main Page]]
+
[[Definition:normctsHahnptilde|Definition:normctsHahnptilde]] >>
+
+drmf_eof +drmf_bof +'''Definition:normctsHahnptilde''' +
+
<< [[Definition:normctsdualHahnStilde|Definition:normctsdualHahnStilde]]
+
[[Main_Page|Main Page]]
+
[[Definition:normJacobiR|Definition:normJacobiR]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\normctsHahnptilde''' represents the normalized continuous Hahn polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\normctsHahnptilde{n}''' produces {\displaystyle \normctsHahnptilde{n}}
+:'''\normctsHahnptilde{n}@{x}{a}{b}{c}{d}''' produces {\displaystyle \normctsHahnptilde{n}@{x}{a}{b}{c}{d}}
+:'''\normctsHahnptilde{n}@@{x}{a}{b}{c}{d}''' produces {\displaystyle \normctsHahnptilde{n}@@{x}{a}{b}{c}{d}}
+ +These are defined by + +\normctsHahnptilde{n}@@{x}{a}{b}{c}{d}:=\normctsHahnptilde{n}@{x}{a}{b}{c}{d}=\frac{n!}{i^n\pochhammer{a+c}{n}\pochhammer{a+d}{n}}\ctsHahn{n}@{x}{a}{b}{c}{d}. + + +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:normctsHahnptilde {\displaystyle {\tilde p}_{n}}] : normalized continuous Hahn polynomial {\displaystyle {\tilde p}} : [http://drmf.wmflabs.org/wiki/Definition:normctsHahnptilde http://drmf.wmflabs.org/wiki/Definition:normctsHahnptilde]
+[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii]
+[http://dlmf.nist.gov/18.19#P2.p1 {\displaystyle p_{n}}] : continuous Hahn polynomial : [http://dlmf.nist.gov/18.19#P2.p1 http://dlmf.nist.gov/18.19#P2.p1] +
+
<< [[Definition:normctsdualHahnStilde|Definition:normctsdualHahnStilde]]
+
[[Main_Page|Main Page]]
+
[[Definition:normJacobiR|Definition:normJacobiR]] >>
+
+drmf_eof +drmf_bof +'''Definition:normJacobiR''' +
+
<< [[Definition:normctsHahnptilde|Definition:normctsHahnptilde]]
+
[[Main_Page|Main Page]]
+
[[Definition:normWilsonWtilde|Definition:normWilsonWtilde]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\normJacobiR''' represents the normalized Jacobi polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\normJacobiR{\alpha}{\beta}{n}''' produces {\displaystyle \normJacobiR{\alpha}{\beta}{n}}
+:'''\normJacobiR{\alpha}{\beta}{n}@{x}''' produces {\displaystyle \normJacobiR{\alpha}{\beta}{n}@{x}}
+ +These are defined by + +{\displaystyle \normJacobiR{\alpha}{\beta}{n}@{x}:= +\frac{\Jacobi{\alpha}{\beta}{n}@{x}}{ +\Jacobi{\alpha}{\beta}{n}@{1}},} + +where + +{\displaystyle \Jacobi{\alpha}{\beta}{n}@1=\frac{\pochhammer{\alpha+1}{n}}{n!}.} + + +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:normJacobiR {\displaystyle R^{(\alpha,\beta)}_{n}}] : normalized Jacobi polynomial : [http://drmf.wmflabs.org/wiki/Definition:normJacobiR http://drmf.wmflabs.org/wiki/Definition:normJacobiR]
+[http://dlmf.nist.gov/18.3#T1.t1.r3 {\displaystyle P^{(\alpha,\beta)}_{n}}] : Jacobi polynomial : [http://dlmf.nist.gov/18.3#T1.t1.r3 http://dlmf.nist.gov/18.3#T1.t1.r3]
+[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii] +
+
<< [[Definition:normctsHahnptilde|Definition:normctsHahnptilde]]
+
[[Main_Page|Main Page]]
+
[[Definition:normWilsonWtilde|Definition:normWilsonWtilde]] >>
+
+drmf_eof +drmf_bof +'''Definition:normWilsonWtilde''' +
+
<< [[Definition:normJacobiR|Definition:normJacobiR]]
+
[[Main_Page|Main Page]]
+
[[Definition:poly|Definition:poly]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\normWilsonWtilde''' represents the normalized Wilson polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\normWilsonWtilde{n}''' produces {\displaystyle \normWilsonWtilde{n}}
+:'''\normWilsonWtilde{n}@{x^2}{a}{b}{c}{d}''' produces {\displaystyle \normWilsonWtilde{n}@{x^2}{a}{b}{c}{d}}
+:'''\normWilsonWtilde{n}@@{x^2}{a}{b}{c}{d}''' produces {\displaystyle \normWilsonWtilde{n}@@{x^2}{a}{b}{c}{d}}
+ +These are defined by + + +{\displaystyle +\normWilsonWtilde{n}@{x^2}{a}{b}{c}{d}:=\frac{\Wilson{n}@{x^2}{a}{b}{c}{d}}{\pochhammer{a+b}{n}\pochhammer{a+c}{n}\pochhammer{a+d}{n}} +} +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:normWilsonWtilde {\displaystyle {\tilde W}_{n}}] : normalized Wilson polynomial {\displaystyle {\tilde W}} : [http://drmf.wmflabs.org/wiki/Definition:normWilsonWtilde http://drmf.wmflabs.org/wiki/Definition:normWilsonWtilde]
+[http://dlmf.nist.gov/18.25#T1.t1.r2 {\displaystyle W_{n}}] : Wilson polynomial : [http://dlmf.nist.gov/18.25#T1.t1.r2 http://dlmf.nist.gov/18.25#T1.t1.r2]
+[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii] +
+
<< [[Definition:normJacobiR|Definition:normJacobiR]]
+
[[Main_Page|Main Page]]
+
[[Definition:poly|Definition:poly]] >>
+
+drmf_eof +drmf_bof +'''Definition:poly''' +
+
<< [[Definition:normWilsonWtilde|Definition:normWilsonWtilde]]
+
[[Main_Page|Main Page]]
+
[[Definition:qBesselPoly|Definition:qBesselPoly]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\poly''' represents the semantic polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\poly{p}{n}''' produces {\displaystyle \poly{p}{n}}
+:'''\poly{p}{n}@{x}''' produces {\displaystyle \poly{p}{n}@{x}}
+ +These are defined by +{\displaystyle +\pseudoJacobi{n}@{x}{\nu}{N}:=\frac{(-2i)^n\pochhammer{-N+i\nu}{n}}{\pochhammer{n-2N-1}{n}}\,\HyperpFq{2}{1}@@{-n,n-2N-1}{-N+i\nu}{\frac{1-ix}{2}} + + +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:poly {\displaystyle {p}_{n}}] : polynomial : [http://drmf.wmflabs.org/wiki/Definition:poly http://drmf.wmflabs.org/wiki/Definition:poly]
+[http://drmf.wmflabs.org/wiki/Definition:pseudoJacobi {\displaystyle P_{n}}] : pseudo Jacobi polynomial : [http://drmf.wmflabs.org/wiki/Definition:pseudoJacobi http://drmf.wmflabs.org/wiki/Definition:pseudoJacobi]
+[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii]
+[http://dlmf.nist.gov/16.2#E1 {\displaystyle {{}_{p}F_{q}}}] : generalized hypergeometric function : [http://dlmf.nist.gov/16.2#E1 http://dlmf.nist.gov/16.2#E1] +
+
<< [[Definition:normWilsonWtilde|Definition:normWilsonWtilde]]
+
[[Main_Page|Main Page]]
+
[[Definition:qBesselPoly|Definition:qBesselPoly]] >>
+
+drmf_eof +drmf_bof +'''Definition:qBesselPoly''' +
+
<< [[Definition:poly|Definition:poly]]
+
[[Main_Page|Main Page]]
+
[[Definition:qCharlier|Definition:qCharlier]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\qBesselPoly''' represents the q-Bessel polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\qBesselPoly{n}''' produces {\displaystyle \qBesselPoly{n}}
+:'''\qBesselPoly{n}@{x}{b}{q}''' produces {\displaystyle \qBesselPoly{n}@{x}{b}{q}}
+ +These are defined by +{\displaystyle +\qBesselPoly{n}@{x}{a}{q}:=\qHyperrphis{2}{1}@@{q^{-n},-aq^n}{0}{q}{qx} + +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:qBessel {\displaystyle y_{n}}] : {\displaystyle q}-Bessel polynomial : [http://drmf.wmflabs.org/wiki/Definition:qBessel http://drmf.wmflabs.org/wiki/Definition:qBessel]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1] +
+
<< [[Definition:poly|Definition:poly]]
+
[[Main_Page|Main Page]]
+
[[Definition:qCharlier|Definition:qCharlier]] >>
+
+drmf_eof +drmf_bof +'''Definition:qCharlier''' +
+
<< [[Definition:qBesselPoly|Definition:qBesselPoly]]
+
[[Main_Page|Main Page]]
+
[[Definition:qDigamma|Definition:qDigamma]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\qCharlier''' represents the q-Charlier polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\qCharlier{n}''' produces {\displaystyle \qCharlier{n}}
+:'''\qCharlier{n}@{x}{c}{q}''' produces {\displaystyle \qCharlier{n}@{x}{c}{q}}
+ +These are defined by +{\displaystyle +\qCharlier{n}@{q^{-x}}{a}{q}:=\qHyperrphis{2}{1}@@{q^{-n},q^{-x}}{0}{q}{-\frac{q^{n+1}}{a}} +} +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:qCharlier {\displaystyle C_{n}}] : {\displaystyle q}-Charlier polynomial : [http://drmf.wmflabs.org/wiki/Definition:qCharlier http://drmf.wmflabs.org/wiki/Definition:qCharlier]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1] +
+
<< [[Definition:qBesselPoly|Definition:qBesselPoly]]
+
[[Main_Page|Main Page]]
+
[[Definition:qDigamma|Definition:qDigamma]] >>
+
+drmf_eof +drmf_bof +'''Definition:qDigamma''' +
+
<< [[Definition:qCharlier|Definition:qCharlier]]
+
[[Main_Page|Main Page]]
+
[[Definition:qHyperrWs|Definition:qHyperrWs]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\qDigamma''' represents the q-gamma function. + +This macro is in the category of real or complex valued functions. + +In math mode, this macro can be called in the following ways: + +:'''\qDigamma{q}''' produces {\displaystyle \qDigamma{q}}
+:'''\qDigamma{q}@{z}''' produces {\displaystyle \qDigamma{q}@{z}}
+ +These are defined by +{\displaystyle +\qDigamma{q}@{z} = \qGamma{q}'@{z} / \qGamma{q}@{z} +} +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:qDigamma {\displaystyle \psi_{q}}] : {\displaystyle q}-digamma function : [http://drmf.wmflabs.org/wiki/Definition:qDigamma http://drmf.wmflabs.org/wiki/Definition:qDigamma]
+[http://dlmf.nist.gov/5.18#E4 {\displaystyle \Gamma_{q}}] : {\displaystyle q}-gamma function : [http://dlmf.nist.gov/5.18#E4 http://dlmf.nist.gov/5.18#E4] +
+
<< [[Definition:qCharlier|Definition:qCharlier]]
+
[[Main_Page|Main Page]]
+
[[Definition:qHyperrWs|Definition:qHyperrWs]] >>
+
+drmf_eof +drmf_bof +'''Definition:qHyperrWs''' +
+
<< [[Definition:qDigamma|Definition:qDigamma]]
+
[[Main_Page|Main Page]]
+
[[Definition:qExpKLS|Definition:qExpKLS]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\qHyperrWs''' represents the q-hypergeometric series. + +This macro is in the category of real or complex valued functions. + +In math mode, this macro can be called in the following ways: + +:'''\qHyperrWs{r}{r+1}''' produces {\displaystyle \qHyperrWs{r}{r+1}}
+:'''\qHyperrWs{r}{r+1}@{a_1}{a_4,a_5,...,a_{r+1}}{q}{z}''' produces {\displaystyle \qHyperrWs{r}{r+1}@{a_1}{a_4,a_5,...,a_{r+1}}{q}{z}}
+:'''\qHyperrWs{r}{r+1}@@{a_1}{a_4,a_5,...,a_{r+1}}{q}{z}''' produces {\displaystyle \qHyperrWs{r}{r+1}@@{a_1}{a_4,a_5,...,a_{r+1}}{q}{z}}
+:'''\qHyperrWs{r}{r+1}@@@{a_1}{a_4,a_5,...,a_{r+1}}{q}{z}''' produces {\displaystyle \qHyperrWs{r}{r+1}@@@{a_1}{a_4,a_5,...,a_{r+1}}{q}{z}}
+ +These are defined by + +\qHyperrWs{r+1}r@ +{a_1}{a_4,a_5,\ldots,a_{r+1}}qz:= +\qHyperrphis{r+1}r@@{a_1,qa_1^\frac12,-qa_1^\frac12,a_4,\ldots,a_{r+1}} +{a_1^\frac12,-a_1^\frac12,qa_1/a_4,\ldots,qa_1/a_{r+1}}qz. + +
+ +== Bibliography == + +Equation (2.1.11) of [[Bibliography#GaR|'''GaR''']]. + +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:qHyperrWs {\displaystyle {{}_{r+1}W_{r}}}] : very-well-poised basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://drmf.wmflabs.org/wiki/Definition:qHyperrWs http://drmf.wmflabs.org/wiki/Definition:qHyperrWs]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1] +
+
<< [[Definition:qDigamma|Definition:qDigamma]]
+
[[Main_Page|Main Page]]
+
[[Definition:qExpKLS|Definition:qExpKLS]] >>
+
+drmf_eof +drmf_bof +'''Definition:qExpKLS''' +
+
<< [[Definition:qHyperrWs|Definition:qHyperrWs]]
+
[[Main_Page|Main Page]]
+
[[Definition:qexpKLS|Definition:qexpKLS]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\qExpKLS''' represents a q-analogue of the +\exp function: \qExpKLS{q}. + +This macro is in the category of real or complex valued functions. + +In math mode, this macro can be called in the following ways: + +:'''\qExpKLS{q}''' produces {\displaystyle \qExpKLS{q}}
+:'''\qExpKLS{q}@{z}''' produces {\displaystyle \qExpKLS{q}@{z}}
+:'''\qExpKLS{q}@@{z}''' produces {\displaystyle \qExpKLS{q}@@{z}}
+ +These are defined by +{\displaystyle +\qExpKLS{q}@{z}:=\qHyperrphis{0}{0}@@{-}{-}{q}{-z}:= +\sum_{n=0}^{\infty}\frac{q^{\binom{n}{2}}}{\qPochhammer{q}{q}{n}}z^n=\qPochhammer{-z}{q}{\infty},\quad 0<|q|<1. +} + +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:qExpKLS {\displaystyle \mathrm{E}_{q}}] : {\displaystyle q}-analogue of the {\displaystyle \exp} function used in KLS: {\displaystyle \mathrm{E}_{q}} : [http://drmf.wmflabs.org/wiki/Definition:qExpKLS http://drmf.wmflabs.org/wiki/Definition:qExpKLS]
+[http://dlmf.nist.gov/4.2#E19 {\displaystyle \mathrm{exp}}] : exponential function : [http://dlmf.nist.gov/4.2#E19 http://dlmf.nist.gov/4.2#E19]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
+[http://drmf.wmflabs.org/wiki/Definition:sum {\displaystyle \Sigma}] : sum : [http://drmf.wmflabs.org/wiki/Definition:sum http://drmf.wmflabs.org/wiki/Definition:sum]
+[http://dlmf.nist.gov/1.2#E1 {\displaystyle \binom{n}{k}}] : binomial coefficient : [http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/1.2#E1] [http://dlmf.nist.gov/26.3#SS1.p1 http://dlmf.nist.gov/26.3#SS1.p1]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1] +
+
<< [[Definition:qHyperrWs|Definition:qHyperrWs]]
+
[[Main_Page|Main Page]]
+
[[Definition:qexpKLS|Definition:qexpKLS]] >>
+
+drmf_eof +drmf_bof +'''Definition:qexpKLS''' +
+
<< [[Definition:qExpKLS|Definition:qExpKLS]]
+
[[Main_Page|Main Page]]
+
[[Definition:qcosKLS|Definition:qcosKLS]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\qexpKLS''' represents a q-analogue of the +\exp function: \qexpKLS{q}. + +This macro is in the category of real or complex valued functions. + +In math mode, this macro can be called in the following ways: + +:'''\qexpKLS{q}''' produces {\displaystyle \qexpKLS{q}}
+:'''\qexpKLS{q}@{z}''' produces {\displaystyle \qexpKLS{q}@{z}}
+:'''\qexpKLS{q}@@{z}''' produces {\displaystyle \qexpKLS{q}@@{z}}
+ +These are defined by +{\displaystyle +\qexpKLS{q}@{z}:=\qHyperrphis{1}{0}@@{0}{-}{q}{z}:=\sum_{n=0}^{\infty}\frac{z^n}{\qPochhammer{q}{q}{n}} +=\frac{1}{\qPochhammer{z}{q}{\infty}},\quad 0<|q|<1 +} + +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:qexpKLS {\displaystyle \mathrm{e}_{q}}] : {\displaystyle q}-analogue of the {\displaystyle \exp} function used in KLS: {\displaystyle \mathrm{e}_{q}} : [http://drmf.wmflabs.org/wiki/Definition:qexpKLS http://drmf.wmflabs.org/wiki/Definition:qexpKLS]
+[http://dlmf.nist.gov/4.2#E19 {\displaystyle \mathrm{exp}}] : exponential function : [http://dlmf.nist.gov/4.2#E19 http://dlmf.nist.gov/4.2#E19]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
+[http://drmf.wmflabs.org/wiki/Definition:sum {\displaystyle \Sigma}] : sum : [http://drmf.wmflabs.org/wiki/Definition:sum http://drmf.wmflabs.org/wiki/Definition:sum]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1] +
+
<< [[Definition:qExpKLS|Definition:qExpKLS]]
+
[[Main_Page|Main Page]]
+
[[Definition:qcosKLS|Definition:qcosKLS]] >>
+
+drmf_eof +drmf_bof +'''Definition:qcosKLS''' +
+
<< [[Definition:qexpKLS|Definition:qexpKLS]]
+
[[Main_Page|Main Page]]
+
[[Definition:qMeixner|Definition:qMeixner]] >>
+
+The LaTeX DLMF and DRMF macro '''\qcosKLS''' represents a q-analogue of the +\cos function:~\qcosKLS{q}. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\qcosKLS{q}''' produces {\displaystyle \qcosKLS{q}}
+:'''\qcosKLS{q}@{z}''' produces {\displaystyle \qcosKLS{q}@{z}}
+:'''\qcosKLS{q}@@{z}''' produces {\displaystyle \qcosKLS{q}@@{z}}
+ +These are defined by +{\displaystyle +\qcosKLS{q}@{z}:=\frac{\qexpKLS{q}@{iz)+{\mathrm e}_q(-iz}}{2}= +\sum_{n=0}^{\infty}\frac{(-1)^nz^{2n}}{\qPochhammer{q}{q}{2n}} +} + +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:qcosKLS {\displaystyle \mathrm{cos}_{q}}] : {\displaystyle q}-analogue of the {\displaystyle \cos} function used in KLS: {\displaystyle \cos_q} : [http://drmf.wmflabs.org/wiki/Definition:qcosKLS http://drmf.wmflabs.org/wiki/Definition:qcosKLS]
+[http://dlmf.nist.gov/4.14#E2 {\displaystyle \mathrm{cos}}] : cosine function : [http://dlmf.nist.gov/4.14#E2 http://dlmf.nist.gov/4.14#E2]
+[http://drmf.wmflabs.org/wiki/Definition:qexpKLS {\displaystyle \mathrm{e}_{q}}] : {\displaystyle q}-analogue of the {\displaystyle \exp} function used in KLS: {\displaystyle \mathrm{e}_{q}} : [http://drmf.wmflabs.org/wiki/Definition:qexpKLS http://drmf.wmflabs.org/wiki/Definition:qexpKLS]
+[http://drmf.wmflabs.org/wiki/Definition:sum {\displaystyle \Sigma}] : sum : [http://drmf.wmflabs.org/wiki/Definition:sum http://drmf.wmflabs.org/wiki/Definition:sum]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1] +
+
<< [[Definition:qexpKLS|Definition:qexpKLS]]
+
[[Main_Page|Main Page]]
+
[[Definition:qMeixner|Definition:qMeixner]] >>
+
+drmf_eof +drmf_bof +'''Definition:qMeixner''' +
+
<< [[Definition:qcosKLS|Definition:qcosKLS]]
+
[[Main_Page|Main Page]]
+
[[Definition:qKrawtchouk|Definition:qKrawtchouk]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\qMeixner''' represents the q-Meixner polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\qMeixner{n}''' produces {\displaystyle \qMeixner{n}}
+:'''\qMeixner{n}@{x}{b}{c}{q}''' produces {\displaystyle \qMeixner{n}@{x}{b}{c}{q}}
+ +These are defined by + +\qMeixner{n}@{q^{-x}}{b}{c}{q}:=\qHyperrphis{2}{1}@@{q^{-n},q^{-x}}{bq}{q}{-\frac{q^{n+1}}{c}}. + + +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:qMeixner {\displaystyle M_{n}}] : {\displaystyle q}-Meixner polynomial : [http://drmf.wmflabs.org/wiki/Definition:qMeixner http://drmf.wmflabs.org/wiki/Definition:qMeixner]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1] +
+
<< [[Definition:qcosKLS|Definition:qcosKLS]]
+
[[Main_Page|Main Page]]
+
[[Definition:qKrawtchouk|Definition:qKrawtchouk]] >>
+
+drmf_eof +drmf_bof +'''Definition:qKrawtchouk''' +
+
<< [[Definition:qMeixner|Definition:qMeixner]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicqHahn|Definition:monicqHahn]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\qKrawtchouk''' represents the q-Krawtchouk polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\qKrawtchouk{n}''' produces {\displaystyle \qKrawtchouk{n}}
+:'''\qKrawtchouk{n}@{q^{-x}}{p}{N}{q}''' produces {\displaystyle \qKrawtchouk{n}@{q^{-x}}{p}{N}{q}}
+:'''\qKrawtchouk{n}@@{q^{-x}}{p}{N}{q}''' produces {\displaystyle \qKrawtchouk{n}@@{q^{-x}}{p}{N}{q}}
+ +These are defined by + +\qKrawtchouk{n}@{q^{-x}}{p}{N}{q}:=\qHyperrphis{3}{2}@@{q^{-n},q^{-x},-pq^n}{q^{-N},0}{q}{q}. +\end{equation} + + +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:qKrawtchouk {\displaystyle K_{n}}] : {\displaystyle q}-Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:qKrawtchouk http://drmf.wmflabs.org/wiki/Definition:qKrawtchouk]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1] +
+
<< [[Definition:qMeixner|Definition:qMeixner]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicqHahn|Definition:monicqHahn]] >>
+
+drmf_eof +drmf_bof +'''Definition:monicqHahn''' +
+
<< [[Definition:qKrawtchouk|Definition:qKrawtchouk]]
+
[[Main_Page|Main Page]]
+
[[Definition:dualqHahn|Definition:dualqHahn]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\monicqHahn''' represents the monic q Hahn polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\monicqHahn{n}''' produces {\displaystyle \monicqHahn{n}}
+:'''\monicqHahn{n}@{q^{-x}}{\alpha}{\beta}{N}''' produces {\displaystyle \monicqHahn{n}@{q^{-x}}{\alpha}{\beta}{N}}
+:'''\monicqHahn{n}@@{q^{-x}}{\alpha}{\beta}{N}''' produces {\displaystyle \monicqHahn{n}@@{q^{-x}}{\alpha}{\beta}{N}}
+ +These are defined by + +{\displaystyle +\qHahn{n}@{q^{-x}}{\alpha}{\beta}{N}{q}=: +\frac{\qPochhammer{\alpha\beta q^{n+1}}{q}{n}}{\qPochhammer{\alpha q,q^{-N}}{q}{n}}\monicqHahn{n}@@{q^{-x}}{\alpha}{\beta}{N}. +} + +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:monicqHahn {\displaystyle {\widehat Q}_{n}}] : monic {\displaystyle q}-Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicqHahn http://drmf.wmflabs.org/wiki/Definition:monicqHahn]
+[http://drmf.wmflabs.org/wiki/Definition:qHahn {\displaystyle Q_{n}}] : {\displaystyle q}-Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:qHahn http://drmf.wmflabs.org/wiki/Definition:qHahn]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1] +
+
<< [[Definition:qKrawtchouk|Definition:qKrawtchouk]]
+
[[Main_Page|Main Page]]
+
[[Definition:dualqHahn|Definition:dualqHahn]] >>
+
+drmf_eof +drmf_bof +'''Definition:dualqHahn''' +
+
<< [[Definition:monicqHahn|Definition:monicqHahn]]
+
[[Main_Page|Main Page]]
+
[[Definition:dualqKrawtchouk|Definition:dualqKrawtchouk]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\dualqHahn''' represents the dual q-Hahn polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\dualqHahn{n}''' produces {\displaystyle \dualqHahn{n}}
+:'''\dualqHahn{n}@{\mu(x)}{\gamma}{\delta}{N}{q}''' produces {\displaystyle \dualqHahn{n}@{\mu(x)}{\gamma}{\delta}{N}{q}}
+:'''\dualqHahn{n}@@{\mu(x)}{\gamma}{\delta}{N}{q}''' produces {\displaystyle \dualqHahn{n}@@{\mu(x)}{\gamma}{\delta}{N}{q}}
+ +These are defined by + +\dualqHahn{n}@{\mu(x)}{\gamma}{\delta}{N}{q}:= +\qHyperrphis32@@{q^{-n},q^{-x},\gamma\delta q^{x+1}} +{\gamma q,q^{-N}}qq, + + +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:dualqHahn {\displaystyle R_{n}}] : dual {\displaystyle q}-Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:dualqHahn http://drmf.wmflabs.org/wiki/Definition:dualqHahn]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1] +
+
<< [[Definition:monicqHahn|Definition:monicqHahn]]
+
[[Main_Page|Main Page]]
+
[[Definition:dualqKrawtchouk|Definition:dualqKrawtchouk]] >>
+
+drmf_eof +drmf_bof +'''Definition:dualqKrawtchouk''' +
+
<< [[Definition:dualqHahn|Definition:dualqHahn]]
+
[[Main_Page|Main Page]]
+
[[Definition:littleqLaguerre|Definition:littleqLaguerre]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\dualqKrawtchouk''' represents the dual q-Krawtchouk. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\dualqKrawtchouk{n}''' produces {\displaystyle \dualqKrawtchouk{n}}
+:'''\dualqKrawtchouk{n}@{\lambda(x)}{c}{N}{q}''' produces {\displaystyle \dualqKrawtchouk{n}@{\lambda(x)}{c}{N}{q}}
+:'''\dualqKrawtchouk{n}@@{\lambda(x)}{c}{N}{q}''' produces {\displaystyle \dualqKrawtchouk{n}@@{\lambda(x)}{c}{N}{q}}
+ +These are defined by +{\displaystyle +\dualqKrawtchouk{n}@{\lambda(x)}{c}{N}{q}:=\qHyperrphis{3}{2}@@{q^{-n},q^{-x},cq^{x-N}}{q^{-N},0}{q}{q} +} + +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:dualqKrawtchouk {\displaystyle K_{n}}] : dual {\displaystyle q}-Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:dualqKrawtchouk http://drmf.wmflabs.org/wiki/Definition:dualqKrawtchouk]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1] +
+
<< [[Definition:dualqHahn|Definition:dualqHahn]]
+
[[Main_Page|Main Page]]
+
[[Definition:littleqLaguerre|Definition:littleqLaguerre]] >>
+
+drmf_eof +drmf_bof +'''Definition:littleqLaguerre''' +
+
<< [[Definition:dualqKrawtchouk|Definition:dualqKrawtchouk]]
+
[[Main_Page|Main Page]]
+
[[Definition:qsinKLS|Definition:qsinKLS]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\littleqLaguerre''' represents the little q-Laguerre / Wall polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\littleqLaguerre{n}''' produces {\displaystyle \littleqLaguerre{n}}
+:'''\littleqLaguerre{n}@{x}{a}{q}''' produces {\displaystyle \littleqLaguerre{n}@{x}{a}{q}}
+ +These are defined by +{\displaystyle +\littleqLaguerre{n}@{x}{a}{q}:=\qHyperrphis{2}{1}@@{q^{-n},0}{aq}{q}{qx}. +} + +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:littleqLaguerre {\displaystyle p_{n}}] : little {\displaystyle q}-Laguerre / Wall polynomial : [http://drmf.wmflabs.org/wiki/Definition:littleqLaguerre http://drmf.wmflabs.org/wiki/Definition:littleqLaguerre]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1] +
+
<< [[Definition:dualqKrawtchouk|Definition:dualqKrawtchouk]]
+
[[Main_Page|Main Page]]
+
[[Definition:qsinKLS|Definition:qsinKLS]] >>
+
+drmf_eof +drmf_bof +'''Definition:qsinKLS''' +
+
<< [[Definition:littleqLaguerre|Definition:littleqLaguerre]]
+
[[Main_Page|Main Page]]
+
[[Definition:qSinKLS|Definition:qSinKLS]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\qsinKLS''' represents a q-analogue of the \sin +function: \qsinKLS{q}. + +This macro is in the category of real or complex valued functions. + +In math mode, this macro can be called in the following ways: + +:'''\qsinKLS{q}''' produces {\displaystyle \qsinKLS{q}}
+:'''\qsinKLS{q}@{z}''' produces {\displaystyle \qsinKLS{q}@{z}}
+:'''\qsinKLS{q}@@{z}''' produces {\displaystyle \qsinKLS{q}@@{z}}
+ +These are defined by +{\displaystyle +\qsinKLS{q}@{z}:=\frac{\qexpKLS{q}@{\iunit z)-{\mathrm e}_q(-\iunit z}}{2\iunit}= +\sum_{n=0}^{\infty}\frac{(-1)^nz^{2n+1}}{\qPochhammer{q}{q}{2n+1}} +} + +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:qsinKLS {\displaystyle \mathrm{sin}_{q}}] : {\displaystyle q}-analogue of the {\displaystyle \sin} function used in KLS: {\displaystyle \sin_q} : [http://drmf.wmflabs.org/wiki/Definition:qsinKLS http://drmf.wmflabs.org/wiki/Definition:qsinKLS]
+[http://dlmf.nist.gov/4.14#E1 {\displaystyle \mathrm{sin}}] : sine function : [http://dlmf.nist.gov/4.14#E1 http://dlmf.nist.gov/4.14#E1]
+[http://drmf.wmflabs.org/wiki/Definition:qexpKLS {\displaystyle \mathrm{e}_{q}}] : {\displaystyle q}-analogue of the {\displaystyle \exp} function used in KLS: {\displaystyle \mathrm{e}_{q}} : [http://drmf.wmflabs.org/wiki/Definition:qexpKLS http://drmf.wmflabs.org/wiki/Definition:qexpKLS]
+[http://dlmf.nist.gov/1.9.i {\displaystyle \mathrm{i}}] : imaginary unit : [http://dlmf.nist.gov/1.9.i http://dlmf.nist.gov/1.9.i]
+[http://drmf.wmflabs.org/wiki/Definition:sum {\displaystyle \Sigma}] : sum : [http://drmf.wmflabs.org/wiki/Definition:sum http://drmf.wmflabs.org/wiki/Definition:sum]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1] +
+
<< [[Definition:littleqLaguerre|Definition:littleqLaguerre]]
+
[[Main_Page|Main Page]]
+
[[Definition:qSinKLS|Definition:qSinKLS]] >>
+
+drmf_eof +drmf_bof +'''Definition:qSinKLS''' +
+
<< [[Definition:qsinKLS|Definition:qsinKLS]]
+
[[Main_Page|Main Page]]
+
[[Definition:qCosKLS|Definition:qCosKLS]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\qSinKLS''' represents a q-analogue of the +\sin function: \qSinKLS{q}. + +This macro is in the category of real or complex valued functions. + +In math mode, this macro can be called in the following ways: + +:'''\qSinKLS{q}''' produces {\displaystyle \qSinKLS{q}}
+:'''\qSinKLS{q}@{z}''' produces {\displaystyle \qSinKLS{q}@{z}}
+:'''\qSinKLS{q}@@{z}''' produces {\displaystyle \qSinKLS{q}@@{z}}
+ +These are defined by +{\displaystyle +\qSinKLS{q}@{z}:=\frac{\qExpKLS{q}@{\iunit z}-\qExpKLS{q}@{-\iunit z}}{2\iunit} +=\sum_{n=0}^{\infty}\frac{(-1)^nq^{\binomial{2n+1}{2}}z^{2n+1}}{\qPochhammer{q}{q}{2n+1}} +} + +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:qSinKLS {\displaystyle \mathrm{Sin}_{q}}] : {\displaystyle q}-analogue of the {\displaystyle \sin} function used in KLS: {\displaystyle \mathrm{Sin}_q} : [http://drmf.wmflabs.org/wiki/Definition:qSinKLS http://drmf.wmflabs.org/wiki/Definition:qSinKLS]
+[http://dlmf.nist.gov/4.14#E1 {\displaystyle \mathrm{sin}}] : sine function : [http://dlmf.nist.gov/4.14#E1 http://dlmf.nist.gov/4.14#E1]
+[http://drmf.wmflabs.org/wiki/Definition:qExpKLS {\displaystyle \mathrm{E}_{q}}] : {\displaystyle q}-analogue of the {\displaystyle \exp} function used in KLS: {\displaystyle \mathrm{E}_{q}} : [http://drmf.wmflabs.org/wiki/Definition:qExpKLS http://drmf.wmflabs.org/wiki/Definition:qExpKLS]
+[http://dlmf.nist.gov/1.9.i {\displaystyle \mathrm{i}}] : imaginary unit : [http://dlmf.nist.gov/1.9.i http://dlmf.nist.gov/1.9.i]
+[http://drmf.wmflabs.org/wiki/Definition:sum {\displaystyle \Sigma}] : sum : [http://drmf.wmflabs.org/wiki/Definition:sum http://drmf.wmflabs.org/wiki/Definition:sum]
+[http://dlmf.nist.gov/1.2#E1 {\displaystyle \binom{n}{k}}] : binomial coefficient : [http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/1.2#E1] [http://dlmf.nist.gov/26.3#SS1.p1 http://dlmf.nist.gov/26.3#SS1.p1]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1] +
+
<< [[Definition:qsinKLS|Definition:qsinKLS]]
+
[[Main_Page|Main Page]]
+
[[Definition:qCosKLS|Definition:qCosKLS]] >>
+
+drmf_eof +drmf_bof +'''Definition:qCosKLS''' +
+
<< [[Definition:qSinKLS|Definition:qSinKLS]]
+
[[Main_Page|Main Page]]
+
[[Definition:Wilson|Definition:Wilson]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\qCosKLS''' represents a q-analogue of the +\cos function: \qCosKLS{q}. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\qCosKLS{q}''' produces {\displaystyle \qCosKLS{q}}
+:'''\qCosKLS{q}@{z}''' produces {\displaystyle \qCosKLS{q}@{z}}
+:'''\qCosKLS{q}@@{z}''' produces {\displaystyle \qCosKLS{q}@@{z}}
+ +These are defined by +{\displaystyle +\qCosKLS{q}@{z}:=\frac{\qExpKLS{q}@{\iunit z}+\qExpKLS{q}@{-\iunit z}}{2} +=\sum_{n=0}^{\infty}\frac{(-1)^nq^{\binomial{2n}{2}}z^{2n}}{\qPochhammer{q}{q}{2n}} +} + +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:qCosKLS {\displaystyle \mathrm{Cos}_{q}}] : {\displaystyle q}-analogue of the {\displaystyle \cos} function used in KLS: {\displaystyle \mathrm{Cos}_{q}} : [http://drmf.wmflabs.org/wiki/Definition:qCosKLS http://drmf.wmflabs.org/wiki/Definition:qCosKLS]
+[http://dlmf.nist.gov/4.14#E2 {\displaystyle \mathrm{cos}}] : cosine function : [http://dlmf.nist.gov/4.14#E2 http://dlmf.nist.gov/4.14#E2]
+[http://drmf.wmflabs.org/wiki/Definition:qExpKLS {\displaystyle \mathrm{E}_{q}}] : {\displaystyle q}-analogue of the {\displaystyle \exp} function used in KLS: {\displaystyle \mathrm{E}_{q}} : [http://drmf.wmflabs.org/wiki/Definition:qExpKLS http://drmf.wmflabs.org/wiki/Definition:qExpKLS]
+[http://dlmf.nist.gov/1.9.i {\displaystyle \mathrm{i}}] : imaginary unit : [http://dlmf.nist.gov/1.9.i http://dlmf.nist.gov/1.9.i]
+[http://drmf.wmflabs.org/wiki/Definition:sum {\displaystyle \Sigma}] : sum : [http://drmf.wmflabs.org/wiki/Definition:sum http://drmf.wmflabs.org/wiki/Definition:sum]
+[http://dlmf.nist.gov/1.2#E1 {\displaystyle \binom{n}{k}}] : binomial coefficient : [http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/1.2#E1] [http://dlmf.nist.gov/26.3#SS1.p1 http://dlmf.nist.gov/26.3#SS1.p1]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1] +
+
<< [[Definition:qSinKLS|Definition:qSinKLS]]
+
[[Main_Page|Main Page]]
+
[[Definition:Wilson|Definition:Wilson]] >>
+
+drmf_eof +drmf_bof +'''Definition:Wilson''' +
+
<< [[Definition:qCosKLS|Definition:qCosKLS]]
+
[[Main_Page|Main Page]]
+
[[Definition:Racah|Definition:Racah]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\Wilson''' represents the Wilson polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\Wilson{n}''' produces {\displaystyle \Wilson{n}}
+:'''\Wilson{n}@{x}{a}{b}{c}{d}''' produces {\displaystyle \Wilson{n}@{x}{a}{b}{c}{d}}
+ +These are defined by +{\displaystyle +\frac{\Wilson{n}@{x^2}{a}{b}{c}{d}}{\pochhammer{a+b}{n}\pochhammer{a+c}{n}\pochhammer{a+d}{n}} +{}:=\HyperpFq{4}{3}@@{-n,n+a+b+c+d-1,a+ix,a-ix}{a+b,a+c,a+d}{1} +} + +== Symbols List == + +[http://dlmf.nist.gov/18.25#T1.t1.r2 {\displaystyle W_{n}}] : Wilson polynomial : [http://dlmf.nist.gov/18.25#T1.t1.r2 http://dlmf.nist.gov/18.25#T1.t1.r2]
+[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii]
+[http://dlmf.nist.gov/16.2#E1 {\displaystyle {{}_{p}F_{q}}}] : generalized hypergeometric function : [http://dlmf.nist.gov/16.2#E1 http://dlmf.nist.gov/16.2#E1] +
+
<< [[Definition:qCosKLS|Definition:qCosKLS]]
+
[[Main_Page|Main Page]]
+
[[Definition:Racah|Definition:Racah]] >>
+
+drmf_eof +drmf_bof +'''Definition:Racah''' +
+
<< [[Definition:Wilson|Definition:Wilson]]
+
[[Main_Page|Main Page]]
+
[[Definition:ctsdualHahn|Definition:ctsdualHahn]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\Racah''' represents the Racah polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\Racah{n}''' produces {\displaystyle \Racah{n}}
+:'''\Racah{n}@{x}{\alpha}{\beta}{\gamma}{\delta}''' produces {\displaystyle \Racah{n}@{x}{\alpha}{\beta}{\gamma}{\delta}}
+ +These are defined by +{\displaystyle +\Racah{n}@{\lambda(x)}{\alpha}{\beta}{\gamma}{\delta} +{}:=\HyperpFq{4}{3}@@{-n,n+\alpha+\beta+1,-x,x+\gamma+\delta+1}{\alpha+1,\beta+\delta+1,\gamma+1}{1}, +} + +where {\displaystyle \lambda(x)=x(x+\gamma+\delta+1), } +and \alpha+1=-N or \beta+\delta+1=-N or \gamma+1=-N +with N a nonnegative integer. + +== Symbols List == + +[http://dlmf.nist.gov/18.25#T1.t1.r4 {\displaystyle R_{n}}] : Racah polynomial : [http://dlmf.nist.gov/18.25#T1.t1.r4 http://dlmf.nist.gov/18.25#T1.t1.r4]
+[http://dlmf.nist.gov/16.2#E1 {\displaystyle {{}_{p}F_{q}}}] : generalized hypergeometric function : [http://dlmf.nist.gov/16.2#E1 http://dlmf.nist.gov/16.2#E1] +
+
<< [[Definition:Wilson|Definition:Wilson]]
+
[[Main_Page|Main Page]]
+
[[Definition:ctsdualHahn|Definition:ctsdualHahn]] >>
+
+drmf_eof +drmf_bof +'''Definition:ctsdualHahn''' +
+
<< [[Definition:Racah|Definition:Racah]]
+
[[Main_Page|Main Page]]
+
[[Definition:ctsHahn|Definition:ctsHahn]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\ctsdualHahn''' represents the Continuous Dual Hahn polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\ctsdualHahn{n}''' produces {\displaystyle \ctsdualHahn{n}}
+:'''\ctsdualHahn{n}@{x^2}{a}{b}{c}''' produces {\displaystyle \ctsdualHahn{n}@{x^2}{a}{b}{c}}
+:'''\ctsdualHahn{n}@@{x^2}{a}{b}{c}''' produces {\displaystyle \ctsdualHahn{n}@@{x^2}{a}{b}{c}}
+ +These are defined by +{\displaystyle +\frac{\ctsdualHahn{n}@{x^2}{a}{b}{c}}{\pochhammer{a+b}{n}\pochhammer{a+c}{n}}:=\HyperpFq{3}{2}@@{-n,a+ix,a-ix}{a+b,a+c}{1} +} + +== Symbols List == + +[http://dlmf.nist.gov/18.25#T1.t1.r3 {\displaystyle S_{n}}] : continuous dual Hahn polynomial : [http://dlmf.nist.gov/18.25#T1.t1.r3 http://dlmf.nist.gov/18.25#T1.t1.r3]
+[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii]
+[http://dlmf.nist.gov/16.2#E1 {\displaystyle {{}_{p}F_{q}}}] : generalized hypergeometric function : [http://dlmf.nist.gov/16.2#E1 http://dlmf.nist.gov/16.2#E1] +
+
<< [[Definition:Racah|Definition:Racah]]
+
[[Main_Page|Main Page]]
+
[[Definition:ctsHahn|Definition:ctsHahn]] >>
+
+drmf_eof +drmf_bof +'''Definition:ctsHahn''' +
+
<< [[Definition:ctsdualHahn|Definition:ctsdualHahn]]
+
[[Main_Page|Main Page]]
+
[[Definition:Hahn|Definition:Hahn]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\ctsHahn''' represents the Continuous Hahn polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\ctsHahn{n}''' produces {\displaystyle \ctsHahn{n}}
+:'''\ctsHahn{n}@{x}{a}{b}{c}{d}''' produces {\displaystyle \ctsHahn{n}@{x}{a}{b}{c}{d}}
+:'''\ctsHahn{n}@@{x}{a}{b}{c}{d}''' produces {\displaystyle \ctsHahn{n}@@{x}{a}{b}{c}{d}}
+ +These are defined by +{\displaystyle +\ctsHahn{n}@{x}{a}{b}{c}{d} +{}:=i^n\frac{\pochhammer{a+c}{n}\pochhammer{a+d}{n}}{n!}\,\HyperpFq{3}{2}@@{-n,n+a+b+c+d-1,a+ix}{a+c,a+d}{1} +} + +== Symbols List == + +[http://dlmf.nist.gov/18.19#P2.p1 {\displaystyle p_{n}}] : continuous Hahn polynomial : [http://dlmf.nist.gov/18.19#P2.p1 http://dlmf.nist.gov/18.19#P2.p1]
+[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii]
+[http://dlmf.nist.gov/16.2#E1 {\displaystyle {{}_{p}F_{q}}}] : generalized hypergeometric function : [http://dlmf.nist.gov/16.2#E1 http://dlmf.nist.gov/16.2#E1] +
+
<< [[Definition:ctsdualHahn|Definition:ctsdualHahn]]
+
[[Main_Page|Main Page]]
+
[[Definition:Hahn|Definition:Hahn]] >>
+
+drmf_eof +drmf_bof +'''Definition:Hahn''' +
+
<< [[Definition:ctsHahn|Definition:ctsHahn]]
+
[[Main_Page|Main Page]]
+
[[Definition:dualHahn|Definition:dualHahn]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\Hahn''' represents the Hahn polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\Hahn{n}''' produces {\displaystyle \Hahn{n}}
+:'''\Hahn{n}@{x}{\alpha}{\beta}{N}''' produces {\displaystyle \Hahn{n}@{x}{\alpha}{\beta}{N}}
+ +These are defined by +{\displaystyle +\Hahn{n}@{x}{\alpha}{\beta}{N}:=\HyperpFq{3}{2}@@{-n,n+\alpha+\beta+1,-x}{\alpha+1,-N}{1} +} + +== Symbols List == + +[http://dlmf.nist.gov/18.19#T1.t1.r3 {\displaystyle Q_{n}}] : Hahn polynomial : [http://dlmf.nist.gov/18.19#T1.t1.r3 http://dlmf.nist.gov/18.19#T1.t1.r3]
+[http://dlmf.nist.gov/16.2#E1 {\displaystyle {{}_{p}F_{q}}}] : generalized hypergeometric function : [http://dlmf.nist.gov/16.2#E1 http://dlmf.nist.gov/16.2#E1] +
+
<< [[Definition:ctsHahn|Definition:ctsHahn]]
+
[[Main_Page|Main Page]]
+
[[Definition:dualHahn|Definition:dualHahn]] >>
+
+drmf_eof +drmf_bof +'''Definition:dualHahn''' +
+
<< [[Definition:Hahn|Definition:Hahn]]
+
[[Main_Page|Main Page]]
+
[[Definition:qRacah|Definition:qRacah]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\dualHahn''' represents the Dual Hahn polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\dualHahn{n}''' produces {\displaystyle \dualHahn{n}}
+:'''\dualHahn{n}@{\lambda(x)}{\gamma}{\delta}{N}''' produces {\displaystyle \dualHahn{n}@{\lambda(x)}{\gamma}{\delta}{N}}
+:'''\dualHahn{n}@@{\lambda(x)}{\gamma}{\delta}{N}''' produces {\displaystyle \dualHahn{n}@@{\lambda(x)}{\gamma}{\delta}{N}}
+ +These are defined by +{\displaystyle +\dualHahn{n}@{\lambda(x)}{\gamma}{\delta}{N}:= +\HyperpFq{3}{2}@@{-n,-x,x+\gamma+\delta+1}{\gamma+1,-N}{1} +} + +== Symbols List == + +[http://dlmf.nist.gov/18.25#T1.t1.r5 {\displaystyle R_{n}}] : dual Hahn polynomial : [http://dlmf.nist.gov/18.25#T1.t1.r5 http://dlmf.nist.gov/18.25#T1.t1.r5]
+[http://dlmf.nist.gov/16.2#E1 {\displaystyle {{}_{p}F_{q}}}] : generalized hypergeometric function : [http://dlmf.nist.gov/16.2#E1 http://dlmf.nist.gov/16.2#E1] +
+
<< [[Definition:Hahn|Definition:Hahn]]
+
[[Main_Page|Main Page]]
+
[[Definition:qRacah|Definition:qRacah]] >>
+
+drmf_eof +drmf_bof +'''Definition:qRacah''' +
+
<< [[Definition:dualHahn|Definition:dualHahn]]
+
[[Main_Page|Main Page]]
+
[[Definition:normctsdualqHahnptilde|Definition:normctsdualqHahnptilde]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\qRacah''' represents the q-Racah polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\qRacah{n}''' produces {\displaystyle \qRacah{n}}
+:'''\qRacah{n}@{x}{\alpha}{\beta}{\gamma}{\delta}{q}''' produces {\displaystyle \qRacah{n}@{x}{\alpha}{\beta}{\gamma}{\delta}{q}}
+ +These are defined by +{\displaystyle +\qRacah{n}@{\mu(x)}{\alpha}{\beta}{\gamma}{\delta}{q} +{}:=\qHyperrphis{4}{3}@@{q^{-n},\alpha\beta q^{n+1},q^{-x},\gamma\delta q^{x+1}}{\alpha q,\beta\delta q,\gamm +} + +== Symbols List == + +[http://dlmf.nist.gov/18.28#E19 {\displaystyle R_{n}}] : {\displaystyle q}-Racah polynomial : [http://dlmf.nist.gov/18.28#E19 http://dlmf.nist.gov/18.28#E19]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1] +
+
<< [[Definition:dualHahn|Definition:dualHahn]]
+
[[Main_Page|Main Page]]
+
[[Definition:normctsdualqHahnptilde|Definition:normctsdualqHahnptilde]] >>
+
+drmf_eof +drmf_bof +'''Definition:normctsdualqHahnptilde''' +
+
<< [[Definition:qRacah|Definition:qRacah]]
+
[[Main_Page|Main Page]]
+
[[Definition:normctsqHahnptilde|Definition:normctsqHahnptilde]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\normctsdualqHahnptilde''' represents the normalized continuous dual q-Hahn tilde polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\normctsdualqHahnptilde{n}''' produces {\displaystyle \normctsdualqHahnptilde{n}}
+:'''\normctsdualqHahnptilde{n}@{x}{a}{b}{c}{q}''' produces {\displaystyle \normctsdualqHahnptilde{n}@{x}{a}{b}{c}{q}}
+:'''\normctsdualqHahnptilde{n}@@{x}{a}{b}{c}{q}''' produces {\displaystyle \normctsdualqHahnptilde{n}@@{x}{a}{b}{c}{q}}
+ +These are defined by +{\displaystyle +{\tilde p}_n(x):=\normctsdualqHahnptilde{n}@{x}{a}{b}{c}{q}=\frac{a^n\ctsdualqHahn{n}@{x}{a}{b}{c}{q}}{\qPochhammer{ab,ac}{q}{n}} +} + +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:normctsdualqHahnptilde {\displaystyle {\tilde p}_{n}}] : normalized continuous dual {\displaystyle q}-Hahn polynomial {\displaystyle {\tilde p}} : [http://drmf.wmflabs.org/wiki/Definition:normctsdualqHahnptilde http://drmf.wmflabs.org/wiki/Definition:normctsdualqHahnptilde]
+[http://drmf.wmflabs.org/wiki/Definition:ctsdualqHahn {\displaystyle p_{n}}] : continuous dual {\displaystyle q}-Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:ctsdualqHahn http://drmf.wmflabs.org/wiki/Definition:ctsdualqHahn]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1] +
+
<< [[Definition:qRacah|Definition:qRacah]]
+
[[Main_Page|Main Page]]
+
[[Definition:normctsqHahnptilde|Definition:normctsqHahnptilde]] >>
+
+drmf_eof +drmf_bof +'''Definition:normctsqHahnptilde''' +
+
<< [[Definition:normctsdualqHahnptilde|Definition:normctsdualqHahnptilde]]
+
[[Main_Page|Main Page]]
+
[[Definition:qHahn|Definition:qHahn]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\normctsqHahnptilde''' represents the normalized continuous q-Hahn tilde polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\normctsqHahnptilde{n}''' produces {\displaystyle \normctsqHahnptilde{n}}
+:'''\normctsqHahnptilde{n}@{x}{a}{b}{c}{d}{q}''' produces {\displaystyle \normctsqHahnptilde{n}@{x}{a}{b}{c}{d}{q}}
+:'''\normctsqHahnptilde{n}@@{x}{a}{b}{c}{d}{q}''' produces {\displaystyle \normctsqHahnptilde{n}@@{x}{a}{b}{c}{d}{q}}
+ +These are defined by +{\displaystyle +{\tilde p}_n(x):=\normctsqHahnptilde{n}@{x}{a}{b}{c}{d}{q}=\frac{(a\expe^{i\phi})^n\ctsqHahn{n}@{x}{a}{b}{c}{d}{q}}{\qPochhammer{ab\expe^{2i\phi},ac,ad}{q}{n}} + +} + +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:normctsqHahnptilde {\displaystyle {\tilde p}_{n}}] : normalized continuous {\displaystyle q}-Hahn polynomial {\displaystyle {\tilde p}} : [http://drmf.wmflabs.org/wiki/Definition:normctsqHahnptilde http://drmf.wmflabs.org/wiki/Definition:normctsqHahnptilde]
+[http://dlmf.nist.gov/4.2.E11 {\displaystyle \mathrm{e}}] : the base of the natural logarithm : [http://dlmf.nist.gov/4.2.E11 http://dlmf.nist.gov/4.2.E11]
+[http://drmf.wmflabs.org/wiki/Definition:ctsqHahn {\displaystyle p_{n}}] : continuous {\displaystyle q}-Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:ctsqHahn http://drmf.wmflabs.org/wiki/Definition:ctsqHahn]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1] +
+
<< [[Definition:normctsdualqHahnptilde|Definition:normctsdualqHahnptilde]]
+
[[Main_Page|Main Page]]
+
[[Definition:qHahn|Definition:qHahn]] >>
+
+drmf_eof +drmf_bof +'''Definition:qHahn''' +
+
<< [[Definition:normctsqHahnptilde|Definition:normctsqHahnptilde]]
+
[[Main_Page|Main Page]]
+
[[Definition:AlSalamChihara|Definition:AlSalamChihara]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\qHahn''' represents the q-Hahn polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\qHahn{n}@{q^{-x}}{\alpha}{\beta}{N}''' produces {\displaystyle \qHahn{n}@{q^{-x}}{\alpha}{\beta}{N}}
+:'''\qHahn{n}@@{q^{-x}}{\alpha}{\beta}{N}''' produces {\displaystyle \qHahn{n}@@{q^{-x}}{\alpha}{\beta}{N}}
+ +These are defined by +{\displaystyle +\qHahn{n}@{q^{-x}}{\alpha}{\beta}{N}{q}:=\qHyperrphis{3}{2}@@{q^{-n},\alpha\beta q^{n+1},q^{-x}}{\alpha q,q^{-N}}{q}{q} +} + +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:qHahn {\displaystyle Q_{n}}] : {\displaystyle q}-Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:qHahn http://drmf.wmflabs.org/wiki/Definition:qHahn]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1] +
+
<< [[Definition:normctsqHahnptilde|Definition:normctsqHahnptilde]]
+
[[Main_Page|Main Page]]
+
[[Definition:AlSalamChihara|Definition:AlSalamChihara]] >>
+
+drmf_eof +drmf_bof +'''Definition:AlSalamChihara''' +
+
<< [[Definition:qHahn|Definition:qHahn]]
+
[[Main_Page|Main Page]]
+
[[Definition:qinvAlSalamChihara|Definition:qinvAlSalamChihara]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\AlSalamChihara''' represents the Al-Salam Chihara polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\AlSalamChihara{n}''' produces {\displaystyle \AlSalamChihara{n}}
+:'''\AlSalamChihara{n}@{x}{a}{b}{q}''' produces {\displaystyle \AlSalamChihara{n}@{x}{a}{b}{q}}
+:'''\AlSalamChihara{n}@@{x}{a}{b}{q}''' produces {\displaystyle \AlSalamChihara{n}@@{x}{a}{b}{q}}
+ +These are defined by +{\displaystyle +\AlSalamChihara{n}@{x}{a}{b}{q}:=\frac{\qPochhammer{ab}{q}{n}}{a^n}\, +\qHyperrphis{3}{2}@@{q^{-n},a\expe^{i\theta},a\expe^{-i\theta}}{ab,0}{q}{q} +} + +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:AlSalamChihara {\displaystyle Q_{n}}] : Al-Salam-Chihara polynomial : [http://drmf.wmflabs.org/wiki/Definition:AlSalamChihara http://drmf.wmflabs.org/wiki/Definition:AlSalamChihara]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
+[http://dlmf.nist.gov/4.2.E11 {\displaystyle \mathrm{e}}] : the base of the natural logarithm : [http://dlmf.nist.gov/4.2.E11 http://dlmf.nist.gov/4.2.E11] +
+
<< [[Definition:qHahn|Definition:qHahn]]
+
[[Main_Page|Main Page]]
+
[[Definition:qinvAlSalamChihara|Definition:qinvAlSalamChihara]] >>
+
+drmf_eof +drmf_bof +'''Definition:qinvAlSalamChihara''' +
+
<< [[Definition:AlSalamChihara|Definition:AlSalamChihara]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicqinvAlSalamChihara|Definition:monicqinvAlSalamChihara]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\qinvAlSalamChihara''' represents the q-inverse of the Al-Salam Chihara polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\qinvAlSalamChihara{n}''' produces {\displaystyle \qinvAlSalamChihara{n}}
+:'''\qinvAlSalamChihara{n}@{x}{a}{b}{q^{-1}}''' produces {\displaystyle \qinvAlSalamChihara{n}@{x}{a}{b}{q^{-1}}}
+ +These are defined by +{\displaystyle +\qinvAlSalamChihara{n}@{\thalf(aq^{-x}+a^{-1}q^x)}{a}{b }{q^{-1}}:= +(-1)^n b^n q^{-\half n(n-1)}\qPochhammer{(ab)^{-1}}{q}{n} +\qHyperrphis{3}{1}@@{q^{-n},q^{-x},a^{-2}q^x}{(ab)^{-1}}{q}{q^nab^{-1}} +} + +== Symbols List == + +[http://dlmf.nist.gov/23.1 {\displaystyle Q_{n}}] : {\displaystyle q}-inverse Al-Salam-Chihara polynomial : [http://dlmf.nist.gov/23.1 http://dlmf.nist.gov/23.1]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1] +
+
<< [[Definition:AlSalamChihara|Definition:AlSalamChihara]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicqinvAlSalamChihara|Definition:monicqinvAlSalamChihara]] >>
+
+drmf_eof +drmf_bof +'''Definition:monicqinvAlSalamChihara''' +
+
<< [[Definition:qinvAlSalamChihara|Definition:qinvAlSalamChihara]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicAffqKrawtchouk|Definition:monicAffqKrawtchouk]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\monicqinvAlSalamChihara''' represents the monic q-inverse of the Al-Salam Chihara polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\monicqinvAlSalamChihara{n}''' produces {\displaystyle \monicqinvAlSalamChihara{n}}
+:'''\monicqinvAlSalamChihara{n}@{x}{a}{b}{q^{-1}}''' produces {\displaystyle \monicqinvAlSalamChihara{n}@{x}{a}{b}{q^{-1}}}
+ +These are defined by +{\displaystyle +x\monicqinvAlSalamChihara{n}@@{x}{a}{b}{q^{-1}=:\monicqinvAlSalamChihara{n+1}@@{x}{a}{b}{q^{-1}+\thalf(a+b)q^{-n} \monicqinvAlSalamChihara{n}@@{x}{a}{b}{q^{-1}+ +\tfrac14(q^{-n}-1)(abq^{-n+1}-1)\monicqinvAlSalamChihara{n-1}@@{x}{a}{b}{q^{-1} +} + +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:monicqinvAlSalamChihara {\displaystyle {\widehat Q}_{n}}] : monic {\displaystyle q}-inverse Al-Salam-Chihara polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicqinvAlSalamChihara http://drmf.wmflabs.org/wiki/Definition:monicqinvAlSalamChihara] +
+
<< [[Definition:qinvAlSalamChihara|Definition:qinvAlSalamChihara]]
+
[[Main_Page|Main Page]]
+
[[Definition:monicAffqKrawtchouk|Definition:monicAffqKrawtchouk]] >>
+
+drmf_eof +drmf_bof +'''Definition:monicAffqKrawtchouk''' +
+
<< [[Definition:monicqinvAlSalamChihara|Definition:monicqinvAlSalamChihara]]
+
[[Main_Page|Main Page]]
+
[[Definition:qMeixnerPollaczek|Definition:qMeixnerPollaczek]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\monicAffqKrawtchouk''' represents the Affine q Krawtchouk polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\monicAffqKrawtchouk{n}''' produces {\displaystyle \monicAffqKrawtchouk{n}}
+:'''\monicAffqKrawtchouk{n}@{q^{-x}}{p}{N}{q}''' produces {\displaystyle \monicAffqKrawtchouk{n}@{q^{-x}}{p}{N}{q}}
+:'''\monicAffqKrawtchouk{n}@@{q^{-x}}{p}{N}{q}''' produces {\displaystyle \monicAffqKrawtchouk{n}@@{q^{-x}}{p}{N}{q}}
+ +These are defined by + +{\displaystyle +\AffqKrawtchouk{n}@{q^{-x}}{p}{N}{q}=:\frac{1}{\qPochhammer{pq,q^{-N}}{q}{n}}\monicAffqKrawtchouk{n}@{q^{-x}}{p}{N}{q}. +} + +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:monicAffqKrawtchouk {\displaystyle \widehat{K}^{\mathrm{Aff}}_{n}}] : monic affine {\displaystyle q}-Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicAffqKrawtchouk http://drmf.wmflabs.org/wiki/Definition:monicAffqKrawtchouk]
+[http://drmf.wmflabs.org/wiki/Definition:AffqKrawtchouk {\displaystyle K^{\mathrm{Aff}}_{n}}] : affine {\displaystyle q}-Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:AffqKrawtchouk http://drmf.wmflabs.org/wiki/Definition:AffqKrawtchouk]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1] +
+
<< [[Definition:monicqinvAlSalamChihara|Definition:monicqinvAlSalamChihara]]
+
[[Main_Page|Main Page]]
+
[[Definition:qMeixnerPollaczek|Definition:qMeixnerPollaczek]] >>
+
+drmf_eof +drmf_bof +'''Definition:qMeixnerPollaczek''' +
+
<< [[Definition:monicAffqKrawtchouk|Definition:monicAffqKrawtchouk]]
+
[[Main_Page|Main Page]]
+
[[Definition:qtmqKrawtchouk|Definition:qtmqKrawtchouk]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\qMeixnerPollaczek''' represents the q-Meixner Pollaczek polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\qMeixnerPollaczek{n}''' produces {\displaystyle \qMeixnerPollaczek{n}}
+:'''\qMeixnerPollaczek{n}@{x}{a}{q}''' produces {\displaystyle \qMeixnerPollaczek{n}@{x}{a}{q}}
+ +These are defined by +{\displaystyle +\qMeixnerPollaczek{n}@{x}{a}{q}:=a^{-n}\expe^{-in\phi}\frac{\qPochhammer{a^2}{q}{n}}{\qPochhammer{q}{q}{n}}\, +\qHyperrphis{3}{2}@@{q^{-n},a\expe^{i(\theta+2\phi)},a\expe^{-i\theta}}{a^2,0}{q}{q} +} + +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:qMeixnerPollaczek {\displaystyle P_{n}}] : {\displaystyle q}-Meixner-Pollaczek polynomial : [http://drmf.wmflabs.org/wiki/Definition:qMeixnerPollaczek http://drmf.wmflabs.org/wiki/Definition:qMeixnerPollaczek]
+[http://dlmf.nist.gov/4.2.E11 {\displaystyle \mathrm{e}}] : the base of the natural logarithm : [http://dlmf.nist.gov/4.2.E11 http://dlmf.nist.gov/4.2.E11]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1] +
+
<< [[Definition:monicAffqKrawtchouk|Definition:monicAffqKrawtchouk]]
+
[[Main_Page|Main Page]]
+
[[Definition:qtmqKrawtchouk|Definition:qtmqKrawtchouk]] >>
+
+drmf_eof +drmf_bof +'''Definition:qtmqKrawtchouk''' +
+
<< [[Definition:qMeixnerPollaczek|Definition:qMeixnerPollaczek]]
+
[[Main_Page|Main Page]]
+
[[Definition:qLaguerre|Definition:qLaguerre]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\qtmqKrawtchouk''' represents the quantum q-Krawtchouk polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\qtmqKrawtchouk{n}''' produces {\displaystyle \qtmqKrawtchouk{n}}
+:'''\qtmqKrawtchouk{n}@{q^{-x}}{p}{N}{q}''' produces {\displaystyle \qtmqKrawtchouk{n}@{q^{-x}}{p}{N}{q}}
+:'''\qtmqKrawtchouk{n}@@{q^{-x}}{p}{N}{q}''' produces {\displaystyle \qtmqKrawtchouk{n}@@{q^{-x}}{p}{N}{q}}
+ +These are defined by +{\displaystyle +\qtmqKrawtchouk{n}@{q^{-x}}{p}{N}{q}:= +\qHyperrphis{2}{1}@@{q^{-n},q^{-x}}{q^{-N}}{q}{pq^{n+1}} +} + +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:qtmqKrawtchouk {\displaystyle K^{\mathrm{qtm}}_{n}}] : quantum {\displaystyle q}-Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:qtmqKrawtchouk http://drmf.wmflabs.org/wiki/Definition:qtmqKrawtchouk]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1] +
+
<< [[Definition:qMeixnerPollaczek|Definition:qMeixnerPollaczek]]
+
[[Main_Page|Main Page]]
+
[[Definition:qLaguerre|Definition:qLaguerre]] >>
+
+drmf_eof +drmf_bof +'''Definition:qLaguerre''' +
+
<< [[Definition:qtmqKrawtchouk|Definition:qtmqKrawtchouk]]
+
[[Main_Page|Main Page]]
+
[[Definition:ctsqHermite|Definition:ctsqHermite]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\qLaguerre''' represents the q-Laguerre polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\qLaguerre{n}''' produces {\displaystyle \qLaguerre{n}}
+:'''\qLaguerre[\alpha]{n}''' produces {\displaystyle \qLaguerre[\alpha]{n}}
+:'''\qLaguerre{n}@{x}{q}''' produces {\displaystyle \qLaguerre{n}@{x}{q}}
+:'''\qLaguerre[\alpha]{n}@{x}{q}''' produces {\displaystyle \qLaguerre[\alpha]{n}@{x}{q}}
+:'''\qLaguerre{n}@@{x}{q}''' produces {\displaystyle \qLaguerre{n}@@{x}{q}}
+:'''\qLaguerre[\alpha]{n}@@{x}{q}''' produces {\displaystyle \qLaguerre[\alpha]{n}@@{x}{q}}
+ +These are defined by +{\displaystyle +\qLaguerre[\alpha]{n}@{x}{q}:=\frac{\qPochhammer{q^{\alpha+1}}{q}{n}}{\qPochhammer{q}{q}{n}}\, +\qHyperrphis{1}{1}@@{q^{-n}}{q^{\alpha+1}}{q}{-q^{n+\alpha+1}x} +} + +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:qLaguerre {\displaystyle L_n^{(\alpha)}}] : {\displaystyle q}-Laguerre polynomial : [http://drmf.wmflabs.org/wiki/Definition:qLaguerre http://drmf.wmflabs.org/wiki/Definition:qLaguerre]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1] +
+
<< [[Definition:qtmqKrawtchouk|Definition:qtmqKrawtchouk]]
+
[[Main_Page|Main Page]]
+
[[Definition:ctsqHermite|Definition:ctsqHermite]] >>
+
+drmf_eof +drmf_bof +'''Definition:ctsqHermite''' +
+
<< [[Definition:qLaguerre|Definition:qLaguerre]]
+
[[Main_Page|Main Page]]
+
[[Definition:StieltjesWigert|Definition:StieltjesWigert]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\ctsqHermite''' represents the continuous q-Hermite polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\ctsqHermite{n}''' produces {\displaystyle \ctsqHermite{n}}
+:'''\ctsqHermite{n}@{x}{q}''' produces {\displaystyle \ctsqHermite{n}@{x}{q}}
+:'''\ctsqHermite{n}@@{x}{q}''' produces {\displaystyle \ctsqHermite{n}@@{x}{q}}
+ +These are defined by +{\displaystyle +\ctsqHermite{n}@{x}{q}:=\expe^{in\theta}\,\qHyperrphis{2}{0}@@{q^{-n},0}{-}{q}{q^n\expe^{-2i\theta}} +} + +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:ctsqHermite {\displaystyle H_{n}}] : continuous {\displaystyle q}-Hermite polynomial : [http://drmf.wmflabs.org/wiki/Definition:ctsqHermite http://drmf.wmflabs.org/wiki/Definition:ctsqHermite]
+[http://dlmf.nist.gov/4.2.E11 {\displaystyle \mathrm{e}}] : the base of the natural logarithm : [http://dlmf.nist.gov/4.2.E11 http://dlmf.nist.gov/4.2.E11]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1] +
+
<< [[Definition:qLaguerre|Definition:qLaguerre]]
+
[[Main_Page|Main Page]]
+
[[Definition:StieltjesWigert|Definition:StieltjesWigert]] >>
+
+drmf_eof +drmf_bof +'''Definition:StieltjesWigert''' +
+
<< [[Definition:ctsqHermite|Definition:ctsqHermite]]
+
[[Main_Page|Main Page]]
+
[[Definition:discrqHermiteI|Definition:discrqHermiteI]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\StieltjesWigert''' represents the Stieltjes Wigert polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\StieltjesWigert{n}''' produces {\displaystyle \StieltjesWigert{n}}
+:'''\StieltjesWigert{n}@{x}{q}''' produces {\displaystyle \StieltjesWigert{n}@{x}{q}}
+:'''\StieltjesWigert{n}@@{x}{q}''' produces {\displaystyle \StieltjesWigert{n}@@{x}{q}}
+ +These are defined by +{\displaystyle +\StieltjesWigert{n}@{x}{q}:=\frac{1}{\qPochhammer{q}{q}{n}}\,\qHyperrphis{1}{1}@@{q^{-n}}{0}{q}{-q^{n+1}x} +} + +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:StieltjesWigert {\displaystyle S_{n}}] : Stieltjes-Wigert polynomial : [http://drmf.wmflabs.org/wiki/Definition:StieltjesWigert http://drmf.wmflabs.org/wiki/Definition:StieltjesWigert]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1] +
+
<< [[Definition:ctsqHermite|Definition:ctsqHermite]]
+
[[Main_Page|Main Page]]
+
[[Definition:discrqHermiteI|Definition:discrqHermiteI]] >>
+
+drmf_eof +drmf_bof +'''Definition:discrqHermiteI''' +
+
<< [[Definition:StieltjesWigert|Definition:StieltjesWigert]]
+
[[Main_Page|Main Page]]
+
[[Definition:discrqHermiteII|Definition:discrqHermiteII]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\discrqHermiteI''' represents the first discrete q Hermite polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\discrqHermiteI{n}''' produces {\displaystyle \discrqHermiteI{n}}
+:'''\discrqHermiteI{n}@{x}{q}''' produces {\displaystyle \discrqHermiteI{n}@{x}{q}}
+:'''\discrqHermiteI{n}@@{x}{q}''' produces {\displaystyle \discrqHermiteI{n}@@{x}{q}}
+ +These are defined by +{\displaystyle +\discrqHermiteI{n}@{x}{q}:=\AlSalamCarlitzI{-1}{n}@{x}{q} +} + +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:discrqHermiteI {\displaystyle h_{n}}] : discrete {\displaystyle q}-Hermite I polynomial : [http://drmf.wmflabs.org/wiki/Definition:discrqHermiteI http://drmf.wmflabs.org/wiki/Definition:discrqHermiteI]
+[http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzI {\displaystyle U^{(n)}_{\alpha}}] : Al-Salam-Carlitz I polynomial : [http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzI http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzI] +
+
<< [[Definition:StieltjesWigert|Definition:StieltjesWigert]]
+
[[Main_Page|Main Page]]
+
[[Definition:discrqHermiteII|Definition:discrqHermiteII]] >>
+
+drmf_eof +drmf_bof +'''Definition:discrqHermiteII''' +
+
<< [[Definition:discrqHermiteI|Definition:discrqHermiteI]]
+
[[Main_Page|Main Page]]
+
[[Definition:StieltjesConstants|Definition:StieltjesConstants]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\discrqHermiteII''' represents the second discrete q Hermite polynomial. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\discrqHermiteII{n}''' produces {\displaystyle \discrqHermiteII{n}}
+:'''\discrqHermiteII{n}@{x}{q}''' produces {\displaystyle \discrqHermiteII{n}@{x}{q}}
+:'''\discrqHermiteII{n}@@{x}{q}''' produces {\displaystyle \discrqHermiteII{n}@@{x}{q}}
+ +These are defined by +{\displaystyle +\discrqHermiteII{n}@{x}{q}:=i^{-n}\AlSalamCarlitzII{-1}{n}@{ix}{q} +} + +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:discrqHermiteII {\displaystyle \tilde{h}_{n}}] : discrete {\displaystyle q}-Hermite II polynomial : [http://drmf.wmflabs.org/wiki/Definition:discrqHermiteII http://drmf.wmflabs.org/wiki/Definition:discrqHermiteII]
+[http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzII {\displaystyle V^{(n)}_{\alpha}}] : Al-Salam-Carlitz II polynomial : [http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzII http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzII] +
+
<< [[Definition:discrqHermiteI|Definition:discrqHermiteI]]
+
[[Main_Page|Main Page]]
+
[[Definition:StieltjesConstants|Definition:StieltjesConstants]] >>
+
+drmf_eof +drmf_bof +'''Definition:StieltjesConstants''' +
+
<< [[Definition:discrqHermiteII|Definition:discrqHermiteII]]
+
[[Main_Page|Main Page]]
+
[[Definition:PolylogarithmS|Definition:PolylogarithmS]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\StieltjesConstants''' represents the Stiltjes constants +\StieltjesConstants{n}. + +This macro is in the category of constants. + +In math mode, this macro can be called in the following way: + +:'''\StieltjesConstants{n}''' produces {\displaystyle \StieltjesConstants{n}}
+ +These are defined by +{\displaystyle \StieltjesConstants{n} := \lim_{m \to \infty} \left( \sum_{k=1}^m +\frac{(\ln@@{k})^n}{k} - \frac{(\ln@@{m})^{n+1}}{n+1} \right). +} + +The Stieltjes constant \StieltjesConstants{0}=0.57721\;56649\;01532\;86... equals \EulerConstant the Euler-Mascheroni constant.
+ +== Bibliography == +Equation (2), Section 2.21 of [[Bibliography#FI|'''FI''']]. + +== URL links == +[http://en.wikipedia.org/wiki/Stieltjes_constants http://en.wikipedia.org/wiki/Stieltjes_constants] + +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:StieltjesConstants {\displaystyle \gamma_{n}}] : Stieltjes constants : [http://drmf.wmflabs.org/wiki/Definition:StieltjesConstants http://drmf.wmflabs.org/wiki/Definition:StieltjesConstants]
+[http://drmf.wmflabs.org/wiki/Definition:sum {\displaystyle \Sigma}] : sum : [http://drmf.wmflabs.org/wiki/Definition:sum http://drmf.wmflabs.org/wiki/Definition:sum]
+[http://dlmf.nist.gov/4.2#E2 {\displaystyle \mathrm{ln}}] : principal branch of logarithm function : [http://dlmf.nist.gov/4.2#E2 http://dlmf.nist.gov/4.2#E2]
+[http://dlmf.nist.gov/5.2#E3 {\displaystyle \gamma}] : Euler's (Euler-Mascheroni) constant : [http://dlmf.nist.gov/5.2#E3 http://dlmf.nist.gov/5.2#E3] +
+
<< [[Definition:discrqHermiteII|Definition:discrqHermiteII]]
+
[[Main_Page|Main Page]]
+
[[Definition:PolylogarithmS|Definition:PolylogarithmS]] >>
+
+drmf_eof +drmf_bof +'''Definition:PolylogarithmS''' +
+
<< [[Definition:StieltjesConstants|Definition:StieltjesConstants]]
+
[[Main_Page|Main Page]]
+
[[Definition:HarmonicNumber|Definition:HarmonicNumber]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\PolylogarithmS''' represents the PolylogarithmS. + +This macro is in the category of real or complex valued functions. + +In math mode, this macro can be called in the following ways: + +:'''\PolylogarithmS{n}{p}''' produces {\displaystyle \PolylogarithmS{n}{p}}
+:'''\PolylogarithmS{n}{p}@{z}''' produces {\displaystyle \PolylogarithmS{n}{p}@{z}}
+ +These are defined by + + + +{\displaystyle + +\PolylogarithmS{n}{p}@{z} + +} + +
+ +== Symbols List == + +[p}$ {\displaystyle S_{n}] : Nielsen generalized polylogarithm function : [p}$ p}$] [http://drmf.wmflabs.org/wiki/Definition:PolylogarithmS http://drmf.wmflabs.org/wiki/Definition:PolylogarithmS] +
+
<< [[Definition:StieltjesConstants|Definition:StieltjesConstants]]
+
[[Main_Page|Main Page]]
+
[[Definition:HarmonicNumber|Definition:HarmonicNumber]] >>
+
+drmf_eof +drmf_bof +'''Definition:HarmonicNumber''' +
+
<< [[Definition:PolylogarithmS|Definition:PolylogarithmS]]
+
[[Main_Page|Main Page]]
+
[[Definition:LucasL|Definition:LucasL]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\HarmonicNumber''' represents the HarmonicNumber. + +This macro is in the category of real or complex valued functions. + +In math mode, this macro can be called in the following ways: + +:'''\HarmonicNumber{n}''' produces {\displaystyle \HarmonicNumber{n}}
+:'''\HarmonicNumber[r]{n}''' produces {\displaystyle \HarmonicNumber[r]{n}}
+:'''\HarmonicNumber{n}''' produces {\displaystyle \HarmonicNumber{n}}
+ +These are defined by + + + +{\displaystyle + +\HarmonicNumber[r]{n} + +} + +
+ +== Symbols List == + +[http://dlmf.nist.gov/25.11#E33 {\displaystyle H_{n}}] : Harmonic number : [http://dlmf.nist.gov/25.11#E33 http://dlmf.nist.gov/25.11#E33] +
+
<< [[Definition:PolylogarithmS|Definition:PolylogarithmS]]
+
[[Main_Page|Main Page]]
+
[[Definition:LucasL|Definition:LucasL]] >>
+
+drmf_eof +drmf_bof +'''Definition:LucasL''' +
+
<< [[Definition:HarmonicNumber|Definition:HarmonicNumber]]
+
[[Main_Page|Main Page]]
+
[[Definition:HurwitzLerchPhi|Definition:HurwitzLerchPhi]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\LucasL''' represents the LucasL. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\LucasL{n}''' produces {\displaystyle \LucasL{n}}
+:'''\LucasL{n}@{x}''' produces {\displaystyle \LucasL{n}@{x}}
+ +These are defined by + + + +{\displaystyle + +\LucasL{n}@{x} + +} + +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:LucasL {\displaystyle L_{n}}] : Lucas polynomial : [http://drmf.wmflabs.org/wiki/Definition:LucasL http://drmf.wmflabs.org/wiki/Definition:LucasL] +
+
<< [[Definition:HarmonicNumber|Definition:HarmonicNumber]]
+
[[Main_Page|Main Page]]
+
[[Definition:HurwitzLerchPhi|Definition:HurwitzLerchPhi]] >>
+
+drmf_eof +drmf_bof +'''Definition:HurwitzLerchPhi''' +
+
<< [[Definition:LucasL|Definition:LucasL]]
+
[[Main_Page|Main Page]]
+
[[Definition:GoldenRatio|Definition:GoldenRatio]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\HurwitzLerchPhi''' represents the HurwitzLerchPhi. + +This macro is in the category of real or complex valued functions. + +In math mode, this macro can be called in the following ways: + +:'''\HurwitzLerchPhi''' produces {\displaystyle \HurwitzLerchPhi}
+:'''\HurwitzLerchPhi@{z}{s}{a}''' produces {\displaystyle \HurwitzLerchPhi@{z}{s}{a}}
+ +These are defined by + + + +{\displaystyle + +\HurwitzLerchPhi@{z}{s}{a} + +} + +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:HurwitzLerchPhi {\displaystyle \Phi}] : Hurwitz-Lerch's transcendent : [http://drmf.wmflabs.org/wiki/Definition:HurwitzLerchPhi http://drmf.wmflabs.org/wiki/Definition:HurwitzLerchPhi] +
+
<< [[Definition:LucasL|Definition:LucasL]]
+
[[Main_Page|Main Page]]
+
[[Definition:GoldenRatio|Definition:GoldenRatio]] >>
+
+drmf_eof +drmf_bof +'''Definition:GoldenRatio''' +
+
<< [[Definition:HurwitzLerchPhi|Definition:HurwitzLerchPhi]]
+
[[Main_Page|Main Page]]
+
[[Definition:WhitPsi|Definition:WhitPsi]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\GoldenRatio''' represents the GoldenRatio. + +This macro is in the category of constants. + +In math mode, this macro can be called in the following way: + +:'''\GoldenRatio''' produces {\displaystyle \GoldenRatio}
+ +These are defined by + + + +{\displaystyle + +\GoldenRatio + +} + +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:GoldenRatio {\displaystyle \phi}] : The golden ratio : [http://drmf.wmflabs.org/wiki/Definition:GoldenRatio http://drmf.wmflabs.org/wiki/Definition:GoldenRatio] +
+
<< [[Definition:HurwitzLerchPhi|Definition:HurwitzLerchPhi]]
+
[[Main_Page|Main Page]]
+
[[Definition:WhitPsi|Definition:WhitPsi]] >>
+
+drmf_eof +drmf_bof +'''Definition:WhitPsi''' +
+
<< [[Definition:GoldenRatio|Definition:GoldenRatio]]
+
[[Main_Page|Main Page]]
+
[[Definition:GompertzConstant|Definition:GompertzConstant]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\WhitPsi''' represents the WhitPsi. + +This macro is in the category of real or complex valued functions. + +In math mode, this macro can be called in the following ways: + +:'''\WhitPsi{\alpha}{\beta}''' produces {\displaystyle \WhitPsi{\alpha}{\beta}}
+:'''\WhitPsi{\alpha}{\beta}@{z}''' produces {\displaystyle \WhitPsi{\alpha}{\beta}@{z}}
+ +These are defined by + + + +{\displaystyle + +\WhitPsi{\alpha}{\beta}@{z} + +} + +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:WhitPsi {\displaystyle \Psi_{\alpha,\beta}}] : Whittaker function {\displaystyle \Psi_{\alpha;\beta}} : [http://drmf.wmflabs.org/wiki/Definition:WhitPsi http://drmf.wmflabs.org/wiki/Definition:WhitPsi] +
+
<< [[Definition:GoldenRatio|Definition:GoldenRatio]]
+
[[Main_Page|Main Page]]
+
[[Definition:GompertzConstant|Definition:GompertzConstant]] >>
+
+drmf_eof +drmf_bof +'''Definition:GompertzConstant''' +
+
<< [[Definition:WhitPsi|Definition:WhitPsi]]
+
[[Main_Page|Main Page]]
+
[[Definition:rabbitConstant|Definition:rabbitConstant]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\GompertzConstant''' represents the GompertzConstant. + +This macro is in the category of constants. + +In math mode, this macro can be called in the following way: + +:'''\GompertzConstant''' produces {\displaystyle \GompertzConstant}
+ +These are defined by + + + +{\displaystyle + +\GompertzConstant + +} + +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:GompertzConstant {\displaystyle G}] : Gompertz's constant : [http://drmf.wmflabs.org/wiki/Definition:GompertzConstant http://drmf.wmflabs.org/wiki/Definition:GompertzConstant] +
+
<< [[Definition:WhitPsi|Definition:WhitPsi]]
+
[[Main_Page|Main Page]]
+
[[Definition:rabbitConstant|Definition:rabbitConstant]] >>
+
+drmf_eof +drmf_bof +'''Definition:rabbitConstant''' +
+
<< [[Definition:GompertzConstant|Definition:GompertzConstant]]
+
[[Main_Page|Main Page]]
+
[[Definition:FibonacciNumber|Definition:FibonacciNumber]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\rabbitConstant''' represents the rabbitConstant. + +This macro is in the category of constants. + +In math mode, this macro can be called in the following way: + +:'''\rabbitConstant''' produces {\displaystyle \rabbitConstant}
+ +These are defined by + + + +{\displaystyle + +\rabbitConstant + +} + +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:RabbitConstant {\displaystyle \rho}] : rabbit constant : [http://drmf.wmflabs.org/wiki/Definition:RabbitConstant http://drmf.wmflabs.org/wiki/Definition:RabbitConstant] +
+
<< [[Definition:GompertzConstant|Definition:GompertzConstant]]
+
[[Main_Page|Main Page]]
+
[[Definition:FibonacciNumber|Definition:FibonacciNumber]] >>
+
+drmf_eof +drmf_bof +'''Definition:FibonacciNumber''' +
+
<< [[Definition:rabbitConstant|Definition:rabbitConstant]]
+
[[Main_Page|Main Page]]
+
[[Definition:Fibonacci|Definition:Fibonacci]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\FibonacciNumber''' represents the FibonacciNumber. + +This macro is in the category of integer valued functions. + +In math mode, this macro can be called in the following way: + +:'''\FibonacciNumber{n}''' produces {\displaystyle \FibonacciNumber{n}}
+ +These are defined by + + + +{\displaystyle + +\FibonacciNumber{n} + +} + +
+ +== Symbols List == + +[http://dlmf.nist.gov/26.11#E5 {\displaystyle F_{n}}] : Fibonacci number : [http://dlmf.nist.gov/26.11#E5 http://dlmf.nist.gov/26.11#E5] +
+
<< [[Definition:rabbitConstant|Definition:rabbitConstant]]
+
[[Main_Page|Main Page]]
+
[[Definition:Fibonacci|Definition:Fibonacci]] >>
+
+drmf_eof +drmf_bof +'''Definition:Fibonacci''' +
+
<< [[Definition:FibonacciNumber|Definition:FibonacciNumber]]
+
[[Main_Page|Main Page]]
+
[[Definition:AffqKrawtchouk|Definition:AffqKrawtchouk]] >>
+
+ +The LaTeX DLMF and DRMF macro '''\Fibonacci''' represents the Fibonacci. + +This macro is in the category of integer valued functions. + +In math mode, this macro can be called in the following way: + +:'''\Fibonacci{n}''' produces {\displaystyle \Fibonacci{n}}
+ +These are defined by + + + +{\displaystyle + +\Fibonacci{n}@{x} + +} + +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:Fibonacci {\displaystyle F_{n}{(x)}}] : Fibonacci polynomial : [http://drmf.wmflabs.org/wiki/Definition:Fibonacci http://drmf.wmflabs.org/wiki/Definition:Fibonacci] +
+
<< [[Definition:FibonacciNumber|Definition:FibonacciNumber]]
+
[[Main_Page|Main Page]]
+
[[Definition:AffqKrawtchouk|Definition:AffqKrawtchouk]] >>
+
+drmf_eof diff --git a/main_page/src/KLS.py b/main_page/src/KLS.py new file mode 100644 index 0000000..32bf360 --- /dev/null +++ b/main_page/src/KLS.py @@ -0,0 +1,19 @@ +__author__ = "Azeem Mohammed" +__status__ = "Development" + +import os + + +# TODO: Actually make the if statements do what they're supposed to. + +if raw_input("Update Main Page? (y or n) ") != "n": + print "List" + os.system("python src/list.py") + print "Symbols List" + os.system("python src/symbols_list.py") + print "Headers" + os.system("python src/headers.py") + print "List Ways" + os.system("python src/mod_main.py") + +print "Done" diff --git a/main_page/src/__init__.py b/main_page/src/__init__.py new file mode 100644 index 0000000..e69de29 diff --git a/main_page/src/headers.py b/main_page/src/headers.py new file mode 100644 index 0000000..f760e99 --- /dev/null +++ b/main_page/src/headers.py @@ -0,0 +1,64 @@ +__author__ = "Azeem Mohammed" +__status__ = "Development" + +import copy + +with open("main_page.mmd", "r") as main_page: + lines = main_page.read() + main_lines = copy.copy(lines) + +start = lines.find("* [[Definition:", lines.find("= Definition Pages =")) +end = lines.find("" in text: + search_string = ">>
\n
" + text = text[text.find(search_string) + len(search_string):] + + if "
" in text: + text = text[:text.find("
")] + + prev = pages[(count - 1) % len(pages)] + cur = "Main Page" + next = pages[(count + 1) % len(pages)] + + delimiter = "\n
<< [[" + prev.replace(" ", "_") + "|" + prev + "]]
" + \ + "\n
[[" + cur.replace(" ", "_") + "|" + cur + "]]
" + \ + "\n
[[" + next.replace(" ", "_") + "|" + next + "]] >>
" + \ + "\n
" + + header = "\n
" + delimiter + footer = "
" + delimiter + + main_page.write("drmf_bof\n" + title + header + text + footer + "\ndrmf_eof\n") + + count += 1 diff --git a/main_page/src/list.py b/main_page/src/list.py new file mode 100644 index 0000000..b65aeb8 --- /dev/null +++ b/main_page/src/list.py @@ -0,0 +1,59 @@ +__author__ = "Azeem Mohammed" +__status__ = "Development" + +import copy +import time + + +def compare_string(a, b): + """Returns whether string a is "less than" string b, up to the pipe symbol.""" + x = a[:a.find("|")] + y = b[:b.find("|")] + + # pipe missing in a string + if not x or not y: + return True + + return x < y + + +def sum_ordinal(string): + """Returns the sum of the ordinal values of all characters in string.""" + return sum((ord(ch) for ch in string)) + + +def sort(strings): + return sorted(copy.deepcopy(strings), cmp=compare_string) + + +def main(): + localtime = time.asctime(time.localtime(time.time())).replace(" ", "_") + with open("main_page.mmd") as main_page: + with open("backups/main_page.mmd" + str(localtime), "w") as main_page_bak: + main_page_bak.write(main_page.read()) + + with open("main_page.mmd") as main_page: + main_text = main_page.read() + + main_lines = main_text.split("\n") + + text = "" + + for line in main_lines: + if "'''Definition:" in line: + definition = line[line.find("'''Definition:") + 3:line.find("'''", line.find("'''Definition:") + 4)] + element = "* [[" + definition + "|" + definition[len("Definition:"):] + "]]" + text += element + "\n" + + s = main_text.find("== Definition Pages ==") + e = main_text.find("= Copyright =") + + text = main_text[:s] + \ + "== Definition Pages ==\n\n
\n" + \ + text + "
\n\n" + main_text[e:] + + with open("main_page.mmd", "w") as main_page: + main_page.write(text) + +if __name__ == '__main__': + main() \ No newline at end of file diff --git a/main_page/src/main.py b/main_page/src/main.py new file mode 100644 index 0000000..19dd4c3 --- /dev/null +++ b/main_page/src/main.py @@ -0,0 +1,8 @@ +__author__ = "Azeem Mohammed" +__status__ = "Development" +__credits__ = ["Azeem Mohammed", "Joon Bang"] + +import os + +if raw_input("Update main page? (y/n): ") != "n": + pass diff --git a/main_page/src/mod_main.py b/main_page/src/mod_main.py new file mode 100644 index 0000000..fab8c86 --- /dev/null +++ b/main_page/src/mod_main.py @@ -0,0 +1,125 @@ +__author__ = "Azeem Mohammed" +__status__ = "Development" + +import copy +import csv + +DEF_STRING = "'''Definition:" + + +def at_sort(strings): + """Sorts by number of @ symbols in string.""" + + return sorted(copy.copy(strings), key=lambda ch: ch.count("@")) + + +def remove_brackets_text(string): + left_brackets = 0 + text = "" + + for ch in string: + if ch == "[": + left_brackets += 1 + elif ch == "]": + left_brackets -= 1 + elif not left_brackets: + text += ch + + return text + + +def find_all_pos(info): + # TODO: what is info? what does pos mean??? + + original = info[0] + key_string = info[3] + + before, key_string = key_string.split("|", 1) + keys = key_string.split(":") + ret = [] + + for key in keys: + if key in ["F", "FO", "O", "O1"]: + if "@" in original: + ret.append(original[:original.find("@")]) + if key == "O": + ret.append(original) + else: + ret.append(original) + elif key == "FnO": # remove optional parameters + if original.find("@") != -1: + ret.append(remove_brackets_text(original[0:original.find("@")])) + else: + ret.append(remove_brackets_text(original)) + elif key in ["P", "PO"]: + ret.append(original.replace("@@", "@")) + elif key == "PnO": # remove optional parameters + ret.append(remove_brackets_text(original.replace("@@", "@"))) + elif key in ["nP", "nPO", "PS", "O2"]: + ret.append(original) + elif key == "nPnO": # remove optional paramters + ret.append(remove_brackets_text(original)) + + if "fo" in key: + ret.append(original.replace("@@@", "@" * int(key[2]))) + + return at_sort(ret), before + +with open("main_page.mmd") as main_page: + lines = main_page.read() + +lines = lines.replace("The LaTeX DLMF macro '''\\", "The LaTeX DLMF and DRMF macro '''\\") + +with open("Categories.txt") as categories: + cats = [category.strip() for category in categories.readlines()] + +text = "" +i = 0 + +while True: + n = lines.find(DEF_STRING, i) + if n == -1: + break + else: + macroN = lines[n + len(DEF_STRING):lines.find("'''", n + len(DEF_STRING))] + r = lines.find("'''\\", n) + q = lines.find("'''", r + len("'''\\")) + + if lines[r:q].find("{") != -1: + lines = lines[:r] + "'''\\" + macroN + lines[q:] + + if lines.find("\nThis macro is in the category of", n) < lines.find(DEF_STRING, n + 15) \ + and "\nThis macro is in the category of" in lines[n:]: + p = lines.find("\nThis macro is in the category of", n) + else: + p = lines.find("\n", lines.find(".", n)) + + text += lines[i:p].rstrip() + "\n\n" + + count = 0 + list_calls = [] + + glossary = csv.reader(open('new.Glossary.csv', 'rb'), delimiter=',', quotechar='\"') + for entry in glossary: + if not entry[0].find("\\" + macroN) and (len(entry[0]) == len(macroN) + 1 or not entry[0][len(macroN) + 1].isalpha()): + q, s = find_all_pos(entry) + list_calls += q + count += len(q) + + for cat in cats: + if s + " -" in cat: + category = "This macro is in the category of" + cat[cat.find("-") + 2:] + "\n\n" + break + + new = category + "In math mode, this macro can be called in the following way" + "s" * (count > 1) + ":\n\n" + + for q in range(0, len(list_calls)): + c = list_calls[q] + new += ":'''" + c + "'''" + " produces {\\displaystyle " + c + "}
\n" + + # Now add the multiple ways \macroname{n}@... produces \macroname{n}@... + text += new + "\n" + i = lines.find("These are defined by", p) + +with open("main_page.mmd", "w") as main_page: + main_page.write(text + lines[i:]) diff --git a/main_page/src/symbols_list.py b/main_page/src/symbols_list.py new file mode 100644 index 0000000..82d8eb9 --- /dev/null +++ b/main_page/src/symbols_list.py @@ -0,0 +1,285 @@ +__author__ = "Azeem Mohammed" +__status__ = "Development" + +import csv + + +def getSym(line): # Gets all symbols on a line for symbols list + line = line.replace("\n", " ") + symList = [] + if line == "": + return symList + symbol = "" + symFlag = False + argFlag = False + cC = 0 + for i in range(0, len(line)): + c = line[i] + if symFlag: + if c == "{" or c == "[": + cC += 1 + argFlag = True + if c != "}" and c != "]": + if argFlag or c.isalpha(): + symbol += c + else: + symFlag = False + argFlag = False + symList.append(symbol) + symList += (getSym(symbol)) + symbol = "" + else: + cC -= 1 + symbol += c + if i + 1 == len(line): + p = "" + else: + p = line[i + 1] + if cC == 0 and p != "{" and p != "[" and p != "@": + symFlag = False + symList.append(symbol) + symList += (getSym(symbol)) + argFlag = False + symbol = "" + + elif c == "\\": + symFlag = True + + symList.append(symbol) + symList += getSym(symbol) + return symList + +mainPage = open("main_page.mmd", "r") +mainLines = mainPage.read() +mainLines = mainLines[mainLines.find("drmf_bof"):mainLines.rfind("drmf_eof") + len("drmf_eof\n")] +mainPage.close() +main = open("main_page.mmd", "w") +mainPages = mainLines.split("drmf_eof") + +if mainPages[-1].strip() == "": + mainPages = mainPages[0:len(mainPages) - 1] + +print len(mainPages) + +glossary = open('new.Glossary.csv', 'rb') +gCSV = csv.reader(glossary, delimiter=',', quotechar='\"') +lGlos = glossary.readlines() +wiki = "" + +for g in mainPages: + if not "\'\'\'Definition:" in g: + h = g.replace("drmf_eof", "") + h = g.replace("drmf_bof", "") + toWrite = "drmf_bof" + h + "drmf_eof\n" + main.write(toWrite) + else: + h = g.replace("drmf_eof", "") + h = g.replace("drmf_bof", "") + h = h.strip("\n") + sflag1 = False + sflag2 = False + if h.find("== Symbols List ==") != -1: + toWrite = h[0:h.find("== Symbols List ==")] + h = toWrite + else: + sflag1 = True + if h.find("drmf_foot") == -1: + toWrite = h + sflag2 = True + else: + toWrite = h[0:h.find("
len(symbol) and (Q[len(symbol)] == "{" or Q[len(symbol)] == "["): + ap = "" + for o in range(len(symbol), len(Q)): + if Q[o] == "{" or z == "[": + argFlag = True + elif Q[o] == "}" or z == "]": + argFlag = False + listArgs.append(ap) + ap = "" + else: + ap += Q[o] + websiteF = "" + web1 = G[5] + for t in range(5, len(G)): + if G[t] != "": + websiteF = websiteF + " [" + G[t] + " " + G[t] + "]" + p1 = G[4].strip("$") + p1 = "{\\displaystyle " + p1 + "}" + new1 = "" + new2 = "" + pause = False + mathF = True + p2 = G[1] + for k in range(0, len(p2)): + if p2[k] == "$": + if mathF: + new2 += "{\\displaystyle " + else: + new2 += "}" + mathF = not mathF + else: + new2 += p2[k] + p2 = new2 + finSym.append(web1 + " " + p1 + "] : " + p2 + " :" + websiteF) + break + if not gFlag: + del newSym[s] + + gFlag = True + for y in finSym: + if gFlag: + gFlag = False + toWrite += ("[" + y) + else: + toWrite += ("
\n[" + y) + + toWrite += "\n
\ndrmf_eof\n\n" + if sflag2: + toWrite += "\n" + main.write(toWrite) diff --git a/main_page/symbolsList.py~ b/main_page/symbolsList.py~ new file mode 100644 index 0000000..b0d1d72 --- /dev/null +++ b/main_page/symbolsList.py~ @@ -0,0 +1,278 @@ +def getSym(line): #Gets all symbols on a line for symbols list + symList=[] + symbol="" + symFlag=False + argFlag=False + cC=0 + for i in range(0,len(line)): + #if "Laguerre[-\\frac{1}{2}]{n}@{x^2}" in line: + #print symbol + c=line[i] + if symFlag: + if c=="{" or c=="[": + cC+=1 + argFlag=True + if c!="}" and c!="]": + if argFlag or c.isalpha(): + symbol+=c + else: + symFlag=False + argFlag=False + symList.append(symbol) + symList+=(getSym(symbol)) + symbol="" + else: + cC-=1 + symbol+=c + if i+1==len(line): + p="" + else: + p=line[i+1] + if cC==0 and p!="{" and p!="[" and p!="@": + symFlag=False + #if "Laguerre[-\\frac{1}{2}]{n}@{x^2}" in line: + #print(symbol,p) + symList.append(symbol) + symList+=(getSym(symbol)) + argFlag=False + symbol="" + + elif c=="\\": + symFlag=True + + #if "Laguerre[-\\frac{1}{2}]{n}@{x^2}" in line: + #print symbol + return(symList) +import os +files_in_dir=os.listdire(".") +for g in files_in_dir: + if not ".py" in g and not ".new" in g: + f=open(g,"r") + w=open(g+".new","w") + t=f.read() + w.write(t) + t=t.replace("\n","") + symbols=getSym(t) + if True: + wiki.write("
\n== Symbols List ==\n
\n") + newSym=[] + #if "09.12:21" in label: + #print(symbols) + for x in symbols: + flagA=True + #if x not in newSym: + cN=0 + cC=0 + ArgCx=0 + for z in x: + if z.isalpha(): + cN+=1 + else: + if z=="{" or z=="[": + cC+=1 + if z=="}" or z=="]": + cC-=1 + if cC==0: + ArgCx+=1 + noA=x[:cN] + for y in newSym: + cN=0 + cC=0 + ArgC=0 + for z in y: + if z.isalpha(): + cN+=1 + else: + if z=="{" or z=="[": + cC+=1 + if z=="}" or z=="]": + cC-=1 + if cC==0: + ArgC+=1 + + + if y[:cN]==noA and ArgC==ArgCx: + flagA=False + #print(x,y) + break + if flagA: + newSym.append(x) + newSym.reverse() + #if "9.12:20" in label: + #print newSym + symF=False + finSym=[] + for s in range(len(newSym)-1,-1,-1): + #print(newSym[s]) + symbolPar="\\"+newSym[s] + ArgCx=0 + parCx=0 + parFlag=False + cC=0 + for z in symbolPar: + if z=="@": + parFlag=True + elif z.isalpha(): + cN+=1 + else: + if z=="{" or z=="[": + cC+=1 + if z=="}" or z=="]": + cC-=1 + if cC==0: + if parFlag: + parCx+=1 + else: + ArgCx+=1 + + if symbolPar.find("{")!=-1 or symbolPar.find("[")!=-1: + if symbolPar.find("[")==-1: + symbol=symbolPar[0:symbolPar.find("{")] + elif symbolPar.find("{")==-1 or symbolPar.find("[")len(symbol) and (G[0][len(symbol)]=="{" or G[0][len(symbol)]=="["): + ap="" + for o in range(len(symbol),len(G[0])): + if G[0][o]=="{" or z=="[": + argFlag=True + elif G[0][o]=="}" or z=="]": + argFlag=False + listArgs.append(ap) + ap="" + else: + ap+=G[0][o] + '''websiteU=g[g.find("http://"):].strip("\n") + k=0 + websites=[] + for r in range(0,len(websiteU)): + if websiteU[r]==",": + if websiteU[k:r].find("http://")!=-1: + websites.append(websiteU[k:r+1].strip(" ")) + k=r + + + if websiteU[k:r].find("http://")!=-1: + websites.append(websiteU[k:r+1].strip(" ")) + websiteF="" + for d in websites: + websiteF=websiteF+" ["+d+" "+d+"]"''' + #websiteF=G[4].strip("\n") + websiteF="" + web1=G[5] + for t in range(5,len(G)): + if G[t]!="": + websiteF=websiteF+" ["+G[t]+" "+G[t]+"]" + p1="{\\displaystyle "+G[0]+"}" + if p1.find("@")!=-1: + p1=p1[:p1.find("@")] + #if checkFlag: + #print(p1) + new1="" + new2="" + pause=False + mathF=True + '''for k in range(0,len(p1)): + if p1[k]=="$": + if mathF: + new1+="{\\displaystyle " + else: + new1+="}" + mathF=not mathF + + elif p1[k]=="#" and p1[k+1].isdigit(): + pause=True + elif pause: + num=int(p1[k]) + #letter=chr(num+96) + letter=listArgs[num-1] + new1+=letter + pause=False + + else: + new1+=p1[k]''' + p2=G[1] + for k in range(0,len(p2)): + if p2[k]=="$": + if mathF: + new2+="{\\displaystyle " + else: + new2+="}" + mathF=not mathF + else: + new2+=p2[k] + #p1=new1 + p2=new2 + finSym.append(web1+" "+p1+"] : "+p2+" :"+websiteF) + break + #gFlag=True + #if not symF: + # symF=True + # wiki.write("[") + #else: + # wiki.write("
\n") + # wiki.write("[") + #wiki.write(g[g.find("http://"):].strip("\n")) + #wiki.write(" ") + #wiki.write(getEq(g[g.find("{$"):]).strip("\n").replace("\\\\","\\")) + #wiki.write("] : ") + #wiki.write(g[g.find(" {")+2:g.find("} ",g.find(" {")+1)]) + #wiki.write(" : [") + #wiki.write(g[g.find("http://"):].strip("\n")) + #wiki.write(" ") + #wiki.write(g[g.find("http://"):].strip("\n")) + #wiki.write("] ")''' + + + + if not gFlag: + del newSym[s] + + gFlag=True + #finSym.reverse() + for y in finSym: + #print(finSym) + if gFlag: + gFlag=False + wiki.write("["+y) + else: + wiki.write("
\n["+y) + + diff --git a/main_page/tex2Wiki.py~ b/main_page/tex2Wiki.py~ new file mode 100644 index 0000000..1cce794 --- /dev/null +++ b/main_page/tex2Wiki.py~ @@ -0,0 +1,966 @@ +#Version 11: Split Sections +#Convert tex to wikiText +import csv #imported for using csv format +import sys #imported for getting args +import os #imported for copying file +def isnumber(char): + return char[0] in "0123456789" +def getString(line): #Gets all data within curly braces on a line + #-------Initialization------- + stringWrite="" + getStr=False + pW="" + #---------------------------- + for c in line: + if c=="{" or c=="}": #if there is a curly brace in the line + getStr=not(getStr) #toggle the getStr flag + if c=="}": #no more info needed + break + elif getStr: #if within curly braces + if not(pW==c and c=="-"): #if there is no double dash (makes single dash) + if c!="$": #if not a $ sign + stringWrite+=c #this character is part of the data + else: #replace $ with '' + stringWrite+="\'\'" #add double ' + pW=c #change last character for finding double dashes + return (stringWrite.rstrip('\n').lstrip()) #return the data without newlines and without leading spaces +def getG(line): #gets equation for symbols list + start=True + final="" + for c in line: + if c=="$" and start: + final+="{\\displaystyle " + start=False + elif c=="$": + final+="}" + start=True + else: + final+=c + return(final) +def getEq(line): #Gets all data within constraints,substitutions + #-------Initialization------- + per=1 + stringWrite="" + fEq=False + count=0 + #---------------------------- + for c in line: #read each character + if count>=0 and per!=0: + per+=1 + if c!=" " and c!="%": + per=0 + if c=="{" and per==0: + if count>0: + stringWrite+=c + count+=1 + elif c=="}" and per==0: + count-=1 + if count>0: + stringWrite+=c + elif c=="$" and per==0: #either begin or end equation + fEq=not(fEq)#toggle fEq flag to know if begin or end equation + if fEq:#if begin + stringWrite+="{\\displaystyle " + else:#if end + stringWrite+="}" + elif c=="\n" and per==0:#if newline + stringWrite=stringWrite.strip()#remove all leading and trailing whitespace + + #should above be rstrip?<-------------------------------------------------------------------CHECK THIS + + stringWrite+=c#add the newline character + per+=1 #watch for % signs + elif count>0 and per==0: #not special character + stringWrite+=c #write the character + + return (stringWrite.rstrip().lstrip()) +def getEqP(line,Flag): #Gets all data within proofs + if Flag: + a=1 + else: + a=0 + per=1 + stringWrite="" + fEq=False + count=0 + length=0 + for c in line: + if fEq and c!=" " and c!="$" and c!="{" and c!="}" and not c.isalpha(): + length+=1 + if count>=0 and per!=0: + per+=1 + if c!=" " and c!="%": + #if c!="%": + per=0 + if c=="{" and per==0: + if count>0: + stringWrite+=c + count+=1 + elif c=="}" and per==0: + count-=1 + if count>0: + stringWrite+=c + elif c=="$" and per==0: + fEq=not(fEq) + if fEq: + stringWrite+="{\\displaystyle " + else: + if length<10: + stringWrite+="}" + else: + stringWrite+="}"+"
" + length=0 + elif c=="\n" and per==0: + stringWrite=stringWrite.strip() + stringWrite+=c + per+=1 + elif count>0 and per==0: + stringWrite+=c + + return (stringWrite.lstrip()) +def getSym(line): #Gets all symbols on a line for symbols list + symList=[] + symbol="" + symFlag=False + argFlag=False + cC=0 + for i in range(0,len(line)-1): + if "[" in line: + print symbol + c=line[i] + if symFlag: + if c=="{" or c=="[": + cC+=1 + argFlag=True + if c!="}" and c!="]": + if argFlag or c.isalpha(): + symbol+=c + else: + symFlag=False + argFlag=False + symList.append(symbol) + symList+=(getSym(symbol)) + symbol="" + else: + cC-=1 + symbol+=c + if cC==0 and line[i+1]!="{" and line[i+1]!="[" and line[i+1]!="@": + symFlag=False + symList.append(symbol) + symList+=(getSym(symbol)) + argFlag=False + symbol="" + + elif c=="\\": + symFlag=True + + return(symList) + '''symList=[] + getStr=False + getPar=False + morePar=False + curStr="" + deeper=False + cC=0 + for c in line: + if c=="\\" and not (getStr): + getStr=True + elif getStr: + if morePar and c!="{": + getPar=False + getStr=False + symList.append(curStr) + curStr="" + morePar=False + else: + morePar=False + if c=="\\": + deeper=True + if c.isalpha(): + curStr=curStr+c + elif not getPar and c!="{" and c!="}": + getStr=False + symList.append(curStr) + curStr="" + elif c=="{": + cC+=1 + getPar=True + curStr=curStr+c + elif c=="}": + cC-=1 + #getPar=False + if cC>0: + curStr=curStr+c + else: + morePar=True + curStr=curStr+c + getPar=False + getStr=False + symList.append(curStr) + curStr="" + #symList.append(curStr) + #curStr="" + else: + curStr=curStr+c + #print(line,symList) + dSym=[] + if deeper: + for sym in symList: + d=getSym(sym) + dSym=dSym+(d) + #print(symList,dSym,line) + symList=symList+dSym + return (symList)''' +def modLabel(label): + isNumer=False + newlabel="" + num="" + for i in range(0,len(label)): + if isNumer and not isnumber(label[i]): + if len(num)>1: + newlabel+=num + isNumer=False + num="" + else: + newlabel+="0"+str(num) + num="" + isNumer=False + if isnumber(label[i]): + isNumer=True + num+=str(label[i]) + else: + isNumer=False + newlabel+=label[i] + if len(num)>1: + newlabel+=num + elif len(num)==1: + newlabel+="0"+num + return(newlabel) +#try: +for jsahlfkjsd in range(0,1): + tex=open(sys.argv[1],'r') + wiki=open(sys.argv[2],'w') + main=open("OrthogonalPolynomials.mmd","r") + mainText=main.read() + mainPrepend="" + mainWrite=open("OrthogonalPolynomials.mmd.new","w") + tester=open("testData.txt",'w') + #glossary=open('Glossary', 'r') + glossary=open('new.Glossary.csv', 'rb') + gCSV=csv.reader(glossary, delimiter=',', quotechar='\"') + #lLinks=open('BruceLabelLinks', 'r') + lGlos=glossary.readlines() + #lLink=lLinks.readlines() + math=False + constraint=False + substitution=False + symbols=[] + lines=tex.readlines() + refLines=[] + sections=[] + labels=[] + eqs=[] + refEqs=[] + parse=False + head=False + #chapRef=[("GA",open("GA.tex",'r')),("ZE",open("ZE.3.tex",'r'))] + refLabels=[] + '''for c in chapRef: + refLines=refLines+(c[1].readlines()) + c[1].close()''' + for i in range(0,len(refLines)): + line=refLines[i] + if "\\begin{equation}" in line: + sLabel=line.find("\\formula{")+9 + eLabel=line.find("}",sLabel) + label=modLabel(line[sLabel:eLabel]) + '''for l in lLink: + if l.find(label)!=-1 and l[len(label)+1]=="=": + rlabel=l[l.find("=>")+3:l.find("\\n")] + rlabel=rlabel.replace("/","") + rlabel=rlabel.replace("#",":") + break''' + refLabels.append(label) + refEqs.append("") + math=True + elif math: + refEqs[len(refEqs)-1]+=line + if "\\end{equation}" in refLines[i+1] or "\\constraint" in refLines[i+1] or "\\substitution" in refLines[i+1] or "\\drmfn" in refLines[i+1]: + math=False + + for i in range(0,len(lines)): + line=lines[i] + if "\\begin{document}" in line: + #wiki.write("drmf_bof\n") + parse=True + elif "\\end{document}" in line and parse: + #wiki.write("
\n") + mainPrepend+="
\n" + mainText=mainPrepend+mainText + mainText=mainText.replace("drmf_bof\n","") + mainText=mainText.replace("drmf_eof\n","") + mainText=mainText.replace("\'\'\'Orthogonal Polynomials\'\'\'\n","") + mainText=mainText[0:mainText.rfind("== Sections ")] + mainText="drmf_bof\n\'\'\'Orthogonal Polynomials\'\'\'\n"+mainText+"\ndrmf_eof" + mainWrite.write(mainText) + mainWrite.close() + main.close() + os.system("cp -f OrthogonalPolynomials.mmd.new OrthogonalPolynomials.mmd") + #wiki.write("\ndrmf_eof\n") + parse=False + elif "\\title" in line and parse: + stringWrite="\'\'\'" + stringWrite+=getString(line)+"\'\'\'\n" + labels.append("Orthogonal Polynomials") + sections.append(["Orthogonal Polynomials",0]) + #wiki.write(stringWrite) + #mainPrepend+=stringWrite + mainPrepend+=("\n== Sections in "+getString(line)+" ==\n\n
\n") + elif "\\part" in line: + if getString(line)=="BOF": + parse=False + elif getString(line)=="EOF": + parse=True + elif parse: + mainPrepend+=("\n
\n= "+getString(line)+" =\n") + head=True + elif "\\section" in line: + mainPrepend+=("* [["+getString(line)+"|"+getString(line)+"]]\n") + sections.append([getString(line)]) + + secCounter=0 + eqCounter=0 + for i in range(0,len(lines)): + line=lines[i] + #line.replace("\\begin{equation}","$") + #line.replace("\\end{equation},"$") + '''if "\\begin{document}" in line: + wiki.write("drmf_bof\n") + parse=True + elif "\\end{document}" in line and parse: + wiki.write("\ndrmf_eof\n") + parse=False + elif "\\title" in line and parse: + stringWrite="\'\'\'" + stringWrite+=getString(line)+"\'\'\'\n" + labels.append("Orthogonal Polynomials") + wiki.write(stringWrite\n) + elif "\\part" in line: + if getString(line)=="BOF": + parse=False + elif getString(line)=="EOF": + parse=True + elif parse: + wiki.write("\n
\n= "+getString(line)+" =\n") + head=True + ''' + if "\\section" in line: + parse=True + secCounter+=1 + wiki.write("drmf_bof\n") + wiki.write("\'\'\'"+getString(line)+"\'\'\'\n") + wiki.write("
\n") + wiki.write("
<< [["+sections[secCounter-1][0]+"|"+sections[secCounter-1][0]+"]]
\n") + wiki.write("
[["+sections[secCounter][0]+"|"+sections[secCounter][0]+"]]
\n") + wiki.write("
[["+sections[(secCounter+1)%len(sections)][0]+"|"+sections[(secCounter+1)%len(sections)][0]+"]] >>
\n
\n\n") + head=True + wiki.write("== "+getString(line)+" ==\n") + elif ("\\section" in lines[(i+1)%len(lines)] or "\\end{document}" in lines[(i+1)%len(lines)]) and parse: + wiki.write("
\n") + wiki.write("
<< [["+sections[secCounter-1][0]+"|"+sections[secCounter-1][0]+"]]
\n") + wiki.write("
[["+sections[secCounter][0]+"|"+sections[secCounter][0]+"]]
\n") + wiki.write("
[["+sections[(secCounter+1)%len(sections)][0]+"|"+sections[(secCounter+1)%len(sections)][0]+"]] >>
\n
\n\n") + wiki.write("drmf_eof\n") + sections[secCounter].append(eqCounter) + eqCounter=0 + + elif "\\subsection" in line and parse: + wiki.write("\n
\n== "+getString(line)+" ==\n") + head=True + elif "\\paragraph" in line and parse: + wiki.write("\n
\n=== "+getString(line)+" ===\n") + head=True + elif "\\begin{equation}" in line and parse: + # symLine="" + if head: + wiki.write("
\n") + head=False + sLabel=line.find("\\formula{")+9 + eLabel=line.find("}",sLabel) + label=modLabel(line[sLabel:eLabel]) + eqCounter+=1 + '''for l in lLink: + if label==l[0:l.find("=")-1]: + rlabel=l[l.find("=>")+3:l.find("\\n")] + rlabel=rlabel.replace("/","") + rlabel=rlabel.replace("#",":") + rlabel=rlabel.replace("!",":") + break''' + labels.append(label) + eqs.append("") + wiki.write("\n\n") + wiki.write("{\displaystyle \n") + math=True + elif "\\begin{equation}" in line and not parse: + sLabel=line.find("\\formula{")+9 + eLabel=line.find("}",sLabel) + label=modLabel(line[sLabel:eLabel]) + '''for l in lLink: + if l.find(label)!=-1: + rlabel=l[l.find("=>")+3:l.find("\\n")] + rlabel=rlabel.replace("/","") + rlabel=rlabel.replace("#",":") + break''' + labels.append("*"+label) #special marker + eqs.append("") + math=True + elif "\\end{equation}" in line: + + math=False + elif "\\constraint" in line and parse: + constraint=True + math=False + conLine="" + elif "\\substitution" in line and parse: + substitution=True + math=False + subLine="" + #wiki.write("
Substitution(s): "+getEq(line)+"

\n") + elif "\\proof" in line and parse: + math=False + elif "\\drmfn" in line and parse: + math=False + if "\\drmfname" in line and parse: + wiki.write("
This formula has the name: "+getString(line)+"

\n") + elif math and parse: + flagM=True + eqs[len(eqs)-1]+=line + + if "\\end{equation}" in lines[i+1] and not "\\subsection" in lines[i+3] and not "\\section" in lines[i+3] and not "\\part" in lines[i+3]: + u=i + flagM2=False + while flagM: + u+=1 + if "\\begin{equation}"in lines[u] in lines[u]: + flagM=False + if "\\section" in lines[u] or "\\subsection" in lines[i] or "\\part" in lines[u] or "\\end{document}" in lines[u]: + flagM=False + flagM2=True + if not(flagM2): + + wiki.write(line.rstrip("\n")) + wiki.write("\n}
\n") + else: + wiki.write(line.rstrip("\n")) + wiki.write("\n}\n") + elif "\\end{equation}" in lines[i+1]: + wiki.write(line.rstrip("\n")) + wiki.write("\n}\n") + elif "\\constraint" in lines[i+1] or "\\substitution" in lines[i+1] or "\\drmfn" in lines[i+1]: + wiki.write(line.rstrip("\n")) + wiki.write("\n}\n") + else: + wiki.write(line) + elif math and not parse: + eqs[len(eqs)-1]+=line + if "\\end{equation}" in lines[i+1] or "\\constraint" in lines[i+1] or "\\substitution" in lines[i+1] or "\\drmfn" in lines[i+1]: + math=False + if substitution and parse: + subLine=subLine+line + if "\\end{equation}" in lines[i+1] or "\\substitution" in lines[i+1] or "\\constraint" in lines[i+1] or "\\drmfn" in lines[i+1] or "\\proof" in lines[i+1]: + substitution=False + wiki.write("
Substitution(s): "+getEq(subLine)+"

\n") + + if constraint and parse: + conLine=conLine+line + if "\\end{equation}" in lines[i+1] or "\\substitution" in lines[i+1] or "\\constraint" in lines[i+1] or "\\drmfn" in lines[i+1] or "\\proof" in lines[i+1]: + constraint=False + wiki.write("
Constraint(s): "+getEq(conLine)+"

\n") + + eqCounter=0 + endNum=len(labels)-1 + '''for n in range(1,len(labels)): + if n+1!=len(labels) and labels[n+1][0]=="*" and labels[n][0]!="*": + labels[n]=""+labels[n] + endNum=n + elif labels[n][0]=="*": + labels[n]=""+labels[n][1:] + else: + labels[n]=""+labels[n]''' + '''for n in range(0,len(refLabels)): + refLabels[n]=""+refLabels[n]''' + parse=False + constraint=False + substitution=False + note=False + hCon=True + hSub=True + hNote=True + hProof=True + proof=False + comToWrite="" + secCount=-1 + newSec=False + for i in range(0,len(lines)): + line=lines[i] + + if "\\section" in line: + secCount+=1 + newSec=True + eqS=0 + + if "\\begin{equation}" in line: + symLine=line.strip("\n") + eqS+=1 + constraint=False + substitution=False + note=False + comToWrite="" + hCon=True + hSub=True + hNote=True + hProof=True + proof=False + parse=True + symbols=[] + eqCounter+=1 + wiki.write("drmf_bof\n") + label=labels[eqCounter] + wiki.write("\'\'\'"+label+"\'\'\'\n\n") + if eqCounter\n") + if newSec: + wiki.write("
<< [["+sections[secCount][0].replace(" ","_")+"|"+sections[secCount][0]+"]]
\n") + else: + wiki.write("
<< [["+labels[eqCounter-1].replace(" ","_")+"|"+labels[eqCounter-1]+"]]
\n") + wiki.write("
[["+labels[0].replace(" ","_")+"#"+label[8:]+"|formula in "+labels[0]+"]]
\n") + if eqS==sections[secCount][1]: + wiki.write("
[["+sections[(secCount+1)%len(sections)][0].replace(" ","_")+"|"+sections[(secCount+1)%len(sections)][0]+"]] >>
\n") + else: + wiki.write("
[["+labels[(eqCounter+1)%(endNum+1)].replace(" ","_")+"|"+labels[(eqCounter+1)%(endNum+1)]+"]] >>
\n") + wiki.write("
\n\n") + elif eqCounter==endNum: + wiki.write("
\n") + if newSec: + newSec=False + wiki.write("
<< [["+sections[secCount][0].replace(" ","_")+"|"+sections[secCount][0]+"]]
\n") + else: + wiki.write("
<< [["+labels[eqCounter-1].replace(" ","_")+"|"+labels[eqCounter-1]+"]]
\n") + wiki.write("
[["+labels[0].replace(" ","_")+"#"+label[8:]+"|formula in "+labels[0]+"]]
\n") + wiki.write("
[["+labels[(eqCounter+1)%(endNum+1)].replace(" ","_")+"|"+labels[(eqCounter+1)%(endNum+1)]+"]]
\n") + wiki.write("
\n\n") + + wiki.write("
{\displaystyle \n") + math=True + elif "\\end{equation}" in line: + wiki.write(comToWrite) + parse=False + math=False + if hProof: + wiki.write("
\n== Proof ==\n
\nWe ask users to provide proof(s), reference(s) to proof(s), or further clarification on the proof(s) in this space.\n") + + wiki.write("
\n== Symbols List ==\n
\n") + newSym=[] + if "09.12:21" in label: + print(symbols) + for x in symbols: + flagA=True + #if x not in newSym: + cN=0 + cC=0 + ArgCx=0 + for z in x: + if z.isalpha(): + cN+=1 + else: + if z=="{": + cC+=1 + if z=="}": + cC-=1 + if cC==0: + ArgCx+=1 + noA=x[:cN] + for y in newSym: + cN=0 + cC=0 + ArgC=0 + for z in y: + if z.isalpha(): + cN+=1 + else: + if z=="{": + cC+=1 + if z=="}": + cC-=1 + if cC==0: + ArgC+=1 + + + if y[:cN]==noA and ArgC==ArgCx: + flagA=False + #print(x,y) + break + if flagA: + newSym.append(x) + newSym.reverse() + symF=False + finSym=[] + for s in range(len(newSym)-1,-1,-1): + #print(newSym[s]) + symbolPar="\\"+newSym[s] + ArgCx=0 + cC=0 + for z in symbolPar: + if z.isalpha(): + cN+=1 + else: + if z=="{": + cC+=1 + if z=="}": + cC-=1 + if cC==0: + ArgCx+=1 + if symbolPar.find("{")!=-1: + symbol=symbolPar[0:symbolPar.find("{")] + else: + symbol=symbolPar + numPar=ArgCx + gFlag=False + gCSV=csv.reader(open('new.Glossary.csv','rb'),delimiter=',', quotechar='\"') + for G in gCSV: + ArgCx=0 + cC=0 + ind=G[0].find("@") + if ind==-1: + ind=len(G[0])-1 + for z in G[0]: + if z.isalpha(): + cN+=1 + else: + if z=="{": + cC+=1 + if z=="}": + cC-=1 + if cC==0: + ArgCx+=1 + checkFlag=False + if G[0].find(symbol)==0 and (len(G[0])==len(symbol) or not G[0][len(symbol)].isalpha()) and numPar!=0: + checkFlag=True + if G[0].find(symbol)==0 and (len(G[0])==len(symbol) or not G[0][len(symbol)].isalpha()) and ArgCx==numPar: + listArgs=[] + if len(G[0])>len(symbol) and G[0][len(symbol)]=="{": + ap="" + for o in range(len(symbol),len(G[0])): + if G[0][o]=="{": + argFlag=True + elif G[0][o]=="}": + argFlag=False + listArgs.append(ap) + ap="" + else: + ap+=G[0][o] + '''websiteU=g[g.find("http://"):].strip("\n") + k=0 + websites=[] + for r in range(0,len(websiteU)): + if websiteU[r]==",": + if websiteU[k:r].find("http://")!=-1: + websites.append(websiteU[k:r+1].strip(" ")) + k=r + + + if websiteU[k:r].find("http://")!=-1: + websites.append(websiteU[k:r+1].strip(" ")) + websiteF="" + for d in websites: + websiteF=websiteF+" ["+d+" "+d+"]"''' + #websiteF=G[4].strip("\n") + websiteF="" + web1=G[5] + for t in range(5,len(G)): + if G[t]!="": + websiteF=websiteF+" ["+G[t]+" "+G[t]+"]" + p1="{\\displaystyle "+G[0]+"}" + if p1.find("@")!=-1: + p1=p1[:p1.find("@")] + #if checkFlag: + #print(p1) + new1="" + new2="" + pause=False + mathF=True + '''for k in range(0,len(p1)): + if p1[k]=="$": + if mathF: + new1+="{\\displaystyle " + else: + new1+="}" + mathF=not mathF + + elif p1[k]=="#" and p1[k+1].isdigit(): + pause=True + elif pause: + num=int(p1[k]) + #letter=chr(num+96) + letter=listArgs[num-1] + new1+=letter + pause=False + + else: + new1+=p1[k]''' + p2=G[1] + for k in range(0,len(p2)): + if p2[k]=="$": + if mathF: + new2+="{\\displaystyle " + else: + new2+="}" + mathF=not mathF + else: + new2+=p2[k] + #p1=new1 + p2=new2 + finSym.append(web1+" "+p1+"] : "+p2+" :"+websiteF) + break + #gFlag=True + #if not symF: + # symF=True + # wiki.write("[") + #else: + # wiki.write("
\n") + # wiki.write("[") + #wiki.write(g[g.find("http://"):].strip("\n")) + #wiki.write(" ") + #wiki.write(getEq(g[g.find("{$"):]).strip("\n").replace("\\\\","\\")) + #wiki.write("] : ") + #wiki.write(g[g.find(" {")+2:g.find("} ",g.find(" {")+1)]) + #wiki.write(" : [") + #wiki.write(g[g.find("http://"):].strip("\n")) + #wiki.write(" ") + #wiki.write(g[g.find("http://"):].strip("\n")) + #wiki.write("] ")''' + + + + if not gFlag: + del newSym[s] + + gFlag=True + #finSym.reverse() + for y in finSym: + #print(finSym) + if gFlag: + gFlag=False + wiki.write("["+y) + else: + wiki.write("
\n["+y) + + + wiki.write("\n
\n") + + wiki.write("== Bibliography==\n
\n")#should there be a space between bibliography and ==? + r=labels[eqCounter] + q=r.find("KLS:")+4 + p=r.find(":",q) + section=r[q:p] + equation=r[p+1:] + if equation.find(":")!=-1: + equation=equation[0:equation.find(":")] + + wiki.write("[HTTP://DLMF.NIST.GOV/"+section+"#"+equation+" Equation in Section "+section+"] of [[Bibliography#KLS|'''KLS''']].\n
\n
\n")#Where should it link to? + wiki.write("== URL links ==\n
\nWe ask users to provide relevant URL links in this space.\n
\n\n") + if eqCounter
\n") + if newSec: + newSec=False + wiki.write("
<< [["+sections[secCount][0].replace(" ","_")+"|"+sections[secCount][0]+"]]
\n") + else: + wiki.write("
<< [["+labels[eqCounter-1].replace(" ","_")+"|"+labels[eqCounter-1]+"]]
\n") + wiki.write("
[["+labels[0].replace(" ","_")+"#"+label[8:]+"|formula in "+labels[0]+"]]
\n") + if eqS==sections[secCount][1]: + wiki.write("
[["+sections[(secCount+1)%len(sections)][0].replace(" ","_")+"|"+sections[(secCount+1)%len(sections)][0]+"]] >>
\n") + else: + wiki.write("
[["+labels[(eqCounter+1)%(endNum+1)].replace(" ","_")+"|"+labels[(eqCounter+1)%(endNum+1)]+"]] >>
\n") + wiki.write("
\n\ndrmf_eof\n") + else: #FOR EXTRA EQUATIONS + wiki.write("
\n") + wiki.write("
<< [["+labels[endNum-1].replace(" ","_")+"|"+labels[endNum-1]+"]]
\n") + wiki.write("
[["+labels[0].replace(" ","_")+"#"+labels[endNum][8:]+"|formula in "+labels[0]+"]]
\n") + wiki.write("
[["+labels[(0)%endNum].replace(" ","_")+"|"+labels[0%endNum]+"]]
\n") + wiki.write("
\n\ndrmf_eof\n") + + + + elif "\\constraint" in line and parse: + #symbols=symbols+getSym(line) + symLine=line.strip("\n") + if hCon: + comToWrite=comToWrite+"
\n== Constraint(s) ==\n
\n" + hCon=False + constraint=True + math=False + conLine="" + #wiki.write("
"+getEq(line)+"

\n") + elif "\\substitution" in line and parse: + #symbols=symbols+getSym(line) + symLine=line.strip("\n") + if hSub: + comToWrite=comToWrite+"
\n== Substitution(s) ==\n
\n" + hSub=False + #wiki.write("
"+getEq(line)+"

\n") + substitution=True + math=False + subLine="" + elif "\\drmfname" in line and parse: + math=False + comToWrite="
\n== Name ==\n
\n
"+getString(line)+"

\n"+comToWrite + + elif "\\drmfnote" in line and parse: + symbols=symbols+getSym(line) + if hNote: + comToWrite=comToWrite+"
\n== Note(s) ==\n
\n" + hNote=False + note=True + math=False + noteLine="" + + elif "\\proof" in line and parse: + #symbols=symbols+getSym(line) + symLine=line.strip("\n") + if hProof: + hProof=False + comToWrite=comToWrite+"
\n== Proof ==\n
\nWe ask users to provide proof(s), reference(s) to proof(s), or further clarification on the proof(s) in this space. \n

\n
" + proof=True + proofLine="" + pause=False + pauseP=False + for ind in range(0,len(line)): + if line[ind:ind+7]=="\\eqref{": + pause=True + eqR=line[ind:line.find("}",ind)+1] + rLab=getString(eqR) + '''for l in lLink: + if rLab==l[0:l.find("=")-1]: + rlabel=l[l.find("=>")+3:l.find("\\n")] + rlabel=rlabel.replace("/","") + rlabel=rlabel.replace("#",":") + rlabel=rlabel.replace("!",":") + break''' + + eInd=refLabels.index(""+label) + z=line[line.find("}",ind+7)+1] + if z=="." or z==",": + pauseP=True + proofLine+=("
\n{\displaystyle \n"+refEqs[eInd]+"}"+z+"
\n") + else: + if z=="}": + proofLine+=("
\n{\displaystyle \n"+refEqs[eInd]+"}
") + else: + proofLine+=("
\n{\displaystyle \n"+refEqs[eInd]+"}
\n") + + + else: + if pause: + if line[ind]=="}": + pause=False + elif pauseP: + pauseP=False + elif line[ind]=="\n" and "\\end{equation}" in lines[i+1]: + pass + else: + proofLine+=(line[ind]) + if "\\end{equation}" in lines[i+1]: + proof=False + #symLine+=line.strip("\n") + wiki.write(comToWrite+getEqP(proofLine,False)+"
\n
\n") + comToWrite="" + symbols=symbols+getSym(symLine) + symLine="" + + #wiki.write(line) + + elif proof: + symLine+=line.strip("\n") + pauseP=False + #symbols=symbols+getSym(line) + for ind in range(0,len(line)): + if line[ind:ind+7]=="\\eqref{": + pause=True + eqR=line[ind:line.find("}",ind)+1] + rLab=getString(eqR) + '''for l in lLink: + if rLab==l[0:l.find("=")-1]: + rlabel=l[l.find("=>")+3:l.find("\\n")] + rlabel=rlabel.replace("/","") + rlabel=rlabel.replace("#",":") + rlabel=rlabel.replace("!",":") + break''' + + eInd=refLabels.index(""+label) + z=line[line.find("}",ind+7)+1] + if z=="." or z==",": + pauseP=True + proofLine+=("
\n{\displaystyle \n"+refEqs[eInd]+"}"+z+"
\n") + else: + proofLine+=("
\n{\displaystyle \n"+refEqs[eInd]+"}
\n") + + else: + if pause: + if line[ind]=="}": + pause=False + elif pauseP: + pauseP=False + elif line[ind]=="\n" and "\\end{equation}" in lines[i+1]: + pass + + else: + proofLine+=(line[ind]) + if "\\end{equation}" in lines[i+1]: + proof=False + #symLine+=line.strip("\n") + wiki.write(comToWrite+getEqP(proofLine,False).rstrip("\n")+"
\n
\n") + comToWrite="" + symbols=symbols+getSym(symLine) + symLine="" + + elif math: + if "\\end{equation}" in lines[i+1] or "\\constraint" in lines[i+1] or "\\substitution" in lines[i+1] or "\\proof" in lines[i+1] or "\\drmfnote" in lines[i+1] or "\\drmfname" in lines[i+1]: + wiki.write(line.rstrip("\n")) + symLine+=line.strip("\n") + symbols=symbols+getSym(symLine) + symLine="" + wiki.write("\n}
\n") + else: + symLine+=line.strip("\n") + wiki.write(line) + if note and parse: + noteLine=noteLine+line + symbols=symbols+getSym(line) + if "\\end{equation}" in lines[i+1] or "\\drmfn" in lines[i+1] or "\\constraint" in lines[i+1] or "\\substitution" in lines[i+1] or "\\proof" in lines[i+1]: + note=False + comToWrite=comToWrite+"
"+getEq(noteLine)+"

\n" + + if constraint and parse: + conLine=conLine+line + symLine+=line.strip("\n") + #symbols=symbols+getSym(line) + if "\\end{equation}" in lines[i+1] or "\\drmfn" in lines[i+1] or "\\constraint" in lines[i+1] or "\\substitution" in lines[i+1] or "\\proof" in lines[i+1]: + constraint=False + symbols=symbols+getSym(symLine) + symLine="" + wiki.write(comToWrite+"
"+getEq(conLine)+"

\n") + comToWrite="" + if substitution and parse: + subLine=subLine+line + symLine+=line.strip("\n") + #symbols=symbols+getSym(line) + if "\\end{equation}" in lines[i+1] or "\\drmfn" in lines[i+1] or "\\substitution" in lines[i+1] or "\\constraint" in lines[i+1] or "\\proof" in lines[i+1]: + substitution=False + symbols=symbols+getSym(symLine) + symLine="" + wiki.write(comToWrite+"
"+getEq(subLine)+"

\n") + comToWrite="" +#except Exception as detail: #If exception occured +# print("Exception",detail) #print details of error +#except: #If anythin else occured... +# print ("ERROR",sys.exc_info()[0])#ERROR with basic info From ab92e47199ec05b6a77dcbf5e6477e1705a0f565 Mon Sep 17 00:00:00 2001 From: Joon Bang Date: Wed, 20 Jul 2016 15:21:23 -0400 Subject: [PATCH 185/402] Fix issues with Azeem's code (2) --- KLS-main-page/KLS_main_page.py | 14 -- KLS-main-page/headers.py | 55 ----- KLS-main-page/list.py | 86 -------- KLS-main-page/modMain.py | 117 ----------- KLS-main-page/symbolsList2.py | 373 --------------------------------- main_page/src/symbols_list.py | 22 +- 6 files changed, 11 insertions(+), 656 deletions(-) delete mode 100644 KLS-main-page/KLS_main_page.py delete mode 100644 KLS-main-page/headers.py delete mode 100644 KLS-main-page/list.py delete mode 100644 KLS-main-page/modMain.py delete mode 100644 KLS-main-page/symbolsList2.py diff --git a/KLS-main-page/KLS_main_page.py b/KLS-main-page/KLS_main_page.py deleted file mode 100644 index fc1722a..0000000 --- a/KLS-main-page/KLS_main_page.py +++ /dev/null @@ -1,14 +0,0 @@ -import os - -inp = raw_input("Update Main Page? (y or n)") -if inp != "n": - print("List") - os.system("python list.py") - print("Symbols List") - os.system("python symbolsList2.py") - print("Headers") - os.system("python headers.py") - print("List Ways") - os.system("python modMain.py") - -print("Done") diff --git a/KLS-main-page/headers.py b/KLS-main-page/headers.py deleted file mode 100644 index bdaac81..0000000 --- a/KLS-main-page/headers.py +++ /dev/null @@ -1,55 +0,0 @@ -file = open("main_page.mmd", "r") -lines = file.read() -listStart = lines.find("= Definition Pages =") -listStart = lines.find("* [[Definition:", listStart) -listEnd = lines.find("" in h: - h = h[h.find(">>
\n
") + len(">> \n"):] - if "
" in h: - h = h[0:h.find("
")] - prev = pages[(x - 1) % len(pages)] - # print h - next = pages[(x + 1) % len(pages)] - cur = "Main Page" - header = "\n
\n" - header += "
<< [[" + prev.replace(" ", "_") + "|" + prev + "]]
\n" - header += "
[[" + cur.replace(" ", "_") + "|" + cur + "]]
\n" - header += "
[[" + next.replace(" ", "_") + "|" + next + "]] >>
\n" - header += "
" - - footer = "
\n" - footer += "
<< [[" + prev.replace(" ", "_") + "|" + prev + "]]
\n" - footer += "
[[" + cur.replace(" ", "_") + "|" + cur + "]]
\n" - footer += "
[[" + next.replace(" ", "_") + "|" + next + "]] >>
\n" - footer += "
" - main.write("drmf_bof\n" + title + header + h + footer + "\ndrmf_eof\n") - x += 1 diff --git a/KLS-main-page/list.py b/KLS-main-page/list.py deleted file mode 100644 index 1e33e9a..0000000 --- a/KLS-main-page/list.py +++ /dev/null @@ -1,86 +0,0 @@ -import os -import time - -localtime = time.asctime(time.localtime(time.time())).replace(" ", "_") -os.system("cp main_page.mmd main_page.mmd" + str(localtime)) - - -def compString(a, b): - x = a[0:a.find("|")] - y = b[0:b.find("|")] - if len(x) > len(y): - max = len(y) - ret = False - else: - max = len(x) - ret = True - if True: - for i in range(0, max): - if ord(x[i].lower()) < ord(y[i].lower()): - return True - elif ord(x[i].lower()) > ord(y[i].lower()): - return False - return ret - - -def sumOrd(x): - sum = 0 - for s in x: - sum += ord(s) - return (sum) - - -def sort(x): - ret = [] - ret.append(x[0]) - for i in range(1, len(x)): - if compString(x[i], ret[0]): - ret = [x[i]] + ret - elif compString(ret[len(ret) - 1], x[i]): - ret.append(x[i]) - else: - for p in range(1, len(ret)): - if compString(x[i], ret[p]): - ret = ret[0:p] + [x[i]] + ret[p:] - break - return (ret) - - -'''listFile=open("list.txt","w") -files_in_dir=os.listdir(".") -toWrite=[] -for g in files_in_dir: - if ".new" in g and "Definition:" in g: - g=g[:g.find(".new")] - element="* [["+g+"|"+g[len("Definition:"):]+"]]" - toWrite.append(element) - toWrite=sort(toWrite) -text="" -for g in toWrite: - text+=(g+"\n") -listFile.close() -listFile=open("list.txt","r") -text=listFile.read()''' -main = open("main_page.mmd", "r") -mainText = main.read() -mainLines = mainText.split("\n") -main.close() -toWrite = [] -for z in mainLines: - if "\'\'\'Definition:" in z: - g = z[z.find("\'\'\'Definition:") + len("\'\'\'"):z.find("\'\'\'", z.find("\'\'\'Definition:") + 4)] - element = "* [[" + g + "|" + g[len("Definition:"):] + "]]" - toWrite.append(element) -main = open("main_page.mmd", "w") -s = mainText.find("== Definition Pages ==") -e = mainText.find("= Copyright =") -text = "" -for g in toWrite: - text += (g + "\n") -# print text -# print mainText[:s] -newText = mainText[ - :s] + "== Definition Pages ==\n\n
\n" + text + "
\n\n" + mainText[ - e:] -main.write(newText) -main.close() diff --git a/KLS-main-page/modMain.py b/KLS-main-page/modMain.py deleted file mode 100644 index 3ec6d8f..0000000 --- a/KLS-main-page/modMain.py +++ /dev/null @@ -1,117 +0,0 @@ -file = open("main_page.mmd", "r") -import csv - -lines = file.read() -lines = lines.replace("The LaTeX DLMF macro \'\'\'\\", "The LaTeX DLMF and DRMF macro \'\'\'\\") -file.close() -file = open("main_page.mmd", "w") -flag = True -gCSV = csv.reader(open('new.Glossary.csv', 'rb'), delimiter=',', quotechar='\"') -toWrite = "" -i = 0 - - -def atSort(l): - return (sorted(l, key=lambda x: x.count("@"))) - - -def remOpt(p): - parse = True - ret = "" - for x in p: - if parse: - if x == "[": - parse = False - else: - ret += x - else: - if x == "]": - parse = True - return ret - - -def findAllPos(g): - key = g[3] - orig = g[0] - r = key[:key.find("|")] - key = key[key.find("|") + 1:] - keys = key.split(":") - ret = [] - for x in keys: - if x == "F" or x == "FO" or x == "O" or x == "O1": - if orig.find("@") != -1: - ret.append(orig[0:orig.find("@")]) - if x == "O": - ret.append(orig) - else: - ret.append(orig) - if x == "FnO": # remove optional parameters - if orig.find("@") != -1: - ret.append(remOpt(orig[0:orig.find("@")])) - else: - ret.append(remOpt(orig)) - if x == "P" or x == "PO": - ret.append(orig.replace("@@", "@")) - if x == "PnO": # remove optional parameters - ret.append(remOpt(orig.replace("@@", "@"))) - if x == "nP" or x == "nPO" or x == "PS" or x == "O2": - ret.append(orig) - if x == "nPnO": # remove optional paramters - ret.append(remOpt(orig)) - if "fo" in x: - ret.append(orig.replace("@@@", "@" * int(x[2]))) - return atSort(ret), r - - -categories = open("Categories.txt", "r") -cats = categories.readlines() -categories.close() -for x in range(0, len(cats)): - cats[x] = cats[x].strip() -while flag: - n = lines.find("\'\'\'Definition:", i) - if n == -1: - flag = False - # elif lines.find("This macro can be called in the following way")1: plural =1; if count=1: plural=0 - if count == 1: plural = 0 - for t in cats: - if s + " -" in t: - category = "This macro is in the category of" + t[t.find("-") + 2:] - break - new = category + "\n\nIn math mode, this macro can be called in the following way" + "s" * plural + ":\n\n" - for q in range(0, len(listCalls)): - c = listCalls[q] - new += ":\'\'\'" + c + "\'\'\'" + " produces {\\displaystyle " + c + "}
\n" - # Now add the multiple ways \macroname{n}@... produces \macroname{n}@... - toWrite += new + "\n" - i = lines.find("These are defined by", p) -file.write(toWrite + lines[i:]) diff --git a/KLS-main-page/symbolsList2.py b/KLS-main-page/symbolsList2.py deleted file mode 100644 index a79c590..0000000 --- a/KLS-main-page/symbolsList2.py +++ /dev/null @@ -1,373 +0,0 @@ -def getSym(line): # Gets all symbols on a line for symbols list - line = line.replace("\n", " ") - symList = [] - if line == "": - return symList - symbol = "" - symFlag = False - argFlag = False - cC = 0 - for i in range(0, len(line)): - # if "MeixnerPollaczek{\\lambda}{n}@{x}{\\phi}=\frac{\\pochhammer{2\\lambda}{n}}" in line: - # print symbol - c = line[i] - if symFlag: - if c == "{" or c == "[": - cC += 1 - argFlag = True - if c != "}" and c != "]": - if argFlag or c.isalpha(): - symbol += c - else: - symFlag = False - argFlag = False - symList.append(symbol) - symList += (getSym(symbol)) - symbol = "" - else: - cC -= 1 - symbol += c - if i + 1 == len(line): - p = "" - else: - p = line[i + 1] - if cC == 0 and p != "{" and p != "[" and p != "@": - symFlag = False - # if "monicAskeyWilson{n+1}@{x}{a}{b}{c}{d}{q}+\\frac{1}{2}" in line: - # print(symbol,p) - symList.append(symbol) - # if symbol=="binom{n}{k}":print line - symList += (getSym(symbol)) - argFlag = False - symbol = "" - - elif c == "\\": - symFlag = True - - # if "MeixnerPollaczek{\\lambda}{n}@{x}{\\phi}=\frac{\\pochhammer{2\\lambda}{n}}" in line: - # print symbol - symList.append(symbol) - # if "ctsbigqHermite" in line: - # print(symList) - symList += getSym(symbol) - return (symList) - - -import csv - -# files_in_dir=os.listdir(".") -mainPage = open("main_page.mmd", "r") -mainLines = mainPage.read() -mainLines = mainLines[mainLines.find("drmf_bof"):mainLines.rfind("drmf_eof") + len("drmf_eof\n")] -mainPage.close() -main = open("main_page.mmd", "w") -mainPages = mainLines.split("drmf_eof") -if mainPages[-1].strip() == "": - mainPages = mainPages[0:len(mainPages) - 1] -print len(mainPages) -glossary = open('new.Glossary.csv', 'rb') -gCSV = csv.reader(glossary, delimiter=',', quotechar='\"') -lGlos = glossary.readlines() -wiki = "" -for g in mainPages: - if not "\'\'\'Definition:" in g: - h = g.replace("drmf_eof", "") - h = g.replace("drmf_bof", "") - toWrite = "drmf_bof" + h + "drmf_eof\n" - main.write(toWrite) - else: - h = g.replace("drmf_eof", "") - h = g.replace("drmf_bof", "") - h = h.strip("\n") - sflag1 = False - sflag2 = False - if h.find("== Symbols List ==") != -1: - toWrite = h[0:h.find("== Symbols List ==")] - h = toWrite - else: - sflag1 = True - if h.find("drmf_foot") == -1: - toWrite = h - sflag2 = True - else: - toWrite = h[0:h.find("
"):]+t[0:t.find("
")] - t = t.replace("\n", " ") - # if "Symbol" in t:print t - symbols = getSym(t) - # print symbols - # if "qsin" in g:print t,symbols - # if "Neumann" in g:print t,"\n",symbols - if True: - toWrite += ("== Symbols List ==\n\n") - newSym = [] - # if "09.07:04" in label: - # print(symbols) - for x in symbols: - flagA = True - # if x not in newSym: - cN = 0 - cC = 0 - flag = True - ArgCx = 0 - for z in x: - if z.isalpha() and flag: - cN += 1 - else: - flag = False - if z == "{" or z == "[": - cC += 1 - if z == "}" or z == "]": - cC -= 1 - if cC == 0: - ArgCx += 1 - noA = x[:cN] - for y in newSym: - cN = 0 - cC = 0 - ArgC = 0 - flag = True - for z in y: - if z.isalpha() and flag: - cN += 1 - else: - flag = False - if z == "{" or z == "[": - cC += 1 - if z == "}" or z == "]": - cC -= 1 - if cC == 0: - ArgC += 1 - - if y[:cN].strip() == noA.strip(): # and ArgC==ArgCx: - flagA = False - # print(x,y) - break - if flagA: - newSym.append(x) - newSym.reverse() - # print newSym - # if "14.27:02" in label: - # print newSym - symF = False - finSym = [] - for s in range(len(newSym) - 1, -1, -1): - # print(newSym[s]) - symbolPar = "\\" + newSym[s] - ArgCx = 0 - parCx = 0 - parFlag = False - cC = 0 - for z in symbolPar: - if z == "@": - parFlag = True - elif z.isalpha(): - cN += 1 - else: - if z == "{" or z == "[": - cC += 1 - if z == "}" or z == "]": - cC -= 1 - if cC == 0: - if parFlag: - parCx += 1 - else: - ArgCx += 1 - - if symbolPar.find("{") != -1 or symbolPar.find("[") != -1: - if symbolPar.find("[") == -1: - symbol = symbolPar[0:symbolPar.find("{")] - elif symbolPar.find("{") == -1 or symbolPar.find("[") < symbolPar.find("{"): - symbol = symbolPar[0:symbolPar.find("[")] - else: - symbol = symbolPar[0:symbolPar.find("{")] - else: - symbol = symbolPar - # if "14.27:02" in label:print symbolPar,symbol - numArg = parCx - numPar = ArgCx - gFlag = False - checkFlag = False - get = False - gCSV = csv.reader(open('new.Glossary.csv', 'rb'), delimiter=',', quotechar='\"') - preG = "" - symbol = symbol.strip("}").strip("]") - for S in gCSV: - G = S - ArgCx = 0 - parCx = 0 - parFlag = False - cC = 0 - ind = G[0].find("@") - if ind == -1: - ind = len(G[0]) - 1 - for z in G[0]: - if z == "@": - parFlag = True - elif z.isalpha(): - cN += 1 - else: - if z == "{" or z == "[": - cC += 1 - if z == "}" or z == "]": - cC -= 1 - if cC == 0: - if parFlag: - parCx += 1 - else: - ArgCx += 1 - if G[0].find(symbol) == 0 and (len(G[0]) == len(symbol) or not G[0][ - len(symbol)].isalpha()): # and (numPar!=0 or numArg!=0): - checkFlag = True - get = True - preG = S - # if "14.27:02" in label:print "ok" + symbol - - # if "9.07:04" in label:print symbol,numPar,numArg,parCx,ArgCx - - elif checkFlag: - get = True - checkFlag = False - if (get): - if get: - G = preG - # print preG - get = False - checkFlag = False - # if "14.27:02" in label:print "yes"+symbol - if True: - if symbolPar.find("@") != -1: - Q = symbolPar[:symbolPar.find("@")] - else: - Q = symbolPar - listArgs = [] - if len(Q) > len(symbol) and (Q[len(symbol)] == "{" or Q[len(symbol)] == "["): - ap = "" - for o in range(len(symbol), len(Q)): - if Q[o] == "{" or z == "[": - argFlag = True - elif Q[o] == "}" or z == "]": - argFlag = False - listArgs.append(ap) - ap = "" - else: - ap += Q[o] - '''websiteU=g[g.find("http://"):].strip("\n") - k=0 - websites=[] - for r in range(0,len(websiteU)): - if websiteU[r]==",": - if websiteU[k:r].find("http://")!=-1: - websites.append(websiteU[k:r+1].strip(" ")) - k=r - - - if websiteU[k:r].find("http://")!=-1: - websites.append(websiteU[k:r+1].strip(" ")) - websiteF="" - for d in websites: - websiteF=websiteF+" ["+d+" "+d+"]"''' - # websiteF=G[4].strip("\n") - websiteF = "" - # print("(",G,")") - web1 = G[5] - for t in range(5, len(G)): - if G[t] != "": - websiteF = websiteF + " [" + G[t] + " " + G[t] + "]" - - # p1=Q - # if Q.find("@")!=-1: - # p1=Q[:Q.find("@")] - p1 = G[4].strip("$") - p1 = "{\\displaystyle " + p1 + "}" - # if checkFlag: - # print(p1) - new1 = "" - new2 = "" - pause = False - mathF = True - '''for k in range(0,len(p1)): - if p1[k]=="$": - if mathF: - new1+="{\\displaystyle " - else: - new1+="}" - mathF=not mathF - - elif p1[k]=="#" and p1[k+1].isdigit(): - pause=True - elif pause: - num=int(p1[k]) - #letter=chr(num+96) - letter=listArgs[num-1] - new1+=letter - pause=False - - else: - new1+=p1[k]''' - p2 = G[1] - for k in range(0, len(p2)): - if p2[k] == "$": - if mathF: - new2 += "{\\displaystyle " - else: - new2 += "}" - mathF = not mathF - else: - new2 += p2[k] - # p1=new1 - p2 = new2 - finSym.append(web1 + " " + p1 + "] : " + p2 + " :" + websiteF) - break - # gFlag=True - # if not symF: - # symF=True - # wiki.write("[") - # else: - # wiki.write("
\n") - # wiki.write("[") - # wiki.write(g[g.find("http://"):].strip("\n")) - # wiki.write(" ") - # wiki.write(getEq(g[g.find("{$"):]).strip("\n").replace("\\\\","\\")) - # wiki.write("] : ") - # wiki.write(g[g.find(" {")+2:g.find("} ",g.find(" {")+1)]) - # wiki.write(" : [") - # wiki.write(g[g.find("http://"):].strip("\n")) - # wiki.write(" ") - # wiki.write(g[g.find("http://"):].strip("\n")) - # wiki.write("] ")''' - - # preG=S - if not gFlag: - del newSym[s] - - gFlag = True - # finSym.reverse() - for y in finSym: - # print(finSym) - if gFlag: - gFlag = False - toWrite += ("[" + y) - else: - toWrite += ("
\n[" + y) - - toWrite += ("\n
\ndrmf_eof\n\n") - if sflag2: - toWrite += "\n" - main.write(toWrite) diff --git a/main_page/src/symbols_list.py b/main_page/src/symbols_list.py index 82d8eb9..bdb8087 100644 --- a/main_page/src/symbols_list.py +++ b/main_page/src/symbols_list.py @@ -16,10 +16,10 @@ def getSym(line): # Gets all symbols on a line for symbols list for i in range(0, len(line)): c = line[i] if symFlag: - if c == "{" or c == "[": + if c in ["{", "["]: cC += 1 argFlag = True - if c != "}" and c != "]": + if c not in ["}", "]"]: if argFlag or c.isalpha(): symbol += c else: @@ -57,7 +57,7 @@ def getSym(line): # Gets all symbols on a line for symbols list mainPages = mainLines.split("drmf_eof") if mainPages[-1].strip() == "": - mainPages = mainPages[0:len(mainPages) - 1] + mainPages = mainPages[0:-1] print len(mainPages) @@ -118,9 +118,9 @@ def getSym(line): # Gets all symbols on a line for symbols list cN += 1 else: flag = False - if z == "{" or z == "[": + if z in ["{", "["]: cC += 1 - if z == "}" or z == "]": + if z in ["}", "]"]: cC -= 1 if cC == 0: ArgCx += 1 @@ -135,9 +135,9 @@ def getSym(line): # Gets all symbols on a line for symbols list cN += 1 else: flag = False - if z == "{" or z == "[": + if z in ["{", "["]: cC += 1 - if z == "}" or z == "]": + if z in ["}", "]"]: cC -= 1 if cC == 0: ArgC += 1 @@ -162,9 +162,9 @@ def getSym(line): # Gets all symbols on a line for symbols list elif z.isalpha(): cN += 1 else: - if z == "{" or z == "[": + if z in ["{", "["]: cC += 1 - if z == "}" or z == "]": + if z in ["}", "]"]: cC -= 1 if cC == 0: if parFlag: @@ -204,9 +204,9 @@ def getSym(line): # Gets all symbols on a line for symbols list elif z.isalpha(): cN += 1 else: - if z == "{" or z == "[": + if z in ["{", "["]: cC += 1 - if z == "}" or z == "]": + if z in ["}", "]"]: cC -= 1 if cC == 0: if parFlag: From 6d2c2de1fb2df8c2696c0f85074f2842743f8b97 Mon Sep 17 00:00:00 2001 From: Oksana Tkach Date: Thu, 21 Jul 2016 10:35:29 -0400 Subject: [PATCH 186/402] Further cover the number of lines tested in replace_special.py Fix \index spacing error. Increase efficiency of replace_special.py --- DLMF_preprocessing/src/math_function.py | 21 +++++++++--------- DLMF_preprocessing/src/replace_special.py | 17 --------------- DLMF_preprocessing/test/test_replace_i.py | 26 ++++++++++++++++++++++- 3 files changed, 36 insertions(+), 28 deletions(-) diff --git a/DLMF_preprocessing/src/math_function.py b/DLMF_preprocessing/src/math_function.py index 2fa1dea..79b198e 100644 --- a/DLMF_preprocessing/src/math_function.py +++ b/DLMF_preprocessing/src/math_function.py @@ -38,7 +38,6 @@ def formatting(file_str): commands = r'\\[a-zA-Z]+' updated = [] - IND_START = r'\index\{' ind_str = "" in_ind = False previous = "" @@ -61,7 +60,7 @@ def formatting(file_str): if '\\begin{equation}' in line: in_eq = True - been_in_eq = False + been_in_eq = True if '\\end{equation}' in line: updated.append(line) in_eq = False @@ -71,15 +70,15 @@ def formatting(file_str): updated.append(line) else: - # not in eq + # not been in eq if not been_in_eq: # if text line != command - if re.match(commands, line) or line == '': - if '\\paragraph' not in line and '\\acknowledgements' not in line: + if re.match(commands, line): + if '\\paragraph' not in line and '\\index' not in line: updated.append(line) else: # not in eq but already been in eq - if '\\index' in line or re.match(section,line) or line == '': + if re.match(section,line) or line == '': updated.append(line) if past_last > ranges[-1][1]: @@ -89,22 +88,24 @@ def formatting(file_str): updated.append(line) # if this line is an index start storing it,or write it if we're done with the indexes - if IND_START in line: + if '\\index{' in line: in_ind = True ind_str += line + "\n" + continue - - if in_ind: + elif in_ind: in_ind = False # add a preceding newline if one is not already present if previous.strip() != "": ind_str = "\n" + ind_str + fullsplit = ind_str.split("\n") + updated.extend(fullsplit) ind_str = "" - previous = line + previous = line wrote = "\n".join(updated) # remove consecutive blank lines and blank lines between \index groups diff --git a/DLMF_preprocessing/src/replace_special.py b/DLMF_preprocessing/src/replace_special.py index 5123917..070584a 100644 --- a/DLMF_preprocessing/src/replace_special.py +++ b/DLMF_preprocessing/src/replace_special.py @@ -96,20 +96,6 @@ def remove_special(content): for lnum, line in enumerate(lines): lnum += 1 - # if this line marks the start of an equation, set the flag - if EQ_START in line: - in_eq = True - - # if this line marks the end of an equation, set the flag - if EQ_END in line: - - # remove other flags too - for flag in inside: - inside[flag][SEEN] = False - - comment_str = "" - - in_eq = True # we need to make the replacements in equations if in_eq: @@ -138,7 +124,6 @@ def remove_special(content): line = _replace_i(line) - # print 'line2', line elif any(info[SEEN] for info in inside.values()): # ^we're in a special block, look for dollar signs to replace "i"s @@ -153,8 +138,6 @@ def remove_special(content): for flag in inside: inside[flag][SEEN] = False - # print(comment_str + "\n") - dollar_locs = [match.start() for match in dollar_pat.finditer(comment_str)] locs_iter = iter(dollar_locs) diff --git a/DLMF_preprocessing/test/test_replace_i.py b/DLMF_preprocessing/test/test_replace_i.py index 725d657..d58d0aa 100644 --- a/DLMF_preprocessing/test/test_replace_i.py +++ b/DLMF_preprocessing/test/test_replace_i.py @@ -1,9 +1,33 @@ from unittest import TestCase -from src.replace_special import remove_special +from replace_special import remove_special +mismatched = """ +%\\frac{q(t)}{p'(t) (p(t) - p(a))^{(\lambda/\mu)-1}}$, using Cauchy's integral +%formula for the residue, and integrating by parts. See also + b_s + = \\frac{1}{\mu} + \Residue_{t=a}\left[\\frac{q(t)}{(p(t) - p(a))^{(\lambda+s)/\mu}}\\right] +% \constraint{$s = 0,1,2,\dots$.} +""" +matched = """ +%\\note{This follows from \eqref{eq:AL.xx}--\eqref{eq:AL.xx} by setting $f(t) = p(t), g(t) = +%\\frac{q(t)}{p'(t) (p(t) - p(a))^{(\lambda/\mu)-1}}$, using Cauchy's integral +%formula for the residue, and integrating by parts. See also +%\citet{Cicuta:1975:RFA}.} + b_s + = \\frac{1}{\mu} + \Residue_{t=a}\left[\\frac{q(t)}{(p(t) - p(a))^{(\lambda+s)/\mu}}\\right] +% \constraint{$s = 0,1,2,\dots$.} +""" class TestReplaceSpecial(TestCase): def test__replace_special(self): tex = "z^a = \\underbrace{z \\cdot z \\cdots z}_{n \n \\text{ times}} = 1 / z^{-a}" result = remove_special(tex) self.assertEqual(tex, result) + def test_dollar_loc(self): + actual = remove_special(mismatched) + self.assertEqual(mismatched, actual) + def test_matching(self): + result = remove_special(matched) + self.assertEqual(matched, result) \ No newline at end of file From 2ced4b0b4e907a089fc10abbde9e82180175d4ce Mon Sep 17 00:00:00 2001 From: Jagan Prem Date: Tue, 2 Aug 2016 08:17:12 -0400 Subject: [PATCH 187/402] Add support for LaTeX comments in math_mode.py and the relevant test cases. --- DLMF_preprocessing/src/math_mode.py | 86 ++++++++++++++--------- DLMF_preprocessing/test/test_math_mode.py | 10 +-- 2 files changed, 59 insertions(+), 37 deletions(-) diff --git a/DLMF_preprocessing/src/math_mode.py b/DLMF_preprocessing/src/math_mode.py index 5017006..1d74000 100644 --- a/DLMF_preprocessing/src/math_mode.py +++ b/DLMF_preprocessing/src/math_mode.py @@ -1,6 +1,8 @@ __author__ = "Jagan Prem" __status__ = "Production" +import re + # Dictionary containing math mode delimiters and their respective endpoints. MATH_START = {"\\[": "\\]", "\\(": "\\)", @@ -87,24 +89,34 @@ def parse_math(string, start, ranges): delim = first_delim(string) i = len(delim) begin = start + i + commented = False while i < len(string): - if string[i:].startswith("\\$"): - i += 1 - elif string[i:].startswith("\\\\]"): - i += 2 - elif string[i:].startswith("\\\\)"): - i += 2 - else: - if does_exit(string[i:]): - if begin != start + i: - ranges.append((begin, start + i)) - i += parse_non_math(string[i:], start + i, ranges) - begin = start + i - i -= 1 - if string[i:].startswith(MATH_START[delim]): - if begin != start + i: - ranges.append((begin, start + i)) - return i + len(MATH_START[delim]) - 1 + if not commented: + if string[i:].startswith("\\%"): + i += 1 + elif string[i] == "%": + i += 1 + commented = True + elif string[i:].startswith("\\$"): + i += 1 + elif string[i:].startswith("\\\\]"): + i += 2 + elif string[i:].startswith("\\\\)"): + i += 2 + else: + if does_exit(string[i:]): + if begin != start + i: + ranges.append((begin, start + i)) + i += parse_non_math(string[i:], start + i, ranges) + begin = start + i + i -= 1 + if string[i:].startswith(MATH_START[delim]): + if begin != start + i: + ranges.append((begin, start + i)) + return i + len(MATH_START[delim]) - 1 + if commented: + if string[i] == "\n": + commented = False i += 1 raise SyntaxError("missing " + MATH_START[delim]) @@ -123,23 +135,33 @@ def parse_non_math(string, start, ranges): delim = "" level = 0 i = len(delim) + commented = False while i < len(string): - if string[i:].startswith("\\$"): - i += 1 - elif string[i:].startswith("\\\\["): - i += 2 - elif string[i:].startswith("\\\\("): - i += 2 - elif does_enter(string[i:]): - i += parse_math(string[i:], start + i, ranges) - elif string[i] == "{": - level += 1 - elif string[i] == "}": - if level == 0 and delim != "": + if not commented: + if string[i:].startswith("\\%"): i += 1 - return i - else: - level -= 1 + elif string[i] == "%": + i += 1 + commented = True + elif string[i:].startswith("\\$"): + i += 1 + elif string[i:].startswith("\\\\["): + i += 2 + elif string[i:].startswith("\\\\("): + i += 2 + elif does_enter(string[i:]): + i += parse_math(string[i:], start + i, ranges) + elif string[i] == "{": + level += 1 + elif string[i] == "}": + if level == 0 and delim != "": + i += 1 + return i + else: + level -= 1 + if commented: + if string[i] == "\n": + commented = False i += 1 if delim == "" and level == 0: return i diff --git a/DLMF_preprocessing/test/test_math_mode.py b/DLMF_preprocessing/test/test_math_mode.py index 2308e51..13eb596 100644 --- a/DLMF_preprocessing/test/test_math_mode.py +++ b/DLMF_preprocessing/test/test_math_mode.py @@ -32,15 +32,15 @@ "output": 18 }, { - "string": "$\\$$", + "string": "$%\n\\$$", "start": 12, - "output": 3 + "output": 5 }, [ (1, 10), (24, 25), (36, 39), - (13, 15) + (13, 17) ] ] @@ -56,9 +56,9 @@ "output": 39 }, { - "string": "\\begin{multline}\n\\left(2\\right)\n\\end{multline}", + "string": "\\begin{multline}\n\\left(2\\right)\n\\end{multline}\n%begin{equation}", "start": 3, - "output": 46 + "output": 63 }, [ (28, 35), From 7e2cd64c2660d4f72a8a1398b20d866f441b2217 Mon Sep 17 00:00:00 2001 From: Jagan Prem Date: Tue, 2 Aug 2016 08:25:43 -0400 Subject: [PATCH 188/402] Removed unnecessary import. --- DLMF_preprocessing/src/math_mode.py | 2 -- DLMF_preprocessing/test/test_math_mode.py | 10 +++++----- 2 files changed, 5 insertions(+), 7 deletions(-) diff --git a/DLMF_preprocessing/src/math_mode.py b/DLMF_preprocessing/src/math_mode.py index 1d74000..35747d4 100644 --- a/DLMF_preprocessing/src/math_mode.py +++ b/DLMF_preprocessing/src/math_mode.py @@ -1,8 +1,6 @@ __author__ = "Jagan Prem" __status__ = "Production" -import re - # Dictionary containing math mode delimiters and their respective endpoints. MATH_START = {"\\[": "\\]", "\\(": "\\)", diff --git a/DLMF_preprocessing/test/test_math_mode.py b/DLMF_preprocessing/test/test_math_mode.py index 13eb596..d8925bf 100644 --- a/DLMF_preprocessing/test/test_math_mode.py +++ b/DLMF_preprocessing/test/test_math_mode.py @@ -32,15 +32,15 @@ "output": 18 }, { - "string": "$%\n\\$$", + "string": "$%\n\\$\\%$", "start": 12, - "output": 5 + "output": 7 }, [ (1, 10), (24, 25), (36, 39), - (13, 17) + (13, 19) ] ] @@ -68,9 +68,9 @@ RANGE_TESTS = [ { - "string": "test \\begin{multline*}\ntest \\hbox{Test}\\end{multline*}", + "string": "test\\% \\begin{multline*}\ntest \\hbox{Test}\\end{multline*}", "output": [ - (22, 28) + (24, 30) ] }, { From 42d9fd70935fa724e67945f679ae48c9e55a80e3 Mon Sep 17 00:00:00 2001 From: Jagan Prem Date: Tue, 2 Aug 2016 08:39:53 -0400 Subject: [PATCH 189/402] Clean up code, fix test, and reduce function complexity. --- DLMF_preprocessing/src/math_mode.py | 61 ++++++++++++----------- DLMF_preprocessing/test/test_math_mode.py | 4 +- 2 files changed, 35 insertions(+), 30 deletions(-) diff --git a/DLMF_preprocessing/src/math_mode.py b/DLMF_preprocessing/src/math_mode.py index 35747d4..d35d714 100644 --- a/DLMF_preprocessing/src/math_mode.py +++ b/DLMF_preprocessing/src/math_mode.py @@ -75,6 +75,13 @@ def does_enter(string): return any(string.startswith(delim) for delim in MATH_START) +def skip_escaped(string): + for escape in ["\\$", "\\\\]", "\\\\)", "\\\\(", "\\\\["]: + if string.startswith(escape): + return len(escape) + return 0 + + def parse_math(string, start, ranges): # type: (str) -> str, int """ @@ -90,31 +97,33 @@ def parse_math(string, start, ranges): commented = False while i < len(string): if not commented: - if string[i:].startswith("\\%"): - i += 1 - elif string[i] == "%": + i += skip_escaped(string[i:]) + if i >= len(string): + break + sub = string[i:] + if sub.startswith("\\%"): i += 1 + elif sub[0] == "%": commented = True - elif string[i:].startswith("\\$"): + elif sub.startswith("\\$"): i += 1 - elif string[i:].startswith("\\\\]"): + elif sub.startswith("\\\\]"): i += 2 - elif string[i:].startswith("\\\\)"): + elif sub.startswith("\\\\)"): i += 2 else: - if does_exit(string[i:]): + if does_exit(sub): if begin != start + i: ranges.append((begin, start + i)) - i += parse_non_math(string[i:], start + i, ranges) + i += parse_non_math(sub, start + i, ranges) begin = start + i i -= 1 - if string[i:].startswith(MATH_START[delim]): + if sub.startswith(MATH_START[delim]): if begin != start + i: ranges.append((begin, start + i)) return i + len(MATH_START[delim]) - 1 - if commented: - if string[i] == "\n": - commented = False + elif commented and string[i] == "\n": + commented = False i += 1 raise SyntaxError("missing " + MATH_START[delim]) @@ -136,30 +145,26 @@ def parse_non_math(string, start, ranges): commented = False while i < len(string): if not commented: - if string[i:].startswith("\\%"): - i += 1 - elif string[i] == "%": + i += skip_escaped(string[i:]) + if i >= len(string): + break + sub = string[i:] + if sub.startswith("\\%"): i += 1 + elif sub[0] == "%": commented = True - elif string[i:].startswith("\\$"): - i += 1 - elif string[i:].startswith("\\\\["): - i += 2 - elif string[i:].startswith("\\\\("): - i += 2 - elif does_enter(string[i:]): - i += parse_math(string[i:], start + i, ranges) - elif string[i] == "{": + elif does_enter(sub): + i += parse_math(sub, start + i, ranges) + elif sub[0] == "{": level += 1 - elif string[i] == "}": + elif sub[0] == "}": if level == 0 and delim != "": i += 1 return i else: level -= 1 - if commented: - if string[i] == "\n": - commented = False + elif commented and string[i] == "\n": + commented = False i += 1 if delim == "" and level == 0: return i diff --git a/DLMF_preprocessing/test/test_math_mode.py b/DLMF_preprocessing/test/test_math_mode.py index d8925bf..225605c 100644 --- a/DLMF_preprocessing/test/test_math_mode.py +++ b/DLMF_preprocessing/test/test_math_mode.py @@ -56,9 +56,9 @@ "output": 39 }, { - "string": "\\begin{multline}\n\\left(2\\right)\n\\end{multline}\n%begin{equation}", + "string": "\\begin{multline}\n\\left(2\\right)\n\\end{multline}\n%begin{equation}\n", "start": 3, - "output": 63 + "output": 64 }, [ (28, 35), From cce831ea84b86f1fac37d72fad35bfe9559bbed8 Mon Sep 17 00:00:00 2001 From: Jagan Prem Date: Tue, 2 Aug 2016 08:43:59 -0400 Subject: [PATCH 190/402] Removed unnecessary conditions. --- DLMF_preprocessing/src/math_mode.py | 8 -------- 1 file changed, 8 deletions(-) diff --git a/DLMF_preprocessing/src/math_mode.py b/DLMF_preprocessing/src/math_mode.py index d35d714..1762bc9 100644 --- a/DLMF_preprocessing/src/math_mode.py +++ b/DLMF_preprocessing/src/math_mode.py @@ -98,19 +98,11 @@ def parse_math(string, start, ranges): while i < len(string): if not commented: i += skip_escaped(string[i:]) - if i >= len(string): - break sub = string[i:] if sub.startswith("\\%"): i += 1 elif sub[0] == "%": commented = True - elif sub.startswith("\\$"): - i += 1 - elif sub.startswith("\\\\]"): - i += 2 - elif sub.startswith("\\\\)"): - i += 2 else: if does_exit(sub): if begin != start + i: From e054e2f80d949ed9b781bb942e8ab985e5702fed Mon Sep 17 00:00:00 2001 From: Jagan Prem Date: Tue, 2 Aug 2016 09:10:17 -0400 Subject: [PATCH 191/402] Add comments. --- DLMF_preprocessing/src/math_mode.py | 6 ++++++ 1 file changed, 6 insertions(+) diff --git a/DLMF_preprocessing/src/math_mode.py b/DLMF_preprocessing/src/math_mode.py index 1762bc9..88ee5b7 100644 --- a/DLMF_preprocessing/src/math_mode.py +++ b/DLMF_preprocessing/src/math_mode.py @@ -76,6 +76,12 @@ def does_enter(string): def skip_escaped(string): + # type: (str) -> int + """ + Returns the distance to skip from escaped delimiters in the string. + :param string: The string to check. + :return: The distance to skip. + """ for escape in ["\\$", "\\\\]", "\\\\)", "\\\\(", "\\\\["]: if string.startswith(escape): return len(escape) From 5a726fc2f922d42008f528b46c23b7b490a98398 Mon Sep 17 00:00:00 2001 From: Jagan Prem Date: Tue, 2 Aug 2016 09:36:07 -0400 Subject: [PATCH 192/402] Clean up and fix more code. --- DLMF_preprocessing/src/math_mode.py | 32 +++++++++++++++++------------ 1 file changed, 19 insertions(+), 13 deletions(-) diff --git a/DLMF_preprocessing/src/math_mode.py b/DLMF_preprocessing/src/math_mode.py index 88ee5b7..1f7bbf6 100644 --- a/DLMF_preprocessing/src/math_mode.py +++ b/DLMF_preprocessing/src/math_mode.py @@ -75,17 +75,27 @@ def does_enter(string): return any(string.startswith(delim) for delim in MATH_START) -def skip_escaped(string): +def skip_escaped(string, loop=True): # type: (str) -> int """ Returns the distance to skip from escaped delimiters in the string. :param string: The string to check. + :param loop: Whether to skip repeatedly or not. :return: The distance to skip. """ - for escape in ["\\$", "\\\\]", "\\\\)", "\\\\(", "\\\\["]: - if string.startswith(escape): - return len(escape) - return 0 + if loop: + s = skip_escaped(string, False) + index = 0 + while s != 0: + string = string[s:] + index += s + s = skip_escaped(string, False) + return index + else: + for escape in ["\\$", "\\\\]", "\\\\)", "\\\\(", "\\\\[", "\\%"]: + if string.startswith(escape): + return len(escape) + return 0 def parse_math(string, start, ranges): @@ -105,9 +115,7 @@ def parse_math(string, start, ranges): if not commented: i += skip_escaped(string[i:]) sub = string[i:] - if sub.startswith("\\%"): - i += 1 - elif sub[0] == "%": + if sub[0] == "%": commented = True else: if does_exit(sub): @@ -120,7 +128,7 @@ def parse_math(string, start, ranges): if begin != start + i: ranges.append((begin, start + i)) return i + len(MATH_START[delim]) - 1 - elif commented and string[i] == "\n": + elif string[i] == "\n": commented = False i += 1 raise SyntaxError("missing " + MATH_START[delim]) @@ -147,9 +155,7 @@ def parse_non_math(string, start, ranges): if i >= len(string): break sub = string[i:] - if sub.startswith("\\%"): - i += 1 - elif sub[0] == "%": + if sub[0] == "%": commented = True elif does_enter(sub): i += parse_math(sub, start + i, ranges) @@ -161,7 +167,7 @@ def parse_non_math(string, start, ranges): return i else: level -= 1 - elif commented and string[i] == "\n": + elif string[i] == "\n": commented = False i += 1 if delim == "" and level == 0: From 12e3147e310ab20c0a01f0d46fbb51055cc06b1f Mon Sep 17 00:00:00 2001 From: Joon Bang Date: Thu, 4 Aug 2016 14:48:16 -0400 Subject: [PATCH 193/402] Clean up some code --- main_page/Categories.txt | 10 + main_page/main_page.mmd | 267 ++++++--- main_page/src/KLS.py | 19 - main_page/src/add_definition.py | 22 + main_page/src/headers.py | 64 --- main_page/src/list.py | 59 -- main_page/src/main.py | 111 +++- main_page/src/mod_main.py | 212 ++++--- main_page/src/symbols_list.py | 395 +++++++------ main_page/symbolsList.py~ | 278 --------- main_page/tex2Wiki.py~ | 966 -------------------------------- 11 files changed, 630 insertions(+), 1773 deletions(-) create mode 100644 main_page/Categories.txt delete mode 100644 main_page/src/KLS.py create mode 100644 main_page/src/add_definition.py delete mode 100644 main_page/src/headers.py delete mode 100644 main_page/src/list.py delete mode 100644 main_page/symbolsList.py~ delete mode 100644 main_page/tex2Wiki.py~ diff --git a/main_page/Categories.txt b/main_page/Categories.txt new file mode 100644 index 0000000..2d2fc4a --- /dev/null +++ b/main_page/Categories.txt @@ -0,0 +1,10 @@ + F - real or complex valued functions. + P - polynomials. + I - integer valued functions. + O - operators. + SM - semantic macros. + Q - quantifiers, set operators, and symbols. + SN - sets of numbers. + C - constants. + L - linear algebra. + D - distributions. diff --git a/main_page/main_page.mmd b/main_page/main_page.mmd index ff6a93c..3e07ca8 100644 --- a/main_page/main_page.mmd +++ b/main_page/main_page.mmd @@ -221,6 +221,10 @@ For more information see [http://link.springer.com/chapter/10.1007%2F978-3-319-0 * [[Definition:rabbitConstant|rabbitConstant]] * [[Definition:FibonacciNumber|FibonacciNumber]] * [[Definition:Fibonacci|Fibonacci]] +* [[Definition:RegGammaP|RegGammaP]] +* [[Definition:RegGammaQ|RegGammaQ]] +* [[Definition:GaussDistF|GaussDistF]] +* [[Definition:GaussDistQ|GaussDistQ]]
= Copyright = @@ -278,7 +282,7 @@ drmf_eof drmf_bof '''Definition:AffqKrawtchouk'''
-
<< [[Definition:Fibonacci|Definition:Fibonacci]]
+
<< [[Definition:GaussDistQ|Definition:GaussDistQ]]
[[Main_Page|Main Page]]
[[Definition:AlSalamCarlitzI|Definition:AlSalamCarlitzI]] >>
@@ -305,7 +309,7 @@ These are defined by [http://drmf.wmflabs.org/wiki/Definition:AffqKrawtchouk {\displaystyle K^{\mathrm{Aff}}_{n}}] : affine {\displaystyle q}-Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:AffqKrawtchouk http://drmf.wmflabs.org/wiki/Definition:AffqKrawtchouk]
[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
-
<< [[Definition:Fibonacci|Definition:Fibonacci]]
+
<< [[Definition:GaussDistQ|Definition:GaussDistQ]]
[[Main_Page|Main Page]]
[[Definition:AlSalamCarlitzI|Definition:AlSalamCarlitzI]] >>
@@ -328,7 +332,7 @@ In math mode, this macro can be called in the following ways: :'''\AlSalamCarlitzI{a}{n}@{x}{q}''' produces {\displaystyle \AlSalamCarlitzI{a}{n}@{x}{q}}
These are defined by -{\displaystyle +{\displaystyle \AlSalamCarlitzI{a}{n}@{x}{q}:=(-a)^nq^{\binom{n}{2}}\,\qHyperrphis{2}{1}@@{q^{-n},x^{-1}}{0}{q}{\frac{qx}{a}}. }
@@ -432,7 +436,7 @@ drmf_bof
[[Definition:BesselPolyIIparam|Definition:BesselPolyIIparam]] >>
-In the DRMF, the LaTeX semantic macro '''\AntiDer''' represents the anti-derivative notation, +In the DRMF, the LaTeX semantic macro '''\AntiDer''' represents the anti-derivative notation, which is defined as follows. This macro is in the category of polynomials. @@ -574,7 +578,7 @@ In math mode, this macro can be called in the following ways: :'''\bigqLaguerre{n}@{x}{a}{b}{q}''' produces {\displaystyle \bigqLaguerre{n}@{x}{a}{b}{q}}
These are defined by -{\displaystyle +{\displaystyle \bigqLaguerre{n}@{x}{a}{b}{q}:=\qHyperrphis{3}{2}@@{q^{-n},0,x}{aq,bq}{q}{q}. }
@@ -1024,7 +1028,7 @@ In math mode, this macro can be called in the following ways: :'''\GottliebLaguerre{n}''' produces {\displaystyle \GottliebLaguerre{n}}
:'''\GottliebLaguerre{n}@{x}''' produces {\displaystyle \GottliebLaguerre{n}@{x}}
-These are defined by +These are defined by {\displaystyle \GottliebLaguerre{n}@{x}:=e^{-n\la} \Meixner{n}@{x}{1}{e^{-\la}}. @@ -1050,7 +1054,7 @@ drmf_bof
[[Definition:JacksonqBesselII|Definition:JacksonqBesselII]] >>
-In the DRMF, the LaTeX semantic macro '''\Int''' represents the definite integral notation, +In the DRMF, the LaTeX semantic macro '''\Int''' represents the definite integral notation, which is defined as follows. This macro is in the category of polynomials. @@ -3376,7 +3380,7 @@ drmf_bof
[[Definition:qexpKLS|Definition:qexpKLS]] >>
-The LaTeX DLMF and DRMF macro '''\qExpKLS''' represents a q-analogue of the +The LaTeX DLMF and DRMF macro '''\qExpKLS''' represents a q-analogue of the \exp function: \qExpKLS{q}. This macro is in the category of real or complex valued functions. @@ -3415,7 +3419,7 @@ drmf_bof
[[Definition:qcosKLS|Definition:qcosKLS]] >>
-The LaTeX DLMF and DRMF macro '''\qexpKLS''' represents a q-analogue of the +The LaTeX DLMF and DRMF macro '''\qexpKLS''' represents a q-analogue of the \exp function: \qexpKLS{q}. This macro is in the category of real or complex valued functions. @@ -3452,7 +3456,7 @@ drmf_bof
[[Main_Page|Main Page]]
[[Definition:qMeixner|Definition:qMeixner]] >>
-The LaTeX DLMF and DRMF macro '''\qcosKLS''' represents a q-analogue of the +The LaTeX DLMF and DRMF macro '''\qcosKLS''' represents a q-analogue of the \cos function:~\qcosKLS{q}. This macro is in the category of polynomials. @@ -3692,7 +3696,7 @@ drmf_bof
[[Definition:qSinKLS|Definition:qSinKLS]] >>
-The LaTeX DLMF and DRMF macro '''\qsinKLS''' represents a q-analogue of the \sin +The LaTeX DLMF and DRMF macro '''\qsinKLS''' represents a q-analogue of the \sin function: \qsinKLS{q}. This macro is in the category of real or complex valued functions. @@ -3731,7 +3735,7 @@ drmf_bof
[[Definition:qCosKLS|Definition:qCosKLS]] >>
-The LaTeX DLMF and DRMF macro '''\qSinKLS''' represents a q-analogue of the +The LaTeX DLMF and DRMF macro '''\qSinKLS''' represents a q-analogue of the \sin function: \qSinKLS{q}. This macro is in the category of real or complex valued functions. @@ -3771,7 +3775,7 @@ drmf_bof
[[Definition:Wilson|Definition:Wilson]] >>
-The LaTeX DLMF and DRMF macro '''\qCosKLS''' represents a q-analogue of the +The LaTeX DLMF and DRMF macro '''\qCosKLS''' represents a q-analogue of the \cos function: \qCosKLS{q}. This macro is in the category of polynomials. @@ -3860,7 +3864,7 @@ These are defined by {}:=\HyperpFq{4}{3}@@{-n,n+\alpha+\beta+1,-x,x+\gamma+\delta+1}{\alpha+1,\beta+\delta+1,\gamma+1}{1}, } -where {\displaystyle \lambda(x)=x(x+\gamma+\delta+1), } +where {\displaystyle \lambda(x)=x(x+\gamma+\delta+1), } and \alpha+1=-N or \beta+\delta+1=-N or \gamma+1=-N with N a nonnegative integer. @@ -4531,7 +4535,7 @@ drmf_bof
[[Definition:PolylogarithmS|Definition:PolylogarithmS]] >>
-The LaTeX DLMF and DRMF macro '''\StieltjesConstants''' represents the Stiltjes constants +The LaTeX DLMF and DRMF macro '''\StieltjesConstants''' represents the Stiltjes constants \StieltjesConstants{n}. This macro is in the category of constants. @@ -4585,18 +4589,16 @@ In math mode, this macro can be called in the following ways: These are defined by - {\displaystyle - -\PolylogarithmS{n}{p}@{z} - +\PolylogarithmS{n}{p}@{z}:=\frac{(-1)^{n+p-1}}{(n-1)!p!}\int_0^1 \frac{(\ln{t})^{n-1}[\ln(1-zt)]^p}{t}\,dt } -
== Symbols List == -[p}$ {\displaystyle S_{n}] : Nielsen generalized polylogarithm function : [p}$ p}$] [http://drmf.wmflabs.org/wiki/Definition:PolylogarithmS http://drmf.wmflabs.org/wiki/Definition:PolylogarithmS] +[p}$ {\displaystyle S_{n}] : Nielsen generalized polylogarithm function : [p}$ p}$] [http://drmf.wmflabs.org/wiki/Definition:PolylogarithmS http://drmf.wmflabs.org/wiki/Definition:PolylogarithmS]
+[http://dlmf.nist.gov/1.4#iv {\displaystyle \int}] : integral : [http://dlmf.nist.gov/1.4#iv http://dlmf.nist.gov/1.4#iv]
+[http://dlmf.nist.gov/4.2#E2 {\displaystyle \mathrm{ln}}] : principal branch of logarithm function : [http://dlmf.nist.gov/4.2#E2 http://dlmf.nist.gov/4.2#E2]
<< [[Definition:StieltjesConstants|Definition:StieltjesConstants]]
[[Main_Page|Main Page]]
@@ -4624,18 +4626,15 @@ In math mode, this macro can be called in the following ways: These are defined by - {\displaystyle - -\HarmonicNumber[r]{n} - +\HarmonicNumber[r]{n}:=\sum_{k=1}^n \frac{1}{k^r} } -
== Symbols List == -[http://dlmf.nist.gov/25.11#E33 {\displaystyle H_{n}}] : Harmonic number : [http://dlmf.nist.gov/25.11#E33 http://dlmf.nist.gov/25.11#E33] +[http://dlmf.nist.gov/25.11#E33 {\displaystyle H_{n}}] : Harmonic number : [http://dlmf.nist.gov/25.11#E33 http://dlmf.nist.gov/25.11#E33]
+[http://drmf.wmflabs.org/wiki/Definition:sum {\displaystyle \Sigma}] : sum : [http://drmf.wmflabs.org/wiki/Definition:sum http://drmf.wmflabs.org/wiki/Definition:sum]
<< [[Definition:PolylogarithmS|Definition:PolylogarithmS]]
[[Main_Page|Main Page]]
@@ -4662,13 +4661,9 @@ In math mode, this macro can be called in the following ways: These are defined by - {\displaystyle - -\LucasL{n}@{x} - +\LucasL{n}@{x}:=2^{-n}\left[\left(x-\sqrt{x^2+4}\right)^n+\left(x+\sqrt{x^2+4}\right)^n\right] } -
== Symbols List == @@ -4700,18 +4695,15 @@ In math mode, this macro can be called in the following ways: These are defined by - {\displaystyle - -\HurwitzLerchPhi@{z}{s}{a} - +\HurwitzLerchPhi@{z}{s}{a}:=\sum_{k=0}^\infty \frac{z^k}{(a+k)^{s}} } -
== Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:HurwitzLerchPhi {\displaystyle \Phi}] : Hurwitz-Lerch's transcendent : [http://drmf.wmflabs.org/wiki/Definition:HurwitzLerchPhi http://drmf.wmflabs.org/wiki/Definition:HurwitzLerchPhi] +[http://drmf.wmflabs.org/wiki/Definition:HurwitzLerchPhi {\displaystyle \Phi}] : Hurwitz-Lerch's transcendent : [http://drmf.wmflabs.org/wiki/Definition:HurwitzLerchPhi http://drmf.wmflabs.org/wiki/Definition:HurwitzLerchPhi]
+[http://drmf.wmflabs.org/wiki/Definition:sum {\displaystyle \Sigma}] : sum : [http://drmf.wmflabs.org/wiki/Definition:sum http://drmf.wmflabs.org/wiki/Definition:sum]
<< [[Definition:LucasL|Definition:LucasL]]
[[Main_Page|Main Page]]
@@ -4737,13 +4729,9 @@ In math mode, this macro can be called in the following way: These are defined by - {\displaystyle - -\GoldenRatio - +\GoldenRatio:=\frac{1}{2}\left(1+\sqrt{5}\right) } -
== Symbols List == @@ -4775,18 +4763,15 @@ In math mode, this macro can be called in the following ways: These are defined by - {\displaystyle - -\WhitPsi{\alpha}{\beta}@{z} - +\WhitPsi{\alpha}{\beta}@{z}:=z^{\frac{\alpha+\beta}{2}-1}e^\frac{z}{2}\WhitW{-\frac{\alpha+\beta}{2}}{\frac{\beta-\alpha}{2}}@{z} } -
== Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:WhitPsi {\displaystyle \Psi_{\alpha,\beta}}] : Whittaker function {\displaystyle \Psi_{\alpha;\beta}} : [http://drmf.wmflabs.org/wiki/Definition:WhitPsi http://drmf.wmflabs.org/wiki/Definition:WhitPsi] +[http://drmf.wmflabs.org/wiki/Definition:WhitPsi {\displaystyle \Psi_{\alpha,\beta}}] : Whittaker function {\displaystyle \Psi_{\alpha;\beta}} : [http://drmf.wmflabs.org/wiki/Definition:WhitPsi http://drmf.wmflabs.org/wiki/Definition:WhitPsi]
+[http://dlmf.nist.gov/13.14#E3 {\displaystyle W_{\kappa,\mu}}] : Whittaker function {\displaystyle W_{\kappa;\mu}} : [http://dlmf.nist.gov/13.14#E3 http://dlmf.nist.gov/13.14#E3]
<< [[Definition:GoldenRatio|Definition:GoldenRatio]]
[[Main_Page|Main Page]]
@@ -4812,18 +4797,15 @@ In math mode, this macro can be called in the following way: These are defined by - {\displaystyle - -\GompertzConstant - +\GompertzConstant:=\int_0^\infty \frac{e^{-u}}{1+u}\,du } -
== Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:GompertzConstant {\displaystyle G}] : Gompertz's constant : [http://drmf.wmflabs.org/wiki/Definition:GompertzConstant http://drmf.wmflabs.org/wiki/Definition:GompertzConstant] +[http://drmf.wmflabs.org/wiki/Definition:GompertzConstant {\displaystyle G}] : Gompertz's constant : [http://drmf.wmflabs.org/wiki/Definition:GompertzConstant http://drmf.wmflabs.org/wiki/Definition:GompertzConstant]
+[http://dlmf.nist.gov/1.4#iv {\displaystyle \int}] : integral : [http://dlmf.nist.gov/1.4#iv http://dlmf.nist.gov/1.4#iv]
<< [[Definition:WhitPsi|Definition:WhitPsi]]
[[Main_Page|Main Page]]
@@ -4849,18 +4831,17 @@ In math mode, this macro can be called in the following way: These are defined by - {\displaystyle - -\rabbitConstant - +\rabbitConstant:=\sum_{i=1}^\infty 2^{-\floor{i\GoldenRatio}} } -
== Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:RabbitConstant {\displaystyle \rho}] : rabbit constant : [http://drmf.wmflabs.org/wiki/Definition:RabbitConstant http://drmf.wmflabs.org/wiki/Definition:RabbitConstant] +[http://drmf.wmflabs.org/wiki/Definition:RabbitConstant {\displaystyle \rho}] : rabbit constant : [http://drmf.wmflabs.org/wiki/Definition:RabbitConstant http://drmf.wmflabs.org/wiki/Definition:RabbitConstant]
+[http://drmf.wmflabs.org/wiki/Definition:sum {\displaystyle \Sigma}] : sum : [http://drmf.wmflabs.org/wiki/Definition:sum http://drmf.wmflabs.org/wiki/Definition:sum]
+[http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r16 {\displaystyle \left\lfloor a\right\rfloor}] : floor : [http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r16 http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r16]
+[http://drmf.wmflabs.org/wiki/Definition:GoldenRatio {\displaystyle \phi}] : The golden ratio : [http://drmf.wmflabs.org/wiki/Definition:GoldenRatio http://drmf.wmflabs.org/wiki/Definition:GoldenRatio]
<< [[Definition:GompertzConstant|Definition:GompertzConstant]]
[[Main_Page|Main Page]]
@@ -4886,13 +4867,9 @@ In math mode, this macro can be called in the following way: These are defined by - {\displaystyle - -\FibonacciNumber{n} - +\FibonacciNumber{n}:=\frac{\left(1+\sqrt{5}\right)^n-\left(1-\sqrt{5}\right)^n}{2^n\sqrt{5}} } -
== Symbols List == @@ -4909,7 +4886,7 @@ drmf_bof
<< [[Definition:FibonacciNumber|Definition:FibonacciNumber]]
[[Main_Page|Main Page]]
-
[[Definition:AffqKrawtchouk|Definition:AffqKrawtchouk]] >>
+
[[Definition:RegGammaP|Definition:RegGammaP]] >>
The LaTeX DLMF and DRMF macro '''\Fibonacci''' represents the Fibonacci. @@ -4922,11 +4899,44 @@ In math mode, this macro can be called in the following way: These are defined by + +{\displaystyle +\Fibonacci{n}@{x}:=\frac{\left(x+\sqrt{x^2+4}\right)^n-\left(x-\sqrt{x^2+4}\right)^n}{2^n\sqrt{x^2+4}} +} +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:Fibonacci {\displaystyle F_{n}{(x)}}] : Fibonacci polynomial : [http://drmf.wmflabs.org/wiki/Definition:Fibonacci http://drmf.wmflabs.org/wiki/Definition:Fibonacci] +
+
<< [[Definition:FibonacciNumber|Definition:FibonacciNumber]]
+
[[Main_Page|Main Page]]
+
[[Definition:RegGammaP|Definition:RegGammaP]] >>
+
+drmf_eof +drmf_bof +'''Definition:RegGammaP''' +
+
<< [[Definition:Fibonacci|Definition:Fibonacci]]
+
[[Main_Page|Main Page]]
+
[[Definition:RegGammaQ|Definition:RegGammaQ]] >>
+
+The LaTeX DLMF and DRMF macro '''\RegGammaP''' represents the RegGammaP. + +This macro is in the category of integer valued functions. + +In math mode, this macro can be called in the following ways: + +:'''\RegGammaP''' produces {\displaystyle \RegGammaP}
+:'''\RegGammaP@{x}{\alpha}{\theta}''' produces {\displaystyle \RegGammaP@{x}{\alpha}{\theta}}
+ +These are defined by + {\displaystyle -\Fibonacci{n}@{x} +\RegGammaP@{x}{\alpha}{\theta}:=\frac{\incgamma@{\alpha}{\frac{x}{\theta}}}{\EulerGamma@{\alpha}} } @@ -4934,9 +4944,126 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:Fibonacci {\displaystyle F_{n}{(x)}}] : Fibonacci polynomial : [http://drmf.wmflabs.org/wiki/Definition:Fibonacci http://drmf.wmflabs.org/wiki/Definition:Fibonacci] +[\theta\right)$ {\displaystyle P\left(x;\alpha}] : Regularized gamma function of incomplete gamma function : [\theta\right)$ \theta\right)$] [http://drmf.wmflabs.org/wiki/Definition:RegGamma http://drmf.wmflabs.org/wiki/Definition:RegGamma]
+[http://dlmf.nist.gov/8.2#E1 {\displaystyle \gamma}] : incomplete gamma function {\displaystyle \gamma} : [http://dlmf.nist.gov/8.2#E1 http://dlmf.nist.gov/8.2#E1]
+[http://dlmf.nist.gov/5.2#E1 {\displaystyle \Gamma}] : Euler's gamma function : [http://dlmf.nist.gov/5.2#E1 http://dlmf.nist.gov/5.2#E1]
-
<< [[Definition:FibonacciNumber|Definition:FibonacciNumber]]
+
<< [[Definition:Fibonacci|Definition:Fibonacci]]
+
[[Main_Page|Main Page]]
+
[[Definition:RegGammaQ|Definition:RegGammaQ]] >>
+
+drmf_eof +drmf_bof +'''Definition:RegGammaQ''' +
+
<< [[Definition:RegGammaP|Definition:RegGammaP]]
+
[[Main_Page|Main Page]]
+
[[Definition:GaussDistF|Definition:GaussDistF]] >>
+
+The LaTeX DLMF and DRMF macro '''\RegGammaQ''' represents the RegGammaQ. + +This macro is in the category of integer valued functions. + +In math mode, this macro can be called in the following ways: + +:'''\RegGammaQ''' produces {\displaystyle \RegGammaQ}
+:'''\RegGammaQ@{x}{\alpha}{\theta}''' produces {\displaystyle \RegGammaQ@{x}{\alpha}{\theta}}
+ +These are defined by + + + +{\displaystyle + +\RegGammaQ@{x}{\alpha}{\theta}:=\frac{\IncGamma@{\alpha}{\frac{x}{\theta}}}{\EulerGamma@{\alpha}} + +} + +
+ +== Symbols List == + +[\theta\right)$ {\displaystyle Q\left(x;\alpha}] : Regularized gamma function of complementary incomplete gamma function : [\theta\right)$ \theta\right)$] [http://drmf.wmflabs.org/wiki/Definition:RegGamma http://drmf.wmflabs.org/wiki/Definition:RegGamma]
+[http://dlmf.nist.gov/8.2#E2 {\displaystyle \Gamma}] : incomplete gamma function {\displaystyle \Gamma} : [http://dlmf.nist.gov/8.2#E2 http://dlmf.nist.gov/8.2#E2]
+[http://dlmf.nist.gov/5.2#E1 {\displaystyle \Gamma}] : Euler's gamma function : [http://dlmf.nist.gov/5.2#E1 http://dlmf.nist.gov/5.2#E1] +
+
<< [[Definition:RegGammaP|Definition:RegGammaP]]
+
[[Main_Page|Main Page]]
+
[[Definition:GaussDistF|Definition:GaussDistF]] >>
+
+drmf_eof +drmf_bof +'''Definition:GaussDistF''' +
+
<< [[Definition:RegGammaQ|Definition:RegGammaQ]]
+
[[Main_Page|Main Page]]
+
[[Definition:GaussDistQ|Definition:GaussDistQ]] >>
+
+The LaTeX DLMF and DRMF macro '''\GaussDistF''' represents the GaussDistF. + +This macro is in the category of integer valued functions. + +In math mode, this macro can be called in the following ways: + +:'''\GaussDistF''' produces {\displaystyle \GaussDistF}
+:'''\GaussDistF@{x}''' produces {\displaystyle \GaussDistF@{x}}
+ +These are defined by + + + +{\displaystyle + +\GaussDistF@{x}:=\frac{1}{2}\left(1+\erf@{\frac{x}{\sqrt{2}}}\right) + +} + +
+ +== Symbols List == + +[0 {\displaystyle F\left(x}] : Gaussian distribution : [0 0] [1\right)$ 1\right)$] [http://drmf.wmflabs.org/wiki/Definition:GaussDist http://drmf.wmflabs.org/wiki/Definition:GaussDist]
+[http://dlmf.nist.gov/7.2#E1 {\displaystyle \mathrm{erf}}] : error function : [http://dlmf.nist.gov/7.2#E1 http://dlmf.nist.gov/7.2#E1] +
+
<< [[Definition:RegGammaQ|Definition:RegGammaQ]]
+
[[Main_Page|Main Page]]
+
[[Definition:GaussDistQ|Definition:GaussDistQ]] >>
+
+drmf_eof +drmf_bof +'''Definition:GaussDistQ''' +
+
<< [[Definition:GaussDistF|Definition:GaussDistF]]
+
[[Main_Page|Main Page]]
+
[[Definition:AffqKrawtchouk|Definition:AffqKrawtchouk]] >>
+
+The LaTeX DLMF and DRMF macro '''\GaussDistQ''' represents the GaussDistQ. + +This macro is in the category of integer valued functions. + +In math mode, this macro can be called in the following ways: + +:'''\GaussDistQ''' produces {\displaystyle \GaussDistQ}
+:'''\GaussDistQ@{x}''' produces {\displaystyle \GaussDistQ@{x}}
+ +These are defined by + + + +{\displaystyle + +\GaussDistQ@{x}:=\frac{1}{2}\erfc@{\frac{x}{\sqrt{2}}} + +} + +
+ +== Symbols List == + +[0 {\displaystyle Q\left(x}] : Related to Gaussian distribution : [0 0] [1\right)$ 1\right)$] [http://drmf.wmflabs.org/wiki/Definition:GaussDist http://drmf.wmflabs.org/wiki/Definition:GaussDist]
+[http://dlmf.nist.gov/7.2#E2 {\displaystyle \mathrm{erfc}}] : complementary error function : [http://dlmf.nist.gov/7.2#E2 http://dlmf.nist.gov/7.2#E2] +
+
<< [[Definition:GaussDistF|Definition:GaussDistF]]
[[Main_Page|Main Page]]
[[Definition:AffqKrawtchouk|Definition:AffqKrawtchouk]] >>
diff --git a/main_page/src/KLS.py b/main_page/src/KLS.py deleted file mode 100644 index 32bf360..0000000 --- a/main_page/src/KLS.py +++ /dev/null @@ -1,19 +0,0 @@ -__author__ = "Azeem Mohammed" -__status__ = "Development" - -import os - - -# TODO: Actually make the if statements do what they're supposed to. - -if raw_input("Update Main Page? (y or n) ") != "n": - print "List" - os.system("python src/list.py") - print "Symbols List" - os.system("python src/symbols_list.py") - print "Headers" - os.system("python src/headers.py") - print "List Ways" - os.system("python src/mod_main.py") - -print "Done" diff --git a/main_page/src/add_definition.py b/main_page/src/add_definition.py new file mode 100644 index 0000000..41c4d7c --- /dev/null +++ b/main_page/src/add_definition.py @@ -0,0 +1,22 @@ +__author__ = "Joon Bang" +__status__ = "Development" + + +def main(): + macro_name = raw_input("Macro name: ") + definition = raw_input("LaTeX definition: ") + + result = "drmf_bof\n" + \ + "'''Definition:" + macro_name + "'''\n" + \ + "The LaTeX DLMF and DRMF macro '''\\" + macro_name + "''' represents the " + macro_name + ".\n\n" + \ + "These are defined by \n\n" + \ + "\n\n{\\displaystyle\n\n" + definition + "\n\n}\n\n
\n\ndrmf_eof\n" + + with open("main_page.mmd", "a") as main_page: + main_page.write("\n" + result) + + print "\n" + result + + +if __name__ == '__main__': + main() diff --git a/main_page/src/headers.py b/main_page/src/headers.py deleted file mode 100644 index f760e99..0000000 --- a/main_page/src/headers.py +++ /dev/null @@ -1,64 +0,0 @@ -__author__ = "Azeem Mohammed" -__status__ = "Development" - -import copy - -with open("main_page.mmd", "r") as main_page: - lines = main_page.read() - main_lines = copy.copy(lines) - -start = lines.find("* [[Definition:", lines.find("= Definition Pages =")) -end = lines.find("" in text: - search_string = ">>
\n
" - text = text[text.find(search_string) + len(search_string):] - - if "
" in text: - text = text[:text.find("
")] - - prev = pages[(count - 1) % len(pages)] - cur = "Main Page" - next = pages[(count + 1) % len(pages)] - - delimiter = "\n
<< [[" + prev.replace(" ", "_") + "|" + prev + "]]
" + \ - "\n
[[" + cur.replace(" ", "_") + "|" + cur + "]]
" + \ - "\n
[[" + next.replace(" ", "_") + "|" + next + "]] >>
" + \ - "\n
" - - header = "\n
" + delimiter - footer = "
" + delimiter - - main_page.write("drmf_bof\n" + title + header + text + footer + "\ndrmf_eof\n") - - count += 1 diff --git a/main_page/src/list.py b/main_page/src/list.py deleted file mode 100644 index b65aeb8..0000000 --- a/main_page/src/list.py +++ /dev/null @@ -1,59 +0,0 @@ -__author__ = "Azeem Mohammed" -__status__ = "Development" - -import copy -import time - - -def compare_string(a, b): - """Returns whether string a is "less than" string b, up to the pipe symbol.""" - x = a[:a.find("|")] - y = b[:b.find("|")] - - # pipe missing in a string - if not x or not y: - return True - - return x < y - - -def sum_ordinal(string): - """Returns the sum of the ordinal values of all characters in string.""" - return sum((ord(ch) for ch in string)) - - -def sort(strings): - return sorted(copy.deepcopy(strings), cmp=compare_string) - - -def main(): - localtime = time.asctime(time.localtime(time.time())).replace(" ", "_") - with open("main_page.mmd") as main_page: - with open("backups/main_page.mmd" + str(localtime), "w") as main_page_bak: - main_page_bak.write(main_page.read()) - - with open("main_page.mmd") as main_page: - main_text = main_page.read() - - main_lines = main_text.split("\n") - - text = "" - - for line in main_lines: - if "'''Definition:" in line: - definition = line[line.find("'''Definition:") + 3:line.find("'''", line.find("'''Definition:") + 4)] - element = "* [[" + definition + "|" + definition[len("Definition:"):] + "]]" - text += element + "\n" - - s = main_text.find("== Definition Pages ==") - e = main_text.find("= Copyright =") - - text = main_text[:s] + \ - "== Definition Pages ==\n\n
\n" + \ - text + "
\n\n" + main_text[e:] - - with open("main_page.mmd", "w") as main_page: - main_page.write(text) - -if __name__ == '__main__': - main() \ No newline at end of file diff --git a/main_page/src/main.py b/main_page/src/main.py index 19dd4c3..d6a93f4 100644 --- a/main_page/src/main.py +++ b/main_page/src/main.py @@ -1,8 +1,111 @@ __author__ = "Azeem Mohammed" __status__ = "Development" -__credits__ = ["Azeem Mohammed", "Joon Bang"] +__credits__ = ["Joon Bang", "Azeem Mohammed"] -import os +import time +import symbols_list +import mod_main -if raw_input("Update main page? (y/n): ") != "n": - pass +""" +Order: +- List +- Symbols List +- Headers +- List Ways +""" + + +def generate_html(tag_name, options, text, spacing=True): + result = "<" + tag_name + if options != "": + result += " " + options + result += ">\n" + text + "\n\n" + + if not spacing: + return result.replace("\n", " ")[:-1] + "\n" + + return result + + +def update_macro_list(text): + lines = text.split("\n") + definitions = list() + + for line in lines: + if "'''Definition:" in line: + macro = line.split("'''")[1][11:] + element = "* [[Definition:" + macro + "|" + macro + "]]" + definitions.append(element) + + def_contents = generate_html("div", "style=\"-moz-column-count:2; column-count:2;-webkit-column-count:2\"", + "\n".join(definitions)) + + start = text.find("== Definition Pages ==") + end = text.find("= Copyright =") + + return text[:start + 23] + "\n" + def_contents + "\n" + text[end:], definitions + + +def update_headers(text, definitions): + pages = text.split("drmf_eof")[:-1] + + macros = list() + formatted = list() + for definition in definitions: + macros.append("Definition:" + definition.split("|")[1][:-2]) + formatted.append("[[" + macros[-1].replace(" ", "_") + "|" + macros[-1] + "]]") + + output = "" + + i = 0 + for page in pages: + if "'''Definition:" not in page: + output += page + "drmf_eof\n" + continue + + if "
" in page: + page = page[page.find(">>
\n
") + len(">>
\n
"):] + + if "
" in page: + page = page[:page.find("
")] + + nav_section = generate_html("div", "id=\"alignleft\"", "<< " + formatted[i - 1], spacing=False) + \ + generate_html("div", "id=\"aligncenter\"", "[[Main_Page|Main Page]]", spacing=False) + \ + generate_html("div", "id=\"alignright\"", formatted[(i + 1) % len(formatted)] + " >>", + spacing=False)[:-1] + + header = generate_html("div", "id=\"drmf_head\"", nav_section)[:-1] + footer = generate_html("div", "id=\"drmf_foot\"", nav_section)[:-1] + + output += "drmf_bof\n" + "'''" + macros[i] + "'''\n" + header + page[:-1] + footer + "\ndrmf_eof\n" + + i += 1 + + return output + + +def main(): + if raw_input("Update main page? (y/n): ") == "n": + return + + # read contents + with open("main_page.mmd") as main_page: + text = main_page.read() + + # create backup + local_time = time.asctime(time.localtime(time.time())).split() + local_time = "_" + '_'.join(local_time[1:3] + local_time[3:][::-1]) + with open("backups/main_page" + local_time + ".mmd.bak", "w") as backup: + backup.write(text) + + text, definitions = update_macro_list(text) + text = symbols_list.main(text) + text = update_headers(text, definitions) + text = mod_main.main(text) + + with open("main_page.mmd", "w") as main_page: + main_page.write(text) + + +if __name__ == '__main__': + main() diff --git a/main_page/src/mod_main.py b/main_page/src/mod_main.py index fab8c86..e7090f7 100644 --- a/main_page/src/mod_main.py +++ b/main_page/src/mod_main.py @@ -1,125 +1,115 @@ __author__ = "Azeem Mohammed" __status__ = "Development" +__credits__ = ["Joon Bang", "Azeem Mohammed"] -import copy import csv -DEF_STRING = "'''Definition:" +def atSort(l): + return (sorted(l, key=lambda x: x.count("@"))) -def at_sort(strings): - """Sorts by number of @ symbols in string.""" - return sorted(copy.copy(strings), key=lambda ch: ch.count("@")) - - -def remove_brackets_text(string): - left_brackets = 0 - text = "" - - for ch in string: - if ch == "[": - left_brackets += 1 - elif ch == "]": - left_brackets -= 1 - elif not left_brackets: - text += ch - - return text - - -def find_all_pos(info): - # TODO: what is info? what does pos mean??? +def remOpt(p): + parse = True + ret = "" + for x in p: + if parse: + if x == "[": + parse = False + else: + ret += x + else: + if x == "]": + parse = True + return ret - original = info[0] - key_string = info[3] - before, key_string = key_string.split("|", 1) - keys = key_string.split(":") +def findAllPos(g): + key = g[3] + orig = g[0] + r = key[:key.find("|")] + key = key[key.find("|") + 1:] + keys = key.split(":") ret = [] - - for key in keys: - if key in ["F", "FO", "O", "O1"]: - if "@" in original: - ret.append(original[:original.find("@")]) - if key == "O": - ret.append(original) + for x in keys: + if x == "F" or x == "FO" or x == "O" or x == "O1": + if orig.find("@") != -1: + ret.append(orig[0:orig.find("@")]) + if x == "O": + ret.append(orig) else: - ret.append(original) - elif key == "FnO": # remove optional parameters - if original.find("@") != -1: - ret.append(remove_brackets_text(original[0:original.find("@")])) + ret.append(orig) + if x == "FnO": # remove optional parameters + if orig.find("@") != -1: + ret.append(remOpt(orig[0:orig.find("@")])) else: - ret.append(remove_brackets_text(original)) - elif key in ["P", "PO"]: - ret.append(original.replace("@@", "@")) - elif key == "PnO": # remove optional parameters - ret.append(remove_brackets_text(original.replace("@@", "@"))) - elif key in ["nP", "nPO", "PS", "O2"]: - ret.append(original) - elif key == "nPnO": # remove optional paramters - ret.append(remove_brackets_text(original)) - - if "fo" in key: - ret.append(original.replace("@@@", "@" * int(key[2]))) - - return at_sort(ret), before - -with open("main_page.mmd") as main_page: - lines = main_page.read() - -lines = lines.replace("The LaTeX DLMF macro '''\\", "The LaTeX DLMF and DRMF macro '''\\") - -with open("Categories.txt") as categories: - cats = [category.strip() for category in categories.readlines()] - -text = "" -i = 0 - -while True: - n = lines.find(DEF_STRING, i) - if n == -1: - break - else: - macroN = lines[n + len(DEF_STRING):lines.find("'''", n + len(DEF_STRING))] - r = lines.find("'''\\", n) - q = lines.find("'''", r + len("'''\\")) - - if lines[r:q].find("{") != -1: - lines = lines[:r] + "'''\\" + macroN + lines[q:] - - if lines.find("\nThis macro is in the category of", n) < lines.find(DEF_STRING, n + 15) \ - and "\nThis macro is in the category of" in lines[n:]: - p = lines.find("\nThis macro is in the category of", n) + ret.append(remOpt(orig)) + if x == "P" or x == "PO": + ret.append(orig.replace("@@", "@")) + if x == "PnO": # remove optional parameters + ret.append(remOpt(orig.replace("@@", "@"))) + if x == "nP" or x == "nPO" or x == "PS" or x == "O2": + ret.append(orig) + if x == "nPnO": # remove optional paramters + ret.append(remOpt(orig)) + if "fo" in x: + ret.append(orig.replace("@@@", "@" * int(x[2]))) + return atSort(ret), r + + +def main(lines): + lines = lines.replace("The LaTeX DLMF macro \'\'\'\\", "The LaTeX DLMF and DRMF macro \'\'\'\\") + flag = True + gCSV = csv.reader(open('new.Glossary.csv', 'rb'), delimiter=',', quotechar='\"') + toWrite = "" + i = 0 + + categories = open("Categories.txt", "r") + cats = categories.readlines() + categories.close() + for x in range(0, len(cats)): + cats[x] = cats[x].strip() + while flag: + n = lines.find("\'\'\'Definition:", i) + if n == -1: + flag = False else: - p = lines.find("\n", lines.find(".", n)) - - text += lines[i:p].rstrip() + "\n\n" - - count = 0 - list_calls = [] - - glossary = csv.reader(open('new.Glossary.csv', 'rb'), delimiter=',', quotechar='\"') - for entry in glossary: - if not entry[0].find("\\" + macroN) and (len(entry[0]) == len(macroN) + 1 or not entry[0][len(macroN) + 1].isalpha()): - q, s = find_all_pos(entry) - list_calls += q - count += len(q) - - for cat in cats: - if s + " -" in cat: - category = "This macro is in the category of" + cat[cat.find("-") + 2:] + "\n\n" - break - - new = category + "In math mode, this macro can be called in the following way" + "s" * (count > 1) + ":\n\n" - - for q in range(0, len(list_calls)): - c = list_calls[q] - new += ":'''" + c + "'''" + " produces {\\displaystyle " + c + "}
\n" - - # Now add the multiple ways \macroname{n}@... produces \macroname{n}@... - text += new + "\n" - i = lines.find("These are defined by", p) - -with open("main_page.mmd", "w") as main_page: - main_page.write(text + lines[i:]) + macroN = lines[n + len("\'\'\'Definition:"):lines.find("\'\'\'", n + len("\'\'\'Definition:"))] + r = lines.find("\'\'\'\\", n) + q = lines.find("\'\'\'", r + len("\'\'\'\\")) + if lines[r:q].find("{") != -1: + lines = lines[0:r] + "\'\'\'\\" + macroN + lines[q:] + + if lines.find("\nThis macro is in the category of", n) < lines.find("\'\'\'Definition:", n + len( + "\'\'\'Definition:") + 1) and lines.find("\nThis macro is in the category of", n) != -1: + # print n + p = lines.find("\nThis macro is in the category of", n) + else: + p = lines.find("\n", lines.find(".", n)) + toWrite += lines[i:p] + toWrite = toWrite.rstrip() + toWrite += "\n\n" + count = 0 + listCalls = [] + gCSV = csv.reader(open('new.Glossary.csv', 'rb'), delimiter=',', quotechar='\"') + for g in gCSV: + if g[0].find("\\" + macroN) == 0 and (len(g[0]) == len(macroN) + 1 + or not g[0][len(macroN) + 1].isalpha()): + q, s = findAllPos(g) + listCalls += q + count += len(q) + plural = 1 # if count>1: plural =1; if count=1: plural=0 + if count == 1: plural = 0 + for t in cats: + if s + " -" in t: + category = "This macro is in the category of" + t[t.find("-") + 2:] + break + new = category + "\n\nIn math mode, this macro can be called in the following way" + "s" * plural + ":\n\n" + for q in range(0, len(listCalls)): + c = listCalls[q] + new += ":\'\'\'" + c + "\'\'\'" + " produces {\\displaystyle " + c + "}
\n" + # Now add the multiple ways \macroname{n}@... produces \macroname{n}@... + toWrite += new + "\n" + i = lines.find("These are defined by", p) + + return toWrite + lines[i:] diff --git a/main_page/src/symbols_list.py b/main_page/src/symbols_list.py index bdb8087..48eef80 100644 --- a/main_page/src/symbols_list.py +++ b/main_page/src/symbols_list.py @@ -1,9 +1,9 @@ __author__ = "Azeem Mohammed" __status__ = "Development" +__credits__ = ["Joon Bang", "Azeem Mohammed"] import csv - def getSym(line): # Gets all symbols on a line for symbols list line = line.replace("\n", " ") symList = [] @@ -16,10 +16,10 @@ def getSym(line): # Gets all symbols on a line for symbols list for i in range(0, len(line)): c = line[i] if symFlag: - if c in ["{", "["]: + if c == "{" or c == "[": cC += 1 argFlag = True - if c not in ["}", "]"]: + if c != "}" and c != "]": if argFlag or c.isalpha(): symbol += c else: @@ -44,242 +44,233 @@ def getSym(line): # Gets all symbols on a line for symbols list elif c == "\\": symFlag = True - symList.append(symbol) symList += getSym(symbol) - return symList - -mainPage = open("main_page.mmd", "r") -mainLines = mainPage.read() -mainLines = mainLines[mainLines.find("drmf_bof"):mainLines.rfind("drmf_eof") + len("drmf_eof\n")] -mainPage.close() -main = open("main_page.mmd", "w") -mainPages = mainLines.split("drmf_eof") - -if mainPages[-1].strip() == "": - mainPages = mainPages[0:-1] + return (symList) -print len(mainPages) -glossary = open('new.Glossary.csv', 'rb') -gCSV = csv.reader(glossary, delimiter=',', quotechar='\"') -lGlos = glossary.readlines() -wiki = "" - -for g in mainPages: - if not "\'\'\'Definition:" in g: - h = g.replace("drmf_eof", "") - h = g.replace("drmf_bof", "") - toWrite = "drmf_bof" + h + "drmf_eof\n" - main.write(toWrite) - else: - h = g.replace("drmf_eof", "") - h = g.replace("drmf_bof", "") - h = h.strip("\n") - sflag1 = False - sflag2 = False - if h.find("== Symbols List ==") != -1: - toWrite = h[0:h.find("== Symbols List ==")] - h = toWrite +def main(mainLines): + mainLines = mainLines[mainLines.find("drmf_bof"):mainLines.rfind("drmf_eof") + len("drmf_eof\n")] + mainPages = mainLines.split("drmf_eof") + if mainPages[-1].strip() == "": + mainPages = mainPages[0:len(mainPages) - 1] + print len(mainPages) + glossary = open('new.Glossary.csv', 'rb') + gCSV = csv.reader(glossary, delimiter=',', quotechar='\"') + lGlos = glossary.readlines() + wiki = "" + result = "" + for g in mainPages: + if not "\'\'\'Definition:" in g: + h = g.replace("drmf_eof", "") + h = g.replace("drmf_bof", "") + toWrite = "drmf_bof" + h + "drmf_eof\n" + result += toWrite else: - sflag1 = True - if h.find("drmf_foot") == -1: - toWrite = h - sflag2 = True - else: - toWrite = h[0:h.find("
len(symbol) and (Q[len(symbol)] == "{" or Q[len(symbol)] == "["): - ap = "" - for o in range(len(symbol), len(Q)): - if Q[o] == "{" or z == "[": - argFlag = True - elif Q[o] == "}" or z == "]": - argFlag = False - listArgs.append(ap) - ap = "" + if z == "{" or z == "[": + cC += 1 + if z == "}" or z == "]": + cC -= 1 + if cC == 0: + if parFlag: + parCx += 1 + else: + ArgCx += 1 + if G[0].find(symbol) == 0 and (len(G[0]) == len(symbol) or not G[0][ + len(symbol)].isalpha()): # and (numPar!=0 or numArg!=0): + checkFlag = True + get = True + preG = S + elif checkFlag: + get = True + checkFlag = False + if (get): + if get: + G = preG + get = False + checkFlag = False + if True: + if symbolPar.find("@") != -1: + Q = symbolPar[:symbolPar.find("@")] else: - ap += Q[o] - websiteF = "" - web1 = G[5] - for t in range(5, len(G)): - if G[t] != "": - websiteF = websiteF + " [" + G[t] + " " + G[t] + "]" - p1 = G[4].strip("$") - p1 = "{\\displaystyle " + p1 + "}" - new1 = "" - new2 = "" - pause = False - mathF = True - p2 = G[1] - for k in range(0, len(p2)): - if p2[k] == "$": - if mathF: - new2 += "{\\displaystyle " + Q = symbolPar + listArgs = [] + if len(Q) > len(symbol) and (Q[len(symbol)] == "{" or Q[len(symbol)] == "["): + ap = "" + for o in range(len(symbol), len(Q)): + if Q[o] == "{" or z == "[": + argFlag = True + elif Q[o] == "}" or z == "]": + argFlag = False + listArgs.append(ap) + ap = "" + else: + ap += Q[o] + websiteF = "" + web1 = G[5] + for t in range(5, len(G)): + if G[t] != "": + websiteF = websiteF + " [" + G[t] + " " + G[t] + "]" + p1 = G[4].strip("$") + p1 = "{\\displaystyle " + p1 + "}" + new1 = "" + new2 = "" + pause = False + mathF = True + p2 = G[1] + for k in range(0, len(p2)): + if p2[k] == "$": + if mathF: + new2 += "{\\displaystyle " + else: + new2 += "}" + mathF = not mathF else: - new2 += "}" - mathF = not mathF - else: - new2 += p2[k] - p2 = new2 - finSym.append(web1 + " " + p1 + "] : " + p2 + " :" + websiteF) - break - if not gFlag: - del newSym[s] + new2 += p2[k] + p2 = new2 + finSym.append(web1 + " " + p1 + "] : " + p2 + " :" + websiteF) + break + if not gFlag: + del newSym[s] - gFlag = True - for y in finSym: - if gFlag: - gFlag = False - toWrite += ("[" + y) - else: - toWrite += ("
\n[" + y) + gFlag = True + for y in finSym: + if gFlag: + gFlag = False + toWrite += ("[" + y) + else: + toWrite += ("
\n[" + y) + + toWrite += ("\n
\ndrmf_eof\n\n") + if sflag2: + toWrite += "\n" + result += toWrite - toWrite += "\n
\ndrmf_eof\n\n" - if sflag2: - toWrite += "\n" - main.write(toWrite) + return result \ No newline at end of file diff --git a/main_page/symbolsList.py~ b/main_page/symbolsList.py~ deleted file mode 100644 index b0d1d72..0000000 --- a/main_page/symbolsList.py~ +++ /dev/null @@ -1,278 +0,0 @@ -def getSym(line): #Gets all symbols on a line for symbols list - symList=[] - symbol="" - symFlag=False - argFlag=False - cC=0 - for i in range(0,len(line)): - #if "Laguerre[-\\frac{1}{2}]{n}@{x^2}" in line: - #print symbol - c=line[i] - if symFlag: - if c=="{" or c=="[": - cC+=1 - argFlag=True - if c!="}" and c!="]": - if argFlag or c.isalpha(): - symbol+=c - else: - symFlag=False - argFlag=False - symList.append(symbol) - symList+=(getSym(symbol)) - symbol="" - else: - cC-=1 - symbol+=c - if i+1==len(line): - p="" - else: - p=line[i+1] - if cC==0 and p!="{" and p!="[" and p!="@": - symFlag=False - #if "Laguerre[-\\frac{1}{2}]{n}@{x^2}" in line: - #print(symbol,p) - symList.append(symbol) - symList+=(getSym(symbol)) - argFlag=False - symbol="" - - elif c=="\\": - symFlag=True - - #if "Laguerre[-\\frac{1}{2}]{n}@{x^2}" in line: - #print symbol - return(symList) -import os -files_in_dir=os.listdire(".") -for g in files_in_dir: - if not ".py" in g and not ".new" in g: - f=open(g,"r") - w=open(g+".new","w") - t=f.read() - w.write(t) - t=t.replace("\n","") - symbols=getSym(t) - if True: - wiki.write("
\n== Symbols List ==\n
\n") - newSym=[] - #if "09.12:21" in label: - #print(symbols) - for x in symbols: - flagA=True - #if x not in newSym: - cN=0 - cC=0 - ArgCx=0 - for z in x: - if z.isalpha(): - cN+=1 - else: - if z=="{" or z=="[": - cC+=1 - if z=="}" or z=="]": - cC-=1 - if cC==0: - ArgCx+=1 - noA=x[:cN] - for y in newSym: - cN=0 - cC=0 - ArgC=0 - for z in y: - if z.isalpha(): - cN+=1 - else: - if z=="{" or z=="[": - cC+=1 - if z=="}" or z=="]": - cC-=1 - if cC==0: - ArgC+=1 - - - if y[:cN]==noA and ArgC==ArgCx: - flagA=False - #print(x,y) - break - if flagA: - newSym.append(x) - newSym.reverse() - #if "9.12:20" in label: - #print newSym - symF=False - finSym=[] - for s in range(len(newSym)-1,-1,-1): - #print(newSym[s]) - symbolPar="\\"+newSym[s] - ArgCx=0 - parCx=0 - parFlag=False - cC=0 - for z in symbolPar: - if z=="@": - parFlag=True - elif z.isalpha(): - cN+=1 - else: - if z=="{" or z=="[": - cC+=1 - if z=="}" or z=="]": - cC-=1 - if cC==0: - if parFlag: - parCx+=1 - else: - ArgCx+=1 - - if symbolPar.find("{")!=-1 or symbolPar.find("[")!=-1: - if symbolPar.find("[")==-1: - symbol=symbolPar[0:symbolPar.find("{")] - elif symbolPar.find("{")==-1 or symbolPar.find("[")len(symbol) and (G[0][len(symbol)]=="{" or G[0][len(symbol)]=="["): - ap="" - for o in range(len(symbol),len(G[0])): - if G[0][o]=="{" or z=="[": - argFlag=True - elif G[0][o]=="}" or z=="]": - argFlag=False - listArgs.append(ap) - ap="" - else: - ap+=G[0][o] - '''websiteU=g[g.find("http://"):].strip("\n") - k=0 - websites=[] - for r in range(0,len(websiteU)): - if websiteU[r]==",": - if websiteU[k:r].find("http://")!=-1: - websites.append(websiteU[k:r+1].strip(" ")) - k=r - - - if websiteU[k:r].find("http://")!=-1: - websites.append(websiteU[k:r+1].strip(" ")) - websiteF="" - for d in websites: - websiteF=websiteF+" ["+d+" "+d+"]"''' - #websiteF=G[4].strip("\n") - websiteF="" - web1=G[5] - for t in range(5,len(G)): - if G[t]!="": - websiteF=websiteF+" ["+G[t]+" "+G[t]+"]" - p1="{\\displaystyle "+G[0]+"}" - if p1.find("@")!=-1: - p1=p1[:p1.find("@")] - #if checkFlag: - #print(p1) - new1="" - new2="" - pause=False - mathF=True - '''for k in range(0,len(p1)): - if p1[k]=="$": - if mathF: - new1+="{\\displaystyle " - else: - new1+="}" - mathF=not mathF - - elif p1[k]=="#" and p1[k+1].isdigit(): - pause=True - elif pause: - num=int(p1[k]) - #letter=chr(num+96) - letter=listArgs[num-1] - new1+=letter - pause=False - - else: - new1+=p1[k]''' - p2=G[1] - for k in range(0,len(p2)): - if p2[k]=="$": - if mathF: - new2+="{\\displaystyle " - else: - new2+="}" - mathF=not mathF - else: - new2+=p2[k] - #p1=new1 - p2=new2 - finSym.append(web1+" "+p1+"] : "+p2+" :"+websiteF) - break - #gFlag=True - #if not symF: - # symF=True - # wiki.write("[") - #else: - # wiki.write("
\n") - # wiki.write("[") - #wiki.write(g[g.find("http://"):].strip("\n")) - #wiki.write(" ") - #wiki.write(getEq(g[g.find("{$"):]).strip("\n").replace("\\\\","\\")) - #wiki.write("] : ") - #wiki.write(g[g.find(" {")+2:g.find("} ",g.find(" {")+1)]) - #wiki.write(" : [") - #wiki.write(g[g.find("http://"):].strip("\n")) - #wiki.write(" ") - #wiki.write(g[g.find("http://"):].strip("\n")) - #wiki.write("] ")''' - - - - if not gFlag: - del newSym[s] - - gFlag=True - #finSym.reverse() - for y in finSym: - #print(finSym) - if gFlag: - gFlag=False - wiki.write("["+y) - else: - wiki.write("
\n["+y) - - diff --git a/main_page/tex2Wiki.py~ b/main_page/tex2Wiki.py~ deleted file mode 100644 index 1cce794..0000000 --- a/main_page/tex2Wiki.py~ +++ /dev/null @@ -1,966 +0,0 @@ -#Version 11: Split Sections -#Convert tex to wikiText -import csv #imported for using csv format -import sys #imported for getting args -import os #imported for copying file -def isnumber(char): - return char[0] in "0123456789" -def getString(line): #Gets all data within curly braces on a line - #-------Initialization------- - stringWrite="" - getStr=False - pW="" - #---------------------------- - for c in line: - if c=="{" or c=="}": #if there is a curly brace in the line - getStr=not(getStr) #toggle the getStr flag - if c=="}": #no more info needed - break - elif getStr: #if within curly braces - if not(pW==c and c=="-"): #if there is no double dash (makes single dash) - if c!="$": #if not a $ sign - stringWrite+=c #this character is part of the data - else: #replace $ with '' - stringWrite+="\'\'" #add double ' - pW=c #change last character for finding double dashes - return (stringWrite.rstrip('\n').lstrip()) #return the data without newlines and without leading spaces -def getG(line): #gets equation for symbols list - start=True - final="" - for c in line: - if c=="$" and start: - final+="{\\displaystyle " - start=False - elif c=="$": - final+="}" - start=True - else: - final+=c - return(final) -def getEq(line): #Gets all data within constraints,substitutions - #-------Initialization------- - per=1 - stringWrite="" - fEq=False - count=0 - #---------------------------- - for c in line: #read each character - if count>=0 and per!=0: - per+=1 - if c!=" " and c!="%": - per=0 - if c=="{" and per==0: - if count>0: - stringWrite+=c - count+=1 - elif c=="}" and per==0: - count-=1 - if count>0: - stringWrite+=c - elif c=="$" and per==0: #either begin or end equation - fEq=not(fEq)#toggle fEq flag to know if begin or end equation - if fEq:#if begin - stringWrite+="{\\displaystyle " - else:#if end - stringWrite+="}" - elif c=="\n" and per==0:#if newline - stringWrite=stringWrite.strip()#remove all leading and trailing whitespace - - #should above be rstrip?<-------------------------------------------------------------------CHECK THIS - - stringWrite+=c#add the newline character - per+=1 #watch for % signs - elif count>0 and per==0: #not special character - stringWrite+=c #write the character - - return (stringWrite.rstrip().lstrip()) -def getEqP(line,Flag): #Gets all data within proofs - if Flag: - a=1 - else: - a=0 - per=1 - stringWrite="" - fEq=False - count=0 - length=0 - for c in line: - if fEq and c!=" " and c!="$" and c!="{" and c!="}" and not c.isalpha(): - length+=1 - if count>=0 and per!=0: - per+=1 - if c!=" " and c!="%": - #if c!="%": - per=0 - if c=="{" and per==0: - if count>0: - stringWrite+=c - count+=1 - elif c=="}" and per==0: - count-=1 - if count>0: - stringWrite+=c - elif c=="$" and per==0: - fEq=not(fEq) - if fEq: - stringWrite+="{\\displaystyle " - else: - if length<10: - stringWrite+="}" - else: - stringWrite+="}"+"
" - length=0 - elif c=="\n" and per==0: - stringWrite=stringWrite.strip() - stringWrite+=c - per+=1 - elif count>0 and per==0: - stringWrite+=c - - return (stringWrite.lstrip()) -def getSym(line): #Gets all symbols on a line for symbols list - symList=[] - symbol="" - symFlag=False - argFlag=False - cC=0 - for i in range(0,len(line)-1): - if "[" in line: - print symbol - c=line[i] - if symFlag: - if c=="{" or c=="[": - cC+=1 - argFlag=True - if c!="}" and c!="]": - if argFlag or c.isalpha(): - symbol+=c - else: - symFlag=False - argFlag=False - symList.append(symbol) - symList+=(getSym(symbol)) - symbol="" - else: - cC-=1 - symbol+=c - if cC==0 and line[i+1]!="{" and line[i+1]!="[" and line[i+1]!="@": - symFlag=False - symList.append(symbol) - symList+=(getSym(symbol)) - argFlag=False - symbol="" - - elif c=="\\": - symFlag=True - - return(symList) - '''symList=[] - getStr=False - getPar=False - morePar=False - curStr="" - deeper=False - cC=0 - for c in line: - if c=="\\" and not (getStr): - getStr=True - elif getStr: - if morePar and c!="{": - getPar=False - getStr=False - symList.append(curStr) - curStr="" - morePar=False - else: - morePar=False - if c=="\\": - deeper=True - if c.isalpha(): - curStr=curStr+c - elif not getPar and c!="{" and c!="}": - getStr=False - symList.append(curStr) - curStr="" - elif c=="{": - cC+=1 - getPar=True - curStr=curStr+c - elif c=="}": - cC-=1 - #getPar=False - if cC>0: - curStr=curStr+c - else: - morePar=True - curStr=curStr+c - getPar=False - getStr=False - symList.append(curStr) - curStr="" - #symList.append(curStr) - #curStr="" - else: - curStr=curStr+c - #print(line,symList) - dSym=[] - if deeper: - for sym in symList: - d=getSym(sym) - dSym=dSym+(d) - #print(symList,dSym,line) - symList=symList+dSym - return (symList)''' -def modLabel(label): - isNumer=False - newlabel="" - num="" - for i in range(0,len(label)): - if isNumer and not isnumber(label[i]): - if len(num)>1: - newlabel+=num - isNumer=False - num="" - else: - newlabel+="0"+str(num) - num="" - isNumer=False - if isnumber(label[i]): - isNumer=True - num+=str(label[i]) - else: - isNumer=False - newlabel+=label[i] - if len(num)>1: - newlabel+=num - elif len(num)==1: - newlabel+="0"+num - return(newlabel) -#try: -for jsahlfkjsd in range(0,1): - tex=open(sys.argv[1],'r') - wiki=open(sys.argv[2],'w') - main=open("OrthogonalPolynomials.mmd","r") - mainText=main.read() - mainPrepend="" - mainWrite=open("OrthogonalPolynomials.mmd.new","w") - tester=open("testData.txt",'w') - #glossary=open('Glossary', 'r') - glossary=open('new.Glossary.csv', 'rb') - gCSV=csv.reader(glossary, delimiter=',', quotechar='\"') - #lLinks=open('BruceLabelLinks', 'r') - lGlos=glossary.readlines() - #lLink=lLinks.readlines() - math=False - constraint=False - substitution=False - symbols=[] - lines=tex.readlines() - refLines=[] - sections=[] - labels=[] - eqs=[] - refEqs=[] - parse=False - head=False - #chapRef=[("GA",open("GA.tex",'r')),("ZE",open("ZE.3.tex",'r'))] - refLabels=[] - '''for c in chapRef: - refLines=refLines+(c[1].readlines()) - c[1].close()''' - for i in range(0,len(refLines)): - line=refLines[i] - if "\\begin{equation}" in line: - sLabel=line.find("\\formula{")+9 - eLabel=line.find("}",sLabel) - label=modLabel(line[sLabel:eLabel]) - '''for l in lLink: - if l.find(label)!=-1 and l[len(label)+1]=="=": - rlabel=l[l.find("=>")+3:l.find("\\n")] - rlabel=rlabel.replace("/","") - rlabel=rlabel.replace("#",":") - break''' - refLabels.append(label) - refEqs.append("") - math=True - elif math: - refEqs[len(refEqs)-1]+=line - if "\\end{equation}" in refLines[i+1] or "\\constraint" in refLines[i+1] or "\\substitution" in refLines[i+1] or "\\drmfn" in refLines[i+1]: - math=False - - for i in range(0,len(lines)): - line=lines[i] - if "\\begin{document}" in line: - #wiki.write("drmf_bof\n") - parse=True - elif "\\end{document}" in line and parse: - #wiki.write("
\n") - mainPrepend+="
\n" - mainText=mainPrepend+mainText - mainText=mainText.replace("drmf_bof\n","") - mainText=mainText.replace("drmf_eof\n","") - mainText=mainText.replace("\'\'\'Orthogonal Polynomials\'\'\'\n","") - mainText=mainText[0:mainText.rfind("== Sections ")] - mainText="drmf_bof\n\'\'\'Orthogonal Polynomials\'\'\'\n"+mainText+"\ndrmf_eof" - mainWrite.write(mainText) - mainWrite.close() - main.close() - os.system("cp -f OrthogonalPolynomials.mmd.new OrthogonalPolynomials.mmd") - #wiki.write("\ndrmf_eof\n") - parse=False - elif "\\title" in line and parse: - stringWrite="\'\'\'" - stringWrite+=getString(line)+"\'\'\'\n" - labels.append("Orthogonal Polynomials") - sections.append(["Orthogonal Polynomials",0]) - #wiki.write(stringWrite) - #mainPrepend+=stringWrite - mainPrepend+=("\n== Sections in "+getString(line)+" ==\n\n
\n") - elif "\\part" in line: - if getString(line)=="BOF": - parse=False - elif getString(line)=="EOF": - parse=True - elif parse: - mainPrepend+=("\n
\n= "+getString(line)+" =\n") - head=True - elif "\\section" in line: - mainPrepend+=("* [["+getString(line)+"|"+getString(line)+"]]\n") - sections.append([getString(line)]) - - secCounter=0 - eqCounter=0 - for i in range(0,len(lines)): - line=lines[i] - #line.replace("\\begin{equation}","$") - #line.replace("\\end{equation},"$") - '''if "\\begin{document}" in line: - wiki.write("drmf_bof\n") - parse=True - elif "\\end{document}" in line and parse: - wiki.write("\ndrmf_eof\n") - parse=False - elif "\\title" in line and parse: - stringWrite="\'\'\'" - stringWrite+=getString(line)+"\'\'\'\n" - labels.append("Orthogonal Polynomials") - wiki.write(stringWrite\n) - elif "\\part" in line: - if getString(line)=="BOF": - parse=False - elif getString(line)=="EOF": - parse=True - elif parse: - wiki.write("\n
\n= "+getString(line)+" =\n") - head=True - ''' - if "\\section" in line: - parse=True - secCounter+=1 - wiki.write("drmf_bof\n") - wiki.write("\'\'\'"+getString(line)+"\'\'\'\n") - wiki.write("
\n") - wiki.write("
<< [["+sections[secCounter-1][0]+"|"+sections[secCounter-1][0]+"]]
\n") - wiki.write("
[["+sections[secCounter][0]+"|"+sections[secCounter][0]+"]]
\n") - wiki.write("
[["+sections[(secCounter+1)%len(sections)][0]+"|"+sections[(secCounter+1)%len(sections)][0]+"]] >>
\n
\n\n") - head=True - wiki.write("== "+getString(line)+" ==\n") - elif ("\\section" in lines[(i+1)%len(lines)] or "\\end{document}" in lines[(i+1)%len(lines)]) and parse: - wiki.write("
\n") - wiki.write("
<< [["+sections[secCounter-1][0]+"|"+sections[secCounter-1][0]+"]]
\n") - wiki.write("
[["+sections[secCounter][0]+"|"+sections[secCounter][0]+"]]
\n") - wiki.write("
[["+sections[(secCounter+1)%len(sections)][0]+"|"+sections[(secCounter+1)%len(sections)][0]+"]] >>
\n
\n\n") - wiki.write("drmf_eof\n") - sections[secCounter].append(eqCounter) - eqCounter=0 - - elif "\\subsection" in line and parse: - wiki.write("\n
\n== "+getString(line)+" ==\n") - head=True - elif "\\paragraph" in line and parse: - wiki.write("\n
\n=== "+getString(line)+" ===\n") - head=True - elif "\\begin{equation}" in line and parse: - # symLine="" - if head: - wiki.write("
\n") - head=False - sLabel=line.find("\\formula{")+9 - eLabel=line.find("}",sLabel) - label=modLabel(line[sLabel:eLabel]) - eqCounter+=1 - '''for l in lLink: - if label==l[0:l.find("=")-1]: - rlabel=l[l.find("=>")+3:l.find("\\n")] - rlabel=rlabel.replace("/","") - rlabel=rlabel.replace("#",":") - rlabel=rlabel.replace("!",":") - break''' - labels.append(label) - eqs.append("") - wiki.write("\n\n") - wiki.write("{\displaystyle \n") - math=True - elif "\\begin{equation}" in line and not parse: - sLabel=line.find("\\formula{")+9 - eLabel=line.find("}",sLabel) - label=modLabel(line[sLabel:eLabel]) - '''for l in lLink: - if l.find(label)!=-1: - rlabel=l[l.find("=>")+3:l.find("\\n")] - rlabel=rlabel.replace("/","") - rlabel=rlabel.replace("#",":") - break''' - labels.append("*"+label) #special marker - eqs.append("") - math=True - elif "\\end{equation}" in line: - - math=False - elif "\\constraint" in line and parse: - constraint=True - math=False - conLine="" - elif "\\substitution" in line and parse: - substitution=True - math=False - subLine="" - #wiki.write("
Substitution(s): "+getEq(line)+"

\n") - elif "\\proof" in line and parse: - math=False - elif "\\drmfn" in line and parse: - math=False - if "\\drmfname" in line and parse: - wiki.write("
This formula has the name: "+getString(line)+"

\n") - elif math and parse: - flagM=True - eqs[len(eqs)-1]+=line - - if "\\end{equation}" in lines[i+1] and not "\\subsection" in lines[i+3] and not "\\section" in lines[i+3] and not "\\part" in lines[i+3]: - u=i - flagM2=False - while flagM: - u+=1 - if "\\begin{equation}"in lines[u] in lines[u]: - flagM=False - if "\\section" in lines[u] or "\\subsection" in lines[i] or "\\part" in lines[u] or "\\end{document}" in lines[u]: - flagM=False - flagM2=True - if not(flagM2): - - wiki.write(line.rstrip("\n")) - wiki.write("\n}
\n") - else: - wiki.write(line.rstrip("\n")) - wiki.write("\n}\n") - elif "\\end{equation}" in lines[i+1]: - wiki.write(line.rstrip("\n")) - wiki.write("\n}\n") - elif "\\constraint" in lines[i+1] or "\\substitution" in lines[i+1] or "\\drmfn" in lines[i+1]: - wiki.write(line.rstrip("\n")) - wiki.write("\n}\n") - else: - wiki.write(line) - elif math and not parse: - eqs[len(eqs)-1]+=line - if "\\end{equation}" in lines[i+1] or "\\constraint" in lines[i+1] or "\\substitution" in lines[i+1] or "\\drmfn" in lines[i+1]: - math=False - if substitution and parse: - subLine=subLine+line - if "\\end{equation}" in lines[i+1] or "\\substitution" in lines[i+1] or "\\constraint" in lines[i+1] or "\\drmfn" in lines[i+1] or "\\proof" in lines[i+1]: - substitution=False - wiki.write("
Substitution(s): "+getEq(subLine)+"

\n") - - if constraint and parse: - conLine=conLine+line - if "\\end{equation}" in lines[i+1] or "\\substitution" in lines[i+1] or "\\constraint" in lines[i+1] or "\\drmfn" in lines[i+1] or "\\proof" in lines[i+1]: - constraint=False - wiki.write("
Constraint(s): "+getEq(conLine)+"

\n") - - eqCounter=0 - endNum=len(labels)-1 - '''for n in range(1,len(labels)): - if n+1!=len(labels) and labels[n+1][0]=="*" and labels[n][0]!="*": - labels[n]=""+labels[n] - endNum=n - elif labels[n][0]=="*": - labels[n]=""+labels[n][1:] - else: - labels[n]=""+labels[n]''' - '''for n in range(0,len(refLabels)): - refLabels[n]=""+refLabels[n]''' - parse=False - constraint=False - substitution=False - note=False - hCon=True - hSub=True - hNote=True - hProof=True - proof=False - comToWrite="" - secCount=-1 - newSec=False - for i in range(0,len(lines)): - line=lines[i] - - if "\\section" in line: - secCount+=1 - newSec=True - eqS=0 - - if "\\begin{equation}" in line: - symLine=line.strip("\n") - eqS+=1 - constraint=False - substitution=False - note=False - comToWrite="" - hCon=True - hSub=True - hNote=True - hProof=True - proof=False - parse=True - symbols=[] - eqCounter+=1 - wiki.write("drmf_bof\n") - label=labels[eqCounter] - wiki.write("\'\'\'"+label+"\'\'\'\n\n") - if eqCounter\n") - if newSec: - wiki.write("
<< [["+sections[secCount][0].replace(" ","_")+"|"+sections[secCount][0]+"]]
\n") - else: - wiki.write("
<< [["+labels[eqCounter-1].replace(" ","_")+"|"+labels[eqCounter-1]+"]]
\n") - wiki.write("
[["+labels[0].replace(" ","_")+"#"+label[8:]+"|formula in "+labels[0]+"]]
\n") - if eqS==sections[secCount][1]: - wiki.write("
[["+sections[(secCount+1)%len(sections)][0].replace(" ","_")+"|"+sections[(secCount+1)%len(sections)][0]+"]] >>
\n") - else: - wiki.write("
[["+labels[(eqCounter+1)%(endNum+1)].replace(" ","_")+"|"+labels[(eqCounter+1)%(endNum+1)]+"]] >>
\n") - wiki.write("
\n\n") - elif eqCounter==endNum: - wiki.write("
\n") - if newSec: - newSec=False - wiki.write("
<< [["+sections[secCount][0].replace(" ","_")+"|"+sections[secCount][0]+"]]
\n") - else: - wiki.write("
<< [["+labels[eqCounter-1].replace(" ","_")+"|"+labels[eqCounter-1]+"]]
\n") - wiki.write("
[["+labels[0].replace(" ","_")+"#"+label[8:]+"|formula in "+labels[0]+"]]
\n") - wiki.write("
[["+labels[(eqCounter+1)%(endNum+1)].replace(" ","_")+"|"+labels[(eqCounter+1)%(endNum+1)]+"]]
\n") - wiki.write("
\n\n") - - wiki.write("
{\displaystyle \n") - math=True - elif "\\end{equation}" in line: - wiki.write(comToWrite) - parse=False - math=False - if hProof: - wiki.write("
\n== Proof ==\n
\nWe ask users to provide proof(s), reference(s) to proof(s), or further clarification on the proof(s) in this space.\n") - - wiki.write("
\n== Symbols List ==\n
\n") - newSym=[] - if "09.12:21" in label: - print(symbols) - for x in symbols: - flagA=True - #if x not in newSym: - cN=0 - cC=0 - ArgCx=0 - for z in x: - if z.isalpha(): - cN+=1 - else: - if z=="{": - cC+=1 - if z=="}": - cC-=1 - if cC==0: - ArgCx+=1 - noA=x[:cN] - for y in newSym: - cN=0 - cC=0 - ArgC=0 - for z in y: - if z.isalpha(): - cN+=1 - else: - if z=="{": - cC+=1 - if z=="}": - cC-=1 - if cC==0: - ArgC+=1 - - - if y[:cN]==noA and ArgC==ArgCx: - flagA=False - #print(x,y) - break - if flagA: - newSym.append(x) - newSym.reverse() - symF=False - finSym=[] - for s in range(len(newSym)-1,-1,-1): - #print(newSym[s]) - symbolPar="\\"+newSym[s] - ArgCx=0 - cC=0 - for z in symbolPar: - if z.isalpha(): - cN+=1 - else: - if z=="{": - cC+=1 - if z=="}": - cC-=1 - if cC==0: - ArgCx+=1 - if symbolPar.find("{")!=-1: - symbol=symbolPar[0:symbolPar.find("{")] - else: - symbol=symbolPar - numPar=ArgCx - gFlag=False - gCSV=csv.reader(open('new.Glossary.csv','rb'),delimiter=',', quotechar='\"') - for G in gCSV: - ArgCx=0 - cC=0 - ind=G[0].find("@") - if ind==-1: - ind=len(G[0])-1 - for z in G[0]: - if z.isalpha(): - cN+=1 - else: - if z=="{": - cC+=1 - if z=="}": - cC-=1 - if cC==0: - ArgCx+=1 - checkFlag=False - if G[0].find(symbol)==0 and (len(G[0])==len(symbol) or not G[0][len(symbol)].isalpha()) and numPar!=0: - checkFlag=True - if G[0].find(symbol)==0 and (len(G[0])==len(symbol) or not G[0][len(symbol)].isalpha()) and ArgCx==numPar: - listArgs=[] - if len(G[0])>len(symbol) and G[0][len(symbol)]=="{": - ap="" - for o in range(len(symbol),len(G[0])): - if G[0][o]=="{": - argFlag=True - elif G[0][o]=="}": - argFlag=False - listArgs.append(ap) - ap="" - else: - ap+=G[0][o] - '''websiteU=g[g.find("http://"):].strip("\n") - k=0 - websites=[] - for r in range(0,len(websiteU)): - if websiteU[r]==",": - if websiteU[k:r].find("http://")!=-1: - websites.append(websiteU[k:r+1].strip(" ")) - k=r - - - if websiteU[k:r].find("http://")!=-1: - websites.append(websiteU[k:r+1].strip(" ")) - websiteF="" - for d in websites: - websiteF=websiteF+" ["+d+" "+d+"]"''' - #websiteF=G[4].strip("\n") - websiteF="" - web1=G[5] - for t in range(5,len(G)): - if G[t]!="": - websiteF=websiteF+" ["+G[t]+" "+G[t]+"]" - p1="{\\displaystyle "+G[0]+"}" - if p1.find("@")!=-1: - p1=p1[:p1.find("@")] - #if checkFlag: - #print(p1) - new1="" - new2="" - pause=False - mathF=True - '''for k in range(0,len(p1)): - if p1[k]=="$": - if mathF: - new1+="{\\displaystyle " - else: - new1+="}" - mathF=not mathF - - elif p1[k]=="#" and p1[k+1].isdigit(): - pause=True - elif pause: - num=int(p1[k]) - #letter=chr(num+96) - letter=listArgs[num-1] - new1+=letter - pause=False - - else: - new1+=p1[k]''' - p2=G[1] - for k in range(0,len(p2)): - if p2[k]=="$": - if mathF: - new2+="{\\displaystyle " - else: - new2+="}" - mathF=not mathF - else: - new2+=p2[k] - #p1=new1 - p2=new2 - finSym.append(web1+" "+p1+"] : "+p2+" :"+websiteF) - break - #gFlag=True - #if not symF: - # symF=True - # wiki.write("[") - #else: - # wiki.write("
\n") - # wiki.write("[") - #wiki.write(g[g.find("http://"):].strip("\n")) - #wiki.write(" ") - #wiki.write(getEq(g[g.find("{$"):]).strip("\n").replace("\\\\","\\")) - #wiki.write("] : ") - #wiki.write(g[g.find(" {")+2:g.find("} ",g.find(" {")+1)]) - #wiki.write(" : [") - #wiki.write(g[g.find("http://"):].strip("\n")) - #wiki.write(" ") - #wiki.write(g[g.find("http://"):].strip("\n")) - #wiki.write("] ")''' - - - - if not gFlag: - del newSym[s] - - gFlag=True - #finSym.reverse() - for y in finSym: - #print(finSym) - if gFlag: - gFlag=False - wiki.write("["+y) - else: - wiki.write("
\n["+y) - - - wiki.write("\n
\n") - - wiki.write("== Bibliography==\n
\n")#should there be a space between bibliography and ==? - r=labels[eqCounter] - q=r.find("KLS:")+4 - p=r.find(":",q) - section=r[q:p] - equation=r[p+1:] - if equation.find(":")!=-1: - equation=equation[0:equation.find(":")] - - wiki.write("[HTTP://DLMF.NIST.GOV/"+section+"#"+equation+" Equation in Section "+section+"] of [[Bibliography#KLS|'''KLS''']].\n
\n
\n")#Where should it link to? - wiki.write("== URL links ==\n
\nWe ask users to provide relevant URL links in this space.\n
\n\n") - if eqCounter
\n") - if newSec: - newSec=False - wiki.write("
<< [["+sections[secCount][0].replace(" ","_")+"|"+sections[secCount][0]+"]]
\n") - else: - wiki.write("
<< [["+labels[eqCounter-1].replace(" ","_")+"|"+labels[eqCounter-1]+"]]
\n") - wiki.write("
[["+labels[0].replace(" ","_")+"#"+label[8:]+"|formula in "+labels[0]+"]]
\n") - if eqS==sections[secCount][1]: - wiki.write("
[["+sections[(secCount+1)%len(sections)][0].replace(" ","_")+"|"+sections[(secCount+1)%len(sections)][0]+"]] >>
\n") - else: - wiki.write("
[["+labels[(eqCounter+1)%(endNum+1)].replace(" ","_")+"|"+labels[(eqCounter+1)%(endNum+1)]+"]] >>
\n") - wiki.write("
\n\ndrmf_eof\n") - else: #FOR EXTRA EQUATIONS - wiki.write("
\n") - wiki.write("
<< [["+labels[endNum-1].replace(" ","_")+"|"+labels[endNum-1]+"]]
\n") - wiki.write("
[["+labels[0].replace(" ","_")+"#"+labels[endNum][8:]+"|formula in "+labels[0]+"]]
\n") - wiki.write("
[["+labels[(0)%endNum].replace(" ","_")+"|"+labels[0%endNum]+"]]
\n") - wiki.write("
\n\ndrmf_eof\n") - - - - elif "\\constraint" in line and parse: - #symbols=symbols+getSym(line) - symLine=line.strip("\n") - if hCon: - comToWrite=comToWrite+"
\n== Constraint(s) ==\n
\n" - hCon=False - constraint=True - math=False - conLine="" - #wiki.write("
"+getEq(line)+"

\n") - elif "\\substitution" in line and parse: - #symbols=symbols+getSym(line) - symLine=line.strip("\n") - if hSub: - comToWrite=comToWrite+"
\n== Substitution(s) ==\n
\n" - hSub=False - #wiki.write("
"+getEq(line)+"

\n") - substitution=True - math=False - subLine="" - elif "\\drmfname" in line and parse: - math=False - comToWrite="
\n== Name ==\n
\n
"+getString(line)+"

\n"+comToWrite - - elif "\\drmfnote" in line and parse: - symbols=symbols+getSym(line) - if hNote: - comToWrite=comToWrite+"
\n== Note(s) ==\n
\n" - hNote=False - note=True - math=False - noteLine="" - - elif "\\proof" in line and parse: - #symbols=symbols+getSym(line) - symLine=line.strip("\n") - if hProof: - hProof=False - comToWrite=comToWrite+"
\n== Proof ==\n
\nWe ask users to provide proof(s), reference(s) to proof(s), or further clarification on the proof(s) in this space. \n

\n
" - proof=True - proofLine="" - pause=False - pauseP=False - for ind in range(0,len(line)): - if line[ind:ind+7]=="\\eqref{": - pause=True - eqR=line[ind:line.find("}",ind)+1] - rLab=getString(eqR) - '''for l in lLink: - if rLab==l[0:l.find("=")-1]: - rlabel=l[l.find("=>")+3:l.find("\\n")] - rlabel=rlabel.replace("/","") - rlabel=rlabel.replace("#",":") - rlabel=rlabel.replace("!",":") - break''' - - eInd=refLabels.index(""+label) - z=line[line.find("}",ind+7)+1] - if z=="." or z==",": - pauseP=True - proofLine+=("
\n{\displaystyle \n"+refEqs[eInd]+"}"+z+"
\n") - else: - if z=="}": - proofLine+=("
\n{\displaystyle \n"+refEqs[eInd]+"}
") - else: - proofLine+=("
\n{\displaystyle \n"+refEqs[eInd]+"}
\n") - - - else: - if pause: - if line[ind]=="}": - pause=False - elif pauseP: - pauseP=False - elif line[ind]=="\n" and "\\end{equation}" in lines[i+1]: - pass - else: - proofLine+=(line[ind]) - if "\\end{equation}" in lines[i+1]: - proof=False - #symLine+=line.strip("\n") - wiki.write(comToWrite+getEqP(proofLine,False)+"
\n
\n") - comToWrite="" - symbols=symbols+getSym(symLine) - symLine="" - - #wiki.write(line) - - elif proof: - symLine+=line.strip("\n") - pauseP=False - #symbols=symbols+getSym(line) - for ind in range(0,len(line)): - if line[ind:ind+7]=="\\eqref{": - pause=True - eqR=line[ind:line.find("}",ind)+1] - rLab=getString(eqR) - '''for l in lLink: - if rLab==l[0:l.find("=")-1]: - rlabel=l[l.find("=>")+3:l.find("\\n")] - rlabel=rlabel.replace("/","") - rlabel=rlabel.replace("#",":") - rlabel=rlabel.replace("!",":") - break''' - - eInd=refLabels.index(""+label) - z=line[line.find("}",ind+7)+1] - if z=="." or z==",": - pauseP=True - proofLine+=("
\n{\displaystyle \n"+refEqs[eInd]+"}"+z+"
\n") - else: - proofLine+=("
\n{\displaystyle \n"+refEqs[eInd]+"}
\n") - - else: - if pause: - if line[ind]=="}": - pause=False - elif pauseP: - pauseP=False - elif line[ind]=="\n" and "\\end{equation}" in lines[i+1]: - pass - - else: - proofLine+=(line[ind]) - if "\\end{equation}" in lines[i+1]: - proof=False - #symLine+=line.strip("\n") - wiki.write(comToWrite+getEqP(proofLine,False).rstrip("\n")+"
\n
\n") - comToWrite="" - symbols=symbols+getSym(symLine) - symLine="" - - elif math: - if "\\end{equation}" in lines[i+1] or "\\constraint" in lines[i+1] or "\\substitution" in lines[i+1] or "\\proof" in lines[i+1] or "\\drmfnote" in lines[i+1] or "\\drmfname" in lines[i+1]: - wiki.write(line.rstrip("\n")) - symLine+=line.strip("\n") - symbols=symbols+getSym(symLine) - symLine="" - wiki.write("\n}
\n") - else: - symLine+=line.strip("\n") - wiki.write(line) - if note and parse: - noteLine=noteLine+line - symbols=symbols+getSym(line) - if "\\end{equation}" in lines[i+1] or "\\drmfn" in lines[i+1] or "\\constraint" in lines[i+1] or "\\substitution" in lines[i+1] or "\\proof" in lines[i+1]: - note=False - comToWrite=comToWrite+"
"+getEq(noteLine)+"

\n" - - if constraint and parse: - conLine=conLine+line - symLine+=line.strip("\n") - #symbols=symbols+getSym(line) - if "\\end{equation}" in lines[i+1] or "\\drmfn" in lines[i+1] or "\\constraint" in lines[i+1] or "\\substitution" in lines[i+1] or "\\proof" in lines[i+1]: - constraint=False - symbols=symbols+getSym(symLine) - symLine="" - wiki.write(comToWrite+"
"+getEq(conLine)+"

\n") - comToWrite="" - if substitution and parse: - subLine=subLine+line - symLine+=line.strip("\n") - #symbols=symbols+getSym(line) - if "\\end{equation}" in lines[i+1] or "\\drmfn" in lines[i+1] or "\\substitution" in lines[i+1] or "\\constraint" in lines[i+1] or "\\proof" in lines[i+1]: - substitution=False - symbols=symbols+getSym(symLine) - symLine="" - wiki.write(comToWrite+"
"+getEq(subLine)+"

\n") - comToWrite="" -#except Exception as detail: #If exception occured -# print("Exception",detail) #print details of error -#except: #If anythin else occured... -# print ("ERROR",sys.exc_info()[0])#ERROR with basic info From 2d6b117358113b88d6455e48e197e8cc4033f910 Mon Sep 17 00:00:00 2001 From: Joon Bang Date: Fri, 5 Aug 2016 09:29:13 -0400 Subject: [PATCH 194/402] Update main_page.mmd, fix bugs with new entries --- main_page/main_page.mmd | 129 +++++++++++++++++++++++++++++++++++++++- main_page/src/main.py | 3 + 2 files changed, 129 insertions(+), 3 deletions(-) diff --git a/main_page/main_page.mmd b/main_page/main_page.mmd index 3e07ca8..f271213 100644 --- a/main_page/main_page.mmd +++ b/main_page/main_page.mmd @@ -225,6 +225,9 @@ For more information see [http://link.springer.com/chapter/10.1007%2F978-3-319-0 * [[Definition:RegGammaQ|RegGammaQ]] * [[Definition:GaussDistF|GaussDistF]] * [[Definition:GaussDistQ|GaussDistQ]] +* [[Definition:RamanujanTauTheta|RamanujanTauTheta]] +* [[Definition:RiemannSiegelTheta|RiemannSiegelTheta]] +* [[Definition:sinc|sinc]]
= Copyright = @@ -282,7 +285,7 @@ drmf_eof drmf_bof '''Definition:AffqKrawtchouk'''
-
<< [[Definition:GaussDistQ|Definition:GaussDistQ]]
+
<< [[Definition:sinc|Definition:sinc]]
[[Main_Page|Main Page]]
[[Definition:AlSalamCarlitzI|Definition:AlSalamCarlitzI]] >>
@@ -309,7 +312,7 @@ These are defined by [http://drmf.wmflabs.org/wiki/Definition:AffqKrawtchouk {\displaystyle K^{\mathrm{Aff}}_{n}}] : affine {\displaystyle q}-Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:AffqKrawtchouk http://drmf.wmflabs.org/wiki/Definition:AffqKrawtchouk]
[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
-
<< [[Definition:GaussDistQ|Definition:GaussDistQ]]
+
<< [[Definition:sinc|Definition:sinc]]
[[Main_Page|Main Page]]
[[Definition:AlSalamCarlitzI|Definition:AlSalamCarlitzI]] >>
@@ -5035,7 +5038,7 @@ drmf_bof
<< [[Definition:GaussDistF|Definition:GaussDistF]]
[[Main_Page|Main Page]]
-
[[Definition:AffqKrawtchouk|Definition:AffqKrawtchouk]] >>
+
[[Definition:RamanujanTauTheta|Definition:RamanujanTauTheta]] >>
The LaTeX DLMF and DRMF macro '''\GaussDistQ''' represents the GaussDistQ. @@ -5065,6 +5068,126 @@ These are defined by
<< [[Definition:GaussDistF|Definition:GaussDistF]]
[[Main_Page|Main Page]]
+
[[Definition:RamanujanTauTheta|Definition:RamanujanTauTheta]] >>
+
+drmf_eof +drmf_bof +'''Definition:RamanujanTauTheta''' +
+
<< [[Definition:GaussDistQ|Definition:GaussDistQ]]
+
[[Main_Page|Main Page]]
+
[[Definition:RiemannSiegelTheta|Definition:RiemannSiegelTheta]] >>
+
+The LaTeX DLMF and DRMF macro '''\RamanujanTauTheta''' represents the RamanujanTauTheta. + +This macro is in the category of real or complex valued functions. + +In math mode, this macro can be called in the following ways: + +:'''\RamanujanTauTheta''' produces {\displaystyle \RamanujanTauTheta}
+:'''\RamanujanTauTheta@{z}''' produces {\displaystyle \RamanujanTauTheta@{z}}
+ +These are defined by + + + +{\displaystyle + +\RamanujanTauTheta@{z}:=-\log\left(2\pi\right)z-\frac{i}{2}\left(\log\EulerGamma@{6+iz}-\log{\EulerGamma@{6-iz}}\right) + +} + +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:RamanujanTauTheta {\displaystyle \tau\theta{z}}] : Ramanujan tau theta function : [http://drmf.wmflabs.org/wiki/Definition:RamanujanTauTheta http://drmf.wmflabs.org/wiki/Definition:RamanujanTauTheta]
+[http://dlmf.nist.gov/4.2#E2 {\displaystyle \mathrm{log}}] : principle branch of natural logarithm : [http://dlmf.nist.gov/4.2#E2 http://dlmf.nist.gov/4.2#E2]
+[http://dlmf.nist.gov/5.2#E1 {\displaystyle \Gamma}] : Euler's gamma function : [http://dlmf.nist.gov/5.2#E1 http://dlmf.nist.gov/5.2#E1] +
+
<< [[Definition:GaussDistQ|Definition:GaussDistQ]]
+
[[Main_Page|Main Page]]
+
[[Definition:RiemannSiegelTheta|Definition:RiemannSiegelTheta]] >>
+
+drmf_eof +drmf_bof +'''Definition:RiemannSiegelTheta''' +
+
<< [[Definition:RamanujanTauTheta|Definition:RamanujanTauTheta]]
+
[[Main_Page|Main Page]]
+
[[Definition:sinc|Definition:sinc]] >>
+
+The LaTeX DLMF and DRMF macro '''\RiemannSiegelTheta''' represents the RiemannSiegelTheta. + +This macro is in the category of real or complex valued functions. + +In math mode, this macro can be called in the following ways: + +:'''\RiemannSiegelTheta''' produces {\displaystyle \RiemannSiegelTheta}
+:'''\RiemannSiegelTheta@{t}''' produces {\displaystyle \RiemannSiegelTheta@{t}}
+ +These are defined by + + + +{\displaystyle + +\RiemannSiegelTheta@{t}:=\arg\left[\EulerGamma@{\frac{1}{4}+\frac{1}{2}it}\right]-\frac{1}{2}t\ln\pi + +} + +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:RiemannSiegelTheta {\displaystyle \vartheta{t}}] : Riemann-Siegel theta function : [http://drmf.wmflabs.org/wiki/Definition:RiemannSiegelTheta http://drmf.wmflabs.org/wiki/Definition:RiemannSiegelTheta]
+[http://dlmf.nist.gov/5.2#E1 {\displaystyle \Gamma}] : Euler's gamma function : [http://dlmf.nist.gov/5.2#E1 http://dlmf.nist.gov/5.2#E1]
+[http://dlmf.nist.gov/4.2#E2 {\displaystyle \mathrm{ln}}] : principal branch of logarithm function : [http://dlmf.nist.gov/4.2#E2 http://dlmf.nist.gov/4.2#E2] +
+
<< [[Definition:RamanujanTauTheta|Definition:RamanujanTauTheta]]
+
[[Main_Page|Main Page]]
+
[[Definition:sinc|Definition:sinc]] >>
+
+drmf_eof +drmf_bof +'''Definition:sinc''' +
+
<< [[Definition:RiemannSiegelTheta|Definition:RiemannSiegelTheta]]
+
[[Main_Page|Main Page]]
+
[[Definition:AffqKrawtchouk|Definition:AffqKrawtchouk]] >>
+
+The LaTeX DLMF and DRMF macro '''\sinc''' represents the sinc. + +This macro is in the category of real or complex valued functions. + +In math mode, this macro can be called in the following ways: + +:'''\sinc''' produces {\displaystyle \sinc}
+:'''\sinc@{x}''' produces {\displaystyle \sinc@{x}}
+ +These are defined by + + + +{\displaystyle + +\sinc@{x}:= +\begin{cases} + \hfill 1 \hfill & \text{ for $x = 0$} \\ + \hfill \frac{\sin{x}}{x} \hfill & \text{ otherwise,} \\ +\end{cases} + +} + +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:Sinc {\displaystyle \mathrm{sinc}{x}}] : sine cardinal function : [http://drmf.wmflabs.org/wiki/Definition:Sinc http://drmf.wmflabs.org/wiki/Definition:Sinc]
+[http://dlmf.nist.gov/4.14#E1 {\displaystyle \mathrm{sin}}] : sine function : [http://dlmf.nist.gov/4.14#E1 http://dlmf.nist.gov/4.14#E1] +
+
<< [[Definition:RiemannSiegelTheta|Definition:RiemannSiegelTheta]]
+
[[Main_Page|Main Page]]
[[Definition:AffqKrawtchouk|Definition:AffqKrawtchouk]] >>
drmf_eof diff --git a/main_page/src/main.py b/main_page/src/main.py index d6a93f4..5524ffe 100644 --- a/main_page/src/main.py +++ b/main_page/src/main.py @@ -63,6 +63,9 @@ def update_headers(text, definitions): output += page + "drmf_eof\n" continue + # remove drmf_bof and definition header to prevent redundancy + page = page[page.find(macros[i]) + len(macros[i]) + 3:] + if "
" in page: page = page[page.find(">>
\n
") + len(">>
\n
"):] From f7e3ad7923d68fddf78f5826e0e90b8160ae699b Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Fri, 5 Aug 2016 10:48:42 -0400 Subject: [PATCH 195/402] Add eCF to travis build checks --- .travis.yml | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/.travis.yml b/.travis.yml index dd5c68f..cbea7fe 100644 --- a/.travis.yml +++ b/.travis.yml @@ -2,6 +2,6 @@ language: python install: - pip install coveralls script: - nosetests --with-coverage tex2Wiki DLMF_preprocessing maple2latex + nosetests --with-coverage tex2Wiki DLMF_preprocessing maple2latex eCF after_success: coveralls From 510cb5709cebafbae0538f2e6cab5e03d6ffeae5 Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Fri, 5 Aug 2016 10:49:07 -0400 Subject: [PATCH 196/402] Add eCF directory --- eCF/.gitignore | 12 + eCF/data/IdentitiesTest.m | 8 + eCF/data/functions | 116 +++++ eCF/src/MathematicaToLaTeX.py | 845 ++++++++++++++++++++++++++++++++++ eCF/src/pythonTest.py | 0 5 files changed, 981 insertions(+) create mode 100644 eCF/.gitignore create mode 100644 eCF/data/IdentitiesTest.m create mode 100644 eCF/data/functions create mode 100644 eCF/src/MathematicaToLaTeX.py create mode 100644 eCF/src/pythonTest.py diff --git a/eCF/.gitignore b/eCF/.gitignore new file mode 100644 index 0000000..975836e --- /dev/null +++ b/eCF/.gitignore @@ -0,0 +1,12 @@ +Divya/ +other/ +test +Identities.* +newIdentities.* +IdentitiesTest.txt +README_Glossary +x.log +tasks.txt +*.sty +*.aux +*.csv diff --git a/eCF/data/IdentitiesTest.m b/eCF/data/IdentitiesTest.m new file mode 100644 index 0000000..64e8ed4 --- /dev/null +++ b/eCF/data/IdentitiesTest.m @@ -0,0 +1,8 @@ +(* {"RamanujanTauTheta", 1}*) +ConditionalExpression[RamanujanTauTheta[z] == (z*(137/60 - EulerGamma - Log[2*Pi]))/(1 + Inactive[ContinuedFractionK][((-1)^k*z^2*PolyGamma[2*k, 6])/((1 + 2*k)!*(KroneckerDelta[1 - k]*Log[2*Pi] + ((-1)^k*PolyGamma[2*(-1 + k), 6])/(-1 + 2*k)!)), 1 - ((-1)^k*z^2*PolyGamma[2*k, 6])/((1 + 2*k)!*(KroneckerDelta[1 - k]*Log[2*Pi] + ((-1)^k*PolyGamma[2*(-1 + k), 6])/(-1 + 2*k)!)), {k, 1, Infinity}]), Element[z, Complexes] && Abs[z] < 1] + +(* {"Sinc", 1}*) +ConditionalExpression[Sinc[z] == (1 + Inactive[ContinuedFractionK][z^2/(2*k*(1 + 2*k)), 1 - z^2/(2*k*(1 + 2*k)), {k, 1, Infinity}])^(-1), Element[z, Complexes]] + +(* {"RiemannSiegelTheta", 1}*) +ConditionalExpression[RiemannSiegelTheta[z] == -(z*(Log[Pi] - PolyGamma[0, 1/4]))/2 - (z^3*PolyGamma[2, 1/4])/(48*(1 + Inactive[ContinuedFractionK][(z^2*PolyGamma[2*(1 + k), 1/4])/(8*(3 + 5*k + 2*k^2)*PolyGamma[2*k, 1/4]), 1 - (z^2*PolyGamma[2*(1 + k), 1/4])/(8*(3 + 5*k + 2*k^2)*PolyGamma[2*k, 1/4]), {k, 1, Infinity}])), Element[z, Complexes] && Abs[z] < 1/2] \ No newline at end of file diff --git a/eCF/data/functions b/eCF/data/functions new file mode 100644 index 0000000..360ce2c --- /dev/null +++ b/eCF/data/functions @@ -0,0 +1,116 @@ + Abs, \abs, ({-}), () + AiryAi, \AiryAi, (@{-}), (AiryAiPrime) + AiryAiPrime, \AiryAi', (@{-}), () + AiryBi, \AiryBi, (@{-}), (AiryBiPrime) + AiryBiPrime, \AiryBi', (@{-}), () + ArcCos, \acos, (@{-}), (ArcCosh) + ArcCosh, \acosh, (@{-}), () + ArcCot, \acot, (@{-}), (ArcCoth) + ArcCoth, \acoth, (@{-}), () + ArcCsc, \acsc, (@{-}), (ArcCsch) + ArcCsch, \acsch, (@{-}), () + ArcSec, \asec, (@{-}), (ArcSech) + ArcSech, \asech, (@{-}), () + ArcSin, \asin, (@{-}), (ArcSinh) + ArcSinh, \asinh, (@{-}), () + ArcTan, \atan, (@{-}), (ArcTanh) + ArcTanh, \atanh, (@{-}), () + Arg, \ph, (@{-}), () + BetaRegularized, \IncI, ({-}@{-}{-}), () + Binomial, \binom, ({-}{-}), () + BernoulliB, \BernoulliB, ({-}, {-}@{-}), () + BesselI, \BesselI, ({-}@{-}), (SphericalBesselI) + BesselJ, \BesselJ, ({-}@{-}), (SphericalBesselJ) + BesselK, \BesselK, ({-}@{-}), (SphericalBesselK) + BesselY, \BesselY, ({-}@{-}), (SphericalBesselY) + ChebyshevT, \ChebyT, ({-}@{-}), () + ChebyshevU, \ChebyU, ({-}@{-}), () + Cos, \cos, (@{-}), (ArcCos, Cosh, CosIntegral) + Cosh, \cosh, (@{-}), (ArcCosh, CoshIntegral) + CoshIntegral, \CoshInt, (@{-}), () + CosIntegral, \CosInt, (@{-}), () + Cot, \cot, (@{-}), (ArcCot, Coth) + Coth, \coth, (@{-}), (ArcCoth) + Csc, \csc, (@{-}), (ArcCsc, Csch) + Csch, \csch, (@{-}), (ArcCsch) + DawsonF, \DawsonsInt, (@{-}), () + D, \deriv, ({-}{-}), (ParabolicCylinderD) + EllipticE, \CompEllIntE, (@{-}), () + EllipticK, \CompEllIntK, (@{-}), () + Erf, \erf, (@{-}), () + EllipticTheta, \JacobiTheta, ({-}@{-}{-}), () + Erfc, \erfc, (@{-}), () + Erfi, \inverf, (@{-}), () + ExpIntegralE, \ExpIntn, ({-}@{-}), () + ExpIntegralEi, \ExpInti, (@{-}), () + Fibonacci, \Fibonacci, (Number{-}, {-}@{-}), () + Floor, \floor, ({-}), () + FresnelC, \FresnelCos, (@{-}), () + FresnelS, \FresnelSin, (@{-}), () + GammaRegularized, \GammaQ, (@{-}{-}), () + GegenbauerC, \Ultra, ({-}{-}@{-}), () + HankelH1, \HankelHi, ({-}@{-}), (SphericalHankelH1) + HankelH2, \HankelHii, ({-}@{-}), (SphericalHankelH2) + HarmonicNumber, \HarmonicNumber, ({-}, [-]{-}), () + HermiteH, \Hermite, ({-}@{-}), (HermiteHe) + HurwitzLerchPhi, \HurwitzLerchPhi, (@{-}{-}{-}), () + HurwitzZeta, \HurwitzZeta, (@{-}{-}), () + Hypergeometric0F1, \HyperpFq, ({0}{1}@@{}{-}{-}), (Hypergeometric0F1Regularized) + Hypergeometric1F1, \HyperpFq, ({1}{1}@@{-}{-}{-}), (Hypergeometric1F1Regularized) + Hypergeometric2F1, \HyperpFq, ({2}{1}@@{-,-}{-}{-}), (Hypergeometric2F1Regularized) + Hypergeometric0F1Regularized, \HyperboldpFq, ({0}{1}@@{}{-}{-}), () + Hypergeometric1F1Regularized, \HyperboldpFq, ({1}{1}@@{-}{-}{-}), () + Hypergeometric2F1Regularized, \HyperboldpFq, ({2}{1}@@{-,-}{-}{-}), () + HypergeometricPFQ, \HyperpFq, ({-}{-}@@--{-}), (QHypergeometricPFQ) + HypergeometricU, \KummerU, (@{-}{-}{-}), () + Im, \imagpart, ({-}), () + JacobiCN, \Jacobicn, (@{-}{-}), () + JacobiDN, \Jacobidn, (@{-}{-}), () + JacobiP, \JacobiP, ({-}{-}{-}@{-}), () + JacobiSN, \Jacobisn, (@{-}{-}), () + KelvinBei, \Kelvinbei, ({-}, {-}@@{-}), () + KelvinBer, \Kelvinber, ({-}, {-}@@{-}), () + KelvinKei, \Kelvinkei, ({-}, {-}@@{-}), () + KelvinKer, \Kelvinker, ({-}, {-}@@{-}), () + KroneckerDelta, \Kronecker, ({-}{0}, {-}{-}), () + LaguerreL, \Laguerre, ({-}@{-}, [-]{-}@{-}), () + LerchPhi, \LerchPhi, (@{-}{-}{-}), (HurwitzLerchPhi) + Log, \log, (@{-}), (PolyLog, LogInt, LogGamma, ProductLog) + LogGamma, \log@@, ({\EulerGamma@{-}}), () + LogIntegral, \LogInt, (@{-}), () + LucasL, \LucasL, ({-}, {-}@{-}), () + Max, \max, ((-,-), (-,-,-)), () + Min, \min, ((-,-), (-,-,-)), () + Mod, , (({-}\bmod{-})), () + ParabolicCylinderD, \WhitD, ({-}@{-}), () + Pochhammer, \pochhammer, ({-}{-}), (QPochhammer) + PolyLog, \Polylogarithm, ({-}@{-}, S{-}{-}@{-}), () + ProductLog, \LambertW, (@{-}), () + QHypergeometricPFQ, \qHyperrphis, ({-}{-}@@--{-}{-}), () + RamanujanTauTheta, \RamanujanTauTheta, (@{-}), () + Re, \realpart, ({-}), (Regularized, Real) + RiemannSiegelTheta, \RiemannSiegelTheta, (@{-}), () + Sec, \sec, (@{-}), (ArcSec, Sech) + Sech, \sech, (@{-}), (ArcSech) + Sinc, \sinc, (@{-}), () + Sign, \sign, (@{-}), () + Sin, \sin, (@{-}), (ArcSin, Sinh, SinIntegral, Sinc) + Sinh, \sinh, (@{-}), (ArcSinh, SinhIntegral) + SinhIntegral, \SinhInt, (@{-}), () + SinIntegral, \SinInt, (@{-}), () + SphericalBesselJ, \SphBesselJ, ({-}@{-}), () + SphericalBesselY, \SphBesselY, ({-}@{-}), () + SphericalHankelH1, \SphHankelHi, ({-}@{-}), () + SphericalHankelH2, \SphHankelHii, ({-}@{-}), () + Sqrt, \sqrt, ({-}), () + StieltjesGamma, \StieltjesConstants, ({-}), () + StirlingS1, \StirlingS, (@{-}{-}), () + StruveH, \StruveH, ({-}@{-}), () + StruveL, \StruveL, ({-}@{-}), () + Subscript, \text, ({-}_{-}), () + Tan, \tan, (@{-}), (ArcTan, Tanh) + Tanh, \tanh, (@{-}), (ArcTanh) + UnitStep, \HeavisideH, (@{-}), () + WhittakerM, \WhitM, ({-}{-}@{-}), () + WhittakerW, \WhitW, ({-}{-}@{-}), () + Zeta, \RiemannZeta, (@{-}, @{-,-}), () \ No newline at end of file diff --git a/eCF/src/MathematicaToLaTeX.py b/eCF/src/MathematicaToLaTeX.py new file mode 100644 index 0000000..14df64b --- /dev/null +++ b/eCF/src/MathematicaToLaTeX.py @@ -0,0 +1,845 @@ +""" + + Started by Divya Gandla, Version Without Regex by Kevin Chen + DRMF Project: Converting Mathematica to LaTeX + + http://www.wolframfoundation.org/programs/eCF_Identities.pdf + +""" + +__author__ = 'Kevin Chen' +__status__ = 'Development' + + +symbols = { + 'Alpha': 'alpha', 'Beta': 'beta', 'Gamma': 'gamma', 'Delta': 'delta', + 'Epsilon': 'epsilon', 'Zeta': 'zeta', 'Eta': 'eta', 'Theta': 'theta', + 'Iota': 'iota', 'Kappa': 'kappa', 'Lambda': 'lambda', 'Mu': 'mu', + 'Nu': 'nu', 'Xi': 'xi', 'Omicron': 'o', 'Pi': 'pi', 'Rho': 'rho', + 'Sigma': 'sigma', 'Tau': 'tau', 'Upsilon': 'upsilon', 'Phi': 'phi', + 'Chi': 'chi', 'Psi': 'phi', 'Omega': 'omega', + + 'CapitalAlpha': ' A', 'CapitalBeta': ' B', 'CapitalGamma': 'Gamma', + 'CapitalDelta': 'Delta', 'CapitalEpsilon': 'E', 'CapitalZeta': ' Z', + 'CapitalEta': ' H', 'CapitalTheta': 'Theta', 'CapitalIota': ' I', + 'CapitalKappa': 'K', 'CapitalLambda': 'Lambda', 'CapitalMu': ' M', + 'CapitalNu': ' N', 'CapitalXi': 'Xi', 'CapitalOmicron': 'O', + 'CapitalPi': 'Pi', 'CapitalRho': ' P', 'CapitalSigma': 'Sigma', + 'CapitalTau': ' T', 'CapitalUpsilon': ' Y', 'CapitalPhi': 'Phi', + 'CapitalChi': ' X', 'CapitalPsi': 'Psi', 'CapitalOmega': 'Omega', + + 'CurlyEpsilon': 'varepsilon', 'CurlyTheta': 'vartheta', + 'CurlyKappa': 'varkappa', 'CurlyPi': 'varpi', 'CurlyRho': 'varrho', + 'FinalSigma': 'varsigma', 'CurlyPhi': 'varphi', + 'CurlyCapitalUpsilon': 'varUpsilon', + + 'Aleph': 'aleph', 'Bet': 'beth', 'Gimel': 'gimel', 'Dalet': 'daleth', + + 'Infinity': 'infty'} + + +def find_surrounding(line, function, ex=(), start=0): + """finds the indices of the beginning and end of a function; this is the + main function that powers the converter. + """ + positions = [0, 0] + line = line[start:] + positions[0] = line.find(function) + + if ex != '' and len(ex) >= 1: + for e in ex: + if (line.find(e) != -1 and + line.find(e) <= positions[0] and + line.find(e) + len(e) >= positions[0] + len(function)): + + return [line.find(e) + len(e) + start, + line.find(e) + len(e) + start] + + count = 0 + + for j in range(positions[0] + len(function), len(line) + 1): + if j == len(line) and count == 0: + positions[1] = positions[0] + break + + if line[j] in list('([{'): + count += 1 + if line[j] in list(')]}'): + count -= 1 + if count == 0: + if j == positions[0] + len(function): + positions[1] = positions[0] + else: + positions[1] = j + 1 + break + + return positions[0] + start, positions[1] + start + + +def arg_split(line, sep): + """Does the same thing as 'split', but does not split when the separator + is inside parentheses, brackets, or braces. Useful for nested statements. + """ + l = list('([{') + r = list(')]}') + args = [] + count = i = 0 + end = len(line) + 1 + line = line + sep + + while i != end: + if i == end: + args.append(line) + break + + if line[i] in l: + count += 1 + if line[i] in r: + count -= 1 + if count == 0 and line[i] == sep: + args.append(line[:i]) + end -= i + 1 + line = line[i + 1:] + i = 0 + else: + i += 1 + + return args + + +def master_function(line, params): + """A master function, reads in the conversion templates from the + 'functions' file and performs the conversion. + """ + m, l, sep, ex = params[:5] + sep = [i.split('-') for i in sep] + + for _ in range(0, line.count(m)): + try: + pos + except NameError: + pos = find_surrounding(line, m, ex=ex) + else: + pos = find_surrounding(line, m, ex=ex, + start=pos[0] + (0, len(l) + 1) + [(1, 0).index(pos[1] == pos[0])]) + + if pos[0] != pos[1]: + args = arg_split(line[pos[0] + len(m) + 1:pos[1] - 1], ',') + + # exceptions: + if m == 'GegenbauerC': + args[0], args[1] = args[1], args[0] + if m == 'HarmonicNumber' and len(args) == 2: + args[0], args[1] = args[1], args[0] + if m == 'LaguerreL' and len(args) == 3: + args[0], args[1] = args[1], args[0] + if m == 'HypergeometricPFQ' or m == 'QHypergeometricPFQ': + if args[1] == '{}': + args.insert(0, 0) + else: + args.insert(0, len(arg_split(args[1][1:-1], ','))) + if args[1] == '{}': + args.insert(0, 0) + else: + args.insert(0, len(arg_split(args[1][1:-1], ','))) + + line = (line[:pos[0]] + l + + '%s'.join(sep[[len(y) for y in sep]. + index(len(args) + 1)]) + line[pos[1]:]) + line %= tuple(args) + + return line + + +def remove_inactive(line): + """Removes the 'Inactive' statement and its surrounding brakets, like in + the beginning of 'ConditionalExpression.' + """ + for _ in range(0, line.count('Inactive')): + pos = find_surrounding(line, 'Inactive') + if pos[0] != pos[1]: + line = line[:pos[0]] + line[pos[0] + 9:pos[1] - 1] + line[pos[1]:] + + return line + + +def remove_conditionalexpression(line): + """Removes 'ConditionalExpression' and its surrounding brakets.""" + for _ in range(0, line.count('ConditionalExpression')): + pos = find_surrounding(line, 'ConditionalExpression') + if pos[0] != pos[1]: + line = line[:pos[0]] + line[pos[0] + 22:pos[1] - 1] + line[pos[1]:] + + return line + + +def remove_symbol(line): + """Removes 'Symbol' and its surrounding brakets.""" + for _ in range(0, line.count('Symbol')): + pos = find_surrounding(line, 'Symbol') + if pos[0] != pos[1]: + line = line[:pos[0]] + line[pos[0] + 7:pos[1] - 1] + line[pos[1]:] + + return line + + +def beta(line): + """Converts Mathematica's 'Beta' function to the equivalent LaTeX macro, + taking into account the variations for the different number of arguments. + """ + for _ in range(0, line.count('Beta')): + try: + pos + except NameError: + pos = find_surrounding(line, 'Beta', + ex=('BetaRegularized', '[Beta]')) + else: + pos = find_surrounding(line, 'Beta', + ex=('BetaRegularized', '[Beta]'), + start=pos[0] + (0, 9) + [(1, 0).index(pos[1] == pos[0])]) + + if pos[0] != pos[1]: + args = arg_split(line[pos[0] + 5:pos[1] - 1], ',') + if len(args) == 2: + line = (line[:pos[0]] + '\\EulerBeta@{%s}{%s}' + % (args[0], args[1]) + line[pos[1]:]) + else: + line = (line[:pos[0]] + '\\IncBeta{%s}@{%s}{%s}' + % (args[0], args[1], args[2]) + line[pos[1]:]) + + return line + + +def carat(line): + """Converts carats ('^') to ones with braces instead of parentheses. e.g: + 'a ^ (b + c)' would only show the first character 'b' as superscript in + LaTeX, but converting it to + 'a ^ {b + c}' would make it look correct in LaTeX, with 'b + c' as the + superscript. + """ + l = list('([{') + r = list(')]}') + sign = list('*/+-=,') + i = 0 + + while i != len(line): + if line[i] == '^': + count = 0 + for k in range(i + 1, len(line)): + if line[k] in l: + count += 1 + if line[k] in r: + count -= 1 + if count == 0 and line[k] in sign: + count -= 1 + if count < 0: + break + if count == 0 and k == len(line) - 1: + k += 1 + break + k -= 1 + + if line[i + 1] == '(' and line[-1] == ')': + line = line[:i] + '^{' + line[i + 2:k] + '}' + line[k + 1:] + else: + line = line[:i] + '^{' + line[i + 1:k + 1] + '}' + line[k + 1:] + + i += 1 + + return line + + +def cfk(line): + """Converts Mathematica's 'ContinuedFractionK' to the equivalent LaTeX + macro. + """ + for _ in range(0, line.count('ContinuedFractionK')): + try: + pos + except NameError: + pos = find_surrounding(line, 'ContinuedFractionK', ex='') + else: + pos = find_surrounding(line, 'ContinuedFractionK', ex='', + start=pos[0] + (0, 5) + [(1, 0).index(pos[1] == pos[0])]) + + if pos[0] != pos[1]: + args = arg_split(line[pos[0] + 19:pos[1] - 1], ',') + moreargs = arg_split(args[-1][1:-1], ',') + if len(args) == 3: + line = (line[:pos[0]] + '\\CFK{%s}{%s}{\\infty}@{%s}{%s}' + % (moreargs[0], moreargs[1], args[0], args[1]) + + line[pos[1]:]) + else: + line = (line[:pos[0]] + '\\CFK{%s}{%s}{\\infty}@{1}{%s}' + % (moreargs[0], moreargs[1], args[0]) + + line[pos[1]:]) + + return line + + +def gamma(line): + """Converts Mathematica's 'Gamma' function to the equivalent LaTeX macro, + taking into account the variations for the different number of arguments. + """ + for _ in range(0, line.count('Gamma')): + try: + pos + except NameError: + pos = find_surrounding(line, 'Gamma', + ex=('PolyGamma', 'CapitalGamma', + 'PolyGamma', 'LogGamma', 'EulerGamma', + 'IncGamma', 'Gamma]', 'GammaQ', + 'GammaRegularized', 'StieltjesGamma')) + else: + pos = find_surrounding(line, 'Gamma', + ex=('PolyGamma', 'CapitalGamma', + 'PolyGamma', 'LogGamma', 'EulerGamma', + 'IncGamma', 'Gamma]', 'GammaQ', + 'GammaRegularized', 'StieltjesGamma'), + start=pos[0] + (0, 11) + [(1, 0).index(pos[1] == pos[0])]) + + if pos[0] != pos[1]: + args = arg_split(line[pos[0] + 6:pos[1] - 1], ',') + if len(args) == 1: + line = (line[:pos[0]] + '\\EulerGamma@{%s}' + % args[0] + line[pos[1]:]) + elif len(args) == 2: + line = (line[:pos[0]] + '\\IncGamma@{%s}{%s}' + % (args[0], args[1]) + line[pos[1]:]) + else: + line = (line[:pos[0]] + '\\Incgamma@{%s}{%s} - ' + % (args[0], args[1]) + '\\Incgamma@{%s}{%s}' + % (args[0], args[2]) + line[pos[1]:]) + + return line.replace('Incgamma', 'IncGamma') + # needed or the addition of an 'IncGamma' would screw up the rest + + +def integrate(line): + """Converts Mathematica's 'Integrate' function to + the equivalent LaTeX macro. + """ + for _ in range(0, line.count('Integrate')): + try: + pos + except NameError: + pos = find_surrounding(line, 'Integrate', ex='') + else: + pos = find_surrounding(line, 'Integrate', ex='', + start=pos[0] + (0, 10) + [(1, 0).index(pos[1] == pos[0])]) + + if pos[0] != pos[1]: + args = arg_split(line[pos[0] + 10:pos[1] - 1], ',') + moreargs = arg_split(args[1][1:-1], ',') + line = (line[:pos[0]] + '\\int_{%s}^{%s}{%s}d{%s}' + % (moreargs[1], moreargs[2], args[0], moreargs[0]) + + line[pos[1]:]) + + return line + + +def legendrep(line): + """Converts Mathematica's 'LegendreP' function to the equivalent LaTeX + macro, taking into account the variations for the different number of + arguments. + """ + for _ in range(0, line.count('LegendreP')): + try: + pos + except NameError: + pos = find_surrounding(line, 'LegendreP', ex='') + else: + pos = find_surrounding(line, 'LegendreP', ex='', + start=pos[0] + (0, 10) + [(1, 0).index(pos[1] == pos[0])]) + + if pos[0] != pos[1]: + args = arg_split(line[pos[0] + 10:pos[1] - 1], ',') + if len(args) == 2: + line = (line[:pos[0]] + '\\LegendreP{%s}@{%s}' + % (args[0], args[1]) + line[pos[1]:]) + else: + # len(args) == 4 + if args[2] in ('1', '2'): + line = (line[:pos[0]] + '\\FerrersP[%s]{%s}@{%s}' + % (args[1], args[0], args[3]) + line[pos[1]:]) + else: + # args[2] == 3 + line = (line[:pos[0]] + '\\LegendreP[%s]{%s}@{%s}' + % (args[1], args[0], args[3]) + line[pos[1]:]) + + return line + + +def legendreq(line): + """Converts Mathematica's 'LegendreQ' function to the equivalent LaTeX + macro, taking into account the variations for the different number of + arguments. + """ + for _ in range(0, line.count('LegendreQ')): + try: + pos + except NameError: + pos = find_surrounding(line, 'LegendreQ', ex='') + else: + pos = find_surrounding(line, 'LegendreQ', ex='', + start=pos[0] + (0, 10) + [(1, 0).index(pos[1] == pos[0])]) + + if pos[0] != pos[1]: + args = arg_split(line[pos[0] + 10:pos[1] - 1], ',') + if len(args) == 2: + line = (line[:pos[0]] + '\\LegendreQ{%s}@{%s}' + % (args[0], args[1]) + line[pos[1]:]) + else: + # len(args) == 4 + if args[2] in ('1', '2'): + line = (line[:pos[0]] + '\\FerrersQ[%s]{%s}@{%s}' + % (args[1], args[0], args[3]) + line[pos[1]:]) + else: + # args[2] == 3 + line = (line[:pos[0]] + '\\LegendreQ[%s]{%s}@{%s}' + % (args[1], args[0], args[3]) + line[pos[1]:]) + + return line + + +def polyeulergamma(line): + """Converts Mathematica's 'Polygamma' function to the equivalent LaTeX + macro, taking into account the variations for the different number of + arguments. + """ + for _ in range(0, line.count('PolyGamma')): + try: + pos + except NameError: + pos = find_surrounding(line, 'PolyGamma', ex='') + else: + pos = find_surrounding(line, 'PolyGamma', ex='', + start=pos[0] + (0, 10) + [(1, 0).index(pos[1] == pos[0])]) + + if pos[0] != pos[1]: + args = arg_split(line[pos[0] + 10:pos[1] - 1], ',') + if len(args) == 2: + line = (line[:pos[0]] + '\\polygamma{%s}@{%s}' + % (args[0], args[1]) + line[pos[1]:]) + else: + line = (line[:pos[0]] + '\\digamma@{%s}' + % args[0] + '}' + line[pos[1]:]) + + return line + + +def product(line): + """Converts Mathematica's product sum function to the equivalent LaTeX + macro. + """ + for _ in range(0, line.count('Product')): + try: + pos + except NameError: + pos = find_surrounding(line, 'Product', ex='') + else: + pos = find_surrounding(line, 'Product', ex='', + start=pos[0] + (0, 6) + [(1, 0).index(pos[1] == pos[0])]) + + if pos[0] != pos[1]: + args = arg_split(line[pos[0] + 8:pos[1] - 1], ',') + moreargs = arg_split(args[-1][1:-1], ',') + line = (line[:pos[0]] + '\\Prod{%s}{%s}{%s}@{%s}' + % (moreargs[0], moreargs[1], moreargs[2], args[0]) + + line[pos[1]:]) + + return line + + +def qpochhammer(line): + """Converts Mathematica's 'QPochhammer' function to the equivalent LaTeX + macro. + """ + for _ in range(0, line.count('QPochhammer')): + try: + pos + except NameError: + pos = find_surrounding(line, 'QPochhammer', ex='') + else: + pos = find_surrounding(line, 'QPochhammer', ex='', + start=pos[0] + (0, 13) + [(1, 0).index(pos[1] == pos[0])]) + + if pos[0] != pos[1]: + args = arg_split(line[pos[0] + 12:pos[1] - 1], ',') + + if len(args) == 1: + line = (line[:pos[0]] + '\\qPochhammer{%s}{%s}{\infty}' + % (args[0], args[0]) + line[pos[1]:]) + elif len(args) == 2: + line = (line[:pos[0]] + '\\qPochhammer{%s}{%s}{\infty}' + % (args[0], args[1]) + line[pos[1]:]) + else: # len(args) = 3 + line = (line[:pos[0]] + '\\qPochhammer{%s}{%s}{%s}' + % (args[0], args[1], args[2]) + line[pos[1]:]) + + return line + + +def summation(line): + """Converts Mathematica's summation function to the equivalent LaTeX + macro. + """ + for _ in range(0, line.count('Sum')): + try: + pos + except NameError: + pos = find_surrounding(line, 'Sum', ex='') + else: + pos = find_surrounding(line, 'Sum', ex='', + start=pos[0] + (0, 5) + [(1, 0).index(pos[1] == pos[0])]) + + if pos[0] != pos[1]: + args = arg_split(line[pos[0] + 4:pos[1] - 1], ',') + moreargs = arg_split(args[-1][1:-1], ',') + line = (line[:pos[0]] + '\\Sum{%s}{%s}{%s}@{%s}' + % (moreargs[0], moreargs[1], moreargs[2], args[0]) + + line[pos[1]:]) + + return line + + +def constraint(line): + """Converts Mathematica's 'Element', 'NotElement', and 'Inequality' + functions to LaTeX formatting using \constraint{}. + """ + sections = arg_split(line, ',') + + if len(sections) == 1: + return line + + constraints = arg_split(sections[-1].replace('&&', '&'), '&') + + for i in range(len(constraints)): + if i == 0: + constraints[i] = ('\n% \\constraint{$' + + constraints[i].replace('&', ' \\land ')) + else: + constraints[i] = ('\n% & $' + + constraints[i].replace('&', ' \\land ')) + if i == len(constraints) - 1: + constraints[i] += '$}' + else: + constraints[i] += '$' + + line = sections[0] + ''.join(constraints) + + mathematica_elements = ('Complexes', 'Wholes', 'Naturals', 'Integers', + 'Irrationals', 'Reals', 'Rational', 'Primes') + latex_elements = ('Complex', 'Whole', 'NatNumber', 'Integer', + 'Irrational', 'Real', 'Rational', 'Prime') + + for _ in range(0, line.count('Element')): + + try: + pos1 + except NameError: + pos1 = find_surrounding(line, 'Element', ex=('NotElement', )) + else: + pos1 = find_surrounding(line, 'Element', ex=('NotElement', ), + start=pos1[0] + (0, 8) + [(1, 0).index(pos1[1] == pos1[0])]) + + if pos1[0] != pos1[1]: + sep = arg_split(line[pos1[0] + 8:pos1[1] - 1], ',') + sep[1] = latex_elements[mathematica_elements.index(sep[1])] + line = (line[:pos1[0]] + sep[0].replace('|', ',') + ' \\in \\' + + sep[1] + line[pos1[1]:]) + + for _ in range(0, line.count('NotElement')): + + try: + pos2 + except NameError: + pos2 = find_surrounding(line, 'NotElement', ex=()) + else: + pos2 = find_surrounding(line, 'NotElement', ex=(), + start=pos2[0] + (0, 11) + [(1, 0).index(pos2[1] == pos2[0])]) + + if pos2[0] != pos2[1]: + sep = arg_split(line[pos2[0] + 11:pos2[1] - 1], ',') + sep[1] = latex_elements[mathematica_elements.index(sep[1])] + line = (line[:pos2[0]] + sep[0].replace('|', ',') + + ' \\notin \\' + sep[1] + line[pos2[1]:]) + + for _ in range(0, line.count('Inequality')): + pos3 = find_surrounding(line, 'Inequality') + line = (line[:pos3[0]] + + ''.join(line[pos3[0] + 11:pos3[1] - 1].split(',')) + + line[pos3[1]:]) + + return line + + +def convert_fraction(line): + """Converts Mathematica fractions, which are only '/', to LaTeX + \frac{}{}-ions. + """ + l = list('([{') + r = list(')]}') + sign = list('*+-=,<>&') + i = 0 + + while i != len(line): + if line[i] == '/': + + # Searching left. + count = 0 + for j in range(i - 1, -1, -1): + if line[j] in r: + count += 1 + if line[j] in l: + count -= 1 + if count == 0 and line[j] in sign: + count -= 1 + if count < 0: + break + if count == 0 and j == 0: + j -= 1 + break + + # Searching right. + count = 0 + for k in range(i + 1, len(line)): + if line[k] in l: + count += 1 + if line[k] in r: + count -= 1 + if count == 0 and line[k] in sign: + count -= 1 + if count < 0: + break + if count == 0 and k == len(line) - 1: + k += 1 + break + k -= 1 + + # Removes extra surrounding parentheses, if there are any. + # This won't work if you're doing "( )( )/( )( )", it will + # incorrectly change it to " )( / )( ", but there are no cases of + # this happening yet, so I have not gone to fixing this yet. + if (line[j + 1] == '(' and line[i - 1] == ')' and + line[i + 1] == '(' and line[k] == ')'): + # ()/() + line = (line[:j + 1] + '\\frac{' + line[j + 2:i - 1] + '}{' + + line[i + 2:k] + '}' + line[k + 1:]) + elif line[j + 1] == '(' and line[i - 1] == ')': + # ()/-- + line = (line[:j + 1] + '\\frac{' + line[j + 2:i - 1] + '}{' + + line[i + 1:k + 1] + '}' + line[k + 1:]) + elif line[i + 1] == '(' and line[k] == ')': + # --/() + line = (line[:j + 1] + '\\frac{' + line[j + 1:i] + '}{' + + line[i + 2:k] + '}' + line[k + 1:]) + else: + # --/-- + line = (line[:j + 1] + '\\frac{' + line[j + 1:i] + '}{' + + line[i + 1:k + 1] + '}' + line[k + 1:]) + + i += 1 + + return line + + +def piecewise(line): + """Converts Mathematica's piecewise function to LaTeX, using 'cases'.""" + for _ in range(0, line.count('Piecewise')): + try: + pos + except NameError: + pos = find_surrounding(line, 'Piecewise', ex='') + else: + pos = find_surrounding(line, 'Piecewise', ex='', + start=pos[0] + (0, 15) + [(1, 0).index(pos[1] == pos[0])]) + + if pos[0] != pos[1]: + args = arg_split(line[pos[0] + 10:pos[1] - 1], ',') + + piece = ' \\\\ '.join([' & '.join(arg_split(i[1:-1], ',')) + for i in arg_split(args[0][1:-1], ',')]) + + if len(args) == 1: + piece += ' \\\\ 0 & \\text{True}' + else: + piece += ' \\\\ ' + args[1] + ' & \\text{True}' + + line = (line[:pos[0]] + '{\\begin{cases} ' + piece + + ' \\end{cases}}' + line[pos[1]:]) + + return line + + +def replace_operators(line): + """Replaces basic operators.""" + line = line.replace('==', '=') + line = line.replace('||', ' \\lor ') + line = line.replace('>=', ' \\geq ') + line = line.replace('<=', ' \\leq ') + line = line.replace('LessEqual', ' \\leq ') + line = line.replace('Less', '<') + line = line.replace('>', ' > ') + line = line.replace('<', ' < ') + line = line.replace('=', ' = ') + line = line.replace('^', ' ^ ') + line = line.replace('*', ' ') + line = line.replace('+', ' + ') + line = line.replace('-', ' - ') + line = line.replace(',', ', ') + + if '%' in line: + parts = (line[:line.index('%')], line[line.index('%'):]) + line = parts[0] + line = line.replace('(', '\\left( ') + line = line.replace(')', ' \\right)') + line = line.replace('&', ' \\land ') + line = line.replace(' ', ' ') + line += parts[1] + + line = line.replace('"a"', 'a') + line = line.replace('Catalan', '\\CatalansConstant') + line = line.replace('GoldenRatio', '\\GoldenRatio') + line = line.replace('Pi', '\\pi') + line = line.replace('CalculateData`Private`nu', + '\\text{CalculateData`Private`nu}') + + return line + + +def replace_vars(line): + """Replaces the easy to convert variables in Mathematica to + its equivalent LaTeX code in the dictionary 'symbols'. + """ + for word in symbols: + if symbols[word][0] == ' ': + line = line.replace('\\[' + word + ']', symbols[word][1:]) + elif word == 'Infinity': + line = line.replace('Infinity', r'\infty') + else: + line = line.replace('[' + word + ']', symbols[word]) + + return line + + +def main(): + """Opens Mathematica file with identities and puts converted lines into + newIdentities.tex. + """ + test = True + + with open('data/newIdentities.tex', 'w') as latex: + with open('data/Identities' + ['', 'Test'][test] + '.m', 'r') \ + as mathematica: + + latex.write('\n\\documentclass{article}\n\n' + '\\usepackage{amsmath}\n' + '\\usepackage{amsthm}\n' + '\\usepackage{amssymb}\n' + '\\usepackage{amsfonts}\n' + '\\usepackage{breqn}\n' + '\\usepackage{DLMFmath}\n' + '\\usepackage{DRMFfcns}\n' + '\\usepackage{DLMFfcns}\n' + '\\usepackage{graphicx}\n' + '\\usepackage[paperwidth=20in, paperheight=20in, ' + 'margin=0.5in]{geometry}\n\n' + '\\begin{document}\n\n\n') + + for line in mathematica: + line = line.replace('\n', '') + + if '(*' in line and '*)' in line: + + if test: + line = line.replace('(*', '%').replace('*)', '%') + else: + line = ('\\begin{equation*} \\tag{' + + line[4:-3].replace('"', '') + '}') + + latex.write(line + '\n') + + else: + + line = line.replace(' ', '') + + line = remove_inactive(line) + line = remove_conditionalexpression(line) + line = remove_symbol(line) + + line = line.replace('EulerGamma', '\\EulerConstant') + + for i in FUNCTION_CONVERSIONS: + line = master_function(line, i) + + line = beta(line) + line = carat(line) + line = cfk(line) + line = gamma(line) + line = integrate(line) + line = legendrep(line) + line = legendreq(line) + line = polyeulergamma(line) + line = product(line) + line = qpochhammer(line) + line = summation(line) + line = convert_fraction(line) + line = constraint(line) + line = piecewise(line) + line = replace_operators(line) + line = replace_vars(line) + + # create macros for: + # Sinc + # RamanujanTauTheta + # RiemannSiegelTheta + + print(line) + + if test and line != '': + line = '\\begin{equation*}\n' + line + + if line != '': + line += '\n\\end{equation*}' + + latex.write(line + '\n') + + latex.write('\n\n\\end{document}\n') + + +with open('data/functions') as functions: + FUNCTION_CONVERSIONS = list(arg_split(line.replace(' ', ''), ',') for line + in functions.read().split('\n') + if (line != '' and '#' not in line)) + +for index in range(len(FUNCTION_CONVERSIONS)): + FUNCTION_CONVERSIONS[index][2] = \ + tuple(arg_split(FUNCTION_CONVERSIONS[index][2][1:-1], ',')) + + if FUNCTION_CONVERSIONS[index][3] == '()': + FUNCTION_CONVERSIONS[index][3] = '' + else: + FUNCTION_CONVERSIONS[index][3] = \ + tuple(FUNCTION_CONVERSIONS[index][3][1:-1].split(',')) + + FUNCTION_CONVERSIONS[index] = tuple(FUNCTION_CONVERSIONS[index]) + +FUNCTION_CONVERSIONS = tuple(FUNCTION_CONVERSIONS) + +if __name__ == '__main__': + main() diff --git a/eCF/src/pythonTest.py b/eCF/src/pythonTest.py new file mode 100644 index 0000000..e69de29 From 96ad76d48e5d099b9049be4f1c787473b414ef04 Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Fri, 5 Aug 2016 10:53:56 -0400 Subject: [PATCH 197/402] Add test files --- eCF/data/IdentitiesTest.m | 8 ----- eCF/src/pythonTest.py | 0 eCF/test/test_arg_split.py | 30 +++++++++++++++++++ eCF/test/test_carat.py | 7 +++++ eCF/test/test_constraint.py | 7 +++++ eCF/test/test_convert_fraction.py | 7 +++++ eCF/test/test_find_surrounding.py | 7 +++++ eCF/test/test_master_function.py | 7 +++++ eCF/test/test_piecewise.py | 7 +++++ eCF/test/test_removal_functions.py | 9 ++++++ eCF/test/test_replace_operators.py | 7 +++++ eCF/test/test_replace_vars.py | 7 +++++ .../test_single_macro_conversion_functions.py | 16 ++++++++++ 13 files changed, 111 insertions(+), 8 deletions(-) delete mode 100644 eCF/data/IdentitiesTest.m delete mode 100644 eCF/src/pythonTest.py create mode 100644 eCF/test/test_arg_split.py create mode 100644 eCF/test/test_carat.py create mode 100644 eCF/test/test_constraint.py create mode 100644 eCF/test/test_convert_fraction.py create mode 100644 eCF/test/test_find_surrounding.py create mode 100644 eCF/test/test_master_function.py create mode 100644 eCF/test/test_piecewise.py create mode 100644 eCF/test/test_removal_functions.py create mode 100644 eCF/test/test_replace_operators.py create mode 100644 eCF/test/test_replace_vars.py create mode 100644 eCF/test/test_single_macro_conversion_functions.py diff --git a/eCF/data/IdentitiesTest.m b/eCF/data/IdentitiesTest.m deleted file mode 100644 index 64e8ed4..0000000 --- a/eCF/data/IdentitiesTest.m +++ /dev/null @@ -1,8 +0,0 @@ -(* {"RamanujanTauTheta", 1}*) -ConditionalExpression[RamanujanTauTheta[z] == (z*(137/60 - EulerGamma - Log[2*Pi]))/(1 + Inactive[ContinuedFractionK][((-1)^k*z^2*PolyGamma[2*k, 6])/((1 + 2*k)!*(KroneckerDelta[1 - k]*Log[2*Pi] + ((-1)^k*PolyGamma[2*(-1 + k), 6])/(-1 + 2*k)!)), 1 - ((-1)^k*z^2*PolyGamma[2*k, 6])/((1 + 2*k)!*(KroneckerDelta[1 - k]*Log[2*Pi] + ((-1)^k*PolyGamma[2*(-1 + k), 6])/(-1 + 2*k)!)), {k, 1, Infinity}]), Element[z, Complexes] && Abs[z] < 1] - -(* {"Sinc", 1}*) -ConditionalExpression[Sinc[z] == (1 + Inactive[ContinuedFractionK][z^2/(2*k*(1 + 2*k)), 1 - z^2/(2*k*(1 + 2*k)), {k, 1, Infinity}])^(-1), Element[z, Complexes]] - -(* {"RiemannSiegelTheta", 1}*) -ConditionalExpression[RiemannSiegelTheta[z] == -(z*(Log[Pi] - PolyGamma[0, 1/4]))/2 - (z^3*PolyGamma[2, 1/4])/(48*(1 + Inactive[ContinuedFractionK][(z^2*PolyGamma[2*(1 + k), 1/4])/(8*(3 + 5*k + 2*k^2)*PolyGamma[2*k, 1/4]), 1 - (z^2*PolyGamma[2*(1 + k), 1/4])/(8*(3 + 5*k + 2*k^2)*PolyGamma[2*k, 1/4]), {k, 1, Infinity}])), Element[z, Complexes] && Abs[z] < 1/2] \ No newline at end of file diff --git a/eCF/src/pythonTest.py b/eCF/src/pythonTest.py deleted file mode 100644 index e69de29..0000000 diff --git a/eCF/test/test_arg_split.py b/eCF/test/test_arg_split.py new file mode 100644 index 0000000..410c58d --- /dev/null +++ b/eCF/test/test_arg_split.py @@ -0,0 +1,30 @@ + +__author__ = 'Kevin Chen' +__status__ = 'Development' + +from unittest import TestCase +from MathematicaToLaTeX import arg_split + + +class TestArgumentSplit(TestCase): + + def test_normal(self): + self.assertEqual(arg_split(','.join(list('abcdef')), ','), ','.join(list('abcdef')).split(',')) + + def test_empty(self): + self.assertEqual(arg_split(',,,,,,', ','), ',,,,,,'.split(',')) + + def test_some_empty(self): + before = 'a,,b,,c,,d' + after = ['a', '', 'b', '', 'c', '', 'd'] + self.assertEqual(arg_split(before, ','), after) + + def test_parens(self): + before = '(a,b),[c,d],{e,(f,g)}' + after = ['(a,b)', '[c,d]', '{e,(f,g)}'] + self.assertEqual(arg_split(before, ','), after) + + def test_combined(self): + before = 'a,(b,c),,[d,e],,,{fg,hi}' + after = ['a', '(b,c)', '', '[d,e]', '', '', '{fg,hi}'] + self.assertEqual(arg_split(before, ','), after) diff --git a/eCF/test/test_carat.py b/eCF/test/test_carat.py new file mode 100644 index 0000000..1cdc8d8 --- /dev/null +++ b/eCF/test/test_carat.py @@ -0,0 +1,7 @@ + +__author__ = 'Kevin Chen' +__status__ = 'Development' + +from unittest import TestCase +from MathematicaToLaTeX import carat + diff --git a/eCF/test/test_constraint.py b/eCF/test/test_constraint.py new file mode 100644 index 0000000..34330f8 --- /dev/null +++ b/eCF/test/test_constraint.py @@ -0,0 +1,7 @@ + +__author__ = 'Kevin Chen' +__status__ = 'Development' + +from unittest import TestCase +from MathematicaToLaTeX import constraint + diff --git a/eCF/test/test_convert_fraction.py b/eCF/test/test_convert_fraction.py new file mode 100644 index 0000000..1d6be0f --- /dev/null +++ b/eCF/test/test_convert_fraction.py @@ -0,0 +1,7 @@ + +__author__ = 'Kevin Chen' +__status__ = 'Development' + +from unittest import TestCase +from MathematicaToLaTeX import convert_fraction + diff --git a/eCF/test/test_find_surrounding.py b/eCF/test/test_find_surrounding.py new file mode 100644 index 0000000..fd545d0 --- /dev/null +++ b/eCF/test/test_find_surrounding.py @@ -0,0 +1,7 @@ + +__author__ = 'Kevin Chen' +__status__ = 'Development' + +from unittest import TestCase +from MathematicaToLaTeX import find_surrounding + diff --git a/eCF/test/test_master_function.py b/eCF/test/test_master_function.py new file mode 100644 index 0000000..427bd1d --- /dev/null +++ b/eCF/test/test_master_function.py @@ -0,0 +1,7 @@ + +__author__ = 'Kevin Chen' +__status__ = 'Development' + +from unittest import TestCase +from MathematicaToLaTeX import master_function + diff --git a/eCF/test/test_piecewise.py b/eCF/test/test_piecewise.py new file mode 100644 index 0000000..5775dde --- /dev/null +++ b/eCF/test/test_piecewise.py @@ -0,0 +1,7 @@ + +__author__ = 'Kevin Chen' +__status__ = 'Development' + +from unittest import TestCase +from MathematicaToLaTeX import piecewise + diff --git a/eCF/test/test_removal_functions.py b/eCF/test/test_removal_functions.py new file mode 100644 index 0000000..3945a15 --- /dev/null +++ b/eCF/test/test_removal_functions.py @@ -0,0 +1,9 @@ + +__author__ = 'Kevin Chen' +__status__ = 'Development' + +from unittest import TestCase +from MathematicaToLaTeX import remove_inactive +from MathematicaToLaTeX import remove_conditionalexpression +from MathematicaToLaTeX import remove_symbol + diff --git a/eCF/test/test_replace_operators.py b/eCF/test/test_replace_operators.py new file mode 100644 index 0000000..b07ce81 --- /dev/null +++ b/eCF/test/test_replace_operators.py @@ -0,0 +1,7 @@ + +__author__ = 'Kevin Chen' +__status__ = 'Development' + +from unittest import TestCase +from MathematicaToLaTeX import replace_operators + diff --git a/eCF/test/test_replace_vars.py b/eCF/test/test_replace_vars.py new file mode 100644 index 0000000..041adc4 --- /dev/null +++ b/eCF/test/test_replace_vars.py @@ -0,0 +1,7 @@ + +__author__ = 'Kevin Chen' +__status__ = 'Development' + +from unittest import TestCase +from MathematicaToLaTeX import replace_vars + diff --git a/eCF/test/test_single_macro_conversion_functions.py b/eCF/test/test_single_macro_conversion_functions.py new file mode 100644 index 0000000..722ba2e --- /dev/null +++ b/eCF/test/test_single_macro_conversion_functions.py @@ -0,0 +1,16 @@ + +__author__ = 'Kevin Chen' +__status__ = 'Development' + +from unittest import TestCase +from MathematicaToLaTeX import beta +from MathematicaToLaTeX import cfk +from MathematicaToLaTeX import gamma +from MathematicaToLaTeX import integrate +from MathematicaToLaTeX import legendrep +from MathematicaToLaTeX import legendreq +from MathematicaToLaTeX import polyeulergamma +from MathematicaToLaTeX import product +from MathematicaToLaTeX import qpochhammer +from MathematicaToLaTeX import summation + From b0748bfe848e87a2fe197aecfd4f1d8627488f56 Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Fri, 5 Aug 2016 11:13:53 -0400 Subject: [PATCH 198/402] Fix path to files for ability to run from any directory --- .gitignore | 11 ++++++++++ eCF/.gitignore | 6 +++--- eCF/data/IdentitiesTest.m | 8 +++++++ eCF/src/MathematicaToLaTeX.py | 40 +++++++++++++++++++++++------------ eCF/src/pythonTest.py | 0 5 files changed, 48 insertions(+), 17 deletions(-) create mode 100644 eCF/data/IdentitiesTest.m create mode 100644 eCF/src/pythonTest.py diff --git a/.gitignore b/.gitignore index 5c0f18c..1d43b2a 100644 --- a/.gitignore +++ b/.gitignore @@ -251,3 +251,14 @@ sympy-plots-for-*.tex/ *.bak *.sav +# ecf project +*.sty +Identities.* +newIdentities.* +*.csv +macros.txt +test.* +*.log +*.aux +ZE.3.* + diff --git a/eCF/.gitignore b/eCF/.gitignore index 975836e..13bc7e0 100644 --- a/eCF/.gitignore +++ b/eCF/.gitignore @@ -1,8 +1,8 @@ Divya/ other/ -test -Identities.* -newIdentities.* +src/test.txt +data/Identities.* +data/newIdentities.* IdentitiesTest.txt README_Glossary x.log diff --git a/eCF/data/IdentitiesTest.m b/eCF/data/IdentitiesTest.m new file mode 100644 index 0000000..64e8ed4 --- /dev/null +++ b/eCF/data/IdentitiesTest.m @@ -0,0 +1,8 @@ +(* {"RamanujanTauTheta", 1}*) +ConditionalExpression[RamanujanTauTheta[z] == (z*(137/60 - EulerGamma - Log[2*Pi]))/(1 + Inactive[ContinuedFractionK][((-1)^k*z^2*PolyGamma[2*k, 6])/((1 + 2*k)!*(KroneckerDelta[1 - k]*Log[2*Pi] + ((-1)^k*PolyGamma[2*(-1 + k), 6])/(-1 + 2*k)!)), 1 - ((-1)^k*z^2*PolyGamma[2*k, 6])/((1 + 2*k)!*(KroneckerDelta[1 - k]*Log[2*Pi] + ((-1)^k*PolyGamma[2*(-1 + k), 6])/(-1 + 2*k)!)), {k, 1, Infinity}]), Element[z, Complexes] && Abs[z] < 1] + +(* {"Sinc", 1}*) +ConditionalExpression[Sinc[z] == (1 + Inactive[ContinuedFractionK][z^2/(2*k*(1 + 2*k)), 1 - z^2/(2*k*(1 + 2*k)), {k, 1, Infinity}])^(-1), Element[z, Complexes]] + +(* {"RiemannSiegelTheta", 1}*) +ConditionalExpression[RiemannSiegelTheta[z] == -(z*(Log[Pi] - PolyGamma[0, 1/4]))/2 - (z^3*PolyGamma[2, 1/4])/(48*(1 + Inactive[ContinuedFractionK][(z^2*PolyGamma[2*(1 + k), 1/4])/(8*(3 + 5*k + 2*k^2)*PolyGamma[2*k, 1/4]), 1 - (z^2*PolyGamma[2*(1 + k), 1/4])/(8*(3 + 5*k + 2*k^2)*PolyGamma[2*k, 1/4]), {k, 1, Infinity}])), Element[z, Complexes] && Abs[z] < 1/2] \ No newline at end of file diff --git a/eCF/src/MathematicaToLaTeX.py b/eCF/src/MathematicaToLaTeX.py index 14df64b..f87b17d 100644 --- a/eCF/src/MathematicaToLaTeX.py +++ b/eCF/src/MathematicaToLaTeX.py @@ -9,7 +9,9 @@ __author__ = 'Kevin Chen' __status__ = 'Development' +__credits__ = ["Divya Gandla", "Kevin Chen"] +import os symbols = { 'Alpha': 'alpha', 'Beta': 'beta', 'Gamma': 'gamma', 'Delta': 'delta', @@ -49,9 +51,9 @@ def find_surrounding(line, function, ex=(), start=0): if ex != '' and len(ex) >= 1: for e in ex: if (line.find(e) != -1 and - line.find(e) <= positions[0] and - line.find(e) + len(e) >= positions[0] + len(function)): - + line.find(e) <= positions[0] and + line.find(e) + len(e) >= positions[0] + len( + function)): return [line.find(e) + len(e) + start, line.find(e) + len(e) + start] @@ -213,7 +215,8 @@ def beta(line): def carat(line): - """Converts carats ('^') to ones with braces instead of parentheses. e.g: + """ + Converts carats ('^') to ones with braces instead of parentheses. e.g: 'a ^ (b + c)' would only show the first character 'b' as superscript in LaTeX, but converting it to 'a ^ {b + c}' would make it look correct in LaTeX, with 'b + c' as the @@ -252,7 +255,8 @@ def carat(line): def cfk(line): - """Converts Mathematica's 'ContinuedFractionK' to the equivalent LaTeX + """ + Converts Mathematica's 'ContinuedFractionK' to the equivalent LaTeX macro. """ for _ in range(0, line.count('ContinuedFractionK')): @@ -281,7 +285,8 @@ def cfk(line): def gamma(line): - """Converts Mathematica's 'Gamma' function to the equivalent LaTeX macro, + """ + Converts Mathematica's 'Gamma' function to the equivalent LaTeX macro, taking into account the variations for the different number of arguments. """ for _ in range(0, line.count('Gamma')): @@ -320,7 +325,8 @@ def gamma(line): def integrate(line): - """Converts Mathematica's 'Integrate' function to + """ + Converts Mathematica's 'Integrate' function to the equivalent LaTeX macro. """ for _ in range(0, line.count('Integrate')): @@ -549,9 +555,9 @@ def constraint(line): try: pos1 except NameError: - pos1 = find_surrounding(line, 'Element', ex=('NotElement', )) + pos1 = find_surrounding(line, 'Element', ex=('NotElement',)) else: - pos1 = find_surrounding(line, 'Element', ex=('NotElement', ), + pos1 = find_surrounding(line, 'Element', ex=('NotElement',), start=pos1[0] + (0, 8) [(1, 0).index(pos1[1] == pos1[0])]) @@ -635,7 +641,7 @@ def convert_fraction(line): # incorrectly change it to " )( / )( ", but there are no cases of # this happening yet, so I have not gone to fixing this yet. if (line[j + 1] == '(' and line[i - 1] == ')' and - line[i + 1] == '(' and line[k] == ')'): + line[i + 1] == '(' and line[k] == ')'): # ()/() line = (line[:j + 1] + '\\frac{' + line[j + 2:i - 1] + '}{' + line[i + 2:k] + '}' + line[k + 1:]) @@ -743,9 +749,14 @@ def main(): """ test = True - with open('data/newIdentities.tex', 'w') as latex: - with open('data/Identities' + ['', 'Test'][test] + '.m', 'r') \ - as mathematica: + with open(os.path.dirname(os.path.realpath(__file__)) + + '/../data/newIdentities.tex', 'w') as latex: + if test: + to_open = 'IdentitiesTest.m' + else: + to_open = 'Identities.m' + with open(os.path.dirname(os.path.realpath(__file__)) + '/../data/' + + to_open, 'r') as mathematica: latex.write('\n\\documentclass{article}\n\n' '\\usepackage{amsmath}\n' @@ -822,7 +833,8 @@ def main(): latex.write('\n\n\\end{document}\n') -with open('data/functions') as functions: +with open(os.path.dirname(os.path.realpath(__file__)) + \ + '/../data/functions') as functions: FUNCTION_CONVERSIONS = list(arg_split(line.replace(' ', ''), ',') for line in functions.read().split('\n') if (line != '' and '#' not in line)) diff --git a/eCF/src/pythonTest.py b/eCF/src/pythonTest.py new file mode 100644 index 0000000..e69de29 From 5820fbfc0b5c215ffea44d1714fa2c53c47fca15 Mon Sep 17 00:00:00 2001 From: Joon Bang Date: Fri, 5 Aug 2016 12:26:45 -0400 Subject: [PATCH 199/402] Clean up more code --- main_page/Categories.txt | 10 - main_page/categories.txt | 10 + main_page/main_page.mmd | 10 +- main_page/src/main.py | 27 +- main_page/src/mod_main.py | 194 ++++++------- main_page/src/symbols_list.py | 496 +++++++++++++++++----------------- 6 files changed, 372 insertions(+), 375 deletions(-) delete mode 100644 main_page/Categories.txt create mode 100644 main_page/categories.txt diff --git a/main_page/Categories.txt b/main_page/Categories.txt deleted file mode 100644 index 2d2fc4a..0000000 --- a/main_page/Categories.txt +++ /dev/null @@ -1,10 +0,0 @@ - F - real or complex valued functions. - P - polynomials. - I - integer valued functions. - O - operators. - SM - semantic macros. - Q - quantifiers, set operators, and symbols. - SN - sets of numbers. - C - constants. - L - linear algebra. - D - distributions. diff --git a/main_page/categories.txt b/main_page/categories.txt new file mode 100644 index 0000000..eee3923 --- /dev/null +++ b/main_page/categories.txt @@ -0,0 +1,10 @@ +F - real or complex valued functions. +P - polynomials. +I - integer valued functions. +O - operators. +SM - semantic macros. +Q - quantifiers, set operators, and symbols. +SN - sets of numbers. +C - constants. +L - linear algebra. +D - distributions. diff --git a/main_page/main_page.mmd b/main_page/main_page.mmd index f271213..65d8685 100644 --- a/main_page/main_page.mmd +++ b/main_page/main_page.mmd @@ -442,7 +442,7 @@ drmf_bof In the DRMF, the LaTeX semantic macro '''\AntiDer''' represents the anti-derivative notation, which is defined as follows. -This macro is in the category of polynomials. +This macro is in the category of semantic macros. In math mode, this macro can be called in the following ways: @@ -925,7 +925,7 @@ drmf_bof The LaTeX DLMF and DRMF macro '''\f''' represents Function. -This macro is in the category of polynomials. +This macro is in the category of semantic macros. In math mode, this macro can be called in the following ways: @@ -1060,7 +1060,7 @@ drmf_bof In the DRMF, the LaTeX semantic macro '''\Int''' represents the definite integral notation, which is defined as follows. -This macro is in the category of polynomials. +This macro is in the category of semantic macros. In math mode, this macro can be called in the following ways: @@ -1200,7 +1200,7 @@ drmf_bof For the sake of compactness, we use the abbreviated '''\lrselection''' notation in a number of formulas. -This macro is in the category of polynomials. +This macro is in the category of semantic macros. In math mode, this macro can be called in the following way: @@ -3208,7 +3208,7 @@ drmf_bof The LaTeX DLMF and DRMF macro '''\poly''' represents the semantic polynomial. -This macro is in the category of polynomials. +This macro is in the category of semantic macros. In math mode, this macro can be called in the following ways: diff --git a/main_page/src/main.py b/main_page/src/main.py index 5524ffe..9c2e117 100644 --- a/main_page/src/main.py +++ b/main_page/src/main.py @@ -3,16 +3,8 @@ __credits__ = ["Joon Bang", "Azeem Mohammed"] import time -import symbols_list import mod_main - -""" -Order: -- List -- Symbols List -- Headers -- List Ways -""" +import symbols_list def generate_html(tag_name, options, text, spacing=True): @@ -87,6 +79,14 @@ def update_headers(text, definitions): return output +def create_backup(): + with open("main_page.mmd") as current: + local_time = time.asctime(time.localtime(time.time())).split() + local_time = "_" + '_'.join(local_time[1:3] + local_time[3:][::-1]) + with open("backups/main_page" + local_time + ".mmd.bak", "w") as backup: + backup.write(current.read()) + + def main(): if raw_input("Update main page? (y/n): ") == "n": return @@ -95,17 +95,14 @@ def main(): with open("main_page.mmd") as main_page: text = main_page.read() - # create backup - local_time = time.asctime(time.localtime(time.time())).split() - local_time = "_" + '_'.join(local_time[1:3] + local_time[3:][::-1]) - with open("backups/main_page" + local_time + ".mmd.bak", "w") as backup: - backup.write(text) - text, definitions = update_macro_list(text) text = symbols_list.main(text) text = update_headers(text, definitions) text = mod_main.main(text) + # only create backup if program did not crash + create_backup() + with open("main_page.mmd", "w") as main_page: main_page.write(text) diff --git a/main_page/src/mod_main.py b/main_page/src/mod_main.py index e7090f7..7d6d40c 100644 --- a/main_page/src/mod_main.py +++ b/main_page/src/mod_main.py @@ -4,112 +4,112 @@ import csv +GLOSSARY_LOCATION = "new.Glossary.csv" -def atSort(l): - return (sorted(l, key=lambda x: x.count("@"))) - -def remOpt(p): +def remove_optional_params(string): parse = True - ret = "" - for x in p: - if parse: - if x == "[": - parse = False - else: - ret += x + text = "" + for ch in string: + if parse and ch == "[": + parse = False + elif parse: + text += ch else: - if x == "]": + if ch == "]": parse = True - return ret - - -def findAllPos(g): - key = g[3] - orig = g[0] - r = key[:key.find("|")] - key = key[key.find("|") + 1:] - keys = key.split(":") - ret = [] - for x in keys: - if x == "F" or x == "FO" or x == "O" or x == "O1": - if orig.find("@") != -1: - ret.append(orig[0:orig.find("@")]) - if x == "O": - ret.append(orig) - else: - ret.append(orig) - if x == "FnO": # remove optional parameters - if orig.find("@") != -1: - ret.append(remOpt(orig[0:orig.find("@")])) - else: - ret.append(remOpt(orig)) - if x == "P" or x == "PO": - ret.append(orig.replace("@@", "@")) - if x == "PnO": # remove optional parameters - ret.append(remOpt(orig.replace("@@", "@"))) - if x == "nP" or x == "nPO" or x == "PS" or x == "O2": - ret.append(orig) - if x == "nPnO": # remove optional paramters - ret.append(remOpt(orig)) - if "fo" in x: - ret.append(orig.replace("@@@", "@" * int(x[2]))) - return atSort(ret), r + + return text + + +def find_all_positions(entry): + macro = entry[0] + category = entry[3].split("|", 1)[0] + keys = entry[3].split("|", 1)[1].split(":") + result = [] + + for key in keys: + if key in ["F", "FO", "O", "O1"]: + result.append(macro.split("@", 1)[0]) + if macro.count("@") > 0 and key == "O": + result.append(macro) + + if key == "FnO": # remove optional parameters + string = macro + if macro.count("@") > 0: + string = string[:string.find("@")] + result.append(remove_optional_params(string)) + + if key in ["P", "PO", "PnO"]: + result.append(macro.replace("@@", "@")) + + if key == "PnO": # remove optional parameters + result[-1] = remove_optional_params(result[-1]) + + if key in ["nP", "nPO", "PS", "O2"]: + result.append(macro) + + if key == "nPnO": # remove optional paramters + result.append(remove_optional_params(macro)) + + if "fo" in key: + result.append(macro.replace("@@@", "@" * int(key[2]))) + + return sorted(result, key=lambda x: x.count("@")), category + + +def macro_match(macro, entry): + """Determines whether the entry refers to the macro.""" + return macro in entry and (len(macro) == len(entry) or not entry[len(macro)].isalpha()) def main(lines): - lines = lines.replace("The LaTeX DLMF macro \'\'\'\\", "The LaTeX DLMF and DRMF macro \'\'\'\\") - flag = True - gCSV = csv.reader(open('new.Glossary.csv', 'rb'), delimiter=',', quotechar='\"') - toWrite = "" + lines = lines.replace("The LaTeX DLMF macro '''\\", "The LaTeX DLMF and DRMF macro '''\\") + to_write = "" i = 0 - categories = open("Categories.txt", "r") - cats = categories.readlines() - categories.close() - for x in range(0, len(cats)): - cats[x] = cats[x].strip() - while flag: - n = lines.find("\'\'\'Definition:", i) + with open("categories.txt") as cats: + categories = cats.readlines() + + while True: + n = lines.find("'''Definition:", i) if n == -1: - flag = False + break + + macro_name = lines[n + len("'''Definition:"):lines.find("'''", n + len("'''Definition:"))] + r = lines.find("'''\\", n) + q = lines.find("'''", r + len("'''\\")) + if lines[r:q].find("{") != -1: + lines = lines[0:r] + "'''\\" + macro_name + lines[q:] + + if lines.find("\nThis macro is in the category of", n) < lines.find("'''Definition:", n + len( + "'''Definition:") + 1) and lines.find("\nThis macro is in the category of", n) != -1: + p = lines.find("\nThis macro is in the category of", n) else: - macroN = lines[n + len("\'\'\'Definition:"):lines.find("\'\'\'", n + len("\'\'\'Definition:"))] - r = lines.find("\'\'\'\\", n) - q = lines.find("\'\'\'", r + len("\'\'\'\\")) - if lines[r:q].find("{") != -1: - lines = lines[0:r] + "\'\'\'\\" + macroN + lines[q:] - - if lines.find("\nThis macro is in the category of", n) < lines.find("\'\'\'Definition:", n + len( - "\'\'\'Definition:") + 1) and lines.find("\nThis macro is in the category of", n) != -1: - # print n - p = lines.find("\nThis macro is in the category of", n) - else: - p = lines.find("\n", lines.find(".", n)) - toWrite += lines[i:p] - toWrite = toWrite.rstrip() - toWrite += "\n\n" - count = 0 - listCalls = [] - gCSV = csv.reader(open('new.Glossary.csv', 'rb'), delimiter=',', quotechar='\"') - for g in gCSV: - if g[0].find("\\" + macroN) == 0 and (len(g[0]) == len(macroN) + 1 - or not g[0][len(macroN) + 1].isalpha()): - q, s = findAllPos(g) - listCalls += q - count += len(q) - plural = 1 # if count>1: plural =1; if count=1: plural=0 - if count == 1: plural = 0 - for t in cats: - if s + " -" in t: - category = "This macro is in the category of" + t[t.find("-") + 2:] - break - new = category + "\n\nIn math mode, this macro can be called in the following way" + "s" * plural + ":\n\n" - for q in range(0, len(listCalls)): - c = listCalls[q] - new += ":\'\'\'" + c + "\'\'\'" + " produces {\\displaystyle " + c + "}
\n" - # Now add the multiple ways \macroname{n}@... produces \macroname{n}@... - toWrite += new + "\n" - i = lines.find("These are defined by", p) - - return toWrite + lines[i:] + p = lines.find(".\n", n) + 1 + to_write += lines[i:p].rstrip() + "\n\n" + + calls = [] + glossary = csv.reader(open(GLOSSARY_LOCATION, 'rb'), delimiter=',', quotechar='\"') + for entry in glossary: + if macro_match("\\" + macro_name, entry[0]): + macro_calls, category = find_all_positions(entry) + calls += macro_calls + + for line in categories: + key, meaning = line.split(" - ") + if key == category: + category_text = "This macro is in the category of " + meaning + break + + text = category_text + "\nIn math mode, this macro can be called in the following way" + "s" * ( + len(calls) > 1) + ":\n\n" + + for call in calls: + text += ":'''" + call + "'''" + " produces {\\displaystyle " + call + "}
\n" + + # Now add the multiple ways \macroname{n}@... produces \macroname{n}@... + to_write += text + "\n" + i = lines.find("These are defined by", p) + + return to_write + lines[i:] diff --git a/main_page/src/symbols_list.py b/main_page/src/symbols_list.py index 48eef80..61a8266 100644 --- a/main_page/src/symbols_list.py +++ b/main_page/src/symbols_list.py @@ -4,273 +4,273 @@ import csv -def getSym(line): # Gets all symbols on a line for symbols list +GLOSSARY_LOCATION = "new.Glossary.csv" + + +def get_symbols(line): + # (str) -> list + """Gets all symbols on a line.""" line = line.replace("\n", " ") - symList = [] + sym_list = [] if line == "": - return symList + return sym_list + symbol = "" - symFlag = False - argFlag = False - cC = 0 - for i in range(0, len(line)): - c = line[i] - if symFlag: - if c == "{" or c == "[": - cC += 1 - argFlag = True - if c != "}" and c != "]": - if argFlag or c.isalpha(): - symbol += c + sym_flag = False + arg_flag = False + count = 0 + + for i in range(len(line)): + ch = line[i] + if sym_flag: + if ch == "{" or ch == "[": + count += 1 + arg_flag = True + + if ch != "}" and ch != "]": + if arg_flag or ch.isalpha(): + symbol += ch else: - symFlag = False - argFlag = False - symList.append(symbol) - symList += (getSym(symbol)) + sym_flag = False + arg_flag = False + sym_list.append(symbol) + sym_list += (get_symbols(symbol)) symbol = "" else: - cC -= 1 - symbol += c + count -= 1 + symbol += ch if i + 1 == len(line): p = "" else: p = line[i + 1] - if cC == 0 and p != "{" and p != "[" and p != "@": - symFlag = False - symList.append(symbol) - symList += (getSym(symbol)) - argFlag = False + if count == 0 and p != "{" and p != "[" and p != "@": + sym_flag = False + sym_list.append(symbol) + sym_list += (get_symbols(symbol)) + arg_flag = False symbol = "" - elif c == "\\": - symFlag = True - symList.append(symbol) - symList += getSym(symbol) - return (symList) - - -def main(mainLines): - mainLines = mainLines[mainLines.find("drmf_bof"):mainLines.rfind("drmf_eof") + len("drmf_eof\n")] - mainPages = mainLines.split("drmf_eof") - if mainPages[-1].strip() == "": - mainPages = mainPages[0:len(mainPages) - 1] - print len(mainPages) - glossary = open('new.Glossary.csv', 'rb') - gCSV = csv.reader(glossary, delimiter=',', quotechar='\"') - lGlos = glossary.readlines() - wiki = "" + elif ch == "\\": + sym_flag = True + + sym_list.append(symbol) + sym_list += get_symbols(symbol) + return sym_list + + +def main(data): + pages = data[data.find("drmf_bof"):].split("drmf_eof")[:-1] + + print "Pages: " + str(len(pages)) + result = "" - for g in mainPages: - if not "\'\'\'Definition:" in g: - h = g.replace("drmf_eof", "") - h = g.replace("drmf_bof", "") - toWrite = "drmf_bof" + h + "drmf_eof\n" - result += toWrite + for page in pages: + # skip over non-definition pages + if "'''Definition:" not in page: + to_write = page + "drmf_eof\n" + result += to_write + continue + + page = page.replace("drmf_bof", "").strip("\n") + add_spacing = False + sflag2 = False + # remove data (to be regenerated later) + if page.find("== Symbols List ==") != -1: + to_write = page.split("== Symbols List ==")[0] + page = to_write else: - h = g.replace("drmf_eof", "") - h = g.replace("drmf_bof", "") - h = h.strip("\n") - sflag1 = False - sflag2 = False - if h.find("== Symbols List ==") != -1: - toWrite = h[0:h.find("== Symbols List ==")] - h = toWrite + add_spacing = True + if page.find("drmf_foot") == -1: + to_write = page + sflag2 = True else: - sflag1 = True - if h.find("drmf_foot") == -1: - toWrite = h - sflag2 = True + to_write = page.split("
")[0] + page = to_write + + to_write = "drmf_bof\n" + to_write + if add_spacing: + to_write += "\n\n" + + to_write += "== Symbols List ==\n\n" + + symbols = get_symbols((' '.join(page.split("\n")) + " ").replace("\n", " ")) + + new_symbols = [] + + for symbol in symbols: + flag_a = True + flag_b = True + c_n = 0 + c_c = 0 + arg_count = 0 + + for ch in symbol: + if ch.isalpha() and flag_a: + c_n += 1 else: - toWrite = h[0:h.find("
len(symbol) and (Q[len(symbol)] == "{" or Q[len(symbol)] == "["): - ap = "" - for o in range(len(symbol), len(Q)): - if Q[o] == "{" or z == "[": - argFlag = True - elif Q[o] == "}" or z == "]": - argFlag = False - listArgs.append(ap) - ap = "" - else: - ap += Q[o] - websiteF = "" - web1 = G[5] - for t in range(5, len(G)): - if G[t] != "": - websiteF = websiteF + " [" + G[t] + " " + G[t] + "]" - p1 = G[4].strip("$") - p1 = "{\\displaystyle " + p1 + "}" - new1 = "" - new2 = "" - pause = False - mathF = True - p2 = G[1] - for k in range(0, len(p2)): - if p2[k] == "$": - if mathF: - new2 += "{\\displaystyle " - else: - new2 += "}" - mathF = not mathF - else: - new2 += p2[k] - p2 = new2 - finSym.append(web1 + " " + p1 + "] : " + p2 + " :" + websiteF) - break - if not gFlag: - del newSym[s] - - gFlag = True - for y in finSym: - if gFlag: - gFlag = False - toWrite += ("[" + y) + arg_count += 1 + if G[0].find(symbol) == 0 and (len(G[0]) == len(symbol) or not G[0][ + len(symbol)].isalpha()): + checkFlag = True + get = True + preG = S + elif checkFlag: + checkFlag = False + get = True + + if get: + G = preG + + if symbolPar.find("@") != -1: + Q = symbolPar[:symbolPar.find("@")] else: - toWrite += ("
\n[" + y) + Q = symbolPar + + arg_list = [] + if len(Q) > len(symbol) and (Q[len(symbol)] == "{" or Q[len(symbol)] == "["): + ap = "" + for o in xrange(len(symbol), len(Q)): + if Q[o] == "}" or ch == "]": + arg_list.append(ap) + ap = "" + elif Q[o] != "{" and ch != "[": + ap += Q[o] + + websiteF = "" + web1 = G[5] + for t in xrange(5, len(G)): + if G[t] != "": + websiteF = websiteF + " [" + G[t] + " " + G[t] + "]" + + p1 = G[4].strip("$") + p1 = "{\\displaystyle " + p1 + "}" + new2 = "" + mathF = True + p2 = G[1] + for k in p2: + if k == "$": + if mathF: + new2 += "{\\displaystyle " + else: + new2 += "}" + mathF = not mathF + else: + new2 += k + p2 = new2 + final_symbols.append(web1 + " " + p1 + "] : " + p2 + " :" + websiteF) + break + + if not gFlag: + new_symbols.pop(i) + + gFlag = True + for y in final_symbols: + if gFlag: + gFlag = False + to_write += ("[" + y) + else: + to_write += ("
\n[" + y) - toWrite += ("\n
\ndrmf_eof\n\n") - if sflag2: - toWrite += "\n" - result += toWrite + to_write += "\n
\ndrmf_eof\n\n" + if sflag2: + to_write += "\n" + result += to_write - return result \ No newline at end of file + return result From f164bbdb35c661a8307c76fd9abd0fba26cda4e5 Mon Sep 17 00:00:00 2001 From: Joon Bang Date: Fri, 5 Aug 2016 12:32:54 -0400 Subject: [PATCH 200/402] Fix QuantifiedCode issues --- main_page/src/symbols_list.py | 38 +++++++++++++++++------------------ 1 file changed, 18 insertions(+), 20 deletions(-) diff --git a/main_page/src/symbols_list.py b/main_page/src/symbols_list.py index 61a8266..196e22e 100644 --- a/main_page/src/symbols_list.py +++ b/main_page/src/symbols_list.py @@ -20,22 +20,20 @@ def get_symbols(line): arg_flag = False count = 0 - for i in range(len(line)): - ch = line[i] + for i, ch in enumerate(line): if sym_flag: - if ch == "{" or ch == "[": + if ch == ["{", "["]: count += 1 arg_flag = True - if ch != "}" and ch != "]": - if arg_flag or ch.isalpha(): - symbol += ch - else: - sym_flag = False - arg_flag = False - sym_list.append(symbol) - sym_list += (get_symbols(symbol)) - symbol = "" + if ch not in ["}", "]"] and arg_flag or ch.isalpha(): + symbol += ch + elif ch not in ["}", "]"]: + sym_flag = False + arg_flag = False + sym_list.append(symbol) + sym_list += (get_symbols(symbol)) + symbol = "" else: count -= 1 symbol += ch @@ -109,9 +107,9 @@ def main(data): c_n += 1 else: flag_a = False - if ch == "{" or ch == "[": + if ch in ["{", "["]: c_c += 1 - if ch == "}" or ch == "]": + if ch in ["}", "]"]: c_c -= 1 if c_c == 0: arg_count += 1 @@ -127,9 +125,9 @@ def main(data): c_n += 1 else: flag_a = False - if ch == "{" or ch == "[": + if ch in ["{", "["]: c_c += 1 - if ch == "}" or ch == "]": + if ch in ["}", "]"]: c_c -= 1 if c_c == 0: arg_count += 1 @@ -155,9 +153,9 @@ def main(data): parFlag = True elif ch.isalpha(): c_n += 1 - elif ch == "{" or ch == "[": + elif ch in ["{", "["]: c_c += 1 - elif ch == "}" or ch == "]": + elif ch in ["}", "]"]: c_c -= 1 if c_c == 0: if parFlag: @@ -197,9 +195,9 @@ def main(data): elif ch.isalpha(): c_n += 1 else: - if ch == "{" or ch == "[": + if ch in ["{", "["]: c_c += 1 - if ch == "}" or ch == "]": + if ch in ["}", "]"]: c_c -= 1 if c_c == 0: if parFlag: From 0aae79a41b73cf0cbb8eb7518b4371f966b46447 Mon Sep 17 00:00:00 2001 From: Joon Bang Date: Mon, 8 Aug 2016 16:12:01 -0400 Subject: [PATCH 201/402] Rewrite symbols_list.py --- main_page/main_page.mmd | 596 ++++++++++++++++------------------ main_page/src/main.py | 99 +++++- main_page/src/symbols_list.py | 12 +- 3 files changed, 379 insertions(+), 328 deletions(-) diff --git a/main_page/main_page.mmd b/main_page/main_page.mmd index 65d8685..7484a68 100644 --- a/main_page/main_page.mmd +++ b/main_page/main_page.mmd @@ -309,8 +309,8 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:AffqKrawtchouk {\displaystyle K^{\mathrm{Aff}}_{n}}] : affine {\displaystyle q}-Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:AffqKrawtchouk http://drmf.wmflabs.org/wiki/Definition:AffqKrawtchouk]
-[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1] +[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or -hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
+[http://drmf.wmflabs.org/wiki/Definition:AffqKrawtchouk {\displaystyle K^{\mathrm{Aff}}_{n}}] : affine -Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:AffqKrawtchouk http://drmf.wmflabs.org/wiki/Definition:AffqKrawtchouk]

<< [[Definition:sinc|Definition:sinc]]
[[Main_Page|Main Page]]
@@ -342,9 +342,9 @@ These are defined by == Symbols List == +[http://dlmf.nist.gov/1.2#E1 {\displaystyle \binom{n}{k}}] : binomial coefficient : [http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/1.2#E1]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or -hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
[http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzI {\displaystyle U^{(n)}_{\alpha}}] : Al-Salam-Carlitz I polynomial : [http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzI http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzI]
-[http://dlmf.nist.gov/1.2#E1 {\displaystyle \binom{n}{k}}] : binomial coefficient : [http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/1.2#E1] [http://dlmf.nist.gov/26.3#SS1.p1 http://dlmf.nist.gov/26.3#SS1.p1]
-[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
<< [[Definition:AffqKrawtchouk|Definition:AffqKrawtchouk]]
[[Main_Page|Main Page]]
@@ -378,8 +378,8 @@ These are defined by == Symbols List == [http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzII {\displaystyle V^{(n)}_{\alpha}}] : Al-Salam-Carlitz II polynomial : [http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzII http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzII]
-[http://dlmf.nist.gov/1.2#E1 {\displaystyle \binom{n}{k}}] : binomial coefficient : [http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/1.2#E1] [http://dlmf.nist.gov/26.3#SS1.p1 http://dlmf.nist.gov/26.3#SS1.p1]
-[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1] +[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or -hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
+[http://dlmf.nist.gov/1.2#E1 {\displaystyle \binom{n}{k}}] : binomial coefficient : [http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/1.2#E1]

<< [[Definition:AlSalamCarlitzI|Definition:AlSalamCarlitzI]]
[[Main_Page|Main Page]]
@@ -423,8 +423,8 @@ and == Symbols List == +[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or -hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
[http://drmf.wmflabs.org/wiki/Definition:AlSalamIsmail {\displaystyle U_{n}}] : Al-Salam Ismail polynomial : [http://drmf.wmflabs.org/wiki/Definition:AlSalamIsmail http://drmf.wmflabs.org/wiki/Definition:AlSalamIsmail]
-[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
<< [[Definition:AlSalamCarlitzII|Definition:AlSalamCarlitzII]]
[[Main_Page|Main Page]]
@@ -457,7 +457,7 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:AntiDer {\displaystyle \int f(x){\mathrm d}\!x}] : semantic antiderivative : [http://drmf.wmflabs.org/wiki/Definition:AntiDer http://drmf.wmflabs.org/wiki/Definition:AntiDer] +[http://drmf.wmflabs.org/wiki/Definition:AntiDer {\displaystyle \int f(x){\mathrm d}\!x}] : semantic antiderivative : [http://drmf.wmflabs.org/wiki/Definition:AntiDer http://drmf.wmflabs.org/wiki/Definition:AntiDer]

<< [[Definition:AlSalamIsmail|Definition:AlSalamIsmail]]
[[Main_Page|Main Page]]
@@ -489,8 +489,8 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:BesselPolyIIparam {\displaystyle y_{n}}] : Bessel polynomial with two parameters {\displaystyle y_n} : [http://drmf.wmflabs.org/wiki/Definition:BesselPolyIIparam http://drmf.wmflabs.org/wiki/Definition:BesselPolyIIparam]
-[http://dlmf.nist.gov/18.34#E1 {\displaystyle y_{n}}] : Bessel polynomial : [http://dlmf.nist.gov/18.34#E1 http://dlmf.nist.gov/18.34#E1] +[http://dlmf.nist.gov/18.34#E1 {\displaystyle y_{n}}] : Bessel polynomial : [http://dlmf.nist.gov/18.34#E1 http://dlmf.nist.gov/18.34#E1]
+[http://drmf.wmflabs.org/wiki/Definition:BesselPolyIIparam {\displaystyle y_{n}}] : Bessel polynomial with two parameters : [http://drmf.wmflabs.org/wiki/Definition:BesselPolyIIparam http://drmf.wmflabs.org/wiki/Definition:BesselPolyIIparam]

<< [[Definition:AntiDer|Definition:AntiDer]]
[[Main_Page|Main Page]]
@@ -522,8 +522,8 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:BesselPolyTheta {\displaystyle \theta_{n}}] : Bessel polynomial with two parameters {\displaystyle \theta_n} : [http://drmf.wmflabs.org/wiki/Definition:BesselPolyTheta http://drmf.wmflabs.org/wiki/Definition:BesselPolyTheta]
-[http://drmf.wmflabs.org/wiki/Definition:BesselPolyIIparam {\displaystyle y_{n}}] : Bessel polynomial with two parameters {\displaystyle y_n} : [http://drmf.wmflabs.org/wiki/Definition:BesselPolyIIparam http://drmf.wmflabs.org/wiki/Definition:BesselPolyIIparam] +[http://drmf.wmflabs.org/wiki/Definition:BesselPolyTheta {\displaystyle \theta_{n}}] : Bessel polynomial with two parameters : [http://drmf.wmflabs.org/wiki/Definition:BesselPolyTheta http://drmf.wmflabs.org/wiki/Definition:BesselPolyTheta]
+[http://drmf.wmflabs.org/wiki/Definition:BesselPolyIIparam {\displaystyle y_{n}}] : Bessel polynomial with two parameters : [http://drmf.wmflabs.org/wiki/Definition:BesselPolyIIparam http://drmf.wmflabs.org/wiki/Definition:BesselPolyIIparam]

<< [[Definition:BesselPolyIIparam|Definition:BesselPolyIIparam]]
[[Main_Page|Main Page]]
@@ -555,8 +555,8 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:bigqJacobiIVparam {\displaystyle P_{n}}] : big {\displaystyle q}-Jacobi polynomial with four parameters : [http://drmf.wmflabs.org/wiki/Definition:bigqJacobiIVparam http://drmf.wmflabs.org/wiki/Definition:bigqJacobiIVparam]
-[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1] +[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or -hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
+[http://drmf.wmflabs.org/wiki/Definition:bigqJacobiIVparam {\displaystyle P_{n}}] : big -Jacobi polynomial with four parameters : [http://drmf.wmflabs.org/wiki/Definition:bigqJacobiIVparam http://drmf.wmflabs.org/wiki/Definition:bigqJacobiIVparam]

<< [[Definition:BesselPolyTheta|Definition:BesselPolyTheta]]
[[Main_Page|Main Page]]
@@ -588,8 +588,8 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:bigqLaguerre {\displaystyle P_{n}}] : big {\displaystyle q}-Laguerre polynomial : [http://drmf.wmflabs.org/wiki/Definition:bigqLaguerre http://drmf.wmflabs.org/wiki/Definition:bigqLaguerre]
-[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1] +[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or -hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
+[http://drmf.wmflabs.org/wiki/Definition:bigqLaguerre {\displaystyle P_{n}}] : big -Laguerre polynomial : [http://drmf.wmflabs.org/wiki/Definition:bigqLaguerre http://drmf.wmflabs.org/wiki/Definition:bigqLaguerre]

<< [[Definition:bigqJacobiIVparam|Definition:bigqJacobiIVparam]]
[[Main_Page|Main Page]]
@@ -621,8 +621,8 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:bigqLegendre {\displaystyle P_{n}}] : big {\displaystyle q}-Legendre polynomial : [http://drmf.wmflabs.org/wiki/Definition:bigqLegendre http://drmf.wmflabs.org/wiki/Definition:bigqLegendre]
-[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1] +[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or -hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
+[http://drmf.wmflabs.org/wiki/Definition:bigqLegendre {\displaystyle P_{n}}] : big -Legendre polynomial : [http://drmf.wmflabs.org/wiki/Definition:bigqLegendre http://drmf.wmflabs.org/wiki/Definition:bigqLegendre]

<< [[Definition:bigqLaguerre|Definition:bigqLaguerre]]
[[Main_Page|Main Page]]
@@ -656,8 +656,8 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:CiglerqChebyT {\displaystyle T_{n}}] : Cigler {\displaystyle q}-Chebyshev polynomial {\displaystyle T} : [http://drmf.wmflabs.org/wiki/Definition:CiglerqChebyT http://drmf.wmflabs.org/wiki/Definition:CiglerqChebyT]
-[http://drmf.wmflabs.org/wiki/Definition:bigqJacobiIVparam {\displaystyle P_{n}}] : big {\displaystyle q}-Jacobi polynomial with four parameters : [http://drmf.wmflabs.org/wiki/Definition:bigqJacobiIVparam http://drmf.wmflabs.org/wiki/Definition:bigqJacobiIVparam] +[http://drmf.wmflabs.org/wiki/Definition:CiglerqChebyT {\displaystyle T_{n}}] : Cigler -Chebyshev polynomial : [http://drmf.wmflabs.org/wiki/Definition:CiglerqChebyT http://drmf.wmflabs.org/wiki/Definition:CiglerqChebyT]
+[http://drmf.wmflabs.org/wiki/Definition:bigqJacobiIVparam {\displaystyle P_{n}}] : big -Jacobi polynomial with four parameters : [http://drmf.wmflabs.org/wiki/Definition:bigqJacobiIVparam http://drmf.wmflabs.org/wiki/Definition:bigqJacobiIVparam]

<< [[Definition:bigqLegendre|Definition:bigqLegendre]]
[[Main_Page|Main Page]]
@@ -690,7 +690,7 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:CiglerqChebyU {\displaystyle U_{n}}] : Cigler {\displaystyle q}-Chebyshev polynomial {\displaystyle U} : [http://drmf.wmflabs.org/wiki/Definition:CiglerqChebyU http://drmf.wmflabs.org/wiki/Definition:CiglerqChebyU] +[http://drmf.wmflabs.org/wiki/Definition:CiglerqChebyU {\displaystyle U_{n}}] : Cigler -Chebyshev polynomial : [http://drmf.wmflabs.org/wiki/Definition:CiglerqChebyU http://drmf.wmflabs.org/wiki/Definition:CiglerqChebyU]

<< [[Definition:CiglerqChebyT|Definition:CiglerqChebyT]]
[[Main_Page|Main Page]]
@@ -722,9 +722,8 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:ctsbigqHermite {\displaystyle H_{n}}] : continuous big {\displaystyle q}-Hermite polynomial : [http://drmf.wmflabs.org/wiki/Definition:ctsbigqHermite http://drmf.wmflabs.org/wiki/Definition:ctsbigqHermite]
-[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
-[http://dlmf.nist.gov/4.2.E11 {\displaystyle \mathrm{e}}] : the base of the natural logarithm : [http://dlmf.nist.gov/4.2.E11 http://dlmf.nist.gov/4.2.E11] +[http://drmf.wmflabs.org/wiki/Definition:ctsbigqHermite {\displaystyle H_{n}}] : continuous big -Hermite polynomial : [http://drmf.wmflabs.org/wiki/Definition:ctsbigqHermite http://drmf.wmflabs.org/wiki/Definition:ctsbigqHermite]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or -hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]

<< [[Definition:CiglerqChebyU|Definition:CiglerqChebyU]]
[[Main_Page|Main Page]]
@@ -757,10 +756,9 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:ctsdualqHahn {\displaystyle p_{n}}] : continuous dual {\displaystyle q}-Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:ctsdualqHahn http://drmf.wmflabs.org/wiki/Definition:ctsdualqHahn]
-[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1]
-[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
-[http://dlmf.nist.gov/4.2.E11 {\displaystyle \mathrm{e}}] : the base of the natural logarithm : [http://dlmf.nist.gov/4.2.E11 http://dlmf.nist.gov/4.2.E11] +[http://drmf.wmflabs.org/wiki/Definition:ctsdualqHahn {\displaystyle p_{n}}] : continuous dual -Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:ctsdualqHahn http://drmf.wmflabs.org/wiki/Definition:ctsdualqHahn]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or -hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]

<< [[Definition:ctsbigqHermite|Definition:ctsbigqHermite]]
[[Main_Page|Main Page]]
@@ -793,10 +791,9 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:ctsqHahn {\displaystyle p_{n}}] : continuous {\displaystyle q}-Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:ctsqHahn http://drmf.wmflabs.org/wiki/Definition:ctsqHahn]
-[http://dlmf.nist.gov/4.2.E11 {\displaystyle \mathrm{e}}] : the base of the natural logarithm : [http://dlmf.nist.gov/4.2.E11 http://dlmf.nist.gov/4.2.E11]
-[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1]
-[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1] +[http://drmf.wmflabs.org/wiki/Definition:ctsqHahn {\displaystyle p_{n}}] : continuous -Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:ctsqHahn http://drmf.wmflabs.org/wiki/Definition:ctsqHahn]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or -hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]

<< [[Definition:ctsdualqHahn|Definition:ctsdualqHahn]]
[[Main_Page|Main Page]]
@@ -833,11 +830,7 @@ with x=\cos@@{\theta}. == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:ctsqJacobi {\displaystyle P^{(\alpha,\beta)}_{n}}] : continuous {\displaystyle q}-Jacobi polynomial : [http://drmf.wmflabs.org/wiki/Definition:ctsqJacobi http://drmf.wmflabs.org/wiki/Definition:ctsqJacobi]
-[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1]
-[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
-[http://dlmf.nist.gov/4.2.E11 {\displaystyle \mathrm{e}}] : the base of the natural logarithm : [http://dlmf.nist.gov/4.2.E11 http://dlmf.nist.gov/4.2.E11]
-[http://dlmf.nist.gov/4.14#E2 {\displaystyle \mathrm{cos}}] : cosine function : [http://dlmf.nist.gov/4.14#E2 http://dlmf.nist.gov/4.14#E2] +
<< [[Definition:ctsqHahn|Definition:ctsqHahn]]
[[Main_Page|Main Page]]
@@ -869,10 +862,9 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:ctsqLaguerre {\displaystyle P^{(n)}_{\alpha}}] : continuous {\displaystyle q}-Laguerre polynomial : [http://drmf.wmflabs.org/wiki/Definition:ctsqLaguerre http://drmf.wmflabs.org/wiki/Definition:ctsqLaguerre]
-[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1]
-[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
-[http://dlmf.nist.gov/4.2.E11 {\displaystyle \mathrm{e}}] : the base of the natural logarithm : [http://dlmf.nist.gov/4.2.E11 http://dlmf.nist.gov/4.2.E11] +[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or -hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]
+[http://drmf.wmflabs.org/wiki/Definition:ctsqLaguerre {\displaystyle P^{(n)}_{\alpha}}] : continuous -Laguerre polynomial : [http://drmf.wmflabs.org/wiki/Definition:ctsqLaguerre http://drmf.wmflabs.org/wiki/Definition:ctsqLaguerre]

<< [[Definition:ctsqJacobi|Definition:ctsqJacobi]]
[[Main_Page|Main Page]]
@@ -906,9 +898,8 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:ctsqLegendre {\displaystyle P_{n}}] : continuous {\displaystyle q}-Legendre polynomial : [http://drmf.wmflabs.org/wiki/Definition:ctsqLegendre http://drmf.wmflabs.org/wiki/Definition:ctsqLegendre]
-[http://drmf.wmflabs.org/wiki/Definition:dualqKrawtchouk {\displaystyle K_{n}}] : dual {\displaystyle q}-Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:dualqKrawtchouk http://drmf.wmflabs.org/wiki/Definition:dualqKrawtchouk]
-[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1] +[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or -hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
+[http://drmf.wmflabs.org/wiki/Definition:dualqKrawtchouk {\displaystyle K_{n}}] : dual -Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:dualqKrawtchouk http://drmf.wmflabs.org/wiki/Definition:dualqKrawtchouk]

<< [[Definition:ctsqLaguerre|Definition:ctsqLaguerre]]
[[Main_Page|Main Page]]
@@ -953,10 +944,8 @@ for m\ne n. == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:f {\displaystyle {f}}] : function : [http://drmf.wmflabs.org/wiki/Definition:f http://drmf.wmflabs.org/wiki/Definition:f]
-[http://drmf.wmflabs.org/wiki/Definition:GenGegenbauer {\displaystyle S^{(\alpha,\beta)}_{n}}] : Generalized Gegenbauer polynomial : [http://drmf.wmflabs.org/wiki/Definition:GenGegenbauer http://drmf.wmflabs.org/wiki/Definition:GenGegenbauer]
[http://dlmf.nist.gov/18.3#T1.t1.r3 {\displaystyle P^{(\alpha,\beta)}_{n}}] : Jacobi polynomial : [http://dlmf.nist.gov/18.3#T1.t1.r3 http://dlmf.nist.gov/18.3#T1.t1.r3]
-[http://dlmf.nist.gov/1.4#iv {\displaystyle \int}] : integral : [http://dlmf.nist.gov/1.4#iv http://dlmf.nist.gov/1.4#iv] +[http://drmf.wmflabs.org/wiki/Definition:GenGegenbauer {\displaystyle S^{(\alpha,\beta)}_{n}}] : Generalized Gegenbauer polynomial : [http://drmf.wmflabs.org/wiki/Definition:GenGegenbauer http://drmf.wmflabs.org/wiki/Definition:GenGegenbauer]

<< [[Definition:ctsqLegendre|Definition:ctsqLegendre]]
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@@ -1007,7 +996,6 @@ Page 156 of [[Bibliography#CHI|'''CHI''']]. [http://drmf.wmflabs.org/wiki/Definition:GenHermite {\displaystyle H_n^{\mu}}] : generalized Hermite polynomial : [http://drmf.wmflabs.org/wiki/Definition:GenHermite http://drmf.wmflabs.org/wiki/Definition:GenHermite]
[http://dlmf.nist.gov/18.3#T1.t1.r27 {\displaystyle L_n^{(\alpha)}}] : Laguerre (or generalized Laguerre) polynomial : [http://dlmf.nist.gov/18.3#T1.t1.r27 http://dlmf.nist.gov/18.3#T1.t1.r27]
-[http://dlmf.nist.gov/1.4#iv {\displaystyle \int}] : integral : [http://dlmf.nist.gov/1.4#iv http://dlmf.nist.gov/1.4#iv]
<< [[Definition:f|Definition:f]]
[[Main_Page|Main Page]]
@@ -1042,7 +1030,7 @@ These are defined by == Symbols List == [http://drmf.wmflabs.org/wiki/Definition:GottliebLaguerre {\displaystyle l_{n}}] : Gottlieb-Laguerre polynomial : [http://drmf.wmflabs.org/wiki/Definition:GottliebLaguerre http://drmf.wmflabs.org/wiki/Definition:GottliebLaguerre]
-[http://dlmf.nist.gov/18.19#T1.t1.r9 {\displaystyle M_{n}}] : Meixner polynomial : [http://dlmf.nist.gov/18.19#T1.t1.r9 http://dlmf.nist.gov/18.19#T1.t1.r9] +[http://dlmf.nist.gov/18.19#T1.t1.r9 {\displaystyle M_{n}}] : Meixner polynomial : [http://dlmf.nist.gov/18.19#T1.t1.r9 http://dlmf.nist.gov/18.19#T1.t1.r9]

<< [[Definition:GenHermite|Definition:GenHermite]]
[[Main_Page|Main Page]]
@@ -1076,10 +1064,9 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:Int {\displaystyle \int_{a}^{b}f(x){\mathrm d}\!x}] : semantic definite integral : [http://drmf.wmflabs.org/wiki/Definition:Int http://drmf.wmflabs.org/wiki/Definition:Int]
-[http://drmf.wmflabs.org/wiki/Definition:JacksonqBesselI {\displaystyle J^{(1)}_{q}}] : Jackson {\displaystyle q}-Bessel function 1 : [http://drmf.wmflabs.org/wiki/Definition:JacksonqBesselI http://drmf.wmflabs.org/wiki/Definition:JacksonqBesselI]
-[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1]
-[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1] +[http://drmf.wmflabs.org/wiki/Definition:JacksonqBesselI {\displaystyle J^{(1)}_{q}}] : Jackson -Bessel function 1 : [http://drmf.wmflabs.org/wiki/Definition:JacksonqBesselI http://drmf.wmflabs.org/wiki/Definition:JacksonqBesselI]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or -hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]

<< [[Definition:GottliebLaguerre|Definition:GottliebLaguerre]]
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@@ -1113,9 +1100,9 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:JacksonqBesselII {\displaystyle J^{(2)}_{q}}] : Jackson {\displaystyle q}-Bessel function 2 : [http://drmf.wmflabs.org/wiki/Definition:JacksonqBesselII http://drmf.wmflabs.org/wiki/Definition:JacksonqBesselII]
-[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1]
-[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1] +[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or -hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
+[http://drmf.wmflabs.org/wiki/Definition:JacksonqBesselII {\displaystyle J^{(2)}_{q}}] : Jackson -Bessel function 2 : [http://drmf.wmflabs.org/wiki/Definition:JacksonqBesselII http://drmf.wmflabs.org/wiki/Definition:JacksonqBesselII]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]

<< [[Definition:Int|Definition:Int]]
[[Main_Page|Main Page]]
@@ -1148,9 +1135,8 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:JacksonqBesselIII {\displaystyle J^{(3)}_{q}}] : Jackson/Hahn-Exton {\displaystyle q}-Bessel function 3 : [http://drmf.wmflabs.org/wiki/Definition:JacksonqBesselIII http://drmf.wmflabs.org/wiki/Definition:JacksonqBesselIII]
-[http://drmf.wmflabs.org/wiki/Definition:littleqLaguerre {\displaystyle p_{n}}] : little {\displaystyle q}-Laguerre / Wall polynomial : [http://drmf.wmflabs.org/wiki/Definition:littleqLaguerre http://drmf.wmflabs.org/wiki/Definition:littleqLaguerre]
-[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1] +[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or -hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
+[http://drmf.wmflabs.org/wiki/Definition:littleqLaguerre {\displaystyle p_{n}}] : little -Laguerre / Wall polynomial : [http://drmf.wmflabs.org/wiki/Definition:littleqLaguerre http://drmf.wmflabs.org/wiki/Definition:littleqLaguerre]

<< [[Definition:JacksonqBesselII|Definition:JacksonqBesselII]]
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@@ -1182,8 +1168,8 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:littleqLegendre {\displaystyle p_{n}}] : little {\displaystyle q}-Legendre polynomial : [http://drmf.wmflabs.org/wiki/Definition:littleqLegendre http://drmf.wmflabs.org/wiki/Definition:littleqLegendre]
-[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1] +[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or -hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
+[http://drmf.wmflabs.org/wiki/Definition:littleqLegendre {\displaystyle p_{n}}] : little -Legendre polynomial : [http://drmf.wmflabs.org/wiki/Definition:littleqLegendre http://drmf.wmflabs.org/wiki/Definition:littleqLegendre]

<< [[Definition:JacksonqBesselIII|Definition:JacksonqBesselIII]]
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@@ -1216,10 +1202,9 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:Lrselection {\displaystyle \left\{\begin{matrix}x\end{matrix}\right\}}] : bracketed generalization of {\displaystyle \pm} : [http://drmf.wmflabs.org/wiki/Definition:Lrselection http://drmf.wmflabs.org/wiki/Definition:Lrselection]
-[http://drmf.wmflabs.org/wiki/Definition:AffqKrawtchouk {\displaystyle K^{\mathrm{Aff}}_{n}}] : affine {\displaystyle q}-Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:AffqKrawtchouk http://drmf.wmflabs.org/wiki/Definition:AffqKrawtchouk]
-[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1]
-[http://drmf.wmflabs.org/wiki/Definition:monicAffqKrawtchouk {\displaystyle \widehat{K}^{\mathrm{Aff}}_{n}}] : monic affine {\displaystyle q}-Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicAffqKrawtchouk http://drmf.wmflabs.org/wiki/Definition:monicAffqKrawtchouk] +[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]
+[http://drmf.wmflabs.org/wiki/Definition:AffqKrawtchouk {\displaystyle K^{\mathrm{Aff}}_{n}}] : affine -Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:AffqKrawtchouk http://drmf.wmflabs.org/wiki/Definition:AffqKrawtchouk]
+[http://drmf.wmflabs.org/wiki/Definition:monicAffqKrawtchouk {\displaystyle \widehat{K}^{\mathrm{Aff}}_{n}}] : monic affine -Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicAffqKrawtchouk http://drmf.wmflabs.org/wiki/Definition:monicAffqKrawtchouk]

<< [[Definition:littleqLegendre|Definition:littleqLegendre]]
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@@ -1254,7 +1239,7 @@ These are defined by == Symbols List == [http://drmf.wmflabs.org/wiki/Definition:monicAlSalamCarlitzI {\displaystyle {\widehat U}^{(\alpha)}_{n}}] : monic Al-Salam-Carlitz I polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicAlSalamCarlitzI http://drmf.wmflabs.org/wiki/Definition:monicAlSalamCarlitzI]
-[http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzI {\displaystyle U^{(n)}_{\alpha}}] : Al-Salam-Carlitz I polynomial : [http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzI http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzI] +[http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzI {\displaystyle U^{(n)}_{\alpha}}] : Al-Salam-Carlitz I polynomial : [http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzI http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzI]

<< [[Definition:lrselection|Definition:lrselection]]
[[Main_Page|Main Page]]
@@ -1289,7 +1274,7 @@ These are defined by == Symbols List == [http://drmf.wmflabs.org/wiki/Definition:monicAlSalamCarlitzII {\displaystyle {\widehat V}^{(n)}_{\alpha}}] : monic Al-Salam-Carlitz II polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicAlSalamCarlitzII http://drmf.wmflabs.org/wiki/Definition:monicAlSalamCarlitzII]
-[http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzII {\displaystyle V^{(n)}_{\alpha}}] : Al-Salam-Carlitz II polynomial : [http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzII http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzII] +[http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzII {\displaystyle V^{(n)}_{\alpha}}] : Al-Salam-Carlitz II polynomial : [http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzII http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzII]

<< [[Definition:monicAlSalamCarlitzI|Definition:monicAlSalamCarlitzI]]
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@@ -1324,7 +1309,7 @@ These are defined by == Symbols List == [http://drmf.wmflabs.org/wiki/Definition:monicAlSalamChihara {\displaystyle {\widehat Q}_{n}}] : monic Al-Salam-Chihara polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicAlSalamChihara http://drmf.wmflabs.org/wiki/Definition:monicAlSalamChihara]
-[http://drmf.wmflabs.org/wiki/Definition:AlSalamChihara {\displaystyle Q_{n}}] : Al-Salam-Chihara polynomial : [http://drmf.wmflabs.org/wiki/Definition:AlSalamChihara http://drmf.wmflabs.org/wiki/Definition:AlSalamChihara] +[http://drmf.wmflabs.org/wiki/Definition:AlSalamChihara {\displaystyle Q_{n}}] : Al-Salam-Chihara polynomial : [http://drmf.wmflabs.org/wiki/Definition:AlSalamChihara http://drmf.wmflabs.org/wiki/Definition:AlSalamChihara]

<< [[Definition:monicAlSalamCarlitzII|Definition:monicAlSalamCarlitzII]]
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@@ -1358,9 +1343,9 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:monicAskeyWilson {\displaystyle {\widehat p}_{n}}] : monic Askey-Wilson polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicAskeyWilson http://drmf.wmflabs.org/wiki/Definition:monicAskeyWilson]
[http://dlmf.nist.gov/18.28#E1 {\displaystyle p_{n}}] : Askey-Wilson polynomial : [http://dlmf.nist.gov/18.28#E1 http://dlmf.nist.gov/18.28#E1]
-[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1] +[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]
+[http://drmf.wmflabs.org/wiki/Definition:monicAskeyWilson {\displaystyle {\widehat p}_{n}}] : monic Askey-Wilson polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicAskeyWilson http://drmf.wmflabs.org/wiki/Definition:monicAskeyWilson]

<< [[Definition:monicAlSalamChihara|Definition:monicAlSalamChihara]]
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@@ -1393,9 +1378,9 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:monicBesselPoly {\displaystyle {\widehat y}_{n}}] : monic Bessel polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicBesselPoly http://drmf.wmflabs.org/wiki/Definition:monicBesselPoly]
[http://dlmf.nist.gov/18.34#E1 {\displaystyle y_{n}}] : Bessel polynomial : [http://dlmf.nist.gov/18.34#E1 http://dlmf.nist.gov/18.34#E1]
-[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii] +[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii]
+[http://drmf.wmflabs.org/wiki/Definition:monicBesselPoly {\displaystyle {\widehat y}_{n}}] : monic Bessel polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicBesselPoly http://drmf.wmflabs.org/wiki/Definition:monicBesselPoly]

<< [[Definition:monicAskeyWilson|Definition:monicAskeyWilson]]
[[Main_Page|Main Page]]
@@ -1429,9 +1414,9 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:monicbigqJacobi {\displaystyle {\widehat P}_{n}}] : monic big {\displaystyle q}-Jacobi polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicbigqJacobi http://drmf.wmflabs.org/wiki/Definition:monicbigqJacobi]
-[http://drmf.wmflabs.org/wiki/Definition:bigqJacobi {\displaystyle P_{n}}] : big {\displaystyle q}-Jacobi polynomial : [http://drmf.wmflabs.org/wiki/Definition:bigqJacobi http://drmf.wmflabs.org/wiki/Definition:bigqJacobi]
-[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1] +[http://drmf.wmflabs.org/wiki/Definition:monicbigqJacobi {\displaystyle {\widehat P}_{n}}] : monic big -Jacobi polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicbigqJacobi http://drmf.wmflabs.org/wiki/Definition:monicbigqJacobi]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]
+[http://drmf.wmflabs.org/wiki/Definition:bigqJacobi {\displaystyle P_{n}}] : big -Jacobi polynomial : [http://drmf.wmflabs.org/wiki/Definition:bigqJacobi http://drmf.wmflabs.org/wiki/Definition:bigqJacobi]

<< [[Definition:monicBesselPoly|Definition:monicBesselPoly]]
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@@ -1465,9 +1450,9 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:monicbigqLaguerre {\displaystyle {\widehat P}_{n}}] : monic big {\displaystyle q}-Laguerre polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicbigqLaguerre http://drmf.wmflabs.org/wiki/Definition:monicbigqLaguerre]
-[http://drmf.wmflabs.org/wiki/Definition:bigqLaguerre {\displaystyle P_{n}}] : big {\displaystyle q}-Laguerre polynomial : [http://drmf.wmflabs.org/wiki/Definition:bigqLaguerre http://drmf.wmflabs.org/wiki/Definition:bigqLaguerre]
-[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1] +[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]
+[http://drmf.wmflabs.org/wiki/Definition:bigqLaguerre {\displaystyle P_{n}}] : big -Laguerre polynomial : [http://drmf.wmflabs.org/wiki/Definition:bigqLaguerre http://drmf.wmflabs.org/wiki/Definition:bigqLaguerre]
+[http://drmf.wmflabs.org/wiki/Definition:monicbigqLaguerre {\displaystyle {\widehat P}_{n}}] : monic big -Laguerre polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicbigqLaguerre http://drmf.wmflabs.org/wiki/Definition:monicbigqLaguerre]

<< [[Definition:monicbigqJacobi|Definition:monicbigqJacobi]]
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@@ -1501,9 +1486,9 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:monicbigqLegendre {\displaystyle {\widehat P}_{n}}] : monic big {\displaystyle q}-Legendre polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicbigqLegendre http://drmf.wmflabs.org/wiki/Definition:monicbigqLegendre]
-[http://drmf.wmflabs.org/wiki/Definition:bigqLegendre {\displaystyle P_{n}}] : big {\displaystyle q}-Legendre polynomial : [http://drmf.wmflabs.org/wiki/Definition:bigqLegendre http://drmf.wmflabs.org/wiki/Definition:bigqLegendre]
-[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1] +[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]
+[http://drmf.wmflabs.org/wiki/Definition:bigqLegendre {\displaystyle P_{n}}] : big -Legendre polynomial : [http://drmf.wmflabs.org/wiki/Definition:bigqLegendre http://drmf.wmflabs.org/wiki/Definition:bigqLegendre]
+[http://drmf.wmflabs.org/wiki/Definition:monicbigqLegendre {\displaystyle {\widehat P}_{n}}] : monic big -Legendre polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicbigqLegendre http://drmf.wmflabs.org/wiki/Definition:monicbigqLegendre]

<< [[Definition:monicbigqLaguerre|Definition:monicbigqLaguerre]]
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@@ -1538,8 +1523,8 @@ These are defined by == Symbols List == +[http://dlmf.nist.gov/18.19#T1.t1.r11 {\displaystyle C_{n}}] : Charlier polynomial : [http://dlmf.nist.gov/18.19#T1.t1.r11 http://dlmf.nist.gov/18.19#T1.t1.r11]
[http://drmf.wmflabs.org/wiki/Definition:monicCharlier {\displaystyle {\widehat C}_{n}}] : monic Charlier polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicCharlier http://drmf.wmflabs.org/wiki/Definition:monicCharlier]
-[http://dlmf.nist.gov/18.19#T1.t1.r11 {\displaystyle C_{n}}] : Charlier polynomial : [http://dlmf.nist.gov/18.19#T1.t1.r11 http://dlmf.nist.gov/18.19#T1.t1.r11]
<< [[Definition:monicbigqLegendre|Definition:monicbigqLegendre]]
[[Main_Page|Main Page]]
@@ -1574,8 +1559,8 @@ for n\ne 1 and \ChebyT{1}@{x}=\monicChebyT{1}@{x}=x. == Symbols List == +[http://dlmf.nist.gov/18.3#T1.t1.r8 {\displaystyle T_{n}}] : Chebyshev polynomial of the first kind : [http://dlmf.nist.gov/18.3#T1.t1.r8 http://dlmf.nist.gov/18.3#T1.t1.r8]
[http://drmf.wmflabs.org/wiki/Definition:monicChebyT {\displaystyle {\widehat T}_{n}}] : monic Chebyshev polynomial of the first kind : [http://drmf.wmflabs.org/wiki/Definition:monicChebyT http://drmf.wmflabs.org/wiki/Definition:monicChebyT]
-[http://dlmf.nist.gov/18.3#T1.t1.r8 {\displaystyle T_{n}}] : Chebyshev polynomial of the first kind : [http://dlmf.nist.gov/18.3#T1.t1.r8 http://dlmf.nist.gov/18.3#T1.t1.r8]
<< [[Definition:monicCharlier|Definition:monicCharlier]]
[[Main_Page|Main Page]]
@@ -1609,8 +1594,8 @@ These are defined by == Symbols List == +[http://dlmf.nist.gov/18.3#T1.t1.r11 {\displaystyle U_{n}}] : Chebyshev polynomial of the second kind : [http://dlmf.nist.gov/18.3#T1.t1.r11 http://dlmf.nist.gov/18.3#T1.t1.r11]
[http://drmf.wmflabs.org/wiki/Definition:monicChebyU {\displaystyle {\widehat U}_{n}}] : monic Chebyshev polynomial of the second kind : [http://drmf.wmflabs.org/wiki/Definition:monicChebyU http://drmf.wmflabs.org/wiki/Definition:monicChebyU]
-[http://dlmf.nist.gov/18.3#T1.t1.r11 {\displaystyle U_{n}}] : Chebyshev polynomial of the second kind : [http://dlmf.nist.gov/18.3#T1.t1.r11 http://dlmf.nist.gov/18.3#T1.t1.r11]
<< [[Definition:monicChebyT|Definition:monicChebyT]]
[[Main_Page|Main Page]]
@@ -1644,8 +1629,8 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:monicctsbigqHermite {\displaystyle {\widehat H}_{n}}] : monic continuous big {\displaystyle q}-Hermite polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicctsbigqHermite http://drmf.wmflabs.org/wiki/Definition:monicctsbigqHermite]
-[http://drmf.wmflabs.org/wiki/Definition:ctsbigqHermite {\displaystyle H_{n}}] : continuous big {\displaystyle q}-Hermite polynomial : [http://drmf.wmflabs.org/wiki/Definition:ctsbigqHermite http://drmf.wmflabs.org/wiki/Definition:ctsbigqHermite] +[http://drmf.wmflabs.org/wiki/Definition:ctsbigqHermite {\displaystyle H_{n}}] : continuous big -Hermite polynomial : [http://drmf.wmflabs.org/wiki/Definition:ctsbigqHermite http://drmf.wmflabs.org/wiki/Definition:ctsbigqHermite]
+[http://drmf.wmflabs.org/wiki/Definition:monicctsbigqHermite {\displaystyle {\widehat H}_{n}}] : monic continuous big -Hermite polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicctsbigqHermite http://drmf.wmflabs.org/wiki/Definition:monicctsbigqHermite]

<< [[Definition:monicChebyU|Definition:monicChebyU]]
[[Main_Page|Main Page]]
@@ -1679,8 +1664,8 @@ These are defined by == Symbols List == +[http://dlmf.nist.gov/18.25#T1.t1.r3 {\displaystyle S_{n}}] : continuous dual Hahn polynomial : [http://dlmf.nist.gov/18.25#T1.t1.r3 http://dlmf.nist.gov/18.25#T1.t1.r3]
[http://drmf.wmflabs.org/wiki/Definition:monicctsdualHahn {\displaystyle {\widehat S}_{n}}] : monic continuous dual Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicctsdualHahn http://drmf.wmflabs.org/wiki/Definition:monicctsdualHahn]
-[http://dlmf.nist.gov/18.25#T1.t1.r3 {\displaystyle S_{n}}] : continuous dual Hahn polynomial : [http://dlmf.nist.gov/18.25#T1.t1.r3 http://dlmf.nist.gov/18.25#T1.t1.r3]
<< [[Definition:monicctsbigqHermite|Definition:monicctsbigqHermite]]
[[Main_Page|Main Page]]
@@ -1715,8 +1700,8 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:monicctsdualqHahn {\displaystyle {\widehat p}_{n}}] : monic continuous dual {\displaystyle q}-Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicctsdualqHahn http://drmf.wmflabs.org/wiki/Definition:monicctsdualqHahn]
-[http://drmf.wmflabs.org/wiki/Definition:ctsdualqHahn {\displaystyle p_{n}}] : continuous dual {\displaystyle q}-Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:ctsdualqHahn http://drmf.wmflabs.org/wiki/Definition:ctsdualqHahn] +[http://drmf.wmflabs.org/wiki/Definition:ctsdualqHahn {\displaystyle p_{n}}] : continuous dual -Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:ctsdualqHahn http://drmf.wmflabs.org/wiki/Definition:ctsdualqHahn]
+[http://drmf.wmflabs.org/wiki/Definition:monicctsdualqHahn {\displaystyle {\widehat p}_{n}}] : monic continuous dual -Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicctsdualqHahn http://drmf.wmflabs.org/wiki/Definition:monicctsdualqHahn]

<< [[Definition:monicctsdualHahn|Definition:monicctsdualHahn]]
[[Main_Page|Main Page]]
@@ -1751,8 +1736,8 @@ These are defined by == Symbols List == [http://drmf.wmflabs.org/wiki/Definition:monicctsHahn {\displaystyle {\widehat p}_{n}}] : monic continuous Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicctsHahn http://drmf.wmflabs.org/wiki/Definition:monicctsHahn]
+[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii]
[http://dlmf.nist.gov/18.19#P2.p1 {\displaystyle p_{n}}] : continuous Hahn polynomial : [http://dlmf.nist.gov/18.19#P2.p1 http://dlmf.nist.gov/18.19#P2.p1]
-[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii]
<< [[Definition:monicctsdualqHahn|Definition:monicctsdualqHahn]]
[[Main_Page|Main Page]]
@@ -1786,9 +1771,9 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:monicctsqHahn {\displaystyle {\widehat p}_{n}}] : monic continuous {\displaystyle q}-Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicctsqHahn http://drmf.wmflabs.org/wiki/Definition:monicctsqHahn]
-[http://drmf.wmflabs.org/wiki/Definition:ctsqHahn {\displaystyle p_{n}}] : continuous {\displaystyle q}-Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:ctsqHahn http://drmf.wmflabs.org/wiki/Definition:ctsqHahn]
-[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1] +[http://drmf.wmflabs.org/wiki/Definition:ctsqHahn {\displaystyle p_{n}}] : continuous -Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:ctsqHahn http://drmf.wmflabs.org/wiki/Definition:ctsqHahn]
+[http://drmf.wmflabs.org/wiki/Definition:monicctsqHahn {\displaystyle {\widehat p}_{n}}] : monic continuous -Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicctsqHahn http://drmf.wmflabs.org/wiki/Definition:monicctsqHahn]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]

<< [[Definition:monicctsHahn|Definition:monicctsHahn]]
[[Main_Page|Main Page]]
@@ -1822,8 +1807,8 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:monicctsqHermite {\displaystyle {\widehat H}_{n}}] : monic continuous {\displaystyle q}-Hermite polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicctsqHermite http://drmf.wmflabs.org/wiki/Definition:monicctsqHermite]
-[http://drmf.wmflabs.org/wiki/Definition:ctsqHermite {\displaystyle H_{n}}] : continuous {\displaystyle q}-Hermite polynomial : [http://drmf.wmflabs.org/wiki/Definition:ctsqHermite http://drmf.wmflabs.org/wiki/Definition:ctsqHermite] +[http://drmf.wmflabs.org/wiki/Definition:ctsqHermite {\displaystyle H_{n}}] : continuous -Hermite polynomial : [http://drmf.wmflabs.org/wiki/Definition:ctsqHermite http://drmf.wmflabs.org/wiki/Definition:ctsqHermite]
+[http://drmf.wmflabs.org/wiki/Definition:monicctsqHermite {\displaystyle {\widehat H}_{n}}] : monic continuous -Hermite polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicctsqHermite http://drmf.wmflabs.org/wiki/Definition:monicctsqHermite]

<< [[Definition:monicctsqHahn|Definition:monicctsqHahn]]
[[Main_Page|Main Page]]
@@ -1858,9 +1843,9 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:monicctsqJacobi {\displaystyle {\widehat P}^{(\alpha,\beta)}_{n}}] : monic continuous {\displaystyle q}-Jacobi polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicctsqJacobi http://drmf.wmflabs.org/wiki/Definition:monicctsqJacobi]
-[http://drmf.wmflabs.org/wiki/Definition:ctsqJacobi {\displaystyle P^{(\alpha,\beta)}_{n}}] : continuous {\displaystyle q}-Jacobi polynomial : [http://drmf.wmflabs.org/wiki/Definition:ctsqJacobi http://drmf.wmflabs.org/wiki/Definition:ctsqJacobi]
-[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1] +[http://drmf.wmflabs.org/wiki/Definition:ctsqJacobi {\displaystyle P^{(\alpha,\beta)}_{n}}] : continuous -Jacobi polynomial : [http://drmf.wmflabs.org/wiki/Definition:ctsqJacobi http://drmf.wmflabs.org/wiki/Definition:ctsqJacobi]
+[http://drmf.wmflabs.org/wiki/Definition:monicctsqJacobi {\displaystyle {\widehat P}^{(\alpha,\beta)}_{n}}] : monic continuous -Jacobi polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicctsqJacobi http://drmf.wmflabs.org/wiki/Definition:monicctsqJacobi]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]

<< [[Definition:monicctsqHermite|Definition:monicctsqHermite]]
[[Main_Page|Main Page]]
@@ -1894,9 +1879,9 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:monicctsqLaguerre {\displaystyle {\widehat P}^{(\alpha)}_{n}}] : monic continuous {\displaystyle q}-Laguerre polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicctsqLaguerre http://drmf.wmflabs.org/wiki/Definition:monicctsqLaguerre]
-[http://drmf.wmflabs.org/wiki/Definition:ctsqLaguerre {\displaystyle P^{(n)}_{\alpha}}] : continuous {\displaystyle q}-Laguerre polynomial : [http://drmf.wmflabs.org/wiki/Definition:ctsqLaguerre http://drmf.wmflabs.org/wiki/Definition:ctsqLaguerre]
-[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1] +[http://drmf.wmflabs.org/wiki/Definition:monicctsqLaguerre {\displaystyle {\widehat P}^{(\alpha)}_{n}}] : monic continuous -Laguerre polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicctsqLaguerre http://drmf.wmflabs.org/wiki/Definition:monicctsqLaguerre]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]
+[http://drmf.wmflabs.org/wiki/Definition:ctsqLaguerre {\displaystyle P^{(n)}_{\alpha}}] : continuous -Laguerre polynomial : [http://drmf.wmflabs.org/wiki/Definition:ctsqLaguerre http://drmf.wmflabs.org/wiki/Definition:ctsqLaguerre]

<< [[Definition:monicctsqJacobi|Definition:monicctsqJacobi]]
[[Main_Page|Main Page]]
@@ -1930,9 +1915,9 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:monicctsqLegendre {\displaystyle {\widehat P}_{n}}] : monic continuous {\displaystyle q}-Legendre polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicctsqLegendre http://drmf.wmflabs.org/wiki/Definition:monicctsqLegendre]
-[http://drmf.wmflabs.org/wiki/Definition:ctsqLegendre {\displaystyle P_{n}}] : continuous {\displaystyle q}-Legendre polynomial : [http://drmf.wmflabs.org/wiki/Definition:ctsqLegendre http://drmf.wmflabs.org/wiki/Definition:ctsqLegendre]
-[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1] +[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]
+[http://drmf.wmflabs.org/wiki/Definition:ctsqLegendre {\displaystyle P_{n}}] : continuous -Legendre polynomial : [http://drmf.wmflabs.org/wiki/Definition:ctsqLegendre http://drmf.wmflabs.org/wiki/Definition:ctsqLegendre]
+[http://drmf.wmflabs.org/wiki/Definition:monicctsqLegendre {\displaystyle {\widehat P}_{n}}] : monic continuous -Legendre polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicctsqLegendre http://drmf.wmflabs.org/wiki/Definition:monicctsqLegendre]

<< [[Definition:monicctsqLaguerre|Definition:monicctsqLaguerre]]
[[Main_Page|Main Page]]
@@ -1966,9 +1951,9 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:monicctsqUltra {\displaystyle {\widehat C}_{n}}] : monic continuous {\displaystyle q}-ultraspherical/Rogers polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicctsqUltra http://drmf.wmflabs.org/wiki/Definition:monicctsqUltra]
-[http://dlmf.nist.gov/18.28#E13 {\displaystyle C_{n}}] : continuous {\displaystyle q}-ultraspherical/Rogers polynomial : [http://dlmf.nist.gov/18.28#E13 http://dlmf.nist.gov/18.28#E13]
-[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1] +[http://drmf.wmflabs.org/wiki/Definition:monicctsqUltra {\displaystyle {\widehat C}_{n}}] : monic continuous -ultraspherical/Rogers polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicctsqUltra http://drmf.wmflabs.org/wiki/Definition:monicctsqUltra]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]
+[http://dlmf.nist.gov/18.28#E13 {\displaystyle C_{n}}] : continuous -ultraspherical/Rogers polynomial : [http://dlmf.nist.gov/18.28#E13 http://dlmf.nist.gov/18.28#E13]

<< [[Definition:monicctsqLegendre|Definition:monicctsqLegendre]]
[[Main_Page|Main Page]]
@@ -2002,8 +1987,8 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:monicdiscrqHermiteI {\displaystyle {\widehat h}_{n}}] : monic discrete {\displaystyle q}-Hermite I polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicdiscrqHermiteI http://drmf.wmflabs.org/wiki/Definition:monicdiscrqHermiteI]
-[http://drmf.wmflabs.org/wiki/Definition:discrqHermiteI {\displaystyle h_{n}}] : discrete {\displaystyle q}-Hermite I polynomial : [http://drmf.wmflabs.org/wiki/Definition:discrqHermiteI http://drmf.wmflabs.org/wiki/Definition:discrqHermiteI] +[http://drmf.wmflabs.org/wiki/Definition:discrqHermiteI {\displaystyle h_{n}}] : discrete -Hermite I polynomial : [http://drmf.wmflabs.org/wiki/Definition:discrqHermiteI http://drmf.wmflabs.org/wiki/Definition:discrqHermiteI]
+[http://drmf.wmflabs.org/wiki/Definition:monicdiscrqHermiteI {\displaystyle {\widehat h}_{n}}] : monic discrete -Hermite I polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicdiscrqHermiteI http://drmf.wmflabs.org/wiki/Definition:monicdiscrqHermiteI]

<< [[Definition:monicctsqUltra|Definition:monicctsqUltra]]
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@@ -2037,8 +2022,8 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:monicdiscrqHermiteII {\displaystyle {\widehat \tilde{p}_{n}}}] : monic discrete {\displaystyle q}-Hermite II polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicdiscrqHermiteII http://drmf.wmflabs.org/wiki/Definition:monicdiscrqHermiteII]
-[http://drmf.wmflabs.org/wiki/Definition:discrqHermiteII {\displaystyle \tilde{h}_{n}}] : discrete {\displaystyle q}-Hermite II polynomial : [http://drmf.wmflabs.org/wiki/Definition:discrqHermiteII http://drmf.wmflabs.org/wiki/Definition:discrqHermiteII] +[http://drmf.wmflabs.org/wiki/Definition:discrqHermiteII {\displaystyle \tilde{h}_{n}}] : discrete -Hermite II polynomial : [http://drmf.wmflabs.org/wiki/Definition:discrqHermiteII http://drmf.wmflabs.org/wiki/Definition:discrqHermiteII]
+[http://drmf.wmflabs.org/wiki/Definition:monicdiscrqHermiteII {\displaystyle {\widehat \tilde{p}_{n}}}] : monic discrete -Hermite II polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicdiscrqHermiteII http://drmf.wmflabs.org/wiki/Definition:monicdiscrqHermiteII]

<< [[Definition:monicdiscrqHermiteI|Definition:monicdiscrqHermiteI]]
[[Main_Page|Main Page]]
@@ -2073,9 +2058,9 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:monicdualHahn {\displaystyle {\widehat R}_{n}}] : monic dual Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicdualHahn http://drmf.wmflabs.org/wiki/Definition:monicdualHahn]
+[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii]
[http://dlmf.nist.gov/18.25#T1.t1.r5 {\displaystyle R_{n}}] : dual Hahn polynomial : [http://dlmf.nist.gov/18.25#T1.t1.r5 http://dlmf.nist.gov/18.25#T1.t1.r5]
-[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii] +[http://drmf.wmflabs.org/wiki/Definition:monicdualHahn {\displaystyle {\widehat R}_{n}}] : monic dual Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicdualHahn http://drmf.wmflabs.org/wiki/Definition:monicdualHahn]

<< [[Definition:monicdiscrqHermiteII|Definition:monicdiscrqHermiteII]]
[[Main_Page|Main Page]]
@@ -2109,9 +2094,9 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:monicdualqHahn {\displaystyle {\widehat R}_{n}}] : monic dual {\displaystyle q}-Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicdualqHahn http://drmf.wmflabs.org/wiki/Definition:monicdualqHahn]
-[http://drmf.wmflabs.org/wiki/Definition:dualqHahn {\displaystyle R_{n}}] : dual {\displaystyle q}-Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:dualqHahn http://drmf.wmflabs.org/wiki/Definition:dualqHahn]
-[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1] +[http://drmf.wmflabs.org/wiki/Definition:dualqHahn {\displaystyle R_{n}}] : dual -Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:dualqHahn http://drmf.wmflabs.org/wiki/Definition:dualqHahn]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]
+[http://drmf.wmflabs.org/wiki/Definition:monicdualqHahn {\displaystyle {\widehat R}_{n}}] : monic dual -Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicdualqHahn http://drmf.wmflabs.org/wiki/Definition:monicdualqHahn]

<< [[Definition:monicdualHahn|Definition:monicdualHahn]]
[[Main_Page|Main Page]]
@@ -2145,9 +2130,9 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:monicdualqKrawtchouk {\displaystyle {\widehat K}_{n}}] : monic dual {\displaystyle q}-Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicdualqKrawtchouk http://drmf.wmflabs.org/wiki/Definition:monicdualqKrawtchouk]
-[http://drmf.wmflabs.org/wiki/Definition:dualqKrawtchouk {\displaystyle K_{n}}] : dual {\displaystyle q}-Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:dualqKrawtchouk http://drmf.wmflabs.org/wiki/Definition:dualqKrawtchouk]
-[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1] +[http://drmf.wmflabs.org/wiki/Definition:monicdualqKrawtchouk {\displaystyle {\widehat K}_{n}}] : monic dual -Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicdualqKrawtchouk http://drmf.wmflabs.org/wiki/Definition:monicdualqKrawtchouk]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]
+[http://drmf.wmflabs.org/wiki/Definition:dualqKrawtchouk {\displaystyle K_{n}}] : dual -Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:dualqKrawtchouk http://drmf.wmflabs.org/wiki/Definition:dualqKrawtchouk]

<< [[Definition:monicdualqHahn|Definition:monicdualqHahn]]
[[Main_Page|Main Page]]
@@ -2181,9 +2166,9 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:monicHahn {\displaystyle {\widehat Q}_{n}}] : monic Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicHahn http://drmf.wmflabs.org/wiki/Definition:monicHahn]
+[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii]
[http://dlmf.nist.gov/18.19#T1.t1.r3 {\displaystyle Q_{n}}] : Hahn polynomial : [http://dlmf.nist.gov/18.19#T1.t1.r3 http://dlmf.nist.gov/18.19#T1.t1.r3]
-[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii] +[http://drmf.wmflabs.org/wiki/Definition:monicHahn {\displaystyle {\widehat Q}_{n}}] : monic Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicHahn http://drmf.wmflabs.org/wiki/Definition:monicHahn]

<< [[Definition:monicdualqKrawtchouk|Definition:monicdualqKrawtchouk]]
[[Main_Page|Main Page]]
@@ -2217,8 +2202,8 @@ These are defined by == Symbols List == +[http://dlmf.nist.gov/18.3#T1.t1.r28 {\displaystyle H_{n}}] : Hermite polynomial : [http://dlmf.nist.gov/18.3#T1.t1.r28 http://dlmf.nist.gov/18.3#T1.t1.r28]
[http://drmf.wmflabs.org/wiki/Definition:monicHermite {\displaystyle {\widehat H}_{n}}] : monic Hermite polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicHermite http://drmf.wmflabs.org/wiki/Definition:monicHermite]
-[http://dlmf.nist.gov/18.3#T1.t1.r28 {\displaystyle H_{n}}] : Hermite polynomial {\displaystyle H_n} : [http://dlmf.nist.gov/18.3#T1.t1.r28 http://dlmf.nist.gov/18.3#T1.t1.r28]
<< [[Definition:monicHahn|Definition:monicHahn]]
[[Main_Page|Main Page]]
@@ -2253,9 +2238,9 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:monicJacobi {\displaystyle {\widehat P}^{(\alpha,\beta)}_{n}}] : monic Jacobi polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicJacobi http://drmf.wmflabs.org/wiki/Definition:monicJacobi]
[http://dlmf.nist.gov/18.3#T1.t1.r3 {\displaystyle P^{(\alpha,\beta)}_{n}}] : Jacobi polynomial : [http://dlmf.nist.gov/18.3#T1.t1.r3 http://dlmf.nist.gov/18.3#T1.t1.r3]
-[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii] +[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii]
+[http://drmf.wmflabs.org/wiki/Definition:monicJacobi {\displaystyle {\widehat P}^{(\alpha,\beta)}_{n}}] : monic Jacobi polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicJacobi http://drmf.wmflabs.org/wiki/Definition:monicJacobi]

<< [[Definition:monicHermite|Definition:monicHermite]]
[[Main_Page|Main Page]]
@@ -2289,9 +2274,9 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:monicKrawtchouk {\displaystyle {\widehat K}_{n}}] : monic Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicKrawtchouk http://drmf.wmflabs.org/wiki/Definition:monicKrawtchouk]
[http://dlmf.nist.gov/18.19#T1.t1.r6 {\displaystyle K_{n}}] : Krawtchouk polynomial : [http://dlmf.nist.gov/18.19#T1.t1.r6 http://dlmf.nist.gov/18.19#T1.t1.r6]
-[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii] +[http://drmf.wmflabs.org/wiki/Definition:monicKrawtchouk {\displaystyle {\widehat K}_{n}}] : monic Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicKrawtchouk http://drmf.wmflabs.org/wiki/Definition:monicKrawtchouk]
+[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii]

<< [[Definition:monicJacobi|Definition:monicJacobi]]
[[Main_Page|Main Page]]
@@ -2326,7 +2311,7 @@ These are defined by == Symbols List == [http://drmf.wmflabs.org/wiki/Definition:monicLaguerre {\displaystyle {\widehat L}^{\alpha}_n}] : monic Laguerre polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicLaguerre http://drmf.wmflabs.org/wiki/Definition:monicLaguerre]
-[http://dlmf.nist.gov/18.3#T1.t1.r27 {\displaystyle L_n^{(\alpha)}}] : Laguerre (or generalized Laguerre) polynomial : [http://dlmf.nist.gov/18.3#T1.t1.r27 http://dlmf.nist.gov/18.3#T1.t1.r27] +[http://dlmf.nist.gov/18.3#T1.t1.r27 {\displaystyle L_n^{(\alpha)}}] : Laguerre (or generalized Laguerre) polynomial : [http://dlmf.nist.gov/18.3#T1.t1.r27 http://dlmf.nist.gov/18.3#T1.t1.r27]

<< [[Definition:monicKrawtchouk|Definition:monicKrawtchouk]]
[[Main_Page|Main Page]]
@@ -2363,7 +2348,7 @@ These are defined by [http://drmf.wmflabs.org/wiki/Definition:monicLegendrePoly {\displaystyle {\widehat P}_{n}}] : monic Legendre polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicLegendrePoly http://drmf.wmflabs.org/wiki/Definition:monicLegendrePoly]
[http://dlmf.nist.gov/18.3#T1.t1.r25 {\displaystyle P_{n}}] : Legendre polynomial : [http://dlmf.nist.gov/18.3#T1.t1.r25 http://dlmf.nist.gov/18.3#T1.t1.r25]
-[http://dlmf.nist.gov/1.2#E1 {\displaystyle \binom{n}{k}}] : binomial coefficient : [http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/1.2#E1] [http://dlmf.nist.gov/26.3#SS1.p1 http://dlmf.nist.gov/26.3#SS1.p1] +[http://dlmf.nist.gov/1.2#E1 {\displaystyle \binom{n}{k}}] : binomial coefficient : [http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/1.2#E1]

<< [[Definition:monicLaguerre|Definition:monicLaguerre]]
[[Main_Page|Main Page]]
@@ -2397,10 +2382,10 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:moniclittleqJacobi {\displaystyle {\widehat p}_{n}}] : monic little {\displaystyle q}-Jacobi polynomial : [http://drmf.wmflabs.org/wiki/Definition:moniclittleqJacobi http://drmf.wmflabs.org/wiki/Definition:moniclittleqJacobi]
-[http://drmf.wmflabs.org/wiki/Definition:littleqJacobi {\displaystyle p_{n}}] : little {\displaystyle q}-Jacobi polynomial : [http://drmf.wmflabs.org/wiki/Definition:littleqJacobi http://drmf.wmflabs.org/wiki/Definition:littleqJacobi]
-[http://dlmf.nist.gov/1.2#E1 {\displaystyle \binom{n}{k}}] : binomial coefficient : [http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/1.2#E1] [http://dlmf.nist.gov/26.3#SS1.p1 http://dlmf.nist.gov/26.3#SS1.p1]
-[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1] +[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]
+[http://dlmf.nist.gov/1.2#E1 {\displaystyle \binom{n}{k}}] : binomial coefficient : [http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/1.2#E1]
+[http://drmf.wmflabs.org/wiki/Definition:moniclittleqJacobi {\displaystyle {\widehat p}_{n}}] : monic little -Jacobi polynomial : [http://drmf.wmflabs.org/wiki/Definition:moniclittleqJacobi http://drmf.wmflabs.org/wiki/Definition:moniclittleqJacobi]
+[http://drmf.wmflabs.org/wiki/Definition:littleqJacobi {\displaystyle p_{n}}] : little -Jacobi polynomial : [http://drmf.wmflabs.org/wiki/Definition:littleqJacobi http://drmf.wmflabs.org/wiki/Definition:littleqJacobi]

<< [[Definition:monicLegendrePoly|Definition:monicLegendrePoly]]
[[Main_Page|Main Page]]
@@ -2433,10 +2418,7 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:moniclittleqLaguerre {\displaystyle {\widehat p}_{n}}] : monic little {\displaystyle q}-Laguerre/Wall polynomial : [http://drmf.wmflabs.org/wiki/Definition:moniclittleqLaguerre http://drmf.wmflabs.org/wiki/Definition:moniclittleqLaguerre]
-[http://drmf.wmflabs.org/wiki/Definition:littleqLaguerre {\displaystyle p_{n}}] : little {\displaystyle q}-Laguerre / Wall polynomial : [http://drmf.wmflabs.org/wiki/Definition:littleqLaguerre http://drmf.wmflabs.org/wiki/Definition:littleqLaguerre]
-[http://dlmf.nist.gov/1.2#E1 {\displaystyle \binom{n}{k}}] : binomial coefficient : [http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/1.2#E1] [http://dlmf.nist.gov/26.3#SS1.p1 http://dlmf.nist.gov/26.3#SS1.p1]
-[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1] +
<< [[Definition:moniclittleqJacobi|Definition:moniclittleqJacobi]]
[[Main_Page|Main Page]]
@@ -2470,10 +2452,10 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:moniclittleqLegendre {\displaystyle {\widehat p}_{n}}] : monic little {\displaystyle q}-Legendre polynomial : [http://drmf.wmflabs.org/wiki/Definition:moniclittleqLegendre http://drmf.wmflabs.org/wiki/Definition:moniclittleqLegendre]
-[http://drmf.wmflabs.org/wiki/Definition:littleqLegendre {\displaystyle p_{n}}] : little {\displaystyle q}-Legendre polynomial : [http://drmf.wmflabs.org/wiki/Definition:littleqLegendre http://drmf.wmflabs.org/wiki/Definition:littleqLegendre]
-[http://dlmf.nist.gov/1.2#E1 {\displaystyle \binom{n}{k}}] : binomial coefficient : [http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/1.2#E1] [http://dlmf.nist.gov/26.3#SS1.p1 http://dlmf.nist.gov/26.3#SS1.p1]
-[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1] +[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]
+[http://dlmf.nist.gov/1.2#E1 {\displaystyle \binom{n}{k}}] : binomial coefficient : [http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/1.2#E1]
+[http://drmf.wmflabs.org/wiki/Definition:littleqLegendre {\displaystyle p_{n}}] : little -Legendre polynomial : [http://drmf.wmflabs.org/wiki/Definition:littleqLegendre http://drmf.wmflabs.org/wiki/Definition:littleqLegendre]
+[http://drmf.wmflabs.org/wiki/Definition:moniclittleqLegendre {\displaystyle {\widehat p}_{n}}] : monic little -Legendre polynomial : [http://drmf.wmflabs.org/wiki/Definition:moniclittleqLegendre http://drmf.wmflabs.org/wiki/Definition:moniclittleqLegendre]

<< [[Definition:moniclittleqLaguerre|Definition:moniclittleqLaguerre]]
[[Main_Page|Main Page]]
@@ -2508,8 +2490,8 @@ These are defined by == Symbols List == [http://drmf.wmflabs.org/wiki/Definition:monicMeixner {\displaystyle {\widehat M}_{n}}] : monic Meixner polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicMeixner http://drmf.wmflabs.org/wiki/Definition:monicMeixner]
+[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii]
[http://dlmf.nist.gov/18.19#T1.t1.r9 {\displaystyle M_{n}}] : Meixner polynomial : [http://dlmf.nist.gov/18.19#T1.t1.r9 http://dlmf.nist.gov/18.19#T1.t1.r9]
-[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii]
<< [[Definition:moniclittleqLegendre|Definition:moniclittleqLegendre]]
[[Main_Page|Main Page]]
@@ -2546,7 +2528,7 @@ These are defined by [http://drmf.wmflabs.org/wiki/Definition:monicMeixnerPollaczek {\displaystyle {\widehat P}^{(\alpha)}_{n}}] : monic Meixner-Pollaczek polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicMeixnerPollaczek http://drmf.wmflabs.org/wiki/Definition:monicMeixnerPollaczek]
[http://dlmf.nist.gov/18.19#P3.p1 {\displaystyle P^{(\alpha)}_{n}}] : Meixner-Pollaczek polynomial : [http://dlmf.nist.gov/18.19#P3.p1 http://dlmf.nist.gov/18.19#P3.p1]
-[http://dlmf.nist.gov/4.14#E1 {\displaystyle \mathrm{sin}}] : sine function : [http://dlmf.nist.gov/4.14#E1 http://dlmf.nist.gov/4.14#E1] +[http://dlmf.nist.gov/4.14#E1 {\displaystyle \mathrm{sin}}] : sine function : [http://dlmf.nist.gov/4.14#E1 http://dlmf.nist.gov/4.14#E1]

<< [[Definition:monicMeixner|Definition:monicMeixner]]
[[Main_Page|Main Page]]
@@ -2581,7 +2563,7 @@ These are defined by == Symbols List == [http://drmf.wmflabs.org/wiki/Definition:monicpseudoJacobi {\displaystyle {\widehat P}_{n}}] : monic pseudo-Jacobi polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicpseudoJacobi http://drmf.wmflabs.org/wiki/Definition:monicpseudoJacobi]
-[http://drmf.wmflabs.org/wiki/Definition:pseudoJacobi {\displaystyle P_{n}}] : pseudo Jacobi polynomial : [http://drmf.wmflabs.org/wiki/Definition:pseudoJacobi http://drmf.wmflabs.org/wiki/Definition:pseudoJacobi] +[http://drmf.wmflabs.org/wiki/Definition:pseudoJacobi {\displaystyle P_{n}}] : pseudo Jacobi polynomial : [http://drmf.wmflabs.org/wiki/Definition:pseudoJacobi http://drmf.wmflabs.org/wiki/Definition:pseudoJacobi]

<< [[Definition:monicMeixnerPollaczek|Definition:monicMeixnerPollaczek]]
[[Main_Page|Main Page]]
@@ -2615,10 +2597,10 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:monicqBessel {\displaystyle {\widehat y}_{n}}] : monic {\displaystyle q}-Bessel polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicqBessel http://drmf.wmflabs.org/wiki/Definition:monicqBessel]
-[http://drmf.wmflabs.org/wiki/Definition:qBessel {\displaystyle y_{n}}] : {\displaystyle q}-Bessel polynomial : [http://drmf.wmflabs.org/wiki/Definition:qBessel http://drmf.wmflabs.org/wiki/Definition:qBessel]
-[http://dlmf.nist.gov/1.2#E1 {\displaystyle \binom{n}{k}}] : binomial coefficient : [http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/1.2#E1] [http://dlmf.nist.gov/26.3#SS1.p1 http://dlmf.nist.gov/26.3#SS1.p1]
-[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1] +[http://drmf.wmflabs.org/wiki/Definition:qBessel {\displaystyle y_{n}}] : -Bessel polynomial : [http://drmf.wmflabs.org/wiki/Definition:qBessel http://drmf.wmflabs.org/wiki/Definition:qBessel]
+[http://dlmf.nist.gov/1.2#E1 {\displaystyle \binom{n}{k}}] : binomial coefficient : [http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/1.2#E1]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]
+[http://drmf.wmflabs.org/wiki/Definition:monicqBessel {\displaystyle {\widehat y}_{n}}] : monic -Bessel polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicqBessel http://drmf.wmflabs.org/wiki/Definition:monicqBessel]

<< [[Definition:monicpseudoJacobi|Definition:monicpseudoJacobi]]
[[Main_Page|Main Page]]
@@ -2653,10 +2635,9 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:monicqCharlier {\displaystyle {\widehat C}_{n}}] : monic {\displaystyle q}-Charlier polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicqCharlier http://drmf.wmflabs.org/wiki/Definition:monicqCharlier]
-[http://drmf.wmflabs.org/wiki/Definition:qHahn {\displaystyle Q_{n}}] : {\displaystyle q}-Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:qHahn http://drmf.wmflabs.org/wiki/Definition:qHahn]
-[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1]
-[http://drmf.wmflabs.org/wiki/Definition:monicqHahn {\displaystyle {\widehat Q}_{n}}] : monic {\displaystyle q}-Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicqHahn http://drmf.wmflabs.org/wiki/Definition:monicqHahn] +[http://drmf.wmflabs.org/wiki/Definition:monicqHahn {\displaystyle {\widehat Q}_{n}}] : monic -Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicqHahn http://drmf.wmflabs.org/wiki/Definition:monicqHahn]
+[http://drmf.wmflabs.org/wiki/Definition:qHahn {\displaystyle Q_{n}}] : -Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:qHahn http://drmf.wmflabs.org/wiki/Definition:qHahn]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]

<< [[Definition:monicqBesselPoly|Definition:monicqBesselPoly]]
[[Main_Page|Main Page]]
@@ -2690,9 +2671,9 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:monicqKrawtchouk {\displaystyle {\widehat K}_{n}}] : monic {\displaystyle q}-Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicqKrawtchouk http://drmf.wmflabs.org/wiki/Definition:monicqKrawtchouk]
-[http://drmf.wmflabs.org/wiki/Definition:qKrawtchouk {\displaystyle K_{n}}] : {\displaystyle q}-Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:qKrawtchouk http://drmf.wmflabs.org/wiki/Definition:qKrawtchouk]
-[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1] +[http://drmf.wmflabs.org/wiki/Definition:monicqKrawtchouk {\displaystyle {\widehat K}_{n}}] : monic -Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicqKrawtchouk http://drmf.wmflabs.org/wiki/Definition:monicqKrawtchouk]
+[http://drmf.wmflabs.org/wiki/Definition:qKrawtchouk {\displaystyle K_{n}}] : -Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:qKrawtchouk http://drmf.wmflabs.org/wiki/Definition:qKrawtchouk]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]

<< [[Definition:monicqCharlier|Definition:monicqCharlier]]
[[Main_Page|Main Page]]
@@ -2726,9 +2707,9 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:monicqLaguerre {\displaystyle {\widehat L}^{(\alpha)}_{n}}] : monic {\displaystyle q}-Laguerre polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicqLaguerre http://drmf.wmflabs.org/wiki/Definition:monicqLaguerre]
-[http://drmf.wmflabs.org/wiki/Definition:qLaguerre {\displaystyle L_n^{(\alpha)}}] : {\displaystyle q}-Laguerre polynomial : [http://drmf.wmflabs.org/wiki/Definition:qLaguerre http://drmf.wmflabs.org/wiki/Definition:qLaguerre]
-[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1] +[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]
+[http://drmf.wmflabs.org/wiki/Definition:monicqLaguerre {\displaystyle {\widehat L}^{(\alpha)}_{n}}] : monic -Laguerre polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicqLaguerre http://drmf.wmflabs.org/wiki/Definition:monicqLaguerre]
+[http://drmf.wmflabs.org/wiki/Definition:qLaguerre {\displaystyle L_n^{(\alpha)}}] : -Laguerre polynomial : [http://drmf.wmflabs.org/wiki/Definition:qLaguerre http://drmf.wmflabs.org/wiki/Definition:qLaguerre]

<< [[Definition:monicqKrawtchouk|Definition:monicqKrawtchouk]]
[[Main_Page|Main Page]]
@@ -2761,9 +2742,9 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:monicqMeixner {\displaystyle {\widehat M}_{n}}] : monic {\displaystyle q}-Meixner polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicqMeixner http://drmf.wmflabs.org/wiki/Definition:monicqMeixner]
-[http://drmf.wmflabs.org/wiki/Definition:qMeixner {\displaystyle M_{n}}] : {\displaystyle q}-Meixner polynomial : [http://drmf.wmflabs.org/wiki/Definition:qMeixner http://drmf.wmflabs.org/wiki/Definition:qMeixner]
-[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1] +[http://drmf.wmflabs.org/wiki/Definition:monicqMeixner {\displaystyle {\widehat M}_{n}}] : monic -Meixner polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicqMeixner http://drmf.wmflabs.org/wiki/Definition:monicqMeixner]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]
+[http://drmf.wmflabs.org/wiki/Definition:qMeixner {\displaystyle M_{n}}] : -Meixner polynomial : [http://drmf.wmflabs.org/wiki/Definition:qMeixner http://drmf.wmflabs.org/wiki/Definition:qMeixner]

<< [[Definition:monicqLaguerre|Definition:monicqLaguerre]]
[[Main_Page|Main Page]]
@@ -2797,9 +2778,9 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:monicqMeixnerPollaczek {\displaystyle {\widehat P}_{n}}] : monic {\displaystyle q}-Meixner-Pollaczek polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicqMeixnerPollaczek http://drmf.wmflabs.org/wiki/Definition:monicqMeixnerPollaczek]
-[http://drmf.wmflabs.org/wiki/Definition:qMeixnerPollaczek {\displaystyle P_{n}}] : {\displaystyle q}-Meixner-Pollaczek polynomial : [http://drmf.wmflabs.org/wiki/Definition:qMeixnerPollaczek http://drmf.wmflabs.org/wiki/Definition:qMeixnerPollaczek]
-[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1] +[http://drmf.wmflabs.org/wiki/Definition:monicqMeixnerPollaczek {\displaystyle {\widehat P}_{n}}] : monic -Meixner-Pollaczek polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicqMeixnerPollaczek http://drmf.wmflabs.org/wiki/Definition:monicqMeixnerPollaczek]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]
+[http://drmf.wmflabs.org/wiki/Definition:qMeixnerPollaczek {\displaystyle P_{n}}] : -Meixner-Pollaczek polynomial : [http://drmf.wmflabs.org/wiki/Definition:qMeixnerPollaczek http://drmf.wmflabs.org/wiki/Definition:qMeixnerPollaczek]

<< [[Definition:monicqMeixner|Definition:monicqMeixner]]
[[Main_Page|Main Page]]
@@ -2834,9 +2815,9 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:monicqRacah {\displaystyle {\widehat R}_{n}}] : monic {\displaystyle q}-Racah polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicqRacah http://drmf.wmflabs.org/wiki/Definition:monicqRacah]
-[http://dlmf.nist.gov/18.28#E19 {\displaystyle R_{n}}] : {\displaystyle q}-Racah polynomial : [http://dlmf.nist.gov/18.28#E19 http://dlmf.nist.gov/18.28#E19]
-[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1] +[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]
+[http://dlmf.nist.gov/18.28#E19 {\displaystyle R_{n}}] : -Racah polynomial : [http://dlmf.nist.gov/18.28#E19 http://dlmf.nist.gov/18.28#E19]
+[http://drmf.wmflabs.org/wiki/Definition:monicqRacah {\displaystyle {\widehat R}_{n}}] : monic -Racah polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicqRacah http://drmf.wmflabs.org/wiki/Definition:monicqRacah]

<< [[Definition:monicqMeixnerPollaczek|Definition:monicqMeixnerPollaczek]]
[[Main_Page|Main Page]]
@@ -2870,9 +2851,9 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:monicqtmqKrawtchouk {\displaystyle {\widehat K^{\mathrm{qtm}}}_{n}}] : monic quantum {\displaystyle q}-Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicqtmqKrawtchouk http://drmf.wmflabs.org/wiki/Definition:monicqtmqKrawtchouk]
-[http://drmf.wmflabs.org/wiki/Definition:qtmqKrawtchouk {\displaystyle K^{\mathrm{qtm}}_{n}}] : quantum {\displaystyle q}-Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:qtmqKrawtchouk http://drmf.wmflabs.org/wiki/Definition:qtmqKrawtchouk]
-[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1] +[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]
+[http://drmf.wmflabs.org/wiki/Definition:qtmqKrawtchouk {\displaystyle K^{\mathrm{qtm}}_{n}}] : quantum -Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:qtmqKrawtchouk http://drmf.wmflabs.org/wiki/Definition:qtmqKrawtchouk]
+[http://drmf.wmflabs.org/wiki/Definition:monicqtmqKrawtchouk {\displaystyle {\widehat K^{\mathrm{qtm}}}_{n}}] : monic quantum -Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicqtmqKrawtchouk http://drmf.wmflabs.org/wiki/Definition:monicqtmqKrawtchouk]

<< [[Definition:monicqRacah|Definition:monicqRacah]]
[[Main_Page|Main Page]]
@@ -2907,9 +2888,9 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:monicRacah {\displaystyle {\widehat R}_{n}}] : monic Racah polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicRacah http://drmf.wmflabs.org/wiki/Definition:monicRacah]
+[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii]
[http://dlmf.nist.gov/18.25#T1.t1.r4 {\displaystyle R_{n}}] : Racah polynomial : [http://dlmf.nist.gov/18.25#T1.t1.r4 http://dlmf.nist.gov/18.25#T1.t1.r4]
-[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii] +[http://drmf.wmflabs.org/wiki/Definition:monicRacah {\displaystyle {\widehat R}_{n}}] : monic Racah polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicRacah http://drmf.wmflabs.org/wiki/Definition:monicRacah]

<< [[Definition:monicqtmqKrawtchouk|Definition:monicqtmqKrawtchouk]]
[[Main_Page|Main Page]]
@@ -2943,9 +2924,9 @@ These are defined by == Symbols List == +[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]
[http://drmf.wmflabs.org/wiki/Definition:monicStieltjesWigert {\displaystyle {\widehat S}_{n}}] : monic Stieltjes-Wigert polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicStieltjesWigert http://drmf.wmflabs.org/wiki/Definition:monicStieltjesWigert]
[http://drmf.wmflabs.org/wiki/Definition:StieltjesWigert {\displaystyle S_{n}}] : Stieltjes-Wigert polynomial : [http://drmf.wmflabs.org/wiki/Definition:StieltjesWigert http://drmf.wmflabs.org/wiki/Definition:StieltjesWigert]
-[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1]
<< [[Definition:monicRacah|Definition:monicRacah]]
[[Main_Page|Main Page]]
@@ -2981,7 +2962,7 @@ These are defined by [http://drmf.wmflabs.org/wiki/Definition:monicUltra {\displaystyle {\widehat C}^{\mu}_{n}}] : monic ultraspherical/Gegenbauer polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicUltra http://drmf.wmflabs.org/wiki/Definition:monicUltra]
[http://dlmf.nist.gov/18.3#T1.t1.r5 {\displaystyle C^{\mu}_{n}}] : ultraspherical/Gegenbauer polynomial : [http://dlmf.nist.gov/18.3#T1.t1.r5 http://dlmf.nist.gov/18.3#T1.t1.r5]
-[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii] +[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii]

<< [[Definition:monicStieltjesWigert|Definition:monicStieltjesWigert]]
[[Main_Page|Main Page]]
@@ -3015,9 +2996,9 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:monicWilson {\displaystyle {\widehat W}_{n}}] : monic Wilson polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicWilson http://drmf.wmflabs.org/wiki/Definition:monicWilson]
[http://dlmf.nist.gov/18.25#T1.t1.r2 {\displaystyle W_{n}}] : Wilson polynomial : [http://dlmf.nist.gov/18.25#T1.t1.r2 http://dlmf.nist.gov/18.25#T1.t1.r2]
-[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii] +[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii]
+[http://drmf.wmflabs.org/wiki/Definition:monicWilson {\displaystyle {\widehat W}_{n}}] : monic Wilson polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicWilson http://drmf.wmflabs.org/wiki/Definition:monicWilson]

<< [[Definition:monicUltra|Definition:monicUltra]]
[[Main_Page|Main Page]]
@@ -3048,7 +3029,7 @@ These are defined by == Symbols List == [http://drmf.wmflabs.org/wiki/Definition:NeumannFactor {\displaystyle \epsilon_{m}}] : Neumann factor : [http://drmf.wmflabs.org/wiki/Definition:NeumannFactor http://drmf.wmflabs.org/wiki/Definition:NeumannFactor]
-[http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r4 {\displaystyle \delta_{m,n}}] : Kronecker delta : [http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r4 http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r4] +[http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r4 {\displaystyle \delta_{m,n}}] : Kronecker delta : [http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r4 http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r4]

<< [[Definition:monicWilson|Definition:monicWilson]]
[[Main_Page|Main Page]]
@@ -3079,9 +3060,9 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:normctsdualHahnStilde {\displaystyle {\tilde S}_{n}}] : normalized continuous dual Hahn polynomial {\displaystyle {\tilde S}} : [http://drmf.wmflabs.org/wiki/Definition:normctsdualHahnStilde http://drmf.wmflabs.org/wiki/Definition:normctsdualHahnStilde]
[http://dlmf.nist.gov/18.25#T1.t1.r3 {\displaystyle S_{n}}] : continuous dual Hahn polynomial : [http://dlmf.nist.gov/18.25#T1.t1.r3 http://dlmf.nist.gov/18.25#T1.t1.r3]
-[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii] +[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii]
+[http://drmf.wmflabs.org/wiki/Definition:normctsdualHahnStilde {\displaystyle {\tilde S}_{n}}] : normalized continuous dual Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:normctsdualHahnStilde http://drmf.wmflabs.org/wiki/Definition:normctsdualHahnStilde]

<< [[Definition:NeumannFactor|Definition:NeumannFactor]]
[[Main_Page|Main Page]]
@@ -3113,9 +3094,9 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:normctsHahnptilde {\displaystyle {\tilde p}_{n}}] : normalized continuous Hahn polynomial {\displaystyle {\tilde p}} : [http://drmf.wmflabs.org/wiki/Definition:normctsHahnptilde http://drmf.wmflabs.org/wiki/Definition:normctsHahnptilde]
+[http://drmf.wmflabs.org/wiki/Definition:normctsHahnptilde {\displaystyle {\tilde p}_{n}}] : normalized continuous Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:normctsHahnptilde http://drmf.wmflabs.org/wiki/Definition:normctsHahnptilde]
+[http://dlmf.nist.gov/18.19#P2.p1 {\displaystyle p_{n}}] : continuous Hahn polynomial : [http://dlmf.nist.gov/18.19#P2.p1 http://dlmf.nist.gov/18.19#P2.p1]
[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii]
-[http://dlmf.nist.gov/18.19#P2.p1 {\displaystyle p_{n}}] : continuous Hahn polynomial : [http://dlmf.nist.gov/18.19#P2.p1 http://dlmf.nist.gov/18.19#P2.p1]
<< [[Definition:normctsdualHahnStilde|Definition:normctsdualHahnStilde]]
[[Main_Page|Main Page]]
@@ -3152,9 +3133,9 @@ where == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:normJacobiR {\displaystyle R^{(\alpha,\beta)}_{n}}] : normalized Jacobi polynomial : [http://drmf.wmflabs.org/wiki/Definition:normJacobiR http://drmf.wmflabs.org/wiki/Definition:normJacobiR]
[http://dlmf.nist.gov/18.3#T1.t1.r3 {\displaystyle P^{(\alpha,\beta)}_{n}}] : Jacobi polynomial : [http://dlmf.nist.gov/18.3#T1.t1.r3 http://dlmf.nist.gov/18.3#T1.t1.r3]
-[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii] +[http://drmf.wmflabs.org/wiki/Definition:normJacobiR {\displaystyle R^{(\alpha,\beta)}_{n}}] : normalized Jacobi polynomial : [http://drmf.wmflabs.org/wiki/Definition:normJacobiR http://drmf.wmflabs.org/wiki/Definition:normJacobiR]
+[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii]

<< [[Definition:normctsHahnptilde|Definition:normctsHahnptilde]]
[[Main_Page|Main Page]]
@@ -3189,9 +3170,9 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:normWilsonWtilde {\displaystyle {\tilde W}_{n}}] : normalized Wilson polynomial {\displaystyle {\tilde W}} : [http://drmf.wmflabs.org/wiki/Definition:normWilsonWtilde http://drmf.wmflabs.org/wiki/Definition:normWilsonWtilde]
[http://dlmf.nist.gov/18.25#T1.t1.r2 {\displaystyle W_{n}}] : Wilson polynomial : [http://dlmf.nist.gov/18.25#T1.t1.r2 http://dlmf.nist.gov/18.25#T1.t1.r2]
-[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii] +[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii]
+[http://drmf.wmflabs.org/wiki/Definition:normWilsonWtilde {\displaystyle {\tilde W}_{n}}] : normalized Wilson polynomial : [http://drmf.wmflabs.org/wiki/Definition:normWilsonWtilde http://drmf.wmflabs.org/wiki/Definition:normWilsonWtilde]

<< [[Definition:normJacobiR|Definition:normJacobiR]]
[[Main_Page|Main Page]]
@@ -3224,10 +3205,9 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:poly {\displaystyle {p}_{n}}] : polynomial : [http://drmf.wmflabs.org/wiki/Definition:poly http://drmf.wmflabs.org/wiki/Definition:poly]
-[http://drmf.wmflabs.org/wiki/Definition:pseudoJacobi {\displaystyle P_{n}}] : pseudo Jacobi polynomial : [http://drmf.wmflabs.org/wiki/Definition:pseudoJacobi http://drmf.wmflabs.org/wiki/Definition:pseudoJacobi]
+[http://dlmf.nist.gov/16.2#E1 {\displaystyle {{}_{p}F_{q}}}] : generalized hypergeometric function : [http://dlmf.nist.gov/16.2#E1 http://dlmf.nist.gov/16.2#E1]
[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii]
-[http://dlmf.nist.gov/16.2#E1 {\displaystyle {{}_{p}F_{q}}}] : generalized hypergeometric function : [http://dlmf.nist.gov/16.2#E1 http://dlmf.nist.gov/16.2#E1] +[http://drmf.wmflabs.org/wiki/Definition:pseudoJacobi {\displaystyle P_{n}}] : pseudo Jacobi polynomial : [http://drmf.wmflabs.org/wiki/Definition:pseudoJacobi http://drmf.wmflabs.org/wiki/Definition:pseudoJacobi]

<< [[Definition:normWilsonWtilde|Definition:normWilsonWtilde]]
[[Main_Page|Main Page]]
@@ -3259,8 +3239,8 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:qBessel {\displaystyle y_{n}}] : {\displaystyle q}-Bessel polynomial : [http://drmf.wmflabs.org/wiki/Definition:qBessel http://drmf.wmflabs.org/wiki/Definition:qBessel]
-[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1] +[http://drmf.wmflabs.org/wiki/Definition:qBessel {\displaystyle y_{n}}] : -Bessel polynomial : [http://drmf.wmflabs.org/wiki/Definition:qBessel http://drmf.wmflabs.org/wiki/Definition:qBessel]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or -hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]

<< [[Definition:poly|Definition:poly]]
[[Main_Page|Main Page]]
@@ -3292,8 +3272,8 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:qCharlier {\displaystyle C_{n}}] : {\displaystyle q}-Charlier polynomial : [http://drmf.wmflabs.org/wiki/Definition:qCharlier http://drmf.wmflabs.org/wiki/Definition:qCharlier]
-[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1] +[http://drmf.wmflabs.org/wiki/Definition:qCharlier {\displaystyle C_{n}}] : -Charlier polynomial : [http://drmf.wmflabs.org/wiki/Definition:qCharlier http://drmf.wmflabs.org/wiki/Definition:qCharlier]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or -hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]

<< [[Definition:qBesselPoly|Definition:qBesselPoly]]
[[Main_Page|Main Page]]
@@ -3325,8 +3305,8 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:qDigamma {\displaystyle \psi_{q}}] : {\displaystyle q}-digamma function : [http://drmf.wmflabs.org/wiki/Definition:qDigamma http://drmf.wmflabs.org/wiki/Definition:qDigamma]
-[http://dlmf.nist.gov/5.18#E4 {\displaystyle \Gamma_{q}}] : {\displaystyle q}-gamma function : [http://dlmf.nist.gov/5.18#E4 http://dlmf.nist.gov/5.18#E4] +[http://dlmf.nist.gov/5.18#E4 {\displaystyle \Gamma_{q}}] : -gamma function : [http://dlmf.nist.gov/5.18#E4 http://dlmf.nist.gov/5.18#E4]
+[http://drmf.wmflabs.org/wiki/Definition:qDigamma {\displaystyle \psi_{q}}] : -digamma function : [http://drmf.wmflabs.org/wiki/Definition:qDigamma http://drmf.wmflabs.org/wiki/Definition:qDigamma]

<< [[Definition:qCharlier|Definition:qCharlier]]
[[Main_Page|Main Page]]
@@ -3367,8 +3347,8 @@ Equation (2.1.11) of [[Bibliography#GaR|'''GaR''']]. == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:qHyperrWs {\displaystyle {{}_{r+1}W_{r}}}] : very-well-poised basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://drmf.wmflabs.org/wiki/Definition:qHyperrWs http://drmf.wmflabs.org/wiki/Definition:qHyperrWs]
-[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1] +[http://drmf.wmflabs.org/wiki/Definition:qHyperrWs {\displaystyle {{}_{r+1}W_{r}}}] : very-well-poised basic hypergeometric (or -hypergeometric) function : [http://drmf.wmflabs.org/wiki/Definition:qHyperrWs http://drmf.wmflabs.org/wiki/Definition:qHyperrWs]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or -hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]

<< [[Definition:qDigamma|Definition:qDigamma]]
[[Main_Page|Main Page]]
@@ -3402,12 +3382,10 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:qExpKLS {\displaystyle \mathrm{E}_{q}}] : {\displaystyle q}-analogue of the {\displaystyle \exp} function used in KLS: {\displaystyle \mathrm{E}_{q}} : [http://drmf.wmflabs.org/wiki/Definition:qExpKLS http://drmf.wmflabs.org/wiki/Definition:qExpKLS]
-[http://dlmf.nist.gov/4.2#E19 {\displaystyle \mathrm{exp}}] : exponential function : [http://dlmf.nist.gov/4.2#E19 http://dlmf.nist.gov/4.2#E19]
-[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
-[http://drmf.wmflabs.org/wiki/Definition:sum {\displaystyle \Sigma}] : sum : [http://drmf.wmflabs.org/wiki/Definition:sum http://drmf.wmflabs.org/wiki/Definition:sum]
-[http://dlmf.nist.gov/1.2#E1 {\displaystyle \binom{n}{k}}] : binomial coefficient : [http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/1.2#E1] [http://dlmf.nist.gov/26.3#SS1.p1 http://dlmf.nist.gov/26.3#SS1.p1]
-[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1] +[http://drmf.wmflabs.org/wiki/Definition:qExpKLS {\displaystyle \mathrm{E}_{q}}] : -analogue of the function used in KLS: : [http://drmf.wmflabs.org/wiki/Definition:qExpKLS http://drmf.wmflabs.org/wiki/Definition:qExpKLS]
+[http://dlmf.nist.gov/1.2#E1 {\displaystyle \binom{n}{k}}] : binomial coefficient : [http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/1.2#E1]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or -hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]

<< [[Definition:qHyperrWs|Definition:qHyperrWs]]
[[Main_Page|Main Page]]
@@ -3441,11 +3419,9 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:qexpKLS {\displaystyle \mathrm{e}_{q}}] : {\displaystyle q}-analogue of the {\displaystyle \exp} function used in KLS: {\displaystyle \mathrm{e}_{q}} : [http://drmf.wmflabs.org/wiki/Definition:qexpKLS http://drmf.wmflabs.org/wiki/Definition:qexpKLS]
-[http://dlmf.nist.gov/4.2#E19 {\displaystyle \mathrm{exp}}] : exponential function : [http://dlmf.nist.gov/4.2#E19 http://dlmf.nist.gov/4.2#E19]
-[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
-[http://drmf.wmflabs.org/wiki/Definition:sum {\displaystyle \Sigma}] : sum : [http://drmf.wmflabs.org/wiki/Definition:sum http://drmf.wmflabs.org/wiki/Definition:sum]
-[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1] +[http://drmf.wmflabs.org/wiki/Definition:qexpKLS {\displaystyle \mathrm{e}_{q}}] : -analogue of the function used in KLS: : [http://drmf.wmflabs.org/wiki/Definition:qexpKLS http://drmf.wmflabs.org/wiki/Definition:qexpKLS]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or -hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]

<< [[Definition:qExpKLS|Definition:qExpKLS]]
[[Main_Page|Main Page]]
@@ -3478,11 +3454,9 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:qcosKLS {\displaystyle \mathrm{cos}_{q}}] : {\displaystyle q}-analogue of the {\displaystyle \cos} function used in KLS: {\displaystyle \cos_q} : [http://drmf.wmflabs.org/wiki/Definition:qcosKLS http://drmf.wmflabs.org/wiki/Definition:qcosKLS]
-[http://dlmf.nist.gov/4.14#E2 {\displaystyle \mathrm{cos}}] : cosine function : [http://dlmf.nist.gov/4.14#E2 http://dlmf.nist.gov/4.14#E2]
-[http://drmf.wmflabs.org/wiki/Definition:qexpKLS {\displaystyle \mathrm{e}_{q}}] : {\displaystyle q}-analogue of the {\displaystyle \exp} function used in KLS: {\displaystyle \mathrm{e}_{q}} : [http://drmf.wmflabs.org/wiki/Definition:qexpKLS http://drmf.wmflabs.org/wiki/Definition:qexpKLS]
-[http://drmf.wmflabs.org/wiki/Definition:sum {\displaystyle \Sigma}] : sum : [http://drmf.wmflabs.org/wiki/Definition:sum http://drmf.wmflabs.org/wiki/Definition:sum]
-[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1] +[http://drmf.wmflabs.org/wiki/Definition:qcosKLS {\displaystyle \mathrm{cos}_{q}}] : -analogue of the function used in KLS: : [http://drmf.wmflabs.org/wiki/Definition:qcosKLS http://drmf.wmflabs.org/wiki/Definition:qcosKLS]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]
+[http://drmf.wmflabs.org/wiki/Definition:qexpKLS {\displaystyle \mathrm{e}_{q}}] : -analogue of the function used in KLS: : [http://drmf.wmflabs.org/wiki/Definition:qexpKLS http://drmf.wmflabs.org/wiki/Definition:qexpKLS]

<< [[Definition:qexpKLS|Definition:qexpKLS]]
[[Main_Page|Main Page]]
@@ -3513,8 +3487,8 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:qMeixner {\displaystyle M_{n}}] : {\displaystyle q}-Meixner polynomial : [http://drmf.wmflabs.org/wiki/Definition:qMeixner http://drmf.wmflabs.org/wiki/Definition:qMeixner]
-[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1] +[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or -hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
+[http://drmf.wmflabs.org/wiki/Definition:qMeixner {\displaystyle M_{n}}] : -Meixner polynomial : [http://drmf.wmflabs.org/wiki/Definition:qMeixner http://drmf.wmflabs.org/wiki/Definition:qMeixner]

<< [[Definition:qcosKLS|Definition:qcosKLS]]
[[Main_Page|Main Page]]
@@ -3547,8 +3521,8 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:qKrawtchouk {\displaystyle K_{n}}] : {\displaystyle q}-Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:qKrawtchouk http://drmf.wmflabs.org/wiki/Definition:qKrawtchouk]
-[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1] +[http://drmf.wmflabs.org/wiki/Definition:qKrawtchouk {\displaystyle K_{n}}] : -Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:qKrawtchouk http://drmf.wmflabs.org/wiki/Definition:qKrawtchouk]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or -hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]

<< [[Definition:qMeixner|Definition:qMeixner]]
[[Main_Page|Main Page]]
@@ -3582,9 +3556,9 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:monicqHahn {\displaystyle {\widehat Q}_{n}}] : monic {\displaystyle q}-Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicqHahn http://drmf.wmflabs.org/wiki/Definition:monicqHahn]
-[http://drmf.wmflabs.org/wiki/Definition:qHahn {\displaystyle Q_{n}}] : {\displaystyle q}-Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:qHahn http://drmf.wmflabs.org/wiki/Definition:qHahn]
-[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1] +[http://drmf.wmflabs.org/wiki/Definition:monicqHahn {\displaystyle {\widehat Q}_{n}}] : monic -Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicqHahn http://drmf.wmflabs.org/wiki/Definition:monicqHahn]
+[http://drmf.wmflabs.org/wiki/Definition:qHahn {\displaystyle Q_{n}}] : -Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:qHahn http://drmf.wmflabs.org/wiki/Definition:qHahn]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]

<< [[Definition:qKrawtchouk|Definition:qKrawtchouk]]
[[Main_Page|Main Page]]
@@ -3618,8 +3592,7 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:dualqHahn {\displaystyle R_{n}}] : dual {\displaystyle q}-Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:dualqHahn http://drmf.wmflabs.org/wiki/Definition:dualqHahn]
-[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1] +[http://drmf.wmflabs.org/wiki/Definition:dualqHahn {\displaystyle R_{n}}] : dual -Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:dualqHahn http://drmf.wmflabs.org/wiki/Definition:dualqHahn]

<< [[Definition:monicqHahn|Definition:monicqHahn]]
[[Main_Page|Main Page]]
@@ -3651,8 +3624,8 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:dualqKrawtchouk {\displaystyle K_{n}}] : dual {\displaystyle q}-Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:dualqKrawtchouk http://drmf.wmflabs.org/wiki/Definition:dualqKrawtchouk]
-[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1] +[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or -hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
+[http://drmf.wmflabs.org/wiki/Definition:dualqKrawtchouk {\displaystyle K_{n}}] : dual -Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:dualqKrawtchouk http://drmf.wmflabs.org/wiki/Definition:dualqKrawtchouk]

<< [[Definition:dualqHahn|Definition:dualqHahn]]
[[Main_Page|Main Page]]
@@ -3683,8 +3656,8 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:littleqLaguerre {\displaystyle p_{n}}] : little {\displaystyle q}-Laguerre / Wall polynomial : [http://drmf.wmflabs.org/wiki/Definition:littleqLaguerre http://drmf.wmflabs.org/wiki/Definition:littleqLaguerre]
-[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1] +[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or -hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
+[http://drmf.wmflabs.org/wiki/Definition:littleqLaguerre {\displaystyle p_{n}}] : little -Laguerre / Wall polynomial : [http://drmf.wmflabs.org/wiki/Definition:littleqLaguerre http://drmf.wmflabs.org/wiki/Definition:littleqLaguerre]

<< [[Definition:dualqKrawtchouk|Definition:dualqKrawtchouk]]
[[Main_Page|Main Page]]
@@ -3718,12 +3691,10 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:qsinKLS {\displaystyle \mathrm{sin}_{q}}] : {\displaystyle q}-analogue of the {\displaystyle \sin} function used in KLS: {\displaystyle \sin_q} : [http://drmf.wmflabs.org/wiki/Definition:qsinKLS http://drmf.wmflabs.org/wiki/Definition:qsinKLS]
-[http://dlmf.nist.gov/4.14#E1 {\displaystyle \mathrm{sin}}] : sine function : [http://dlmf.nist.gov/4.14#E1 http://dlmf.nist.gov/4.14#E1]
-[http://drmf.wmflabs.org/wiki/Definition:qexpKLS {\displaystyle \mathrm{e}_{q}}] : {\displaystyle q}-analogue of the {\displaystyle \exp} function used in KLS: {\displaystyle \mathrm{e}_{q}} : [http://drmf.wmflabs.org/wiki/Definition:qexpKLS http://drmf.wmflabs.org/wiki/Definition:qexpKLS]
+[http://drmf.wmflabs.org/wiki/Definition:qexpKLS {\displaystyle \mathrm{e}_{q}}] : -analogue of the function used in KLS: : [http://drmf.wmflabs.org/wiki/Definition:qexpKLS http://drmf.wmflabs.org/wiki/Definition:qexpKLS]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]
+[http://drmf.wmflabs.org/wiki/Definition:qsinKLS {\displaystyle \mathrm{sin}_{q}}] : -analogue of the function used in KLS: : [http://drmf.wmflabs.org/wiki/Definition:qsinKLS http://drmf.wmflabs.org/wiki/Definition:qsinKLS]
[http://dlmf.nist.gov/1.9.i {\displaystyle \mathrm{i}}] : imaginary unit : [http://dlmf.nist.gov/1.9.i http://dlmf.nist.gov/1.9.i]
-[http://drmf.wmflabs.org/wiki/Definition:sum {\displaystyle \Sigma}] : sum : [http://drmf.wmflabs.org/wiki/Definition:sum http://drmf.wmflabs.org/wiki/Definition:sum]
-[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1]
<< [[Definition:littleqLaguerre|Definition:littleqLaguerre]]
[[Main_Page|Main Page]]
@@ -3757,13 +3728,11 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:qSinKLS {\displaystyle \mathrm{Sin}_{q}}] : {\displaystyle q}-analogue of the {\displaystyle \sin} function used in KLS: {\displaystyle \mathrm{Sin}_q} : [http://drmf.wmflabs.org/wiki/Definition:qSinKLS http://drmf.wmflabs.org/wiki/Definition:qSinKLS]
-[http://dlmf.nist.gov/4.14#E1 {\displaystyle \mathrm{sin}}] : sine function : [http://dlmf.nist.gov/4.14#E1 http://dlmf.nist.gov/4.14#E1]
-[http://drmf.wmflabs.org/wiki/Definition:qExpKLS {\displaystyle \mathrm{E}_{q}}] : {\displaystyle q}-analogue of the {\displaystyle \exp} function used in KLS: {\displaystyle \mathrm{E}_{q}} : [http://drmf.wmflabs.org/wiki/Definition:qExpKLS http://drmf.wmflabs.org/wiki/Definition:qExpKLS]
+[http://drmf.wmflabs.org/wiki/Definition:qExpKLS {\displaystyle \mathrm{E}_{q}}] : -analogue of the function used in KLS: : [http://drmf.wmflabs.org/wiki/Definition:qExpKLS http://drmf.wmflabs.org/wiki/Definition:qExpKLS]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]
+[http://dlmf.nist.gov/1.2#E1 {\displaystyle \binom{n}{k}}] : binomial coefficient : [http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/1.2#E1]
+[http://drmf.wmflabs.org/wiki/Definition:qSinKLS {\displaystyle \mathrm{Sin}_{q}}] : -analogue of the function used in KLS: : [http://drmf.wmflabs.org/wiki/Definition:qSinKLS http://drmf.wmflabs.org/wiki/Definition:qSinKLS]
[http://dlmf.nist.gov/1.9.i {\displaystyle \mathrm{i}}] : imaginary unit : [http://dlmf.nist.gov/1.9.i http://dlmf.nist.gov/1.9.i]
-[http://drmf.wmflabs.org/wiki/Definition:sum {\displaystyle \Sigma}] : sum : [http://drmf.wmflabs.org/wiki/Definition:sum http://drmf.wmflabs.org/wiki/Definition:sum]
-[http://dlmf.nist.gov/1.2#E1 {\displaystyle \binom{n}{k}}] : binomial coefficient : [http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/1.2#E1] [http://dlmf.nist.gov/26.3#SS1.p1 http://dlmf.nist.gov/26.3#SS1.p1]
-[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1]
<< [[Definition:qsinKLS|Definition:qsinKLS]]
[[Main_Page|Main Page]]
@@ -3797,13 +3766,11 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:qCosKLS {\displaystyle \mathrm{Cos}_{q}}] : {\displaystyle q}-analogue of the {\displaystyle \cos} function used in KLS: {\displaystyle \mathrm{Cos}_{q}} : [http://drmf.wmflabs.org/wiki/Definition:qCosKLS http://drmf.wmflabs.org/wiki/Definition:qCosKLS]
-[http://dlmf.nist.gov/4.14#E2 {\displaystyle \mathrm{cos}}] : cosine function : [http://dlmf.nist.gov/4.14#E2 http://dlmf.nist.gov/4.14#E2]
-[http://drmf.wmflabs.org/wiki/Definition:qExpKLS {\displaystyle \mathrm{E}_{q}}] : {\displaystyle q}-analogue of the {\displaystyle \exp} function used in KLS: {\displaystyle \mathrm{E}_{q}} : [http://drmf.wmflabs.org/wiki/Definition:qExpKLS http://drmf.wmflabs.org/wiki/Definition:qExpKLS]
+[http://drmf.wmflabs.org/wiki/Definition:qCosKLS {\displaystyle \mathrm{Cos}_{q}}] : -analogue of the function used in KLS: : [http://drmf.wmflabs.org/wiki/Definition:qCosKLS http://drmf.wmflabs.org/wiki/Definition:qCosKLS]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]
+[http://dlmf.nist.gov/1.2#E1 {\displaystyle \binom{n}{k}}] : binomial coefficient : [http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/1.2#E1]
[http://dlmf.nist.gov/1.9.i {\displaystyle \mathrm{i}}] : imaginary unit : [http://dlmf.nist.gov/1.9.i http://dlmf.nist.gov/1.9.i]
-[http://drmf.wmflabs.org/wiki/Definition:sum {\displaystyle \Sigma}] : sum : [http://drmf.wmflabs.org/wiki/Definition:sum http://drmf.wmflabs.org/wiki/Definition:sum]
-[http://dlmf.nist.gov/1.2#E1 {\displaystyle \binom{n}{k}}] : binomial coefficient : [http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/1.2#E1] [http://dlmf.nist.gov/26.3#SS1.p1 http://dlmf.nist.gov/26.3#SS1.p1]
-[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1] +[http://drmf.wmflabs.org/wiki/Definition:qExpKLS {\displaystyle \mathrm{E}_{q}}] : -analogue of the function used in KLS: : [http://drmf.wmflabs.org/wiki/Definition:qExpKLS http://drmf.wmflabs.org/wiki/Definition:qExpKLS]

<< [[Definition:qSinKLS|Definition:qSinKLS]]
[[Main_Page|Main Page]]
@@ -3835,9 +3802,9 @@ These are defined by == Symbols List == +[http://dlmf.nist.gov/16.2#E1 {\displaystyle {{}_{p}F_{q}}}] : generalized hypergeometric function : [http://dlmf.nist.gov/16.2#E1 http://dlmf.nist.gov/16.2#E1]
[http://dlmf.nist.gov/18.25#T1.t1.r2 {\displaystyle W_{n}}] : Wilson polynomial : [http://dlmf.nist.gov/18.25#T1.t1.r2 http://dlmf.nist.gov/18.25#T1.t1.r2]
[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii]
-[http://dlmf.nist.gov/16.2#E1 {\displaystyle {{}_{p}F_{q}}}] : generalized hypergeometric function : [http://dlmf.nist.gov/16.2#E1 http://dlmf.nist.gov/16.2#E1]
<< [[Definition:qCosKLS|Definition:qCosKLS]]
[[Main_Page|Main Page]]
@@ -3873,8 +3840,8 @@ with N a nonnegative integer. == Symbols List == +[http://dlmf.nist.gov/16.2#E1 {\displaystyle {{}_{p}F_{q}}}] : generalized hypergeometric function : [http://dlmf.nist.gov/16.2#E1 http://dlmf.nist.gov/16.2#E1]
[http://dlmf.nist.gov/18.25#T1.t1.r4 {\displaystyle R_{n}}] : Racah polynomial : [http://dlmf.nist.gov/18.25#T1.t1.r4 http://dlmf.nist.gov/18.25#T1.t1.r4]
-[http://dlmf.nist.gov/16.2#E1 {\displaystyle {{}_{p}F_{q}}}] : generalized hypergeometric function : [http://dlmf.nist.gov/16.2#E1 http://dlmf.nist.gov/16.2#E1]
<< [[Definition:Wilson|Definition:Wilson]]
[[Main_Page|Main Page]]
@@ -3906,9 +3873,9 @@ These are defined by == Symbols List == +[http://dlmf.nist.gov/16.2#E1 {\displaystyle {{}_{p}F_{q}}}] : generalized hypergeometric function : [http://dlmf.nist.gov/16.2#E1 http://dlmf.nist.gov/16.2#E1]
[http://dlmf.nist.gov/18.25#T1.t1.r3 {\displaystyle S_{n}}] : continuous dual Hahn polynomial : [http://dlmf.nist.gov/18.25#T1.t1.r3 http://dlmf.nist.gov/18.25#T1.t1.r3]
[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii]
-[http://dlmf.nist.gov/16.2#E1 {\displaystyle {{}_{p}F_{q}}}] : generalized hypergeometric function : [http://dlmf.nist.gov/16.2#E1 http://dlmf.nist.gov/16.2#E1]
<< [[Definition:Racah|Definition:Racah]]
[[Main_Page|Main Page]]
@@ -3941,9 +3908,9 @@ These are defined by == Symbols List == -[http://dlmf.nist.gov/18.19#P2.p1 {\displaystyle p_{n}}] : continuous Hahn polynomial : [http://dlmf.nist.gov/18.19#P2.p1 http://dlmf.nist.gov/18.19#P2.p1]
+[http://dlmf.nist.gov/16.2#E1 {\displaystyle {{}_{p}F_{q}}}] : generalized hypergeometric function : [http://dlmf.nist.gov/16.2#E1 http://dlmf.nist.gov/16.2#E1]
[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii]
-[http://dlmf.nist.gov/16.2#E1 {\displaystyle {{}_{p}F_{q}}}] : generalized hypergeometric function : [http://dlmf.nist.gov/16.2#E1 http://dlmf.nist.gov/16.2#E1] +[http://dlmf.nist.gov/18.19#P2.p1 {\displaystyle p_{n}}] : continuous Hahn polynomial : [http://dlmf.nist.gov/18.19#P2.p1 http://dlmf.nist.gov/18.19#P2.p1]

<< [[Definition:ctsdualHahn|Definition:ctsdualHahn]]
[[Main_Page|Main Page]]
@@ -3974,8 +3941,8 @@ These are defined by == Symbols List == +[http://dlmf.nist.gov/16.2#E1 {\displaystyle {{}_{p}F_{q}}}] : generalized hypergeometric function : [http://dlmf.nist.gov/16.2#E1 http://dlmf.nist.gov/16.2#E1]
[http://dlmf.nist.gov/18.19#T1.t1.r3 {\displaystyle Q_{n}}] : Hahn polynomial : [http://dlmf.nist.gov/18.19#T1.t1.r3 http://dlmf.nist.gov/18.19#T1.t1.r3]
-[http://dlmf.nist.gov/16.2#E1 {\displaystyle {{}_{p}F_{q}}}] : generalized hypergeometric function : [http://dlmf.nist.gov/16.2#E1 http://dlmf.nist.gov/16.2#E1]
<< [[Definition:ctsHahn|Definition:ctsHahn]]
[[Main_Page|Main Page]]
@@ -4008,8 +3975,8 @@ These are defined by == Symbols List == +[http://dlmf.nist.gov/16.2#E1 {\displaystyle {{}_{p}F_{q}}}] : generalized hypergeometric function : [http://dlmf.nist.gov/16.2#E1 http://dlmf.nist.gov/16.2#E1]
[http://dlmf.nist.gov/18.25#T1.t1.r5 {\displaystyle R_{n}}] : dual Hahn polynomial : [http://dlmf.nist.gov/18.25#T1.t1.r5 http://dlmf.nist.gov/18.25#T1.t1.r5]
-[http://dlmf.nist.gov/16.2#E1 {\displaystyle {{}_{p}F_{q}}}] : generalized hypergeometric function : [http://dlmf.nist.gov/16.2#E1 http://dlmf.nist.gov/16.2#E1]
<< [[Definition:Hahn|Definition:Hahn]]
[[Main_Page|Main Page]]
@@ -4041,8 +4008,8 @@ These are defined by == Symbols List == -[http://dlmf.nist.gov/18.28#E19 {\displaystyle R_{n}}] : {\displaystyle q}-Racah polynomial : [http://dlmf.nist.gov/18.28#E19 http://dlmf.nist.gov/18.28#E19]
-[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1] +[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or -hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
+[http://dlmf.nist.gov/18.28#E19 {\displaystyle R_{n}}] : -Racah polynomial : [http://dlmf.nist.gov/18.28#E19 http://dlmf.nist.gov/18.28#E19]

<< [[Definition:dualHahn|Definition:dualHahn]]
[[Main_Page|Main Page]]
@@ -4074,9 +4041,9 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:normctsdualqHahnptilde {\displaystyle {\tilde p}_{n}}] : normalized continuous dual {\displaystyle q}-Hahn polynomial {\displaystyle {\tilde p}} : [http://drmf.wmflabs.org/wiki/Definition:normctsdualqHahnptilde http://drmf.wmflabs.org/wiki/Definition:normctsdualqHahnptilde]
-[http://drmf.wmflabs.org/wiki/Definition:ctsdualqHahn {\displaystyle p_{n}}] : continuous dual {\displaystyle q}-Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:ctsdualqHahn http://drmf.wmflabs.org/wiki/Definition:ctsdualqHahn]
-[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1] +[http://drmf.wmflabs.org/wiki/Definition:ctsdualqHahn {\displaystyle p_{n}}] : continuous dual -Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:ctsdualqHahn http://drmf.wmflabs.org/wiki/Definition:ctsdualqHahn]
+[http://drmf.wmflabs.org/wiki/Definition:normctsdualqHahnptilde {\displaystyle {\tilde p}_{n}}] : normalized continuous dual -Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:normctsdualqHahnptilde http://drmf.wmflabs.org/wiki/Definition:normctsdualqHahnptilde]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]

<< [[Definition:qRacah|Definition:qRacah]]
[[Main_Page|Main Page]]
@@ -4109,10 +4076,9 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:normctsqHahnptilde {\displaystyle {\tilde p}_{n}}] : normalized continuous {\displaystyle q}-Hahn polynomial {\displaystyle {\tilde p}} : [http://drmf.wmflabs.org/wiki/Definition:normctsqHahnptilde http://drmf.wmflabs.org/wiki/Definition:normctsqHahnptilde]
-[http://dlmf.nist.gov/4.2.E11 {\displaystyle \mathrm{e}}] : the base of the natural logarithm : [http://dlmf.nist.gov/4.2.E11 http://dlmf.nist.gov/4.2.E11]
-[http://drmf.wmflabs.org/wiki/Definition:ctsqHahn {\displaystyle p_{n}}] : continuous {\displaystyle q}-Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:ctsqHahn http://drmf.wmflabs.org/wiki/Definition:ctsqHahn]
-[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1] +[http://drmf.wmflabs.org/wiki/Definition:normctsqHahnptilde {\displaystyle {\tilde p}_{n}}] : normalized continuous -Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:normctsqHahnptilde http://drmf.wmflabs.org/wiki/Definition:normctsqHahnptilde]
+[http://drmf.wmflabs.org/wiki/Definition:ctsqHahn {\displaystyle p_{n}}] : continuous -Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:ctsqHahn http://drmf.wmflabs.org/wiki/Definition:ctsqHahn]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]

<< [[Definition:normctsdualqHahnptilde|Definition:normctsdualqHahnptilde]]
[[Main_Page|Main Page]]
@@ -4143,8 +4109,8 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:qHahn {\displaystyle Q_{n}}] : {\displaystyle q}-Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:qHahn http://drmf.wmflabs.org/wiki/Definition:qHahn]
-[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1] +[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or -hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
+[http://drmf.wmflabs.org/wiki/Definition:qHahn {\displaystyle Q_{n}}] : -Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:qHahn http://drmf.wmflabs.org/wiki/Definition:qHahn]

<< [[Definition:normctsqHahnptilde|Definition:normctsqHahnptilde]]
[[Main_Page|Main Page]]
@@ -4177,10 +4143,9 @@ These are defined by == Symbols List == +[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or -hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]
[http://drmf.wmflabs.org/wiki/Definition:AlSalamChihara {\displaystyle Q_{n}}] : Al-Salam-Chihara polynomial : [http://drmf.wmflabs.org/wiki/Definition:AlSalamChihara http://drmf.wmflabs.org/wiki/Definition:AlSalamChihara]
-[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1]
-[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
-[http://dlmf.nist.gov/4.2.E11 {\displaystyle \mathrm{e}}] : the base of the natural logarithm : [http://dlmf.nist.gov/4.2.E11 http://dlmf.nist.gov/4.2.E11]
<< [[Definition:qHahn|Definition:qHahn]]
[[Main_Page|Main Page]]
@@ -4213,9 +4178,9 @@ These are defined by == Symbols List == -[http://dlmf.nist.gov/23.1 {\displaystyle Q_{n}}] : {\displaystyle q}-inverse Al-Salam-Chihara polynomial : [http://dlmf.nist.gov/23.1 http://dlmf.nist.gov/23.1]
-[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1]
-[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1] +[http://dlmf.nist.gov/23.1 {\displaystyle Q_{n}}] : -inverse Al-Salam-Chihara polynomial : [http://dlmf.nist.gov/23.1 http://dlmf.nist.gov/23.1]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or -hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]

<< [[Definition:AlSalamChihara|Definition:AlSalamChihara]]
[[Main_Page|Main Page]]
@@ -4247,7 +4212,7 @@ x\monicqinvAlSalamChihara{n}@@{x}{a}{b}{q^{-1}=:\monicqinvAlSalamChihara{n+1}@@{ == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:monicqinvAlSalamChihara {\displaystyle {\widehat Q}_{n}}] : monic {\displaystyle q}-inverse Al-Salam-Chihara polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicqinvAlSalamChihara http://drmf.wmflabs.org/wiki/Definition:monicqinvAlSalamChihara] +[http://drmf.wmflabs.org/wiki/Definition:monicqinvAlSalamChihara {\displaystyle {\widehat Q}_{n}}] : monic -inverse Al-Salam-Chihara polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicqinvAlSalamChihara http://drmf.wmflabs.org/wiki/Definition:monicqinvAlSalamChihara]

<< [[Definition:qinvAlSalamChihara|Definition:qinvAlSalamChihara]]
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@@ -4280,9 +4245,9 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:monicAffqKrawtchouk {\displaystyle \widehat{K}^{\mathrm{Aff}}_{n}}] : monic affine {\displaystyle q}-Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicAffqKrawtchouk http://drmf.wmflabs.org/wiki/Definition:monicAffqKrawtchouk]
-[http://drmf.wmflabs.org/wiki/Definition:AffqKrawtchouk {\displaystyle K^{\mathrm{Aff}}_{n}}] : affine {\displaystyle q}-Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:AffqKrawtchouk http://drmf.wmflabs.org/wiki/Definition:AffqKrawtchouk]
-[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1] +[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]
+[http://drmf.wmflabs.org/wiki/Definition:AffqKrawtchouk {\displaystyle K^{\mathrm{Aff}}_{n}}] : affine -Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:AffqKrawtchouk http://drmf.wmflabs.org/wiki/Definition:AffqKrawtchouk]
+[http://drmf.wmflabs.org/wiki/Definition:monicAffqKrawtchouk {\displaystyle \widehat{K}^{\mathrm{Aff}}_{n}}] : monic affine -Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicAffqKrawtchouk http://drmf.wmflabs.org/wiki/Definition:monicAffqKrawtchouk]

<< [[Definition:monicqinvAlSalamChihara|Definition:monicqinvAlSalamChihara]]
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@@ -4314,10 +4279,9 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:qMeixnerPollaczek {\displaystyle P_{n}}] : {\displaystyle q}-Meixner-Pollaczek polynomial : [http://drmf.wmflabs.org/wiki/Definition:qMeixnerPollaczek http://drmf.wmflabs.org/wiki/Definition:qMeixnerPollaczek]
-[http://dlmf.nist.gov/4.2.E11 {\displaystyle \mathrm{e}}] : the base of the natural logarithm : [http://dlmf.nist.gov/4.2.E11 http://dlmf.nist.gov/4.2.E11]
-[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1]
-[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1] +[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or -hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]
+[http://drmf.wmflabs.org/wiki/Definition:qMeixnerPollaczek {\displaystyle P_{n}}] : -Meixner-Pollaczek polynomial : [http://drmf.wmflabs.org/wiki/Definition:qMeixnerPollaczek http://drmf.wmflabs.org/wiki/Definition:qMeixnerPollaczek]

<< [[Definition:monicAffqKrawtchouk|Definition:monicAffqKrawtchouk]]
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@@ -4350,8 +4314,8 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:qtmqKrawtchouk {\displaystyle K^{\mathrm{qtm}}_{n}}] : quantum {\displaystyle q}-Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:qtmqKrawtchouk http://drmf.wmflabs.org/wiki/Definition:qtmqKrawtchouk]
-[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1] +[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or -hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
+[http://drmf.wmflabs.org/wiki/Definition:qtmqKrawtchouk {\displaystyle K^{\mathrm{qtm}}_{n}}] : quantum -Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:qtmqKrawtchouk http://drmf.wmflabs.org/wiki/Definition:qtmqKrawtchouk]

<< [[Definition:qMeixnerPollaczek|Definition:qMeixnerPollaczek]]
[[Main_Page|Main Page]]
@@ -4387,9 +4351,9 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:qLaguerre {\displaystyle L_n^{(\alpha)}}] : {\displaystyle q}-Laguerre polynomial : [http://drmf.wmflabs.org/wiki/Definition:qLaguerre http://drmf.wmflabs.org/wiki/Definition:qLaguerre]
-[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1]
-[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1] +[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or -hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]
+[http://drmf.wmflabs.org/wiki/Definition:qLaguerre {\displaystyle L_n^{(\alpha)}}] : -Laguerre polynomial : [http://drmf.wmflabs.org/wiki/Definition:qLaguerre http://drmf.wmflabs.org/wiki/Definition:qLaguerre]

<< [[Definition:qtmqKrawtchouk|Definition:qtmqKrawtchouk]]
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@@ -4421,9 +4385,8 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:ctsqHermite {\displaystyle H_{n}}] : continuous {\displaystyle q}-Hermite polynomial : [http://drmf.wmflabs.org/wiki/Definition:ctsqHermite http://drmf.wmflabs.org/wiki/Definition:ctsqHermite]
-[http://dlmf.nist.gov/4.2.E11 {\displaystyle \mathrm{e}}] : the base of the natural logarithm : [http://dlmf.nist.gov/4.2.E11 http://dlmf.nist.gov/4.2.E11]
-[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1] +[http://drmf.wmflabs.org/wiki/Definition:ctsqHermite {\displaystyle H_{n}}] : continuous -Hermite polynomial : [http://drmf.wmflabs.org/wiki/Definition:ctsqHermite http://drmf.wmflabs.org/wiki/Definition:ctsqHermite]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or -hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]

<< [[Definition:qLaguerre|Definition:qLaguerre]]
[[Main_Page|Main Page]]
@@ -4455,9 +4418,9 @@ These are defined by == Symbols List == +[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or -hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]
[http://drmf.wmflabs.org/wiki/Definition:StieltjesWigert {\displaystyle S_{n}}] : Stieltjes-Wigert polynomial : [http://drmf.wmflabs.org/wiki/Definition:StieltjesWigert http://drmf.wmflabs.org/wiki/Definition:StieltjesWigert]
-[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1]
-[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
<< [[Definition:ctsqHermite|Definition:ctsqHermite]]
[[Main_Page|Main Page]]
@@ -4489,8 +4452,8 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:discrqHermiteI {\displaystyle h_{n}}] : discrete {\displaystyle q}-Hermite I polynomial : [http://drmf.wmflabs.org/wiki/Definition:discrqHermiteI http://drmf.wmflabs.org/wiki/Definition:discrqHermiteI]
-[http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzI {\displaystyle U^{(n)}_{\alpha}}] : Al-Salam-Carlitz I polynomial : [http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzI http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzI] +[http://drmf.wmflabs.org/wiki/Definition:discrqHermiteI {\displaystyle h_{n}}] : discrete -Hermite I polynomial : [http://drmf.wmflabs.org/wiki/Definition:discrqHermiteI http://drmf.wmflabs.org/wiki/Definition:discrqHermiteI]
+[http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzI {\displaystyle U^{(n)}_{\alpha}}] : Al-Salam-Carlitz I polynomial : [http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzI http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzI]

<< [[Definition:StieltjesWigert|Definition:StieltjesWigert]]
[[Main_Page|Main Page]]
@@ -4522,8 +4485,8 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:discrqHermiteII {\displaystyle \tilde{h}_{n}}] : discrete {\displaystyle q}-Hermite II polynomial : [http://drmf.wmflabs.org/wiki/Definition:discrqHermiteII http://drmf.wmflabs.org/wiki/Definition:discrqHermiteII]
-[http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzII {\displaystyle V^{(n)}_{\alpha}}] : Al-Salam-Carlitz II polynomial : [http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzII http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzII] +[http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzII {\displaystyle V^{(n)}_{\alpha}}] : Al-Salam-Carlitz II polynomial : [http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzII http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzII]
+[http://drmf.wmflabs.org/wiki/Definition:discrqHermiteII {\displaystyle \tilde{h}_{n}}] : discrete -Hermite II polynomial : [http://drmf.wmflabs.org/wiki/Definition:discrqHermiteII http://drmf.wmflabs.org/wiki/Definition:discrqHermiteII]

<< [[Definition:discrqHermiteI|Definition:discrqHermiteI]]
[[Main_Page|Main Page]]
@@ -4563,9 +4526,7 @@ Equation (2), Section 2.21 of [[Bibliography#FI|'''FI''']]. == Symbols List == [http://drmf.wmflabs.org/wiki/Definition:StieltjesConstants {\displaystyle \gamma_{n}}] : Stieltjes constants : [http://drmf.wmflabs.org/wiki/Definition:StieltjesConstants http://drmf.wmflabs.org/wiki/Definition:StieltjesConstants]
-[http://drmf.wmflabs.org/wiki/Definition:sum {\displaystyle \Sigma}] : sum : [http://drmf.wmflabs.org/wiki/Definition:sum http://drmf.wmflabs.org/wiki/Definition:sum]
[http://dlmf.nist.gov/4.2#E2 {\displaystyle \mathrm{ln}}] : principal branch of logarithm function : [http://dlmf.nist.gov/4.2#E2 http://dlmf.nist.gov/4.2#E2]
-[http://dlmf.nist.gov/5.2#E3 {\displaystyle \gamma}] : Euler's (Euler-Mascheroni) constant : [http://dlmf.nist.gov/5.2#E3 http://dlmf.nist.gov/5.2#E3]
<< [[Definition:discrqHermiteII|Definition:discrqHermiteII]]
[[Main_Page|Main Page]]
@@ -4599,9 +4560,8 @@ These are defined by == Symbols List == -[p}$ {\displaystyle S_{n}] : Nielsen generalized polylogarithm function : [p}$ p}$] [http://drmf.wmflabs.org/wiki/Definition:PolylogarithmS http://drmf.wmflabs.org/wiki/Definition:PolylogarithmS]
-[http://dlmf.nist.gov/1.4#iv {\displaystyle \int}] : integral : [http://dlmf.nist.gov/1.4#iv http://dlmf.nist.gov/1.4#iv]
-[http://dlmf.nist.gov/4.2#E2 {\displaystyle \mathrm{ln}}] : principal branch of logarithm function : [http://dlmf.nist.gov/4.2#E2 http://dlmf.nist.gov/4.2#E2] +[http://dlmf.nist.gov/4.2#E2 {\displaystyle \mathrm{ln}}] : principal branch of logarithm function : [http://dlmf.nist.gov/4.2#E2 http://dlmf.nist.gov/4.2#E2]
+[http://drmf.wmflabs.org/wiki/Definition:PolylogarithmS {\displaystyle S_{n}] : Nielsen generalized polylogarithm function : [http://drmf.wmflabs.org/wiki/Definition:PolylogarithmS http://drmf.wmflabs.org/wiki/Definition:PolylogarithmS]

<< [[Definition:StieltjesConstants|Definition:StieltjesConstants]]
[[Main_Page|Main Page]]
@@ -4636,8 +4596,7 @@ These are defined by == Symbols List == -[http://dlmf.nist.gov/25.11#E33 {\displaystyle H_{n}}] : Harmonic number : [http://dlmf.nist.gov/25.11#E33 http://dlmf.nist.gov/25.11#E33]
-[http://drmf.wmflabs.org/wiki/Definition:sum {\displaystyle \Sigma}] : sum : [http://drmf.wmflabs.org/wiki/Definition:sum http://drmf.wmflabs.org/wiki/Definition:sum] +[http://drmf.wmflabs.org/wiki/Definition:HarmonicNumber {\displaystyle H_{n}^{r}}] : Harmonic number : [http://drmf.wmflabs.org/wiki/Definition:HarmonicNumber http://drmf.wmflabs.org/wiki/Definition:HarmonicNumber]

<< [[Definition:PolylogarithmS|Definition:PolylogarithmS]]
[[Main_Page|Main Page]]
@@ -4671,7 +4630,7 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:LucasL {\displaystyle L_{n}}] : Lucas polynomial : [http://drmf.wmflabs.org/wiki/Definition:LucasL http://drmf.wmflabs.org/wiki/Definition:LucasL] +[http://drmf.wmflabs.org/wiki/Definition:LucasL {\displaystyle L_{n}}] : Lucas polynomial : [http://drmf.wmflabs.org/wiki/Definition:LucasL http://drmf.wmflabs.org/wiki/Definition:LucasL]

<< [[Definition:HarmonicNumber|Definition:HarmonicNumber]]
[[Main_Page|Main Page]]
@@ -4706,7 +4665,6 @@ These are defined by == Symbols List == [http://drmf.wmflabs.org/wiki/Definition:HurwitzLerchPhi {\displaystyle \Phi}] : Hurwitz-Lerch's transcendent : [http://drmf.wmflabs.org/wiki/Definition:HurwitzLerchPhi http://drmf.wmflabs.org/wiki/Definition:HurwitzLerchPhi]
-[http://drmf.wmflabs.org/wiki/Definition:sum {\displaystyle \Sigma}] : sum : [http://drmf.wmflabs.org/wiki/Definition:sum http://drmf.wmflabs.org/wiki/Definition:sum]
<< [[Definition:LucasL|Definition:LucasL]]
[[Main_Page|Main Page]]
@@ -4739,7 +4697,7 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:GoldenRatio {\displaystyle \phi}] : The golden ratio : [http://drmf.wmflabs.org/wiki/Definition:GoldenRatio http://drmf.wmflabs.org/wiki/Definition:GoldenRatio] +
<< [[Definition:HurwitzLerchPhi|Definition:HurwitzLerchPhi]]
[[Main_Page|Main Page]]
@@ -4773,8 +4731,8 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:WhitPsi {\displaystyle \Psi_{\alpha,\beta}}] : Whittaker function {\displaystyle \Psi_{\alpha;\beta}} : [http://drmf.wmflabs.org/wiki/Definition:WhitPsi http://drmf.wmflabs.org/wiki/Definition:WhitPsi]
-[http://dlmf.nist.gov/13.14#E3 {\displaystyle W_{\kappa,\mu}}] : Whittaker function {\displaystyle W_{\kappa;\mu}} : [http://dlmf.nist.gov/13.14#E3 http://dlmf.nist.gov/13.14#E3] +[http://dlmf.nist.gov/13.14#E3 {\displaystyle W_{\kappa,\mu}}] : Whittaker function : [http://dlmf.nist.gov/13.14#E3 http://dlmf.nist.gov/13.14#E3]
+[http://drmf.wmflabs.org/wiki/Definition:WhitPsi {\displaystyle \Psi_{\alpha,\beta}}] : Whittaker function : [http://drmf.wmflabs.org/wiki/Definition:WhitPsi http://drmf.wmflabs.org/wiki/Definition:WhitPsi]

<< [[Definition:GoldenRatio|Definition:GoldenRatio]]
[[Main_Page|Main Page]]
@@ -4807,8 +4765,7 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:GompertzConstant {\displaystyle G}] : Gompertz's constant : [http://drmf.wmflabs.org/wiki/Definition:GompertzConstant http://drmf.wmflabs.org/wiki/Definition:GompertzConstant]
-[http://dlmf.nist.gov/1.4#iv {\displaystyle \int}] : integral : [http://dlmf.nist.gov/1.4#iv http://dlmf.nist.gov/1.4#iv] +
<< [[Definition:WhitPsi|Definition:WhitPsi]]
[[Main_Page|Main Page]]
@@ -4841,10 +4798,7 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:RabbitConstant {\displaystyle \rho}] : rabbit constant : [http://drmf.wmflabs.org/wiki/Definition:RabbitConstant http://drmf.wmflabs.org/wiki/Definition:RabbitConstant]
-[http://drmf.wmflabs.org/wiki/Definition:sum {\displaystyle \Sigma}] : sum : [http://drmf.wmflabs.org/wiki/Definition:sum http://drmf.wmflabs.org/wiki/Definition:sum]
[http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r16 {\displaystyle \left\lfloor a\right\rfloor}] : floor : [http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r16 http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r16]
-[http://drmf.wmflabs.org/wiki/Definition:GoldenRatio {\displaystyle \phi}] : The golden ratio : [http://drmf.wmflabs.org/wiki/Definition:GoldenRatio http://drmf.wmflabs.org/wiki/Definition:GoldenRatio]
<< [[Definition:GompertzConstant|Definition:GompertzConstant]]
[[Main_Page|Main Page]]
@@ -4877,7 +4831,7 @@ These are defined by == Symbols List == -[http://dlmf.nist.gov/26.11#E5 {\displaystyle F_{n}}] : Fibonacci number : [http://dlmf.nist.gov/26.11#E5 http://dlmf.nist.gov/26.11#E5] +[http://dlmf.nist.gov/26.11#E5 {\displaystyle F_{n}}] : Fibonacci number : [http://dlmf.nist.gov/26.11#E5 http://dlmf.nist.gov/26.11#E5]

<< [[Definition:rabbitConstant|Definition:rabbitConstant]]
[[Main_Page|Main Page]]
@@ -4910,7 +4864,7 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:Fibonacci {\displaystyle F_{n}{(x)}}] : Fibonacci polynomial : [http://drmf.wmflabs.org/wiki/Definition:Fibonacci http://drmf.wmflabs.org/wiki/Definition:Fibonacci] +[http://drmf.wmflabs.org/wiki/Definition:Fibonacci {\displaystyle F_{n}{(x)}}] : Fibonacci polynomial : [http://drmf.wmflabs.org/wiki/Definition:Fibonacci http://drmf.wmflabs.org/wiki/Definition:Fibonacci]

<< [[Definition:FibonacciNumber|Definition:FibonacciNumber]]
[[Main_Page|Main Page]]
@@ -4947,9 +4901,9 @@ These are defined by == Symbols List == -[\theta\right)$ {\displaystyle P\left(x;\alpha}] : Regularized gamma function of incomplete gamma function : [\theta\right)$ \theta\right)$] [http://drmf.wmflabs.org/wiki/Definition:RegGamma http://drmf.wmflabs.org/wiki/Definition:RegGamma]
-[http://dlmf.nist.gov/8.2#E1 {\displaystyle \gamma}] : incomplete gamma function {\displaystyle \gamma} : [http://dlmf.nist.gov/8.2#E1 http://dlmf.nist.gov/8.2#E1]
-[http://dlmf.nist.gov/5.2#E1 {\displaystyle \Gamma}] : Euler's gamma function : [http://dlmf.nist.gov/5.2#E1 http://dlmf.nist.gov/5.2#E1] +[http://dlmf.nist.gov/5.2#E1 {\displaystyle \Gamma}] : Euler's gamma function : [http://dlmf.nist.gov/5.2#E1 http://dlmf.nist.gov/5.2#E1]
+[http://drmf.wmflabs.org/wiki/Definition:RegGamma {\displaystyle P\left(x;\alpha}] : Regularized gamma function of incomplete gamma function : [http://drmf.wmflabs.org/wiki/Definition:RegGamma http://drmf.wmflabs.org/wiki/Definition:RegGamma]
+[http://dlmf.nist.gov/8.2#E1 {\displaystyle \gamma}] : incomplete gamma function : [http://dlmf.nist.gov/8.2#E1 http://dlmf.nist.gov/8.2#E1]

<< [[Definition:Fibonacci|Definition:Fibonacci]]
[[Main_Page|Main Page]]
@@ -4986,9 +4940,9 @@ These are defined by == Symbols List == -[\theta\right)$ {\displaystyle Q\left(x;\alpha}] : Regularized gamma function of complementary incomplete gamma function : [\theta\right)$ \theta\right)$] [http://drmf.wmflabs.org/wiki/Definition:RegGamma http://drmf.wmflabs.org/wiki/Definition:RegGamma]
-[http://dlmf.nist.gov/8.2#E2 {\displaystyle \Gamma}] : incomplete gamma function {\displaystyle \Gamma} : [http://dlmf.nist.gov/8.2#E2 http://dlmf.nist.gov/8.2#E2]
-[http://dlmf.nist.gov/5.2#E1 {\displaystyle \Gamma}] : Euler's gamma function : [http://dlmf.nist.gov/5.2#E1 http://dlmf.nist.gov/5.2#E1] +[http://dlmf.nist.gov/5.2#E1 {\displaystyle \Gamma}] : Euler's gamma function : [http://dlmf.nist.gov/5.2#E1 http://dlmf.nist.gov/5.2#E1]
+[http://drmf.wmflabs.org/wiki/Definition:RegGamma {\displaystyle Q\left(x;\alpha}] : Regularized gamma function of complementary incomplete gamma function : [http://drmf.wmflabs.org/wiki/Definition:RegGamma http://drmf.wmflabs.org/wiki/Definition:RegGamma]
+[http://dlmf.nist.gov/8.2#E2 {\displaystyle \Gamma}] : incomplete gamma function : [http://dlmf.nist.gov/8.2#E2 http://dlmf.nist.gov/8.2#E2]

<< [[Definition:RegGammaP|Definition:RegGammaP]]
[[Main_Page|Main Page]]
@@ -5025,8 +4979,8 @@ These are defined by == Symbols List == -[0 {\displaystyle F\left(x}] : Gaussian distribution : [0 0] [1\right)$ 1\right)$] [http://drmf.wmflabs.org/wiki/Definition:GaussDist http://drmf.wmflabs.org/wiki/Definition:GaussDist]
-[http://dlmf.nist.gov/7.2#E1 {\displaystyle \mathrm{erf}}] : error function : [http://dlmf.nist.gov/7.2#E1 http://dlmf.nist.gov/7.2#E1] +[http://drmf.wmflabs.org/wiki/Definition:GaussDist {\displaystyle F\left(x}] : Gaussian distribution : [http://drmf.wmflabs.org/wiki/Definition:GaussDist http://drmf.wmflabs.org/wiki/Definition:GaussDist]
+[http://dlmf.nist.gov/7.2#E1 {\displaystyle \mathrm{erf}}] : error function : [http://dlmf.nist.gov/7.2#E1 http://dlmf.nist.gov/7.2#E1]

<< [[Definition:RegGammaQ|Definition:RegGammaQ]]
[[Main_Page|Main Page]]
@@ -5063,8 +5017,8 @@ These are defined by == Symbols List == -[0 {\displaystyle Q\left(x}] : Related to Gaussian distribution : [0 0] [1\right)$ 1\right)$] [http://drmf.wmflabs.org/wiki/Definition:GaussDist http://drmf.wmflabs.org/wiki/Definition:GaussDist]
-[http://dlmf.nist.gov/7.2#E2 {\displaystyle \mathrm{erfc}}] : complementary error function : [http://dlmf.nist.gov/7.2#E2 http://dlmf.nist.gov/7.2#E2] +[http://dlmf.nist.gov/7.2#E2 {\displaystyle \mathrm{erfc}}] : complementary error function : [http://dlmf.nist.gov/7.2#E2 http://dlmf.nist.gov/7.2#E2]
+[http://drmf.wmflabs.org/wiki/Definition:GaussDist {\displaystyle Q\left(x}] : Related to Gaussian distribution : [http://drmf.wmflabs.org/wiki/Definition:GaussDist http://drmf.wmflabs.org/wiki/Definition:GaussDist]

<< [[Definition:GaussDistF|Definition:GaussDistF]]
[[Main_Page|Main Page]]
@@ -5101,9 +5055,9 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:RamanujanTauTheta {\displaystyle \tau\theta{z}}] : Ramanujan tau theta function : [http://drmf.wmflabs.org/wiki/Definition:RamanujanTauTheta http://drmf.wmflabs.org/wiki/Definition:RamanujanTauTheta]
+[http://dlmf.nist.gov/5.2#E1 {\displaystyle \Gamma}] : Euler's gamma function : [http://dlmf.nist.gov/5.2#E1 http://dlmf.nist.gov/5.2#E1]
[http://dlmf.nist.gov/4.2#E2 {\displaystyle \mathrm{log}}] : principle branch of natural logarithm : [http://dlmf.nist.gov/4.2#E2 http://dlmf.nist.gov/4.2#E2]
-[http://dlmf.nist.gov/5.2#E1 {\displaystyle \Gamma}] : Euler's gamma function : [http://dlmf.nist.gov/5.2#E1 http://dlmf.nist.gov/5.2#E1] +[http://drmf.wmflabs.org/wiki/Definition:RamanujanTauTheta {\displaystyle \tau\theta{z}}] : Ramanujan tau theta function : [http://drmf.wmflabs.org/wiki/Definition:RamanujanTauTheta http://drmf.wmflabs.org/wiki/Definition:RamanujanTauTheta]

<< [[Definition:GaussDistQ|Definition:GaussDistQ]]
[[Main_Page|Main Page]]
@@ -5140,9 +5094,9 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:RiemannSiegelTheta {\displaystyle \vartheta{t}}] : Riemann-Siegel theta function : [http://drmf.wmflabs.org/wiki/Definition:RiemannSiegelTheta http://drmf.wmflabs.org/wiki/Definition:RiemannSiegelTheta]
[http://dlmf.nist.gov/5.2#E1 {\displaystyle \Gamma}] : Euler's gamma function : [http://dlmf.nist.gov/5.2#E1 http://dlmf.nist.gov/5.2#E1]
-[http://dlmf.nist.gov/4.2#E2 {\displaystyle \mathrm{ln}}] : principal branch of logarithm function : [http://dlmf.nist.gov/4.2#E2 http://dlmf.nist.gov/4.2#E2] +[http://drmf.wmflabs.org/wiki/Definition:RiemannSiegelTheta {\displaystyle \vartheta{t}}] : Riemann-Siegel theta function : [http://drmf.wmflabs.org/wiki/Definition:RiemannSiegelTheta http://drmf.wmflabs.org/wiki/Definition:RiemannSiegelTheta]
+[http://dlmf.nist.gov/4.2#E2 {\displaystyle \mathrm{ln}}] : principal branch of logarithm function : [http://dlmf.nist.gov/4.2#E2 http://dlmf.nist.gov/4.2#E2]

<< [[Definition:RamanujanTauTheta|Definition:RamanujanTauTheta]]
[[Main_Page|Main Page]]
@@ -5183,8 +5137,8 @@ These are defined by == Symbols List == +[http://dlmf.nist.gov/4.14#E1 {\displaystyle \mathrm{sin}}] : sine function : [http://dlmf.nist.gov/4.14#E1 http://dlmf.nist.gov/4.14#E1]
[http://drmf.wmflabs.org/wiki/Definition:Sinc {\displaystyle \mathrm{sinc}{x}}] : sine cardinal function : [http://drmf.wmflabs.org/wiki/Definition:Sinc http://drmf.wmflabs.org/wiki/Definition:Sinc]
-[http://dlmf.nist.gov/4.14#E1 {\displaystyle \mathrm{sin}}] : sine function : [http://dlmf.nist.gov/4.14#E1 http://dlmf.nist.gov/4.14#E1]
<< [[Definition:RiemannSiegelTheta|Definition:RiemannSiegelTheta]]
[[Main_Page|Main Page]]
diff --git a/main_page/src/main.py b/main_page/src/main.py index 9c2e117..2e4e42e 100644 --- a/main_page/src/main.py +++ b/main_page/src/main.py @@ -2,9 +2,11 @@ __status__ = "Development" __credits__ = ["Joon Bang", "Azeem Mohammed"] +import csv import time import mod_main -import symbols_list + +GLOSSARY_LOCATION = "new.Glossary.csv" def generate_html(tag_name, options, text, spacing=True): @@ -79,6 +81,97 @@ def update_headers(text, definitions): return output +def get_macro_name(macro): + macro_name = "" + for ch in macro: + if ch.isalpha(): + macro_name += ch + elif ch in ["@", "{", "["]: + break + + return macro_name + + +def get_symbols(text, glossary): + symbols = list() + for keyword in list(glossary): + for space in ["{", "@", "[", "\\", " "]: + if "\\" + keyword + space in text: + symbols.append(keyword) + + span_text = "" + for symbol in set(sorted(symbols)): + t = "" + link = glossary[symbol][-2] + meaning = list(glossary[symbol][1]) + + count = 0 + for i, ch in enumerate(meaning): + if ch == "$" and count % 2 == 0: + meaning[i] = "[" + link + " {\\displaystyle " + appearance + \ + "}] : " + ''.join(meaning) + " : [" + link + " " + link + "]
\n" + + return span_text[:-1] + + +def add_symbols_data(data): + glossary = dict() + with open(GLOSSARY_LOCATION, "rb") as csv_file: + glossary_file = csv.reader(csv_file, delimiter=',', quotechar='\"') + for row in glossary_file: + glossary[get_macro_name(row[0])] = row + + pages = data[data.find("drmf_bof"):].split("drmf_eof")[:-1] + + print "Pages: " + str(len(pages)) + + result = "" + for page in pages: + # skip over non-definition pages + if "'''Definition:" not in page: + to_write = page + "drmf_eof\n" + result += to_write + continue + + page = page.replace("drmf_bof", "").strip("\n") + add_spacing = False + sflag2 = False + # remove data (to be regenerated later) + if page.find("== Symbols List ==") != -1: + to_write = page.split("== Symbols List ==")[0] + page = to_write + else: + add_spacing = True + if page.find("drmf_foot") == -1: + to_write = page + sflag2 = True + else: + to_write = page.split("
")[0] + page = to_write + + to_write = "drmf_bof\n" + to_write + if add_spacing: + to_write += "\n\n" + + to_write += "== Symbols List ==\n\n" + to_parse = page.split("defined by")[1].split("
")[0] + to_write += get_symbols(to_parse, glossary) + to_write += "\n
\ndrmf_eof\n\n" + if sflag2: + to_write += "\n" + result += to_write + + return result + + def create_backup(): with open("main_page.mmd") as current: local_time = time.asctime(time.localtime(time.time())).split() @@ -96,12 +189,12 @@ def main(): text = main_page.read() text, definitions = update_macro_list(text) - text = symbols_list.main(text) + text = add_symbols_data(text) text = update_headers(text, definitions) text = mod_main.main(text) # only create backup if program did not crash - create_backup() + # create_backup() with open("main_page.mmd", "w") as main_page: main_page.write(text) diff --git a/main_page/src/symbols_list.py b/main_page/src/symbols_list.py index 196e22e..b44936c 100644 --- a/main_page/src/symbols_list.py +++ b/main_page/src/symbols_list.py @@ -10,10 +10,11 @@ def get_symbols(line): # (str) -> list """Gets all symbols on a line.""" + if line == "": + return [] + line = line.replace("\n", " ") sym_list = [] - if line == "": - return sym_list symbol = "" sym_flag = False @@ -26,14 +27,16 @@ def get_symbols(line): count += 1 arg_flag = True - if ch not in ["}", "]"] and arg_flag or ch.isalpha(): + if ch not in ["}", "]"] and (arg_flag or ch.isalpha()): symbol += ch + elif ch not in ["}", "]"]: sym_flag = False arg_flag = False sym_list.append(symbol) sym_list += (get_symbols(symbol)) symbol = "" + else: count -= 1 symbol += ch @@ -41,7 +44,7 @@ def get_symbols(line): p = "" else: p = line[i + 1] - if count == 0 and p != "{" and p != "[" and p != "@": + if count == 0 and p not in ["{", "[", "@"]: sym_flag = False sym_list.append(symbol) sym_list += (get_symbols(symbol)) @@ -92,6 +95,7 @@ def main(data): to_write += "== Symbols List ==\n\n" symbols = get_symbols((' '.join(page.split("\n")) + " ").replace("\n", " ")) + print symbols new_symbols = [] From 3584993b855f237f7fbceaf81619a4bc4d14d9fc Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Tue, 9 Aug 2016 15:27:19 -0400 Subject: [PATCH 202/402] Add tests for remove_inactive, remove_conditionalexpression, and remove_symbol functions in MathematicaToLaTeX --- eCF/data/IdentitiesTest.m | 4 +- eCF/src/MathematicaToLaTeX.py | 128 +++++++++++++++-------------- eCF/test/test_removal_functions.py | 50 +++++++++++ 3 files changed, 119 insertions(+), 63 deletions(-) diff --git a/eCF/data/IdentitiesTest.m b/eCF/data/IdentitiesTest.m index 64e8ed4..512c469 100644 --- a/eCF/data/IdentitiesTest.m +++ b/eCF/data/IdentitiesTest.m @@ -5,4 +5,6 @@ ConditionalExpression[Sinc[z] == (1 + Inactive[ContinuedFractionK][z^2/(2*k*(1 + 2*k)), 1 - z^2/(2*k*(1 + 2*k)), {k, 1, Infinity}])^(-1), Element[z, Complexes]] (* {"RiemannSiegelTheta", 1}*) -ConditionalExpression[RiemannSiegelTheta[z] == -(z*(Log[Pi] - PolyGamma[0, 1/4]))/2 - (z^3*PolyGamma[2, 1/4])/(48*(1 + Inactive[ContinuedFractionK][(z^2*PolyGamma[2*(1 + k), 1/4])/(8*(3 + 5*k + 2*k^2)*PolyGamma[2*k, 1/4]), 1 - (z^2*PolyGamma[2*(1 + k), 1/4])/(8*(3 + 5*k + 2*k^2)*PolyGamma[2*k, 1/4]), {k, 1, Infinity}])), Element[z, Complexes] && Abs[z] < 1/2] \ No newline at end of file +ConditionalExpression[RiemannSiegelTheta[z] == -(z*(Log[Pi] - PolyGamma[0, 1/4]))/2 - (z^3*PolyGamma[2, 1/4])/(48*(1 + Inactive[ContinuedFractionK][(z^2*PolyGamma[2*(1 + k), 1/4])/(8*(3 + 5*k + 2*k^2)*PolyGamma[2*k, 1/4]), 1 - (z^2*PolyGamma[2*(1 + k), 1/4])/(8*(3 + 5*k + 2*k^2)*PolyGamma[2*k, 1/4]), {k, 1, Infinity}])), Element[z, Complexes] && Abs[z] < 1/2] + +testInactive[testInactive[test]test]test \ No newline at end of file diff --git a/eCF/src/MathematicaToLaTeX.py b/eCF/src/MathematicaToLaTeX.py index f87b17d..7cbbe62 100644 --- a/eCF/src/MathematicaToLaTeX.py +++ b/eCF/src/MathematicaToLaTeX.py @@ -41,8 +41,9 @@ def find_surrounding(line, function, ex=(), start=0): - """finds the indices of the beginning and end of a function; this is the - main function that powers the converter. + """ + Finds the indices of the beginning and end of a function; this is the main + function that powers the converter. """ positions = [0, 0] line = line[start:] @@ -51,14 +52,13 @@ def find_surrounding(line, function, ex=(), start=0): if ex != '' and len(ex) >= 1: for e in ex: if (line.find(e) != -1 and - line.find(e) <= positions[0] and - line.find(e) + len(e) >= positions[0] + len( + line.find(e) <= positions[0] and + line.find(e) + len(e) >= positions[0] + len( function)): return [line.find(e) + len(e) + start, line.find(e) + len(e) + start] count = 0 - for j in range(positions[0] + len(function), len(line) + 1): if j == len(line) and count == 0: positions[1] = positions[0] @@ -79,8 +79,9 @@ def find_surrounding(line, function, ex=(), start=0): def arg_split(line, sep): - """Does the same thing as 'split', but does not split when the separator - is inside parentheses, brackets, or braces. Useful for nested statements. + """ + Does the same thing as 'split', but does not split when the separator is + inside parentheses, brackets, or braces. Useful for nested statements. """ l = list('([{') r = list(')]}') @@ -110,13 +111,14 @@ def arg_split(line, sep): def master_function(line, params): - """A master function, reads in the conversion templates from the - 'functions' file and performs the conversion. + """ + A master function, reads in the conversion templates from the 'functions' + file and performs the conversion. """ m, l, sep, ex = params[:5] sep = [i.split('-') for i in sep] - for _ in range(0, line.count(m)): + for _ in range(line.count(m)): try: pos except NameError: @@ -155,10 +157,8 @@ def master_function(line, params): def remove_inactive(line): - """Removes the 'Inactive' statement and its surrounding brakets, like in - the beginning of 'ConditionalExpression.' - """ - for _ in range(0, line.count('Inactive')): + """Removes 'Inactive' and its surrounding brackets.""" + for _ in range(line.count('Inactive')): pos = find_surrounding(line, 'Inactive') if pos[0] != pos[1]: line = line[:pos[0]] + line[pos[0] + 9:pos[1] - 1] + line[pos[1]:] @@ -167,8 +167,8 @@ def remove_inactive(line): def remove_conditionalexpression(line): - """Removes 'ConditionalExpression' and its surrounding brakets.""" - for _ in range(0, line.count('ConditionalExpression')): + """Removes 'ConditionalExpression' and its surrounding brackets.""" + for _ in range(line.count('ConditionalExpression')): pos = find_surrounding(line, 'ConditionalExpression') if pos[0] != pos[1]: line = line[:pos[0]] + line[pos[0] + 22:pos[1] - 1] + line[pos[1]:] @@ -177,8 +177,8 @@ def remove_conditionalexpression(line): def remove_symbol(line): - """Removes 'Symbol' and its surrounding brakets.""" - for _ in range(0, line.count('Symbol')): + """Removes 'Symbol' and its surrounding brackets.""" + for _ in range(line.count('Symbol')): pos = find_surrounding(line, 'Symbol') if pos[0] != pos[1]: line = line[:pos[0]] + line[pos[0] + 7:pos[1] - 1] + line[pos[1]:] @@ -187,10 +187,11 @@ def remove_symbol(line): def beta(line): - """Converts Mathematica's 'Beta' function to the equivalent LaTeX macro, + """ + Converts Mathematica's 'Beta' function to the equivalent LaTeX macro, taking into account the variations for the different number of arguments. """ - for _ in range(0, line.count('Beta')): + for _ in range(line.count('Beta')): try: pos except NameError: @@ -256,10 +257,9 @@ def carat(line): def cfk(line): """ - Converts Mathematica's 'ContinuedFractionK' to the equivalent LaTeX - macro. + Converts Mathematica's 'ContinuedFractionK' to the equivalent LaTeX macro. """ - for _ in range(0, line.count('ContinuedFractionK')): + for _ in range(line.count('ContinuedFractionK')): try: pos except NameError: @@ -289,7 +289,7 @@ def gamma(line): Converts Mathematica's 'Gamma' function to the equivalent LaTeX macro, taking into account the variations for the different number of arguments. """ - for _ in range(0, line.count('Gamma')): + for _ in range(line.count('Gamma')): try: pos except NameError: @@ -326,10 +326,9 @@ def gamma(line): def integrate(line): """ - Converts Mathematica's 'Integrate' function to - the equivalent LaTeX macro. + Converts Mathematica's 'Integrate' function to the equivalent LaTeX macro. """ - for _ in range(0, line.count('Integrate')): + for _ in range(line.count('Integrate')): try: pos except NameError: @@ -350,11 +349,11 @@ def integrate(line): def legendrep(line): - """Converts Mathematica's 'LegendreP' function to the equivalent LaTeX - macro, taking into account the variations for the different number of - arguments. """ - for _ in range(0, line.count('LegendreP')): + Converts Mathematica's 'LegendreP' function to the equivalent LaTeX macro, + taking into account the variations for the different number of arguments. + """ + for _ in range(line.count('LegendreP')): try: pos except NameError: @@ -383,11 +382,11 @@ def legendrep(line): def legendreq(line): - """Converts Mathematica's 'LegendreQ' function to the equivalent LaTeX - macro, taking into account the variations for the different number of - arguments. """ - for _ in range(0, line.count('LegendreQ')): + Converts Mathematica's 'LegendreQ' function to the equivalent LaTeX macro, + taking into account the variations for the different number of arguments. + """ + for _ in range(line.count('LegendreQ')): try: pos except NameError: @@ -416,11 +415,11 @@ def legendreq(line): def polyeulergamma(line): - """Converts Mathematica's 'Polygamma' function to the equivalent LaTeX - macro, taking into account the variations for the different number of - arguments. """ - for _ in range(0, line.count('PolyGamma')): + Converts Mathematica's 'Polygamma' function to the equivalent LaTeX macro, + taking into account the variations for the different number of arguments. + """ + for _ in range(line.count('PolyGamma')): try: pos except NameError: @@ -443,10 +442,10 @@ def polyeulergamma(line): def product(line): - """Converts Mathematica's product sum function to the equivalent LaTeX - macro. """ - for _ in range(0, line.count('Product')): + Converts Mathematica's product sum function to the equivalent LaTeX macro. + """ + for _ in range(line.count('Product')): try: pos except NameError: @@ -467,10 +466,11 @@ def product(line): def qpochhammer(line): - """Converts Mathematica's 'QPochhammer' function to the equivalent LaTeX + """ + Converts Mathematica's 'QPochhammer' function to the equivalent LaTeX macro. """ - for _ in range(0, line.count('QPochhammer')): + for _ in range(line.count('QPochhammer')): try: pos except NameError: @@ -497,10 +497,10 @@ def qpochhammer(line): def summation(line): - """Converts Mathematica's summation function to the equivalent LaTeX - macro. """ - for _ in range(0, line.count('Sum')): + Converts Mathematica's summation function to the equivalent LaTeX macro. + """ + for _ in range(line.count('Sum')): try: pos except NameError: @@ -521,8 +521,9 @@ def summation(line): def constraint(line): - """Converts Mathematica's 'Element', 'NotElement', and 'Inequality' - functions to LaTeX formatting using \constraint{}. + """ + Converts Mathematica's 'Element', 'NotElement', and 'Inequality' functions + to LaTeX formatting using \constraint{}. """ sections = arg_split(line, ',') @@ -550,7 +551,7 @@ def constraint(line): latex_elements = ('Complex', 'Whole', 'NatNumber', 'Integer', 'Irrational', 'Real', 'Rational', 'Prime') - for _ in range(0, line.count('Element')): + for _ in range(line.count('Element')): try: pos1 @@ -567,7 +568,7 @@ def constraint(line): line = (line[:pos1[0]] + sep[0].replace('|', ',') + ' \\in \\' + sep[1] + line[pos1[1]:]) - for _ in range(0, line.count('NotElement')): + for _ in range(line.count('NotElement')): try: pos2 @@ -584,7 +585,7 @@ def constraint(line): line = (line[:pos2[0]] + sep[0].replace('|', ',') + ' \\notin \\' + sep[1] + line[pos2[1]:]) - for _ in range(0, line.count('Inequality')): + for _ in range(line.count('Inequality')): pos3 = find_surrounding(line, 'Inequality') line = (line[:pos3[0]] + ''.join(line[pos3[0] + 11:pos3[1] - 1].split(',')) + @@ -594,7 +595,8 @@ def constraint(line): def convert_fraction(line): - """Converts Mathematica fractions, which are only '/', to LaTeX + """ + Converts Mathematica fractions, which are only '/', to LaTeX \frac{}{}-ions. """ l = list('([{') @@ -641,7 +643,7 @@ def convert_fraction(line): # incorrectly change it to " )( / )( ", but there are no cases of # this happening yet, so I have not gone to fixing this yet. if (line[j + 1] == '(' and line[i - 1] == ')' and - line[i + 1] == '(' and line[k] == ')'): + line[i + 1] == '(' and line[k] == ')'): # ()/() line = (line[:j + 1] + '\\frac{' + line[j + 2:i - 1] + '}{' + line[i + 2:k] + '}' + line[k + 1:]) @@ -665,7 +667,7 @@ def convert_fraction(line): def piecewise(line): """Converts Mathematica's piecewise function to LaTeX, using 'cases'.""" - for _ in range(0, line.count('Piecewise')): + for _ in range(line.count('Piecewise')): try: pos except NameError: @@ -729,8 +731,9 @@ def replace_operators(line): def replace_vars(line): - """Replaces the easy to convert variables in Mathematica to - its equivalent LaTeX code in the dictionary 'symbols'. + """ + Replaces the easy to convert variables in Mathematica to its equivalent + LaTeX code in the dictionary 'symbols'. """ for word in symbols: if symbols[word][0] == ' ': @@ -744,19 +747,20 @@ def replace_vars(line): def main(): - """Opens Mathematica file with identities and puts converted lines into + """ + Opens Mathematica file with identities and puts converted lines into newIdentities.tex. """ test = True with open(os.path.dirname(os.path.realpath(__file__)) + - '/../data/newIdentities.tex', 'w') as latex: + '/../data/newIdentities.tex', 'w') as latex: if test: to_open = 'IdentitiesTest.m' else: to_open = 'Identities.m' with open(os.path.dirname(os.path.realpath(__file__)) + '/../data/' + - to_open, 'r') as mathematica: + to_open, 'r') as mathematica: latex.write('\n\\documentclass{article}\n\n' '\\usepackage{amsmath}\n' @@ -833,8 +837,8 @@ def main(): latex.write('\n\n\\end{document}\n') -with open(os.path.dirname(os.path.realpath(__file__)) + \ - '/../data/functions') as functions: +with open(os.path.dirname(os.path.realpath(__file__)) + + '/../data/functions') as functions: FUNCTION_CONVERSIONS = list(arg_split(line.replace(' ', ''), ',') for line in functions.read().split('\n') if (line != '' and '#' not in line)) diff --git a/eCF/test/test_removal_functions.py b/eCF/test/test_removal_functions.py index 3945a15..1803517 100644 --- a/eCF/test/test_removal_functions.py +++ b/eCF/test/test_removal_functions.py @@ -7,3 +7,53 @@ from MathematicaToLaTeX import remove_conditionalexpression from MathematicaToLaTeX import remove_symbol +BEFORE1 = ('Inactive[test]', + 'testInactive[test]test', + 'Inactive[testInactive[test]test]', + 'testInactive[testInactive[test]test]test') +BEFORE2 = ('ConditionalExpression[test]', + 'testConditionalExpression[test]test', + 'ConditionalExpression[testConditionalExpression[test]test]', + 'testConditionalExpression[testConditionalExpression[test]test]test') +BEFORE3 = ('Symbol[test]', + 'testSymbol[test]test', + 'Symbol[testSymbol[test]test]', + 'testSymbol[testSymbol[test]test]test') +AFTER = ('test', + 'testtesttest', + 'testtesttest', + 'testtesttesttesttest') + +class TestRemovalFunctions(TestCase): + + def test_single(self): + self.assertEqual(remove_inactive( + BEFORE1[0]), AFTER[0]) + self.assertEqual(remove_conditionalexpression( + BEFORE2[0]), AFTER[0]) + self.assertEqual(remove_symbol( + BEFORE3[0]), AFTER[0]) + + def test_single_withsurrounding(self): + self.assertEqual(remove_inactive( + BEFORE1[1]), AFTER[1]) + self.assertEqual(remove_conditionalexpression( + BEFORE2[1]), AFTER[1]) + self.assertEqual(remove_symbol( + BEFORE3[1]), AFTER[1]) + + def test_nested_inactive(self): + self.assertEqual(remove_inactive( + BEFORE1[2]), AFTER[2]) + self.assertEqual(remove_conditionalexpression( + BEFORE2[2]), AFTER[2]) + self.assertEqual(remove_symbol( + BEFORE3[2]), AFTER[2]) + + def test_nested_withsurrounding(self): + self.assertEqual(remove_inactive( + BEFORE1[3]), AFTER[3]) + self.assertEqual(remove_conditionalexpression( + BEFORE2[3]), AFTER[3]) + self.assertEqual(remove_symbol( + BEFORE3[3]), AFTER[3]) From c3cb6bd8d9e680786253d5300debf28f99cc7d00 Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Tue, 9 Aug 2016 15:45:41 -0400 Subject: [PATCH 203/402] Add another test for arg_split function --- eCF/test/test_arg_split.py | 3 +++ 1 file changed, 3 insertions(+) diff --git a/eCF/test/test_arg_split.py b/eCF/test/test_arg_split.py index 410c58d..a5a4961 100644 --- a/eCF/test/test_arg_split.py +++ b/eCF/test/test_arg_split.py @@ -28,3 +28,6 @@ def test_combined(self): before = 'a,(b,c),,[d,e],,,{fg,hi}' after = ['a', '(b,c)', '', '[d,e]', '', '', '{fg,hi}'] self.assertEqual(arg_split(before, ','), after) + + def test_empty(self): + self.assertEqual(arg_split('noseperator', ','), ['noseperator']) From 444169863e56c8ed2b8bce27e7f8cfead4179745 Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Tue, 9 Aug 2016 16:07:05 -0400 Subject: [PATCH 204/402] Remove a few unused lines in MathematicaToLaTeX.py --- eCF/src/MathematicaToLaTeX.py | 6 +----- 1 file changed, 1 insertion(+), 5 deletions(-) diff --git a/eCF/src/MathematicaToLaTeX.py b/eCF/src/MathematicaToLaTeX.py index 7cbbe62..0b634fb 100644 --- a/eCF/src/MathematicaToLaTeX.py +++ b/eCF/src/MathematicaToLaTeX.py @@ -91,10 +91,6 @@ def arg_split(line, sep): line = line + sep while i != end: - if i == end: - args.append(line) - break - if line[i] in l: count += 1 if line[i] in r: @@ -751,7 +747,7 @@ def main(): Opens Mathematica file with identities and puts converted lines into newIdentities.tex. """ - test = True + test = False with open(os.path.dirname(os.path.realpath(__file__)) + '/../data/newIdentities.tex', 'w') as latex: From 015b40016a0a1b71915540da87a3ba92a2082c50 Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Tue, 9 Aug 2016 16:40:11 -0400 Subject: [PATCH 205/402] Change string format from using the old percent symbol to the new and preferred braces --- eCF/src/MathematicaToLaTeX.py | 92 +++++++++++++++++------------------ 1 file changed, 46 insertions(+), 46 deletions(-) diff --git a/eCF/src/MathematicaToLaTeX.py b/eCF/src/MathematicaToLaTeX.py index 0b634fb..fa7e780 100644 --- a/eCF/src/MathematicaToLaTeX.py +++ b/eCF/src/MathematicaToLaTeX.py @@ -202,11 +202,11 @@ def beta(line): if pos[0] != pos[1]: args = arg_split(line[pos[0] + 5:pos[1] - 1], ',') if len(args) == 2: - line = (line[:pos[0]] + '\\EulerBeta@{%s}{%s}' - % (args[0], args[1]) + line[pos[1]:]) + line = (line[:pos[0]] + '\\EulerBeta@{{0}}{{1}}' + .format(args[0], args[1]) + line[pos[1]:]) else: - line = (line[:pos[0]] + '\\IncBeta{%s}@{%s}{%s}' - % (args[0], args[1], args[2]) + line[pos[1]:]) + line = (line[:pos[0]] + '\\IncBeta{{0}}@{{1}}{{2}}' + .format(args[0], args[1], args[2]) + line[pos[1]:]) return line @@ -269,13 +269,13 @@ def cfk(line): args = arg_split(line[pos[0] + 19:pos[1] - 1], ',') moreargs = arg_split(args[-1][1:-1], ',') if len(args) == 3: - line = (line[:pos[0]] + '\\CFK{%s}{%s}{\\infty}@{%s}{%s}' - % (moreargs[0], moreargs[1], args[0], args[1]) + - line[pos[1]:]) + line = (line[:pos[0]] + '\\CFK{{0}}{{1}}{2}@{{3}}{{4}}' + .format(moreargs[0], moreargs[1],'{\\infty}', + args[0], args[1]) + line[pos[1]:]) else: - line = (line[:pos[0]] + '\\CFK{%s}{%s}{\\infty}@{1}{%s}' - % (moreargs[0], moreargs[1], args[0]) + - line[pos[1]:]) + line = (line[:pos[0]] + '\\CFK{{0}}{{1}}{2}@{3}{{4}}' + .format(moreargs[0], moreargs[1], '{\\infty}', '{1}', + args[0]) + line[pos[1]:]) return line @@ -306,15 +306,15 @@ def gamma(line): if pos[0] != pos[1]: args = arg_split(line[pos[0] + 6:pos[1] - 1], ',') if len(args) == 1: - line = (line[:pos[0]] + '\\EulerGamma@{%s}' - % args[0] + line[pos[1]:]) + line = (line[:pos[0]] + '\\EulerGamma@{{0}}' + .format(args[0]) + line[pos[1]:]) elif len(args) == 2: - line = (line[:pos[0]] + '\\IncGamma@{%s}{%s}' - % (args[0], args[1]) + line[pos[1]:]) + line = (line[:pos[0]] + '\\IncGamma@{{0}}{{1}}' + .format(args[0], args[1]) + line[pos[1]:]) else: - line = (line[:pos[0]] + '\\Incgamma@{%s}{%s} - ' - % (args[0], args[1]) + '\\Incgamma@{%s}{%s}' - % (args[0], args[2]) + line[pos[1]:]) + line = (line[:pos[0]] + '\\Incgamma@{{0}}{{1}} - ' + .format(args[0], args[1]) + '\\Incgamma@{{0}}{{1}}' + .format(args[0], args[2]) + line[pos[1]:]) return line.replace('Incgamma', 'IncGamma') # needed or the addition of an 'IncGamma' would screw up the rest @@ -337,8 +337,8 @@ def integrate(line): if pos[0] != pos[1]: args = arg_split(line[pos[0] + 10:pos[1] - 1], ',') moreargs = arg_split(args[1][1:-1], ',') - line = (line[:pos[0]] + '\\int_{%s}^{%s}{%s}d{%s}' - % (moreargs[1], moreargs[2], args[0], moreargs[0]) + + line = (line[:pos[0]] + '\\int_{{1}}^{{2}}{{3}}d{{0}}' + .format(moreargs[0], moreargs[1], moreargs[2], args[0]) + line[pos[1]:]) return line @@ -362,17 +362,17 @@ def legendrep(line): if pos[0] != pos[1]: args = arg_split(line[pos[0] + 10:pos[1] - 1], ',') if len(args) == 2: - line = (line[:pos[0]] + '\\LegendreP{%s}@{%s}' - % (args[0], args[1]) + line[pos[1]:]) + line = (line[:pos[0]] + '\\LegendreP{{0}}@{{1}}' + .format(args[0], args[1]) + line[pos[1]:]) else: # len(args) == 4 if args[2] in ('1', '2'): - line = (line[:pos[0]] + '\\FerrersP[%s]{%s}@{%s}' - % (args[1], args[0], args[3]) + line[pos[1]:]) + line = (line[:pos[0]] + '\\FerrersP[{1}]{{0}}@{{2}}' + .format(args[0], args[1], args[3]) + line[pos[1]:]) else: # args[2] == 3 - line = (line[:pos[0]] + '\\LegendreP[%s]{%s}@{%s}' - % (args[1], args[0], args[3]) + line[pos[1]:]) + line = (line[:pos[0]] + '\\LegendreP[{1}]{{0}}@{{2}}' + .format(args[0], args[1], args[3]) + line[pos[1]:]) return line @@ -395,17 +395,17 @@ def legendreq(line): if pos[0] != pos[1]: args = arg_split(line[pos[0] + 10:pos[1] - 1], ',') if len(args) == 2: - line = (line[:pos[0]] + '\\LegendreQ{%s}@{%s}' - % (args[0], args[1]) + line[pos[1]:]) + line = (line[:pos[0]] + '\\LegendreQ{{0}}@{{1}}' + .format(args[0], args[1]) + line[pos[1]:]) else: # len(args) == 4 if args[2] in ('1', '2'): - line = (line[:pos[0]] + '\\FerrersQ[%s]{%s}@{%s}' - % (args[1], args[0], args[3]) + line[pos[1]:]) + line = (line[:pos[0]] + '\\FerrersQ[{1}]{{0}}@{{2}}' + .format(args[0], args[1], args[3]) + line[pos[1]:]) else: # args[2] == 3 - line = (line[:pos[0]] + '\\LegendreQ[%s]{%s}@{%s}' - % (args[1], args[0], args[3]) + line[pos[1]:]) + line = (line[:pos[0]] + '\\LegendreQ[{1}]{{0}}@{{2}}' + .format(args[0], args[1], args[3]) + line[pos[1]:]) return line @@ -428,10 +428,10 @@ def polyeulergamma(line): if pos[0] != pos[1]: args = arg_split(line[pos[0] + 10:pos[1] - 1], ',') if len(args) == 2: - line = (line[:pos[0]] + '\\polygamma{%s}@{%s}' - % (args[0], args[1]) + line[pos[1]:]) + line = (line[:pos[0]] + '\\polygamma{{0}}@{{1}}' + .format(args[0], args[1]) + line[pos[1]:]) else: - line = (line[:pos[0]] + '\\digamma@{%s}' + line = (line[:pos[0]] + '\\digamma@{{0}}' % args[0] + '}' + line[pos[1]:]) return line @@ -454,8 +454,8 @@ def product(line): if pos[0] != pos[1]: args = arg_split(line[pos[0] + 8:pos[1] - 1], ',') moreargs = arg_split(args[-1][1:-1], ',') - line = (line[:pos[0]] + '\\Prod{%s}{%s}{%s}@{%s}' - % (moreargs[0], moreargs[1], moreargs[2], args[0]) + + line = (line[:pos[0]] + '\\Prod{{0}}{{1}}{{2}}@{{3}}' + .format(moreargs[0], moreargs[1], moreargs[2], args[0]) + line[pos[1]:]) return line @@ -480,14 +480,14 @@ def qpochhammer(line): args = arg_split(line[pos[0] + 12:pos[1] - 1], ',') if len(args) == 1: - line = (line[:pos[0]] + '\\qPochhammer{%s}{%s}{\infty}' - % (args[0], args[0]) + line[pos[1]:]) + line = (line[:pos[0]] + '\\qPochhammer{{0}}{{1}}{2}' + .format(args[0], args[0], '{\\infty}') + line[pos[1]:]) elif len(args) == 2: - line = (line[:pos[0]] + '\\qPochhammer{%s}{%s}{\infty}' - % (args[0], args[1]) + line[pos[1]:]) + line = (line[:pos[0]] + '\\qPochhammer{{0}}{{1}}{2}' + .format(args[0], args[1], '{\\infty}') + line[pos[1]:]) else: # len(args) = 3 - line = (line[:pos[0]] + '\\qPochhammer{%s}{%s}{%s}' - % (args[0], args[1], args[2]) + line[pos[1]:]) + line = (line[:pos[0]] + '\\qPochhammer{{0}}{{1}}{{2}}' + .format(args[0], args[1], args[2]) + line[pos[1]:]) return line @@ -509,8 +509,8 @@ def summation(line): if pos[0] != pos[1]: args = arg_split(line[pos[0] + 4:pos[1] - 1], ',') moreargs = arg_split(args[-1][1:-1], ',') - line = (line[:pos[0]] + '\\Sum{%s}{%s}{%s}@{%s}' - % (moreargs[0], moreargs[1], moreargs[2], args[0]) + + line = (line[:pos[0]] + '\\Sum{{0}}{{1}}{{2}}@{{3}}' + .format(moreargs[0], moreargs[1], moreargs[2], args[0]) + line[pos[1]:]) return line @@ -528,7 +528,7 @@ def constraint(line): constraints = arg_split(sections[-1].replace('&&', '&'), '&') - for i in range(len(constraints)): + for i, item, in enumerate(constraints): if i == 0: constraints[i] = ('\n% \\constraint{$' + constraints[i].replace('&', ' \\land ')) @@ -839,7 +839,7 @@ def main(): in functions.read().split('\n') if (line != '' and '#' not in line)) -for index in range(len(FUNCTION_CONVERSIONS)): +for index, item in enumerate(FUNCTION_CONVERSIONS): FUNCTION_CONVERSIONS[index][2] = \ tuple(arg_split(FUNCTION_CONVERSIONS[index][2][1:-1], ',')) From b3e1f50d6d9f91f290e25b90a7872dfca4dc83da Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Tue, 9 Aug 2016 16:41:40 -0400 Subject: [PATCH 206/402] Fix one line to use format --- eCF/src/MathematicaToLaTeX.py | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/eCF/src/MathematicaToLaTeX.py b/eCF/src/MathematicaToLaTeX.py index fa7e780..dc9b16f 100644 --- a/eCF/src/MathematicaToLaTeX.py +++ b/eCF/src/MathematicaToLaTeX.py @@ -432,7 +432,7 @@ def polyeulergamma(line): .format(args[0], args[1]) + line[pos[1]:]) else: line = (line[:pos[0]] + '\\digamma@{{0}}' - % args[0] + '}' + line[pos[1]:]) + .format(args[0]) + '}' + line[pos[1]:]) return line From 681a22c67c99b3e57598d3361e727408770e4657 Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Tue, 9 Aug 2016 16:43:25 -0400 Subject: [PATCH 207/402] Avoid unnecessary name repetition in equality comparison --- eCF/src/MathematicaToLaTeX.py | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/eCF/src/MathematicaToLaTeX.py b/eCF/src/MathematicaToLaTeX.py index dc9b16f..de5e4a7 100644 --- a/eCF/src/MathematicaToLaTeX.py +++ b/eCF/src/MathematicaToLaTeX.py @@ -134,7 +134,7 @@ def master_function(line, params): args[0], args[1] = args[1], args[0] if m == 'LaguerreL' and len(args) == 3: args[0], args[1] = args[1], args[0] - if m == 'HypergeometricPFQ' or m == 'QHypergeometricPFQ': + if m in ('HypergeometricPFQ', 'QHypergeometricPFQ'): if args[1] == '{}': args.insert(0, 0) else: From b7dc5e03c5f993639d087a32e8fae7b9c7da3aa3 Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Wed, 10 Aug 2016 10:00:46 -0400 Subject: [PATCH 208/402] Add test cases for Beta function --- eCF/src/MathematicaToLaTeX.py | 138 ++++++++++-------- eCF/test/test_arg_split.py | 2 +- eCF/test/test_removal_functions.py | 2 + .../test_single_macro_conversion_functions.py | 77 ++++++++++ 4 files changed, 159 insertions(+), 60 deletions(-) diff --git a/eCF/src/MathematicaToLaTeX.py b/eCF/src/MathematicaToLaTeX.py index de5e4a7..fbf3211 100644 --- a/eCF/src/MathematicaToLaTeX.py +++ b/eCF/src/MathematicaToLaTeX.py @@ -144,9 +144,10 @@ def master_function(line, params): else: args.insert(0, len(arg_split(args[1][1:-1], ','))) - line = (line[:pos[0]] + l + - '%s'.join(sep[[len(y) for y in sep]. - index(len(args) + 1)]) + line[pos[1]:]) + line = (line[:pos[0]] + l + '%s'.join(sep[[len(y) for y in sep] + .index(len(args) + 1)]) + + line[pos[1]:]) + line %= tuple(args) return line @@ -182,35 +183,6 @@ def remove_symbol(line): return line -def beta(line): - """ - Converts Mathematica's 'Beta' function to the equivalent LaTeX macro, - taking into account the variations for the different number of arguments. - """ - for _ in range(line.count('Beta')): - try: - pos - except NameError: - pos = find_surrounding(line, 'Beta', - ex=('BetaRegularized', '[Beta]')) - else: - pos = find_surrounding(line, 'Beta', - ex=('BetaRegularized', '[Beta]'), - start=pos[0] + (0, 9) - [(1, 0).index(pos[1] == pos[0])]) - - if pos[0] != pos[1]: - args = arg_split(line[pos[0] + 5:pos[1] - 1], ',') - if len(args) == 2: - line = (line[:pos[0]] + '\\EulerBeta@{{0}}{{1}}' - .format(args[0], args[1]) + line[pos[1]:]) - else: - line = (line[:pos[0]] + '\\IncBeta{{0}}@{{1}}{{2}}' - .format(args[0], args[1], args[2]) + line[pos[1]:]) - - return line - - def carat(line): """ Converts carats ('^') to ones with braces instead of parentheses. e.g: @@ -251,6 +223,35 @@ def carat(line): return line +def beta(line): + """ + Converts Mathematica's 'Beta' function to the equivalent LaTeX macro, + taking into account the variations for the different number of arguments. + """ + for _ in range(line.count('Beta')): + try: + pos + except NameError: + pos = find_surrounding(line, 'Beta', + ex=('BetaRegularized', '[Beta]')) + else: + pos = find_surrounding(line, 'Beta', + ex=('BetaRegularized', '[Beta]'), + start=pos[0] + (0, 9) + [(1, 0).index(pos[1] == pos[0])]) + + if pos[0] != pos[1]: + args = arg_split(line[pos[0] + 5:pos[1] - 1], ',') + if len(args) == 2: + line = (line[:pos[0]] + '\\EulerBeta@{{{0}}}{{{1}}}' + .format(args[0], args[1]) + line[pos[1]:]) + else: + line = (line[:pos[0]] + '\\IncBeta{{{0}}}@{{{1}}}{{{2}}}' + .format(args[0], args[1], args[2]) + line[pos[1]:]) + + return line + + def cfk(line): """ Converts Mathematica's 'ContinuedFractionK' to the equivalent LaTeX macro. @@ -269,11 +270,11 @@ def cfk(line): args = arg_split(line[pos[0] + 19:pos[1] - 1], ',') moreargs = arg_split(args[-1][1:-1], ',') if len(args) == 3: - line = (line[:pos[0]] + '\\CFK{{0}}{{1}}{2}@{{3}}{{4}}' + line = (line[:pos[0]] + '\\CFK{{{0}}}{{{1}}}{2}@{{{3}}}{{{4}}}' .format(moreargs[0], moreargs[1],'{\\infty}', args[0], args[1]) + line[pos[1]:]) else: - line = (line[:pos[0]] + '\\CFK{{0}}{{1}}{2}@{3}{{4}}' + line = (line[:pos[0]] + '\\CFK{{{0}}}{{{1}}}{2}@{3}{{{4}}}' .format(moreargs[0], moreargs[1], '{\\infty}', '{1}', args[0]) + line[pos[1]:]) @@ -306,14 +307,14 @@ def gamma(line): if pos[0] != pos[1]: args = arg_split(line[pos[0] + 6:pos[1] - 1], ',') if len(args) == 1: - line = (line[:pos[0]] + '\\EulerGamma@{{0}}' + line = (line[:pos[0]] + '\\EulerGamma@{{{0}}}' .format(args[0]) + line[pos[1]:]) elif len(args) == 2: - line = (line[:pos[0]] + '\\IncGamma@{{0}}{{1}}' + line = (line[:pos[0]] + '\\IncGamma@{{{0}}}{{{1}}}' .format(args[0], args[1]) + line[pos[1]:]) else: - line = (line[:pos[0]] + '\\Incgamma@{{0}}{{1}} - ' - .format(args[0], args[1]) + '\\Incgamma@{{0}}{{1}}' + line = (line[:pos[0]] + '\\Incgamma@{{{0}}}{{{1}}} - ' + .format(args[0], args[1]) + '\\Incgamma@{{{0}}}{{{1}}}' .format(args[0], args[2]) + line[pos[1]:]) return line.replace('Incgamma', 'IncGamma') @@ -337,7 +338,7 @@ def integrate(line): if pos[0] != pos[1]: args = arg_split(line[pos[0] + 10:pos[1] - 1], ',') moreargs = arg_split(args[1][1:-1], ',') - line = (line[:pos[0]] + '\\int_{{1}}^{{2}}{{3}}d{{0}}' + line = (line[:pos[0]] + '\\int_{{{1}}}^{{{2}}}{{{3}}}d{{{0}}}' .format(moreargs[0], moreargs[1], moreargs[2], args[0]) + line[pos[1]:]) @@ -362,16 +363,16 @@ def legendrep(line): if pos[0] != pos[1]: args = arg_split(line[pos[0] + 10:pos[1] - 1], ',') if len(args) == 2: - line = (line[:pos[0]] + '\\LegendreP{{0}}@{{1}}' + line = (line[:pos[0]] + '\\LegendreP{{{0}}}@{{{1}}}' .format(args[0], args[1]) + line[pos[1]:]) else: # len(args) == 4 if args[2] in ('1', '2'): - line = (line[:pos[0]] + '\\FerrersP[{1}]{{0}}@{{2}}' + line = (line[:pos[0]] + '\\FerrersP[{1}]{{{0}}}@{{{2}}}' .format(args[0], args[1], args[3]) + line[pos[1]:]) else: # args[2] == 3 - line = (line[:pos[0]] + '\\LegendreP[{1}]{{0}}@{{2}}' + line = (line[:pos[0]] + '\\LegendreP[{1}]{{{0}}}@{{{2}}}' .format(args[0], args[1], args[3]) + line[pos[1]:]) return line @@ -395,16 +396,16 @@ def legendreq(line): if pos[0] != pos[1]: args = arg_split(line[pos[0] + 10:pos[1] - 1], ',') if len(args) == 2: - line = (line[:pos[0]] + '\\LegendreQ{{0}}@{{1}}' + line = (line[:pos[0]] + '\\LegendreQ{{{0}}}@{{{1}}}' .format(args[0], args[1]) + line[pos[1]:]) else: # len(args) == 4 if args[2] in ('1', '2'): - line = (line[:pos[0]] + '\\FerrersQ[{1}]{{0}}@{{2}}' + line = (line[:pos[0]] + '\\FerrersQ[{1}]{{{0}}}@{{{2}}}' .format(args[0], args[1], args[3]) + line[pos[1]:]) else: # args[2] == 3 - line = (line[:pos[0]] + '\\LegendreQ[{1}]{{0}}@{{2}}' + line = (line[:pos[0]] + '\\LegendreQ[{1}]{{{0}}}@{{{2}}}' .format(args[0], args[1], args[3]) + line[pos[1]:]) return line @@ -428,10 +429,10 @@ def polyeulergamma(line): if pos[0] != pos[1]: args = arg_split(line[pos[0] + 10:pos[1] - 1], ',') if len(args) == 2: - line = (line[:pos[0]] + '\\polygamma{{0}}@{{1}}' + line = (line[:pos[0]] + '\\polygamma{{{0}}}@{{{1}}}' .format(args[0], args[1]) + line[pos[1]:]) else: - line = (line[:pos[0]] + '\\digamma@{{0}}' + line = (line[:pos[0]] + '\\digamma@{{{0}}}' .format(args[0]) + '}' + line[pos[1]:]) return line @@ -454,7 +455,7 @@ def product(line): if pos[0] != pos[1]: args = arg_split(line[pos[0] + 8:pos[1] - 1], ',') moreargs = arg_split(args[-1][1:-1], ',') - line = (line[:pos[0]] + '\\Prod{{0}}{{1}}{{2}}@{{3}}' + line = (line[:pos[0]] + '\\Prod{{{0}}}{{{1}}}{{{2}}}@{{{3}}}' .format(moreargs[0], moreargs[1], moreargs[2], args[0]) + line[pos[1]:]) @@ -480,13 +481,13 @@ def qpochhammer(line): args = arg_split(line[pos[0] + 12:pos[1] - 1], ',') if len(args) == 1: - line = (line[:pos[0]] + '\\qPochhammer{{0}}{{1}}{2}' + line = (line[:pos[0]] + '\\qPochhammer{{{0}}}{{{1}}}{2}' .format(args[0], args[0], '{\\infty}') + line[pos[1]:]) elif len(args) == 2: - line = (line[:pos[0]] + '\\qPochhammer{{0}}{{1}}{2}' + line = (line[:pos[0]] + '\\qPochhammer{{{0}}}{{{1}}}{2}' .format(args[0], args[1], '{\\infty}') + line[pos[1]:]) else: # len(args) = 3 - line = (line[:pos[0]] + '\\qPochhammer{{0}}{{1}}{{2}}' + line = (line[:pos[0]] + '\\qPochhammer{{{0}}}{{{1}}}{{{2}}}' .format(args[0], args[1], args[2]) + line[pos[1]:]) return line @@ -509,7 +510,7 @@ def summation(line): if pos[0] != pos[1]: args = arg_split(line[pos[0] + 4:pos[1] - 1], ',') moreargs = arg_split(args[-1][1:-1], ',') - line = (line[:pos[0]] + '\\Sum{{0}}{{1}}{{2}}@{{3}}' + line = (line[:pos[0]] + '\\Sum{{{0}}}{{{1}}}{{{2}}}@{{{3}}}' .format(moreargs[0], moreargs[1], moreargs[2], args[0]) + line[pos[1]:]) @@ -639,7 +640,7 @@ def convert_fraction(line): # incorrectly change it to " )( / )( ", but there are no cases of # this happening yet, so I have not gone to fixing this yet. if (line[j + 1] == '(' and line[i - 1] == ')' and - line[i + 1] == '(' and line[k] == ')'): + line[i + 1] == '(' and line[k] == ')'): # ()/() line = (line[:j + 1] + '\\frac{' + line[j + 2:i - 1] + '}{' + line[i + 2:k] + '}' + line[k + 1:]) @@ -656,6 +657,28 @@ def convert_fraction(line): line = (line[:j + 1] + '\\frac{' + line[j + 1:i] + '}{' + line[i + 1:k + 1] + '}' + line[k + 1:]) + '''if (line[j + 1] == '(' and line[i - 1] == ')' and + line[i + 1] == '(' and line[k] == ')'): + # ()/() + line = '{0}\\frac{{{1}}}{{{2}}}{3}'\ + .format(line[:j + 1], line[j + 2:i - 1], + line[i + 2:k], line[k + 1:]) + elif line[j + 1] == '(' and line[i - 1] == ')': + # ()/-- + line = '{0}\\frac{{{1}}}{{{2}}}{3}'\ + .format(line[:j + 1], line[j + 2:i - 1], + line[i + 1:k + 1], line[k + 1:]) + elif line[i + 1] == '(' and line[k] == ')': + # --/() + line = '{0}\\frac{{{1}}}{{{2}}}{3}'\ + .format(line[:j + 1], line[j + 1:i], + line[i + 2:k], line[k + 1:]) + else: + # --/-- + line = '{0}\\frac{{{1}}}{{{2}}}{3}'\ + .format(line[:j + 1], line[j + 1:i], + line[i + 1:k + 1], line[k + 1:])''' + i += 1 return line @@ -798,8 +821,9 @@ def main(): for i in FUNCTION_CONVERSIONS: line = master_function(line, i) - line = beta(line) line = carat(line) + + line = beta(line) line = cfk(line) line = gamma(line) line = integrate(line) @@ -809,17 +833,13 @@ def main(): line = product(line) line = qpochhammer(line) line = summation(line) + line = convert_fraction(line) line = constraint(line) line = piecewise(line) line = replace_operators(line) line = replace_vars(line) - # create macros for: - # Sinc - # RamanujanTauTheta - # RiemannSiegelTheta - print(line) if test and line != '': diff --git a/eCF/test/test_arg_split.py b/eCF/test/test_arg_split.py index a5a4961..51b704f 100644 --- a/eCF/test/test_arg_split.py +++ b/eCF/test/test_arg_split.py @@ -29,5 +29,5 @@ def test_combined(self): after = ['a', '(b,c)', '', '[d,e]', '', '', '{fg,hi}'] self.assertEqual(arg_split(before, ','), after) - def test_empty(self): + def test_none(self): self.assertEqual(arg_split('noseperator', ','), ['noseperator']) diff --git a/eCF/test/test_removal_functions.py b/eCF/test/test_removal_functions.py index 1803517..eefbd21 100644 --- a/eCF/test/test_removal_functions.py +++ b/eCF/test/test_removal_functions.py @@ -7,6 +7,7 @@ from MathematicaToLaTeX import remove_conditionalexpression from MathematicaToLaTeX import remove_symbol + BEFORE1 = ('Inactive[test]', 'testInactive[test]test', 'Inactive[testInactive[test]test]', @@ -24,6 +25,7 @@ 'testtesttest', 'testtesttesttesttest') + class TestRemovalFunctions(TestCase): def test_single(self): diff --git a/eCF/test/test_single_macro_conversion_functions.py b/eCF/test/test_single_macro_conversion_functions.py index 722ba2e..833648d 100644 --- a/eCF/test/test_single_macro_conversion_functions.py +++ b/eCF/test/test_single_macro_conversion_functions.py @@ -14,3 +14,80 @@ from MathematicaToLaTeX import qpochhammer from MathematicaToLaTeX import summation + +class TestBeta(TestCase): + + def test_single(self): + self.assertEqual(beta('Beta[a,b]'), '\\EulerBeta@{a}{b}') + self.assertEqual(beta('--Beta[a,b]--'), '--\\EulerBeta@{a}{b}--') + self.assertEqual(beta('Beta[z,a,b]'), '\\IncBeta{z}@{a}{b}') + self.assertEqual(beta('--Beta[z,a,b]--'), '--\\IncBeta{z}@{a}{b}--') + self.assertEqual(beta('Beta[a,b]Beta[z,a,b]'), '\\EulerBeta@{a}{b}\\IncBeta{z}@{a}{b}') + self.assertEqual(beta('--Beta[a,b]--Beta[z,a,b]--'), '--\\EulerBeta@{a}{b}--\\IncBeta{z}@{a}{b}--') + + def test_nested(self): + self.assertEqual(beta('Beta[Beta[a,b],Beta[a,b]]'), '\\EulerBeta@{\\EulerBeta@{a}{b}}{\\EulerBeta@{a}{b}}') + self.assertEqual(beta('--Beta[Beta[a,b],Beta[a,b]]--'), '--\\EulerBeta@{\\EulerBeta@{a}{b}}{\\EulerBeta@{a}{b}}--') + self.assertEqual(beta('Beta[Beta[z,a,b],Beta[z,a,b],Beta[z,a,b]]'), '\\IncBeta{\\IncBeta{z}@{a}{b}}@{\\IncBeta{z}@{a}{b}}{\\IncBeta{z}@{a}{b}}') + self.assertEqual(beta('--Beta[Beta[z,a,b],Beta[z,a,b],Beta[z,a,b]]--'), '--\\IncBeta{\\IncBeta{z}@{a}{b}}@{\\IncBeta{z}@{a}{b}}{\\IncBeta{z}@{a}{b}}--') + + def test_exceptions(self): + self.assertEqual(beta('BetaRegularized[z,a,b]'), 'BetaRegularized[z,a,b]') + self.assertEqual(beta('\\[Beta]'), '\\[Beta]') + + def test_none(self): + self.assertEqual(beta('none'), 'none') + + +class TestCFK(TestCase): + + def test_none(self): + self.assertEqual(cfk('none'), 'none') + + +class TestGamma(TestCase): + + def test_none(self): + self.assertEqual(gamma('none'), 'none') + + +class TestIntegrate(TestCase): + + def test_none(self): + self.assertEqual(integrate('none'), 'none') + + +class TestLegendreP(TestCase): + + def test_none(self): + self.assertEqual(legendrep('none'), 'none') + + +class TestLegendreQ(TestCase): + + def test_none(self): + self.assertEqual(legendreq('none'), 'none') + + +class TestPolyEulergamma(TestCase): + + def test_none(self): + self.assertEqual(polyeulergamma('none'), 'none') + + +class TestProduct(TestCase): + + def test_none(self): + self.assertEqual(product('none'), 'none') + + +class TestQPochhammer(TestCase): + + def test_none(self): + self.assertEqual(qpochhammer('none'), 'none') + + +class TestSummation(TestCase): + + def test_none(self): + self.assertEqual(summation('none'), 'none') \ No newline at end of file From f7f953012c0271dfdc16bb58068c1f5f29143239 Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Wed, 10 Aug 2016 10:16:22 -0400 Subject: [PATCH 209/402] Add test cases for cfk function --- eCF/src/MathematicaToLaTeX.py | 10 ++++++---- eCF/test/test_single_macro_conversion_functions.py | 13 +++++++++++++ 2 files changed, 19 insertions(+), 4 deletions(-) diff --git a/eCF/src/MathematicaToLaTeX.py b/eCF/src/MathematicaToLaTeX.py index fbf3211..0adedab 100644 --- a/eCF/src/MathematicaToLaTeX.py +++ b/eCF/src/MathematicaToLaTeX.py @@ -270,12 +270,14 @@ def cfk(line): args = arg_split(line[pos[0] + 19:pos[1] - 1], ',') moreargs = arg_split(args[-1][1:-1], ',') if len(args) == 3: - line = (line[:pos[0]] + '\\CFK{{{0}}}{{{1}}}{2}@{{{3}}}{{{4}}}' - .format(moreargs[0], moreargs[1],'{\\infty}', + line = (line[:pos[0]] + + '\\CFK{{{0}}}{{{1}}}{{{2}}}@{{{3}}}{{{4}}}' + .format(moreargs[0], moreargs[1], moreargs[2], args[0], args[1]) + line[pos[1]:]) else: - line = (line[:pos[0]] + '\\CFK{{{0}}}{{{1}}}{2}@{3}{{{4}}}' - .format(moreargs[0], moreargs[1], '{\\infty}', '{1}', + line = (line[:pos[0]] + + '\\CFK{{{0}}}{{{1}}}{{{2}}}@{{1}}{{{3}}}' + .format(moreargs[0], moreargs[1], moreargs[2], args[0]) + line[pos[1]:]) return line diff --git a/eCF/test/test_single_macro_conversion_functions.py b/eCF/test/test_single_macro_conversion_functions.py index 833648d..f901f4d 100644 --- a/eCF/test/test_single_macro_conversion_functions.py +++ b/eCF/test/test_single_macro_conversion_functions.py @@ -41,6 +41,19 @@ def test_none(self): class TestCFK(TestCase): + def test_single(self): + self.assertEqual(cfk('ContinuedFractionK[f,g,{i,imin,imax}]'), '\\CFK{i}{imin}{imax}@{f}{g}') + self.assertEqual(cfk('--ContinuedFractionK[f,g,{i,imin,imax}]--'), '--\\CFK{i}{imin}{imax}@{f}{g}--') + self.assertEqual(cfk('ContinuedFractionK[g,{i,imin,imax}]'), '\\CFK{i}{imin}{imax}@{1}{g}') + self.assertEqual(cfk('--ContinuedFractionK[g,{i,imin,imax}]--'), '--\\CFK{i}{imin}{imax}@{1}{g}--') + self.assertEqual(cfk('ContinuedFractionK[f,g,{i,imin,imax}]ContinuedFractionK[g,{i,imin,imax}]'), '\\CFK{i}{imin}{imax}@{f}{g}\\CFK{i}{imin}{imax}@{1}{g}') + self.assertEqual(cfk('--ContinuedFractionK[f,g,{i,imin,imax}]--ContinuedFractionK[g,{i,imin,imax}]--'), '--\\CFK{i}{imin}{imax}@{f}{g}--\\CFK{i}{imin}{imax}@{1}{g}--') + + def test_nested(self): + self.assertEqual(cfk('ContinuedFractionK[ContinuedFractionK[f,g,{i,imin,imax}],ContinuedFractionK[f,g,{i,imin,imax}],{i,imin,imax}]'), '\\CFK{i}{imin}{imax}@{\\CFK{i}{imin}{imax}@{f}{g}}{\\CFK{i}{imin}{imax}@{f}{g}}') + self.assertEqual(cfk('--ContinuedFractionK[ContinuedFractionK[f,g,{i,imin,imax}],ContinuedFractionK[f,g,{i,imin,imax}],{i,imin,imax}]--'), '--\\CFK{i}{imin}{imax}@{\\CFK{i}{imin}{imax}@{f}{g}}{\\CFK{i}{imin}{imax}@{f}{g}}--') + self.assertEqual(cfk('ContinuedFractionK[ContinuedFractionK[g,{i,imin,imax}],{i,imin,imax}]'), '\\CFK{i}{imin}{imax}@{1}{\\CFK{i}{imin}{imax}@{1}{g}}') + def test_none(self): self.assertEqual(cfk('none'), 'none') From e088a4f5d8d3e7bf23601bb3f1c4ce4e475d389d Mon Sep 17 00:00:00 2001 From: Joon Bang Date: Wed, 10 Aug 2016 13:23:25 -0400 Subject: [PATCH 210/402] Fix symbols list generation --- main_page/main_page.mmd | 578 ++++++++++++++++++++++------------------ main_page/src/main.py | 55 ++-- 2 files changed, 349 insertions(+), 284 deletions(-) diff --git a/main_page/main_page.mmd b/main_page/main_page.mmd index 7484a68..0e87ea3 100644 --- a/main_page/main_page.mmd +++ b/main_page/main_page.mmd @@ -309,8 +309,8 @@ These are defined by == Symbols List == -[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or -hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
-[http://drmf.wmflabs.org/wiki/Definition:AffqKrawtchouk {\displaystyle K^{\mathrm{Aff}}_{n}}] : affine -Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:AffqKrawtchouk http://drmf.wmflabs.org/wiki/Definition:AffqKrawtchouk]
+[http://drmf.wmflabs.org/wiki/Definition:AffqKrawtchouk {\displaystyle K^{\mathrm{Aff}}_{n}}] : affine {\displaystyle q}-Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:AffqKrawtchouk http://drmf.wmflabs.org/wiki/Definition:AffqKrawtchouk]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
<< [[Definition:sinc|Definition:sinc]]
[[Main_Page|Main Page]]
@@ -342,9 +342,9 @@ These are defined by == Symbols List == -[http://dlmf.nist.gov/1.2#E1 {\displaystyle \binom{n}{k}}] : binomial coefficient : [http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/1.2#E1]
-[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or -hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
[http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzI {\displaystyle U^{(n)}_{\alpha}}] : Al-Salam-Carlitz I polynomial : [http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzI http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzI]
+[http://dlmf.nist.gov/1.2#E1 {\displaystyle \binom{n}{k}}] : binomial coefficient : [http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/1.2#E1] [http://dlmf.nist.gov/26.3#SS1.p1 http://dlmf.nist.gov/26.3#SS1.p1]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
<< [[Definition:AffqKrawtchouk|Definition:AffqKrawtchouk]]
[[Main_Page|Main Page]]
@@ -378,8 +378,8 @@ These are defined by == Symbols List == [http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzII {\displaystyle V^{(n)}_{\alpha}}] : Al-Salam-Carlitz II polynomial : [http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzII http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzII]
-[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or -hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
-[http://dlmf.nist.gov/1.2#E1 {\displaystyle \binom{n}{k}}] : binomial coefficient : [http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/1.2#E1]
+[http://dlmf.nist.gov/1.2#E1 {\displaystyle \binom{n}{k}}] : binomial coefficient : [http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/1.2#E1] [http://dlmf.nist.gov/26.3#SS1.p1 http://dlmf.nist.gov/26.3#SS1.p1]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
<< [[Definition:AlSalamCarlitzI|Definition:AlSalamCarlitzI]]
[[Main_Page|Main Page]]
@@ -423,8 +423,8 @@ and == Symbols List == -[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or -hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
[http://drmf.wmflabs.org/wiki/Definition:AlSalamIsmail {\displaystyle U_{n}}] : Al-Salam Ismail polynomial : [http://drmf.wmflabs.org/wiki/Definition:AlSalamIsmail http://drmf.wmflabs.org/wiki/Definition:AlSalamIsmail]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
<< [[Definition:AlSalamCarlitzII|Definition:AlSalamCarlitzII]]
[[Main_Page|Main Page]]
@@ -457,7 +457,7 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:AntiDer {\displaystyle \int f(x){\mathrm d}\!x}] : semantic antiderivative : [http://drmf.wmflabs.org/wiki/Definition:AntiDer http://drmf.wmflabs.org/wiki/Definition:AntiDer]
+[http://drmf.wmflabs.org/wiki/Definition:AntiDer {\displaystyle \int f(x){\mathrm d}\!x}] : semantic antiderivative : [http://drmf.wmflabs.org/wiki/Definition:AntiDer http://drmf.wmflabs.org/wiki/Definition:AntiDer]
<< [[Definition:AlSalamIsmail|Definition:AlSalamIsmail]]
[[Main_Page|Main Page]]
@@ -490,7 +490,7 @@ These are defined by == Symbols List == [http://dlmf.nist.gov/18.34#E1 {\displaystyle y_{n}}] : Bessel polynomial : [http://dlmf.nist.gov/18.34#E1 http://dlmf.nist.gov/18.34#E1]
-[http://drmf.wmflabs.org/wiki/Definition:BesselPolyIIparam {\displaystyle y_{n}}] : Bessel polynomial with two parameters : [http://drmf.wmflabs.org/wiki/Definition:BesselPolyIIparam http://drmf.wmflabs.org/wiki/Definition:BesselPolyIIparam]
+[http://drmf.wmflabs.org/wiki/Definition:BesselPolyIIparam {\displaystyle y_{n}}] : Bessel polynomial with two parameters {\displaystyle y_n} : [http://drmf.wmflabs.org/wiki/Definition:BesselPolyIIparam http://drmf.wmflabs.org/wiki/Definition:BesselPolyIIparam]
<< [[Definition:AntiDer|Definition:AntiDer]]
[[Main_Page|Main Page]]
@@ -522,8 +522,8 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:BesselPolyTheta {\displaystyle \theta_{n}}] : Bessel polynomial with two parameters : [http://drmf.wmflabs.org/wiki/Definition:BesselPolyTheta http://drmf.wmflabs.org/wiki/Definition:BesselPolyTheta]
-[http://drmf.wmflabs.org/wiki/Definition:BesselPolyIIparam {\displaystyle y_{n}}] : Bessel polynomial with two parameters : [http://drmf.wmflabs.org/wiki/Definition:BesselPolyIIparam http://drmf.wmflabs.org/wiki/Definition:BesselPolyIIparam]
+[http://drmf.wmflabs.org/wiki/Definition:BesselPolyIIparam {\displaystyle y_{n}}] : Bessel polynomial with two parameters {\displaystyle y_n} : [http://drmf.wmflabs.org/wiki/Definition:BesselPolyIIparam http://drmf.wmflabs.org/wiki/Definition:BesselPolyIIparam]
+[http://drmf.wmflabs.org/wiki/Definition:BesselPolyTheta {\displaystyle \theta_{n}}] : Bessel polynomial with two parameters {\displaystyle \theta_n} : [http://drmf.wmflabs.org/wiki/Definition:BesselPolyTheta http://drmf.wmflabs.org/wiki/Definition:BesselPolyTheta]
<< [[Definition:BesselPolyIIparam|Definition:BesselPolyIIparam]]
[[Main_Page|Main Page]]
@@ -555,8 +555,8 @@ These are defined by == Symbols List == -[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or -hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
-[http://drmf.wmflabs.org/wiki/Definition:bigqJacobiIVparam {\displaystyle P_{n}}] : big -Jacobi polynomial with four parameters : [http://drmf.wmflabs.org/wiki/Definition:bigqJacobiIVparam http://drmf.wmflabs.org/wiki/Definition:bigqJacobiIVparam]
+[http://drmf.wmflabs.org/wiki/Definition:bigqJacobiIVparam {\displaystyle P_{n}}] : big {\displaystyle q}-Jacobi polynomial with four parameters : [http://drmf.wmflabs.org/wiki/Definition:bigqJacobiIVparam http://drmf.wmflabs.org/wiki/Definition:bigqJacobiIVparam]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
<< [[Definition:BesselPolyTheta|Definition:BesselPolyTheta]]
[[Main_Page|Main Page]]
@@ -588,8 +588,8 @@ These are defined by == Symbols List == -[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or -hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
-[http://drmf.wmflabs.org/wiki/Definition:bigqLaguerre {\displaystyle P_{n}}] : big -Laguerre polynomial : [http://drmf.wmflabs.org/wiki/Definition:bigqLaguerre http://drmf.wmflabs.org/wiki/Definition:bigqLaguerre]
+[http://drmf.wmflabs.org/wiki/Definition:bigqLaguerre {\displaystyle P_{n}}] : big {\displaystyle q}-Laguerre polynomial : [http://drmf.wmflabs.org/wiki/Definition:bigqLaguerre http://drmf.wmflabs.org/wiki/Definition:bigqLaguerre]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
<< [[Definition:bigqJacobiIVparam|Definition:bigqJacobiIVparam]]
[[Main_Page|Main Page]]
@@ -621,8 +621,8 @@ These are defined by == Symbols List == -[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or -hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
-[http://drmf.wmflabs.org/wiki/Definition:bigqLegendre {\displaystyle P_{n}}] : big -Legendre polynomial : [http://drmf.wmflabs.org/wiki/Definition:bigqLegendre http://drmf.wmflabs.org/wiki/Definition:bigqLegendre]
+[http://drmf.wmflabs.org/wiki/Definition:bigqLegendre {\displaystyle P_{n}}] : big {\displaystyle q}-Legendre polynomial : [http://drmf.wmflabs.org/wiki/Definition:bigqLegendre http://drmf.wmflabs.org/wiki/Definition:bigqLegendre]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
<< [[Definition:bigqLaguerre|Definition:bigqLaguerre]]
[[Main_Page|Main Page]]
@@ -656,8 +656,8 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:CiglerqChebyT {\displaystyle T_{n}}] : Cigler -Chebyshev polynomial : [http://drmf.wmflabs.org/wiki/Definition:CiglerqChebyT http://drmf.wmflabs.org/wiki/Definition:CiglerqChebyT]
-[http://drmf.wmflabs.org/wiki/Definition:bigqJacobiIVparam {\displaystyle P_{n}}] : big -Jacobi polynomial with four parameters : [http://drmf.wmflabs.org/wiki/Definition:bigqJacobiIVparam http://drmf.wmflabs.org/wiki/Definition:bigqJacobiIVparam]
+[http://drmf.wmflabs.org/wiki/Definition:bigqJacobiIVparam {\displaystyle P_{n}}] : big {\displaystyle q}-Jacobi polynomial with four parameters : [http://drmf.wmflabs.org/wiki/Definition:bigqJacobiIVparam http://drmf.wmflabs.org/wiki/Definition:bigqJacobiIVparam]
+[http://drmf.wmflabs.org/wiki/Definition:CiglerqChebyT {\displaystyle T_{n}}] : Cigler {\displaystyle q}-Chebyshev polynomial {\displaystyle T} : [http://drmf.wmflabs.org/wiki/Definition:CiglerqChebyT http://drmf.wmflabs.org/wiki/Definition:CiglerqChebyT]
<< [[Definition:bigqLegendre|Definition:bigqLegendre]]
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@@ -690,7 +690,7 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:CiglerqChebyU {\displaystyle U_{n}}] : Cigler -Chebyshev polynomial : [http://drmf.wmflabs.org/wiki/Definition:CiglerqChebyU http://drmf.wmflabs.org/wiki/Definition:CiglerqChebyU]
+[http://drmf.wmflabs.org/wiki/Definition:CiglerqChebyU {\displaystyle U_{n}}] : Cigler {\displaystyle q}-Chebyshev polynomial {\displaystyle U} : [http://drmf.wmflabs.org/wiki/Definition:CiglerqChebyU http://drmf.wmflabs.org/wiki/Definition:CiglerqChebyU]
<< [[Definition:CiglerqChebyT|Definition:CiglerqChebyT]]
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@@ -722,8 +722,9 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:ctsbigqHermite {\displaystyle H_{n}}] : continuous big -Hermite polynomial : [http://drmf.wmflabs.org/wiki/Definition:ctsbigqHermite http://drmf.wmflabs.org/wiki/Definition:ctsbigqHermite]
-[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or -hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
+[http://drmf.wmflabs.org/wiki/Definition:ctsbigqHermite {\displaystyle H_{n}}] : continuous big {\displaystyle q}-Hermite polynomial : [http://drmf.wmflabs.org/wiki/Definition:ctsbigqHermite http://drmf.wmflabs.org/wiki/Definition:ctsbigqHermite]
+[http://dlmf.nist.gov/4.2.E11 {\displaystyle \mathrm{e}}] : the base of the natural logarithm : [http://dlmf.nist.gov/4.2.E11 http://dlmf.nist.gov/4.2.E11]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
<< [[Definition:CiglerqChebyU|Definition:CiglerqChebyU]]
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@@ -756,9 +757,10 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:ctsdualqHahn {\displaystyle p_{n}}] : continuous dual -Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:ctsdualqHahn http://drmf.wmflabs.org/wiki/Definition:ctsdualqHahn]
-[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or -hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
-[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]
+[http://drmf.wmflabs.org/wiki/Definition:ctsdualqHahn {\displaystyle p_{n}}] : continuous dual {\displaystyle q}-Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:ctsdualqHahn http://drmf.wmflabs.org/wiki/Definition:ctsdualqHahn]
+[http://dlmf.nist.gov/4.2.E11 {\displaystyle \mathrm{e}}] : the base of the natural logarithm : [http://dlmf.nist.gov/4.2.E11 http://dlmf.nist.gov/4.2.E11]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1]
<< [[Definition:ctsbigqHermite|Definition:ctsbigqHermite]]
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@@ -791,9 +793,10 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:ctsqHahn {\displaystyle p_{n}}] : continuous -Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:ctsqHahn http://drmf.wmflabs.org/wiki/Definition:ctsqHahn]
-[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or -hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
-[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]
+[http://drmf.wmflabs.org/wiki/Definition:ctsqHahn {\displaystyle p_{n}}] : continuous {\displaystyle q}-Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:ctsqHahn http://drmf.wmflabs.org/wiki/Definition:ctsqHahn]
+[http://dlmf.nist.gov/4.2.E11 {\displaystyle \mathrm{e}}] : the base of the natural logarithm : [http://dlmf.nist.gov/4.2.E11 http://dlmf.nist.gov/4.2.E11]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1]
<< [[Definition:ctsdualqHahn|Definition:ctsdualqHahn]]
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@@ -830,7 +833,11 @@ with x=\cos@@{\theta}. == Symbols List == - +[http://dlmf.nist.gov/4.14#E2 {\displaystyle \mathrm{cos}}] : cosine function : [http://dlmf.nist.gov/4.14#E2 http://dlmf.nist.gov/4.14#E2]
+[http://drmf.wmflabs.org/wiki/Definition:ctsqJacobi {\displaystyle P^{(\alpha,\beta)}_{n}}] : continuous {\displaystyle q}-Jacobi polynomial : [http://drmf.wmflabs.org/wiki/Definition:ctsqJacobi http://drmf.wmflabs.org/wiki/Definition:ctsqJacobi]
+[http://dlmf.nist.gov/4.2.E11 {\displaystyle \mathrm{e}}] : the base of the natural logarithm : [http://dlmf.nist.gov/4.2.E11 http://dlmf.nist.gov/4.2.E11]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1]
<< [[Definition:ctsqHahn|Definition:ctsqHahn]]
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@@ -862,9 +869,10 @@ These are defined by == Symbols List == -[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or -hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
-[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]
-[http://drmf.wmflabs.org/wiki/Definition:ctsqLaguerre {\displaystyle P^{(n)}_{\alpha}}] : continuous -Laguerre polynomial : [http://drmf.wmflabs.org/wiki/Definition:ctsqLaguerre http://drmf.wmflabs.org/wiki/Definition:ctsqLaguerre]
+[http://drmf.wmflabs.org/wiki/Definition:ctsqLaguerre {\displaystyle P^{(n)}_{\alpha}}] : continuous {\displaystyle q}-Laguerre polynomial : [http://drmf.wmflabs.org/wiki/Definition:ctsqLaguerre http://drmf.wmflabs.org/wiki/Definition:ctsqLaguerre]
+[http://dlmf.nist.gov/4.2.E11 {\displaystyle \mathrm{e}}] : the base of the natural logarithm : [http://dlmf.nist.gov/4.2.E11 http://dlmf.nist.gov/4.2.E11]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1]
<< [[Definition:ctsqJacobi|Definition:ctsqJacobi]]
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@@ -898,8 +906,9 @@ These are defined by == Symbols List == -[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or -hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
-[http://drmf.wmflabs.org/wiki/Definition:dualqKrawtchouk {\displaystyle K_{n}}] : dual -Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:dualqKrawtchouk http://drmf.wmflabs.org/wiki/Definition:dualqKrawtchouk]
+[http://drmf.wmflabs.org/wiki/Definition:ctsqLegendre {\displaystyle P_{n}}] : continuous {\displaystyle q}-Legendre polynomial : [http://drmf.wmflabs.org/wiki/Definition:ctsqLegendre http://drmf.wmflabs.org/wiki/Definition:ctsqLegendre]
+[http://drmf.wmflabs.org/wiki/Definition:dualqKrawtchouk {\displaystyle K_{n}}] : dual {\displaystyle q}-Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:dualqKrawtchouk http://drmf.wmflabs.org/wiki/Definition:dualqKrawtchouk]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
<< [[Definition:ctsqLaguerre|Definition:ctsqLaguerre]]
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@@ -944,8 +953,10 @@ for m\ne n. == Symbols List == -[http://dlmf.nist.gov/18.3#T1.t1.r3 {\displaystyle P^{(\alpha,\beta)}_{n}}] : Jacobi polynomial : [http://dlmf.nist.gov/18.3#T1.t1.r3 http://dlmf.nist.gov/18.3#T1.t1.r3]
+[http://drmf.wmflabs.org/wiki/Definition:f {\displaystyle {f}}] : function : [http://drmf.wmflabs.org/wiki/Definition:f http://drmf.wmflabs.org/wiki/Definition:f]
[http://drmf.wmflabs.org/wiki/Definition:GenGegenbauer {\displaystyle S^{(\alpha,\beta)}_{n}}] : Generalized Gegenbauer polynomial : [http://drmf.wmflabs.org/wiki/Definition:GenGegenbauer http://drmf.wmflabs.org/wiki/Definition:GenGegenbauer]
+[http://dlmf.nist.gov/1.4#iv {\displaystyle \int}] : integral : [http://dlmf.nist.gov/1.4#iv http://dlmf.nist.gov/1.4#iv]
+[http://dlmf.nist.gov/18.3#T1.t1.r3 {\displaystyle P^{(\alpha,\beta)}_{n}}] : Jacobi polynomial : [http://dlmf.nist.gov/18.3#T1.t1.r3 http://dlmf.nist.gov/18.3#T1.t1.r3]
<< [[Definition:ctsqLegendre|Definition:ctsqLegendre]]
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@@ -995,7 +1006,8 @@ Page 156 of [[Bibliography#CHI|'''CHI''']]. == Symbols List == [http://drmf.wmflabs.org/wiki/Definition:GenHermite {\displaystyle H_n^{\mu}}] : generalized Hermite polynomial : [http://drmf.wmflabs.org/wiki/Definition:GenHermite http://drmf.wmflabs.org/wiki/Definition:GenHermite]
-[http://dlmf.nist.gov/18.3#T1.t1.r27 {\displaystyle L_n^{(\alpha)}}] : Laguerre (or generalized Laguerre) polynomial : [http://dlmf.nist.gov/18.3#T1.t1.r27 http://dlmf.nist.gov/18.3#T1.t1.r27]
+[http://dlmf.nist.gov/1.4#iv {\displaystyle \int}] : integral : [http://dlmf.nist.gov/1.4#iv http://dlmf.nist.gov/1.4#iv]
+[http://dlmf.nist.gov/18.3#T1.t1.r27 {\displaystyle L_n^{(\alpha)}}] : Laguerre (or generalized Laguerre) polynomial : [http://dlmf.nist.gov/18.3#T1.t1.r27 http://dlmf.nist.gov/18.3#T1.t1.r27]
<< [[Definition:f|Definition:f]]
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@@ -1030,7 +1042,7 @@ These are defined by == Symbols List == [http://drmf.wmflabs.org/wiki/Definition:GottliebLaguerre {\displaystyle l_{n}}] : Gottlieb-Laguerre polynomial : [http://drmf.wmflabs.org/wiki/Definition:GottliebLaguerre http://drmf.wmflabs.org/wiki/Definition:GottliebLaguerre]
-[http://dlmf.nist.gov/18.19#T1.t1.r9 {\displaystyle M_{n}}] : Meixner polynomial : [http://dlmf.nist.gov/18.19#T1.t1.r9 http://dlmf.nist.gov/18.19#T1.t1.r9]
+[http://dlmf.nist.gov/18.19#T1.t1.r9 {\displaystyle M_{n}}] : Meixner polynomial : [http://dlmf.nist.gov/18.19#T1.t1.r9 http://dlmf.nist.gov/18.19#T1.t1.r9]
<< [[Definition:GenHermite|Definition:GenHermite]]
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@@ -1064,9 +1076,10 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:JacksonqBesselI {\displaystyle J^{(1)}_{q}}] : Jackson -Bessel function 1 : [http://drmf.wmflabs.org/wiki/Definition:JacksonqBesselI http://drmf.wmflabs.org/wiki/Definition:JacksonqBesselI]
-[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or -hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
-[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]
+[http://drmf.wmflabs.org/wiki/Definition:Int {\displaystyle \int_{a}^{b}f(x){\mathrm d}\!x}] : semantic definite integral : [http://drmf.wmflabs.org/wiki/Definition:Int http://drmf.wmflabs.org/wiki/Definition:Int]
+[http://drmf.wmflabs.org/wiki/Definition:JacksonqBesselI {\displaystyle J^{(1)}_{q}}] : Jackson {\displaystyle q}-Bessel function 1 : [http://drmf.wmflabs.org/wiki/Definition:JacksonqBesselI http://drmf.wmflabs.org/wiki/Definition:JacksonqBesselI]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1]
<< [[Definition:GottliebLaguerre|Definition:GottliebLaguerre]]
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@@ -1100,9 +1113,9 @@ These are defined by == Symbols List == -[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or -hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
-[http://drmf.wmflabs.org/wiki/Definition:JacksonqBesselII {\displaystyle J^{(2)}_{q}}] : Jackson -Bessel function 2 : [http://drmf.wmflabs.org/wiki/Definition:JacksonqBesselII http://drmf.wmflabs.org/wiki/Definition:JacksonqBesselII]
-[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]
+[http://drmf.wmflabs.org/wiki/Definition:JacksonqBesselII {\displaystyle J^{(2)}_{q}}] : Jackson {\displaystyle q}-Bessel function 2 : [http://drmf.wmflabs.org/wiki/Definition:JacksonqBesselII http://drmf.wmflabs.org/wiki/Definition:JacksonqBesselII]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1]
<< [[Definition:Int|Definition:Int]]
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@@ -1135,8 +1148,9 @@ These are defined by == Symbols List == -[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or -hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
-[http://drmf.wmflabs.org/wiki/Definition:littleqLaguerre {\displaystyle p_{n}}] : little -Laguerre / Wall polynomial : [http://drmf.wmflabs.org/wiki/Definition:littleqLaguerre http://drmf.wmflabs.org/wiki/Definition:littleqLaguerre]
+[http://drmf.wmflabs.org/wiki/Definition:JacksonqBesselIII {\displaystyle J^{(3)}_{q}}] : Jackson/Hahn-Exton {\displaystyle q}-Bessel function 3 : [http://drmf.wmflabs.org/wiki/Definition:JacksonqBesselIII http://drmf.wmflabs.org/wiki/Definition:JacksonqBesselIII]
+[http://drmf.wmflabs.org/wiki/Definition:littleqLaguerre {\displaystyle p_{n}}] : little {\displaystyle q}-Laguerre / Wall polynomial : [http://drmf.wmflabs.org/wiki/Definition:littleqLaguerre http://drmf.wmflabs.org/wiki/Definition:littleqLaguerre]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
<< [[Definition:JacksonqBesselII|Definition:JacksonqBesselII]]
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@@ -1168,8 +1182,8 @@ These are defined by == Symbols List == -[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or -hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
-[http://drmf.wmflabs.org/wiki/Definition:littleqLegendre {\displaystyle p_{n}}] : little -Legendre polynomial : [http://drmf.wmflabs.org/wiki/Definition:littleqLegendre http://drmf.wmflabs.org/wiki/Definition:littleqLegendre]
+[http://drmf.wmflabs.org/wiki/Definition:littleqLegendre {\displaystyle p_{n}}] : little {\displaystyle q}-Legendre polynomial : [http://drmf.wmflabs.org/wiki/Definition:littleqLegendre http://drmf.wmflabs.org/wiki/Definition:littleqLegendre]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
<< [[Definition:JacksonqBesselIII|Definition:JacksonqBesselIII]]
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@@ -1202,9 +1216,10 @@ These are defined by == Symbols List == -[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]
-[http://drmf.wmflabs.org/wiki/Definition:AffqKrawtchouk {\displaystyle K^{\mathrm{Aff}}_{n}}] : affine -Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:AffqKrawtchouk http://drmf.wmflabs.org/wiki/Definition:AffqKrawtchouk]
-[http://drmf.wmflabs.org/wiki/Definition:monicAffqKrawtchouk {\displaystyle \widehat{K}^{\mathrm{Aff}}_{n}}] : monic affine -Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicAffqKrawtchouk http://drmf.wmflabs.org/wiki/Definition:monicAffqKrawtchouk]
+[http://drmf.wmflabs.org/wiki/Definition:AffqKrawtchouk {\displaystyle K^{\mathrm{Aff}}_{n}}] : affine {\displaystyle q}-Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:AffqKrawtchouk http://drmf.wmflabs.org/wiki/Definition:AffqKrawtchouk]
+[http://drmf.wmflabs.org/wiki/Definition:Lrselection {\displaystyle \left\{\begin{matrix}x\end{matrix}\right\}}] : bracketed generalization of {\displaystyle \pm} : [http://drmf.wmflabs.org/wiki/Definition:Lrselection http://drmf.wmflabs.org/wiki/Definition:Lrselection]
+[http://drmf.wmflabs.org/wiki/Definition:monicAffqKrawtchouk {\displaystyle \widehat{K}^{\mathrm{Aff}}_{n}}] : monic affine {\displaystyle q}-Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicAffqKrawtchouk http://drmf.wmflabs.org/wiki/Definition:monicAffqKrawtchouk]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1]
<< [[Definition:littleqLegendre|Definition:littleqLegendre]]
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@@ -1238,8 +1253,8 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:monicAlSalamCarlitzI {\displaystyle {\widehat U}^{(\alpha)}_{n}}] : monic Al-Salam-Carlitz I polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicAlSalamCarlitzI http://drmf.wmflabs.org/wiki/Definition:monicAlSalamCarlitzI]
[http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzI {\displaystyle U^{(n)}_{\alpha}}] : Al-Salam-Carlitz I polynomial : [http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzI http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzI]
+[http://drmf.wmflabs.org/wiki/Definition:monicAlSalamCarlitzI {\displaystyle {\widehat U}^{(\alpha)}_{n}}] : monic Al-Salam-Carlitz I polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicAlSalamCarlitzI http://drmf.wmflabs.org/wiki/Definition:monicAlSalamCarlitzI]
<< [[Definition:lrselection|Definition:lrselection]]
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@@ -1273,8 +1288,8 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:monicAlSalamCarlitzII {\displaystyle {\widehat V}^{(n)}_{\alpha}}] : monic Al-Salam-Carlitz II polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicAlSalamCarlitzII http://drmf.wmflabs.org/wiki/Definition:monicAlSalamCarlitzII]
[http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzII {\displaystyle V^{(n)}_{\alpha}}] : Al-Salam-Carlitz II polynomial : [http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzII http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzII]
+[http://drmf.wmflabs.org/wiki/Definition:monicAlSalamCarlitzII {\displaystyle {\widehat V}^{(n)}_{\alpha}}] : monic Al-Salam-Carlitz II polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicAlSalamCarlitzII http://drmf.wmflabs.org/wiki/Definition:monicAlSalamCarlitzII]
<< [[Definition:monicAlSalamCarlitzI|Definition:monicAlSalamCarlitzI]]
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@@ -1308,8 +1323,8 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:monicAlSalamChihara {\displaystyle {\widehat Q}_{n}}] : monic Al-Salam-Chihara polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicAlSalamChihara http://drmf.wmflabs.org/wiki/Definition:monicAlSalamChihara]
[http://drmf.wmflabs.org/wiki/Definition:AlSalamChihara {\displaystyle Q_{n}}] : Al-Salam-Chihara polynomial : [http://drmf.wmflabs.org/wiki/Definition:AlSalamChihara http://drmf.wmflabs.org/wiki/Definition:AlSalamChihara]
+[http://drmf.wmflabs.org/wiki/Definition:monicAlSalamChihara {\displaystyle {\widehat Q}_{n}}] : monic Al-Salam-Chihara polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicAlSalamChihara http://drmf.wmflabs.org/wiki/Definition:monicAlSalamChihara]
<< [[Definition:monicAlSalamCarlitzII|Definition:monicAlSalamCarlitzII]]
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@@ -1344,8 +1359,8 @@ These are defined by == Symbols List == [http://dlmf.nist.gov/18.28#E1 {\displaystyle p_{n}}] : Askey-Wilson polynomial : [http://dlmf.nist.gov/18.28#E1 http://dlmf.nist.gov/18.28#E1]
-[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]
[http://drmf.wmflabs.org/wiki/Definition:monicAskeyWilson {\displaystyle {\widehat p}_{n}}] : monic Askey-Wilson polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicAskeyWilson http://drmf.wmflabs.org/wiki/Definition:monicAskeyWilson]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1]
<< [[Definition:monicAlSalamChihara|Definition:monicAlSalamChihara]]
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@@ -1379,8 +1394,8 @@ These are defined by == Symbols List == [http://dlmf.nist.gov/18.34#E1 {\displaystyle y_{n}}] : Bessel polynomial : [http://dlmf.nist.gov/18.34#E1 http://dlmf.nist.gov/18.34#E1]
-[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii]
[http://drmf.wmflabs.org/wiki/Definition:monicBesselPoly {\displaystyle {\widehat y}_{n}}] : monic Bessel polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicBesselPoly http://drmf.wmflabs.org/wiki/Definition:monicBesselPoly]
+[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii]
<< [[Definition:monicAskeyWilson|Definition:monicAskeyWilson]]
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@@ -1414,9 +1429,9 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:monicbigqJacobi {\displaystyle {\widehat P}_{n}}] : monic big -Jacobi polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicbigqJacobi http://drmf.wmflabs.org/wiki/Definition:monicbigqJacobi]
-[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]
-[http://drmf.wmflabs.org/wiki/Definition:bigqJacobi {\displaystyle P_{n}}] : big -Jacobi polynomial : [http://drmf.wmflabs.org/wiki/Definition:bigqJacobi http://drmf.wmflabs.org/wiki/Definition:bigqJacobi]
+[http://drmf.wmflabs.org/wiki/Definition:bigqJacobi {\displaystyle P_{n}}] : big {\displaystyle q}-Jacobi polynomial : [http://drmf.wmflabs.org/wiki/Definition:bigqJacobi http://drmf.wmflabs.org/wiki/Definition:bigqJacobi]
+[http://drmf.wmflabs.org/wiki/Definition:monicbigqJacobi {\displaystyle {\widehat P}_{n}}] : monic big {\displaystyle q}-Jacobi polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicbigqJacobi http://drmf.wmflabs.org/wiki/Definition:monicbigqJacobi]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1]
<< [[Definition:monicBesselPoly|Definition:monicBesselPoly]]
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@@ -1450,9 +1465,9 @@ These are defined by == Symbols List == -[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]
-[http://drmf.wmflabs.org/wiki/Definition:bigqLaguerre {\displaystyle P_{n}}] : big -Laguerre polynomial : [http://drmf.wmflabs.org/wiki/Definition:bigqLaguerre http://drmf.wmflabs.org/wiki/Definition:bigqLaguerre]
-[http://drmf.wmflabs.org/wiki/Definition:monicbigqLaguerre {\displaystyle {\widehat P}_{n}}] : monic big -Laguerre polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicbigqLaguerre http://drmf.wmflabs.org/wiki/Definition:monicbigqLaguerre]
+[http://drmf.wmflabs.org/wiki/Definition:bigqLaguerre {\displaystyle P_{n}}] : big {\displaystyle q}-Laguerre polynomial : [http://drmf.wmflabs.org/wiki/Definition:bigqLaguerre http://drmf.wmflabs.org/wiki/Definition:bigqLaguerre]
+[http://drmf.wmflabs.org/wiki/Definition:monicbigqLaguerre {\displaystyle {\widehat P}_{n}}] : monic big {\displaystyle q}-Laguerre polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicbigqLaguerre http://drmf.wmflabs.org/wiki/Definition:monicbigqLaguerre]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1]
<< [[Definition:monicbigqJacobi|Definition:monicbigqJacobi]]
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@@ -1486,9 +1501,9 @@ These are defined by == Symbols List == -[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]
-[http://drmf.wmflabs.org/wiki/Definition:bigqLegendre {\displaystyle P_{n}}] : big -Legendre polynomial : [http://drmf.wmflabs.org/wiki/Definition:bigqLegendre http://drmf.wmflabs.org/wiki/Definition:bigqLegendre]
-[http://drmf.wmflabs.org/wiki/Definition:monicbigqLegendre {\displaystyle {\widehat P}_{n}}] : monic big -Legendre polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicbigqLegendre http://drmf.wmflabs.org/wiki/Definition:monicbigqLegendre]
+[http://drmf.wmflabs.org/wiki/Definition:bigqLegendre {\displaystyle P_{n}}] : big {\displaystyle q}-Legendre polynomial : [http://drmf.wmflabs.org/wiki/Definition:bigqLegendre http://drmf.wmflabs.org/wiki/Definition:bigqLegendre]
+[http://drmf.wmflabs.org/wiki/Definition:monicbigqLegendre {\displaystyle {\widehat P}_{n}}] : monic big {\displaystyle q}-Legendre polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicbigqLegendre http://drmf.wmflabs.org/wiki/Definition:monicbigqLegendre]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1]
<< [[Definition:monicbigqLaguerre|Definition:monicbigqLaguerre]]
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@@ -1524,7 +1539,7 @@ These are defined by == Symbols List == [http://dlmf.nist.gov/18.19#T1.t1.r11 {\displaystyle C_{n}}] : Charlier polynomial : [http://dlmf.nist.gov/18.19#T1.t1.r11 http://dlmf.nist.gov/18.19#T1.t1.r11]
-[http://drmf.wmflabs.org/wiki/Definition:monicCharlier {\displaystyle {\widehat C}_{n}}] : monic Charlier polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicCharlier http://drmf.wmflabs.org/wiki/Definition:monicCharlier]
+[http://drmf.wmflabs.org/wiki/Definition:monicCharlier {\displaystyle {\widehat C}_{n}}] : monic Charlier polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicCharlier http://drmf.wmflabs.org/wiki/Definition:monicCharlier]
<< [[Definition:monicbigqLegendre|Definition:monicbigqLegendre]]
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@@ -1560,7 +1575,7 @@ for n\ne 1 and \ChebyT{1}@{x}=\monicChebyT{1}@{x}=x. == Symbols List == [http://dlmf.nist.gov/18.3#T1.t1.r8 {\displaystyle T_{n}}] : Chebyshev polynomial of the first kind : [http://dlmf.nist.gov/18.3#T1.t1.r8 http://dlmf.nist.gov/18.3#T1.t1.r8]
-[http://drmf.wmflabs.org/wiki/Definition:monicChebyT {\displaystyle {\widehat T}_{n}}] : monic Chebyshev polynomial of the first kind : [http://drmf.wmflabs.org/wiki/Definition:monicChebyT http://drmf.wmflabs.org/wiki/Definition:monicChebyT]
+[http://drmf.wmflabs.org/wiki/Definition:monicChebyT {\displaystyle {\widehat T}_{n}}] : monic Chebyshev polynomial of the first kind : [http://drmf.wmflabs.org/wiki/Definition:monicChebyT http://drmf.wmflabs.org/wiki/Definition:monicChebyT]
<< [[Definition:monicCharlier|Definition:monicCharlier]]
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@@ -1595,7 +1610,7 @@ These are defined by == Symbols List == [http://dlmf.nist.gov/18.3#T1.t1.r11 {\displaystyle U_{n}}] : Chebyshev polynomial of the second kind : [http://dlmf.nist.gov/18.3#T1.t1.r11 http://dlmf.nist.gov/18.3#T1.t1.r11]
-[http://drmf.wmflabs.org/wiki/Definition:monicChebyU {\displaystyle {\widehat U}_{n}}] : monic Chebyshev polynomial of the second kind : [http://drmf.wmflabs.org/wiki/Definition:monicChebyU http://drmf.wmflabs.org/wiki/Definition:monicChebyU]
+[http://drmf.wmflabs.org/wiki/Definition:monicChebyU {\displaystyle {\widehat U}_{n}}] : monic Chebyshev polynomial of the second kind : [http://drmf.wmflabs.org/wiki/Definition:monicChebyU http://drmf.wmflabs.org/wiki/Definition:monicChebyU]
<< [[Definition:monicChebyT|Definition:monicChebyT]]
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@@ -1629,8 +1644,8 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:ctsbigqHermite {\displaystyle H_{n}}] : continuous big -Hermite polynomial : [http://drmf.wmflabs.org/wiki/Definition:ctsbigqHermite http://drmf.wmflabs.org/wiki/Definition:ctsbigqHermite]
-[http://drmf.wmflabs.org/wiki/Definition:monicctsbigqHermite {\displaystyle {\widehat H}_{n}}] : monic continuous big -Hermite polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicctsbigqHermite http://drmf.wmflabs.org/wiki/Definition:monicctsbigqHermite]
+[http://drmf.wmflabs.org/wiki/Definition:ctsbigqHermite {\displaystyle H_{n}}] : continuous big {\displaystyle q}-Hermite polynomial : [http://drmf.wmflabs.org/wiki/Definition:ctsbigqHermite http://drmf.wmflabs.org/wiki/Definition:ctsbigqHermite]
+[http://drmf.wmflabs.org/wiki/Definition:monicctsbigqHermite {\displaystyle {\widehat H}_{n}}] : monic continuous big {\displaystyle q}-Hermite polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicctsbigqHermite http://drmf.wmflabs.org/wiki/Definition:monicctsbigqHermite]
<< [[Definition:monicChebyU|Definition:monicChebyU]]
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@@ -1665,7 +1680,7 @@ These are defined by == Symbols List == [http://dlmf.nist.gov/18.25#T1.t1.r3 {\displaystyle S_{n}}] : continuous dual Hahn polynomial : [http://dlmf.nist.gov/18.25#T1.t1.r3 http://dlmf.nist.gov/18.25#T1.t1.r3]
-[http://drmf.wmflabs.org/wiki/Definition:monicctsdualHahn {\displaystyle {\widehat S}_{n}}] : monic continuous dual Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicctsdualHahn http://drmf.wmflabs.org/wiki/Definition:monicctsdualHahn]
+[http://drmf.wmflabs.org/wiki/Definition:monicctsdualHahn {\displaystyle {\widehat S}_{n}}] : monic continuous dual Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicctsdualHahn http://drmf.wmflabs.org/wiki/Definition:monicctsdualHahn]
<< [[Definition:monicctsbigqHermite|Definition:monicctsbigqHermite]]
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@@ -1700,8 +1715,8 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:ctsdualqHahn {\displaystyle p_{n}}] : continuous dual -Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:ctsdualqHahn http://drmf.wmflabs.org/wiki/Definition:ctsdualqHahn]
-[http://drmf.wmflabs.org/wiki/Definition:monicctsdualqHahn {\displaystyle {\widehat p}_{n}}] : monic continuous dual -Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicctsdualqHahn http://drmf.wmflabs.org/wiki/Definition:monicctsdualqHahn]
+[http://drmf.wmflabs.org/wiki/Definition:ctsdualqHahn {\displaystyle p_{n}}] : continuous dual {\displaystyle q}-Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:ctsdualqHahn http://drmf.wmflabs.org/wiki/Definition:ctsdualqHahn]
+[http://drmf.wmflabs.org/wiki/Definition:monicctsdualqHahn {\displaystyle {\widehat p}_{n}}] : monic continuous dual {\displaystyle q}-Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicctsdualqHahn http://drmf.wmflabs.org/wiki/Definition:monicctsdualqHahn]
<< [[Definition:monicctsdualHahn|Definition:monicctsdualHahn]]
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@@ -1735,9 +1750,9 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:monicctsHahn {\displaystyle {\widehat p}_{n}}] : monic continuous Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicctsHahn http://drmf.wmflabs.org/wiki/Definition:monicctsHahn]
-[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii]
[http://dlmf.nist.gov/18.19#P2.p1 {\displaystyle p_{n}}] : continuous Hahn polynomial : [http://dlmf.nist.gov/18.19#P2.p1 http://dlmf.nist.gov/18.19#P2.p1]
+[http://drmf.wmflabs.org/wiki/Definition:monicctsHahn {\displaystyle {\widehat p}_{n}}] : monic continuous Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicctsHahn http://drmf.wmflabs.org/wiki/Definition:monicctsHahn]
+[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii]
<< [[Definition:monicctsdualqHahn|Definition:monicctsdualqHahn]]
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@@ -1771,9 +1786,9 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:ctsqHahn {\displaystyle p_{n}}] : continuous -Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:ctsqHahn http://drmf.wmflabs.org/wiki/Definition:ctsqHahn]
-[http://drmf.wmflabs.org/wiki/Definition:monicctsqHahn {\displaystyle {\widehat p}_{n}}] : monic continuous -Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicctsqHahn http://drmf.wmflabs.org/wiki/Definition:monicctsqHahn]
-[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]
+[http://drmf.wmflabs.org/wiki/Definition:ctsqHahn {\displaystyle p_{n}}] : continuous {\displaystyle q}-Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:ctsqHahn http://drmf.wmflabs.org/wiki/Definition:ctsqHahn]
+[http://drmf.wmflabs.org/wiki/Definition:monicctsqHahn {\displaystyle {\widehat p}_{n}}] : monic continuous {\displaystyle q}-Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicctsqHahn http://drmf.wmflabs.org/wiki/Definition:monicctsqHahn]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1]
<< [[Definition:monicctsHahn|Definition:monicctsHahn]]
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@@ -1807,8 +1822,8 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:ctsqHermite {\displaystyle H_{n}}] : continuous -Hermite polynomial : [http://drmf.wmflabs.org/wiki/Definition:ctsqHermite http://drmf.wmflabs.org/wiki/Definition:ctsqHermite]
-[http://drmf.wmflabs.org/wiki/Definition:monicctsqHermite {\displaystyle {\widehat H}_{n}}] : monic continuous -Hermite polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicctsqHermite http://drmf.wmflabs.org/wiki/Definition:monicctsqHermite]
+[http://drmf.wmflabs.org/wiki/Definition:ctsqHermite {\displaystyle H_{n}}] : continuous {\displaystyle q}-Hermite polynomial : [http://drmf.wmflabs.org/wiki/Definition:ctsqHermite http://drmf.wmflabs.org/wiki/Definition:ctsqHermite]
+[http://drmf.wmflabs.org/wiki/Definition:monicctsqHermite {\displaystyle {\widehat H}_{n}}] : monic continuous {\displaystyle q}-Hermite polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicctsqHermite http://drmf.wmflabs.org/wiki/Definition:monicctsqHermite]
<< [[Definition:monicctsqHahn|Definition:monicctsqHahn]]
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@@ -1843,9 +1858,9 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:ctsqJacobi {\displaystyle P^{(\alpha,\beta)}_{n}}] : continuous -Jacobi polynomial : [http://drmf.wmflabs.org/wiki/Definition:ctsqJacobi http://drmf.wmflabs.org/wiki/Definition:ctsqJacobi]
-[http://drmf.wmflabs.org/wiki/Definition:monicctsqJacobi {\displaystyle {\widehat P}^{(\alpha,\beta)}_{n}}] : monic continuous -Jacobi polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicctsqJacobi http://drmf.wmflabs.org/wiki/Definition:monicctsqJacobi]
-[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]
+[http://drmf.wmflabs.org/wiki/Definition:ctsqJacobi {\displaystyle P^{(\alpha,\beta)}_{n}}] : continuous {\displaystyle q}-Jacobi polynomial : [http://drmf.wmflabs.org/wiki/Definition:ctsqJacobi http://drmf.wmflabs.org/wiki/Definition:ctsqJacobi]
+[http://drmf.wmflabs.org/wiki/Definition:monicctsqJacobi {\displaystyle {\widehat P}^{(\alpha,\beta)}_{n}}] : monic continuous {\displaystyle q}-Jacobi polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicctsqJacobi http://drmf.wmflabs.org/wiki/Definition:monicctsqJacobi]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1]
<< [[Definition:monicctsqHermite|Definition:monicctsqHermite]]
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@@ -1879,9 +1894,9 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:monicctsqLaguerre {\displaystyle {\widehat P}^{(\alpha)}_{n}}] : monic continuous -Laguerre polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicctsqLaguerre http://drmf.wmflabs.org/wiki/Definition:monicctsqLaguerre]
-[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]
-[http://drmf.wmflabs.org/wiki/Definition:ctsqLaguerre {\displaystyle P^{(n)}_{\alpha}}] : continuous -Laguerre polynomial : [http://drmf.wmflabs.org/wiki/Definition:ctsqLaguerre http://drmf.wmflabs.org/wiki/Definition:ctsqLaguerre]
+[http://drmf.wmflabs.org/wiki/Definition:ctsqLaguerre {\displaystyle P^{(n)}_{\alpha}}] : continuous {\displaystyle q}-Laguerre polynomial : [http://drmf.wmflabs.org/wiki/Definition:ctsqLaguerre http://drmf.wmflabs.org/wiki/Definition:ctsqLaguerre]
+[http://drmf.wmflabs.org/wiki/Definition:monicctsqLaguerre {\displaystyle {\widehat P}^{(\alpha)}_{n}}] : monic continuous {\displaystyle q}-Laguerre polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicctsqLaguerre http://drmf.wmflabs.org/wiki/Definition:monicctsqLaguerre]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1]
<< [[Definition:monicctsqJacobi|Definition:monicctsqJacobi]]
[[Main_Page|Main Page]]
@@ -1915,9 +1930,9 @@ These are defined by == Symbols List == -[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]
-[http://drmf.wmflabs.org/wiki/Definition:ctsqLegendre {\displaystyle P_{n}}] : continuous -Legendre polynomial : [http://drmf.wmflabs.org/wiki/Definition:ctsqLegendre http://drmf.wmflabs.org/wiki/Definition:ctsqLegendre]
-[http://drmf.wmflabs.org/wiki/Definition:monicctsqLegendre {\displaystyle {\widehat P}_{n}}] : monic continuous -Legendre polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicctsqLegendre http://drmf.wmflabs.org/wiki/Definition:monicctsqLegendre]
+[http://drmf.wmflabs.org/wiki/Definition:ctsqLegendre {\displaystyle P_{n}}] : continuous {\displaystyle q}-Legendre polynomial : [http://drmf.wmflabs.org/wiki/Definition:ctsqLegendre http://drmf.wmflabs.org/wiki/Definition:ctsqLegendre]
+[http://drmf.wmflabs.org/wiki/Definition:monicctsqLegendre {\displaystyle {\widehat P}_{n}}] : monic continuous {\displaystyle q}-Legendre polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicctsqLegendre http://drmf.wmflabs.org/wiki/Definition:monicctsqLegendre]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1]
<< [[Definition:monicctsqLaguerre|Definition:monicctsqLaguerre]]
[[Main_Page|Main Page]]
@@ -1951,9 +1966,9 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:monicctsqUltra {\displaystyle {\widehat C}_{n}}] : monic continuous -ultraspherical/Rogers polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicctsqUltra http://drmf.wmflabs.org/wiki/Definition:monicctsqUltra]
-[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]
-[http://dlmf.nist.gov/18.28#E13 {\displaystyle C_{n}}] : continuous -ultraspherical/Rogers polynomial : [http://dlmf.nist.gov/18.28#E13 http://dlmf.nist.gov/18.28#E13]
+[http://dlmf.nist.gov/18.28#E13 {\displaystyle C_{n}}] : continuous {\displaystyle q}-ultraspherical/Rogers polynomial : [http://dlmf.nist.gov/18.28#E13 http://dlmf.nist.gov/18.28#E13]
+[http://drmf.wmflabs.org/wiki/Definition:monicctsqUltra {\displaystyle {\widehat C}_{n}}] : monic continuous {\displaystyle q}-ultraspherical/Rogers polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicctsqUltra http://drmf.wmflabs.org/wiki/Definition:monicctsqUltra]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1]
<< [[Definition:monicctsqLegendre|Definition:monicctsqLegendre]]
[[Main_Page|Main Page]]
@@ -1987,8 +2002,8 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:discrqHermiteI {\displaystyle h_{n}}] : discrete -Hermite I polynomial : [http://drmf.wmflabs.org/wiki/Definition:discrqHermiteI http://drmf.wmflabs.org/wiki/Definition:discrqHermiteI]
-[http://drmf.wmflabs.org/wiki/Definition:monicdiscrqHermiteI {\displaystyle {\widehat h}_{n}}] : monic discrete -Hermite I polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicdiscrqHermiteI http://drmf.wmflabs.org/wiki/Definition:monicdiscrqHermiteI]
+[http://drmf.wmflabs.org/wiki/Definition:discrqHermiteI {\displaystyle h_{n}}] : discrete {\displaystyle q}-Hermite I polynomial : [http://drmf.wmflabs.org/wiki/Definition:discrqHermiteI http://drmf.wmflabs.org/wiki/Definition:discrqHermiteI]
+[http://drmf.wmflabs.org/wiki/Definition:monicdiscrqHermiteI {\displaystyle {\widehat h}_{n}}] : monic discrete {\displaystyle q}-Hermite I polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicdiscrqHermiteI http://drmf.wmflabs.org/wiki/Definition:monicdiscrqHermiteI]
<< [[Definition:monicctsqUltra|Definition:monicctsqUltra]]
[[Main_Page|Main Page]]
@@ -2022,8 +2037,8 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:discrqHermiteII {\displaystyle \tilde{h}_{n}}] : discrete -Hermite II polynomial : [http://drmf.wmflabs.org/wiki/Definition:discrqHermiteII http://drmf.wmflabs.org/wiki/Definition:discrqHermiteII]
-[http://drmf.wmflabs.org/wiki/Definition:monicdiscrqHermiteII {\displaystyle {\widehat \tilde{p}_{n}}}] : monic discrete -Hermite II polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicdiscrqHermiteII http://drmf.wmflabs.org/wiki/Definition:monicdiscrqHermiteII]
+[http://drmf.wmflabs.org/wiki/Definition:discrqHermiteII {\displaystyle \tilde{h}_{n}}] : discrete {\displaystyle q}-Hermite II polynomial : [http://drmf.wmflabs.org/wiki/Definition:discrqHermiteII http://drmf.wmflabs.org/wiki/Definition:discrqHermiteII]
+[http://drmf.wmflabs.org/wiki/Definition:monicdiscrqHermiteII {\displaystyle {\widehat \tilde{p}_{n}}}] : monic discrete {\displaystyle q}-Hermite II polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicdiscrqHermiteII http://drmf.wmflabs.org/wiki/Definition:monicdiscrqHermiteII]
<< [[Definition:monicdiscrqHermiteI|Definition:monicdiscrqHermiteI]]
[[Main_Page|Main Page]]
@@ -2058,9 +2073,9 @@ These are defined by == Symbols List == -[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii]
[http://dlmf.nist.gov/18.25#T1.t1.r5 {\displaystyle R_{n}}] : dual Hahn polynomial : [http://dlmf.nist.gov/18.25#T1.t1.r5 http://dlmf.nist.gov/18.25#T1.t1.r5]
[http://drmf.wmflabs.org/wiki/Definition:monicdualHahn {\displaystyle {\widehat R}_{n}}] : monic dual Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicdualHahn http://drmf.wmflabs.org/wiki/Definition:monicdualHahn]
+[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii]
<< [[Definition:monicdiscrqHermiteII|Definition:monicdiscrqHermiteII]]
[[Main_Page|Main Page]]
@@ -2094,9 +2109,9 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:dualqHahn {\displaystyle R_{n}}] : dual -Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:dualqHahn http://drmf.wmflabs.org/wiki/Definition:dualqHahn]
-[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]
-[http://drmf.wmflabs.org/wiki/Definition:monicdualqHahn {\displaystyle {\widehat R}_{n}}] : monic dual -Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicdualqHahn http://drmf.wmflabs.org/wiki/Definition:monicdualqHahn]
+[http://drmf.wmflabs.org/wiki/Definition:dualqHahn {\displaystyle R_{n}}] : dual {\displaystyle q}-Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:dualqHahn http://drmf.wmflabs.org/wiki/Definition:dualqHahn]
+[http://drmf.wmflabs.org/wiki/Definition:monicdualqHahn {\displaystyle {\widehat R}_{n}}] : monic dual {\displaystyle q}-Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicdualqHahn http://drmf.wmflabs.org/wiki/Definition:monicdualqHahn]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1]
<< [[Definition:monicdualHahn|Definition:monicdualHahn]]
[[Main_Page|Main Page]]
@@ -2130,9 +2145,9 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:monicdualqKrawtchouk {\displaystyle {\widehat K}_{n}}] : monic dual -Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicdualqKrawtchouk http://drmf.wmflabs.org/wiki/Definition:monicdualqKrawtchouk]
-[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]
-[http://drmf.wmflabs.org/wiki/Definition:dualqKrawtchouk {\displaystyle K_{n}}] : dual -Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:dualqKrawtchouk http://drmf.wmflabs.org/wiki/Definition:dualqKrawtchouk]
+[http://drmf.wmflabs.org/wiki/Definition:dualqKrawtchouk {\displaystyle K_{n}}] : dual {\displaystyle q}-Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:dualqKrawtchouk http://drmf.wmflabs.org/wiki/Definition:dualqKrawtchouk]
+[http://drmf.wmflabs.org/wiki/Definition:monicdualqKrawtchouk {\displaystyle {\widehat K}_{n}}] : monic dual {\displaystyle q}-Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicdualqKrawtchouk http://drmf.wmflabs.org/wiki/Definition:monicdualqKrawtchouk]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1]
<< [[Definition:monicdualqHahn|Definition:monicdualqHahn]]
[[Main_Page|Main Page]]
@@ -2166,9 +2181,9 @@ These are defined by == Symbols List == -[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii]
[http://dlmf.nist.gov/18.19#T1.t1.r3 {\displaystyle Q_{n}}] : Hahn polynomial : [http://dlmf.nist.gov/18.19#T1.t1.r3 http://dlmf.nist.gov/18.19#T1.t1.r3]
[http://drmf.wmflabs.org/wiki/Definition:monicHahn {\displaystyle {\widehat Q}_{n}}] : monic Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicHahn http://drmf.wmflabs.org/wiki/Definition:monicHahn]
+[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii]
<< [[Definition:monicdualqKrawtchouk|Definition:monicdualqKrawtchouk]]
[[Main_Page|Main Page]]
@@ -2202,8 +2217,8 @@ These are defined by == Symbols List == -[http://dlmf.nist.gov/18.3#T1.t1.r28 {\displaystyle H_{n}}] : Hermite polynomial : [http://dlmf.nist.gov/18.3#T1.t1.r28 http://dlmf.nist.gov/18.3#T1.t1.r28]
-[http://drmf.wmflabs.org/wiki/Definition:monicHermite {\displaystyle {\widehat H}_{n}}] : monic Hermite polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicHermite http://drmf.wmflabs.org/wiki/Definition:monicHermite]
+[http://dlmf.nist.gov/18.3#T1.t1.r28 {\displaystyle H_{n}}] : Hermite polynomial {\displaystyle H_n} : [http://dlmf.nist.gov/18.3#T1.t1.r28 http://dlmf.nist.gov/18.3#T1.t1.r28]
+[http://drmf.wmflabs.org/wiki/Definition:monicHermite {\displaystyle {\widehat H}_{n}}] : monic Hermite polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicHermite http://drmf.wmflabs.org/wiki/Definition:monicHermite]
<< [[Definition:monicHahn|Definition:monicHahn]]
[[Main_Page|Main Page]]
@@ -2239,8 +2254,8 @@ These are defined by == Symbols List == [http://dlmf.nist.gov/18.3#T1.t1.r3 {\displaystyle P^{(\alpha,\beta)}_{n}}] : Jacobi polynomial : [http://dlmf.nist.gov/18.3#T1.t1.r3 http://dlmf.nist.gov/18.3#T1.t1.r3]
-[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii]
[http://drmf.wmflabs.org/wiki/Definition:monicJacobi {\displaystyle {\widehat P}^{(\alpha,\beta)}_{n}}] : monic Jacobi polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicJacobi http://drmf.wmflabs.org/wiki/Definition:monicJacobi]
+[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii]
<< [[Definition:monicHermite|Definition:monicHermite]]
[[Main_Page|Main Page]]
@@ -2276,7 +2291,7 @@ These are defined by [http://dlmf.nist.gov/18.19#T1.t1.r6 {\displaystyle K_{n}}] : Krawtchouk polynomial : [http://dlmf.nist.gov/18.19#T1.t1.r6 http://dlmf.nist.gov/18.19#T1.t1.r6]
[http://drmf.wmflabs.org/wiki/Definition:monicKrawtchouk {\displaystyle {\widehat K}_{n}}] : monic Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicKrawtchouk http://drmf.wmflabs.org/wiki/Definition:monicKrawtchouk]
-[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii]
+[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii]
<< [[Definition:monicJacobi|Definition:monicJacobi]]
[[Main_Page|Main Page]]
@@ -2310,8 +2325,8 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:monicLaguerre {\displaystyle {\widehat L}^{\alpha}_n}] : monic Laguerre polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicLaguerre http://drmf.wmflabs.org/wiki/Definition:monicLaguerre]
[http://dlmf.nist.gov/18.3#T1.t1.r27 {\displaystyle L_n^{(\alpha)}}] : Laguerre (or generalized Laguerre) polynomial : [http://dlmf.nist.gov/18.3#T1.t1.r27 http://dlmf.nist.gov/18.3#T1.t1.r27]
+[http://drmf.wmflabs.org/wiki/Definition:monicLaguerre {\displaystyle {\widehat L}^{\alpha}_n}] : monic Laguerre polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicLaguerre http://drmf.wmflabs.org/wiki/Definition:monicLaguerre]
<< [[Definition:monicKrawtchouk|Definition:monicKrawtchouk]]
[[Main_Page|Main Page]]
@@ -2346,9 +2361,9 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:monicLegendrePoly {\displaystyle {\widehat P}_{n}}] : monic Legendre polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicLegendrePoly http://drmf.wmflabs.org/wiki/Definition:monicLegendrePoly]
+[http://dlmf.nist.gov/1.2#E1 {\displaystyle \binom{n}{k}}] : binomial coefficient : [http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/1.2#E1] [http://dlmf.nist.gov/26.3#SS1.p1 http://dlmf.nist.gov/26.3#SS1.p1]
[http://dlmf.nist.gov/18.3#T1.t1.r25 {\displaystyle P_{n}}] : Legendre polynomial : [http://dlmf.nist.gov/18.3#T1.t1.r25 http://dlmf.nist.gov/18.3#T1.t1.r25]
-[http://dlmf.nist.gov/1.2#E1 {\displaystyle \binom{n}{k}}] : binomial coefficient : [http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/1.2#E1]
+[http://drmf.wmflabs.org/wiki/Definition:monicLegendrePoly {\displaystyle {\widehat P}_{n}}] : monic Legendre polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicLegendrePoly http://drmf.wmflabs.org/wiki/Definition:monicLegendrePoly]
<< [[Definition:monicLaguerre|Definition:monicLaguerre]]
[[Main_Page|Main Page]]
@@ -2382,10 +2397,10 @@ These are defined by == Symbols List == -[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]
-[http://dlmf.nist.gov/1.2#E1 {\displaystyle \binom{n}{k}}] : binomial coefficient : [http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/1.2#E1]
-[http://drmf.wmflabs.org/wiki/Definition:moniclittleqJacobi {\displaystyle {\widehat p}_{n}}] : monic little -Jacobi polynomial : [http://drmf.wmflabs.org/wiki/Definition:moniclittleqJacobi http://drmf.wmflabs.org/wiki/Definition:moniclittleqJacobi]
-[http://drmf.wmflabs.org/wiki/Definition:littleqJacobi {\displaystyle p_{n}}] : little -Jacobi polynomial : [http://drmf.wmflabs.org/wiki/Definition:littleqJacobi http://drmf.wmflabs.org/wiki/Definition:littleqJacobi]
+[http://dlmf.nist.gov/1.2#E1 {\displaystyle \binom{n}{k}}] : binomial coefficient : [http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/1.2#E1] [http://dlmf.nist.gov/26.3#SS1.p1 http://dlmf.nist.gov/26.3#SS1.p1]
+[http://drmf.wmflabs.org/wiki/Definition:littleqJacobi {\displaystyle p_{n}}] : little {\displaystyle q}-Jacobi polynomial : [http://drmf.wmflabs.org/wiki/Definition:littleqJacobi http://drmf.wmflabs.org/wiki/Definition:littleqJacobi]
+[http://drmf.wmflabs.org/wiki/Definition:moniclittleqJacobi {\displaystyle {\widehat p}_{n}}] : monic little {\displaystyle q}-Jacobi polynomial : [http://drmf.wmflabs.org/wiki/Definition:moniclittleqJacobi http://drmf.wmflabs.org/wiki/Definition:moniclittleqJacobi]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1]
<< [[Definition:monicLegendrePoly|Definition:monicLegendrePoly]]
[[Main_Page|Main Page]]
@@ -2418,7 +2433,10 @@ These are defined by == Symbols List == - +[http://dlmf.nist.gov/1.2#E1 {\displaystyle \binom{n}{k}}] : binomial coefficient : [http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/1.2#E1] [http://dlmf.nist.gov/26.3#SS1.p1 http://dlmf.nist.gov/26.3#SS1.p1]
+[http://drmf.wmflabs.org/wiki/Definition:littleqLaguerre {\displaystyle p_{n}}] : little {\displaystyle q}-Laguerre / Wall polynomial : [http://drmf.wmflabs.org/wiki/Definition:littleqLaguerre http://drmf.wmflabs.org/wiki/Definition:littleqLaguerre]
+[http://drmf.wmflabs.org/wiki/Definition:moniclittleqLaguerre {\displaystyle {\widehat p}_{n}}] : monic little {\displaystyle q}-Laguerre/Wall polynomial : [http://drmf.wmflabs.org/wiki/Definition:moniclittleqLaguerre http://drmf.wmflabs.org/wiki/Definition:moniclittleqLaguerre]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1]
<< [[Definition:moniclittleqJacobi|Definition:moniclittleqJacobi]]
[[Main_Page|Main Page]]
@@ -2452,10 +2470,10 @@ These are defined by == Symbols List == -[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]
-[http://dlmf.nist.gov/1.2#E1 {\displaystyle \binom{n}{k}}] : binomial coefficient : [http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/1.2#E1]
-[http://drmf.wmflabs.org/wiki/Definition:littleqLegendre {\displaystyle p_{n}}] : little -Legendre polynomial : [http://drmf.wmflabs.org/wiki/Definition:littleqLegendre http://drmf.wmflabs.org/wiki/Definition:littleqLegendre]
-[http://drmf.wmflabs.org/wiki/Definition:moniclittleqLegendre {\displaystyle {\widehat p}_{n}}] : monic little -Legendre polynomial : [http://drmf.wmflabs.org/wiki/Definition:moniclittleqLegendre http://drmf.wmflabs.org/wiki/Definition:moniclittleqLegendre]
+[http://dlmf.nist.gov/1.2#E1 {\displaystyle \binom{n}{k}}] : binomial coefficient : [http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/1.2#E1] [http://dlmf.nist.gov/26.3#SS1.p1 http://dlmf.nist.gov/26.3#SS1.p1]
+[http://drmf.wmflabs.org/wiki/Definition:littleqLegendre {\displaystyle p_{n}}] : little {\displaystyle q}-Legendre polynomial : [http://drmf.wmflabs.org/wiki/Definition:littleqLegendre http://drmf.wmflabs.org/wiki/Definition:littleqLegendre]
+[http://drmf.wmflabs.org/wiki/Definition:moniclittleqLegendre {\displaystyle {\widehat p}_{n}}] : monic little {\displaystyle q}-Legendre polynomial : [http://drmf.wmflabs.org/wiki/Definition:moniclittleqLegendre http://drmf.wmflabs.org/wiki/Definition:moniclittleqLegendre]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1]
<< [[Definition:moniclittleqLaguerre|Definition:moniclittleqLaguerre]]
[[Main_Page|Main Page]]
@@ -2489,9 +2507,9 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:monicMeixner {\displaystyle {\widehat M}_{n}}] : monic Meixner polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicMeixner http://drmf.wmflabs.org/wiki/Definition:monicMeixner]
-[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii]
[http://dlmf.nist.gov/18.19#T1.t1.r9 {\displaystyle M_{n}}] : Meixner polynomial : [http://dlmf.nist.gov/18.19#T1.t1.r9 http://dlmf.nist.gov/18.19#T1.t1.r9]
+[http://drmf.wmflabs.org/wiki/Definition:monicMeixner {\displaystyle {\widehat M}_{n}}] : monic Meixner polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicMeixner http://drmf.wmflabs.org/wiki/Definition:monicMeixner]
+[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii]
<< [[Definition:moniclittleqLegendre|Definition:moniclittleqLegendre]]
[[Main_Page|Main Page]]
@@ -2526,9 +2544,9 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:monicMeixnerPollaczek {\displaystyle {\widehat P}^{(\alpha)}_{n}}] : monic Meixner-Pollaczek polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicMeixnerPollaczek http://drmf.wmflabs.org/wiki/Definition:monicMeixnerPollaczek]
[http://dlmf.nist.gov/18.19#P3.p1 {\displaystyle P^{(\alpha)}_{n}}] : Meixner-Pollaczek polynomial : [http://dlmf.nist.gov/18.19#P3.p1 http://dlmf.nist.gov/18.19#P3.p1]
-[http://dlmf.nist.gov/4.14#E1 {\displaystyle \mathrm{sin}}] : sine function : [http://dlmf.nist.gov/4.14#E1 http://dlmf.nist.gov/4.14#E1]
+[http://drmf.wmflabs.org/wiki/Definition:monicMeixnerPollaczek {\displaystyle {\widehat P}^{(\alpha)}_{n}}] : monic Meixner-Pollaczek polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicMeixnerPollaczek http://drmf.wmflabs.org/wiki/Definition:monicMeixnerPollaczek]
+[http://dlmf.nist.gov/4.14#E1 {\displaystyle \mathrm{sin}}] : sine function : [http://dlmf.nist.gov/4.14#E1 http://dlmf.nist.gov/4.14#E1]
<< [[Definition:monicMeixner|Definition:monicMeixner]]
[[Main_Page|Main Page]]
@@ -2563,7 +2581,7 @@ These are defined by == Symbols List == [http://drmf.wmflabs.org/wiki/Definition:monicpseudoJacobi {\displaystyle {\widehat P}_{n}}] : monic pseudo-Jacobi polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicpseudoJacobi http://drmf.wmflabs.org/wiki/Definition:monicpseudoJacobi]
-[http://drmf.wmflabs.org/wiki/Definition:pseudoJacobi {\displaystyle P_{n}}] : pseudo Jacobi polynomial : [http://drmf.wmflabs.org/wiki/Definition:pseudoJacobi http://drmf.wmflabs.org/wiki/Definition:pseudoJacobi]
+[http://drmf.wmflabs.org/wiki/Definition:pseudoJacobi {\displaystyle P_{n}}] : pseudo Jacobi polynomial : [http://drmf.wmflabs.org/wiki/Definition:pseudoJacobi http://drmf.wmflabs.org/wiki/Definition:pseudoJacobi]
<< [[Definition:monicMeixnerPollaczek|Definition:monicMeixnerPollaczek]]
[[Main_Page|Main Page]]
@@ -2597,10 +2615,10 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:qBessel {\displaystyle y_{n}}] : -Bessel polynomial : [http://drmf.wmflabs.org/wiki/Definition:qBessel http://drmf.wmflabs.org/wiki/Definition:qBessel]
-[http://dlmf.nist.gov/1.2#E1 {\displaystyle \binom{n}{k}}] : binomial coefficient : [http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/1.2#E1]
-[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]
-[http://drmf.wmflabs.org/wiki/Definition:monicqBessel {\displaystyle {\widehat y}_{n}}] : monic -Bessel polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicqBessel http://drmf.wmflabs.org/wiki/Definition:monicqBessel]
+[http://dlmf.nist.gov/1.2#E1 {\displaystyle \binom{n}{k}}] : binomial coefficient : [http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/1.2#E1] [http://dlmf.nist.gov/26.3#SS1.p1 http://dlmf.nist.gov/26.3#SS1.p1]
+[http://drmf.wmflabs.org/wiki/Definition:monicqBessel {\displaystyle {\widehat y}_{n}}] : monic {\displaystyle q}-Bessel polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicqBessel http://drmf.wmflabs.org/wiki/Definition:monicqBessel]
+[http://drmf.wmflabs.org/wiki/Definition:qBessel {\displaystyle y_{n}}] : {\displaystyle q}-Bessel polynomial : [http://drmf.wmflabs.org/wiki/Definition:qBessel http://drmf.wmflabs.org/wiki/Definition:qBessel]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1]
<< [[Definition:monicpseudoJacobi|Definition:monicpseudoJacobi]]
[[Main_Page|Main Page]]
@@ -2635,9 +2653,10 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:monicqHahn {\displaystyle {\widehat Q}_{n}}] : monic -Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicqHahn http://drmf.wmflabs.org/wiki/Definition:monicqHahn]
-[http://drmf.wmflabs.org/wiki/Definition:qHahn {\displaystyle Q_{n}}] : -Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:qHahn http://drmf.wmflabs.org/wiki/Definition:qHahn]
-[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]
+[http://drmf.wmflabs.org/wiki/Definition:monicqCharlier {\displaystyle {\widehat C}_{n}}] : monic {\displaystyle q}-Charlier polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicqCharlier http://drmf.wmflabs.org/wiki/Definition:monicqCharlier]
+[http://drmf.wmflabs.org/wiki/Definition:monicqHahn {\displaystyle {\widehat Q}_{n}}] : monic {\displaystyle q}-Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicqHahn http://drmf.wmflabs.org/wiki/Definition:monicqHahn]
+[http://drmf.wmflabs.org/wiki/Definition:qHahn {\displaystyle Q_{n}}] : {\displaystyle q}-Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:qHahn http://drmf.wmflabs.org/wiki/Definition:qHahn]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1]
<< [[Definition:monicqBesselPoly|Definition:monicqBesselPoly]]
[[Main_Page|Main Page]]
@@ -2671,9 +2690,9 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:monicqKrawtchouk {\displaystyle {\widehat K}_{n}}] : monic -Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicqKrawtchouk http://drmf.wmflabs.org/wiki/Definition:monicqKrawtchouk]
-[http://drmf.wmflabs.org/wiki/Definition:qKrawtchouk {\displaystyle K_{n}}] : -Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:qKrawtchouk http://drmf.wmflabs.org/wiki/Definition:qKrawtchouk]
-[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]
+[http://drmf.wmflabs.org/wiki/Definition:monicqKrawtchouk {\displaystyle {\widehat K}_{n}}] : monic {\displaystyle q}-Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicqKrawtchouk http://drmf.wmflabs.org/wiki/Definition:monicqKrawtchouk]
+[http://drmf.wmflabs.org/wiki/Definition:qKrawtchouk {\displaystyle K_{n}}] : {\displaystyle q}-Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:qKrawtchouk http://drmf.wmflabs.org/wiki/Definition:qKrawtchouk]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1]
<< [[Definition:monicqCharlier|Definition:monicqCharlier]]
[[Main_Page|Main Page]]
@@ -2707,9 +2726,9 @@ These are defined by == Symbols List == -[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]
-[http://drmf.wmflabs.org/wiki/Definition:monicqLaguerre {\displaystyle {\widehat L}^{(\alpha)}_{n}}] : monic -Laguerre polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicqLaguerre http://drmf.wmflabs.org/wiki/Definition:monicqLaguerre]
-[http://drmf.wmflabs.org/wiki/Definition:qLaguerre {\displaystyle L_n^{(\alpha)}}] : -Laguerre polynomial : [http://drmf.wmflabs.org/wiki/Definition:qLaguerre http://drmf.wmflabs.org/wiki/Definition:qLaguerre]
+[http://drmf.wmflabs.org/wiki/Definition:monicqLaguerre {\displaystyle {\widehat L}^{(\alpha)}_{n}}] : monic {\displaystyle q}-Laguerre polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicqLaguerre http://drmf.wmflabs.org/wiki/Definition:monicqLaguerre]
+[http://drmf.wmflabs.org/wiki/Definition:qLaguerre {\displaystyle L_n^{(\alpha)}}] : {\displaystyle q}-Laguerre polynomial : [http://drmf.wmflabs.org/wiki/Definition:qLaguerre http://drmf.wmflabs.org/wiki/Definition:qLaguerre]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1]
<< [[Definition:monicqKrawtchouk|Definition:monicqKrawtchouk]]
[[Main_Page|Main Page]]
@@ -2742,9 +2761,9 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:monicqMeixner {\displaystyle {\widehat M}_{n}}] : monic -Meixner polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicqMeixner http://drmf.wmflabs.org/wiki/Definition:monicqMeixner]
-[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]
-[http://drmf.wmflabs.org/wiki/Definition:qMeixner {\displaystyle M_{n}}] : -Meixner polynomial : [http://drmf.wmflabs.org/wiki/Definition:qMeixner http://drmf.wmflabs.org/wiki/Definition:qMeixner]
+[http://drmf.wmflabs.org/wiki/Definition:monicqMeixner {\displaystyle {\widehat M}_{n}}] : monic {\displaystyle q}-Meixner polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicqMeixner http://drmf.wmflabs.org/wiki/Definition:monicqMeixner]
+[http://drmf.wmflabs.org/wiki/Definition:qMeixner {\displaystyle M_{n}}] : {\displaystyle q}-Meixner polynomial : [http://drmf.wmflabs.org/wiki/Definition:qMeixner http://drmf.wmflabs.org/wiki/Definition:qMeixner]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1]
<< [[Definition:monicqLaguerre|Definition:monicqLaguerre]]
[[Main_Page|Main Page]]
@@ -2778,9 +2797,9 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:monicqMeixnerPollaczek {\displaystyle {\widehat P}_{n}}] : monic -Meixner-Pollaczek polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicqMeixnerPollaczek http://drmf.wmflabs.org/wiki/Definition:monicqMeixnerPollaczek]
-[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]
-[http://drmf.wmflabs.org/wiki/Definition:qMeixnerPollaczek {\displaystyle P_{n}}] : -Meixner-Pollaczek polynomial : [http://drmf.wmflabs.org/wiki/Definition:qMeixnerPollaczek http://drmf.wmflabs.org/wiki/Definition:qMeixnerPollaczek]
+[http://drmf.wmflabs.org/wiki/Definition:monicqMeixnerPollaczek {\displaystyle {\widehat P}_{n}}] : monic {\displaystyle q}-Meixner-Pollaczek polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicqMeixnerPollaczek http://drmf.wmflabs.org/wiki/Definition:monicqMeixnerPollaczek]
+[http://drmf.wmflabs.org/wiki/Definition:qMeixnerPollaczek {\displaystyle P_{n}}] : {\displaystyle q}-Meixner-Pollaczek polynomial : [http://drmf.wmflabs.org/wiki/Definition:qMeixnerPollaczek http://drmf.wmflabs.org/wiki/Definition:qMeixnerPollaczek]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1]
<< [[Definition:monicqMeixner|Definition:monicqMeixner]]
[[Main_Page|Main Page]]
@@ -2815,9 +2834,9 @@ These are defined by == Symbols List == -[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]
-[http://dlmf.nist.gov/18.28#E19 {\displaystyle R_{n}}] : -Racah polynomial : [http://dlmf.nist.gov/18.28#E19 http://dlmf.nist.gov/18.28#E19]
-[http://drmf.wmflabs.org/wiki/Definition:monicqRacah {\displaystyle {\widehat R}_{n}}] : monic -Racah polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicqRacah http://drmf.wmflabs.org/wiki/Definition:monicqRacah]
+[http://drmf.wmflabs.org/wiki/Definition:monicqRacah {\displaystyle {\widehat R}_{n}}] : monic {\displaystyle q}-Racah polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicqRacah http://drmf.wmflabs.org/wiki/Definition:monicqRacah]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1]
+[http://dlmf.nist.gov/18.28#E19 {\displaystyle R_{n}}] : {\displaystyle q}-Racah polynomial : [http://dlmf.nist.gov/18.28#E19 http://dlmf.nist.gov/18.28#E19]
<< [[Definition:monicqMeixnerPollaczek|Definition:monicqMeixnerPollaczek]]
[[Main_Page|Main Page]]
@@ -2851,9 +2870,9 @@ These are defined by == Symbols List == -[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]
-[http://drmf.wmflabs.org/wiki/Definition:qtmqKrawtchouk {\displaystyle K^{\mathrm{qtm}}_{n}}] : quantum -Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:qtmqKrawtchouk http://drmf.wmflabs.org/wiki/Definition:qtmqKrawtchouk]
-[http://drmf.wmflabs.org/wiki/Definition:monicqtmqKrawtchouk {\displaystyle {\widehat K^{\mathrm{qtm}}}_{n}}] : monic quantum -Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicqtmqKrawtchouk http://drmf.wmflabs.org/wiki/Definition:monicqtmqKrawtchouk]
+[http://drmf.wmflabs.org/wiki/Definition:monicqtmqKrawtchouk {\displaystyle {\widehat K^{\mathrm{qtm}}}_{n}}] : monic quantum {\displaystyle q}-Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicqtmqKrawtchouk http://drmf.wmflabs.org/wiki/Definition:monicqtmqKrawtchouk]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1]
+[http://drmf.wmflabs.org/wiki/Definition:qtmqKrawtchouk {\displaystyle K^{\mathrm{qtm}}_{n}}] : quantum {\displaystyle q}-Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:qtmqKrawtchouk http://drmf.wmflabs.org/wiki/Definition:qtmqKrawtchouk]
<< [[Definition:monicqRacah|Definition:monicqRacah]]
[[Main_Page|Main Page]]
@@ -2888,9 +2907,9 @@ These are defined by == Symbols List == -[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii]
-[http://dlmf.nist.gov/18.25#T1.t1.r4 {\displaystyle R_{n}}] : Racah polynomial : [http://dlmf.nist.gov/18.25#T1.t1.r4 http://dlmf.nist.gov/18.25#T1.t1.r4]
[http://drmf.wmflabs.org/wiki/Definition:monicRacah {\displaystyle {\widehat R}_{n}}] : monic Racah polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicRacah http://drmf.wmflabs.org/wiki/Definition:monicRacah]
+[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii]
+[http://dlmf.nist.gov/18.25#T1.t1.r4 {\displaystyle R_{n}}] : Racah polynomial : [http://dlmf.nist.gov/18.25#T1.t1.r4 http://dlmf.nist.gov/18.25#T1.t1.r4]
<< [[Definition:monicqtmqKrawtchouk|Definition:monicqtmqKrawtchouk]]
[[Main_Page|Main Page]]
@@ -2924,9 +2943,9 @@ These are defined by == Symbols List == -[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]
[http://drmf.wmflabs.org/wiki/Definition:monicStieltjesWigert {\displaystyle {\widehat S}_{n}}] : monic Stieltjes-Wigert polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicStieltjesWigert http://drmf.wmflabs.org/wiki/Definition:monicStieltjesWigert]
-[http://drmf.wmflabs.org/wiki/Definition:StieltjesWigert {\displaystyle S_{n}}] : Stieltjes-Wigert polynomial : [http://drmf.wmflabs.org/wiki/Definition:StieltjesWigert http://drmf.wmflabs.org/wiki/Definition:StieltjesWigert]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1]
+[http://drmf.wmflabs.org/wiki/Definition:StieltjesWigert {\displaystyle S_{n}}] : Stieltjes-Wigert polynomial : [http://drmf.wmflabs.org/wiki/Definition:StieltjesWigert http://drmf.wmflabs.org/wiki/Definition:StieltjesWigert]
<< [[Definition:monicRacah|Definition:monicRacah]]
[[Main_Page|Main Page]]
@@ -2961,8 +2980,8 @@ These are defined by == Symbols List == [http://drmf.wmflabs.org/wiki/Definition:monicUltra {\displaystyle {\widehat C}^{\mu}_{n}}] : monic ultraspherical/Gegenbauer polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicUltra http://drmf.wmflabs.org/wiki/Definition:monicUltra]
-[http://dlmf.nist.gov/18.3#T1.t1.r5 {\displaystyle C^{\mu}_{n}}] : ultraspherical/Gegenbauer polynomial : [http://dlmf.nist.gov/18.3#T1.t1.r5 http://dlmf.nist.gov/18.3#T1.t1.r5]
[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii]
+[http://dlmf.nist.gov/18.3#T1.t1.r5 {\displaystyle C^{\mu}_{n}}] : ultraspherical/Gegenbauer polynomial : [http://dlmf.nist.gov/18.3#T1.t1.r5 http://dlmf.nist.gov/18.3#T1.t1.r5]
<< [[Definition:monicStieltjesWigert|Definition:monicStieltjesWigert]]
[[Main_Page|Main Page]]
@@ -2996,9 +3015,9 @@ These are defined by == Symbols List == -[http://dlmf.nist.gov/18.25#T1.t1.r2 {\displaystyle W_{n}}] : Wilson polynomial : [http://dlmf.nist.gov/18.25#T1.t1.r2 http://dlmf.nist.gov/18.25#T1.t1.r2]
-[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii]
[http://drmf.wmflabs.org/wiki/Definition:monicWilson {\displaystyle {\widehat W}_{n}}] : monic Wilson polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicWilson http://drmf.wmflabs.org/wiki/Definition:monicWilson]
+[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii]
+[http://dlmf.nist.gov/18.25#T1.t1.r2 {\displaystyle W_{n}}] : Wilson polynomial : [http://dlmf.nist.gov/18.25#T1.t1.r2 http://dlmf.nist.gov/18.25#T1.t1.r2]
<< [[Definition:monicUltra|Definition:monicUltra]]
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@@ -3028,8 +3047,8 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:NeumannFactor {\displaystyle \epsilon_{m}}] : Neumann factor : [http://drmf.wmflabs.org/wiki/Definition:NeumannFactor http://drmf.wmflabs.org/wiki/Definition:NeumannFactor]
[http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r4 {\displaystyle \delta_{m,n}}] : Kronecker delta : [http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r4 http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r4]
+[http://drmf.wmflabs.org/wiki/Definition:NeumannFactor {\displaystyle \epsilon_{m}}] : Neumann factor : [http://drmf.wmflabs.org/wiki/Definition:NeumannFactor http://drmf.wmflabs.org/wiki/Definition:NeumannFactor]
<< [[Definition:monicWilson|Definition:monicWilson]]
[[Main_Page|Main Page]]
@@ -3061,8 +3080,8 @@ These are defined by == Symbols List == [http://dlmf.nist.gov/18.25#T1.t1.r3 {\displaystyle S_{n}}] : continuous dual Hahn polynomial : [http://dlmf.nist.gov/18.25#T1.t1.r3 http://dlmf.nist.gov/18.25#T1.t1.r3]
-[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii]
-[http://drmf.wmflabs.org/wiki/Definition:normctsdualHahnStilde {\displaystyle {\tilde S}_{n}}] : normalized continuous dual Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:normctsdualHahnStilde http://drmf.wmflabs.org/wiki/Definition:normctsdualHahnStilde]
+[http://drmf.wmflabs.org/wiki/Definition:normctsdualHahnStilde {\displaystyle {\tilde S}_{n}}] : normalized continuous dual Hahn polynomial {\displaystyle {\tilde S}} : [http://drmf.wmflabs.org/wiki/Definition:normctsdualHahnStilde http://drmf.wmflabs.org/wiki/Definition:normctsdualHahnStilde]
+[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii]
<< [[Definition:NeumannFactor|Definition:NeumannFactor]]
[[Main_Page|Main Page]]
@@ -3094,9 +3113,9 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:normctsHahnptilde {\displaystyle {\tilde p}_{n}}] : normalized continuous Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:normctsHahnptilde http://drmf.wmflabs.org/wiki/Definition:normctsHahnptilde]
[http://dlmf.nist.gov/18.19#P2.p1 {\displaystyle p_{n}}] : continuous Hahn polynomial : [http://dlmf.nist.gov/18.19#P2.p1 http://dlmf.nist.gov/18.19#P2.p1]
-[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii]
+[http://drmf.wmflabs.org/wiki/Definition:normctsHahnptilde {\displaystyle {\tilde p}_{n}}] : normalized continuous Hahn polynomial {\displaystyle {\tilde p}} : [http://drmf.wmflabs.org/wiki/Definition:normctsHahnptilde http://drmf.wmflabs.org/wiki/Definition:normctsHahnptilde]
+[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii]
<< [[Definition:normctsdualHahnStilde|Definition:normctsdualHahnStilde]]
[[Main_Page|Main Page]]
@@ -3135,7 +3154,7 @@ where [http://dlmf.nist.gov/18.3#T1.t1.r3 {\displaystyle P^{(\alpha,\beta)}_{n}}] : Jacobi polynomial : [http://dlmf.nist.gov/18.3#T1.t1.r3 http://dlmf.nist.gov/18.3#T1.t1.r3]
[http://drmf.wmflabs.org/wiki/Definition:normJacobiR {\displaystyle R^{(\alpha,\beta)}_{n}}] : normalized Jacobi polynomial : [http://drmf.wmflabs.org/wiki/Definition:normJacobiR http://drmf.wmflabs.org/wiki/Definition:normJacobiR]
-[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii]
+[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii]
<< [[Definition:normctsHahnptilde|Definition:normctsHahnptilde]]
[[Main_Page|Main Page]]
@@ -3170,9 +3189,9 @@ These are defined by == Symbols List == -[http://dlmf.nist.gov/18.25#T1.t1.r2 {\displaystyle W_{n}}] : Wilson polynomial : [http://dlmf.nist.gov/18.25#T1.t1.r2 http://dlmf.nist.gov/18.25#T1.t1.r2]
+[http://drmf.wmflabs.org/wiki/Definition:normWilsonWtilde {\displaystyle {\tilde W}_{n}}] : normalized Wilson polynomial {\displaystyle {\tilde W}} : [http://drmf.wmflabs.org/wiki/Definition:normWilsonWtilde http://drmf.wmflabs.org/wiki/Definition:normWilsonWtilde]
[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii]
-[http://drmf.wmflabs.org/wiki/Definition:normWilsonWtilde {\displaystyle {\tilde W}_{n}}] : normalized Wilson polynomial : [http://drmf.wmflabs.org/wiki/Definition:normWilsonWtilde http://drmf.wmflabs.org/wiki/Definition:normWilsonWtilde]
+[http://dlmf.nist.gov/18.25#T1.t1.r2 {\displaystyle W_{n}}] : Wilson polynomial : [http://dlmf.nist.gov/18.25#T1.t1.r2 http://dlmf.nist.gov/18.25#T1.t1.r2]
<< [[Definition:normJacobiR|Definition:normJacobiR]]
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@@ -3207,7 +3226,8 @@ These are defined by [http://dlmf.nist.gov/16.2#E1 {\displaystyle {{}_{p}F_{q}}}] : generalized hypergeometric function : [http://dlmf.nist.gov/16.2#E1 http://dlmf.nist.gov/16.2#E1]
[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii]
-[http://drmf.wmflabs.org/wiki/Definition:pseudoJacobi {\displaystyle P_{n}}] : pseudo Jacobi polynomial : [http://drmf.wmflabs.org/wiki/Definition:pseudoJacobi http://drmf.wmflabs.org/wiki/Definition:pseudoJacobi]
+[http://drmf.wmflabs.org/wiki/Definition:poly {\displaystyle {p}_{n}}] : polynomial : [http://drmf.wmflabs.org/wiki/Definition:poly http://drmf.wmflabs.org/wiki/Definition:poly]
+[http://drmf.wmflabs.org/wiki/Definition:pseudoJacobi {\displaystyle P_{n}}] : pseudo Jacobi polynomial : [http://drmf.wmflabs.org/wiki/Definition:pseudoJacobi http://drmf.wmflabs.org/wiki/Definition:pseudoJacobi]
<< [[Definition:normWilsonWtilde|Definition:normWilsonWtilde]]
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@@ -3239,8 +3259,8 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:qBessel {\displaystyle y_{n}}] : -Bessel polynomial : [http://drmf.wmflabs.org/wiki/Definition:qBessel http://drmf.wmflabs.org/wiki/Definition:qBessel]
-[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or -hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
+[http://drmf.wmflabs.org/wiki/Definition:qBessel {\displaystyle y_{n}}] : {\displaystyle q}-Bessel polynomial : [http://drmf.wmflabs.org/wiki/Definition:qBessel http://drmf.wmflabs.org/wiki/Definition:qBessel]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
<< [[Definition:poly|Definition:poly]]
[[Main_Page|Main Page]]
@@ -3272,8 +3292,8 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:qCharlier {\displaystyle C_{n}}] : -Charlier polynomial : [http://drmf.wmflabs.org/wiki/Definition:qCharlier http://drmf.wmflabs.org/wiki/Definition:qCharlier]
-[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or -hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
+[http://drmf.wmflabs.org/wiki/Definition:qCharlier {\displaystyle C_{n}}] : {\displaystyle q}-Charlier polynomial : [http://drmf.wmflabs.org/wiki/Definition:qCharlier http://drmf.wmflabs.org/wiki/Definition:qCharlier]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
<< [[Definition:qBesselPoly|Definition:qBesselPoly]]
[[Main_Page|Main Page]]
@@ -3305,8 +3325,8 @@ These are defined by == Symbols List == -[http://dlmf.nist.gov/5.18#E4 {\displaystyle \Gamma_{q}}] : -gamma function : [http://dlmf.nist.gov/5.18#E4 http://dlmf.nist.gov/5.18#E4]
-[http://drmf.wmflabs.org/wiki/Definition:qDigamma {\displaystyle \psi_{q}}] : -digamma function : [http://drmf.wmflabs.org/wiki/Definition:qDigamma http://drmf.wmflabs.org/wiki/Definition:qDigamma]
+[http://drmf.wmflabs.org/wiki/Definition:qDigamma {\displaystyle \psi_{q}}] : {\displaystyle q}-digamma function : [http://drmf.wmflabs.org/wiki/Definition:qDigamma http://drmf.wmflabs.org/wiki/Definition:qDigamma]
+[http://dlmf.nist.gov/5.18#E4 {\displaystyle \Gamma_{q}}] : {\displaystyle q}-gamma function : [http://dlmf.nist.gov/5.18#E4 http://dlmf.nist.gov/5.18#E4]
<< [[Definition:qCharlier|Definition:qCharlier]]
[[Main_Page|Main Page]]
@@ -3347,8 +3367,8 @@ Equation (2.1.11) of [[Bibliography#GaR|'''GaR''']]. == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:qHyperrWs {\displaystyle {{}_{r+1}W_{r}}}] : very-well-poised basic hypergeometric (or -hypergeometric) function : [http://drmf.wmflabs.org/wiki/Definition:qHyperrWs http://drmf.wmflabs.org/wiki/Definition:qHyperrWs]
-[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or -hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
+[http://drmf.wmflabs.org/wiki/Definition:qHyperrWs {\displaystyle {{}_{r+1}W_{r}}}] : very-well-poised basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://drmf.wmflabs.org/wiki/Definition:qHyperrWs http://drmf.wmflabs.org/wiki/Definition:qHyperrWs]
<< [[Definition:qDigamma|Definition:qDigamma]]
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@@ -3382,10 +3402,12 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:qExpKLS {\displaystyle \mathrm{E}_{q}}] : -analogue of the function used in KLS: : [http://drmf.wmflabs.org/wiki/Definition:qExpKLS http://drmf.wmflabs.org/wiki/Definition:qExpKLS]
-[http://dlmf.nist.gov/1.2#E1 {\displaystyle \binom{n}{k}}] : binomial coefficient : [http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/1.2#E1]
-[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or -hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
-[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]
+[http://dlmf.nist.gov/1.2#E1 {\displaystyle \binom{n}{k}}] : binomial coefficient : [http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/1.2#E1] [http://dlmf.nist.gov/26.3#SS1.p1 http://dlmf.nist.gov/26.3#SS1.p1]
+[http://dlmf.nist.gov/4.2#E19 {\displaystyle \mathrm{exp}}] : exponential function : [http://dlmf.nist.gov/4.2#E19 http://dlmf.nist.gov/4.2#E19]
+[http://drmf.wmflabs.org/wiki/Definition:qExpKLS {\displaystyle \mathrm{E}_{q}}] : {\displaystyle q}-analogue of the {\displaystyle \exp} function used in KLS: {\displaystyle \mathrm{E}_{q}} : [http://drmf.wmflabs.org/wiki/Definition:qExpKLS http://drmf.wmflabs.org/wiki/Definition:qExpKLS]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1]
+[http://drmf.wmflabs.org/wiki/Definition:sum {\displaystyle \Sigma}] : sum : [http://drmf.wmflabs.org/wiki/Definition:sum http://drmf.wmflabs.org/wiki/Definition:sum]
<< [[Definition:qHyperrWs|Definition:qHyperrWs]]
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@@ -3419,9 +3441,11 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:qexpKLS {\displaystyle \mathrm{e}_{q}}] : -analogue of the function used in KLS: : [http://drmf.wmflabs.org/wiki/Definition:qexpKLS http://drmf.wmflabs.org/wiki/Definition:qexpKLS]
-[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or -hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
-[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]
+[http://dlmf.nist.gov/4.2#E19 {\displaystyle \mathrm{exp}}] : exponential function : [http://dlmf.nist.gov/4.2#E19 http://dlmf.nist.gov/4.2#E19]
+[http://drmf.wmflabs.org/wiki/Definition:qexpKLS {\displaystyle \mathrm{e}_{q}}] : {\displaystyle q}-analogue of the {\displaystyle \exp} function used in KLS: {\displaystyle \mathrm{e}_{q}} : [http://drmf.wmflabs.org/wiki/Definition:qexpKLS http://drmf.wmflabs.org/wiki/Definition:qexpKLS]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1]
+[http://drmf.wmflabs.org/wiki/Definition:sum {\displaystyle \Sigma}] : sum : [http://drmf.wmflabs.org/wiki/Definition:sum http://drmf.wmflabs.org/wiki/Definition:sum]
<< [[Definition:qExpKLS|Definition:qExpKLS]]
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@@ -3454,9 +3478,11 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:qcosKLS {\displaystyle \mathrm{cos}_{q}}] : -analogue of the function used in KLS: : [http://drmf.wmflabs.org/wiki/Definition:qcosKLS http://drmf.wmflabs.org/wiki/Definition:qcosKLS]
-[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]
-[http://drmf.wmflabs.org/wiki/Definition:qexpKLS {\displaystyle \mathrm{e}_{q}}] : -analogue of the function used in KLS: : [http://drmf.wmflabs.org/wiki/Definition:qexpKLS http://drmf.wmflabs.org/wiki/Definition:qexpKLS]
+[http://dlmf.nist.gov/4.14#E2 {\displaystyle \mathrm{cos}}] : cosine function : [http://dlmf.nist.gov/4.14#E2 http://dlmf.nist.gov/4.14#E2]
+[http://drmf.wmflabs.org/wiki/Definition:qcosKLS {\displaystyle \mathrm{cos}_{q}}] : {\displaystyle q}-analogue of the {\displaystyle \cos} function used in KLS: {\displaystyle \cos_q} : [http://drmf.wmflabs.org/wiki/Definition:qcosKLS http://drmf.wmflabs.org/wiki/Definition:qcosKLS]
+[http://drmf.wmflabs.org/wiki/Definition:qexpKLS {\displaystyle \mathrm{e}_{q}}] : {\displaystyle q}-analogue of the {\displaystyle \exp} function used in KLS: {\displaystyle \mathrm{e}_{q}} : [http://drmf.wmflabs.org/wiki/Definition:qexpKLS http://drmf.wmflabs.org/wiki/Definition:qexpKLS]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1]
+[http://drmf.wmflabs.org/wiki/Definition:sum {\displaystyle \Sigma}] : sum : [http://drmf.wmflabs.org/wiki/Definition:sum http://drmf.wmflabs.org/wiki/Definition:sum]
<< [[Definition:qexpKLS|Definition:qexpKLS]]
[[Main_Page|Main Page]]
@@ -3487,8 +3513,8 @@ These are defined by == Symbols List == -[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or -hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
-[http://drmf.wmflabs.org/wiki/Definition:qMeixner {\displaystyle M_{n}}] : -Meixner polynomial : [http://drmf.wmflabs.org/wiki/Definition:qMeixner http://drmf.wmflabs.org/wiki/Definition:qMeixner]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
+[http://drmf.wmflabs.org/wiki/Definition:qMeixner {\displaystyle M_{n}}] : {\displaystyle q}-Meixner polynomial : [http://drmf.wmflabs.org/wiki/Definition:qMeixner http://drmf.wmflabs.org/wiki/Definition:qMeixner]
<< [[Definition:qcosKLS|Definition:qcosKLS]]
[[Main_Page|Main Page]]
@@ -3521,8 +3547,8 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:qKrawtchouk {\displaystyle K_{n}}] : -Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:qKrawtchouk http://drmf.wmflabs.org/wiki/Definition:qKrawtchouk]
-[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or -hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
+[http://drmf.wmflabs.org/wiki/Definition:qKrawtchouk {\displaystyle K_{n}}] : {\displaystyle q}-Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:qKrawtchouk http://drmf.wmflabs.org/wiki/Definition:qKrawtchouk]
<< [[Definition:qMeixner|Definition:qMeixner]]
[[Main_Page|Main Page]]
@@ -3556,9 +3582,9 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:monicqHahn {\displaystyle {\widehat Q}_{n}}] : monic -Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicqHahn http://drmf.wmflabs.org/wiki/Definition:monicqHahn]
-[http://drmf.wmflabs.org/wiki/Definition:qHahn {\displaystyle Q_{n}}] : -Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:qHahn http://drmf.wmflabs.org/wiki/Definition:qHahn]
-[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]
+[http://drmf.wmflabs.org/wiki/Definition:monicqHahn {\displaystyle {\widehat Q}_{n}}] : monic {\displaystyle q}-Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicqHahn http://drmf.wmflabs.org/wiki/Definition:monicqHahn]
+[http://drmf.wmflabs.org/wiki/Definition:qHahn {\displaystyle Q_{n}}] : {\displaystyle q}-Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:qHahn http://drmf.wmflabs.org/wiki/Definition:qHahn]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1]
<< [[Definition:qKrawtchouk|Definition:qKrawtchouk]]
[[Main_Page|Main Page]]
@@ -3592,7 +3618,8 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:dualqHahn {\displaystyle R_{n}}] : dual -Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:dualqHahn http://drmf.wmflabs.org/wiki/Definition:dualqHahn]
+[http://drmf.wmflabs.org/wiki/Definition:dualqHahn {\displaystyle R_{n}}] : dual {\displaystyle q}-Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:dualqHahn http://drmf.wmflabs.org/wiki/Definition:dualqHahn]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
<< [[Definition:monicqHahn|Definition:monicqHahn]]
[[Main_Page|Main Page]]
@@ -3624,8 +3651,8 @@ These are defined by == Symbols List == -[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or -hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
-[http://drmf.wmflabs.org/wiki/Definition:dualqKrawtchouk {\displaystyle K_{n}}] : dual -Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:dualqKrawtchouk http://drmf.wmflabs.org/wiki/Definition:dualqKrawtchouk]
+[http://drmf.wmflabs.org/wiki/Definition:dualqKrawtchouk {\displaystyle K_{n}}] : dual {\displaystyle q}-Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:dualqKrawtchouk http://drmf.wmflabs.org/wiki/Definition:dualqKrawtchouk]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
<< [[Definition:dualqHahn|Definition:dualqHahn]]
[[Main_Page|Main Page]]
@@ -3656,8 +3683,8 @@ These are defined by == Symbols List == -[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or -hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
-[http://drmf.wmflabs.org/wiki/Definition:littleqLaguerre {\displaystyle p_{n}}] : little -Laguerre / Wall polynomial : [http://drmf.wmflabs.org/wiki/Definition:littleqLaguerre http://drmf.wmflabs.org/wiki/Definition:littleqLaguerre]
+[http://drmf.wmflabs.org/wiki/Definition:littleqLaguerre {\displaystyle p_{n}}] : little {\displaystyle q}-Laguerre / Wall polynomial : [http://drmf.wmflabs.org/wiki/Definition:littleqLaguerre http://drmf.wmflabs.org/wiki/Definition:littleqLaguerre]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
<< [[Definition:dualqKrawtchouk|Definition:dualqKrawtchouk]]
[[Main_Page|Main Page]]
@@ -3691,10 +3718,12 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:qexpKLS {\displaystyle \mathrm{e}_{q}}] : -analogue of the function used in KLS: : [http://drmf.wmflabs.org/wiki/Definition:qexpKLS http://drmf.wmflabs.org/wiki/Definition:qexpKLS]
-[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]
-[http://drmf.wmflabs.org/wiki/Definition:qsinKLS {\displaystyle \mathrm{sin}_{q}}] : -analogue of the function used in KLS: : [http://drmf.wmflabs.org/wiki/Definition:qsinKLS http://drmf.wmflabs.org/wiki/Definition:qsinKLS]
[http://dlmf.nist.gov/1.9.i {\displaystyle \mathrm{i}}] : imaginary unit : [http://dlmf.nist.gov/1.9.i http://dlmf.nist.gov/1.9.i]
+[http://drmf.wmflabs.org/wiki/Definition:qexpKLS {\displaystyle \mathrm{e}_{q}}] : {\displaystyle q}-analogue of the {\displaystyle \exp} function used in KLS: {\displaystyle \mathrm{e}_{q}} : [http://drmf.wmflabs.org/wiki/Definition:qexpKLS http://drmf.wmflabs.org/wiki/Definition:qexpKLS]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1]
+[http://drmf.wmflabs.org/wiki/Definition:qsinKLS {\displaystyle \mathrm{sin}_{q}}] : {\displaystyle q}-analogue of the {\displaystyle \sin} function used in KLS: {\displaystyle \sin_q} : [http://drmf.wmflabs.org/wiki/Definition:qsinKLS http://drmf.wmflabs.org/wiki/Definition:qsinKLS]
+[http://dlmf.nist.gov/4.14#E1 {\displaystyle \mathrm{sin}}] : sine function : [http://dlmf.nist.gov/4.14#E1 http://dlmf.nist.gov/4.14#E1]
+[http://drmf.wmflabs.org/wiki/Definition:sum {\displaystyle \Sigma}] : sum : [http://drmf.wmflabs.org/wiki/Definition:sum http://drmf.wmflabs.org/wiki/Definition:sum]
<< [[Definition:littleqLaguerre|Definition:littleqLaguerre]]
[[Main_Page|Main Page]]
@@ -3728,11 +3757,13 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:qExpKLS {\displaystyle \mathrm{E}_{q}}] : -analogue of the function used in KLS: : [http://drmf.wmflabs.org/wiki/Definition:qExpKLS http://drmf.wmflabs.org/wiki/Definition:qExpKLS]
-[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]
-[http://dlmf.nist.gov/1.2#E1 {\displaystyle \binom{n}{k}}] : binomial coefficient : [http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/1.2#E1]
-[http://drmf.wmflabs.org/wiki/Definition:qSinKLS {\displaystyle \mathrm{Sin}_{q}}] : -analogue of the function used in KLS: : [http://drmf.wmflabs.org/wiki/Definition:qSinKLS http://drmf.wmflabs.org/wiki/Definition:qSinKLS]
+[http://dlmf.nist.gov/1.2#E1 {\displaystyle \binom{n}{k}}] : binomial coefficient : [http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/1.2#E1] [http://dlmf.nist.gov/26.3#SS1.p1 http://dlmf.nist.gov/26.3#SS1.p1]
[http://dlmf.nist.gov/1.9.i {\displaystyle \mathrm{i}}] : imaginary unit : [http://dlmf.nist.gov/1.9.i http://dlmf.nist.gov/1.9.i]
+[http://drmf.wmflabs.org/wiki/Definition:qExpKLS {\displaystyle \mathrm{E}_{q}}] : {\displaystyle q}-analogue of the {\displaystyle \exp} function used in KLS: {\displaystyle \mathrm{E}_{q}} : [http://drmf.wmflabs.org/wiki/Definition:qExpKLS http://drmf.wmflabs.org/wiki/Definition:qExpKLS]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1]
+[http://drmf.wmflabs.org/wiki/Definition:qSinKLS {\displaystyle \mathrm{Sin}_{q}}] : {\displaystyle q}-analogue of the {\displaystyle \sin} function used in KLS: {\displaystyle \mathrm{Sin}_q} : [http://drmf.wmflabs.org/wiki/Definition:qSinKLS http://drmf.wmflabs.org/wiki/Definition:qSinKLS]
+[http://dlmf.nist.gov/4.14#E1 {\displaystyle \mathrm{sin}}] : sine function : [http://dlmf.nist.gov/4.14#E1 http://dlmf.nist.gov/4.14#E1]
+[http://drmf.wmflabs.org/wiki/Definition:sum {\displaystyle \Sigma}] : sum : [http://drmf.wmflabs.org/wiki/Definition:sum http://drmf.wmflabs.org/wiki/Definition:sum]
<< [[Definition:qsinKLS|Definition:qsinKLS]]
[[Main_Page|Main Page]]
@@ -3766,11 +3797,13 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:qCosKLS {\displaystyle \mathrm{Cos}_{q}}] : -analogue of the function used in KLS: : [http://drmf.wmflabs.org/wiki/Definition:qCosKLS http://drmf.wmflabs.org/wiki/Definition:qCosKLS]
-[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]
-[http://dlmf.nist.gov/1.2#E1 {\displaystyle \binom{n}{k}}] : binomial coefficient : [http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/1.2#E1]
+[http://dlmf.nist.gov/1.2#E1 {\displaystyle \binom{n}{k}}] : binomial coefficient : [http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/1.2#E1] [http://dlmf.nist.gov/26.3#SS1.p1 http://dlmf.nist.gov/26.3#SS1.p1]
+[http://dlmf.nist.gov/4.14#E2 {\displaystyle \mathrm{cos}}] : cosine function : [http://dlmf.nist.gov/4.14#E2 http://dlmf.nist.gov/4.14#E2]
[http://dlmf.nist.gov/1.9.i {\displaystyle \mathrm{i}}] : imaginary unit : [http://dlmf.nist.gov/1.9.i http://dlmf.nist.gov/1.9.i]
-[http://drmf.wmflabs.org/wiki/Definition:qExpKLS {\displaystyle \mathrm{E}_{q}}] : -analogue of the function used in KLS: : [http://drmf.wmflabs.org/wiki/Definition:qExpKLS http://drmf.wmflabs.org/wiki/Definition:qExpKLS]
+[http://drmf.wmflabs.org/wiki/Definition:qCosKLS {\displaystyle \mathrm{Cos}_{q}}] : {\displaystyle q}-analogue of the {\displaystyle \cos} function used in KLS: {\displaystyle \mathrm{Cos}_{q}} : [http://drmf.wmflabs.org/wiki/Definition:qCosKLS http://drmf.wmflabs.org/wiki/Definition:qCosKLS]
+[http://drmf.wmflabs.org/wiki/Definition:qExpKLS {\displaystyle \mathrm{E}_{q}}] : {\displaystyle q}-analogue of the {\displaystyle \exp} function used in KLS: {\displaystyle \mathrm{E}_{q}} : [http://drmf.wmflabs.org/wiki/Definition:qExpKLS http://drmf.wmflabs.org/wiki/Definition:qExpKLS]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1]
+[http://drmf.wmflabs.org/wiki/Definition:sum {\displaystyle \Sigma}] : sum : [http://drmf.wmflabs.org/wiki/Definition:sum http://drmf.wmflabs.org/wiki/Definition:sum]
<< [[Definition:qSinKLS|Definition:qSinKLS]]
[[Main_Page|Main Page]]
@@ -3803,8 +3836,8 @@ These are defined by == Symbols List == [http://dlmf.nist.gov/16.2#E1 {\displaystyle {{}_{p}F_{q}}}] : generalized hypergeometric function : [http://dlmf.nist.gov/16.2#E1 http://dlmf.nist.gov/16.2#E1]
-[http://dlmf.nist.gov/18.25#T1.t1.r2 {\displaystyle W_{n}}] : Wilson polynomial : [http://dlmf.nist.gov/18.25#T1.t1.r2 http://dlmf.nist.gov/18.25#T1.t1.r2]
[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii]
+[http://dlmf.nist.gov/18.25#T1.t1.r2 {\displaystyle W_{n}}] : Wilson polynomial : [http://dlmf.nist.gov/18.25#T1.t1.r2 http://dlmf.nist.gov/18.25#T1.t1.r2]
<< [[Definition:qCosKLS|Definition:qCosKLS]]
[[Main_Page|Main Page]]
@@ -3841,7 +3874,7 @@ with N a nonnegative integer. == Symbols List == [http://dlmf.nist.gov/16.2#E1 {\displaystyle {{}_{p}F_{q}}}] : generalized hypergeometric function : [http://dlmf.nist.gov/16.2#E1 http://dlmf.nist.gov/16.2#E1]
-[http://dlmf.nist.gov/18.25#T1.t1.r4 {\displaystyle R_{n}}] : Racah polynomial : [http://dlmf.nist.gov/18.25#T1.t1.r4 http://dlmf.nist.gov/18.25#T1.t1.r4]
+[http://dlmf.nist.gov/18.25#T1.t1.r4 {\displaystyle R_{n}}] : Racah polynomial : [http://dlmf.nist.gov/18.25#T1.t1.r4 http://dlmf.nist.gov/18.25#T1.t1.r4]
<< [[Definition:Wilson|Definition:Wilson]]
[[Main_Page|Main Page]]
@@ -3873,9 +3906,9 @@ These are defined by == Symbols List == -[http://dlmf.nist.gov/16.2#E1 {\displaystyle {{}_{p}F_{q}}}] : generalized hypergeometric function : [http://dlmf.nist.gov/16.2#E1 http://dlmf.nist.gov/16.2#E1]
[http://dlmf.nist.gov/18.25#T1.t1.r3 {\displaystyle S_{n}}] : continuous dual Hahn polynomial : [http://dlmf.nist.gov/18.25#T1.t1.r3 http://dlmf.nist.gov/18.25#T1.t1.r3]
-[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii]
+[http://dlmf.nist.gov/16.2#E1 {\displaystyle {{}_{p}F_{q}}}] : generalized hypergeometric function : [http://dlmf.nist.gov/16.2#E1 http://dlmf.nist.gov/16.2#E1]
+[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii]
<< [[Definition:Racah|Definition:Racah]]
[[Main_Page|Main Page]]
@@ -3908,9 +3941,9 @@ These are defined by == Symbols List == -[http://dlmf.nist.gov/16.2#E1 {\displaystyle {{}_{p}F_{q}}}] : generalized hypergeometric function : [http://dlmf.nist.gov/16.2#E1 http://dlmf.nist.gov/16.2#E1]
-[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii]
[http://dlmf.nist.gov/18.19#P2.p1 {\displaystyle p_{n}}] : continuous Hahn polynomial : [http://dlmf.nist.gov/18.19#P2.p1 http://dlmf.nist.gov/18.19#P2.p1]
+[http://dlmf.nist.gov/16.2#E1 {\displaystyle {{}_{p}F_{q}}}] : generalized hypergeometric function : [http://dlmf.nist.gov/16.2#E1 http://dlmf.nist.gov/16.2#E1]
+[http://dlmf.nist.gov/5.2#iii {\displaystyle (a)_n}] : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii]
<< [[Definition:ctsdualHahn|Definition:ctsdualHahn]]
[[Main_Page|Main Page]]
@@ -3941,8 +3974,8 @@ These are defined by == Symbols List == -[http://dlmf.nist.gov/16.2#E1 {\displaystyle {{}_{p}F_{q}}}] : generalized hypergeometric function : [http://dlmf.nist.gov/16.2#E1 http://dlmf.nist.gov/16.2#E1]
[http://dlmf.nist.gov/18.19#T1.t1.r3 {\displaystyle Q_{n}}] : Hahn polynomial : [http://dlmf.nist.gov/18.19#T1.t1.r3 http://dlmf.nist.gov/18.19#T1.t1.r3]
+[http://dlmf.nist.gov/16.2#E1 {\displaystyle {{}_{p}F_{q}}}] : generalized hypergeometric function : [http://dlmf.nist.gov/16.2#E1 http://dlmf.nist.gov/16.2#E1]
<< [[Definition:ctsHahn|Definition:ctsHahn]]
[[Main_Page|Main Page]]
@@ -3975,8 +4008,8 @@ These are defined by == Symbols List == -[http://dlmf.nist.gov/16.2#E1 {\displaystyle {{}_{p}F_{q}}}] : generalized hypergeometric function : [http://dlmf.nist.gov/16.2#E1 http://dlmf.nist.gov/16.2#E1]
[http://dlmf.nist.gov/18.25#T1.t1.r5 {\displaystyle R_{n}}] : dual Hahn polynomial : [http://dlmf.nist.gov/18.25#T1.t1.r5 http://dlmf.nist.gov/18.25#T1.t1.r5]
+[http://dlmf.nist.gov/16.2#E1 {\displaystyle {{}_{p}F_{q}}}] : generalized hypergeometric function : [http://dlmf.nist.gov/16.2#E1 http://dlmf.nist.gov/16.2#E1]
<< [[Definition:Hahn|Definition:Hahn]]
[[Main_Page|Main Page]]
@@ -4008,8 +4041,8 @@ These are defined by == Symbols List == -[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or -hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
-[http://dlmf.nist.gov/18.28#E19 {\displaystyle R_{n}}] : -Racah polynomial : [http://dlmf.nist.gov/18.28#E19 http://dlmf.nist.gov/18.28#E19]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
+[http://dlmf.nist.gov/18.28#E19 {\displaystyle R_{n}}] : {\displaystyle q}-Racah polynomial : [http://dlmf.nist.gov/18.28#E19 http://dlmf.nist.gov/18.28#E19]
<< [[Definition:dualHahn|Definition:dualHahn]]
[[Main_Page|Main Page]]
@@ -4041,9 +4074,9 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:ctsdualqHahn {\displaystyle p_{n}}] : continuous dual -Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:ctsdualqHahn http://drmf.wmflabs.org/wiki/Definition:ctsdualqHahn]
-[http://drmf.wmflabs.org/wiki/Definition:normctsdualqHahnptilde {\displaystyle {\tilde p}_{n}}] : normalized continuous dual -Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:normctsdualqHahnptilde http://drmf.wmflabs.org/wiki/Definition:normctsdualqHahnptilde]
-[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]
+[http://drmf.wmflabs.org/wiki/Definition:ctsdualqHahn {\displaystyle p_{n}}] : continuous dual {\displaystyle q}-Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:ctsdualqHahn http://drmf.wmflabs.org/wiki/Definition:ctsdualqHahn]
+[http://drmf.wmflabs.org/wiki/Definition:normctsdualqHahnptilde {\displaystyle {\tilde p}_{n}}] : normalized continuous dual {\displaystyle q}-Hahn polynomial {\displaystyle {\tilde p}} : [http://drmf.wmflabs.org/wiki/Definition:normctsdualqHahnptilde http://drmf.wmflabs.org/wiki/Definition:normctsdualqHahnptilde]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1]
<< [[Definition:qRacah|Definition:qRacah]]
[[Main_Page|Main Page]]
@@ -4076,9 +4109,10 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:normctsqHahnptilde {\displaystyle {\tilde p}_{n}}] : normalized continuous -Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:normctsqHahnptilde http://drmf.wmflabs.org/wiki/Definition:normctsqHahnptilde]
-[http://drmf.wmflabs.org/wiki/Definition:ctsqHahn {\displaystyle p_{n}}] : continuous -Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:ctsqHahn http://drmf.wmflabs.org/wiki/Definition:ctsqHahn]
-[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]
+[http://drmf.wmflabs.org/wiki/Definition:ctsqHahn {\displaystyle p_{n}}] : continuous {\displaystyle q}-Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:ctsqHahn http://drmf.wmflabs.org/wiki/Definition:ctsqHahn]
+[http://dlmf.nist.gov/4.2.E11 {\displaystyle \mathrm{e}}] : the base of the natural logarithm : [http://dlmf.nist.gov/4.2.E11 http://dlmf.nist.gov/4.2.E11]
+[http://drmf.wmflabs.org/wiki/Definition:normctsqHahnptilde {\displaystyle {\tilde p}_{n}}] : normalized continuous {\displaystyle q}-Hahn polynomial {\displaystyle {\tilde p}} : [http://drmf.wmflabs.org/wiki/Definition:normctsqHahnptilde http://drmf.wmflabs.org/wiki/Definition:normctsqHahnptilde]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1]
<< [[Definition:normctsdualqHahnptilde|Definition:normctsdualqHahnptilde]]
[[Main_Page|Main Page]]
@@ -4109,8 +4143,8 @@ These are defined by == Symbols List == -[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or -hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
-[http://drmf.wmflabs.org/wiki/Definition:qHahn {\displaystyle Q_{n}}] : -Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:qHahn http://drmf.wmflabs.org/wiki/Definition:qHahn]
+[http://drmf.wmflabs.org/wiki/Definition:qHahn {\displaystyle Q_{n}}] : {\displaystyle q}-Hahn polynomial : [http://drmf.wmflabs.org/wiki/Definition:qHahn http://drmf.wmflabs.org/wiki/Definition:qHahn]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
<< [[Definition:normctsqHahnptilde|Definition:normctsqHahnptilde]]
[[Main_Page|Main Page]]
@@ -4143,9 +4177,10 @@ These are defined by == Symbols List == -[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or -hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
-[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]
[http://drmf.wmflabs.org/wiki/Definition:AlSalamChihara {\displaystyle Q_{n}}] : Al-Salam-Chihara polynomial : [http://drmf.wmflabs.org/wiki/Definition:AlSalamChihara http://drmf.wmflabs.org/wiki/Definition:AlSalamChihara]
+[http://dlmf.nist.gov/4.2.E11 {\displaystyle \mathrm{e}}] : the base of the natural logarithm : [http://dlmf.nist.gov/4.2.E11 http://dlmf.nist.gov/4.2.E11]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1]
<< [[Definition:qHahn|Definition:qHahn]]
[[Main_Page|Main Page]]
@@ -4178,9 +4213,9 @@ These are defined by == Symbols List == -[http://dlmf.nist.gov/23.1 {\displaystyle Q_{n}}] : -inverse Al-Salam-Chihara polynomial : [http://dlmf.nist.gov/23.1 http://dlmf.nist.gov/23.1]
-[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or -hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
-[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
+[http://dlmf.nist.gov/23.1 {\displaystyle Q_{n}}] : {\displaystyle q}-inverse Al-Salam-Chihara polynomial : [http://dlmf.nist.gov/23.1 http://dlmf.nist.gov/23.1]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1]
<< [[Definition:AlSalamChihara|Definition:AlSalamChihara]]
[[Main_Page|Main Page]]
@@ -4212,7 +4247,7 @@ x\monicqinvAlSalamChihara{n}@@{x}{a}{b}{q^{-1}=:\monicqinvAlSalamChihara{n+1}@@{ == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:monicqinvAlSalamChihara {\displaystyle {\widehat Q}_{n}}] : monic -inverse Al-Salam-Chihara polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicqinvAlSalamChihara http://drmf.wmflabs.org/wiki/Definition:monicqinvAlSalamChihara]
+[http://drmf.wmflabs.org/wiki/Definition:monicqinvAlSalamChihara {\displaystyle {\widehat Q}_{n}}] : monic {\displaystyle q}-inverse Al-Salam-Chihara polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicqinvAlSalamChihara http://drmf.wmflabs.org/wiki/Definition:monicqinvAlSalamChihara]
<< [[Definition:qinvAlSalamChihara|Definition:qinvAlSalamChihara]]
[[Main_Page|Main Page]]
@@ -4245,9 +4280,9 @@ These are defined by == Symbols List == -[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]
-[http://drmf.wmflabs.org/wiki/Definition:AffqKrawtchouk {\displaystyle K^{\mathrm{Aff}}_{n}}] : affine -Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:AffqKrawtchouk http://drmf.wmflabs.org/wiki/Definition:AffqKrawtchouk]
-[http://drmf.wmflabs.org/wiki/Definition:monicAffqKrawtchouk {\displaystyle \widehat{K}^{\mathrm{Aff}}_{n}}] : monic affine -Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicAffqKrawtchouk http://drmf.wmflabs.org/wiki/Definition:monicAffqKrawtchouk]
+[http://drmf.wmflabs.org/wiki/Definition:AffqKrawtchouk {\displaystyle K^{\mathrm{Aff}}_{n}}] : affine {\displaystyle q}-Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:AffqKrawtchouk http://drmf.wmflabs.org/wiki/Definition:AffqKrawtchouk]
+[http://drmf.wmflabs.org/wiki/Definition:monicAffqKrawtchouk {\displaystyle \widehat{K}^{\mathrm{Aff}}_{n}}] : monic affine {\displaystyle q}-Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicAffqKrawtchouk http://drmf.wmflabs.org/wiki/Definition:monicAffqKrawtchouk]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1]
<< [[Definition:monicqinvAlSalamChihara|Definition:monicqinvAlSalamChihara]]
[[Main_Page|Main Page]]
@@ -4279,9 +4314,10 @@ These are defined by == Symbols List == -[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or -hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
-[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]
-[http://drmf.wmflabs.org/wiki/Definition:qMeixnerPollaczek {\displaystyle P_{n}}] : -Meixner-Pollaczek polynomial : [http://drmf.wmflabs.org/wiki/Definition:qMeixnerPollaczek http://drmf.wmflabs.org/wiki/Definition:qMeixnerPollaczek]
+[http://dlmf.nist.gov/4.2.E11 {\displaystyle \mathrm{e}}] : the base of the natural logarithm : [http://dlmf.nist.gov/4.2.E11 http://dlmf.nist.gov/4.2.E11]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
+[http://drmf.wmflabs.org/wiki/Definition:qMeixnerPollaczek {\displaystyle P_{n}}] : {\displaystyle q}-Meixner-Pollaczek polynomial : [http://drmf.wmflabs.org/wiki/Definition:qMeixnerPollaczek http://drmf.wmflabs.org/wiki/Definition:qMeixnerPollaczek]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1]
<< [[Definition:monicAffqKrawtchouk|Definition:monicAffqKrawtchouk]]
[[Main_Page|Main Page]]
@@ -4314,8 +4350,8 @@ These are defined by == Symbols List == -[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or -hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
-[http://drmf.wmflabs.org/wiki/Definition:qtmqKrawtchouk {\displaystyle K^{\mathrm{qtm}}_{n}}] : quantum -Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:qtmqKrawtchouk http://drmf.wmflabs.org/wiki/Definition:qtmqKrawtchouk]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
+[http://drmf.wmflabs.org/wiki/Definition:qtmqKrawtchouk {\displaystyle K^{\mathrm{qtm}}_{n}}] : quantum {\displaystyle q}-Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:qtmqKrawtchouk http://drmf.wmflabs.org/wiki/Definition:qtmqKrawtchouk]
<< [[Definition:qMeixnerPollaczek|Definition:qMeixnerPollaczek]]
[[Main_Page|Main Page]]
@@ -4351,9 +4387,9 @@ These are defined by == Symbols List == -[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or -hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
-[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]
-[http://drmf.wmflabs.org/wiki/Definition:qLaguerre {\displaystyle L_n^{(\alpha)}}] : -Laguerre polynomial : [http://drmf.wmflabs.org/wiki/Definition:qLaguerre http://drmf.wmflabs.org/wiki/Definition:qLaguerre]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
+[http://drmf.wmflabs.org/wiki/Definition:qLaguerre {\displaystyle L_n^{(\alpha)}}] : {\displaystyle q}-Laguerre polynomial : [http://drmf.wmflabs.org/wiki/Definition:qLaguerre http://drmf.wmflabs.org/wiki/Definition:qLaguerre]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1]
<< [[Definition:qtmqKrawtchouk|Definition:qtmqKrawtchouk]]
[[Main_Page|Main Page]]
@@ -4385,8 +4421,9 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:ctsqHermite {\displaystyle H_{n}}] : continuous -Hermite polynomial : [http://drmf.wmflabs.org/wiki/Definition:ctsqHermite http://drmf.wmflabs.org/wiki/Definition:ctsqHermite]
-[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or -hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
+[http://drmf.wmflabs.org/wiki/Definition:ctsqHermite {\displaystyle H_{n}}] : continuous {\displaystyle q}-Hermite polynomial : [http://drmf.wmflabs.org/wiki/Definition:ctsqHermite http://drmf.wmflabs.org/wiki/Definition:ctsqHermite]
+[http://dlmf.nist.gov/4.2.E11 {\displaystyle \mathrm{e}}] : the base of the natural logarithm : [http://dlmf.nist.gov/4.2.E11 http://dlmf.nist.gov/4.2.E11]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
<< [[Definition:qLaguerre|Definition:qLaguerre]]
[[Main_Page|Main Page]]
@@ -4418,9 +4455,9 @@ These are defined by == Symbols List == -[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or -hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
-[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : -Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i]
-[http://drmf.wmflabs.org/wiki/Definition:StieltjesWigert {\displaystyle S_{n}}] : Stieltjes-Wigert polynomial : [http://drmf.wmflabs.org/wiki/Definition:StieltjesWigert http://drmf.wmflabs.org/wiki/Definition:StieltjesWigert]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
+[http://dlmf.nist.gov/5.18#i {\displaystyle (a;q)_n}] : {\displaystyle q}-Pochhammer symbol : [http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/5.18#i] [http://dlmf.nist.gov/17.2#SS1.p1 http://dlmf.nist.gov/17.2#SS1.p1]
+[http://drmf.wmflabs.org/wiki/Definition:StieltjesWigert {\displaystyle S_{n}}] : Stieltjes-Wigert polynomial : [http://drmf.wmflabs.org/wiki/Definition:StieltjesWigert http://drmf.wmflabs.org/wiki/Definition:StieltjesWigert]
<< [[Definition:ctsqHermite|Definition:ctsqHermite]]
[[Main_Page|Main Page]]
@@ -4452,8 +4489,8 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:discrqHermiteI {\displaystyle h_{n}}] : discrete -Hermite I polynomial : [http://drmf.wmflabs.org/wiki/Definition:discrqHermiteI http://drmf.wmflabs.org/wiki/Definition:discrqHermiteI]
[http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzI {\displaystyle U^{(n)}_{\alpha}}] : Al-Salam-Carlitz I polynomial : [http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzI http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzI]
+[http://drmf.wmflabs.org/wiki/Definition:discrqHermiteI {\displaystyle h_{n}}] : discrete {\displaystyle q}-Hermite I polynomial : [http://drmf.wmflabs.org/wiki/Definition:discrqHermiteI http://drmf.wmflabs.org/wiki/Definition:discrqHermiteI]
<< [[Definition:StieltjesWigert|Definition:StieltjesWigert]]
[[Main_Page|Main Page]]
@@ -4486,7 +4523,7 @@ These are defined by == Symbols List == [http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzII {\displaystyle V^{(n)}_{\alpha}}] : Al-Salam-Carlitz II polynomial : [http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzII http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzII]
-[http://drmf.wmflabs.org/wiki/Definition:discrqHermiteII {\displaystyle \tilde{h}_{n}}] : discrete -Hermite II polynomial : [http://drmf.wmflabs.org/wiki/Definition:discrqHermiteII http://drmf.wmflabs.org/wiki/Definition:discrqHermiteII]
+[http://drmf.wmflabs.org/wiki/Definition:discrqHermiteII {\displaystyle \tilde{h}_{n}}] : discrete {\displaystyle q}-Hermite II polynomial : [http://drmf.wmflabs.org/wiki/Definition:discrqHermiteII http://drmf.wmflabs.org/wiki/Definition:discrqHermiteII]
<< [[Definition:discrqHermiteI|Definition:discrqHermiteI]]
[[Main_Page|Main Page]]
@@ -4525,8 +4562,10 @@ Equation (2), Section 2.21 of [[Bibliography#FI|'''FI''']]. == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:StieltjesConstants {\displaystyle \gamma_{n}}] : Stieltjes constants : [http://drmf.wmflabs.org/wiki/Definition:StieltjesConstants http://drmf.wmflabs.org/wiki/Definition:StieltjesConstants]
+[http://dlmf.nist.gov/5.2#E3 {\displaystyle \gamma}] : Euler's (Euler-Mascheroni) constant : [http://dlmf.nist.gov/5.2#E3 http://dlmf.nist.gov/5.2#E3]
[http://dlmf.nist.gov/4.2#E2 {\displaystyle \mathrm{ln}}] : principal branch of logarithm function : [http://dlmf.nist.gov/4.2#E2 http://dlmf.nist.gov/4.2#E2]
+[http://drmf.wmflabs.org/wiki/Definition:StieltjesConstants {\displaystyle \gamma_{n}}] : Stieltjes constants : [http://drmf.wmflabs.org/wiki/Definition:StieltjesConstants http://drmf.wmflabs.org/wiki/Definition:StieltjesConstants]
+[http://drmf.wmflabs.org/wiki/Definition:sum {\displaystyle \Sigma}] : sum : [http://drmf.wmflabs.org/wiki/Definition:sum http://drmf.wmflabs.org/wiki/Definition:sum]
<< [[Definition:discrqHermiteII|Definition:discrqHermiteII]]
[[Main_Page|Main Page]]
@@ -4560,8 +4599,9 @@ These are defined by == Symbols List == +[http://dlmf.nist.gov/1.4#iv {\displaystyle \int}] : integral : [http://dlmf.nist.gov/1.4#iv http://dlmf.nist.gov/1.4#iv]
[http://dlmf.nist.gov/4.2#E2 {\displaystyle \mathrm{ln}}] : principal branch of logarithm function : [http://dlmf.nist.gov/4.2#E2 http://dlmf.nist.gov/4.2#E2]
-[http://drmf.wmflabs.org/wiki/Definition:PolylogarithmS {\displaystyle S_{n}] : Nielsen generalized polylogarithm function : [http://drmf.wmflabs.org/wiki/Definition:PolylogarithmS http://drmf.wmflabs.org/wiki/Definition:PolylogarithmS]
+[http://drmf.wmflabs.org/wiki/Definition:PolylogarithmS {\displaystyle S_{n}] : Nielsen generalized polylogarithm function : [http://drmf.wmflabs.org/wiki/Definition:PolylogarithmS http://drmf.wmflabs.org/wiki/Definition:PolylogarithmS]
<< [[Definition:StieltjesConstants|Definition:StieltjesConstants]]
[[Main_Page|Main Page]]
@@ -4597,6 +4637,7 @@ These are defined by == Symbols List == [http://drmf.wmflabs.org/wiki/Definition:HarmonicNumber {\displaystyle H_{n}^{r}}] : Harmonic number : [http://drmf.wmflabs.org/wiki/Definition:HarmonicNumber http://drmf.wmflabs.org/wiki/Definition:HarmonicNumber]
+[http://drmf.wmflabs.org/wiki/Definition:sum {\displaystyle \Sigma}] : sum : [http://drmf.wmflabs.org/wiki/Definition:sum http://drmf.wmflabs.org/wiki/Definition:sum]
<< [[Definition:PolylogarithmS|Definition:PolylogarithmS]]
[[Main_Page|Main Page]]
@@ -4630,7 +4671,7 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:LucasL {\displaystyle L_{n}}] : Lucas polynomial : [http://drmf.wmflabs.org/wiki/Definition:LucasL http://drmf.wmflabs.org/wiki/Definition:LucasL]
+[http://drmf.wmflabs.org/wiki/Definition:LucasL {\displaystyle L_{n}}] : Lucas polynomial : [http://drmf.wmflabs.org/wiki/Definition:LucasL http://drmf.wmflabs.org/wiki/Definition:LucasL]
<< [[Definition:HarmonicNumber|Definition:HarmonicNumber]]
[[Main_Page|Main Page]]
@@ -4665,6 +4706,7 @@ These are defined by == Symbols List == [http://drmf.wmflabs.org/wiki/Definition:HurwitzLerchPhi {\displaystyle \Phi}] : Hurwitz-Lerch's transcendent : [http://drmf.wmflabs.org/wiki/Definition:HurwitzLerchPhi http://drmf.wmflabs.org/wiki/Definition:HurwitzLerchPhi]
+[http://drmf.wmflabs.org/wiki/Definition:sum {\displaystyle \Sigma}] : sum : [http://drmf.wmflabs.org/wiki/Definition:sum http://drmf.wmflabs.org/wiki/Definition:sum]
<< [[Definition:LucasL|Definition:LucasL]]
[[Main_Page|Main Page]]
@@ -4697,7 +4739,7 @@ These are defined by == Symbols List == - +[http://drmf.wmflabs.org/wiki/Definition:GoldenRatio {\displaystyle \phi}] : The golden ratio : [http://drmf.wmflabs.org/wiki/Definition:GoldenRatio http://drmf.wmflabs.org/wiki/Definition:GoldenRatio]
<< [[Definition:HurwitzLerchPhi|Definition:HurwitzLerchPhi]]
[[Main_Page|Main Page]]
@@ -4731,8 +4773,8 @@ These are defined by == Symbols List == -[http://dlmf.nist.gov/13.14#E3 {\displaystyle W_{\kappa,\mu}}] : Whittaker function : [http://dlmf.nist.gov/13.14#E3 http://dlmf.nist.gov/13.14#E3]
-[http://drmf.wmflabs.org/wiki/Definition:WhitPsi {\displaystyle \Psi_{\alpha,\beta}}] : Whittaker function : [http://drmf.wmflabs.org/wiki/Definition:WhitPsi http://drmf.wmflabs.org/wiki/Definition:WhitPsi]
+[http://drmf.wmflabs.org/wiki/Definition:WhitPsi {\displaystyle \Psi_{\alpha,\beta}}] : Whittaker function {\displaystyle \Psi_{\alpha;\beta}} : [http://drmf.wmflabs.org/wiki/Definition:WhitPsi http://drmf.wmflabs.org/wiki/Definition:WhitPsi]
+[http://dlmf.nist.gov/13.14#E3 {\displaystyle W_{\kappa,\mu}}] : Whittaker function {\displaystyle W_{\kappa;\mu}} : [http://dlmf.nist.gov/13.14#E3 http://dlmf.nist.gov/13.14#E3]
<< [[Definition:GoldenRatio|Definition:GoldenRatio]]
[[Main_Page|Main Page]]
@@ -4765,7 +4807,8 @@ These are defined by == Symbols List == - +[http://drmf.wmflabs.org/wiki/Definition:GompertzConstant {\displaystyle G}] : Gompertz's constant : [http://drmf.wmflabs.org/wiki/Definition:GompertzConstant http://drmf.wmflabs.org/wiki/Definition:GompertzConstant]
+[http://dlmf.nist.gov/1.4#iv {\displaystyle \int}] : integral : [http://dlmf.nist.gov/1.4#iv http://dlmf.nist.gov/1.4#iv]
<< [[Definition:WhitPsi|Definition:WhitPsi]]
[[Main_Page|Main Page]]
@@ -4799,6 +4842,9 @@ These are defined by == Symbols List == [http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r16 {\displaystyle \left\lfloor a\right\rfloor}] : floor : [http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r16 http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r16]
+[http://drmf.wmflabs.org/wiki/Definition:GoldenRatio {\displaystyle \phi}] : The golden ratio : [http://drmf.wmflabs.org/wiki/Definition:GoldenRatio http://drmf.wmflabs.org/wiki/Definition:GoldenRatio]
+[http://drmf.wmflabs.org/wiki/Definition:RabbitConstant {\displaystyle \rho}] : rabbit constant : [http://drmf.wmflabs.org/wiki/Definition:RabbitConstant http://drmf.wmflabs.org/wiki/Definition:RabbitConstant]
+[http://drmf.wmflabs.org/wiki/Definition:sum {\displaystyle \Sigma}] : sum : [http://drmf.wmflabs.org/wiki/Definition:sum http://drmf.wmflabs.org/wiki/Definition:sum]
<< [[Definition:GompertzConstant|Definition:GompertzConstant]]
[[Main_Page|Main Page]]
@@ -4831,7 +4877,7 @@ These are defined by == Symbols List == -[http://dlmf.nist.gov/26.11#E5 {\displaystyle F_{n}}] : Fibonacci number : [http://dlmf.nist.gov/26.11#E5 http://dlmf.nist.gov/26.11#E5]
+[http://dlmf.nist.gov/26.11#E5 {\displaystyle F_{n}}] : Fibonacci number : [http://dlmf.nist.gov/26.11#E5 http://dlmf.nist.gov/26.11#E5]
<< [[Definition:rabbitConstant|Definition:rabbitConstant]]
[[Main_Page|Main Page]]
@@ -4864,7 +4910,7 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:Fibonacci {\displaystyle F_{n}{(x)}}] : Fibonacci polynomial : [http://drmf.wmflabs.org/wiki/Definition:Fibonacci http://drmf.wmflabs.org/wiki/Definition:Fibonacci]
+[http://drmf.wmflabs.org/wiki/Definition:Fibonacci {\displaystyle F_{n}{(x)}}] : Fibonacci polynomial : [http://drmf.wmflabs.org/wiki/Definition:Fibonacci http://drmf.wmflabs.org/wiki/Definition:Fibonacci]
<< [[Definition:FibonacciNumber|Definition:FibonacciNumber]]
[[Main_Page|Main Page]]
@@ -4902,8 +4948,8 @@ These are defined by == Symbols List == [http://dlmf.nist.gov/5.2#E1 {\displaystyle \Gamma}] : Euler's gamma function : [http://dlmf.nist.gov/5.2#E1 http://dlmf.nist.gov/5.2#E1]
-[http://drmf.wmflabs.org/wiki/Definition:RegGamma {\displaystyle P\left(x;\alpha}] : Regularized gamma function of incomplete gamma function : [http://drmf.wmflabs.org/wiki/Definition:RegGamma http://drmf.wmflabs.org/wiki/Definition:RegGamma]
-[http://dlmf.nist.gov/8.2#E1 {\displaystyle \gamma}] : incomplete gamma function : [http://dlmf.nist.gov/8.2#E1 http://dlmf.nist.gov/8.2#E1]
+[http://dlmf.nist.gov/8.2#E1 {\displaystyle \gamma}] : incomplete gamma function {\displaystyle \gamma} : [http://dlmf.nist.gov/8.2#E1 http://dlmf.nist.gov/8.2#E1]
+[http://drmf.wmflabs.org/wiki/Definition:RegGamma {\displaystyle P\left(x;\alpha}] : Regularized gamma function of incomplete gamma function : [http://drmf.wmflabs.org/wiki/Definition:RegGamma http://drmf.wmflabs.org/wiki/Definition:RegGamma]
<< [[Definition:Fibonacci|Definition:Fibonacci]]
[[Main_Page|Main Page]]
@@ -4941,8 +4987,8 @@ These are defined by == Symbols List == [http://dlmf.nist.gov/5.2#E1 {\displaystyle \Gamma}] : Euler's gamma function : [http://dlmf.nist.gov/5.2#E1 http://dlmf.nist.gov/5.2#E1]
-[http://drmf.wmflabs.org/wiki/Definition:RegGamma {\displaystyle Q\left(x;\alpha}] : Regularized gamma function of complementary incomplete gamma function : [http://drmf.wmflabs.org/wiki/Definition:RegGamma http://drmf.wmflabs.org/wiki/Definition:RegGamma]
-[http://dlmf.nist.gov/8.2#E2 {\displaystyle \Gamma}] : incomplete gamma function : [http://dlmf.nist.gov/8.2#E2 http://dlmf.nist.gov/8.2#E2]
+[http://dlmf.nist.gov/8.2#E2 {\displaystyle \Gamma}] : incomplete gamma function {\displaystyle \Gamma} : [http://dlmf.nist.gov/8.2#E2 http://dlmf.nist.gov/8.2#E2]
+[http://drmf.wmflabs.org/wiki/Definition:RegGamma {\displaystyle Q\left(x;\alpha}] : Regularized gamma function of complementary incomplete gamma function : [http://drmf.wmflabs.org/wiki/Definition:RegGamma http://drmf.wmflabs.org/wiki/Definition:RegGamma]
<< [[Definition:RegGammaP|Definition:RegGammaP]]
[[Main_Page|Main Page]]
@@ -4979,8 +5025,8 @@ These are defined by == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:GaussDist {\displaystyle F\left(x}] : Gaussian distribution : [http://drmf.wmflabs.org/wiki/Definition:GaussDist http://drmf.wmflabs.org/wiki/Definition:GaussDist]
[http://dlmf.nist.gov/7.2#E1 {\displaystyle \mathrm{erf}}] : error function : [http://dlmf.nist.gov/7.2#E1 http://dlmf.nist.gov/7.2#E1]
+[http://drmf.wmflabs.org/wiki/Definition:GaussDist {\displaystyle F\left(x}] : Gaussian distribution : [http://drmf.wmflabs.org/wiki/Definition:GaussDist http://drmf.wmflabs.org/wiki/Definition:GaussDist]
<< [[Definition:RegGammaQ|Definition:RegGammaQ]]
[[Main_Page|Main Page]]
@@ -5018,7 +5064,7 @@ These are defined by == Symbols List == [http://dlmf.nist.gov/7.2#E2 {\displaystyle \mathrm{erfc}}] : complementary error function : [http://dlmf.nist.gov/7.2#E2 http://dlmf.nist.gov/7.2#E2]
-[http://drmf.wmflabs.org/wiki/Definition:GaussDist {\displaystyle Q\left(x}] : Related to Gaussian distribution : [http://drmf.wmflabs.org/wiki/Definition:GaussDist http://drmf.wmflabs.org/wiki/Definition:GaussDist]
+[http://drmf.wmflabs.org/wiki/Definition:GaussDist {\displaystyle Q\left(x}] : Related to Gaussian distribution : [http://drmf.wmflabs.org/wiki/Definition:GaussDist http://drmf.wmflabs.org/wiki/Definition:GaussDist]
<< [[Definition:GaussDistF|Definition:GaussDistF]]
[[Main_Page|Main Page]]
@@ -5057,7 +5103,7 @@ These are defined by [http://dlmf.nist.gov/5.2#E1 {\displaystyle \Gamma}] : Euler's gamma function : [http://dlmf.nist.gov/5.2#E1 http://dlmf.nist.gov/5.2#E1]
[http://dlmf.nist.gov/4.2#E2 {\displaystyle \mathrm{log}}] : principle branch of natural logarithm : [http://dlmf.nist.gov/4.2#E2 http://dlmf.nist.gov/4.2#E2]
-[http://drmf.wmflabs.org/wiki/Definition:RamanujanTauTheta {\displaystyle \tau\theta{z}}] : Ramanujan tau theta function : [http://drmf.wmflabs.org/wiki/Definition:RamanujanTauTheta http://drmf.wmflabs.org/wiki/Definition:RamanujanTauTheta]
+[http://drmf.wmflabs.org/wiki/Definition:RamanujanTauTheta {\displaystyle \tau\theta{z}}] : Ramanujan tau theta function : [http://drmf.wmflabs.org/wiki/Definition:RamanujanTauTheta http://drmf.wmflabs.org/wiki/Definition:RamanujanTauTheta]
<< [[Definition:GaussDistQ|Definition:GaussDistQ]]
[[Main_Page|Main Page]]
@@ -5095,8 +5141,8 @@ These are defined by == Symbols List == [http://dlmf.nist.gov/5.2#E1 {\displaystyle \Gamma}] : Euler's gamma function : [http://dlmf.nist.gov/5.2#E1 http://dlmf.nist.gov/5.2#E1]
-[http://drmf.wmflabs.org/wiki/Definition:RiemannSiegelTheta {\displaystyle \vartheta{t}}] : Riemann-Siegel theta function : [http://drmf.wmflabs.org/wiki/Definition:RiemannSiegelTheta http://drmf.wmflabs.org/wiki/Definition:RiemannSiegelTheta]
[http://dlmf.nist.gov/4.2#E2 {\displaystyle \mathrm{ln}}] : principal branch of logarithm function : [http://dlmf.nist.gov/4.2#E2 http://dlmf.nist.gov/4.2#E2]
+[http://drmf.wmflabs.org/wiki/Definition:RiemannSiegelTheta {\displaystyle \vartheta{t}}] : Riemann-Siegel theta function : [http://drmf.wmflabs.org/wiki/Definition:RiemannSiegelTheta http://drmf.wmflabs.org/wiki/Definition:RiemannSiegelTheta]
<< [[Definition:RamanujanTauTheta|Definition:RamanujanTauTheta]]
[[Main_Page|Main Page]]
@@ -5138,7 +5184,7 @@ These are defined by == Symbols List == [http://dlmf.nist.gov/4.14#E1 {\displaystyle \mathrm{sin}}] : sine function : [http://dlmf.nist.gov/4.14#E1 http://dlmf.nist.gov/4.14#E1]
-[http://drmf.wmflabs.org/wiki/Definition:Sinc {\displaystyle \mathrm{sinc}{x}}] : sine cardinal function : [http://drmf.wmflabs.org/wiki/Definition:Sinc http://drmf.wmflabs.org/wiki/Definition:Sinc]
+[http://drmf.wmflabs.org/wiki/Definition:Sinc {\displaystyle \mathrm{sinc}{x}}] : sine cardinal function : [http://drmf.wmflabs.org/wiki/Definition:Sinc http://drmf.wmflabs.org/wiki/Definition:Sinc]
<< [[Definition:RiemannSiegelTheta|Definition:RiemannSiegelTheta]]
[[Main_Page|Main Page]]
diff --git a/main_page/src/main.py b/main_page/src/main.py index 2e4e42e..3d5bb35 100644 --- a/main_page/src/main.py +++ b/main_page/src/main.py @@ -84,7 +84,7 @@ def update_headers(text, definitions): def get_macro_name(macro): macro_name = "" for ch in macro: - if ch.isalpha(): + if ch.isalpha() or ch == "\\": macro_name += ch elif ch in ["@", "{", "["]: break @@ -92,34 +92,54 @@ def get_macro_name(macro): return macro_name +def find_all(pattern, string): + """Yields all positions of where pattern is present in string.""" + + i = string.find(pattern) + while i != -1: + yield i + i = string.find(pattern, i + 1) + + def get_symbols(text, glossary): - symbols = list() - for keyword in list(glossary): - for space in ["{", "@", "[", "\\", " "]: - if "\\" + keyword + space in text: - symbols.append(keyword) + symbols = set() + + for keyword in glossary: + for index in find_all(keyword, text): + # if the macro is present in the text + if index != -1: + index += len(keyword) # now index of next character + + if index >= len(text) or not text[index].isalpha(): + symbols.add(keyword) span_text = "" - for symbol in set(sorted(symbols)): - t = "" - link = glossary[symbol][-2] + for symbol in sorted(symbols, key=str.lower): + links = list() + for cell in glossary[symbol]: + if "http://" in cell or "https://" in cell: + links.append(cell) + + id_link = links[0] + links = ["[" + link + " " + link + "]" for link in links] + meaning = list(glossary[symbol][1]) count = 0 for i, ch in enumerate(meaning): if ch == "$" and count % 2 == 0: - meaning[i] = "[" + link + " {\\displaystyle " + appearance + \ - "}] : " + ''.join(meaning) + " : [" + link + " " + link + "]
\n" + span_text += "[" + id_link + " {\\displaystyle " + appearance + \ + "}] : " + ''.join(meaning) + " : " + " ".join(links) + "
\n" - return span_text[:-1] + return span_text[:-7] # slice off the extra br and endline def add_symbols_data(data): @@ -145,7 +165,7 @@ def add_symbols_data(data): add_spacing = False sflag2 = False # remove data (to be regenerated later) - if page.find("== Symbols List ==") != -1: + if "== Symbols List ==" in page: to_write = page.split("== Symbols List ==")[0] page = to_write else: @@ -162,8 +182,7 @@ def add_symbols_data(data): to_write += "\n\n" to_write += "== Symbols List ==\n\n" - to_parse = page.split("defined by")[1].split("
")[0] - to_write += get_symbols(to_parse, glossary) + to_write += get_symbols(page, glossary) to_write += "\n
\ndrmf_eof\n\n" if sflag2: to_write += "\n" @@ -194,7 +213,7 @@ def main(): text = mod_main.main(text) # only create backup if program did not crash - # create_backup() + create_backup() with open("main_page.mmd", "w") as main_page: main_page.write(text) From 7968ea50769e6c108c7f9f20f92d57c364544665 Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Wed, 10 Aug 2016 14:06:19 -0400 Subject: [PATCH 211/402] Add test for Gamma, Integrate, LegendreP, LegendreQ, PolyEulerGamma, Product, QPochhammer, and Summation functions --- eCF/.gitignore | 1 + eCF/data/IdentitiesTest.m | 11 +- eCF/src/MathematicaToLaTeX.py | 22 ++-- .../test_single_macro_conversion_functions.py | 115 ++++++++++++++++++ 4 files changed, 128 insertions(+), 21 deletions(-) diff --git a/eCF/.gitignore b/eCF/.gitignore index 13bc7e0..a695525 100644 --- a/eCF/.gitignore +++ b/eCF/.gitignore @@ -3,6 +3,7 @@ other/ src/test.txt data/Identities.* data/newIdentities.* +data/IdentitiesTest.m IdentitiesTest.txt README_Glossary x.log diff --git a/eCF/data/IdentitiesTest.m b/eCF/data/IdentitiesTest.m index 512c469..a064cfc 100644 --- a/eCF/data/IdentitiesTest.m +++ b/eCF/data/IdentitiesTest.m @@ -1,10 +1 @@ -(* {"RamanujanTauTheta", 1}*) -ConditionalExpression[RamanujanTauTheta[z] == (z*(137/60 - EulerGamma - Log[2*Pi]))/(1 + Inactive[ContinuedFractionK][((-1)^k*z^2*PolyGamma[2*k, 6])/((1 + 2*k)!*(KroneckerDelta[1 - k]*Log[2*Pi] + ((-1)^k*PolyGamma[2*(-1 + k), 6])/(-1 + 2*k)!)), 1 - ((-1)^k*z^2*PolyGamma[2*k, 6])/((1 + 2*k)!*(KroneckerDelta[1 - k]*Log[2*Pi] + ((-1)^k*PolyGamma[2*(-1 + k), 6])/(-1 + 2*k)!)), {k, 1, Infinity}]), Element[z, Complexes] && Abs[z] < 1] - -(* {"Sinc", 1}*) -ConditionalExpression[Sinc[z] == (1 + Inactive[ContinuedFractionK][z^2/(2*k*(1 + 2*k)), 1 - z^2/(2*k*(1 + 2*k)), {k, 1, Infinity}])^(-1), Element[z, Complexes]] - -(* {"RiemannSiegelTheta", 1}*) -ConditionalExpression[RiemannSiegelTheta[z] == -(z*(Log[Pi] - PolyGamma[0, 1/4]))/2 - (z^3*PolyGamma[2, 1/4])/(48*(1 + Inactive[ContinuedFractionK][(z^2*PolyGamma[2*(1 + k), 1/4])/(8*(3 + 5*k + 2*k^2)*PolyGamma[2*k, 1/4]), 1 - (z^2*PolyGamma[2*(1 + k), 1/4])/(8*(3 + 5*k + 2*k^2)*PolyGamma[2*k, 1/4]), {k, 1, Infinity}])), Element[z, Complexes] && Abs[z] < 1/2] - -testInactive[testInactive[test]test]test \ No newline at end of file +QPochhammer[QPochhammer[q]] \ No newline at end of file diff --git a/eCF/src/MathematicaToLaTeX.py b/eCF/src/MathematicaToLaTeX.py index 0adedab..beb0364 100644 --- a/eCF/src/MathematicaToLaTeX.py +++ b/eCF/src/MathematicaToLaTeX.py @@ -293,16 +293,16 @@ def gamma(line): pos except NameError: pos = find_surrounding(line, 'Gamma', - ex=('PolyGamma', 'CapitalGamma', - 'PolyGamma', 'LogGamma', 'EulerGamma', - 'IncGamma', 'Gamma]', 'GammaQ', - 'GammaRegularized', 'StieltjesGamma')) + ex=('PolyGamma', 'CapitalGamma', 'LogGamma', + 'EulerGamma', 'IncGamma', 'Gamma]', + 'GammaQ', 'GammaRegularized', + 'StieltjesGamma')) else: pos = find_surrounding(line, 'Gamma', - ex=('PolyGamma', 'CapitalGamma', - 'PolyGamma', 'LogGamma', 'EulerGamma', - 'IncGamma', 'Gamma]', 'GammaQ', - 'GammaRegularized', 'StieltjesGamma'), + ex=('PolyGamma', 'CapitalGamma', 'LogGamma', + 'EulerGamma', 'IncGamma', 'Gamma]', + 'GammaQ', 'GammaRegularized', + 'StieltjesGamma'), start=pos[0] + (0, 11) [(1, 0).index(pos[1] == pos[0])]) @@ -435,7 +435,7 @@ def polyeulergamma(line): .format(args[0], args[1]) + line[pos[1]:]) else: line = (line[:pos[0]] + '\\digamma@{{{0}}}' - .format(args[0]) + '}' + line[pos[1]:]) + .format(args[0]) + line[pos[1]:]) return line @@ -526,7 +526,7 @@ def constraint(line): """ sections = arg_split(line, ',') - if len(sections) == 1: + if len(sections) in (1, 0): return line constraints = arg_split(sections[-1].replace('&&', '&'), '&') @@ -772,7 +772,7 @@ def main(): Opens Mathematica file with identities and puts converted lines into newIdentities.tex. """ - test = False + test = True with open(os.path.dirname(os.path.realpath(__file__)) + '/../data/newIdentities.tex', 'w') as latex: diff --git a/eCF/test/test_single_macro_conversion_functions.py b/eCF/test/test_single_macro_conversion_functions.py index f901f4d..ace02b8 100644 --- a/eCF/test/test_single_macro_conversion_functions.py +++ b/eCF/test/test_single_macro_conversion_functions.py @@ -60,47 +60,162 @@ def test_none(self): class TestGamma(TestCase): + def test_single(self): + self.assertEqual(gamma('Gamma[z]'), '\\EulerGamma@{z}') + self.assertEqual(gamma('--Gamma[z]--'), '--\\EulerGamma@{z}--') + self.assertEqual(gamma('Gamma[a,z]'), '\\IncGamma@{a}{z}') + self.assertEqual(gamma('--Gamma[a,z]--'), '--\\IncGamma@{a}{z}--') + self.assertEqual(gamma('Gamma[a,z0,z1]'), '\\IncGamma@{a}{z0} - \\IncGamma@{a}{z1}') + self.assertEqual(gamma('--Gamma[a,z0,z1]--'), '--\\IncGamma@{a}{z0} - \\IncGamma@{a}{z1}--') + + def test_nested(self): + self.assertEqual(gamma('Gamma[Gamma[z]]'), '\\EulerGamma@{\\EulerGamma@{z}}') + self.assertEqual(gamma('--Gamma[Gamma[z]]--'), '--\\EulerGamma@{\\EulerGamma@{z}}--') + self.assertEqual(gamma('Gamma[Gamma[a,z],Gamma[a,z]]'), '\\IncGamma@{\\IncGamma@{a}{z}}{\\IncGamma@{a}{z}}') + self.assertEqual(gamma('--Gamma[Gamma[a,z],Gamma[a,z]]--'), '--\\IncGamma@{\\IncGamma@{a}{z}}{\\IncGamma@{a}{z}}--') + + def test_exceptions(self): + self.assertEqual(gamma('PolyGamma[n,z]'), 'PolyGamma[n,z]') + self.assertEqual(gamma('\\[CapitalGamma]'), '\\[CapitalGamma]') + self.assertEqual(gamma('LogGamma[z]'), 'LogGamma[z]') + self.assertEqual(gamma('EulerGamma'), 'EulerGamma') + self.assertEqual(gamma('\\IncGamma@{a}{b}'), '\\IncGamma@{a}{b}') + self.assertEqual(gamma('\\[Gamma]'), '\\[Gamma]') + self.assertEqual(gamma('\\GammaQ@{a}{z}'), '\\GammaQ@{a}{z}') + self.assertEqual(gamma('GammaRegularized[a,z]'), 'GammaRegularized[a,z]') + self.assertEqual(gamma('StieltjesGamma[k]'), 'StieltjesGamma[k]') + def test_none(self): self.assertEqual(gamma('none'), 'none') class TestIntegrate(TestCase): + def test_single(self): + self.assertEqual(integrate('Integrate[f,{x,xmin,xmax}]'), '\\int_{xmin}^{xmax}{f}d{x}') + self.assertEqual(integrate('--Integrate[f,{x,xmin,xmax}]--'), '--\\int_{xmin}^{xmax}{f}d{x}--') + + def test_nested(self): + self.assertEqual(integrate('Integrate[Integrate[f,{x,xmin,xmax}],{Integrate[f,{x,xmin,xmax}],xmin,xmax}]'), '\\int_{xmin}^{xmax}{\\int_{xmin}^{xmax}{f}d{x}}d{\\int_{xmin}^{xmax}{f}d{x}}') + self.assertEqual(integrate('--Integrate[Integrate[f,{x,xmin,xmax}],{Integrate[f,{x,xmin,xmax}],xmin,xmax}]--'), '--\\int_{xmin}^{xmax}{\\int_{xmin}^{xmax}{f}d{x}}d{\\int_{xmin}^{xmax}{f}d{x}}--') + def test_none(self): self.assertEqual(integrate('none'), 'none') class TestLegendreP(TestCase): + def test_single(self): + self.assertEqual(legendrep('LegendreP[n,x]'), '\\LegendreP{n}@{x}') + self.assertEqual(legendrep('--LegendreP[n,x]--'), '--\\LegendreP{n}@{x}--') + self.assertEqual(legendrep('LegendreP[n,m,1,x]'), '\\FerrersP[m]{n}@{x}') + self.assertEqual(legendrep('--LegendreP[n,m,1,x]--'), '--\\FerrersP[m]{n}@{x}--') + self.assertEqual(legendrep('LegendreP[n,m,2,x]'), '\\FerrersP[m]{n}@{x}') + self.assertEqual(legendrep('--LegendreP[n,m,2,x]--'), '--\\FerrersP[m]{n}@{x}--') + self.assertEqual(legendrep('LegendreP[n,m,3,x]'), '\\LegendreP[m]{n}@{x}') + self.assertEqual(legendrep('--LegendreP[n,m,3,x]--'), '--\\LegendreP[m]{n}@{x}--') + + def test_nested(self): + self.assertEqual(legendrep('LegendreP[LegendreP[n,x],LegendreP[n,x]]'), '\\LegendreP{\\LegendreP{n}@{x}}@{\\LegendreP{n}@{x}}') + self.assertEqual(legendrep('--LegendreP[LegendreP[n,x],LegendreP[n,x]]--'), '--\\LegendreP{\\LegendreP{n}@{x}}@{\\LegendreP{n}@{x}}--') + self.assertEqual(legendrep('LegendreP[LegendreP[n,m,1,x],LegendreP[n,m,1,x],1,LegendreP[n,m,1,x]]'), '\\FerrersP[\\FerrersP[m]{n}@{x}]{\\FerrersP[m]{n}@{x}}@{\\FerrersP[m]{n}@{x}}') + self.assertEqual(legendrep('--LegendreP[LegendreP[n,m,1,x],LegendreP[n,m,1,x],1,LegendreP[n,m,1,x]]--'), '--\\FerrersP[\\FerrersP[m]{n}@{x}]{\\FerrersP[m]{n}@{x}}@{\\FerrersP[m]{n}@{x}}--') + self.assertEqual(legendrep('LegendreP[LegendreP[n,m,3,x],LegendreP[n,m,3,x],3,LegendreP[n,m,3,x]]'), '\\LegendreP[\\LegendreP[m]{n}@{x}]{\\LegendreP[m]{n}@{x}}@{\\LegendreP[m]{n}@{x}}') + self.assertEqual(legendrep('--LegendreP[LegendreP[n,m,3,x],LegendreP[n,m,3,x],3,LegendreP[n,m,3,x]]--'), '--\\LegendreP[\\LegendreP[m]{n}@{x}]{\\LegendreP[m]{n}@{x}}@{\\LegendreP[m]{n}@{x}}--') + def test_none(self): self.assertEqual(legendrep('none'), 'none') class TestLegendreQ(TestCase): + def test_single(self): + self.assertEqual(legendreq('LegendreQ[n,x]'), '\\LegendreQ{n}@{x}') + self.assertEqual(legendreq('--LegendreQ[n,x]--'), '--\\LegendreQ{n}@{x}--') + self.assertEqual(legendreq('LegendreQ[n,m,1,x]'), '\\FerrersQ[m]{n}@{x}') + self.assertEqual(legendreq('--LegendreQ[n,m,1,x]--'), '--\\FerrersQ[m]{n}@{x}--') + self.assertEqual(legendreq('LegendreQ[n,m,2,x]'), '\\FerrersQ[m]{n}@{x}') + self.assertEqual(legendreq('--LegendreQ[n,m,2,x]--'), '--\\FerrersQ[m]{n}@{x}--') + self.assertEqual(legendreq('LegendreQ[n,m,3,x]'), '\\LegendreQ[m]{n}@{x}') + self.assertEqual(legendreq('--LegendreQ[n,m,3,x]--'), '--\\LegendreQ[m]{n}@{x}--') + + def test_nested(self): + self.assertEqual(legendreq('LegendreQ[LegendreQ[n,x],LegendreQ[n,x]]'), '\\LegendreQ{\\LegendreQ{n}@{x}}@{\\LegendreQ{n}@{x}}') + self.assertEqual(legendreq('--LegendreQ[LegendreQ[n,x],LegendreQ[n,x]]--'), '--\\LegendreQ{\\LegendreQ{n}@{x}}@{\\LegendreQ{n}@{x}}--') + self.assertEqual(legendreq('LegendreQ[LegendreQ[n,m,1,x],LegendreQ[n,m,1,x],1,LegendreQ[n,m,1,x]]'), '\\FerrersQ[\\FerrersQ[m]{n}@{x}]{\\FerrersQ[m]{n}@{x}}@{\\FerrersQ[m]{n}@{x}}') + self.assertEqual(legendreq('--LegendreQ[LegendreQ[n,m,1,x],LegendreQ[n,m,1,x],1,LegendreQ[n,m,1,x]]--'), '--\\FerrersQ[\\FerrersQ[m]{n}@{x}]{\\FerrersQ[m]{n}@{x}}@{\\FerrersQ[m]{n}@{x}}--') + self.assertEqual(legendreq('LegendreQ[LegendreQ[n,m,3,x],LegendreQ[n,m,3,x],3,LegendreQ[n,m,3,x]]'), '\\LegendreQ[\\LegendreQ[m]{n}@{x}]{\\LegendreQ[m]{n}@{x}}@{\\LegendreQ[m]{n}@{x}}') + self.assertEqual(legendreq('--LegendreQ[LegendreQ[n,m,3,x],LegendreQ[n,m,3,x],3,LegendreQ[n,m,3,x]]--'), '--\\LegendreQ[\\LegendreQ[m]{n}@{x}]{\\LegendreQ[m]{n}@{x}}@{\\LegendreQ[m]{n}@{x}}--') + def test_none(self): self.assertEqual(legendreq('none'), 'none') class TestPolyEulergamma(TestCase): + def test_single(self): + self.assertEqual(polyeulergamma('PolyGamma[z]'), '\\digamma@{z}') + self.assertEqual(polyeulergamma('--PolyGamma[z]--'), '--\\digamma@{z}--') + self.assertEqual(polyeulergamma('PolyGamma[n,z]'), '\\polygamma{n}@{z}') + self.assertEqual(polyeulergamma('--PolyGamma[n,z]--'), '--\\polygamma{n}@{z}--') + self.assertEqual(polyeulergamma('PolyGamma[z]PolyGamma[n,z]'), '\\digamma@{z}\\polygamma{n}@{z}') + self.assertEqual(polyeulergamma('--PolyGamma[z]--PolyGamma[n,z]--'), '--\\digamma@{z}--\\polygamma{n}@{z}--') + + def test_nested(self): + self.assertEqual(polyeulergamma('PolyGamma[PolyGamma[z]]'), '\\digamma@{\\digamma@{z}}') + self.assertEqual(polyeulergamma('--PolyGamma[PolyGamma[z]]--'), '--\\digamma@{\\digamma@{z}}--') + self.assertEqual(polyeulergamma('PolyGamma[PolyGamma[n,z],PolyGamma[n,z]]'), '\\polygamma{\\polygamma{n}@{z}}@{\\polygamma{n}@{z}}') + self.assertEqual(polyeulergamma('--PolyGamma[PolyGamma[n,z],PolyGamma[n,z]]--'), '--\\polygamma{\\polygamma{n}@{z}}@{\\polygamma{n}@{z}}--') + def test_none(self): self.assertEqual(polyeulergamma('none'), 'none') class TestProduct(TestCase): + def test_single(self): + self.assertEqual(product('Product[f,{i,imin,imax}]'), '\\Prod{i}{imin}{imax}@{f}') + self.assertEqual(product('--Product[f,{i,imin,imax}]--'), '--\\Prod{i}{imin}{imax}@{f}--') + + def test_nested(self): + self.assertEqual(product('Product[Product[f,{i,imin,imax}],{Product[f,{i,imin,imax}],imin,imax}]'), '\\Prod{\\Prod{i}{imin}{imax}@{f}}{imin}{imax}@{\\Prod{i}{imin}{imax}@{f}}') + self.assertEqual(product('--Product[Product[f,{i,imin,imax}],{Product[f,{i,imin,imax}],imin,imax}]--'), '--\\Prod{\\Prod{i}{imin}{imax}@{f}}{imin}{imax}@{\\Prod{i}{imin}{imax}@{f}}--') + def test_none(self): self.assertEqual(product('none'), 'none') class TestQPochhammer(TestCase): + def test_single(self): + self.assertEqual(qpochhammer('QPochhammer[a,q,n]'), '\\qPochhammer{a}{q}{n}') + self.assertEqual(qpochhammer('--QPochhammer[a,q,n]--'), '--\\qPochhammer{a}{q}{n}--') + self.assertEqual(qpochhammer('QPochhammer[a,q]'), '\\qPochhammer{a}{q}{\\infty}') + self.assertEqual(qpochhammer('--QPochhammer[a,q]--'), '--\\qPochhammer{a}{q}{\\infty}--') + self.assertEqual(qpochhammer('QPochhammer[q]'), '\\qPochhammer{q}{q}{\\infty}') + self.assertEqual(qpochhammer('--QPochhammer[q]--'), '--\\qPochhammer{q}{q}{\\infty}--') + self.assertEqual(qpochhammer('QPochhammer[a,q,n]QPochhammer[a,q]QPochhammer[q]'), '\\qPochhammer{a}{q}{n}\\qPochhammer{a}{q}{\\infty}\\qPochhammer{q}{q}{\\infty}') + self.assertEqual(qpochhammer('--QPochhammer[a,q,n]--QPochhammer[a,q]--QPochhammer[q]--'), '--\\qPochhammer{a}{q}{n}--\\qPochhammer{a}{q}{\\infty}--\\qPochhammer{q}{q}{\\infty}--') + + def test_nested(self): + self.assertEqual(qpochhammer('QPochhammer[QPochhammer[q],QPochhammer[q],QPochhammer[q]]'), '\\qPochhammer{\\qPochhammer{q}{q}{\\infty}}{\\qPochhammer{q}{q}{\\infty}}{\\qPochhammer{q}{q}{\\infty}}') + self.assertEqual(qpochhammer('--QPochhammer[QPochhammer[q],QPochhammer[q],QPochhammer[q]]--'), '--\\qPochhammer{\\qPochhammer{q}{q}{\\infty}}{\\qPochhammer{q}{q}{\\infty}}{\\qPochhammer{q}{q}{\\infty}}--') + self.assertEqual(qpochhammer('QPochhammer[QPochhammer[q],QPochhammer[q]]'), '\\qPochhammer{\\qPochhammer{q}{q}{\\infty}}{\\qPochhammer{q}{q}{\\infty}}{\\infty}') + self.assertEqual(qpochhammer('--QPochhammer[QPochhammer[q],QPochhammer[q]]--'), '--\\qPochhammer{\\qPochhammer{q}{q}{\\infty}}{\\qPochhammer{q}{q}{\\infty}}{\\infty}--') + def test_none(self): self.assertEqual(qpochhammer('none'), 'none') class TestSummation(TestCase): + def test_single(self): + self.assertEqual(summation('Sum[f,{i,imin,imax}]'), '\\Sum{i}{imin}{imax}@{f}') + self.assertEqual(summation('--Sum[f,{i,imin,imax}]--'), '--\\Sum{i}{imin}{imax}@{f}--') + + def test_nested(self): + self.assertEqual(summation('Sum[Sum[f,{i,imin,imax}],{Sum[f,{i,imin,imax}],imin,imax}]'), '\\Sum{\\Sum{i}{imin}{imax}@{f}}{imin}{imax}@{\\Sum{i}{imin}{imax}@{f}}') + self.assertEqual(summation('--Sum[Sum[f,{i,imin,imax}],{Sum[f,{i,imin,imax}],imin,imax}]--'), '--\\Sum{\\Sum{i}{imin}{imax}@{f}}{imin}{imax}@{\\Sum{i}{imin}{imax}@{f}}--') + def test_none(self): self.assertEqual(summation('none'), 'none') \ No newline at end of file From 4c81613c6c208979c78cd7858deac084c0fccfe4 Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Wed, 10 Aug 2016 14:13:05 -0400 Subject: [PATCH 212/402] Remove IdentitiesTest.m --- eCF/data/IdentitiesTest.m | 1 - 1 file changed, 1 deletion(-) delete mode 100644 eCF/data/IdentitiesTest.m diff --git a/eCF/data/IdentitiesTest.m b/eCF/data/IdentitiesTest.m deleted file mode 100644 index a064cfc..0000000 --- a/eCF/data/IdentitiesTest.m +++ /dev/null @@ -1 +0,0 @@ -QPochhammer[QPochhammer[q]] \ No newline at end of file From bb95b565a1a4a33e2d35ac9b9a78dccca5105d3e Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Wed, 10 Aug 2016 14:58:58 -0400 Subject: [PATCH 213/402] Add tests for replace_operators function --- eCF/src/MathematicaToLaTeX.py | 11 ++++++--- eCF/test/test_replace_operators.py | 39 ++++++++++++++++++++++++++++++ 2 files changed, 47 insertions(+), 3 deletions(-) diff --git a/eCF/src/MathematicaToLaTeX.py b/eCF/src/MathematicaToLaTeX.py index beb0364..81c8510 100644 --- a/eCF/src/MathematicaToLaTeX.py +++ b/eCF/src/MathematicaToLaTeX.py @@ -534,10 +534,10 @@ def constraint(line): for i, item, in enumerate(constraints): if i == 0: constraints[i] = ('\n% \\constraint{$' + - constraints[i].replace('&', ' \\land ')) + item.replace('&', ' \\land ')) else: constraints[i] = ('\n% & $' + - constraints[i].replace('&', ' \\land ')) + item.replace('&', ' \\land ')) if i == len(constraints) - 1: constraints[i] += '$}' else: @@ -740,6 +740,11 @@ def replace_operators(line): line = line.replace('&', ' \\land ') line = line.replace(' ', ' ') line += parts[1] + else: + line = line.replace('(', '\\left( ') + line = line.replace(')', ' \\right)') + line = line.replace('&', ' \\land ') + line = line.replace(' ', ' ') line = line.replace('"a"', 'a') line = line.replace('Catalan', '\\CatalansConstant') @@ -772,7 +777,7 @@ def main(): Opens Mathematica file with identities and puts converted lines into newIdentities.tex. """ - test = True + test = False with open(os.path.dirname(os.path.realpath(__file__)) + '/../data/newIdentities.tex', 'w') as latex: diff --git a/eCF/test/test_replace_operators.py b/eCF/test/test_replace_operators.py index b07ce81..ee87043 100644 --- a/eCF/test/test_replace_operators.py +++ b/eCF/test/test_replace_operators.py @@ -5,3 +5,42 @@ from unittest import TestCase from MathematicaToLaTeX import replace_operators + +class TestReplaceOperators(TestCase): + + def test_without_percent(self): + self.assertEqual(replace_operators('=='), ' = ') + self.assertEqual(replace_operators('||'), ' \\lor ') + self.assertEqual(replace_operators('>='), ' \\geq ') + self.assertEqual(replace_operators('<='), ' \\leq ') + self.assertEqual(replace_operators('LessEqual'), ' \\leq ') + self.assertEqual(replace_operators('Less'), ' < ') + self.assertEqual(replace_operators('>'), ' > ') + self.assertEqual(replace_operators('<'), ' < ') + self.assertEqual(replace_operators('='), ' = ') + self.assertEqual(replace_operators('^'), ' ^ ') + self.assertEqual(replace_operators('*'), ' ') + self.assertEqual(replace_operators('+'), ' + ') + self.assertEqual(replace_operators('-'), ' - ') + self.assertEqual(replace_operators(','), ', ') + self.assertEqual(replace_operators('('), '\\left( ') + self.assertEqual(replace_operators(')'), ' \\right)') + self.assertEqual(replace_operators('&'), ' \\land ') + self.assertEqual(replace_operators(' '), ' ') + self.assertEqual(replace_operators('"a"'), 'a') + + def test_constants(self): + self.assertEqual(replace_operators('Catalan'), '\\CatalansConstant') + self.assertEqual(replace_operators('GoldenRatio'), '\\GoldenRatio') + self.assertEqual(replace_operators('Pi'), '\\pi') + self.assertEqual(replace_operators('CalculateData`Private`nu'), '\\text{CalculateData`Private`nu}') + + def test_with_percent(self): + self.assertEqual(replace_operators('(%('), '\\left( %(') + self.assertEqual(replace_operators(')%)'), ' \\right)%)') + self.assertEqual(replace_operators('&%&'), ' \\land %&') + self.assertEqual(replace_operators(' % '), ' % ') + + def test_none(self): + self.assertEqual(replace_operators(''), '') + self.assertEqual(replace_operators('%'), '%') \ No newline at end of file From 0d1515d749467f312cb9d8604314654de4add01c Mon Sep 17 00:00:00 2001 From: Joon Bang Date: Thu, 11 Aug 2016 10:50:07 -0400 Subject: [PATCH 214/402] Consolidate all code into one file --- main_page/src/add_definition.py | 4 +- main_page/src/main.py | 151 ++++++++++++++--- main_page/src/mod_main.py | 115 ------------- main_page/src/symbols_list.py | 278 -------------------------------- 4 files changed, 131 insertions(+), 417 deletions(-) delete mode 100644 main_page/src/mod_main.py delete mode 100644 main_page/src/symbols_list.py diff --git a/main_page/src/add_definition.py b/main_page/src/add_definition.py index 41c4d7c..bc7bc77 100644 --- a/main_page/src/add_definition.py +++ b/main_page/src/add_definition.py @@ -9,8 +9,8 @@ def main(): result = "drmf_bof\n" + \ "'''Definition:" + macro_name + "'''\n" + \ "The LaTeX DLMF and DRMF macro '''\\" + macro_name + "''' represents the " + macro_name + ".\n\n" + \ - "These are defined by \n\n" + \ - "\n\n{\\displaystyle\n\n" + definition + "\n\n}\n\n
\n\ndrmf_eof\n" + "These are defined by \n" + \ + "{\\displaystyle\n" + definition + "\n}\n
\n\ndrmf_eof\n" with open("main_page.mmd", "a") as main_page: main_page.write("\n" + result) diff --git a/main_page/src/main.py b/main_page/src/main.py index 3d5bb35..940e063 100644 --- a/main_page/src/main.py +++ b/main_page/src/main.py @@ -4,7 +4,6 @@ import csv import time -import mod_main GLOSSARY_LOCATION = "new.Glossary.csv" @@ -21,6 +20,82 @@ def generate_html(tag_name, options, text, spacing=True): return result +def get_macro_name(macro): + macro_name = "" + for ch in macro: + if ch.isalpha() or ch == "\\": + macro_name += ch + elif ch in ["@", "{", "["]: + break + + return macro_name + + +def find_all(pattern, string): + """Yields all positions of where pattern is present in string.""" + + i = string.find(pattern) + while i != -1: + yield i + i = string.find(pattern, i + 1) + + +def remove_optional_params(string): + parse = True + text = "" + for ch in string: + if parse and ch == "[": + parse = False + elif parse: + text += ch + else: + if ch == "]": + parse = True + + return text + + +def find_all_positions(entry): + macro = entry[0] + category = entry[3].split("|", 1)[0] + keys = entry[3].split("|", 1)[1].split(":") + result = [] + + for key in keys: + if key in ["F", "FO", "O", "O1"]: + result.append(macro.split("@", 1)[0]) + if macro.count("@") > 0 and key == "O": + result.append(macro) + + if key == "FnO": # remove optional parameters + string = macro + if macro.count("@") > 0: + string = string[:string.find("@")] + result.append(remove_optional_params(string)) + + if key in ["P", "PO", "PnO"]: + result.append(macro.replace("@@", "@")) + + if key == "PnO": # remove optional parameters + result[-1] = remove_optional_params(result[-1]) + + if key in ["nP", "nPO", "PS", "O2"]: + result.append(macro) + + if key == "nPnO": # remove optional paramters + result.append(remove_optional_params(macro)) + + if "fo" in key: + result.append(macro.replace("@@@", "@" * int(key[2]))) + + return sorted(result, key=lambda x: x.count("@")), category + + +def macro_match(macro, entry): + """Determines whether the entry refers to the macro.""" + return macro in entry and (len(macro) == len(entry) or not entry[len(macro)].isalpha()) + + def update_macro_list(text): lines = text.split("\n") definitions = list() @@ -81,26 +156,6 @@ def update_headers(text, definitions): return output -def get_macro_name(macro): - macro_name = "" - for ch in macro: - if ch.isalpha() or ch == "\\": - macro_name += ch - elif ch in ["@", "{", "["]: - break - - return macro_name - - -def find_all(pattern, string): - """Yields all positions of where pattern is present in string.""" - - i = string.find(pattern) - while i != -1: - yield i - i = string.find(pattern, i + 1) - - def get_symbols(text, glossary): symbols = set() @@ -191,6 +246,58 @@ def add_symbols_data(data): return result +def add_usage(lines): + with open("categories.txt") as cats: + categories = cats.readlines() + + to_write = "" + i = 0 + + while True: + n = lines.find("'''Definition:", i) + if n == -1: + break + + macro_name = lines[n:].split("'''")[1][11:] + left = lines.find("'''\\", n) + right = lines.find("'''", left + len("'''\\")) + + if lines[left:right].find("{") != -1: + lines = lines[:left] + "'''\\" + macro_name + lines[right:] + + if lines.find("\nThis macro is in the category of", n) < lines.find("'''Definition:", n + len( + "'''Definition:") + 1) and lines.find("\nThis macro is in the category of", n) != -1: + p = lines.find("\nThis macro is in the category of", n) + else: + p = lines.find(".\n", n) + 1 + to_write += lines[i:p].rstrip() + "\n\n" + + calls = [] + glossary = csv.reader(open(GLOSSARY_LOCATION, 'rb'), delimiter=',', quotechar='\"') + for entry in glossary: + if macro_match("\\" + macro_name, entry[0]): + macro_calls, category = find_all_positions(entry) + calls += macro_calls + + for line in categories: + key, meaning = line.split(" - ") + if key == category: + category_text = "This macro is in the category of " + meaning + break + + text = category_text + "\nIn math mode, this macro can be called in the following way" + "s" * ( + len(calls) > 1) + ":\n\n" + + for call in calls: + text += ":'''" + call + "'''" + " produces {\\displaystyle " + call + "}
\n" + + # Now add the multiple ways \macroname{n}@... produces \macroname{n}@... + to_write += text + "\n" + i = lines.find("These are defined by", p) + + return to_write + lines[i:] + + def create_backup(): with open("main_page.mmd") as current: local_time = time.asctime(time.localtime(time.time())).split() @@ -210,7 +317,7 @@ def main(): text, definitions = update_macro_list(text) text = add_symbols_data(text) text = update_headers(text, definitions) - text = mod_main.main(text) + text = add_usage(text) # only create backup if program did not crash create_backup() diff --git a/main_page/src/mod_main.py b/main_page/src/mod_main.py deleted file mode 100644 index 7d6d40c..0000000 --- a/main_page/src/mod_main.py +++ /dev/null @@ -1,115 +0,0 @@ -__author__ = "Azeem Mohammed" -__status__ = "Development" -__credits__ = ["Joon Bang", "Azeem Mohammed"] - -import csv - -GLOSSARY_LOCATION = "new.Glossary.csv" - - -def remove_optional_params(string): - parse = True - text = "" - for ch in string: - if parse and ch == "[": - parse = False - elif parse: - text += ch - else: - if ch == "]": - parse = True - - return text - - -def find_all_positions(entry): - macro = entry[0] - category = entry[3].split("|", 1)[0] - keys = entry[3].split("|", 1)[1].split(":") - result = [] - - for key in keys: - if key in ["F", "FO", "O", "O1"]: - result.append(macro.split("@", 1)[0]) - if macro.count("@") > 0 and key == "O": - result.append(macro) - - if key == "FnO": # remove optional parameters - string = macro - if macro.count("@") > 0: - string = string[:string.find("@")] - result.append(remove_optional_params(string)) - - if key in ["P", "PO", "PnO"]: - result.append(macro.replace("@@", "@")) - - if key == "PnO": # remove optional parameters - result[-1] = remove_optional_params(result[-1]) - - if key in ["nP", "nPO", "PS", "O2"]: - result.append(macro) - - if key == "nPnO": # remove optional paramters - result.append(remove_optional_params(macro)) - - if "fo" in key: - result.append(macro.replace("@@@", "@" * int(key[2]))) - - return sorted(result, key=lambda x: x.count("@")), category - - -def macro_match(macro, entry): - """Determines whether the entry refers to the macro.""" - return macro in entry and (len(macro) == len(entry) or not entry[len(macro)].isalpha()) - - -def main(lines): - lines = lines.replace("The LaTeX DLMF macro '''\\", "The LaTeX DLMF and DRMF macro '''\\") - to_write = "" - i = 0 - - with open("categories.txt") as cats: - categories = cats.readlines() - - while True: - n = lines.find("'''Definition:", i) - if n == -1: - break - - macro_name = lines[n + len("'''Definition:"):lines.find("'''", n + len("'''Definition:"))] - r = lines.find("'''\\", n) - q = lines.find("'''", r + len("'''\\")) - if lines[r:q].find("{") != -1: - lines = lines[0:r] + "'''\\" + macro_name + lines[q:] - - if lines.find("\nThis macro is in the category of", n) < lines.find("'''Definition:", n + len( - "'''Definition:") + 1) and lines.find("\nThis macro is in the category of", n) != -1: - p = lines.find("\nThis macro is in the category of", n) - else: - p = lines.find(".\n", n) + 1 - to_write += lines[i:p].rstrip() + "\n\n" - - calls = [] - glossary = csv.reader(open(GLOSSARY_LOCATION, 'rb'), delimiter=',', quotechar='\"') - for entry in glossary: - if macro_match("\\" + macro_name, entry[0]): - macro_calls, category = find_all_positions(entry) - calls += macro_calls - - for line in categories: - key, meaning = line.split(" - ") - if key == category: - category_text = "This macro is in the category of " + meaning - break - - text = category_text + "\nIn math mode, this macro can be called in the following way" + "s" * ( - len(calls) > 1) + ":\n\n" - - for call in calls: - text += ":'''" + call + "'''" + " produces {\\displaystyle " + call + "}
\n" - - # Now add the multiple ways \macroname{n}@... produces \macroname{n}@... - to_write += text + "\n" - i = lines.find("These are defined by", p) - - return to_write + lines[i:] diff --git a/main_page/src/symbols_list.py b/main_page/src/symbols_list.py deleted file mode 100644 index b44936c..0000000 --- a/main_page/src/symbols_list.py +++ /dev/null @@ -1,278 +0,0 @@ -__author__ = "Azeem Mohammed" -__status__ = "Development" -__credits__ = ["Joon Bang", "Azeem Mohammed"] - -import csv - -GLOSSARY_LOCATION = "new.Glossary.csv" - - -def get_symbols(line): - # (str) -> list - """Gets all symbols on a line.""" - if line == "": - return [] - - line = line.replace("\n", " ") - sym_list = [] - - symbol = "" - sym_flag = False - arg_flag = False - count = 0 - - for i, ch in enumerate(line): - if sym_flag: - if ch == ["{", "["]: - count += 1 - arg_flag = True - - if ch not in ["}", "]"] and (arg_flag or ch.isalpha()): - symbol += ch - - elif ch not in ["}", "]"]: - sym_flag = False - arg_flag = False - sym_list.append(symbol) - sym_list += (get_symbols(symbol)) - symbol = "" - - else: - count -= 1 - symbol += ch - if i + 1 == len(line): - p = "" - else: - p = line[i + 1] - if count == 0 and p not in ["{", "[", "@"]: - sym_flag = False - sym_list.append(symbol) - sym_list += (get_symbols(symbol)) - arg_flag = False - symbol = "" - - elif ch == "\\": - sym_flag = True - - sym_list.append(symbol) - sym_list += get_symbols(symbol) - return sym_list - - -def main(data): - pages = data[data.find("drmf_bof"):].split("drmf_eof")[:-1] - - print "Pages: " + str(len(pages)) - - result = "" - for page in pages: - # skip over non-definition pages - if "'''Definition:" not in page: - to_write = page + "drmf_eof\n" - result += to_write - continue - - page = page.replace("drmf_bof", "").strip("\n") - add_spacing = False - sflag2 = False - # remove data (to be regenerated later) - if page.find("== Symbols List ==") != -1: - to_write = page.split("== Symbols List ==")[0] - page = to_write - else: - add_spacing = True - if page.find("drmf_foot") == -1: - to_write = page - sflag2 = True - else: - to_write = page.split("
")[0] - page = to_write - - to_write = "drmf_bof\n" + to_write - if add_spacing: - to_write += "\n\n" - - to_write += "== Symbols List ==\n\n" - - symbols = get_symbols((' '.join(page.split("\n")) + " ").replace("\n", " ")) - print symbols - - new_symbols = [] - - for symbol in symbols: - flag_a = True - flag_b = True - c_n = 0 - c_c = 0 - arg_count = 0 - - for ch in symbol: - if ch.isalpha() and flag_a: - c_n += 1 - else: - flag_a = False - if ch in ["{", "["]: - c_c += 1 - if ch in ["}", "]"]: - c_c -= 1 - if c_c == 0: - arg_count += 1 - noA = symbol[:c_n] - - for y in new_symbols: - c_n = 0 - c_c = 0 - arg_count = 0 - flag_a = True - for ch in y: - if ch.isalpha() and flag_a: - c_n += 1 - else: - flag_a = False - if ch in ["{", "["]: - c_c += 1 - if ch in ["}", "]"]: - c_c -= 1 - if c_c == 0: - arg_count += 1 - - if y[:c_n].strip() == noA.strip(): - flag_b = False - break - - if flag_b: - new_symbols.append(symbol) - - new_symbols.reverse() - final_symbols = [] - - for i in xrange(len(new_symbols) - 1, -1, -1): - symbolPar = "\\" + new_symbols[i] - arg_count = 0 - par_count = 0 - parFlag = False - c_c = 0 - for ch in symbolPar: - if ch == "@": - parFlag = True - elif ch.isalpha(): - c_n += 1 - elif ch in ["{", "["]: - c_c += 1 - elif ch in ["}", "]"]: - c_c -= 1 - if c_c == 0: - if parFlag: - par_count += 1 - else: - arg_count += 1 - - if symbolPar.find("{") != -1 or symbolPar.find("[") != -1: - if symbolPar.find("[") == -1: - symbol = symbolPar[0:symbolPar.find("{")] - elif symbolPar.find("{") == -1 or symbolPar.find("[") < symbolPar.find("{"): - symbol = symbolPar[0:symbolPar.find("[")] - else: - symbol = symbolPar[0:symbolPar.find("{")] - else: - symbol = symbolPar - gFlag = False - checkFlag = False - get = False - - glossary = csv.reader(open(GLOSSARY_LOCATION, "rb"), delimiter=',', quotechar='\"') - - preG = "" - symbol = symbol.strip("}").strip("]") - for S in glossary: - G = S - arg_count = 0 - par_count = 0 - parFlag = False - c_c = 0 - ind = G[0].find("@") - if ind == -1: - ind = len(G[0]) - 1 - for ch in G[0]: - if ch == "@": - parFlag = True - elif ch.isalpha(): - c_n += 1 - else: - if ch in ["{", "["]: - c_c += 1 - if ch in ["}", "]"]: - c_c -= 1 - if c_c == 0: - if parFlag: - par_count += 1 - else: - arg_count += 1 - if G[0].find(symbol) == 0 and (len(G[0]) == len(symbol) or not G[0][ - len(symbol)].isalpha()): - checkFlag = True - get = True - preG = S - elif checkFlag: - checkFlag = False - get = True - - if get: - G = preG - - if symbolPar.find("@") != -1: - Q = symbolPar[:symbolPar.find("@")] - else: - Q = symbolPar - - arg_list = [] - if len(Q) > len(symbol) and (Q[len(symbol)] == "{" or Q[len(symbol)] == "["): - ap = "" - for o in xrange(len(symbol), len(Q)): - if Q[o] == "}" or ch == "]": - arg_list.append(ap) - ap = "" - elif Q[o] != "{" and ch != "[": - ap += Q[o] - - websiteF = "" - web1 = G[5] - for t in xrange(5, len(G)): - if G[t] != "": - websiteF = websiteF + " [" + G[t] + " " + G[t] + "]" - - p1 = G[4].strip("$") - p1 = "{\\displaystyle " + p1 + "}" - new2 = "" - mathF = True - p2 = G[1] - for k in p2: - if k == "$": - if mathF: - new2 += "{\\displaystyle " - else: - new2 += "}" - mathF = not mathF - else: - new2 += k - p2 = new2 - final_symbols.append(web1 + " " + p1 + "] : " + p2 + " :" + websiteF) - break - - if not gFlag: - new_symbols.pop(i) - - gFlag = True - for y in final_symbols: - if gFlag: - gFlag = False - to_write += ("[" + y) - else: - to_write += ("
\n[" + y) - - to_write += "\n
\ndrmf_eof\n\n" - if sflag2: - to_write += "\n" - result += to_write - - return result From 106bdd9393138a1c3ffab42ba71916df2e26a522 Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Thu, 11 Aug 2016 11:26:31 -0400 Subject: [PATCH 215/402] Change convert_fraction function to use better string formatting --- eCF/src/MathematicaToLaTeX.py | 26 ++++---------------------- 1 file changed, 4 insertions(+), 22 deletions(-) diff --git a/eCF/src/MathematicaToLaTeX.py b/eCF/src/MathematicaToLaTeX.py index 81c8510..286092a 100644 --- a/eCF/src/MathematicaToLaTeX.py +++ b/eCF/src/MathematicaToLaTeX.py @@ -642,25 +642,7 @@ def convert_fraction(line): # incorrectly change it to " )( / )( ", but there are no cases of # this happening yet, so I have not gone to fixing this yet. if (line[j + 1] == '(' and line[i - 1] == ')' and - line[i + 1] == '(' and line[k] == ')'): - # ()/() - line = (line[:j + 1] + '\\frac{' + line[j + 2:i - 1] + '}{' + - line[i + 2:k] + '}' + line[k + 1:]) - elif line[j + 1] == '(' and line[i - 1] == ')': - # ()/-- - line = (line[:j + 1] + '\\frac{' + line[j + 2:i - 1] + '}{' + - line[i + 1:k + 1] + '}' + line[k + 1:]) - elif line[i + 1] == '(' and line[k] == ')': - # --/() - line = (line[:j + 1] + '\\frac{' + line[j + 1:i] + '}{' + - line[i + 2:k] + '}' + line[k + 1:]) - else: - # --/-- - line = (line[:j + 1] + '\\frac{' + line[j + 1:i] + '}{' + - line[i + 1:k + 1] + '}' + line[k + 1:]) - - '''if (line[j + 1] == '(' and line[i - 1] == ')' and - line[i + 1] == '(' and line[k] == ')'): + line[i + 1] == '(' and line[k] == ')'): # ()/() line = '{0}\\frac{{{1}}}{{{2}}}{3}'\ .format(line[:j + 1], line[j + 2:i - 1], @@ -679,7 +661,7 @@ def convert_fraction(line): # --/-- line = '{0}\\frac{{{1}}}{{{2}}}{3}'\ .format(line[:j + 1], line[j + 1:i], - line[i + 1:k + 1], line[k + 1:])''' + line[i + 1:k + 1], line[k + 1:]) i += 1 @@ -860,8 +842,8 @@ def main(): latex.write('\n\n\\end{document}\n') -with open(os.path.dirname(os.path.realpath(__file__)) + - '/../data/functions') as functions: +with open(os.path.dirname(os.path.realpath(__file__)) + '/../data/functions') \ + as functions: FUNCTION_CONVERSIONS = list(arg_split(line.replace(' ', ''), ',') for line in functions.read().split('\n') if (line != '' and '#' not in line)) From 6a91131ed11454668651719adb38a4e347dd4687 Mon Sep 17 00:00:00 2001 From: Joon Bang Date: Thu, 11 Aug 2016 12:52:35 -0400 Subject: [PATCH 216/402] Add unit tests --- .travis.yml | 2 +- main_page/src/add_definition.py | 2 +- main_page/src/functions.py | 294 +++++++++++++++++ main_page/src/main.py | 310 +----------------- main_page/test/__init__.py | 0 .../test/add_symbols_data_tests/exp.1.txt | 26 ++ .../test/add_symbols_data_tests/res.1.txt | 39 +++ main_page/test/add_usage_tests/exp.1.txt | 30 ++ main_page/test/add_usage_tests/res.1.txt | 38 +++ main_page/test/macro_list_tests/exp.1.txt | 13 + main_page/test/macro_list_tests/res.1.txt | 18 + main_page/test/test_functions.py | 64 ++++ main_page/test/update_headers_tests/exp.1.txt | 28 ++ main_page/test/update_headers_tests/res.1.txt | 45 +++ 14 files changed, 606 insertions(+), 303 deletions(-) create mode 100644 main_page/src/functions.py create mode 100644 main_page/test/__init__.py create mode 100644 main_page/test/add_symbols_data_tests/exp.1.txt create mode 100644 main_page/test/add_symbols_data_tests/res.1.txt create mode 100644 main_page/test/add_usage_tests/exp.1.txt create mode 100644 main_page/test/add_usage_tests/res.1.txt create mode 100644 main_page/test/macro_list_tests/exp.1.txt create mode 100644 main_page/test/macro_list_tests/res.1.txt create mode 100644 main_page/test/test_functions.py create mode 100644 main_page/test/update_headers_tests/exp.1.txt create mode 100644 main_page/test/update_headers_tests/res.1.txt diff --git a/.travis.yml b/.travis.yml index dd5c68f..7b43807 100644 --- a/.travis.yml +++ b/.travis.yml @@ -2,6 +2,6 @@ language: python install: - pip install coveralls script: - nosetests --with-coverage tex2Wiki DLMF_preprocessing maple2latex + nosetests --with-coverage tex2Wiki DLMF_preprocessing maple2latex main_page after_success: coveralls diff --git a/main_page/src/add_definition.py b/main_page/src/add_definition.py index bc7bc77..2d0bb5e 100644 --- a/main_page/src/add_definition.py +++ b/main_page/src/add_definition.py @@ -12,7 +12,7 @@ def main(): "These are defined by \n" + \ "{\\displaystyle\n" + definition + "\n}\n
\n\ndrmf_eof\n" - with open("main_page.mmd", "a") as main_page: + with open("main_page/main_page.mmd", "a") as main_page: main_page.write("\n" + result) print "\n" + result diff --git a/main_page/src/functions.py b/main_page/src/functions.py new file mode 100644 index 0000000..67abba4 --- /dev/null +++ b/main_page/src/functions.py @@ -0,0 +1,294 @@ +__author__ = "Joon Bang" +__status__ = "Development" + +import csv + +GLOSSARY_LOCATION = "main_page/new.Glossary.csv" + + +def generate_html(tag_name, options, text, spacing=True): + result = "<" + tag_name + if options != "": + result += " " + options + result += ">\n" + text + "\n\n" + + if not spacing: + return result.replace("\n", " ")[:-1] + "\n" + + return result + + +def get_macro_name(macro): + macro_name = "" + for ch in macro: + if ch.isalpha() or ch == "\\": + macro_name += ch + elif ch in ["@", "{", "["]: + break + + return macro_name + + +def find_all(pattern, string): + """Yields all positions of where pattern is present in string.""" + + i = string.find(pattern) + while i != -1: + yield i + i = string.find(pattern, i + 1) + + +def remove_optional_params(string): + parse = True + text = "" + for ch in string: + if parse and ch == "[": + parse = False + elif parse: + text += ch + else: + if ch == "]": + parse = True + + return text + + +def find_all_positions(entry): + macro = entry[0] + category = entry[3].split("|", 1)[0] + keys = entry[3].split("|", 1)[1].split(":") + result = [] + + for key in keys: + if key in ["F", "FO", "O", "O1"]: + result.append(macro.split("@", 1)[0]) + if macro.count("@") > 0 and key == "O": + result.append(macro) + + if key == "FnO": # remove optional parameters + string = macro + if macro.count("@") > 0: + string = string[:string.find("@")] + result.append(remove_optional_params(string)) + + if key in ["P", "PO", "PnO"]: + result.append(macro.replace("@@", "@")) + + if key == "PnO": # remove optional parameters + result[-1] = remove_optional_params(result[-1]) + + if key in ["nP", "nPO", "PS", "O2"]: + result.append(macro) + + if key == "nPnO": # remove optional paramters + result.append(remove_optional_params(macro)) + + if "fo" in key: + result.append(macro.replace("@@@", "@" * int(key[2]))) + + return sorted(result, key=lambda x: x.count("@")), category + + +def macro_match(macro, entry): + """Determines whether the entry refers to the macro.""" + return macro in entry and (len(macro) == len(entry) or not entry[len(macro)].isalpha()) + + +def update_macro_list(text): + lines = text.split("\n") + definitions = list() + + for line in lines: + if "'''Definition:" in line: + macro = line.split("'''")[1][11:] + element = "* [[Definition:" + macro + "|" + macro + "]]" + definitions.append(element) + + def_contents = generate_html("div", "style=\"-moz-column-count:2; column-count:2;-webkit-column-count:2\"", + "\n".join(definitions)) + + start = text.find("== Definition Pages ==") + end = text.find("= Copyright =") + + return text[:start + 23] + "\n" + def_contents + "\n" + text[end:], definitions + + +def update_headers(text, definitions): + pages = text.split("drmf_eof")[:-1] + + macros = list() + formatted = list() + for definition in definitions: + macros.append("Definition:" + definition.split("|")[1][:-2]) + formatted.append("[[" + macros[-1].replace(" ", "_") + "|" + macros[-1] + "]]") + + output = "" + + i = 0 + for page in pages: + if "'''Definition:" not in page: + output += page + "drmf_eof\n" + continue + + # remove drmf_bof and definition header to prevent redundancy + page = page[page.find(macros[i]) + len(macros[i]) + 3:] + + if "
" in page: + page = page[page.find(">>
\n
") + len(">>
\n
"):] + + if "
" in page: + page = page[:page.find("
")] + + nav_section = generate_html("div", "id=\"alignleft\"", "<< " + formatted[i - 1], spacing=False) + \ + generate_html("div", "id=\"aligncenter\"", "[[Main_Page|Main Page]]", spacing=False) + \ + generate_html("div", "id=\"alignright\"", formatted[(i + 1) % len(formatted)] + " >>", + spacing=False)[:-1] + + header = generate_html("div", "id=\"drmf_head\"", nav_section)[:-1] + footer = generate_html("div", "id=\"drmf_foot\"", nav_section)[:-1] + + output += "drmf_bof\n" + "'''" + macros[i] + "'''\n" + header + page[:-1] + footer + "\ndrmf_eof\n" + + i += 1 + + return output + + +def get_symbols(text, glossary): + symbols = set() + + for keyword in glossary: + for index in find_all(keyword, text): + # if the macro is present in the text + if index != -1: + index += len(keyword) # now index of next character + + if index >= len(text) or not text[index].isalpha(): + symbols.add(keyword) + + span_text = "" + for symbol in sorted(symbols, key=str.lower): + links = list() + for cell in glossary[symbol]: + if "http://" in cell or "https://" in cell: + links.append(cell) + + id_link = links[0] + links = ["[" + link + " " + link + "]" for link in links] + + meaning = list(glossary[symbol][1]) + + count = 0 + for i, ch in enumerate(meaning): + if ch == "$" and count % 2 == 0: + meaning[i] = "{\\displaystyle " + count += 1 + elif ch == "$": + meaning[i] = "}" + count += 1 + + appearance = glossary[symbol][4].strip("$") + + span_text += "[" + id_link + " {\\displaystyle " + appearance + \ + "}] : " + ''.join(meaning) + " : " + " ".join(links) + "
\n" + + return span_text[:-7] # slice off the extra br and endline + + +def add_symbols_data(data): + glossary = dict() + with open(GLOSSARY_LOCATION, "rb") as csv_file: + glossary_file = csv.reader(csv_file, delimiter=',', quotechar='\"') + for row in glossary_file: + glossary[get_macro_name(row[0])] = row + + pages = data[data.find("drmf_bof"):].split("drmf_eof")[:-1] + + print "Pages: " + str(len(pages)) + + result = "" + for page in pages: + # skip over non-definition pages + if "'''Definition:" not in page: + result += page + "drmf_eof\n" + continue + + page = page.replace("drmf_bof", "").strip("\n") + add_spacing = False + # remove data (to be regenerated later) + if "== Symbols List ==" in page: + to_write = page.split("== Symbols List ==")[0] + page = to_write + else: + add_spacing = True + if page.find("drmf_foot") == -1: + to_write = page + else: + to_write = page.split("
")[0].rstrip("\n") + page = to_write + + to_write = "drmf_bof\n" + to_write + if add_spacing: + to_write += "\n\n" + + to_write += "== Symbols List ==\n\n" + to_write += get_symbols(page, glossary) + to_write += "\n
\ndrmf_eof\n" + result += to_write + + return result + + +def add_usage(lines): + with open("main_page/categories.txt") as cats: + categories = cats.readlines() + + to_write = "" + i = 0 + + while True: + n = lines.find("'''Definition:", i) + if n == -1: + break + + macro_name = lines[n:].split("'''")[1][11:] + left = lines.find("'''\\", n) + right = lines.find("'''", left + len("'''\\")) + + if lines[left:right].find("{") != -1: + lines = lines[:left] + "'''\\" + macro_name + lines[right:] + + if lines.find("\nThis macro is in the category of", n) < lines.find("'''Definition:", n + len( + "'''Definition:") + 1) and lines.find("\nThis macro is in the category of", n) != -1: + p = lines.find("\nThis macro is in the category of", n) + else: + p = lines.find(".\n", n) + 1 + + to_write += lines[i:p].rstrip() + "\n\n" + + calls = [] + glossary = csv.reader(open(GLOSSARY_LOCATION, 'rb'), delimiter=',', quotechar='\"') + for entry in glossary: + if macro_match("\\" + macro_name, entry[0]): + macro_calls, category = find_all_positions(entry) + calls += macro_calls + + for line in categories: + key, meaning = line.split(" - ") + if key == category: + category_text = "This macro is in the category of " + meaning + break + else: + category_text = "This macro is in the category of integer valued functions.\n" + + text = category_text + "\nIn math mode, this macro can be called in the following way" + "s" * ( + len(calls) > 1) + ":\n\n" + + for call in calls: + text += ":'''" + call + "'''" + " produces {\\displaystyle " + call + "}
\n" + + # Now add the multiple ways \macroname{n}@... produces \macroname{n}@... + to_write += text + "\n" + i = lines.find("These are defined by", p) + + return to_write + lines[i:] diff --git a/main_page/src/main.py b/main_page/src/main.py index 940e063..096b4db 100644 --- a/main_page/src/main.py +++ b/main_page/src/main.py @@ -2,307 +2,15 @@ __status__ = "Development" __credits__ = ["Joon Bang", "Azeem Mohammed"] -import csv import time - -GLOSSARY_LOCATION = "new.Glossary.csv" - - -def generate_html(tag_name, options, text, spacing=True): - result = "<" + tag_name - if options != "": - result += " " + options - result += ">\n" + text + "\n\n" - - if not spacing: - return result.replace("\n", " ")[:-1] + "\n" - - return result - - -def get_macro_name(macro): - macro_name = "" - for ch in macro: - if ch.isalpha() or ch == "\\": - macro_name += ch - elif ch in ["@", "{", "["]: - break - - return macro_name - - -def find_all(pattern, string): - """Yields all positions of where pattern is present in string.""" - - i = string.find(pattern) - while i != -1: - yield i - i = string.find(pattern, i + 1) - - -def remove_optional_params(string): - parse = True - text = "" - for ch in string: - if parse and ch == "[": - parse = False - elif parse: - text += ch - else: - if ch == "]": - parse = True - - return text - - -def find_all_positions(entry): - macro = entry[0] - category = entry[3].split("|", 1)[0] - keys = entry[3].split("|", 1)[1].split(":") - result = [] - - for key in keys: - if key in ["F", "FO", "O", "O1"]: - result.append(macro.split("@", 1)[0]) - if macro.count("@") > 0 and key == "O": - result.append(macro) - - if key == "FnO": # remove optional parameters - string = macro - if macro.count("@") > 0: - string = string[:string.find("@")] - result.append(remove_optional_params(string)) - - if key in ["P", "PO", "PnO"]: - result.append(macro.replace("@@", "@")) - - if key == "PnO": # remove optional parameters - result[-1] = remove_optional_params(result[-1]) - - if key in ["nP", "nPO", "PS", "O2"]: - result.append(macro) - - if key == "nPnO": # remove optional paramters - result.append(remove_optional_params(macro)) - - if "fo" in key: - result.append(macro.replace("@@@", "@" * int(key[2]))) - - return sorted(result, key=lambda x: x.count("@")), category - - -def macro_match(macro, entry): - """Determines whether the entry refers to the macro.""" - return macro in entry and (len(macro) == len(entry) or not entry[len(macro)].isalpha()) - - -def update_macro_list(text): - lines = text.split("\n") - definitions = list() - - for line in lines: - if "'''Definition:" in line: - macro = line.split("'''")[1][11:] - element = "* [[Definition:" + macro + "|" + macro + "]]" - definitions.append(element) - - def_contents = generate_html("div", "style=\"-moz-column-count:2; column-count:2;-webkit-column-count:2\"", - "\n".join(definitions)) - - start = text.find("== Definition Pages ==") - end = text.find("= Copyright =") - - return text[:start + 23] + "\n" + def_contents + "\n" + text[end:], definitions - - -def update_headers(text, definitions): - pages = text.split("drmf_eof")[:-1] - - macros = list() - formatted = list() - for definition in definitions: - macros.append("Definition:" + definition.split("|")[1][:-2]) - formatted.append("[[" + macros[-1].replace(" ", "_") + "|" + macros[-1] + "]]") - - output = "" - - i = 0 - for page in pages: - if "'''Definition:" not in page: - output += page + "drmf_eof\n" - continue - - # remove drmf_bof and definition header to prevent redundancy - page = page[page.find(macros[i]) + len(macros[i]) + 3:] - - if "
" in page: - page = page[page.find(">>
\n
") + len(">>
\n
"):] - - if "
" in page: - page = page[:page.find("
")] - - nav_section = generate_html("div", "id=\"alignleft\"", "<< " + formatted[i - 1], spacing=False) + \ - generate_html("div", "id=\"aligncenter\"", "[[Main_Page|Main Page]]", spacing=False) + \ - generate_html("div", "id=\"alignright\"", formatted[(i + 1) % len(formatted)] + " >>", - spacing=False)[:-1] - - header = generate_html("div", "id=\"drmf_head\"", nav_section)[:-1] - footer = generate_html("div", "id=\"drmf_foot\"", nav_section)[:-1] - - output += "drmf_bof\n" + "'''" + macros[i] + "'''\n" + header + page[:-1] + footer + "\ndrmf_eof\n" - - i += 1 - - return output - - -def get_symbols(text, glossary): - symbols = set() - - for keyword in glossary: - for index in find_all(keyword, text): - # if the macro is present in the text - if index != -1: - index += len(keyword) # now index of next character - - if index >= len(text) or not text[index].isalpha(): - symbols.add(keyword) - - span_text = "" - for symbol in sorted(symbols, key=str.lower): - links = list() - for cell in glossary[symbol]: - if "http://" in cell or "https://" in cell: - links.append(cell) - - id_link = links[0] - links = ["[" + link + " " + link + "]" for link in links] - - meaning = list(glossary[symbol][1]) - - count = 0 - for i, ch in enumerate(meaning): - if ch == "$" and count % 2 == 0: - meaning[i] = "{\\displaystyle " - count += 1 - elif ch == "$": - meaning[i] = "}" - count += 1 - - appearance = glossary[symbol][4].strip("$") - - span_text += "[" + id_link + " {\\displaystyle " + appearance + \ - "}] : " + ''.join(meaning) + " : " + " ".join(links) + "
\n" - - return span_text[:-7] # slice off the extra br and endline - - -def add_symbols_data(data): - glossary = dict() - with open(GLOSSARY_LOCATION, "rb") as csv_file: - glossary_file = csv.reader(csv_file, delimiter=',', quotechar='\"') - for row in glossary_file: - glossary[get_macro_name(row[0])] = row - - pages = data[data.find("drmf_bof"):].split("drmf_eof")[:-1] - - print "Pages: " + str(len(pages)) - - result = "" - for page in pages: - # skip over non-definition pages - if "'''Definition:" not in page: - to_write = page + "drmf_eof\n" - result += to_write - continue - - page = page.replace("drmf_bof", "").strip("\n") - add_spacing = False - sflag2 = False - # remove data (to be regenerated later) - if "== Symbols List ==" in page: - to_write = page.split("== Symbols List ==")[0] - page = to_write - else: - add_spacing = True - if page.find("drmf_foot") == -1: - to_write = page - sflag2 = True - else: - to_write = page.split("
")[0] - page = to_write - - to_write = "drmf_bof\n" + to_write - if add_spacing: - to_write += "\n\n" - - to_write += "== Symbols List ==\n\n" - to_write += get_symbols(page, glossary) - to_write += "\n
\ndrmf_eof\n\n" - if sflag2: - to_write += "\n" - result += to_write - - return result - - -def add_usage(lines): - with open("categories.txt") as cats: - categories = cats.readlines() - - to_write = "" - i = 0 - - while True: - n = lines.find("'''Definition:", i) - if n == -1: - break - - macro_name = lines[n:].split("'''")[1][11:] - left = lines.find("'''\\", n) - right = lines.find("'''", left + len("'''\\")) - - if lines[left:right].find("{") != -1: - lines = lines[:left] + "'''\\" + macro_name + lines[right:] - - if lines.find("\nThis macro is in the category of", n) < lines.find("'''Definition:", n + len( - "'''Definition:") + 1) and lines.find("\nThis macro is in the category of", n) != -1: - p = lines.find("\nThis macro is in the category of", n) - else: - p = lines.find(".\n", n) + 1 - to_write += lines[i:p].rstrip() + "\n\n" - - calls = [] - glossary = csv.reader(open(GLOSSARY_LOCATION, 'rb'), delimiter=',', quotechar='\"') - for entry in glossary: - if macro_match("\\" + macro_name, entry[0]): - macro_calls, category = find_all_positions(entry) - calls += macro_calls - - for line in categories: - key, meaning = line.split(" - ") - if key == category: - category_text = "This macro is in the category of " + meaning - break - - text = category_text + "\nIn math mode, this macro can be called in the following way" + "s" * ( - len(calls) > 1) + ":\n\n" - - for call in calls: - text += ":'''" + call + "'''" + " produces {\\displaystyle " + call + "}
\n" - - # Now add the multiple ways \macroname{n}@... produces \macroname{n}@... - to_write += text + "\n" - i = lines.find("These are defined by", p) - - return to_write + lines[i:] +import functions def create_backup(): - with open("main_page.mmd") as current: + with open("main_page/main_page.mmd") as current: local_time = time.asctime(time.localtime(time.time())).split() local_time = "_" + '_'.join(local_time[1:3] + local_time[3:][::-1]) - with open("backups/main_page" + local_time + ".mmd.bak", "w") as backup: + with open("main_page/backups/main_page" + local_time + ".mmd.bak", "w") as backup: backup.write(current.read()) @@ -311,18 +19,18 @@ def main(): return # read contents - with open("main_page.mmd") as main_page: + with open("main_page/main_page.mmd") as main_page: text = main_page.read() - text, definitions = update_macro_list(text) - text = add_symbols_data(text) - text = update_headers(text, definitions) - text = add_usage(text) + text, definitions = functions.update_macro_list(text) + text = functions.add_symbols_data(text) + text = functions.update_headers(text, definitions) + text = functions.add_usage(text) # only create backup if program did not crash create_backup() - with open("main_page.mmd", "w") as main_page: + with open("main_page/main_page.mmd", "w") as main_page: main_page.write(text) diff --git a/main_page/test/__init__.py b/main_page/test/__init__.py new file mode 100644 index 0000000..e69de29 diff --git a/main_page/test/add_symbols_data_tests/exp.1.txt b/main_page/test/add_symbols_data_tests/exp.1.txt new file mode 100644 index 0000000..7e5ab7c --- /dev/null +++ b/main_page/test/add_symbols_data_tests/exp.1.txt @@ -0,0 +1,26 @@ +drmf_bof +'''Main Page''' +drmf_eof +drmf_bof +'''Definition:Definition1''' + +\AlSalamCarlitzII{a}{n}@{x}{q}:=(-a)^nq^{-\binom{n}{2}}\,\qHyperrphis{2}{0}@@{q^{-n},x}{-}{q}{\frac{q^n}{a}} + +== Symbols List == + +drmf_eof +drmf_bof +'''Definition:Definition2''' + +\RiemannSiegelTheta@{t}:=\arg\left[\EulerGamma@{\frac{1}{4}+\frac{1}{2}it}\right]-\frac{1}{2}t\ln\pi + +
+
+ +drmf_eof +drmf_bof +'''Definition:Definition3''' + +\RamanujanTauTheta@{z}:=-\log\left(2\pi\right)z-\frac{i}{2}\left(\log\EulerGamma@{6+iz}-\log{\EulerGamma@{6-iz}}\right) + +drmf_eof diff --git a/main_page/test/add_symbols_data_tests/res.1.txt b/main_page/test/add_symbols_data_tests/res.1.txt new file mode 100644 index 0000000..11bfbae --- /dev/null +++ b/main_page/test/add_symbols_data_tests/res.1.txt @@ -0,0 +1,39 @@ +drmf_bof +'''Main Page''' +drmf_eof +drmf_bof +'''Definition:Definition1''' + +\AlSalamCarlitzII{a}{n}@{x}{q}:=(-a)^nq^{-\binom{n}{2}}\,\qHyperrphis{2}{0}@@{q^{-n},x}{-}{q}{\frac{q^n}{a}} + +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzII {\displaystyle V^{(n)}_{\alpha}}] : Al-Salam-Carlitz II polynomial : [http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzII http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzII]
+[http://dlmf.nist.gov/1.2#E1 {\displaystyle \binom{n}{k}}] : binomial coefficient : [http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/1.2#E1] [http://dlmf.nist.gov/26.3#SS1.p1 http://dlmf.nist.gov/26.3#SS1.p1]
+[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1] +
+drmf_eof +drmf_bof +'''Definition:Definition2''' + +\RiemannSiegelTheta@{t}:=\arg\left[\EulerGamma@{\frac{1}{4}+\frac{1}{2}it}\right]-\frac{1}{2}t\ln\pi + +== Symbols List == + +[http://dlmf.nist.gov/5.2#E1 {\displaystyle \Gamma}] : Euler's gamma function : [http://dlmf.nist.gov/5.2#E1 http://dlmf.nist.gov/5.2#E1]
+[http://dlmf.nist.gov/4.2#E2 {\displaystyle \mathrm{ln}}] : principal branch of logarithm function : [http://dlmf.nist.gov/4.2#E2 http://dlmf.nist.gov/4.2#E2]
+[http://drmf.wmflabs.org/wiki/Definition:RiemannSiegelTheta {\displaystyle \vartheta{t}}] : Riemann-Siegel theta function : [http://drmf.wmflabs.org/wiki/Definition:RiemannSiegelTheta http://drmf.wmflabs.org/wiki/Definition:RiemannSiegelTheta] +
+drmf_eof +drmf_bof +'''Definition:Definition3''' + +\RamanujanTauTheta@{z}:=-\log\left(2\pi\right)z-\frac{i}{2}\left(\log\EulerGamma@{6+iz}-\log{\EulerGamma@{6-iz}}\right) + +== Symbols List == + +[http://dlmf.nist.gov/5.2#E1 {\displaystyle \Gamma}] : Euler's gamma function : [http://dlmf.nist.gov/5.2#E1 http://dlmf.nist.gov/5.2#E1]
+[http://dlmf.nist.gov/4.2#E2 {\displaystyle \mathrm{log}}] : principle branch of natural logarithm : [http://dlmf.nist.gov/4.2#E2 http://dlmf.nist.gov/4.2#E2]
+[http://drmf.wmflabs.org/wiki/Definition:RamanujanTauTheta {\displaystyle \tau\theta{z}}] : Ramanujan tau theta function : [http://drmf.wmflabs.org/wiki/Definition:RamanujanTauTheta http://drmf.wmflabs.org/wiki/Definition:RamanujanTauTheta] +
+drmf_eof diff --git a/main_page/test/add_usage_tests/exp.1.txt b/main_page/test/add_usage_tests/exp.1.txt new file mode 100644 index 0000000..0b3d64b --- /dev/null +++ b/main_page/test/add_usage_tests/exp.1.txt @@ -0,0 +1,30 @@ +drmf_bof +'''Main Page''' +drmf_eof +drmf_bof +'''Definition:normctsdualqHahnptilde''' +The LaTeX DLMF and DRMF macro '''\normctsdualqHahnptilde''' represents the normctsdualqHahnptilde. + +These are defined by +{\displaystyle +{\tilde p}_n(x):=\normctsdualqHahnptilde{n}@{x}{a}{b}{c}{q}=\frac{a^n\ctsdualqHahn{n}@{x}{a}{b}{c}{q}}{\qPochhammer{ab,ac}{q}{n}} +} +
+drmf_eof +drmf_bof +'''Definition:GaussDistQ''' +The LaTeX DLMF and DRMF macro '''\GaussDistQ''' represents the GaussDistQ. + +This macro is in the category of integer valued functions. + +In math mode, this macro can be called in the following ways: + +:'''\GaussDistQ''' produces {\displaystyle \GaussDistQ}
+:'''\GaussDistQ@{x}''' produces {\displaystyle \GaussDistQ@{x}}
+ +These are defined by +{\displaystyle +\GaussDistQ@{x}:=\frac{1}{2}\erfc@{\frac{x}{\sqrt{2}}} +} +
+drmf_eof diff --git a/main_page/test/add_usage_tests/res.1.txt b/main_page/test/add_usage_tests/res.1.txt new file mode 100644 index 0000000..6b5407e --- /dev/null +++ b/main_page/test/add_usage_tests/res.1.txt @@ -0,0 +1,38 @@ +drmf_bof +'''Main Page''' +drmf_eof +drmf_bof +'''Definition:normctsdualqHahnptilde''' +The LaTeX DLMF and DRMF macro '''\normctsdualqHahnptilde''' represents the normctsdualqHahnptilde. + +This macro is in the category of polynomials. + +In math mode, this macro can be called in the following ways: + +:'''\normctsdualqHahnptilde{n}''' produces {\displaystyle \normctsdualqHahnptilde{n}}
+:'''\normctsdualqHahnptilde{n}@{x}{a}{b}{c}{q}''' produces {\displaystyle \normctsdualqHahnptilde{n}@{x}{a}{b}{c}{q}}
+:'''\normctsdualqHahnptilde{n}@@{x}{a}{b}{c}{q}''' produces {\displaystyle \normctsdualqHahnptilde{n}@@{x}{a}{b}{c}{q}}
+ +These are defined by +{\displaystyle +{\tilde p}_n(x):=\normctsdualqHahnptilde{n}@{x}{a}{b}{c}{q}=\frac{a^n\ctsdualqHahn{n}@{x}{a}{b}{c}{q}}{\qPochhammer{ab,ac}{q}{n}} +} +
+drmf_eof +drmf_bof +'''Definition:GaussDistQ''' +The LaTeX DLMF and DRMF macro '''\GaussDistQ''' represents the GaussDistQ. + +This macro is in the category of integer valued functions. + +In math mode, this macro can be called in the following ways: + +:'''\GaussDistQ''' produces {\displaystyle \GaussDistQ}
+:'''\GaussDistQ@{x}''' produces {\displaystyle \GaussDistQ@{x}}
+ +These are defined by +{\displaystyle +\GaussDistQ@{x}:=\frac{1}{2}\erfc@{\frac{x}{\sqrt{2}}} +} +
+drmf_eof diff --git a/main_page/test/macro_list_tests/exp.1.txt b/main_page/test/macro_list_tests/exp.1.txt new file mode 100644 index 0000000..89f50fe --- /dev/null +++ b/main_page/test/macro_list_tests/exp.1.txt @@ -0,0 +1,13 @@ +drmf_bof + +== Definition Pages == + += Copyright = + +drmf_eof +drmf_bof +'''Definition:Definition1''' +drmf_eof +drmf_bof +'''Definition:Definition2''' +drmf_eof diff --git a/main_page/test/macro_list_tests/res.1.txt b/main_page/test/macro_list_tests/res.1.txt new file mode 100644 index 0000000..bfd0dbc --- /dev/null +++ b/main_page/test/macro_list_tests/res.1.txt @@ -0,0 +1,18 @@ +drmf_bof + +== Definition Pages == + +
+* [[Definition:Definition1|Definition1]] +* [[Definition:Definition2|Definition2]] +
+ += Copyright = + +drmf_eof +drmf_bof +'''Definition:Definition1''' +drmf_eof +drmf_bof +'''Definition:Definition2''' +drmf_eof diff --git a/main_page/test/test_functions.py b/main_page/test/test_functions.py new file mode 100644 index 0000000..f2a4d1c --- /dev/null +++ b/main_page/test/test_functions.py @@ -0,0 +1,64 @@ +#!/usr/bin/env python + +__author__ = "Joon Bang" +__status__ = "Development" + +from unittest import TestCase +import src.functions as m + +update_macro_list_test_cases = [ + { + "exp": open("main_page/test/macro_list_tests/exp.1.txt").read(), + "res": ( + open("main_page/test/macro_list_tests/res.1.txt").read(), + ["* [[Definition:Definition1|Definition1]]", "* [[Definition:Definition2|Definition2]]"] + ) + } +] + +update_headers_test_cases = [ + { + "exp": open("main_page/test/update_headers_tests/exp.1.txt").read(), + "def": ["* [[Definition:Definition1|Definition1]]", "* [[Definition:Definition2|Definition2]]", + "* [[Definition:Definition3|Definition3]]"], + "res": open("main_page/test/update_headers_tests/res.1.txt").read() + } +] + +add_symbols_data_test_cases = [ + { + "exp": open("main_page/test/add_symbols_data_tests/exp.1.txt").read(), + "res": open("main_page/test/add_symbols_data_tests/res.1.txt").read() + } +] + +add_usage_test_cases = [ + { + "exp": open("main_page/test/add_usage_tests/exp.1.txt").read(), + "res": open("main_page/test/add_usage_tests/res.1.txt").read() + } +] + + +class TestUpdateMacroList(TestCase): + def test_update_macro_list(self): + for case in update_macro_list_test_cases: + self.assertEqual(m.update_macro_list(case["exp"]), case["res"]) + + +class TestUpdateHeaders(TestCase): + def test_update_headers(self): + for case in update_headers_test_cases: + self.assertEqual(m.update_headers(case["exp"], case["def"]), case["res"]) + + +class TestAddSymbolsData(TestCase): + def test_add_symbols_data(self): + for case in add_symbols_data_test_cases: + self.assertEqual(m.add_symbols_data(case["exp"]), case["res"]) + + +class TestAddUsage(TestCase): + def test_add_usage(self): + for case in add_usage_test_cases: + self.assertEqual(m.add_usage(case["exp"]), case["res"]) diff --git a/main_page/test/update_headers_tests/exp.1.txt b/main_page/test/update_headers_tests/exp.1.txt new file mode 100644 index 0000000..09cfcb8 --- /dev/null +++ b/main_page/test/update_headers_tests/exp.1.txt @@ -0,0 +1,28 @@ +drmf_bof +'''Main Page''' +drmf_eof +drmf_bof +'''Definition:Definition1''' +
+
>>
+
+Definition1 Text. +
+
+
+drmf_eof +drmf_bof +'''Definition:Definition2''' +
+
>>
+
+Definition2 Text. +
+
+
+drmf_eof +drmf_bof +'''Definition:Definition3''' +Definition3 Text. +
+drmf_eof diff --git a/main_page/test/update_headers_tests/res.1.txt b/main_page/test/update_headers_tests/res.1.txt new file mode 100644 index 0000000..ea47d21 --- /dev/null +++ b/main_page/test/update_headers_tests/res.1.txt @@ -0,0 +1,45 @@ +drmf_bof +'''Main Page''' +drmf_eof +drmf_bof +'''Definition:Definition1''' +
+
<< [[Definition:Definition3|Definition:Definition3]]
+
[[Main_Page|Main Page]]
+
[[Definition:Definition2|Definition:Definition2]] >>
+
+Definition1 Text. +
+
<< [[Definition:Definition3|Definition:Definition3]]
+
[[Main_Page|Main Page]]
+
[[Definition:Definition2|Definition:Definition2]] >>
+
+drmf_eof +drmf_bof +'''Definition:Definition2''' +
+
<< [[Definition:Definition1|Definition:Definition1]]
+
[[Main_Page|Main Page]]
+
[[Definition:Definition3|Definition:Definition3]] >>
+
+Definition2 Text. +
+
<< [[Definition:Definition1|Definition:Definition1]]
+
[[Main_Page|Main Page]]
+
[[Definition:Definition3|Definition:Definition3]] >>
+
+drmf_eof +drmf_bof +'''Definition:Definition3''' +
+
<< [[Definition:Definition2|Definition:Definition2]]
+
[[Main_Page|Main Page]]
+
[[Definition:Definition1|Definition:Definition1]] >>
+
+Definition3 Text. +
+
<< [[Definition:Definition2|Definition:Definition2]]
+
[[Main_Page|Main Page]]
+
[[Definition:Definition1|Definition:Definition1]] >>
+
+drmf_eof From 272ed9aefcdfb533c8f16f3d56a30e2ea94048ed Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Thu, 11 Aug 2016 14:15:28 -0400 Subject: [PATCH 217/402] Update .gitignore, add tests for carat function --- eCF/.gitignore | 3 +++ eCF/src/MathematicaToLaTeX.py | 2 +- eCF/test/test_carat.py | 24 ++++++++++++++++++++++++ 3 files changed, 28 insertions(+), 1 deletion(-) diff --git a/eCF/.gitignore b/eCF/.gitignore index a695525..9eb163b 100644 --- a/eCF/.gitignore +++ b/eCF/.gitignore @@ -4,6 +4,9 @@ src/test.txt data/Identities.* data/newIdentities.* data/IdentitiesTest.m +src/Glossary.csv +src/new.Glossary.csv +src/new.Glossary.updated.csv IdentitiesTest.txt README_Glossary x.log diff --git a/eCF/src/MathematicaToLaTeX.py b/eCF/src/MathematicaToLaTeX.py index 286092a..5e3ac96 100644 --- a/eCF/src/MathematicaToLaTeX.py +++ b/eCF/src/MathematicaToLaTeX.py @@ -193,7 +193,7 @@ def carat(line): """ l = list('([{') r = list(')]}') - sign = list('*/+-=,') + sign = list('*/+-=, ') i = 0 while i != len(line): diff --git a/eCF/test/test_carat.py b/eCF/test/test_carat.py index 1cdc8d8..6edcfe9 100644 --- a/eCF/test/test_carat.py +++ b/eCF/test/test_carat.py @@ -5,3 +5,27 @@ from unittest import TestCase from MathematicaToLaTeX import carat + +class TestCarat(TestCase): + + def test_seperate(self): + self.assertEqual(carat('a^b'), 'a^{b}') + self.assertEqual(carat('a^b*c'), 'a^{b}*c') + self.assertEqual(carat('a^b/c'), 'a^{b}/c') + self.assertEqual(carat('a^b+c'), 'a^{b}+c') + self.assertEqual(carat('a^b-c'), 'a^{b}-c') + self.assertEqual(carat('a^b=c'), 'a^{b}=c') + self.assertEqual(carat('a^b,c'), 'a^{b},c') + self.assertEqual(carat('a^b c'), 'a^{b} c') + + def test_parentheses(self): + self.assertEqual(carat('a^(b+c)'), 'a^{b+c}') + self.assertEqual(carat('a^(b+c)*d'), 'a^{(b+c)}*d') + self.assertEqual(carat('a^(b+c)/d'), 'a^{(b+c)}/d') + self.assertEqual(carat('a^(b+c)+d'), 'a^{(b+c)}+d') + self.assertEqual(carat('a^(b+c)-d'), 'a^{(b+c)}-d') + self.assertEqual(carat('a^(b+c)=d'), 'a^{(b+c)}=d') + self.assertEqual(carat('a^(b+c),d'), 'a^{(b+c)},d') + + def test_none(self): + self.assertEqual(carat('nocarat'), 'nocarat') From d5ab83dd4eb62a56c3e562d3596ebb24db40feeb Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Thu, 11 Aug 2016 14:17:35 -0400 Subject: [PATCH 218/402] Remove pythonTest.py --- eCF/.gitignore | 1 + 1 file changed, 1 insertion(+) diff --git a/eCF/.gitignore b/eCF/.gitignore index 9eb163b..7f9c223 100644 --- a/eCF/.gitignore +++ b/eCF/.gitignore @@ -7,6 +7,7 @@ data/IdentitiesTest.m src/Glossary.csv src/new.Glossary.csv src/new.Glossary.updated.csv +src/pythonTest.py IdentitiesTest.txt README_Glossary x.log From 923ace05efd75c1b0e184a4eda560833f104d3cf Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Thu, 11 Aug 2016 14:20:27 -0400 Subject: [PATCH 219/402] Remove pythonTest.py --- eCF/src/pythonTest.py | 0 1 file changed, 0 insertions(+), 0 deletions(-) delete mode 100644 eCF/src/pythonTest.py diff --git a/eCF/src/pythonTest.py b/eCF/src/pythonTest.py deleted file mode 100644 index e69de29..0000000 From e6579815a9753d184b90ae102a494ca8c86d0d2b Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Thu, 11 Aug 2016 16:14:24 -0400 Subject: [PATCH 220/402] Fix error with constraint function and change ContinuedFractionK to appear with parentheses around the fraction --- eCF/src/MathematicaToLaTeX.py | 13 +++++++++---- eCF/test/test_constraint.py | 9 +++++++++ 2 files changed, 18 insertions(+), 4 deletions(-) diff --git a/eCF/src/MathematicaToLaTeX.py b/eCF/src/MathematicaToLaTeX.py index 5e3ac96..c58a88e 100644 --- a/eCF/src/MathematicaToLaTeX.py +++ b/eCF/src/MathematicaToLaTeX.py @@ -227,6 +227,11 @@ def beta(line): """ Converts Mathematica's 'Beta' function to the equivalent LaTeX macro, taking into account the variations for the different number of arguments. + + :type line: str + :param line: line to be converted + :rtype: str + :returns: converted line """ for _ in range(line.count('Beta')): try: @@ -271,12 +276,12 @@ def cfk(line): moreargs = arg_split(args[-1][1:-1], ',') if len(args) == 3: line = (line[:pos[0]] + - '\\CFK{{{0}}}{{{1}}}{{{2}}}@{{{3}}}{{{4}}}' + '\\CFK{{{0}}}{{{1}}}{{{2}}}@@{{{3}}}{{{4}}}' .format(moreargs[0], moreargs[1], moreargs[2], args[0], args[1]) + line[pos[1]:]) else: line = (line[:pos[0]] + - '\\CFK{{{0}}}{{{1}}}{{{2}}}@{{1}}{{{3}}}' + '\\CFK{{{0}}}{{{1}}}{{{2}}}@@{{1}}{{{3}}}' .format(moreargs[0], moreargs[1], moreargs[2], args[0]) + line[pos[1]:]) @@ -719,13 +724,13 @@ def replace_operators(line): line = parts[0] line = line.replace('(', '\\left( ') line = line.replace(')', ' \\right)') - line = line.replace('&', ' \\land ') + #line = line.replace('&', ' \\land ') line = line.replace(' ', ' ') line += parts[1] else: line = line.replace('(', '\\left( ') line = line.replace(')', ' \\right)') - line = line.replace('&', ' \\land ') + #line = line.replace('&', ' \\land ') line = line.replace(' ', ' ') line = line.replace('"a"', 'a') diff --git a/eCF/test/test_constraint.py b/eCF/test/test_constraint.py index 34330f8..f22fd6a 100644 --- a/eCF/test/test_constraint.py +++ b/eCF/test/test_constraint.py @@ -5,3 +5,12 @@ from unittest import TestCase from MathematicaToLaTeX import constraint +class TestConstraint(TestCase): + +\ + def test_single(self): + + def test_multiple(self): + + def test_none(self): + self.assertEqual(constraint('noconstraint'), 'noconstraint') From 55c2c323d629ccb1631f42fb306e862a35404f72 Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Thu, 11 Aug 2016 16:15:35 -0400 Subject: [PATCH 221/402] Remove two unused lines in replace_operators function --- eCF/src/MathematicaToLaTeX.py | 2 -- 1 file changed, 2 deletions(-) diff --git a/eCF/src/MathematicaToLaTeX.py b/eCF/src/MathematicaToLaTeX.py index c58a88e..4bc54e9 100644 --- a/eCF/src/MathematicaToLaTeX.py +++ b/eCF/src/MathematicaToLaTeX.py @@ -724,13 +724,11 @@ def replace_operators(line): line = parts[0] line = line.replace('(', '\\left( ') line = line.replace(')', ' \\right)') - #line = line.replace('&', ' \\land ') line = line.replace(' ', ' ') line += parts[1] else: line = line.replace('(', '\\left( ') line = line.replace(')', ' \\right)') - #line = line.replace('&', ' \\land ') line = line.replace(' ', ' ') line = line.replace('"a"', 'a') From 9a81f55a87b99df4729604485c54262c5aaf0a71 Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Thu, 11 Aug 2016 16:42:55 -0400 Subject: [PATCH 222/402] Add some tests for constraint function --- eCF/src/MathematicaToLaTeX.py | 2 +- eCF/test/test_constraint.py | 12 ++++++++++-- eCF/test/test_replace_operators.py | 2 -- .../test_single_macro_conversion_functions.py | 18 +++++++++--------- 4 files changed, 20 insertions(+), 14 deletions(-) diff --git a/eCF/src/MathematicaToLaTeX.py b/eCF/src/MathematicaToLaTeX.py index 4bc54e9..7d753a4 100644 --- a/eCF/src/MathematicaToLaTeX.py +++ b/eCF/src/MathematicaToLaTeX.py @@ -762,7 +762,7 @@ def main(): Opens Mathematica file with identities and puts converted lines into newIdentities.tex. """ - test = False + test = True with open(os.path.dirname(os.path.realpath(__file__)) + '/../data/newIdentities.tex', 'w') as latex: diff --git a/eCF/test/test_constraint.py b/eCF/test/test_constraint.py index f22fd6a..8b433c4 100644 --- a/eCF/test/test_constraint.py +++ b/eCF/test/test_constraint.py @@ -5,12 +5,20 @@ from unittest import TestCase from MathematicaToLaTeX import constraint + class TestConstraint(TestCase): -\ def test_single(self): + self.assertEqual(constraint('text,Element[z,Complexes]'), 'text\n% \constraint{$z \in \Complex$}') + self.assertEqual(constraint('text,Element[z,Wholes]'), 'text\n% \constraint{$z \in \Whole$}') + self.assertEqual(constraint('text,Element[z,Naturals]'), 'text\n% \constraint{$z \in \NatNumber$}') + self.assertEqual(constraint('text,Element[z,Integers]'), 'text\n% \constraint{$z \in \Integer$}') + self.assertEqual(constraint('text,Element[z,Irrationals]'), 'text\n% \constraint{$z \in \Irrational$}') + self.assertEqual(constraint('text,Element[z,Reals]'), 'text\n% \constraint{$z \in \Real$}') + self.assertEqual(constraint('text,Element[z,Rational]'), 'text\n% \constraint{$z \in \Rational$}') + self.assertEqual(constraint('text,Element[z,Primes]'), 'text\n% \constraint{$z \in \Prime$}') - def test_multiple(self): + # def test_multiple(self): def test_none(self): self.assertEqual(constraint('noconstraint'), 'noconstraint') diff --git a/eCF/test/test_replace_operators.py b/eCF/test/test_replace_operators.py index ee87043..d8eaf3e 100644 --- a/eCF/test/test_replace_operators.py +++ b/eCF/test/test_replace_operators.py @@ -25,7 +25,6 @@ def test_without_percent(self): self.assertEqual(replace_operators(','), ', ') self.assertEqual(replace_operators('('), '\\left( ') self.assertEqual(replace_operators(')'), ' \\right)') - self.assertEqual(replace_operators('&'), ' \\land ') self.assertEqual(replace_operators(' '), ' ') self.assertEqual(replace_operators('"a"'), 'a') @@ -38,7 +37,6 @@ def test_constants(self): def test_with_percent(self): self.assertEqual(replace_operators('(%('), '\\left( %(') self.assertEqual(replace_operators(')%)'), ' \\right)%)') - self.assertEqual(replace_operators('&%&'), ' \\land %&') self.assertEqual(replace_operators(' % '), ' % ') def test_none(self): diff --git a/eCF/test/test_single_macro_conversion_functions.py b/eCF/test/test_single_macro_conversion_functions.py index ace02b8..4ea7693 100644 --- a/eCF/test/test_single_macro_conversion_functions.py +++ b/eCF/test/test_single_macro_conversion_functions.py @@ -42,17 +42,17 @@ def test_none(self): class TestCFK(TestCase): def test_single(self): - self.assertEqual(cfk('ContinuedFractionK[f,g,{i,imin,imax}]'), '\\CFK{i}{imin}{imax}@{f}{g}') - self.assertEqual(cfk('--ContinuedFractionK[f,g,{i,imin,imax}]--'), '--\\CFK{i}{imin}{imax}@{f}{g}--') - self.assertEqual(cfk('ContinuedFractionK[g,{i,imin,imax}]'), '\\CFK{i}{imin}{imax}@{1}{g}') - self.assertEqual(cfk('--ContinuedFractionK[g,{i,imin,imax}]--'), '--\\CFK{i}{imin}{imax}@{1}{g}--') - self.assertEqual(cfk('ContinuedFractionK[f,g,{i,imin,imax}]ContinuedFractionK[g,{i,imin,imax}]'), '\\CFK{i}{imin}{imax}@{f}{g}\\CFK{i}{imin}{imax}@{1}{g}') - self.assertEqual(cfk('--ContinuedFractionK[f,g,{i,imin,imax}]--ContinuedFractionK[g,{i,imin,imax}]--'), '--\\CFK{i}{imin}{imax}@{f}{g}--\\CFK{i}{imin}{imax}@{1}{g}--') + self.assertEqual(cfk('ContinuedFractionK[f,g,{i,imin,imax}]'), '\\CFK{i}{imin}{imax}@@{f}{g}') + self.assertEqual(cfk('--ContinuedFractionK[f,g,{i,imin,imax}]--'), '--\\CFK{i}{imin}{imax}@@{f}{g}--') + self.assertEqual(cfk('ContinuedFractionK[g,{i,imin,imax}]'), '\\CFK{i}{imin}{imax}@@{1}{g}') + self.assertEqual(cfk('--ContinuedFractionK[g,{i,imin,imax}]--'), '--\\CFK{i}{imin}{imax}@@{1}{g}--') + self.assertEqual(cfk('ContinuedFractionK[f,g,{i,imin,imax}]ContinuedFractionK[g,{i,imin,imax}]'), '\\CFK{i}{imin}{imax}@@{f}{g}\\CFK{i}{imin}{imax}@@{1}{g}') + self.assertEqual(cfk('--ContinuedFractionK[f,g,{i,imin,imax}]--ContinuedFractionK[g,{i,imin,imax}]--'), '--\\CFK{i}{imin}{imax}@@{f}{g}--\\CFK{i}{imin}{imax}@@{1}{g}--') def test_nested(self): - self.assertEqual(cfk('ContinuedFractionK[ContinuedFractionK[f,g,{i,imin,imax}],ContinuedFractionK[f,g,{i,imin,imax}],{i,imin,imax}]'), '\\CFK{i}{imin}{imax}@{\\CFK{i}{imin}{imax}@{f}{g}}{\\CFK{i}{imin}{imax}@{f}{g}}') - self.assertEqual(cfk('--ContinuedFractionK[ContinuedFractionK[f,g,{i,imin,imax}],ContinuedFractionK[f,g,{i,imin,imax}],{i,imin,imax}]--'), '--\\CFK{i}{imin}{imax}@{\\CFK{i}{imin}{imax}@{f}{g}}{\\CFK{i}{imin}{imax}@{f}{g}}--') - self.assertEqual(cfk('ContinuedFractionK[ContinuedFractionK[g,{i,imin,imax}],{i,imin,imax}]'), '\\CFK{i}{imin}{imax}@{1}{\\CFK{i}{imin}{imax}@{1}{g}}') + self.assertEqual(cfk('ContinuedFractionK[ContinuedFractionK[f,g,{i,imin,imax}],ContinuedFractionK[f,g,{i,imin,imax}],{i,imin,imax}]'), '\\CFK{i}{imin}{imax}@@{\\CFK{i}{imin}{imax}@@{f}{g}}{\\CFK{i}{imin}{imax}@@{f}{g}}') + self.assertEqual(cfk('--ContinuedFractionK[ContinuedFractionK[f,g,{i,imin,imax}],ContinuedFractionK[f,g,{i,imin,imax}],{i,imin,imax}]--'), '--\\CFK{i}{imin}{imax}@@{\\CFK{i}{imin}{imax}@@{f}{g}}{\\CFK{i}{imin}{imax}@@{f}{g}}--') + self.assertEqual(cfk('ContinuedFractionK[ContinuedFractionK[g,{i,imin,imax}],{i,imin,imax}]'), '\\CFK{i}{imin}{imax}@@{1}{\\CFK{i}{imin}{imax}@@{1}{g}}') def test_none(self): self.assertEqual(cfk('none'), 'none') From f38bee3b7963b693d517f946aa59c206188e9aca Mon Sep 17 00:00:00 2001 From: Joon Bang Date: Fri, 12 Aug 2016 11:16:25 -0400 Subject: [PATCH 223/402] Add entry for Binet function --- main_page/main_page.mmd | 41 ++++++++++++++++++++++++++++++++++++++--- 1 file changed, 38 insertions(+), 3 deletions(-) diff --git a/main_page/main_page.mmd b/main_page/main_page.mmd index 0e87ea3..7cbe107 100644 --- a/main_page/main_page.mmd +++ b/main_page/main_page.mmd @@ -228,6 +228,7 @@ For more information see [http://link.springer.com/chapter/10.1007%2F978-3-319-0 * [[Definition:RamanujanTauTheta|RamanujanTauTheta]] * [[Definition:RiemannSiegelTheta|RiemannSiegelTheta]] * [[Definition:sinc|sinc]] +* [[Definition:Binet|Binet]]
= Copyright = @@ -285,7 +286,7 @@ drmf_eof drmf_bof '''Definition:AffqKrawtchouk'''
-
<< [[Definition:sinc|Definition:sinc]]
+
<< [[Definition:Binet|Definition:Binet]]
[[Main_Page|Main Page]]
[[Definition:AlSalamCarlitzI|Definition:AlSalamCarlitzI]] >>
@@ -312,7 +313,7 @@ These are defined by [http://drmf.wmflabs.org/wiki/Definition:AffqKrawtchouk {\displaystyle K^{\mathrm{Aff}}_{n}}] : affine {\displaystyle q}-Krawtchouk polynomial : [http://drmf.wmflabs.org/wiki/Definition:AffqKrawtchouk http://drmf.wmflabs.org/wiki/Definition:AffqKrawtchouk]
[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
-
<< [[Definition:sinc|Definition:sinc]]
+
<< [[Definition:Binet|Definition:Binet]]
[[Main_Page|Main Page]]
[[Definition:AlSalamCarlitzI|Definition:AlSalamCarlitzI]] >>
@@ -5154,7 +5155,7 @@ drmf_bof
<< [[Definition:RiemannSiegelTheta|Definition:RiemannSiegelTheta]]
[[Main_Page|Main Page]]
-
[[Definition:AffqKrawtchouk|Definition:AffqKrawtchouk]] >>
+
[[Definition:Binet|Definition:Binet]] >>
The LaTeX DLMF and DRMF macro '''\sinc''' represents the sinc. @@ -5188,6 +5189,40 @@ These are defined by
<< [[Definition:RiemannSiegelTheta|Definition:RiemannSiegelTheta]]
[[Main_Page|Main Page]]
+
[[Definition:Binet|Definition:Binet]] >>
+
+drmf_eof +drmf_bof +'''Definition:Binet''' +
+
<< [[Definition:sinc|Definition:sinc]]
+
[[Main_Page|Main Page]]
+
[[Definition:AffqKrawtchouk|Definition:AffqKrawtchouk]] >>
+
+The LaTeX DLMF and DRMF macro '''\Binet''' represents the Binet. + +This macro is in the category of real or complex valued functions. + +In math mode, this macro can be called in the following ways: + +:'''\Binet''' produces {\displaystyle \Binet}
+:'''\Binet@{z}''' produces {\displaystyle \Binet@{z}}
+ +These are defined by +{\displaystyle +\Binet@{z}:=\ln@{\EulerGamma@{z}}-\left(z-\frac{1}{2}\right)\ln@{z}+z-\ln@{\sqrt{2\cpi}} +} +
+ +== Symbols List == + +[http://drmf.wmflabs.org/wiki/Definition:Binet {\displaystyle J\left(x\right)}] : Binet function : [http://drmf.wmflabs.org/wiki/Definition:Binet http://drmf.wmflabs.org/wiki/Definition:Binet]
+[http://dlmf.nist.gov/5.19.E4 {\displaystyle \pi}] : ratio of a circle's circumference to its diameter : [http://dlmf.nist.gov/5.19.E4 http://dlmf.nist.gov/5.19.E4]
+[http://dlmf.nist.gov/5.2#E1 {\displaystyle \Gamma}] : Euler's gamma function : [http://dlmf.nist.gov/5.2#E1 http://dlmf.nist.gov/5.2#E1]
+[http://dlmf.nist.gov/4.2#E2 {\displaystyle \mathrm{ln}}] : principal branch of logarithm function : [http://dlmf.nist.gov/4.2#E2 http://dlmf.nist.gov/4.2#E2] +
+
<< [[Definition:sinc|Definition:sinc]]
+
[[Main_Page|Main Page]]
[[Definition:AffqKrawtchouk|Definition:AffqKrawtchouk]] >>
drmf_eof From d81dd01c05fde96347a9cd12d417b82933257bbe Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Fri, 12 Aug 2016 11:22:36 -0400 Subject: [PATCH 224/402] Finish tests for constraint function --- eCF/.gitignore | 1 + eCF/src/MathematicaToLaTeX.py | 2 +- eCF/test/test_constraint.py | 158 +++++++++++++++++++++++++++++++--- 3 files changed, 148 insertions(+), 13 deletions(-) diff --git a/eCF/.gitignore b/eCF/.gitignore index 7f9c223..e6caac9 100644 --- a/eCF/.gitignore +++ b/eCF/.gitignore @@ -4,6 +4,7 @@ src/test.txt data/Identities.* data/newIdentities.* data/IdentitiesTest.m +data/References.txt src/Glossary.csv src/new.Glossary.csv src/new.Glossary.updated.csv diff --git a/eCF/src/MathematicaToLaTeX.py b/eCF/src/MathematicaToLaTeX.py index 7d753a4..4bc54e9 100644 --- a/eCF/src/MathematicaToLaTeX.py +++ b/eCF/src/MathematicaToLaTeX.py @@ -762,7 +762,7 @@ def main(): Opens Mathematica file with identities and puts converted lines into newIdentities.tex. """ - test = True + test = False with open(os.path.dirname(os.path.realpath(__file__)) + '/../data/newIdentities.tex', 'w') as latex: diff --git a/eCF/test/test_constraint.py b/eCF/test/test_constraint.py index 8b433c4..c881a08 100644 --- a/eCF/test/test_constraint.py +++ b/eCF/test/test_constraint.py @@ -8,17 +8,151 @@ class TestConstraint(TestCase): - def test_single(self): - self.assertEqual(constraint('text,Element[z,Complexes]'), 'text\n% \constraint{$z \in \Complex$}') - self.assertEqual(constraint('text,Element[z,Wholes]'), 'text\n% \constraint{$z \in \Whole$}') - self.assertEqual(constraint('text,Element[z,Naturals]'), 'text\n% \constraint{$z \in \NatNumber$}') - self.assertEqual(constraint('text,Element[z,Integers]'), 'text\n% \constraint{$z \in \Integer$}') - self.assertEqual(constraint('text,Element[z,Irrationals]'), 'text\n% \constraint{$z \in \Irrational$}') - self.assertEqual(constraint('text,Element[z,Reals]'), 'text\n% \constraint{$z \in \Real$}') - self.assertEqual(constraint('text,Element[z,Rational]'), 'text\n% \constraint{$z \in \Rational$}') - self.assertEqual(constraint('text,Element[z,Primes]'), 'text\n% \constraint{$z \in \Prime$}') - - # def test_multiple(self): + def test_element_singlevar(self): + self.assertEqual(constraint('text,Element[a,Complexes]'), + 'text\n' + '% \constraint{$a \in \Complex$}') + self.assertEqual(constraint('text,Element[a,Wholes]'), + 'text\n' + '% \constraint{$a \in \Whole$}') + self.assertEqual(constraint('text,Element[a,Naturals]'), + 'text\n' + '% \constraint{$a \in \\NatNumber$}') + self.assertEqual(constraint('text,Element[a,Integers]'), + 'text\n' + '% \constraint{$a \in \Integer$}') + self.assertEqual(constraint('text,Element[a,Irrationals]'), + 'text\n' + '% \constraint{$a \in \Irrational$}') + self.assertEqual(constraint('text,Element[a,Reals]'), + 'text\n' + '% \constraint{$a \in \Real$}') + self.assertEqual(constraint('text,Element[a,Rational]'), + 'text\n' + '% \constraint{$a \in \Rational$}') + self.assertEqual(constraint('text,Element[a,Primes]'), + 'text\n' + '% \constraint{$a \in \Prime$}') + + def test_element_multivar(self): + self.assertEqual(constraint('text,Element[a|b|c|d,Complexes]'), + 'text\n' + '% \constraint{$a,b,c,d \in \Complex$}') + self.assertEqual(constraint('text,Element[a|b|c|d,Wholes]'), + 'text\n' + '% \constraint{$a,b,c,d \in \Whole$}') + self.assertEqual(constraint('text,Element[a|b|c|d,Naturals]'), + 'text\n' + '% \constraint{$a,b,c,d \in \\NatNumber$}') + self.assertEqual(constraint('text,Element[a|b|c|d,Integers]'), + 'text\n' + '% \constraint{$a,b,c,d \in \Integer$}') + self.assertEqual(constraint('text,Element[a|b|c|d,Irrationals]'), + 'text\n' + '% \constraint{$a,b,c,d \in \Irrational$}') + self.assertEqual(constraint('text,Element[a|b|c|d,Reals]'), + 'text\n' + '% \constraint{$a,b,c,d \in \Real$}') + self.assertEqual(constraint('text,Element[a|b|c|d,Rational]'), + 'text\n' + '% \constraint{$a,b,c,d \in \Rational$}') + self.assertEqual(constraint('text,Element[a|b|c|d,Primes]'), + 'text\n' + '% \constraint{$a,b,c,d \in \Prime$}') + + def test_notelement_singlevar(self): + self.assertEqual(constraint('text,NotElement[a,Complexes]'), + 'text\n' + '% \\constraint{$a \\notin \\Complex$}') + self.assertEqual(constraint('text,NotElement[a,Wholes]'), + 'text\n' + '% \\constraint{$a \\notin \\Whole$}') + self.assertEqual(constraint('text,NotElement[a,Naturals]'), + 'text\n' + '% \\constraint{$a \\notin \\NatNumber$}') + self.assertEqual(constraint('text,NotElement[a,Integers]'), + 'text\n' + '% \\constraint{$a \\notin \\Integer$}') + self.assertEqual(constraint('text,NotElement[a,Irrationals]'), + 'text\n' + '% \\constraint{$a \\notin \\Irrational$}') + self.assertEqual(constraint('text,NotElement[a,Reals]'), + 'text\n' + '% \\constraint{$a \\notin \\Real$}') + self.assertEqual(constraint('text,NotElement[a,Rational]'), + 'text\n' + '% \\constraint{$a \\notin \\Rational$}') + self.assertEqual(constraint('text,NotElement[a,Primes]'), + 'text\n' + '% \\constraint{$a \\notin \\Prime$}') + + def test_notelement_multivar(self): + self.assertEqual(constraint('text,NotElement[a|b|c|d,Complexes]'), + 'text\n' + '% \\constraint{$a,b,c,d \\notin \\Complex$}') + self.assertEqual(constraint('text,NotElement[a|b|c|d,Wholes]'), + 'text\n' + '% \\constraint{$a,b,c,d \\notin \\Whole$}') + self.assertEqual(constraint('text,NotElement[a|b|c|d,Naturals]'), + 'text\n' + '% \\constraint{$a,b,c,d \\notin \\NatNumber$}') + self.assertEqual(constraint('text,NotElement[a|b|c|d,Integers]'), + 'text\n' + '% \\constraint{$a,b,c,d \\notin \\Integer$}') + self.assertEqual(constraint('text,NotElement[a|b|c|d,Irrationals]'), + 'text\n' + '% \\constraint{$a,b,c,d \\notin \\Irrational$}') + self.assertEqual(constraint('text,NotElement[a|b|c|d,Reals]'), + 'text\n' + '% \\constraint{$a,b,c,d \\notin \\Real$}') + self.assertEqual(constraint('text,NotElement[a|b|c|d,Rational]'), + 'text\n' + '% \\constraint{$a,b,c,d \\notin \\Rational$}') + self.assertEqual(constraint('text,NotElement[a|b|c|d,Primes]'), + 'text\n' + '% \\constraint{$a,b,c,d \\notin \\Prime$}') + + def test_inequality(self): + self.assertEqual(constraint('text,Inequality[a,Less,b,LessEqual,c]'), + 'text\n' + '% \\constraint{$aLessbLessEqualc$}') + + def test_multiple(self): + self.assertEqual(constraint('text,Element[a,Complexes]&&Element[b,Wholes]'), + 'text\n' + '% \\constraint{$a \\in \\Complex$\n' + '% & $b \\in \\Whole$}') + self.assertEqual(constraint('text,Element[a,Complexes]&&Element[b,Wholes]&&Element[c,Naturals]'), + 'text\n' + '% \\constraint{$a \\in \\Complex$\n' + '% & $b \\in \\Whole$\n' + '% & $c \\in \\NatNumber$}') + + self.assertEqual(constraint('text,NotElement[a,Complexes]&&NotElement[b,Wholes]'), + 'text\n' + '% \\constraint{$a \\notin \\Complex$\n' + '% & $b \\notin \\Whole$}') + self.assertEqual(constraint('text,NotElement[a,Complexes]&&NotElement[b,Wholes]&&NotElement[c,Naturals]'), + 'text\n' + '% \\constraint{$a \\notin \\Complex$\n' + '% & $b \\notin \\Whole$\n' + '% & $c \\notin \\NatNumber$}') + + self.assertEqual(constraint('text,Inequality[a,Less,b]&&Inequality[b,Less,c]'), + 'text\n' + '% \\constraint{$aLessb$\n' + '% & $bLessc$}') + self.assertEqual(constraint('text,Inequality[a,Less,b]&&Inequality[b,Less,c]&&Inequality[c,Less,d]'), + 'text\n' + '% \\constraint{$aLessb$\n' + '% & $bLessc$\n' + '% & $cLessd$}') + + self.assertEqual(constraint('text,Element[a|b|c,Complexes]&&NotElement[d|e|f,Primes]&&Inequality[g,Less,h,Less,i]'), + 'text\n' + '% \\constraint{$a,b,c \\in \\Complex$\n' + '% & $d,e,f \\notin \\Prime$\n' + '% & $gLesshLessi$}') def test_none(self): - self.assertEqual(constraint('noconstraint'), 'noconstraint') + self.assertEqual(constraint('noconstraint'), 'noconstraint') \ No newline at end of file From c9fb6969ab937aee3d2fcdc882ced761b2f937d1 Mon Sep 17 00:00:00 2001 From: Joon Bang Date: Fri, 12 Aug 2016 11:57:18 -0400 Subject: [PATCH 225/402] Add fake new.Glossary.csv to fix unit tests --- main_page/new.Glossary.csv | 3 ++ main_page/src/functions.py | 5 +- .../test/add_symbols_data_tests/exp.1.txt | 12 ++--- .../test/add_symbols_data_tests/res.1.txt | 25 ++++------ main_page/test/add_usage_tests/exp.1.txt | 39 ++++++++++----- main_page/test/add_usage_tests/res.1.txt | 47 ++++++++++++------- 6 files changed, 79 insertions(+), 52 deletions(-) create mode 100644 main_page/new.Glossary.csv diff --git a/main_page/new.Glossary.csv b/main_page/new.Glossary.csv new file mode 100644 index 0000000..30db2f0 --- /dev/null +++ b/main_page/new.Glossary.csv @@ -0,0 +1,3 @@ +\DefinitionOne@{x},sample definition 1,sample-definition-1,F|P,$\mathrm{Def1}{x}$,http://fake-hyperlink.com/Definition1, +\DefinitionTwo@{x},sample definition 2,sample-definition-2,F|P,$\mathrm{Def2}{x}$,http://fake-hyperlink.com/Definition2, +\DefinitionTwoTwo,sample definition 22,sample-definition-22,C|P,$\mathrm{twenty two}$,http://fake-hyperlink.com/Definition22, \ No newline at end of file diff --git a/main_page/src/functions.py b/main_page/src/functions.py index 67abba4..23b54c8 100644 --- a/main_page/src/functions.py +++ b/main_page/src/functions.py @@ -26,6 +26,8 @@ def get_macro_name(macro): elif ch in ["@", "{", "["]: break + print macro_name + return macro_name @@ -204,8 +206,6 @@ def add_symbols_data(data): pages = data[data.find("drmf_bof"):].split("drmf_eof")[:-1] - print "Pages: " + str(len(pages)) - result = "" for page in pages: # skip over non-definition pages @@ -288,6 +288,7 @@ def add_usage(lines): text += ":'''" + call + "'''" + " produces {\\displaystyle " + call + "}
\n" # Now add the multiple ways \macroname{n}@... produces \macroname{n}@... + print text to_write += text + "\n" i = lines.find("These are defined by", p) diff --git a/main_page/test/add_symbols_data_tests/exp.1.txt b/main_page/test/add_symbols_data_tests/exp.1.txt index 7e5ab7c..aba079a 100644 --- a/main_page/test/add_symbols_data_tests/exp.1.txt +++ b/main_page/test/add_symbols_data_tests/exp.1.txt @@ -2,25 +2,25 @@ drmf_bof '''Main Page''' drmf_eof drmf_bof -'''Definition:Definition1''' +'''Definition:DefinitionOne''' -\AlSalamCarlitzII{a}{n}@{x}{q}:=(-a)^nq^{-\binom{n}{2}}\,\qHyperrphis{2}{0}@@{q^{-n},x}{-}{q}{\frac{q^n}{a}} +\DefinitionOne@{x}:=\DefinitionTwo@{x}+1 == Symbols List == drmf_eof drmf_bof -'''Definition:Definition2''' +'''Definition:DefinitionTwo''' -\RiemannSiegelTheta@{t}:=\arg\left[\EulerGamma@{\frac{1}{4}+\frac{1}{2}it}\right]-\frac{1}{2}t\ln\pi +\DefinitionTwo@{x}:=\frac{6}{127}
drmf_eof drmf_bof -'''Definition:Definition3''' +'''Definition:DefinitionTwoTwo''' -\RamanujanTauTheta@{z}:=-\log\left(2\pi\right)z-\frac{i}{2}\left(\log\EulerGamma@{6+iz}-\log{\EulerGamma@{6-iz}}\right) +\DefinitionTwoTwo:=22 drmf_eof diff --git a/main_page/test/add_symbols_data_tests/res.1.txt b/main_page/test/add_symbols_data_tests/res.1.txt index 11bfbae..a494846 100644 --- a/main_page/test/add_symbols_data_tests/res.1.txt +++ b/main_page/test/add_symbols_data_tests/res.1.txt @@ -2,38 +2,33 @@ drmf_bof '''Main Page''' drmf_eof drmf_bof -'''Definition:Definition1''' +'''Definition:DefinitionOne''' -\AlSalamCarlitzII{a}{n}@{x}{q}:=(-a)^nq^{-\binom{n}{2}}\,\qHyperrphis{2}{0}@@{q^{-n},x}{-}{q}{\frac{q^n}{a}} +\DefinitionOne@{x}:=\DefinitionTwo@{x}+1 == Symbols List == -[http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzII {\displaystyle V^{(n)}_{\alpha}}] : Al-Salam-Carlitz II polynomial : [http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzII http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzII]
-[http://dlmf.nist.gov/1.2#E1 {\displaystyle \binom{n}{k}}] : binomial coefficient : [http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/1.2#E1] [http://dlmf.nist.gov/26.3#SS1.p1 http://dlmf.nist.gov/26.3#SS1.p1]
-[http://dlmf.nist.gov/17.4#E1 {\displaystyle {{}_{r}\phi_{s}}}] : basic hypergeometric (or {\displaystyle q}-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1] +[http://fake-hyperlink.com/Definition1 {\displaystyle \mathrm{Def1}{x}}] : sample definition 1 : [http://fake-hyperlink.com/Definition1 http://fake-hyperlink.com/Definition1]
+[http://fake-hyperlink.com/Definition2 {\displaystyle \mathrm{Def2}{x}}] : sample definition 2 : [http://fake-hyperlink.com/Definition2 http://fake-hyperlink.com/Definition2]
drmf_eof drmf_bof -'''Definition:Definition2''' +'''Definition:DefinitionTwo''' -\RiemannSiegelTheta@{t}:=\arg\left[\EulerGamma@{\frac{1}{4}+\frac{1}{2}it}\right]-\frac{1}{2}t\ln\pi +\DefinitionTwo@{x}:=\frac{6}{127} == Symbols List == -[http://dlmf.nist.gov/5.2#E1 {\displaystyle \Gamma}] : Euler's gamma function : [http://dlmf.nist.gov/5.2#E1 http://dlmf.nist.gov/5.2#E1]
-[http://dlmf.nist.gov/4.2#E2 {\displaystyle \mathrm{ln}}] : principal branch of logarithm function : [http://dlmf.nist.gov/4.2#E2 http://dlmf.nist.gov/4.2#E2]
-[http://drmf.wmflabs.org/wiki/Definition:RiemannSiegelTheta {\displaystyle \vartheta{t}}] : Riemann-Siegel theta function : [http://drmf.wmflabs.org/wiki/Definition:RiemannSiegelTheta http://drmf.wmflabs.org/wiki/Definition:RiemannSiegelTheta] +[http://fake-hyperlink.com/Definition2 {\displaystyle \mathrm{Def2}{x}}] : sample definition 2 : [http://fake-hyperlink.com/Definition2 http://fake-hyperlink.com/Definition2]
drmf_eof drmf_bof -'''Definition:Definition3''' +'''Definition:DefinitionTwoTwo''' -\RamanujanTauTheta@{z}:=-\log\left(2\pi\right)z-\frac{i}{2}\left(\log\EulerGamma@{6+iz}-\log{\EulerGamma@{6-iz}}\right) +\DefinitionTwoTwo:=22 == Symbols List == -[http://dlmf.nist.gov/5.2#E1 {\displaystyle \Gamma}] : Euler's gamma function : [http://dlmf.nist.gov/5.2#E1 http://dlmf.nist.gov/5.2#E1]
-[http://dlmf.nist.gov/4.2#E2 {\displaystyle \mathrm{log}}] : principle branch of natural logarithm : [http://dlmf.nist.gov/4.2#E2 http://dlmf.nist.gov/4.2#E2]
-[http://drmf.wmflabs.org/wiki/Definition:RamanujanTauTheta {\displaystyle \tau\theta{z}}] : Ramanujan tau theta function : [http://drmf.wmflabs.org/wiki/Definition:RamanujanTauTheta http://drmf.wmflabs.org/wiki/Definition:RamanujanTauTheta] +[http://fake-hyperlink.com/Definition22 {\displaystyle \mathrm{twenty two}}] : sample definition 22 : [http://fake-hyperlink.com/Definition22 http://fake-hyperlink.com/Definition22]
drmf_eof diff --git a/main_page/test/add_usage_tests/exp.1.txt b/main_page/test/add_usage_tests/exp.1.txt index 0b3d64b..ad1e151 100644 --- a/main_page/test/add_usage_tests/exp.1.txt +++ b/main_page/test/add_usage_tests/exp.1.txt @@ -2,29 +2,44 @@ drmf_bof '''Main Page''' drmf_eof drmf_bof -'''Definition:normctsdualqHahnptilde''' -The LaTeX DLMF and DRMF macro '''\normctsdualqHahnptilde''' represents the normctsdualqHahnptilde. +'''Definition:DefinitionOne''' +The LaTeX DLMF and DRMF macro '''\DefinitionOne''' represents the DefinitionOne. + +This macro is in the category of real or complex valued functions. + +In math mode, this macro can be called in the following way: + +:'''\DefinitionOne@{x}''' produces {\displaystyle \DefinitionOne@{x}}
These are defined by -{\displaystyle -{\tilde p}_n(x):=\normctsdualqHahnptilde{n}@{x}{a}{b}{c}{q}=\frac{a^n\ctsdualqHahn{n}@{x}{a}{b}{c}{q}}{\qPochhammer{ab,ac}{q}{n}} +\displaystyle{ +\DefinitionOne@{x}:=\DefinitionTwo@{x}+1 }
drmf_eof drmf_bof -'''Definition:GaussDistQ''' -The LaTeX DLMF and DRMF macro '''\GaussDistQ''' represents the GaussDistQ. +'''Definition:DefinitionTwo''' +The LaTeX DLMF and DRMF macro '''\DefinitionTwo''' represents the DefinitionTwo. -This macro is in the category of integer valued functions. +This macro is in the category of real or complex valued functions. -In math mode, this macro can be called in the following ways: +In math mode, this macro can be called in the following way: -:'''\GaussDistQ''' produces {\displaystyle \GaussDistQ}
-:'''\GaussDistQ@{x}''' produces {\displaystyle \GaussDistQ@{x}}
+:'''\DefinitionTwo@{x}''' produces {\displaystyle \DefinitionTwo@{x}}
These are defined by -{\displaystyle -\GaussDistQ@{x}:=\frac{1}{2}\erfc@{\frac{x}{\sqrt{2}}} +\displaystyle{ +\DefinitionTwo@{x}:=\frac{6}{127} }
drmf_eof +drmf_bof +'''Definition:DefinitionTwoTwo''' +The LaTeX DLMF and DRMF macro '''\DefinitionTwoTwo''' represents the DefinitionTwoTwo. + +These are defined by +\displaystyle{ +\DefinitionTwoTwo:=22 +
+ +drmf_eof diff --git a/main_page/test/add_usage_tests/res.1.txt b/main_page/test/add_usage_tests/res.1.txt index 6b5407e..2ce81ca 100644 --- a/main_page/test/add_usage_tests/res.1.txt +++ b/main_page/test/add_usage_tests/res.1.txt @@ -2,37 +2,50 @@ drmf_bof '''Main Page''' drmf_eof drmf_bof -'''Definition:normctsdualqHahnptilde''' -The LaTeX DLMF and DRMF macro '''\normctsdualqHahnptilde''' represents the normctsdualqHahnptilde. +'''Definition:DefinitionOne''' +The LaTeX DLMF and DRMF macro '''\DefinitionOne''' represents the DefinitionOne. -This macro is in the category of polynomials. +This macro is in the category of real or complex valued functions. -In math mode, this macro can be called in the following ways: +In math mode, this macro can be called in the following way: -:'''\normctsdualqHahnptilde{n}''' produces {\displaystyle \normctsdualqHahnptilde{n}}
-:'''\normctsdualqHahnptilde{n}@{x}{a}{b}{c}{q}''' produces {\displaystyle \normctsdualqHahnptilde{n}@{x}{a}{b}{c}{q}}
-:'''\normctsdualqHahnptilde{n}@@{x}{a}{b}{c}{q}''' produces {\displaystyle \normctsdualqHahnptilde{n}@@{x}{a}{b}{c}{q}}
+:'''\DefinitionOne@{x}''' produces {\displaystyle \DefinitionOne@{x}}
These are defined by -{\displaystyle -{\tilde p}_n(x):=\normctsdualqHahnptilde{n}@{x}{a}{b}{c}{q}=\frac{a^n\ctsdualqHahn{n}@{x}{a}{b}{c}{q}}{\qPochhammer{ab,ac}{q}{n}} +\displaystyle{ +\DefinitionOne@{x}:=\DefinitionTwo@{x}+1 }
drmf_eof drmf_bof -'''Definition:GaussDistQ''' -The LaTeX DLMF and DRMF macro '''\GaussDistQ''' represents the GaussDistQ. +'''Definition:DefinitionTwo''' +The LaTeX DLMF and DRMF macro '''\DefinitionTwo''' represents the DefinitionTwo. -This macro is in the category of integer valued functions. +This macro is in the category of real or complex valued functions. -In math mode, this macro can be called in the following ways: +In math mode, this macro can be called in the following way: -:'''\GaussDistQ''' produces {\displaystyle \GaussDistQ}
-:'''\GaussDistQ@{x}''' produces {\displaystyle \GaussDistQ@{x}}
+:'''\DefinitionTwo@{x}''' produces {\displaystyle \DefinitionTwo@{x}}
These are defined by -{\displaystyle -\GaussDistQ@{x}:=\frac{1}{2}\erfc@{\frac{x}{\sqrt{2}}} +\displaystyle{ +\DefinitionTwo@{x}:=\frac{6}{127} }
drmf_eof +drmf_bof +'''Definition:DefinitionTwoTwo''' +The LaTeX DLMF and DRMF macro '''\DefinitionTwoTwo''' represents the DefinitionTwoTwo. + +This macro is in the category of constants. + +In math mode, this macro can be called in the following way: + +:'''\DefinitionTwoTwo''' produces {\displaystyle \DefinitionTwoTwo}
+ +These are defined by +\displaystyle{ +\DefinitionTwoTwo:=22 +
+ +drmf_eof From c7f5932fd199b68f5e8adf8142b9348e153d1ec5 Mon Sep 17 00:00:00 2001 From: Joon Bang Date: Fri, 12 Aug 2016 12:05:21 -0400 Subject: [PATCH 226/402] Rename new.Glossary.csv to fake.Glossary.csv --- main_page/src/functions.py | 8 ++++---- main_page/{new.Glossary.csv => test/fake.Glossary.csv} | 0 main_page/test/test_functions.py | 6 ++++-- 3 files changed, 8 insertions(+), 6 deletions(-) rename main_page/{new.Glossary.csv => test/fake.Glossary.csv} (100%) diff --git a/main_page/src/functions.py b/main_page/src/functions.py index 23b54c8..c680e84 100644 --- a/main_page/src/functions.py +++ b/main_page/src/functions.py @@ -197,9 +197,9 @@ def get_symbols(text, glossary): return span_text[:-7] # slice off the extra br and endline -def add_symbols_data(data): +def add_symbols_data(data,glossary_location=GLOSSARY_LOCATION): glossary = dict() - with open(GLOSSARY_LOCATION, "rb") as csv_file: + with open(glossary_location, "rb") as csv_file: glossary_file = csv.reader(csv_file, delimiter=',', quotechar='\"') for row in glossary_file: glossary[get_macro_name(row[0])] = row @@ -239,7 +239,7 @@ def add_symbols_data(data): return result -def add_usage(lines): +def add_usage(lines,glossary_location=GLOSSARY_LOCATION): with open("main_page/categories.txt") as cats: categories = cats.readlines() @@ -267,7 +267,7 @@ def add_usage(lines): to_write += lines[i:p].rstrip() + "\n\n" calls = [] - glossary = csv.reader(open(GLOSSARY_LOCATION, 'rb'), delimiter=',', quotechar='\"') + glossary = csv.reader(open(glossary_location, 'rb'), delimiter=',', quotechar='\"') for entry in glossary: if macro_match("\\" + macro_name, entry[0]): macro_calls, category = find_all_positions(entry) diff --git a/main_page/new.Glossary.csv b/main_page/test/fake.Glossary.csv similarity index 100% rename from main_page/new.Glossary.csv rename to main_page/test/fake.Glossary.csv diff --git a/main_page/test/test_functions.py b/main_page/test/test_functions.py index f2a4d1c..7e712f5 100644 --- a/main_page/test/test_functions.py +++ b/main_page/test/test_functions.py @@ -55,10 +55,12 @@ def test_update_headers(self): class TestAddSymbolsData(TestCase): def test_add_symbols_data(self): for case in add_symbols_data_test_cases: - self.assertEqual(m.add_symbols_data(case["exp"]), case["res"]) + self.assertEqual(m.add_symbols_data(case["exp"], glossary_location="main_page/test/fake.Glossary.csv"), + case["res"]) class TestAddUsage(TestCase): def test_add_usage(self): for case in add_usage_test_cases: - self.assertEqual(m.add_usage(case["exp"]), case["res"]) + self.assertEqual(m.add_usage(case["exp"], glossary_location="main_page/test/fake.Glossary.csv"), + case["res"]) From 1ae0b291e44b62b0330d642d23c1564fa109374e Mon Sep 17 00:00:00 2001 From: Joon Bang Date: Fri, 12 Aug 2016 12:24:14 -0400 Subject: [PATCH 227/402] Add docstrings and parameter/return types --- main_page/src/functions.py | 32 +++++++++++++++++++++++++------- 1 file changed, 25 insertions(+), 7 deletions(-) diff --git a/main_page/src/functions.py b/main_page/src/functions.py index c680e84..618eb8a 100644 --- a/main_page/src/functions.py +++ b/main_page/src/functions.py @@ -7,6 +7,8 @@ def generate_html(tag_name, options, text, spacing=True): + # (str, str, str(, bool)) -> str + """Generates an html tag, with optional html parameters.""" result = "<" + tag_name if options != "": result += " " + options @@ -19,6 +21,8 @@ def generate_html(tag_name, options, text, spacing=True): def get_macro_name(macro): + # (str) -> str + """Obtains the macro name.""" macro_name = "" for ch in macro: if ch.isalpha() or ch == "\\": @@ -26,13 +30,12 @@ def get_macro_name(macro): elif ch in ["@", "{", "["]: break - print macro_name - return macro_name def find_all(pattern, string): - """Yields all positions of where pattern is present in string.""" + # (str, str) -> generator + """Finds all instances of pattern in string.""" i = string.find(pattern) while i != -1: @@ -41,6 +44,8 @@ def find_all(pattern, string): def remove_optional_params(string): + # (str) -> str + """Removes optional parameters from a LaTeX semantic macro.""" parse = True text = "" for ch in string: @@ -55,7 +60,9 @@ def remove_optional_params(string): return text -def find_all_positions(entry): +def get_all_variants(entry): + # (str) -> str + """Returns string containing info on all variants of a macro.""" macro = entry[0] category = entry[3].split("|", 1)[0] keys = entry[3].split("|", 1)[1].split(":") @@ -92,11 +99,14 @@ def find_all_positions(entry): def macro_match(macro, entry): + # (str, str) -> bool """Determines whether the entry refers to the macro.""" return macro in entry and (len(macro) == len(entry) or not entry[len(macro)].isalpha()) def update_macro_list(text): + # (str) -> str + """Updates the list of macros found in the main page of main_page.mmd.""" lines = text.split("\n") definitions = list() @@ -116,6 +126,8 @@ def update_macro_list(text): def update_headers(text, definitions): + # (str, list) -> str + """Updates the headers used for navigation for every page.""" pages = text.split("drmf_eof")[:-1] macros = list() @@ -157,6 +169,8 @@ def update_headers(text, definitions): def get_symbols(text, glossary): + # (str, dict) -> str + """Generates span text based on symbols present in text.""" symbols = set() for keyword in glossary: @@ -197,7 +211,9 @@ def get_symbols(text, glossary): return span_text[:-7] # slice off the extra br and endline -def add_symbols_data(data,glossary_location=GLOSSARY_LOCATION): +def add_symbols_data(data, glossary_location=GLOSSARY_LOCATION): + # (str(, str)) -> str + """Adds list of symbols present in page under symbols list section.""" glossary = dict() with open(glossary_location, "rb") as csv_file: glossary_file = csv.reader(csv_file, delimiter=',', quotechar='\"') @@ -239,7 +255,9 @@ def add_symbols_data(data,glossary_location=GLOSSARY_LOCATION): return result -def add_usage(lines,glossary_location=GLOSSARY_LOCATION): +def add_usage(lines, glossary_location=GLOSSARY_LOCATION): + # (str(, str)) -> str + """Adds the macro category information and calling sequences.""" with open("main_page/categories.txt") as cats: categories = cats.readlines() @@ -270,7 +288,7 @@ def add_usage(lines,glossary_location=GLOSSARY_LOCATION): glossary = csv.reader(open(glossary_location, 'rb'), delimiter=',', quotechar='\"') for entry in glossary: if macro_match("\\" + macro_name, entry[0]): - macro_calls, category = find_all_positions(entry) + macro_calls, category = get_all_variants(entry) calls += macro_calls for line in categories: From 3d5caa471e13d3b54e1b825a5d6e1b7aeea56900 Mon Sep 17 00:00:00 2001 From: Edward Bian Date: Mon, 18 Apr 2016 17:31:12 -0400 Subject: [PATCH 228/402] Hardcoded in a fix for the paragraph placement --- KLSadd_insertion/src/updateChapters.py | 279 ++++++++----------------- 1 file changed, 88 insertions(+), 191 deletions(-) diff --git a/KLSadd_insertion/src/updateChapters.py b/KLSadd_insertion/src/updateChapters.py index f072dc6..c364b29 100644 --- a/KLSadd_insertion/src/updateChapters.py +++ b/KLSadd_insertion/src/updateChapters.py @@ -1,7 +1,3 @@ -__author__ = "Rahul Shah" -__status__ = "Development" -__credits__ = ["Rahul Shah", "Edward Bian"] - """ Rahul Shah self-taught python don't kill me I know its bad @@ -21,199 +17,130 @@ -rewrite for "smarter" edits (ex. add new limit relations straight to the limit relations section in the section itself, not at the end) Current update status: incomplete Goals:change the book chapter files to include paragraphs from the addendum, and after that insert some edits so they are intelligently integrated into the chapter - -Edward Bian -Currently under heavy modification; sections may not work and/or look inefficient/confusing """ -# start out by reading KLSadd.tex to get all of the paragraphs that must be added to the chapter files -# also keep track of which chapter each one is in +#start out by reading KLSadd.tex to get all of the paragraphs that must be added to the chapter files +#also keep track of which chapter each one is in -# variables: chapNums for the chapter number each section belongs to paras will hold the sections that will be copied over -# chapNums is needed to know which file to open (9 or 14) -# mathPeople is just the name for the sections that are used, like Wilson, -# Racah, etc. I know some are not people, its just a var name +#variables: chapNums for the chapter number each section belongs to paras will hold the sections that will be copied over +#chapNums is needed to know which file to open (9 or 14) +#mathPeople is just the name for the sections that are used, like Wilson, Racah, etc. I know some are not people, its just a var name +#yes I like lists chapNums = [] paras = [] -klsparas = [] mathPeople = [] -newCommands = [] # used to hold the indexes of the commands -ref9II = [] # Hold section search indexes -ref14II = [] -ref9III = [] # Holds all indexes -ref14III = [] -specref9 = [] -specref14 = [] -comms = "" # holds the ACTUAL STRINGS of the commands -# 2/18/16 this method addresses the goal of hardcoding in the necessary packages to let the chapter files run as pdf's. -# Currently only works with chapter 9, ask Dr. Cohl to help port your -# chapter 14 output file into a pdf - +newCommands = [] #used to hold the indexes of the commands +comms = "" #holds the ACTUAL STRINGS of the commands +#2/18/16 this method addresses the goal of hardcoding in the necessary packages to let the chapter files run as pdf's. +#Currently only works with chapter 9, ask Dr. Cohl to help port your chapter 14 output file into a pdf def prepareForPDF(chap): footmiscIndex = 0 index = 0 for word in chap: - index += 1 + index+=1 if("footmisc" in word): - footmiscIndex += index - # edits the chapter string sent to include hyperref, xparse, and cite packages + footmiscIndex+= index + #edits the chapter string sent to include hyperref, xparse, and cite packages #str[footmiscIndex] += "\\usepackage[pdftex]{hyperref} \n\\usepackage {xparse} \n\\usepackage{cite} \n" - chap.insert( - footmiscIndex, - "\\usepackage[pdftex]{hyperref} \n\\usepackage {xparse} \n\\usepackage{cite} \n") + chap.insert(footmiscIndex, "\\usepackage[pdftex]{hyperref} \n\\usepackage {xparse} \n\\usepackage{cite} \n") return chap -# 2/18/16 this method reads in relevant commands that are in KLSadd.tex -# and returns them as a list - +#2/18/16 this method reads in relevant commands that are in KLSadd.tex and returns them as a list def getCommands(kls): index = 0 for word in kls: - index += 1 + index+=1 if("smallskipamount" in word): - newCommands.append(index - 1) + newCommands.append(index-1) if("mybibitem[1]" in word): newCommands.append(index) comms = kls[newCommands[0]:newCommands[1]] return comms -# 2/18/16 this method addresses the goal of hardcoding in the necessary -# commands to let the chapter files run as pdf's. Currently only works -# with chapter 9 - +#2/18/16 this method addresses the goal of hardcoding in the necessary commands to let the chapter files run as pdf's. Currently only works with chapter 9 def insertCommands(kls, chap, cms): - # reads in the newCommands[] and puts them in chap - # the index of the "begin document" keyphrase, this is where the new - # commands need to be inserted. - beginIndex = -1 - # find index of begin document in KLSadd + #reads in the newCommands[] and puts them in chap + beginIndex = -1 #the index of the "begin document" keyphrase, this is where the new commands need to be inserted. + #find index of begin document in KLSadd index = 0 for word in kls: - index += 1 + index+=1 if("begin{document}" in word): beginIndex += index tempIndex = 0 for i in cms: - chap.insert(beginIndex + tempIndex, i) - tempIndex += 1 + chap.insert(beginIndex+tempIndex,i) + tempIndex +=1 return chap -# method to find the indices of the reference paragraphs - - +#method to find the indices of the reference paragraphs def findReferences(chapter): references = [] index = -1 - # chaptercheck designates which chapter is being searched for references chaptercheck = 0 + #chaptercheck designates which chapter is being searched for references if chapticker == 0: chaptercheck = str(9) elif chapticker == 1: chaptercheck = str(14) - # canAdd tells the program whether the next section is a reference canAdd = False + #canAdd tells the program whether the next section is a reference for word in chapter: - index += 1 - # check sections and subsections + index+=1 + #check sections and subsections if("section{" in word or "subsection*{" in word) and ("subsubsection*{" not in word): - w = word[word.find("{") + 1: word.find("}")] - ws = word[word.find("{") + 1: word.find("~")] + w = word[word.find("{")+1: word.find("}")] + ws = word[word.find("{")+1: word.find("~")] for unit in mathPeople: - subunit = unit[unit.find(" ") + 1: unit.find("#")] - # System of checks that verifies if section is in chapter - if ((w in subunit) or (ws in subunit)) and (chaptercheck in unit) and (len(w) == len( - subunit)) or (("Pseudo Jacobi" in w) and ("Pseudo Jacobi (or Routh-Romanovski)" in subunit)): + subunit = unit[unit.find(" ")+1: unit.find("#")] + if ((w in subunit) or (ws in subunit)) and (chaptercheck in unit) and (len(w) == len(subunit)) or (("Pseudo Jacobi" in w) and ("Pseudo Jacobi (or Routh-Romanovski)" in subunit)): canAdd = True - if chapticker == 0: - ref9II.append(index) - ref9III.append(index) - elif chapticker == 1: - ref14II.append(index) - ref14III.append(index) - if("\\subsection*{References}" in word) and (canAdd): - # Appends valid locations + #System of checks that verifies if section is in chapter + if("\\subsection*{References}" in word) and (canAdd == True): references.append(index) - if chapticker == 0: - ref9II.append(index) - ref9III.append(index) - elif chapticker == 1: - ref14II.append(index) - ref14III.append(index) canAdd = False - if ("subsection*{" in word and "References" not in word): - if chapticker == 0: - ref9III.append(index) - elif chapticker == 1: - ref14III.append(index) - print(ref9II) - print (ref14II) - print(ref9III) - print(ref14III) + #Appends valid locations return references - -def referencePlacer(chap, references, p, kls, refII, refIII, specref): - # count is used to represent the values in count - count = 0 - # Tells which chapter it's on - designator = 0 +#method to change file string(actually a list right now), returns string to be written to file +#If you write a method that changes something, it is preffered that you call the method in here +def fixChapter(chap, references, p, kls): + #chap is the file string(actually a list), references is the specific references for the file, + #and p is the paras variable(not sure if needed) kls is the KLSadd.tex as a list + count = 0 #count is used to represent the values in count + designator = 0 #Tells which chapter it's on if chapticker2 == 0: designator = "9." elif chapticker2 == 1: designator = "14." for i in references: - # Place before References paragraph + #Place before References paragraph word1 = str(p[count]) - if (designator in word1[word1.find( - "\\subsection*{") + 1: word1.find("}")]): - chap[i - 2] += "%Begin KLSadd additions" - chap[i - 2] += p[count] - chap[i - 2] += "%End of KLSadd additions" + if (designator in word1[word1.find("\\subsection*{") + 1: word1.find("}") ]): + chap[i-2] += "%Begin KLSadd additions" + chap[i-2] += p[count] + chap[i-2] += "%End of KLSadd additions" count += 1 else: - while (designator not in word1[word1.find( - "\\subsection*{") + 1: word1.find("}")]): + while (designator not in word1[word1.find("\\subsection*{") + 1: word1.find("}") ]): word1 = str(p[count]) - if (designator in word1[word1.find( - "\\subsection*{") + 1: word1.find("}")]): + if (designator in word1[word1.find("\\subsection*{") + 1: word1.find("}") ]): chap[i - 2] += "%Begin KLSadd additions" chap[i - 2] += p[count] chap[i - 2] += "%End of KLSadd additions" count += 1 else: - count += 1 - - -def referencePlacerII(chap, references, p, kls, refII, refIII, specref): - if chapticker2 == 0: - designator = "9." - elif chapticker2 == 1: - designator = "14." - count = 0 - sectionnuma = specref[0] - sectionnumb = specref[1] - for i in specref: - pass - - -# method to change file string(actually a list right now), returns string to be written to file -# If you write a method that changes something, it is preffered that you -# call the method in here -def fixChapter(chap, references, p, kls, refII, refIII, specref): - # chap is the file string(actually a list), references is the specific references for the file, - # and p is the paras variable(not sure if needed) kls is the KLSadd.tex as - # a list - referencePlacer(chap, references, p, kls, refII, refIII, specref) + count+=1 + #Unfinished code segment that inserts the collected paras into the chapter chap = prepareForPDF(chap) cms = getCommands(kls) - chap = insertCommands(kls, chap, cms) + chap = insertCommands(kls,chap, cms) commentticker = 0 - # Hard coded command remover for word in chap: - word2 = chap[chap.index(word) - 1] + word2 = chap[chap.index(word)-1] if ("\\newcommand{\qhypK}[5]{\,\mbox{}_{#1}\phi_{#2}\!\left(" not in word2): if ("\\newcommand\\half{\\frac12}" in word): wordtoadd = "%" + word @@ -228,108 +155,78 @@ def fixChapter(chap, references, p, kls, refII, refIII, specref): wordtoadd = "%" + word chap[commentticker] = wordtoadd commentticker += 1 + #Hard coded command remover ticker1 = 0 - # Formatting to make the Latex file run while ticker1 < len(chap): if ('\\myciteKLS' in chap[ticker1]): chap[ticker1] = chap[ticker1].replace('\\myciteKLS', '\\cite') ticker1 += 1 + #Formatting to make the Latex file run return chap -# open the KLSadd file to do things with +#open the KLSadd file to do things with with open("KLSadd.tex", "r") as add: - # store the file as a string + #store the file as a string addendum = add.readlines() - # Makes sections look like other sections for word in addendum: if ("paragraph{" in word): lenword = len(word) - 1 - temp = word[0:word.find( - "{") + 1] + "\large\\bf KLSadd: " + word[word.find("{") + 1: lenword] + temp = word[0:word.find("{") + 1] + "\large\\bf KLSadd: " + word[word.find("{") + 1: lenword] addendum[addendum.index(word)] = temp if ("subsubsection*{" in word): lenword = len(word) - 1 - addendum[addendum.index(word)] = word[0:word.find( - "{") + 1] + "\large\\bf KLSadd: " + word[word.find("{") + 1: lenword] + addendum[addendum.index(word)] = word[0:word.find("{") + 1] + "\large\\bf KLSadd: " + word[word.find("{") + 1: lenword] + #Font processing index = 0 indexes = [] - # Designates sections that need stuff added - # get the index for word in addendum: - index += 1 + index+=1 if("." in word and "\\subsection*{" in word): if ("9." in word): chapNums.append(9) - name = word[word.find("{") + 1: word.find("}")] + name = word[word.find("{") + 1: word.find("}") ] mathPeople.append(name + "#") - specref9.append(index - 1) if("14." in word): chapNums.append(14) - name = word[word.find("{") + 1: word.find("}")] + name = word[word.find("{") + 1: word.find("}") ] mathPeople.append(name + "#") - specref14.append(index - 1) - indexes.append(index - 1) - if ("paragraph{" in word) and (index > 313): - klsparas.append(index - 1) - print(indexes) - print(specref9) - print(specref14) - print(klsparas) - print(mathPeople) - # now indexes holds all of the places there is a section - # using these indexes, get all of the words in between and add that to the - # paras[] - for i in range(len(indexes) - 1): - box = ''.join(addendum[indexes[i]: indexes[i + 1] - 1]) + #Designates sections that need stuff added + #get the index + indexes.append(index-1) + #now indexes holds all of the places there is a section + #using these indexes, get all of the words in between and add that to the paras[] + for i in range(len(indexes)-1): + box = ''.join(addendum[indexes[i]: indexes[i+1]-1]) paras.append(box) box2 = ''.join(addendum[indexes[35]: 2245]) paras.append(box2) - # paras now holds the paragraphs that need to go into the chapter files, but they need to go in the appropriate - # section(like Wilson, Racah, Hahn, etc.) so we use the mathPeople variable - # we can use the section names to place the relevant paragraphs in the - # right place + #paras now holds the paragraphs that need to go into the chapter files, but they need to go in the appropriate + #section(like Wilson, Racah, Hahn, etc.) so we use the mathPeople variable + #we can use the section names to place the relevant paragraphs in the right place - # as of 2/8/16 the paragraphs will go before the References paragraph of the relevant section - # parse both files 9 and 14 as strings + #as of 2/8/16 the paragraphs will go before the References paragraph of the relevant section + #parse both files 9 and 14 as strings - # chapter 9 + #chapter 9 with open("chap09.tex", "r") as ch9: - entire9 = ch9.readlines() # reads in as a list of strings + entire9 = ch9.readlines() #reads in as a list of strings ch9.close() - # chapter 14 + #chapter 14 with open("chap14.tex", "r") as ch14: entire14 = ch14.readlines() ch14.close() - # call the findReferences method to find the index of the References paragraph in the two file strings - # two variables for the references lists one for chapter 9 one for chapter - # 14 + #call the findReferences method to find the index of the References paragraph in the two file strings + #two variables for the references lists one for chapter 9 one for chapter 14 chapticker = 0 references9 = findReferences(entire9) chapticker += 1 references14 = findReferences(entire14) - # call the fixChapter method to get a list with the addendum paragraphs - # added in + #call the fixChapter method to get a list with the addendum paragraphs added in chapticker2 = 0 - str9 = ''.join( - fixChapter( - entire9, - references9, - paras, - addendum, - ref9II, - ref9III, - specref9)) + str9 = ''.join(fixChapter(entire9, references9, paras, addendum)) chapticker2 += 1 - str14 = ''.join( - fixChapter( - entire14, - references14, - paras, - addendum, - ref14II, - ref14III, - specref14)) + str14 = ''.join(fixChapter(entire14, references14, paras, addendum)) """ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ @@ -339,10 +236,10 @@ def fixChapter(chap, references, p, kls, refII, refIII, specref): """ -# write to files -# new output files for safety -with open("updated9.tex", "w") as temp9: +#write to files +#new output files for safety +with open("updated9.tex","w") as temp9: temp9.write(str9) with open("updated14.tex", "w") as temp14: - temp14.write(str14) + temp14.write(str14) \ No newline at end of file From a3487ebe7b2ac0a2e431690b68d5f120c918e1f7 Mon Sep 17 00:00:00 2001 From: Edward Bian Date: Wed, 4 May 2016 17:41:42 -0400 Subject: [PATCH 229/402] New indexes --- KLSadd_insertion/src/updateChapters.py | 112 ++++++++++++++++++------- 1 file changed, 84 insertions(+), 28 deletions(-) diff --git a/KLSadd_insertion/src/updateChapters.py b/KLSadd_insertion/src/updateChapters.py index c364b29..6867c00 100644 --- a/KLSadd_insertion/src/updateChapters.py +++ b/KLSadd_insertion/src/updateChapters.py @@ -17,6 +17,9 @@ -rewrite for "smarter" edits (ex. add new limit relations straight to the limit relations section in the section itself, not at the end) Current update status: incomplete Goals:change the book chapter files to include paragraphs from the addendum, and after that insert some edits so they are intelligently integrated into the chapter + +Edward Bian +Currently under heavy modification; sections may not work and/or look inefficient/confusing """ #start out by reading KLSadd.tex to get all of the paragraphs that must be added to the chapter files @@ -25,11 +28,17 @@ #variables: chapNums for the chapter number each section belongs to paras will hold the sections that will be copied over #chapNums is needed to know which file to open (9 or 14) #mathPeople is just the name for the sections that are used, like Wilson, Racah, etc. I know some are not people, its just a var name -#yes I like lists chapNums = [] paras = [] +klsparas = [] mathPeople = [] newCommands = [] #used to hold the indexes of the commands +ref9II = [] #Hold section search indexes +ref14II = [] +ref9III = [] #Holds all indexes +ref14III = [] +specref9 = [] +specref14 = [] comms = "" #holds the ACTUAL STRINGS of the commands #2/18/16 this method addresses the goal of hardcoding in the necessary packages to let the chapter files run as pdf's. #Currently only works with chapter 9, ask Dr. Cohl to help port your chapter 14 output file into a pdf @@ -80,14 +89,14 @@ def insertCommands(kls, chap, cms): def findReferences(chapter): references = [] index = -1 - chaptercheck = 0 #chaptercheck designates which chapter is being searched for references + chaptercheck = 0 if chapticker == 0: chaptercheck = str(9) elif chapticker == 1: chaptercheck = str(14) - canAdd = False #canAdd tells the program whether the next section is a reference + canAdd = False for word in chapter: index+=1 #check sections and subsections @@ -96,49 +105,88 @@ def findReferences(chapter): ws = word[word.find("{")+1: word.find("~")] for unit in mathPeople: subunit = unit[unit.find(" ")+1: unit.find("#")] + # System of checks that verifies if section is in chapter if ((w in subunit) or (ws in subunit)) and (chaptercheck in unit) and (len(w) == len(subunit)) or (("Pseudo Jacobi" in w) and ("Pseudo Jacobi (or Routh-Romanovski)" in subunit)): canAdd = True - #System of checks that verifies if section is in chapter + if chapticker == 0: + ref9II.append(index) + ref9III.append(index) + elif chapticker == 1: + ref14II.append(index) + ref14III.append(index) if("\\subsection*{References}" in word) and (canAdd == True): + # Appends valid locations references.append(index) + if chapticker == 0: + ref9II.append(index) + ref9III.append(index) + elif chapticker == 1: + ref14II.append(index) + ref14III.append(index) canAdd = False - #Appends valid locations + if ("subsection*{" in word and "References" not in word): + if chapticker == 0: + ref9III.append(index) + elif chapticker == 1: + ref14III.append(index) + print(ref9II) + print (ref14II) + print(ref9III) + print(ref14III) return references -#method to change file string(actually a list right now), returns string to be written to file -#If you write a method that changes something, it is preffered that you call the method in here -def fixChapter(chap, references, p, kls): - #chap is the file string(actually a list), references is the specific references for the file, - #and p is the paras variable(not sure if needed) kls is the KLSadd.tex as a list - count = 0 #count is used to represent the values in count - designator = 0 #Tells which chapter it's on +def referencePlacer(chap, references, p, kls,refII,refIII,specref): + # count is used to represent the values in count + count = 0 + # Tells which chapter it's on + designator = 0 if chapticker2 == 0: designator = "9." elif chapticker2 == 1: designator = "14." for i in references: - #Place before References paragraph + # Place before References paragraph word1 = str(p[count]) - if (designator in word1[word1.find("\\subsection*{") + 1: word1.find("}") ]): - chap[i-2] += "%Begin KLSadd additions" - chap[i-2] += p[count] - chap[i-2] += "%End of KLSadd additions" + if (designator in word1[word1.find("\\subsection*{") + 1: word1.find("}")]): + chap[i - 2] += "%Begin KLSadd additions" + chap[i - 2] += p[count] + chap[i - 2] += "%End of KLSadd additions" count += 1 else: - while (designator not in word1[word1.find("\\subsection*{") + 1: word1.find("}") ]): + while (designator not in word1[word1.find("\\subsection*{") + 1: word1.find("}")]): word1 = str(p[count]) - if (designator in word1[word1.find("\\subsection*{") + 1: word1.find("}") ]): + if (designator in word1[word1.find("\\subsection*{") + 1: word1.find("}")]): chap[i - 2] += "%Begin KLSadd additions" chap[i - 2] += p[count] chap[i - 2] += "%End of KLSadd additions" count += 1 else: - count+=1 - #Unfinished code segment that inserts the collected paras into the chapter + count += 1 + +def referencePlacerII(chap, references, p, kls,refII,refIII,specref): + if chapticker2 == 0: + designator = "9." + elif chapticker2 == 1: + designator = "14." + count = 0 + sectionnuma = specref[0] + sectionnumb = specref[1] + for i in specref: + pass + + + +#method to change file string(actually a list right now), returns string to be written to file +#If you write a method that changes something, it is preffered that you call the method in here +def fixChapter(chap, references, p, kls,refII,refIII,specref): + #chap is the file string(actually a list), references is the specific references for the file, + #and p is the paras variable(not sure if needed) kls is the KLSadd.tex as a list + referencePlacer(chap, references, p, kls, refII,refIII,specref) chap = prepareForPDF(chap) cms = getCommands(kls) chap = insertCommands(kls,chap, cms) commentticker = 0 + # Hard coded command remover for word in chap: word2 = chap[chap.index(word)-1] if ("\\newcommand{\qhypK}[5]{\,\mbox{}_{#1}\phi_{#2}\!\left(" not in word2): @@ -155,19 +203,19 @@ def fixChapter(chap, references, p, kls): wordtoadd = "%" + word chap[commentticker] = wordtoadd commentticker += 1 - #Hard coded command remover ticker1 = 0 + # Formatting to make the Latex file run while ticker1 < len(chap): if ('\\myciteKLS' in chap[ticker1]): chap[ticker1] = chap[ticker1].replace('\\myciteKLS', '\\cite') ticker1 += 1 - #Formatting to make the Latex file run return chap #open the KLSadd file to do things with with open("KLSadd.tex", "r") as add: #store the file as a string addendum = add.readlines() + #Makes sections look like other sections for word in addendum: if ("paragraph{" in word): lenword = len(word) - 1 @@ -176,9 +224,10 @@ def fixChapter(chap, references, p, kls): if ("subsubsection*{" in word): lenword = len(word) - 1 addendum[addendum.index(word)] = word[0:word.find("{") + 1] + "\large\\bf KLSadd: " + word[word.find("{") + 1: lenword] - #Font processing index = 0 indexes = [] + # Designates sections that need stuff added + # get the index for word in addendum: index+=1 if("." in word and "\\subsection*{" in word): @@ -186,13 +235,20 @@ def fixChapter(chap, references, p, kls): chapNums.append(9) name = word[word.find("{") + 1: word.find("}") ] mathPeople.append(name + "#") + specref9.append(index-1) if("14." in word): chapNums.append(14) name = word[word.find("{") + 1: word.find("}") ] mathPeople.append(name + "#") - #Designates sections that need stuff added - #get the index + specref14.append(index - 1) indexes.append(index-1) + if ("paragraph{" in word) and (index > 313): + klsparas.append(index-1) + print(indexes) + print(specref9) + print(specref14) + print(klsparas) + print(mathPeople) #now indexes holds all of the places there is a section #using these indexes, get all of the words in between and add that to the paras[] for i in range(len(indexes)-1): @@ -224,9 +280,9 @@ def fixChapter(chap, references, p, kls): references14 = findReferences(entire14) #call the fixChapter method to get a list with the addendum paragraphs added in chapticker2 = 0 - str9 = ''.join(fixChapter(entire9, references9, paras, addendum)) + str9 = ''.join(fixChapter(entire9, references9, paras, addendum,ref9II,ref9III,specref9)) chapticker2 += 1 - str14 = ''.join(fixChapter(entire14, references14, paras, addendum)) + str14 = ''.join(fixChapter(entire14, references14, paras, addendum,ref14II,ref14III,specref14)) """ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ From cde28cdda75affcb17e477889998bd11f508fd6f Mon Sep 17 00:00:00 2001 From: Edward Bian Date: Tue, 5 Jul 2016 13:29:54 -0400 Subject: [PATCH 230/402] Fixes for Hypergeometric Representation and non working other functions --- KLSadd_insertion/src/updateChapters.py | 380 +++++++++++++++++++++++-- 1 file changed, 355 insertions(+), 25 deletions(-) diff --git a/KLSadd_insertion/src/updateChapters.py b/KLSadd_insertion/src/updateChapters.py index 6867c00..dfa760b 100644 --- a/KLSadd_insertion/src/updateChapters.py +++ b/KLSadd_insertion/src/updateChapters.py @@ -30,19 +30,220 @@ #mathPeople is just the name for the sections that are used, like Wilson, Racah, etc. I know some are not people, its just a var name chapNums = [] paras = [] -klsparas = [] + +#klsaddparas contains the locations of everything with "paragraph{" in KLSadd, these are the subsections that need to be sorted +klsaddparas = [] mathPeople = [] newCommands = [] #used to hold the indexes of the commands + +#ref9II holds the beginnings and ends of each chapter 9 section, provided and area to search ref9II = [] #Hold section search indexes + +#ref14II holds the beginnings and ends of each chapter 14 section, provided and area to search ref14II = [] + +#ref9III holds all subsections in each chapter 9 section along with the what ref9II holds ref9III = [] #Holds all indexes + +#ref14III holds all subsections in each chapter 14 section along with the what ref14II holds ref14III = [] + +#specref9 is the KLS indexes that go in chapter 9 specref9 = [] + +#specref14 is the KLS indexes that in chapter 14 specref14 = [] + +sectorschap = [] +sectorskls = [] +insertindices = [] +insertloc = [] +irrelevant = ["Relations","Functions","Transform","In","For","To","Is","And","Of","relations","functions","transform","in","for","to","is","and","of"] + +check = [] +check2 = [] + comms = "" #holds the ACTUAL STRINGS of the commands #2/18/16 this method addresses the goal of hardcoding in the necessary packages to let the chapter files run as pdf's. #Currently only works with chapter 9, ask Dr. Cohl to help port your chapter 14 output file into a pdf +#5/5/16 this method implements the Hypergeometric paragraphs into relevant sections for practicality +# Later versions should attempt to: fix paragraphs with variations on this name like +# Basic Hypergeometric Representation, q-Hypergeometric Representation + +def HyperGeoFix(kls, chap): + canAdd = False + # keep track of sections and subsections from chap + hyperHeaders = [] + hyperSubs = [] + index = 0 + + #using Edward's reference paragraphs! + #now get the section names of all of the KLSadd + kHyperHeader = [] + kHyperSub = [] + index = 0 + for i in kls: + index +=1 + line = str(i) + if "\\subsection" in line: + temp = line[line.find(" ", 12)+1: line.find("}", 12)] #get just the name (like mathpeople) + if "hypergeometric representation" in line.lower(): + for i in klsaddparas: + if index < i: + klsloc = klsaddparas.index(i) + break + t = ''.join(kls[index: klsaddparas[klsloc]]) + kHyperSub.append(t)#append the whole paragraph, pray every paragraph ends with a % comment + kHyperHeader.append(temp)#append the name of subsection + kHyperHeader[8] = "Discrete $q$-Hermite~II" + print(kHyperHeader) + print(len(kHyperHeader)) + print(kHyperSub) + print(len(kHyperSub)) + print(klsaddparas) + kHypIndex = 0 + offset = 0 + if len(check) == 0: + tempref = ref9III + global check + print(len(check)) + print(check) + check.append("honhonhonbaguetteouioui") + else: + tempref = ref14III + offset = 4 + print("baguette") + print(tempref) + for d in range(0, len(tempref)): # check every section and subsection line + i = tempref[d] + line = str(chap[i]) + if "\\section{" in line or "\subsection{" in line: + if "\\subsection{" in line: + temp = line[12:line.find("}", 7)] + else: + temp = line[9:line.find("}", 7)] + if "hypergeometric representation" in line.lower(): + hyperSubs.append([tempref[d + 1]]) # appends the index for the line before following subsection + hyperHeaders.append(temp) # appends the name of the section the hypergeo subsection is in so we can compare + + if temp in kHyperHeader: + try: + chap[tempref[d + 1] - 2] += "\paragraph{\\bf KLS Addendum: }" + chap[tempref[d + 1] - 2] += kHyperSub[kHypIndex + offset] + print(kHyperSub[kHypIndex + offset]) + #print(chap[tempref[d + 1] - 2]) + kHypIndex+=1 + print(kHypIndex) + except IndexError: + print("QED") + +def SpecialValue(kls, chap): + canAdd = False + # keep track of sections and subsections from chap + svhyperHeaders = [] + svhyperSubs = [] + index = 0 + + # using Edward's reference paragraphs! + # now get the section names of all of the KLSadd + svkHyperHeader = [] + svkHyperSub = [] + index = 0 + for i in kls: + index +=1 + line = str(i) + if "\\subsection" in line: + temp = line[line.find(" ", 12)+1: line.find("}", 12)] #get just the name (like mathpeople) + if "special value" in line.lower(): + for i in klsaddparas: + if index < i: + svklsloc = klsaddparas.index(i) + break + t = ''.join(kls[index: klsaddparas[svklsloc]]) + svkHyperSub.append(t) # append the whole paragraph, pray every paragraph ends with a % comment + svkHyperHeader.append(temp) # append the name of subsection + print("svk") + print(svkHyperHeader) + svkHypIndex = 0 + offset = 0 + if len(check2) == 0: + tempref = ref9III + check2.append("test") + else: + tempref = ref14III + offset = 6 + for d in range(0, len(tempref)): # check every section and subsection line + i = tempref[d] + line = str(chap[i]) + if "\\section{" in line or "\\subsection{" in line: + if "\\subsection{" in line: + temp = line[12:line.find("}", 7)] + else: + temp = line[9:line.find("}", 7)] + if "hypergeometric representation" in line.lower(): + svhyperSubs.append([tempref[d + 1]]) # appends the index for the line before following subsection + svhyperHeaders.append(temp) # appends the name of the section the hypergeo subsection is in so we can compare + if temp in svkHyperHeader: + chap[tempref[d + 1] - 2] += "\paragraph{\\bf KLS Addendum Special Value: }" + chap[tempref[d + 1] - 2] += svkHyperSub[svkHypIndex + offset] + svkHypIndex += 1 + +def LimRelFix(kls, chap): + canAdd = False + # keep track of sections and subsections from chap + hyperHeaders = [] + hyperSubs = [] + index = 0 + + #using Edward's reference paragraphs! + #now get the section names of all of the KLSadd + limrelHyperHeader = [] + limrelHyperSub = [] + index = 0 + for i in kls: + index +=1 + line = str(i) + if "\\subsection" in line: + temp = line[line.find(" ", 12)+1: line.find("}", 12)] #get just the name (like mathpeople) + if "limit relations" in line.lower(): + for i in klsaddparas: + if index < i: + klsloc = klsaddparas.index(i) + break + t = ''.join(kls[index: klsaddparas[klsloc]]) + limrelHyperSub.append(t)#append the whole paragraph, pray every paragraph ends with a % comment + limrelHyperHeader.append(temp)#append the name of subsection + + limrelHypIndex = 0 + offset = 0 + if len(check) == 0: + tempref = ref9III + check.append("test") + else: + tempref = ref14III + offset = 2 + for d in range(0, len(tempref)): # check every section and subsection line + i = tempref[d] + line = str(chap[i]) + if "\\section{" in line or "\\subsection{" in line: + if "\\subsection{" in line: + temp = line[12:line.find("}", 7)] + else: + temp = line[9:line.find("}", 7)] + #print(temp) + if "limit relations" in line.lower(): + hyperSubs.append([tempref[d + 1]]) # appends the index for the line before following subsection + hyperHeaders.append(temp) # appends the name of the section the hypergeo subsection is in so we can compare + + if temp in limrelHyperHeader: + #print(temp) + chap[tempref[d + 1] - 2] += "\paragraph{\\bf KLS Addendum Limit Relations: }" + chap[tempref[d + 1] - 2] += limrelHyperSub[limrelHypIndex + offset] + limrelHypIndex+=1 + #print(limrelHyperSub) + #print(limrelHyperHeader) + def prepareForPDF(chap): footmiscIndex = 0 index = 0 @@ -99,6 +300,7 @@ def findReferences(chapter): canAdd = False for word in chapter: index+=1 + specialdetector = 1 #check sections and subsections if("section{" in word or "subsection*{" in word) and ("subsubsection*{" not in word): w = word[word.find("{")+1: word.find("}")] @@ -106,7 +308,7 @@ def findReferences(chapter): for unit in mathPeople: subunit = unit[unit.find(" ")+1: unit.find("#")] # System of checks that verifies if section is in chapter - if ((w in subunit) or (ws in subunit)) and (chaptercheck in unit) and (len(w) == len(subunit)) or (("Pseudo Jacobi" in w) and ("Pseudo Jacobi (or Routh-Romanovski)" in subunit)): + if ((w in subunit) or (ws in subunit)) and (chaptercheck in unit) and (len(w) == len(subunit)) or (("Pseudo Jacobi" in w) and ("Pseudo Jacobi (or Routh-Romanovski)" in subunit)) and ("Hypergeometric representation" not in w) and ("Special value" not in w): canAdd = True if chapticker == 0: ref9II.append(index) @@ -114,6 +316,7 @@ def findReferences(chapter): elif chapticker == 1: ref14II.append(index) ref14III.append(index) + specialdetector = 0 if("\\subsection*{References}" in word) and (canAdd == True): # Appends valid locations references.append(index) @@ -129,13 +332,28 @@ def findReferences(chapter): ref9III.append(index) elif chapticker == 1: ref14III.append(index) - print(ref9II) - print (ref14II) - print(ref9III) - print(ref14III) + if "\\section{" in word and specialdetector == 1 and chaptercheck == "14": + w2 = word[word.find("{") + 1: word.find("}")] + if "Bessel" not in w2: + ref14III.append(index) + besselcheck = 1 + bqlegendrecheck = 2 + for i in ref9III: + if i == 2646: + besselcheck = 0 + if besselcheck == 1: + ref9III.insert(230, 2646) + for i in ref14III: + if i == 1217: + bqlegendrecheck -= 1 + if bqlegendrecheck > 0: + ref14III.insert(87, 1217) + ref14III.append(3053) + ref14III.append(2554) + return references -def referencePlacer(chap, references, p, kls,refII,refIII,specref): +def referencePlacer(chap, references, p, kls,refII,refIII,specref,klsaddparas): # count is used to represent the values in count count = 0 # Tells which chapter it's on @@ -163,25 +381,127 @@ def referencePlacer(chap, references, p, kls,refII,refIII,specref): else: count += 1 -def referencePlacerII(chap, references, p, kls,refII,refIII,specref): +def referencePlacerII(chap, references, p, kls,refII,refIII,specref,klsaddparas,sectorskls,sectorschap,insertindices,insertloc): if chapticker2 == 0: designator = "9." elif chapticker2 == 1: designator = "14." count = 0 - sectionnuma = specref[0] - sectionnumb = specref[1] - for i in specref: - pass + count2 = 0 + count3 = 0 + secchaploc = 0 + secklsloc = 0 + sectionnuma = specref[count] + sectionnumb = specref[count+1] + sectionnumc = refII[count2] + sectionnumd = refII[count2 + 1] + while count < len(specref)-1: + for word in klsaddparas: + if word <> "@": + if word > sectionnuma and word < sectionnumb: + lenwordII = len(kls[word])-1 + #That 29 is where the KLS formatting ends + if "subsub" in kls[word]: + temp = kls[word][34: lenwordII] + else: + temp = kls[word][29 : lenwordII] + #with open("keywords.tex", "a") as log: + # log.write(temp + " Location: " + str(word) + "\n") + #log.close() + #print(temp) + #print(word) + sectorskls = temp.split(' ') + for i in sectorskls: + for a in irrelevant: + if i == a: + sectorskls[sectorskls.index(i)] = "@@" + i == "@@" + #print(sectorskls) + while count2 < len(refII) - 1: + for word2 in refIII: + if word2 > sectionnumc and word2 < sectionnumd and word2 <> "@": + lenwordIII = len(chap[word2])-3 + # That 13 is where the KLS formatting ends + if "subsub" in chap[word2]: + temp2 = chap[word2][16: lenwordIII] + else: + temp2 = chap[word2][13: lenwordIII] + #if designator == "14.": + # with open("keywords.tex", "a") as log: + # log.write(temp2 + " Location: " + str(word2) + "\n") + # log.close() + sectorschap = temp2.split(' ') + for i in sectorschap: + for a in irrelevant: + if i == a: + i == "@@" + #print("temp2") + #print(temp2) + #print(word2) + ticker1 = 1 + for i in sectorskls: + for j in sectorschap: + if (i in j) or (j in i): + if ticker1 == 1: + try: + locationdata = klsaddparas[klsaddparas.index(word)],"+" + moredata = str(klsaddparas[klsaddparas.index(word)]) + insertindices.append(locationdata) + insertloc.append(refIII[refIII.index(word2)+1]) + evenmoredata = str(refIII[refIII.index(word2)+1]) + with open("keywordsII.tex", "a") as log: + log.write("Pair "+"\n") + log.write(moredata + "\n") + log.write(evenmoredata+"\n") + log.close() + klsaddparas[klsaddparas.index(word)] = "@" + refIII[refIII.index(word2)] = "@" + ticker1 = 0 + except ValueError: + pass + count2 += 1 + sectionnumc = refII[count2] + if count2 + 1 == len(refII): + sectionnumd = refII[count2] + else: + sectionnumd = refII[count2 + 1] + count2 = 0 + count += 1 + sectionnuma = specref[count] + if count + 1 == len(specref): + sectionnumb = specref[count] + else: + sectionnumb = specref[count + 1] + #print(insertindices) + #print(insertloc) + #print(klsaddparas) + #print(refIII) #method to change file string(actually a list right now), returns string to be written to file #If you write a method that changes something, it is preffered that you call the method in here -def fixChapter(chap, references, p, kls,refII,refIII,specref): +def fixChapter(chap, references, p, kls,refII,refIII,specref,klsaddparas,sectorskls,sectorschap,insertindices,insertloc): #chap is the file string(actually a list), references is the specific references for the file, #and p is the paras variable(not sure if needed) kls is the KLSadd.tex as a list - referencePlacer(chap, references, p, kls, refII,refIII,specref) + chapticker3 = 0 + + HyperGeoFix(kls, chap) + + global check + print(len(check)) + #check = [] + + SpecialValue(kls,chap) + + #global check + #print(len(check)) + #check = [] + + #LimRelFix(kls, chap) + + referencePlacer(chap, references, p, kls, refII,refIII,specref,klsaddparas) + #referencePlacerII(chap, references, p, kls, refII, refIII, specref, klsaddparas,sectorskls,sectorschap,insertindices,insertloc) chap = prepareForPDF(chap) cms = getCommands(kls) chap = insertCommands(kls,chap, cms) @@ -212,7 +532,7 @@ def fixChapter(chap, references, p, kls,refII,refIII,specref): return chap #open the KLSadd file to do things with -with open("KLSadd.tex", "r") as add: +with open("KLSadd(3).tex", "r") as add: #store the file as a string addendum = add.readlines() #Makes sections look like other sections @@ -240,15 +560,20 @@ def fixChapter(chap, references, p, kls,refII,refIII,specref): chapNums.append(14) name = word[word.find("{") + 1: word.find("}") ] mathPeople.append(name + "#") + if len(specref14) == 0: + specref9.append(index - 1) specref14.append(index - 1) indexes.append(index-1) if ("paragraph{" in word) and (index > 313): - klsparas.append(index-1) - print(indexes) - print(specref9) - print(specref14) - print(klsparas) - print(mathPeople) + klsaddparas.append(index-1) + if ("subsub" in word) and (index > 313): + klsaddparas.append(index - 1) + klsaddparas.append(2246) + #print(indexes) + #print(specref9) + #print(specref14) + #print(klsaddparas) + #print(mathPeople) #now indexes holds all of the places there is a section #using these indexes, get all of the words in between and add that to the paras[] for i in range(len(indexes)-1): @@ -276,14 +601,19 @@ def fixChapter(chap, references, p, kls,refII,refIII,specref): #two variables for the references lists one for chapter 9 one for chapter 14 chapticker = 0 references9 = findReferences(entire9) + print(ref14III) + print("memes") chapticker += 1 references14 = findReferences(entire14) + print(ref14III) + print("memes") #call the fixChapter method to get a list with the addendum paragraphs added in chapticker2 = 0 - str9 = ''.join(fixChapter(entire9, references9, paras, addendum,ref9II,ref9III,specref9)) + str9 = ''.join(fixChapter(entire9, references9, paras, addendum,ref9II,ref9III,specref9,klsaddparas,sectorskls,sectorschap,insertindices,insertloc)) chapticker2 += 1 - str14 = ''.join(fixChapter(entire14, references14, paras, addendum,ref14II,ref14III,specref14)) - + str14 = ''.join(fixChapter(entire14, references14, paras, addendum,ref14II,ref14III,specref14,klsaddparas,sectorskls,sectorschap,insertindices,insertloc)) + #print(insertindices) + #print(insertloc) """ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ If you are writing something that will make a change to the chapter files, write it BEFORE this line, this part @@ -298,4 +628,4 @@ def fixChapter(chap, references, p, kls,refII,refIII,specref): temp9.write(str9) with open("updated14.tex", "w") as temp14: - temp14.write(str14) \ No newline at end of file + temp14.write(str14) From eeb7e59bf0805dcf5d4bd9f593f8a6ced8c60e88 Mon Sep 17 00:00:00 2001 From: Edward Bian Date: Tue, 5 Jul 2016 16:25:33 -0400 Subject: [PATCH 231/402] Special Value and HyperGeo for 9 and 14 fixed --- KLSadd_insertion/src/updateChapters.py | 73 ++++---------------------- 1 file changed, 11 insertions(+), 62 deletions(-) diff --git a/KLSadd_insertion/src/updateChapters.py b/KLSadd_insertion/src/updateChapters.py index dfa760b..f17e62f 100644 --- a/KLSadd_insertion/src/updateChapters.py +++ b/KLSadd_insertion/src/updateChapters.py @@ -109,7 +109,7 @@ def HyperGeoFix(kls, chap): global check print(len(check)) print(check) - check.append("honhonhonbaguetteouioui") + check.append("Words") else: tempref = ref14III offset = 4 @@ -129,7 +129,7 @@ def HyperGeoFix(kls, chap): if temp in kHyperHeader: try: - chap[tempref[d + 1] - 2] += "\paragraph{\\bf KLS Addendum: }" + chap[tempref[d + 1] - 2] += "\paragraph{\\bf KLS Addendum: Hypergeometric Representation}" chap[tempref[d + 1] - 2] += kHyperSub[kHypIndex + offset] print(kHyperSub[kHypIndex + offset]) #print(chap[tempref[d + 1] - 2]) @@ -172,7 +172,7 @@ def SpecialValue(kls, chap): check2.append("test") else: tempref = ref14III - offset = 6 + offset = 8 for d in range(0, len(tempref)): # check every section and subsection line i = tempref[d] line = str(chap[i]) @@ -185,65 +185,10 @@ def SpecialValue(kls, chap): svhyperSubs.append([tempref[d + 1]]) # appends the index for the line before following subsection svhyperHeaders.append(temp) # appends the name of the section the hypergeo subsection is in so we can compare if temp in svkHyperHeader: - chap[tempref[d + 1] - 2] += "\paragraph{\\bf KLS Addendum Special Value: }" + chap[tempref[d + 1] - 2] += "\paragraph{\\bf KLS Addendum: Special Value}" chap[tempref[d + 1] - 2] += svkHyperSub[svkHypIndex + offset] svkHypIndex += 1 -def LimRelFix(kls, chap): - canAdd = False - # keep track of sections and subsections from chap - hyperHeaders = [] - hyperSubs = [] - index = 0 - - #using Edward's reference paragraphs! - #now get the section names of all of the KLSadd - limrelHyperHeader = [] - limrelHyperSub = [] - index = 0 - for i in kls: - index +=1 - line = str(i) - if "\\subsection" in line: - temp = line[line.find(" ", 12)+1: line.find("}", 12)] #get just the name (like mathpeople) - if "limit relations" in line.lower(): - for i in klsaddparas: - if index < i: - klsloc = klsaddparas.index(i) - break - t = ''.join(kls[index: klsaddparas[klsloc]]) - limrelHyperSub.append(t)#append the whole paragraph, pray every paragraph ends with a % comment - limrelHyperHeader.append(temp)#append the name of subsection - - limrelHypIndex = 0 - offset = 0 - if len(check) == 0: - tempref = ref9III - check.append("test") - else: - tempref = ref14III - offset = 2 - for d in range(0, len(tempref)): # check every section and subsection line - i = tempref[d] - line = str(chap[i]) - if "\\section{" in line or "\\subsection{" in line: - if "\\subsection{" in line: - temp = line[12:line.find("}", 7)] - else: - temp = line[9:line.find("}", 7)] - #print(temp) - if "limit relations" in line.lower(): - hyperSubs.append([tempref[d + 1]]) # appends the index for the line before following subsection - hyperHeaders.append(temp) # appends the name of the section the hypergeo subsection is in so we can compare - - if temp in limrelHyperHeader: - #print(temp) - chap[tempref[d + 1] - 2] += "\paragraph{\\bf KLS Addendum Limit Relations: }" - chap[tempref[d + 1] - 2] += limrelHyperSub[limrelHypIndex + offset] - limrelHypIndex+=1 - #print(limrelHyperSub) - #print(limrelHyperHeader) - def prepareForPDF(chap): footmiscIndex = 0 index = 0 @@ -347,9 +292,7 @@ def findReferences(chapter): if i == 1217: bqlegendrecheck -= 1 if bqlegendrecheck > 0: - ref14III.insert(87, 1217) - ref14III.append(3053) - ref14III.append(2554) + pass return references @@ -605,6 +548,9 @@ def fixChapter(chap, references, p, kls,refII,refIII,specref,klsaddparas,sectors print("memes") chapticker += 1 references14 = findReferences(entire14) + ref14III.insert(88, 1217) + ref14III.insert(244, 3053) + ref14III.insert(198, 2554) print(ref14III) print("memes") #call the fixChapter method to get a list with the addendum paragraphs added in @@ -629,3 +575,6 @@ def fixChapter(chap, references, p, kls,refII,refIII,specref,klsaddparas,sectors with open("updated14.tex", "w") as temp14: temp14.write(str14) +print(ref14III) +print(ref14III[87]) + From 1861c633d198c755f9ee3fd1c8d0f76188e158d1 Mon Sep 17 00:00:00 2001 From: Edward Bian Date: Wed, 6 Jul 2016 16:23:34 -0400 Subject: [PATCH 232/402] Fix for Limit Relations --- KLSadd_insertion/src/updateChapters.py | 95 +++++++++++++++++++------- 1 file changed, 70 insertions(+), 25 deletions(-) diff --git a/KLSadd_insertion/src/updateChapters.py b/KLSadd_insertion/src/updateChapters.py index f17e62f..c8775ec 100644 --- a/KLSadd_insertion/src/updateChapters.py +++ b/KLSadd_insertion/src/updateChapters.py @@ -62,6 +62,7 @@ check = [] check2 = [] +check3 = [] comms = "" #holds the ACTUAL STRINGS of the commands #2/18/16 this method addresses the goal of hardcoding in the necessary packages to let the chapter files run as pdf's. @@ -97,24 +98,14 @@ def HyperGeoFix(kls, chap): kHyperSub.append(t)#append the whole paragraph, pray every paragraph ends with a % comment kHyperHeader.append(temp)#append the name of subsection kHyperHeader[8] = "Discrete $q$-Hermite~II" - print(kHyperHeader) - print(len(kHyperHeader)) - print(kHyperSub) - print(len(kHyperSub)) - print(klsaddparas) kHypIndex = 0 offset = 0 if len(check) == 0: tempref = ref9III - global check - print(len(check)) - print(check) check.append("Words") else: tempref = ref14III offset = 4 - print("baguette") - print(tempref) for d in range(0, len(tempref)): # check every section and subsection line i = tempref[d] line = str(chap[i]) @@ -131,13 +122,15 @@ def HyperGeoFix(kls, chap): try: chap[tempref[d + 1] - 2] += "\paragraph{\\bf KLS Addendum: Hypergeometric Representation}" chap[tempref[d + 1] - 2] += kHyperSub[kHypIndex + offset] - print(kHyperSub[kHypIndex + offset]) #print(chap[tempref[d + 1] - 2]) kHypIndex+=1 - print(kHypIndex) except IndexError: - print("QED") + print("Warning! Code has encountered some sort of error involving section identification for 'Hypergeometric Representation'. Problems may occur.") + #print(kHyperHeader) + #print(hyperHeaders) + #print(kHyperSub) + #print(hyperSubs) def SpecialValue(kls, chap): canAdd = False # keep track of sections and subsections from chap @@ -163,8 +156,6 @@ def SpecialValue(kls, chap): t = ''.join(kls[index: klsaddparas[svklsloc]]) svkHyperSub.append(t) # append the whole paragraph, pray every paragraph ends with a % comment svkHyperHeader.append(temp) # append the name of subsection - print("svk") - print(svkHyperHeader) svkHypIndex = 0 offset = 0 if len(check2) == 0: @@ -183,12 +174,73 @@ def SpecialValue(kls, chap): temp = line[9:line.find("}", 7)] if "hypergeometric representation" in line.lower(): svhyperSubs.append([tempref[d + 1]]) # appends the index for the line before following subsection - svhyperHeaders.append(temp) # appends the name of the section the hypergeo subsection is in so we can compare + svhyperHeaders.append(temp) #appends the name of the section the hypergeo subsection is in so we can compare if temp in svkHyperHeader: chap[tempref[d + 1] - 2] += "\paragraph{\\bf KLS Addendum: Special Value}" chap[tempref[d + 1] - 2] += svkHyperSub[svkHypIndex + offset] svkHypIndex += 1 +def LimRelFix(kls, chap): + canAdd = False + lrhyperHeaders = [] + lrhyperSubs = [] + index = 0 + + lrkHyperHeader = [] + lrkHyperSub = [] + index = 0 + for i in kls: + index +=1 + line = str(i) + if "\\subsection" in line: + temp = line[line.find(" ", 12)+1: line.find("}", 12)] + if "limit relation" in line.lower(): + for i in klsaddparas: + if index < i: + klsloc = klsaddparas.index(i) + break + t = ''.join(kls[index: klsaddparas[klsloc]]) + lrkHyperSub.append(t)#append the whole paragraph, pray every paragraph ends with a % comment + lrkHyperHeader.append(temp)#append the name of subsection + print(lrkHyperHeader) + lrkHyperHeader[0] = "Pseudo Jacobi" + print(lrkHyperHeader) + print(lrkHyperSub) + lrkHypIndex = 0 + offset = 0 + if len(check3) == 0: + tempref = ref9III + check3.append("Words") + else: + tempref = ref14III + offset = 1 + print("baguetteII") + print(tempref) + for d in range(0, len(tempref)): # check every section and subsection line + i = tempref[d] + line = str(chap[i]) + if "\\section{" in line or "\subsection{" in line: + if "\\subsection{" in line: + temp = line[12:line.find("}", 7)] + else: + temp = line[9:line.find("}", 7)] + if "limit relation" in line.lower(): + lrhyperSubs.append([tempref[d + 1]]) # appends the index for the line before following subsection + lrhyperHeaders.append(temp) # appends the name of the section the hypergeo subsection is in so we can compare + + if temp in lrkHyperHeader: + try: + chap[tempref[d + 1] - 2] += "\paragraph{\\bf KLS Addendum: Limit Relation(s)}" + chap[tempref[d + 1] - 2] += lrkHyperSub[lrkHypIndex + offset] + print(lrkHyperSub[lrkHypIndex + offset]) + #print(chap[tempref[d + 1] - 2]) + lrkHypIndex+=1 + print(lrkHypIndex) + except IndexError: + print("Warning! Code has encountered some sort of error involving section identification for 'Limit Relations'. Problems may occur.") + print(lrhyperHeaders) + print(lrhyperSubs) + def prepareForPDF(chap): footmiscIndex = 0 index = 0 @@ -441,7 +493,7 @@ def fixChapter(chap, references, p, kls,refII,refIII,specref,klsaddparas,sectors #print(len(check)) #check = [] - #LimRelFix(kls, chap) + LimRelFix(kls, chap) referencePlacer(chap, references, p, kls, refII,refIII,specref,klsaddparas) #referencePlacerII(chap, references, p, kls, refII, refIII, specref, klsaddparas,sectorskls,sectorschap,insertindices,insertloc) @@ -544,22 +596,17 @@ def fixChapter(chap, references, p, kls,refII,refIII,specref,klsaddparas,sectors #two variables for the references lists one for chapter 9 one for chapter 14 chapticker = 0 references9 = findReferences(entire9) - print(ref14III) - print("memes") chapticker += 1 references14 = findReferences(entire14) ref14III.insert(88, 1217) ref14III.insert(244, 3053) ref14III.insert(198, 2554) - print(ref14III) - print("memes") #call the fixChapter method to get a list with the addendum paragraphs added in chapticker2 = 0 str9 = ''.join(fixChapter(entire9, references9, paras, addendum,ref9II,ref9III,specref9,klsaddparas,sectorskls,sectorschap,insertindices,insertloc)) chapticker2 += 1 str14 = ''.join(fixChapter(entire14, references14, paras, addendum,ref14II,ref14III,specref14,klsaddparas,sectorskls,sectorschap,insertindices,insertloc)) - #print(insertindices) - #print(insertloc) + """ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ If you are writing something that will make a change to the chapter files, write it BEFORE this line, this part @@ -575,6 +622,4 @@ def fixChapter(chap, references, p, kls,refII,refIII,specref,klsaddparas,sectors with open("updated14.tex", "w") as temp14: temp14.write(str14) -print(ref14III) -print(ref14III[87]) From ba5956f6c2c2a7c0614dd592d0b4705dbab9bec2 Mon Sep 17 00:00:00 2001 From: Edward Bian Date: Tue, 2 Aug 2016 11:35:07 -0400 Subject: [PATCH 233/402] General Chapter Sorter --- KLSadd_insertion/src/updateChapters.py | 328 ++++++++++++++++--------- 1 file changed, 213 insertions(+), 115 deletions(-) diff --git a/KLSadd_insertion/src/updateChapters.py b/KLSadd_insertion/src/updateChapters.py index c8775ec..1c397ea 100644 --- a/KLSadd_insertion/src/updateChapters.py +++ b/KLSadd_insertion/src/updateChapters.py @@ -63,6 +63,14 @@ check = [] check2 = [] check3 = [] +check4 = [] +klslist = [] +klslistII = [] +klslistIII = [] + +w, h = 2, 9999 +sorterIIIcheck = [[0 for x in range(w)] for y in range(h)] +print(sorterIIIcheck) comms = "" #holds the ACTUAL STRINGS of the commands #2/18/16 this method addresses the goal of hardcoding in the necessary packages to let the chapter files run as pdf's. @@ -71,6 +79,184 @@ #5/5/16 this method implements the Hypergeometric paragraphs into relevant sections for practicality # Later versions should attempt to: fix paragraphs with variations on this name like # Basic Hypergeometric Representation, q-Hypergeometric Representation +def NewKeywords(kls): + for i in kls: + if kls.index(i) > 313: + if ("paragraph{") in i: + lenword = len(i) - 1 + temp = i[i.find("{") + 1: lenword-1] + klslist.append(temp) + klslistII.append(temp+'\n') + if ("subsubsection*{") in i: + lenword = len(i) - 1 + temp = i[i.find("{") + 1: lenword - 1] + klslist.append(temp) + klslistII.append(temp + '\n') + print(klslist) + i = 0 + while i < len(klslist): + if ("reference" in klslist[i].lower()) or ("hypergeometric representation" in klslist[i].lower()) or ("limit relation" in klslist[i].lower()) or ("special value" in klslist[i].lower()): + del klslist[i] + else: + i += 1 + liststr = ''.join(klslistII) + with open("keywordsIII.tex", "w") as kw: + kw.write(liststr) + for i in klslist: + add = True + for j in klslistIII: + if i == j: + add = False + if add == True: + klslistIII.append(i) + print(klslist) + liststr2 = '\n'.join(klslistIII) + with open("trunckeywords.tex", "w") as kw: + kw.write(liststr2) + print(klslistIII) + print(klslistIII.index("Orthogonality")) + #del klslistIII[35] + print(klslistIII) +""" +--------------------------------------------------------- +""" +def FixChapterIII(kls,chap,word,sortloc): + canAdd = False + # keep track of sections and subsections from chap + hyperHeadersIII = [] + hyperSubsIII = [] + index = 0 + sep1 = 0 + nameIII = word.lower() + if nameIII == "orthogonality": + sep1 = 1 + #print(nameIII) + #using Edward's reference paragraphs! + #now get the section names of all of the KLSadd + kHyperHeaderIII = [] + kHyperSubIII = [] + index = 0 + for i in kls: + index +=1 + line = str(i) + if "\\subsection" in line: + temp = line[line.find(" ", 12)+1: line.find("}", 12)] #get just the name (like mathpeople) + if sep1 == 0: + if nameIII in line.lower() and kls.index(i) > 313 and "\\paragraph{" in line: + #print(i) + #print(kls.index(i)) + for i in klsaddparas: + if index < i: + klsloc = klsaddparas.index(i) + break + t = ''.join(kls[index: klsaddparas[klsloc]]) + kHyperSubIII.append(t)#append the whole paragraph, pray every paragraph ends with a % comment + kHyperHeaderIII.append(temp)#append the name of subsection + else: + if nameIII in line.lower() and kls.index(i) > 313 and "\\paragraph{" in line and "orthogonality relation" not in line.lower(): + # print(i) + # print(kls.index(i)) + for i in klsaddparas: + if index < i: + klsloc = klsaddparas.index(i) + break + t = ''.join(kls[index: klsaddparas[klsloc]]) + kHyperSubIII.append(t) # append the whole paragraph, pray every paragraph ends with a % comment + kHyperHeaderIII.append(temp) # append the name of subsection + kHypIndexIII = 0 + offset = 0 + + i = 0 + while i < len(kHyperHeaderIII): + try: + if kHyperHeaderIII[i] == kHyperHeaderIII[i+1]: + kHyperSubIII[i+1] = kHyperSubIII[i] + "\paragraph{\\bf KLS Addendum: " + word + "}\n" + kHyperSubIII[i+1] + kHyperHeaderIII[i] = "memes" + kHyperSubIII[i] = "memes" + else: + i += 1 + except IndexError: + i += 1 + a = 0 + #print(kHyperHeaderIII) + #print(kHyperSubIII) + while a < len(kHyperHeaderIII): + if kHyperHeaderIII[a] == "memes": + del kHyperHeaderIII[a] + del kHyperSubIII[a] + else: + a += 1 + #print(kHyperSubIII) + #print(len(kHyperSubIII)) + #print(kHyperHeaderIII) + #print(len(kHyperHeaderIII)) + global sorterIIIcheck + #print(sortloc) + chap9 = 0 + if sorterIIIcheck[sortloc][0] == 0: + tempref = ref9III + sorterIIIcheck[sortloc][0] += 1 + chap9 = 1 + else: + tempref = ref14III + offset = sorterIIIcheck[sortloc][1] + print("offset") + print(offset) + if sep1 == 1: + offset = 8 + for d in range(0, len(tempref)): # check every section and subsection line + i = tempref[d] + line = str(chap[i]) + if "\\section{" in line or "\\subsection{" in line: + if "\\subsection{" in line: + temp = line[12:line.find("}", 7)] + else: + temp = line[9:line.find("}", 7)] + if sep1 == 0: + if nameIII in line.lower(): + hyperSubsIII.append([tempref[d + 1]]) # appends the index for the line before following subsection + hyperHeadersIII.append(temp) # appends the name of the section the hypergeo subsection is in so we can compare + + if temp in kHyperHeaderIII: + try: + chap[tempref[d + 1] - 1] += "\paragraph{\\bf KLS Addendum: " + word + "}" + chap[tempref[d + 1] - 1] += kHyperSubIII[kHypIndexIII + offset] + if chap9 == 1: + sorterIIIcheck[sortloc][1] += 1 + #print(chap[tempref[d + 1] - 2]) + kHypIndexIII+=1 + except IndexError: + print("Warning! Code has encountered some sort of error involving section identification for '" + nameIII + "'. Problems may occur.") + print("If you see this message, bad things are happening.") + else: + if nameIII in line.lower() and "orthogonality relation" not in line: + hyperSubsIII.append([tempref[d + 1]]) # appends the index for the line before following subsection + hyperHeadersIII.append( + temp) # appends the name of the section the hypergeo subsection is in so we can compare + + if temp in kHyperHeaderIII: + try: + chap[tempref[d + 1] - 1] += "\paragraph{\\bf KLS Addendum: " + word + "}" + chap[tempref[d + 1] - 1] += kHyperSubIII[kHypIndexIII + offset] + # print(chap[tempref[d + 1] - 2]) + kHypIndexIII += 1 + except IndexError: + print("Warning! Code has encountered some sort of error involving section identification for '" + nameIII + "'. Problems may occur.") + print("If you see this message, bad things are happening.") + + if len(hyperHeadersIII) != 0 and len(hyperSubsIII) != 0: + print("Chapter Values:") + print(word) + print(kHyperHeaderIII) + print(hyperHeadersIII) + print(len(hyperHeadersIII)) + #print(kHyperSubIII) + print(kHyperSubIII) + print(hyperSubsIII) + print(len)(hyperSubsIII) + print("End") + print("sortloc") + print(sorterIIIcheck[sortloc][1]) def HyperGeoFix(kls, chap): canAdd = False @@ -126,6 +312,7 @@ def HyperGeoFix(kls, chap): kHypIndex+=1 except IndexError: print("Warning! Code has encountered some sort of error involving section identification for 'Hypergeometric Representation'. Problems may occur.") + print("If you see this message, bad things are happening.") #print(kHyperHeader) #print(hyperHeaders) @@ -156,6 +343,8 @@ def SpecialValue(kls, chap): t = ''.join(kls[index: klsaddparas[svklsloc]]) svkHyperSub.append(t) # append the whole paragraph, pray every paragraph ends with a % comment svkHyperHeader.append(temp) # append the name of subsection + print(svkHyperHeader) + print(svkHyperSub) svkHypIndex = 0 offset = 0 if len(check2) == 0: @@ -202,10 +391,7 @@ def LimRelFix(kls, chap): t = ''.join(kls[index: klsaddparas[klsloc]]) lrkHyperSub.append(t)#append the whole paragraph, pray every paragraph ends with a % comment lrkHyperHeader.append(temp)#append the name of subsection - print(lrkHyperHeader) lrkHyperHeader[0] = "Pseudo Jacobi" - print(lrkHyperHeader) - print(lrkHyperSub) lrkHypIndex = 0 offset = 0 if len(check3) == 0: @@ -214,8 +400,6 @@ def LimRelFix(kls, chap): else: tempref = ref14III offset = 1 - print("baguetteII") - print(tempref) for d in range(0, len(tempref)): # check every section and subsection line i = tempref[d] line = str(chap[i]) @@ -232,14 +416,10 @@ def LimRelFix(kls, chap): try: chap[tempref[d + 1] - 2] += "\paragraph{\\bf KLS Addendum: Limit Relation(s)}" chap[tempref[d + 1] - 2] += lrkHyperSub[lrkHypIndex + offset] - print(lrkHyperSub[lrkHypIndex + offset]) - #print(chap[tempref[d + 1] - 2]) lrkHypIndex+=1 - print(lrkHypIndex) except IndexError: print("Warning! Code has encountered some sort of error involving section identification for 'Limit Relations'. Problems may occur.") - print(lrhyperHeaders) - print(lrhyperSubs) + print("If you see this message, bad things are happening.") def prepareForPDF(chap): footmiscIndex = 0 @@ -376,124 +556,33 @@ def referencePlacer(chap, references, p, kls,refII,refIII,specref,klsaddparas): else: count += 1 -def referencePlacerII(chap, references, p, kls,refII,refIII,specref,klsaddparas,sectorskls,sectorschap,insertindices,insertloc): - if chapticker2 == 0: - designator = "9." - elif chapticker2 == 1: - designator = "14." - count = 0 - count2 = 0 - count3 = 0 - secchaploc = 0 - secklsloc = 0 - sectionnuma = specref[count] - sectionnumb = specref[count+1] - sectionnumc = refII[count2] - sectionnumd = refII[count2 + 1] - while count < len(specref)-1: - for word in klsaddparas: - if word <> "@": - if word > sectionnuma and word < sectionnumb: - - lenwordII = len(kls[word])-1 - #That 29 is where the KLS formatting ends - if "subsub" in kls[word]: - temp = kls[word][34: lenwordII] - else: - temp = kls[word][29 : lenwordII] - #with open("keywords.tex", "a") as log: - # log.write(temp + " Location: " + str(word) + "\n") - #log.close() - #print(temp) - #print(word) - sectorskls = temp.split(' ') - for i in sectorskls: - for a in irrelevant: - if i == a: - sectorskls[sectorskls.index(i)] = "@@" - i == "@@" - #print(sectorskls) - - while count2 < len(refII) - 1: - for word2 in refIII: - if word2 > sectionnumc and word2 < sectionnumd and word2 <> "@": - lenwordIII = len(chap[word2])-3 - # That 13 is where the KLS formatting ends - if "subsub" in chap[word2]: - temp2 = chap[word2][16: lenwordIII] - else: - temp2 = chap[word2][13: lenwordIII] - #if designator == "14.": - # with open("keywords.tex", "a") as log: - # log.write(temp2 + " Location: " + str(word2) + "\n") - # log.close() - sectorschap = temp2.split(' ') - for i in sectorschap: - for a in irrelevant: - if i == a: - i == "@@" - #print("temp2") - #print(temp2) - #print(word2) - ticker1 = 1 - for i in sectorskls: - for j in sectorschap: - if (i in j) or (j in i): - if ticker1 == 1: - try: - locationdata = klsaddparas[klsaddparas.index(word)],"+" - moredata = str(klsaddparas[klsaddparas.index(word)]) - insertindices.append(locationdata) - insertloc.append(refIII[refIII.index(word2)+1]) - evenmoredata = str(refIII[refIII.index(word2)+1]) - with open("keywordsII.tex", "a") as log: - log.write("Pair "+"\n") - log.write(moredata + "\n") - log.write(evenmoredata+"\n") - log.close() - klsaddparas[klsaddparas.index(word)] = "@" - refIII[refIII.index(word2)] = "@" - ticker1 = 0 - except ValueError: - pass - count2 += 1 - sectionnumc = refII[count2] - if count2 + 1 == len(refII): - sectionnumd = refII[count2] - else: - sectionnumd = refII[count2 + 1] - count2 = 0 - count += 1 - sectionnuma = specref[count] - if count + 1 == len(specref): - sectionnumb = specref[count] - else: - sectionnumb = specref[count + 1] - #print(insertindices) - #print(insertloc) - #print(klsaddparas) - #print(refIII) - #method to change file string(actually a list right now), returns string to be written to file #If you write a method that changes something, it is preffered that you call the method in here -def fixChapter(chap, references, p, kls,refII,refIII,specref,klsaddparas,sectorskls,sectorschap,insertindices,insertloc): +def fixChapter(chap, references, p, kls,refII,refIII,specref,klsaddparas,sectorskls,sectorschap,insertindices,insertloc,klslist): #chap is the file string(actually a list), references is the specific references for the file, #and p is the paras variable(not sure if needed) kls is the KLSadd.tex as a list chapticker3 = 0 - HyperGeoFix(kls, chap) + #HyperGeoFix(kls, chap) global check print(len(check)) #check = [] - SpecialValue(kls,chap) + #SpecialValue(kls,chap) #global check #print(len(check)) #check = [] - LimRelFix(kls, chap) + #LimRelFix(kls, chap) + sortloc = 0 + #FixChapterII(kls,chap,klslist) + bob = ['Uniqueness of orthogonality measure'] + for i in klslistIII: + #print(i) + FixChapterIII(kls, chap, i, sortloc) + sortloc += 1 referencePlacer(chap, references, p, kls, refII,refIII,specref,klsaddparas) #referencePlacerII(chap, references, p, kls, refII, refIII, specref, klsaddparas,sectorskls,sectorschap,insertindices,insertloc) @@ -530,6 +619,7 @@ def fixChapter(chap, references, p, kls,refII,refIII,specref,klsaddparas,sectors with open("KLSadd(3).tex", "r") as add: #store the file as a string addendum = add.readlines() + NewKeywords(addendum) #Makes sections look like other sections for word in addendum: if ("paragraph{" in word): @@ -563,6 +653,8 @@ def fixChapter(chap, references, p, kls,refII,refIII,specref,klsaddparas,sectors klsaddparas.append(index-1) if ("subsub" in word) and (index > 313): klsaddparas.append(index - 1) + if ("\subsection*{" in word) and (index > 313): + klsaddparas.append(index - 1) klsaddparas.append(2246) #print(indexes) #print(specref9) @@ -603,9 +695,13 @@ def fixChapter(chap, references, p, kls,refII,refIII,specref,klsaddparas,sectors ref14III.insert(198, 2554) #call the fixChapter method to get a list with the addendum paragraphs added in chapticker2 = 0 - str9 = ''.join(fixChapter(entire9, references9, paras, addendum,ref9II,ref9III,specref9,klsaddparas,sectorskls,sectorschap,insertindices,insertloc)) + str9 = ''.join(fixChapter(entire9, references9, paras, addendum,ref9II,ref9III,specref9,klsaddparas,sectorskls,sectorschap,insertindices,insertloc,klslist)) + #with open("updated9.tex", "w") as temp9: + # temp9.write(str9) + #print("dignsakfibasjksdjhsfs") + #check4.append("memes") chapticker2 += 1 - str14 = ''.join(fixChapter(entire14, references14, paras, addendum,ref14II,ref14III,specref14,klsaddparas,sectorskls,sectorschap,insertindices,insertloc)) + str14 = ''.join(fixChapter(entire14, references14, paras, addendum,ref14II,ref14III,specref14,klsaddparas,sectorskls,sectorschap,insertindices,insertloc,klslist)) """ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ @@ -622,4 +718,6 @@ def fixChapter(chap, references, p, kls,refII,refIII,specref,klsaddparas,sectors with open("updated14.tex", "w") as temp14: temp14.write(str14) - +print(klsaddparas) +print(sorterIIIcheck) +print(ref9III) \ No newline at end of file From c0e4db90bb12c21c0c0c758034079d10df4cc182 Mon Sep 17 00:00:00 2001 From: Edward Bian Date: Thu, 11 Aug 2016 16:48:14 -0400 Subject: [PATCH 234/402] Now cleaned and with better section sorter --- KLSadd_insertion/src/updateChapters.py | 990 ++++++++++++------------- 1 file changed, 459 insertions(+), 531 deletions(-) diff --git a/KLSadd_insertion/src/updateChapters.py b/KLSadd_insertion/src/updateChapters.py index 1c397ea..4dc9f89 100644 --- a/KLSadd_insertion/src/updateChapters.py +++ b/KLSadd_insertion/src/updateChapters.py @@ -1,209 +1,197 @@ -""" -Rahul Shah -self-taught python don't kill me I know its bad -2/5/16 -Re-write of linetest.py so people other than me can read it -This project was/is being written to streamline the update process of the book used -for the online repository, updated via the KLSadd addendum file which only affects chapters -9 and 14 -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -NOTE! AS OF 2/26/16 THIS FILE IS NOT YET UPDATED TO THE FULL CAPACITY OF THE PREVIOUS FILE. IF FURTHER DEVELOPMENT IS NEEDED, REFER TO THE -linetest.py FILE IF THIS FILE DOES NOT ADDRESS THE NEW/EXTRA GOALS. This means that XCITE PARSE etc HAS NOT BEEN ADDED -Additional goals (already addressed in linetest.py): --insert correct usepackages --insert correct commands found in KLSadd.tex to chapter files --insert editor's initials into sections (ie %RS: begin addition, %RS: end addition) -Additional goals (not already addressed in linetest.py): --rewrite for "smarter" edits (ex. add new limit relations straight to the limit relations section in the section itself, not at the end) -Current update status: incomplete -Goals:change the book chapter files to include paragraphs from the addendum, and after that insert some edits so they are intelligently integrated into the chapter - -Edward Bian -Currently under heavy modification; sections may not work and/or look inefficient/confusing -""" - -#start out by reading KLSadd.tex to get all of the paragraphs that must be added to the chapter files -#also keep track of which chapter each one is in - -#variables: chapNums for the chapter number each section belongs to paras will hold the sections that will be copied over -#chapNums is needed to know which file to open (9 or 14) -#mathPeople is just the name for the sections that are used, like Wilson, Racah, etc. I know some are not people, its just a var name -chapNums = [] +__author__ = "Rahul Shah" +__status__ = "Development" +__credits__ = ["Rahul Shah", "Edward Bian"] + +# start out by reading KLSadd.tex to get all of the paragraphs that must be added to the chapter files +# also keep track of which chapter each one is in + + +# chapNums is needed to know which file to open (9 or 14) +# mathPeople is just the name for the sections that are used, like Wilson, Racah, etc. +chap_nums = [] paras = [] -#klsaddparas contains the locations of everything with "paragraph{" in KLSadd, these are the subsections that need to be sorted +# klsaddparas contains the locations of things with "paragraph{" in KLSadd, these are subsections that need to be sorted klsaddparas = [] -mathPeople = [] -newCommands = [] #used to hold the indexes of the commands +math_people = [] +new_commands = [] # used to hold the indexes of the commands -#ref9II holds the beginnings and ends of each chapter 9 section, provided and area to search -ref9II = [] #Hold section search indexes +# ref9II holds the beginnings and ends of each chapter 9 section, provided and area to search +ref9_2 = [] -#ref14II holds the beginnings and ends of each chapter 14 section, provided and area to search -ref14II = [] +# ref14II holds the beginnings and ends of each chapter 14 section, provided and area to search +ref14_2 = [] -#ref9III holds all subsections in each chapter 9 section along with the what ref9II holds -ref9III = [] #Holds all indexes +# ref9III holds all subsections in each chapter 9 section along with the what ref9II holds +ref9_3 = [] -#ref14III holds all subsections in each chapter 14 section along with the what ref14II holds -ref14III = [] +# ref14III holds all subsections in each chapter 14 section along with the what ref14II holds +ref14_3 = [] -#specref9 is the KLS indexes that go in chapter 9 +# specref9 is the KLS indexes that go in chapter 9 specref9 = [] -#specref14 is the KLS indexes that in chapter 14 +# specref14 is the KLS indexes that in chapter 14 specref14 = [] -sectorschap = [] -sectorskls = [] -insertindices = [] -insertloc = [] -irrelevant = ["Relations","Functions","Transform","In","For","To","Is","And","Of","relations","functions","transform","in","for","to","is","and","of"] - check = [] check2 = [] check3 = [] -check4 = [] -klslist = [] -klslistII = [] -klslistIII = [] +kls_list = [] +sortmatch = [] -w, h = 2, 9999 +w, h = 3, 9999 sorterIIIcheck = [[0 for x in range(w)] for y in range(h)] -print(sorterIIIcheck) -comms = "" #holds the ACTUAL STRINGS of the commands -#2/18/16 this method addresses the goal of hardcoding in the necessary packages to let the chapter files run as pdf's. -#Currently only works with chapter 9, ask Dr. Cohl to help port your chapter 14 output file into a pdf +sortmatch_2 = [] -#5/5/16 this method implements the Hypergeometric paragraphs into relevant sections for practicality + +# 2/18/16 this method addresses the goal of hardcoding in the necessary packages to let the chapter files run as pdf's. +# Currently only works with chapter 9, ask Dr. Cohl to help port your chapter 14 output file into a pdf + +# 5/5/16 this method implements the Hypergeometric paragraphs into relevant sections for practicality # Later versions should attempt to: fix paragraphs with variations on this name like # Basic Hypergeometric Representation, q-Hypergeometric Representation -def NewKeywords(kls): + + +def new_keywords(kls): + """ + This section checks through the Addendum and identifies words that are valid keywords of sections + It then takes that list and removes duplicates, as well as printing the output to another file. + + :param kls: The addendum that provides the words. + :return: Inputs results into a global list. + """ + + kls_list_chap = [] for i in kls: if kls.index(i) > 313: - if ("paragraph{") in i: - lenword = len(i) - 1 - temp = i[i.find("{") + 1: lenword-1] - klslist.append(temp) - klslistII.append(temp+'\n') - if ("subsubsection*{") in i: - lenword = len(i) - 1 - temp = i[i.find("{") + 1: lenword - 1] - klslist.append(temp) - klslistII.append(temp + '\n') - print(klslist) + if "paragraph{" in i or "subsubsection*{" in i: + kls_list.append(i[i.find("{") + 1: len(i) - 2]) + i = 0 - while i < len(klslist): - if ("reference" in klslist[i].lower()) or ("hypergeometric representation" in klslist[i].lower()) or ("limit relation" in klslist[i].lower()) or ("special value" in klslist[i].lower()): - del klslist[i] + while i < len(kls_list): + if "reference" in kls_list[i].lower() or "limit relation" in kls_list[i].lower(): + + del kls_list[i] else: i += 1 - liststr = ''.join(klslistII) - with open("keywordsIII.tex", "w") as kw: - kw.write(liststr) - for i in klslist: + kls_list.append("Limit Relation") + for i in kls_list: add = True - for j in klslistIII: + for j in kls_list_chap: if i == j: add = False - if add == True: - klslistIII.append(i) - print(klslist) - liststr2 = '\n'.join(klslistIII) + if add: + kls_list_chap.append(i) + liststr2 = '\n'.join(kls_list_chap) with open("trunckeywords.tex", "w") as kw: kw.write(liststr2) - print(klslistIII) - print(klslistIII.index("Orthogonality")) - #del klslistIII[35] - print(klslistIII) -""" ---------------------------------------------------------- -""" -def FixChapterIII(kls,chap,word,sortloc): - canAdd = False - # keep track of sections and subsections from chap - hyperHeadersIII = [] - hyperSubsIII = [] - index = 0 + print kls_list_chap + return kls_list_chap + + +def fix_chapter_sort(kls, chap, word, sortloc): + """ + This function sorts through the input files and identifies and places sections from the addendum after their + respective subsections in the chapter. + + :param kls: The addendum where the sections to be inserted are found. + :param chap: The destination of inserted sections. + :param word: The keyword that is being processed. + :param sortloc: The location in the list of words, of the word being sorted. + :return: This function outputs the processed chapter into a larger function for further processing. + """ + hyper_headers_chap = [] + hyper_subs_chap = [] + sep1 = 0 - nameIII = word.lower() - if nameIII == "orthogonality": + name_chap = word.lower() + if name_chap == "orthogonality": sep1 = 1 - #print(nameIII) - #using Edward's reference paragraphs! - #now get the section names of all of the KLSadd - kHyperHeaderIII = [] - kHyperSubIII = [] + elif name_chap == "special value": + sep1 = 2 + + khyper_header_chap = [] + k_hyper_sub_chap = [] index = 0 + for i in kls: - index +=1 + index += 1 line = str(i) if "\\subsection" in line: - temp = line[line.find(" ", 12)+1: line.find("}", 12)] #get just the name (like mathpeople) + temp = line[line.find(" ", 12)+1: line.find("}", 12)] # get just the name (like mathpeople) + if sep1 == 0: - if nameIII in line.lower() and kls.index(i) > 313 and "\\paragraph{" in line: - #print(i) - #print(kls.index(i)) + if name_chap in line.lower() and kls.index(i) > 313 and ("\\paragraph{" in line or + "\\subsubsection*{" in line): for i in klsaddparas: if index < i: klsloc = klsaddparas.index(i) break t = ''.join(kls[index: klsaddparas[klsloc]]) - kHyperSubIII.append(t)#append the whole paragraph, pray every paragraph ends with a % comment - kHyperHeaderIII.append(temp)#append the name of subsection - else: - if nameIII in line.lower() and kls.index(i) > 313 and "\\paragraph{" in line and "orthogonality relation" not in line.lower(): - # print(i) - # print(kls.index(i)) + k_hyper_sub_chap.append(t) # append the whole paragraph, pray every paragraph ends with a % comment + khyper_header_chap.append(temp) # append the name of subsection + elif sep1 == 1: + if name_chap in line.lower() and kls.index(i) > 313 and ("\\paragraph{" in line or + "\\subsubsection*{" in line) and "orthogonality relation" not in line.lower(): for i in klsaddparas: if index < i: klsloc = klsaddparas.index(i) break t = ''.join(kls[index: klsaddparas[klsloc]]) - kHyperSubIII.append(t) # append the whole paragraph, pray every paragraph ends with a % comment - kHyperHeaderIII.append(temp) # append the name of subsection - kHypIndexIII = 0 + k_hyper_sub_chap.append(t) # append the whole paragraph, pray every paragraph ends with a % comment + khyper_header_chap.append(temp) # append the name of subsection + elif sep1 == 2: + if name_chap in line.lower() and kls.index(i) > 313 and ("\\paragraph{" in line or + "\\subsubsection*{" in line): + for i in klsaddparas: + if index < i: + klsloc = klsaddparas.index(i) + break + t = ''.join(kls[index: klsaddparas[klsloc]]) + k_hyper_sub_chap.append(t) # append the whole paragraph, pray every paragraph ends with a % comment + khyper_header_chap.append(temp) # append the name of subsection + for i in khyper_header_chap: + if i == "Pseudo Jacobi (or Routh-Romanovski)": + khyper_header_chap[khyper_header_chap.index(i)] = "Pseudo Jacobi" + k_hyp_index_iii = 0 offset = 0 i = 0 - while i < len(kHyperHeaderIII): + while i < len(khyper_header_chap): try: - if kHyperHeaderIII[i] == kHyperHeaderIII[i+1]: - kHyperSubIII[i+1] = kHyperSubIII[i] + "\paragraph{\\bf KLS Addendum: " + word + "}\n" + kHyperSubIII[i+1] - kHyperHeaderIII[i] = "memes" - kHyperSubIII[i] = "memes" + if khyper_header_chap[i] == khyper_header_chap[i+1]: + k_hyper_sub_chap[i + 1] = k_hyper_sub_chap[i] + "\paragraph{\\bf KLS Addendum: " + \ + word + "}\n" + k_hyper_sub_chap[i + 1] + khyper_header_chap[i] = "memes" + k_hyper_sub_chap[i] = "memes" else: i += 1 except IndexError: i += 1 + a = 0 - #print(kHyperHeaderIII) - #print(kHyperSubIII) - while a < len(kHyperHeaderIII): - if kHyperHeaderIII[a] == "memes": - del kHyperHeaderIII[a] - del kHyperSubIII[a] + while a < len(khyper_header_chap): + if khyper_header_chap[a] == "memes": + del khyper_header_chap[a] + del k_hyper_sub_chap[a] else: a += 1 - #print(kHyperSubIII) - #print(len(kHyperSubIII)) - #print(kHyperHeaderIII) - #print(len(kHyperHeaderIII)) + global sorterIIIcheck - #print(sortloc) chap9 = 0 + if sorterIIIcheck[sortloc][0] == 0: - tempref = ref9III + tempref = ref9_3 sorterIIIcheck[sortloc][0] += 1 chap9 = 1 else: - tempref = ref14III + tempref = ref14_3 offset = sorterIIIcheck[sortloc][1] - print("offset") - print(offset) if sep1 == 1: offset = 8 + for d in range(0, len(tempref)): # check every section and subsection line i = tempref[d] line = str(chap[i]) @@ -212,315 +200,226 @@ def FixChapterIII(kls,chap,word,sortloc): temp = line[12:line.find("}", 7)] else: temp = line[9:line.find("}", 7)] + if sep1 == 0: - if nameIII in line.lower(): - hyperSubsIII.append([tempref[d + 1]]) # appends the index for the line before following subsection - hyperHeadersIII.append(temp) # appends the name of the section the hypergeo subsection is in so we can compare + if name_chap in line.lower(): + hyper_subs_chap.append([tempref[d + 1]]) # appends the index for the line before following subsection + hyper_headers_chap.append(temp) # appends the name of the section the hypergeo subsection is in - if temp in kHyperHeaderIII: + if temp in khyper_header_chap: try: chap[tempref[d + 1] - 1] += "\paragraph{\\bf KLS Addendum: " + word + "}" - chap[tempref[d + 1] - 1] += kHyperSubIII[kHypIndexIII + offset] + chap[tempref[d + 1] - 1] += k_hyper_sub_chap[k_hyp_index_iii + offset] if chap9 == 1: sorterIIIcheck[sortloc][1] += 1 - #print(chap[tempref[d + 1] - 2]) - kHypIndexIII+=1 + k_hyp_index_iii += 1 except IndexError: - print("Warning! Code has encountered some sort of error involving section identification for '" + nameIII + "'. Problems may occur.") - print("If you see this message, bad things are happening.") - else: - if nameIII in line.lower() and "orthogonality relation" not in line: - hyperSubsIII.append([tempref[d + 1]]) # appends the index for the line before following subsection - hyperHeadersIII.append( - temp) # appends the name of the section the hypergeo subsection is in so we can compare + print("Warning! Code has found an error involving section finding for '" + name_chap + "'.") + elif sep1 == 1: + if name_chap in line.lower() and "orthogonality relation" not in line: + hyper_subs_chap.append([tempref[d + 1]]) # appends the index for the line before following subsection + hyper_headers_chap.append(temp) # appends the name of the section the hypergeo subsection is in - if temp in kHyperHeaderIII: + if temp in khyper_header_chap: try: chap[tempref[d + 1] - 1] += "\paragraph{\\bf KLS Addendum: " + word + "}" - chap[tempref[d + 1] - 1] += kHyperSubIII[kHypIndexIII + offset] - # print(chap[tempref[d + 1] - 2]) - kHypIndexIII += 1 + chap[tempref[d + 1] - 1] += k_hyper_sub_chap[k_hyp_index_iii + offset] + k_hyp_index_iii += 1 except IndexError: - print("Warning! Code has encountered some sort of error involving section identification for '" + nameIII + "'. Problems may occur.") - print("If you see this message, bad things are happening.") - - if len(hyperHeadersIII) != 0 and len(hyperSubsIII) != 0: - print("Chapter Values:") - print(word) - print(kHyperHeaderIII) - print(hyperHeadersIII) - print(len(hyperHeadersIII)) - #print(kHyperSubIII) - print(kHyperSubIII) - print(hyperSubsIII) - print(len)(hyperSubsIII) - print("End") - print("sortloc") - print(sorterIIIcheck[sortloc][1]) - -def HyperGeoFix(kls, chap): - canAdd = False - # keep track of sections and subsections from chap - hyperHeaders = [] - hyperSubs = [] - index = 0 + print("Warning! Code has found an error involving section finding for '" + name_chap + "'.") + elif sep1 == 2: + if "hypergeometric representation" in line.lower(): + hyper_subs_chap.append([tempref[d + 1]]) # appends the index for the line before following subsection + hyper_headers_chap.append( + temp) # appends the name of the section the hypergeo subsection is in so we can compare - #using Edward's reference paragraphs! - #now get the section names of all of the KLSadd - kHyperHeader = [] - kHyperSub = [] - index = 0 - for i in kls: - index +=1 - line = str(i) - if "\\subsection" in line: - temp = line[line.find(" ", 12)+1: line.find("}", 12)] #get just the name (like mathpeople) - if "hypergeometric representation" in line.lower(): - for i in klsaddparas: - if index < i: - klsloc = klsaddparas.index(i) - break - t = ''.join(kls[index: klsaddparas[klsloc]]) - kHyperSub.append(t)#append the whole paragraph, pray every paragraph ends with a % comment - kHyperHeader.append(temp)#append the name of subsection - kHyperHeader[8] = "Discrete $q$-Hermite~II" - kHypIndex = 0 - offset = 0 - if len(check) == 0: - tempref = ref9III - check.append("Words") - else: - tempref = ref14III - offset = 4 - for d in range(0, len(tempref)): # check every section and subsection line - i = tempref[d] - line = str(chap[i]) - if "\\section{" in line or "\subsection{" in line: - if "\\subsection{" in line: - temp = line[12:line.find("}", 7)] - else: - temp = line[9:line.find("}", 7)] - if "hypergeometric representation" in line.lower(): - hyperSubs.append([tempref[d + 1]]) # appends the index for the line before following subsection - hyperHeaders.append(temp) # appends the name of the section the hypergeo subsection is in so we can compare - - if temp in kHyperHeader: - try: - chap[tempref[d + 1] - 2] += "\paragraph{\\bf KLS Addendum: Hypergeometric Representation}" - chap[tempref[d + 1] - 2] += kHyperSub[kHypIndex + offset] - #print(chap[tempref[d + 1] - 2]) - kHypIndex+=1 - except IndexError: - print("Warning! Code has encountered some sort of error involving section identification for 'Hypergeometric Representation'. Problems may occur.") - print("If you see this message, bad things are happening.") - - #print(kHyperHeader) - #print(hyperHeaders) - #print(kHyperSub) - #print(hyperSubs) -def SpecialValue(kls, chap): - canAdd = False - # keep track of sections and subsections from chap - svhyperHeaders = [] - svhyperSubs = [] - index = 0 + if temp in khyper_header_chap: + try: + chap[tempref[d + 1] - 1] += "\paragraph{\\bf KLS Addendum: Special Value }" + chap[tempref[d + 1] - 1] += k_hyper_sub_chap[k_hyp_index_iii + offset] + k_hyp_index_iii += 1 + except IndexError: + print("Warning! Code has found an error involving section finding for '" + name_chap + "'.") + + if len(hyper_headers_chap) != 0: + print "Stuff for checking" + print k_hyper_sub_chap + print khyper_header_chap + print hyper_headers_chap + print word + sortmatch.append(word) + sortmatch_2.append(k_hyper_sub_chap) + sorterIIIcheck[sortloc][2] = word + + +def cutwords(word_to_find, word_to_search_in): + """ + This function checks through the outputs of later sections and removes duplicates, so that the output file does + not have duplicates. + + IN DEVELOPMENT + + :param word_to_find: Word that is being found + :param word_to_search_in: The big word that is being searched + :return: The big word without the word that was searched for + """ + a = word_to_find + b = word_to_search_in + precheck = 1 + checkloop = False + if a in b: + if "\paragraph{\large\bf KLSadd: " or "\subsubsection*{\large\bf KLSadd: " in b: + while checkloop == False: + if "\\paragraph{\\large\\bf KLSadd: " in b[b.find(a)-precheck:b.find(a)]: + cut = b[:b.find(a) - precheck] + b[b.find(a) + len(a):] + checkloop = True + return cut + elif "\\subsubsection*{\\large\\bf KLSadd: " in b[b.find(a)-precheck:b.find(a)]: + cut = b[:b.find(a) - precheck] + b[b.find(a) + len(a):] + checkloop = True + return cut + else: + print "ding2" + precheck += 1 - # using Edward's reference paragraphs! - # now get the section names of all of the KLSadd - svkHyperHeader = [] - svkHyperSub = [] - index = 0 - for i in kls: - index +=1 - line = str(i) - if "\\subsection" in line: - temp = line[line.find(" ", 12)+1: line.find("}", 12)] #get just the name (like mathpeople) - if "special value" in line.lower(): - for i in klsaddparas: - if index < i: - svklsloc = klsaddparas.index(i) - break - t = ''.join(kls[index: klsaddparas[svklsloc]]) - svkHyperSub.append(t) # append the whole paragraph, pray every paragraph ends with a % comment - svkHyperHeader.append(temp) # append the name of subsection - print(svkHyperHeader) - print(svkHyperSub) - svkHypIndex = 0 - offset = 0 - if len(check2) == 0: - tempref = ref9III - check2.append("test") + else: + cut = b[:b.find(a)] + b[b.find(a) + len(a):] + return cut else: - tempref = ref14III - offset = 8 - for d in range(0, len(tempref)): # check every section and subsection line - i = tempref[d] - line = str(chap[i]) - if "\\section{" in line or "\\subsection{" in line: - if "\\subsection{" in line: - temp = line[12:line.find("}", 7)] - else: - temp = line[9:line.find("}", 7)] - if "hypergeometric representation" in line.lower(): - svhyperSubs.append([tempref[d + 1]]) # appends the index for the line before following subsection - svhyperHeaders.append(temp) #appends the name of the section the hypergeo subsection is in so we can compare - if temp in svkHyperHeader: - chap[tempref[d + 1] - 2] += "\paragraph{\\bf KLS Addendum: Special Value}" - chap[tempref[d + 1] - 2] += svkHyperSub[svkHypIndex + offset] - svkHypIndex += 1 - -def LimRelFix(kls, chap): - canAdd = False - lrhyperHeaders = [] - lrhyperSubs = [] - index = 0 + return b - lrkHyperHeader = [] - lrkHyperSub = [] - index = 0 - for i in kls: - index +=1 - line = str(i) - if "\\subsection" in line: - temp = line[line.find(" ", 12)+1: line.find("}", 12)] - if "limit relation" in line.lower(): - for i in klsaddparas: - if index < i: - klsloc = klsaddparas.index(i) - break - t = ''.join(kls[index: klsaddparas[klsloc]]) - lrkHyperSub.append(t)#append the whole paragraph, pray every paragraph ends with a % comment - lrkHyperHeader.append(temp)#append the name of subsection - lrkHyperHeader[0] = "Pseudo Jacobi" - lrkHypIndex = 0 - offset = 0 - if len(check3) == 0: - tempref = ref9III - check3.append("Words") - else: - tempref = ref14III - offset = 1 - for d in range(0, len(tempref)): # check every section and subsection line - i = tempref[d] - line = str(chap[i]) - if "\\section{" in line or "\subsection{" in line: - if "\\subsection{" in line: - temp = line[12:line.find("}", 7)] - else: - temp = line[9:line.find("}", 7)] - if "limit relation" in line.lower(): - lrhyperSubs.append([tempref[d + 1]]) # appends the index for the line before following subsection - lrhyperHeaders.append(temp) # appends the name of the section the hypergeo subsection is in so we can compare - - if temp in lrkHyperHeader: - try: - chap[tempref[d + 1] - 2] += "\paragraph{\\bf KLS Addendum: Limit Relation(s)}" - chap[tempref[d + 1] - 2] += lrkHyperSub[lrkHypIndex + offset] - lrkHypIndex+=1 - except IndexError: - print("Warning! Code has encountered some sort of error involving section identification for 'Limit Relations'. Problems may occur.") - print("If you see this message, bad things are happening.") - -def prepareForPDF(chap): - footmiscIndex = 0 +def prepare_for_PDF(chap): + """ + Edits the chapter string sent to include hyperref, xparse, and cite packages + + :param chap: The chapter (9 or 14) that is being processed + :return: + """ + foot_misc_index = 0 index = 0 for word in chap: - index+=1 - if("footmisc" in word): - footmiscIndex+= index - #edits the chapter string sent to include hyperref, xparse, and cite packages - #str[footmiscIndex] += "\\usepackage[pdftex]{hyperref} \n\\usepackage {xparse} \n\\usepackage{cite} \n" - chap.insert(footmiscIndex, "\\usepackage[pdftex]{hyperref} \n\\usepackage {xparse} \n\\usepackage{cite} \n") + index += 1 + if "footmisc" in word: + foot_misc_index += index + # str[footmiscIndex] += "\\usepackage[pdftex]{hyperref} \n\\usepackage {xparse} \n\\usepackage{cite} \n" + chap.insert(foot_misc_index, "\\usepackage[pdftex]{hyperref} \n\\usepackage {xparse} \n\\usepackage{cite} \n") return chap -#2/18/16 this method reads in relevant commands that are in KLSadd.tex and returns them as a list -def getCommands(kls): +def get_commands(kls): + """ + this method reads in relevant commands that are in KLSadd.tex and returns them as a list + + :param kls: The addendum is the source of what is being found + :return: + """ index = 0 for word in kls: - index+=1 - if("smallskipamount" in word): - newCommands.append(index-1) - if("mybibitem[1]" in word): - newCommands.append(index) - comms = kls[newCommands[0]:newCommands[1]] - return comms - -#2/18/16 this method addresses the goal of hardcoding in the necessary commands to let the chapter files run as pdf's. Currently only works with chapter 9 - -def insertCommands(kls, chap, cms): - #reads in the newCommands[] and puts them in chap - beginIndex = -1 #the index of the "begin document" keyphrase, this is where the new commands need to be inserted. - #find index of begin document in KLSadd + index += 1 + if "smallskipamount" in word: + new_commands.append(index - 1) + if "mybibitem[1]" in word: + new_commands.append(index) + return kls[new_commands[0]:new_commands[1]] + + +# 2/18/16 this method addresses the goal of hardcoding in the necessary commands to let the chapter files run as pdf's. +# Currently only works with chapter 9 +def insert_commands(kls, chap, cms): + """ + Inserts commands identified in previous functions. + + :param kls: Addendum to look through + :param chap: The chapter that is receiving commands (9 or 14) + :param cms: Commands to be inserted + :return: + """ + # reads in the newCommands[] and puts them in chap + begin_index = -1 # the index of the "begin document" keyphrase, this is where the new commands need to be inserted. + # find index of begin document in KLSadd index = 0 + for word in kls: - index+=1 - if("begin{document}" in word): - beginIndex += index + index += 1 + if "begin{document}" in word: + begin_index += index tempIndex = 0 + for i in cms: - chap.insert(beginIndex+tempIndex,i) - tempIndex +=1 + chap.insert(begin_index + tempIndex, i) + tempIndex += 1 return chap -#method to find the indices of the reference paragraphs -def findReferences(chapter): + +# method to find the indices of the reference paragraphs +def find_references(chapter,chapticker): + """ + This function searches the chapters and locates potential destinations of additions from the addendum. + It puts these destinations in a list. + + :param chapter: The chapter being searched. + :param chapticker: Which chapter is being searched (9 or 14). + :return: List of references. + """ references = [] index = -1 - #chaptercheck designates which chapter is being searched for references + # chaptercheck designates which chapter is being searched for references chaptercheck = 0 if chapticker == 0: chaptercheck = str(9) elif chapticker == 1: chaptercheck = str(14) - #canAdd tells the program whether the next section is a reference - canAdd = False + # canAdd tells the program whether the next section is a reference + canadd = False + for word in chapter: - index+=1 + index += 1 specialdetector = 1 - #check sections and subsections + # check sections and subsections if("section{" in word or "subsection*{" in word) and ("subsubsection*{" not in word): w = word[word.find("{")+1: word.find("}")] ws = word[word.find("{")+1: word.find("~")] - for unit in mathPeople: + for unit in math_people: subunit = unit[unit.find(" ")+1: unit.find("#")] # System of checks that verifies if section is in chapter - if ((w in subunit) or (ws in subunit)) and (chaptercheck in unit) and (len(w) == len(subunit)) or (("Pseudo Jacobi" in w) and ("Pseudo Jacobi (or Routh-Romanovski)" in subunit)) and ("Hypergeometric representation" not in w) and ("Special value" not in w): - canAdd = True + if (w in subunit or ws in subunit) and (chaptercheck in unit) and len(w) == len(subunit) or\ + ("Pseudo Jacobi" in w and "Pseudo Jacobi (or Routh-Romanovski)" in subunit): + canadd = True if chapticker == 0: - ref9II.append(index) - ref9III.append(index) + ref9_2.append(index) + ref9_3.append(index) elif chapticker == 1: - ref14II.append(index) - ref14III.append(index) + ref14_2.append(index) + ref14_3.append(index) specialdetector = 0 - if("\\subsection*{References}" in word) and (canAdd == True): + if"\\subsection*{References}" in word and canadd: # Appends valid locations references.append(index) if chapticker == 0: - ref9II.append(index) - ref9III.append(index) + ref9_2.append(index) + ref9_3.append(index) elif chapticker == 1: - ref14II.append(index) - ref14III.append(index) - canAdd = False - if ("subsection*{" in word and "References" not in word): + ref14_2.append(index) + ref14_3.append(index) + canadd = False + if "subsection*{" in word and "References" not in word: if chapticker == 0: - ref9III.append(index) + ref9_3.append(index) elif chapticker == 1: - ref14III.append(index) + ref14_3.append(index) if "\\section{" in word and specialdetector == 1 and chaptercheck == "14": w2 = word[word.find("{") + 1: word.find("}")] if "Bessel" not in w2: - ref14III.append(index) + ref14_3.append(index) besselcheck = 1 bqlegendrecheck = 2 - for i in ref9III: + + for i in ref9_3: if i == 2646: besselcheck = 0 if besselcheck == 1: - ref9III.insert(230, 2646) - for i in ref14III: + ref9_3.insert(230, 2646) + for i in ref14_3: if i == 1217: bqlegendrecheck -= 1 if bqlegendrecheck > 0: @@ -528,7 +427,17 @@ def findReferences(chapter): return references -def referencePlacer(chap, references, p, kls,refII,refIII,specref,klsaddparas): + +def reference_placer(chap, references, p, chapticker2): + """ + Places identified additions from the addendum into the chapter. + + :param chap: Chapter receiving the additions. + :param references: List containing references + :param p: List containing additions to be added. + :param chapticker2: Which chapter is being searched (9 or 14). + :return: + """ # count is used to represent the values in count count = 0 # Tells which chapter it's on @@ -537,18 +446,19 @@ def referencePlacer(chap, references, p, kls,refII,refIII,specref,klsaddparas): designator = "9." elif chapticker2 == 1: designator = "14." + for i in references: # Place before References paragraph word1 = str(p[count]) - if (designator in word1[word1.find("\\subsection*{") + 1: word1.find("}")]): + if designator in word1[word1.find("\\subsection*{") + 1: word1.find("}")]: chap[i - 2] += "%Begin KLSadd additions" chap[i - 2] += p[count] chap[i - 2] += "%End of KLSadd additions" count += 1 else: - while (designator not in word1[word1.find("\\subsection*{") + 1: word1.find("}")]): + while designator not in word1[word1.find("\\subsection*{") + 1: word1.find("}")]: word1 = str(p[count]) - if (designator in word1[word1.find("\\subsection*{") + 1: word1.find("}")]): + if designator in word1[word1.find("\\subsection*{") + 1: word1.find("}")]: chap[i - 2] += "%Begin KLSadd additions" chap[i - 2] += p[count] chap[i - 2] += "%End of KLSadd additions" @@ -556,168 +466,186 @@ def referencePlacer(chap, references, p, kls,refII,refIII,specref,klsaddparas): else: count += 1 -#method to change file string(actually a list right now), returns string to be written to file -#If you write a method that changes something, it is preffered that you call the method in here -def fixChapter(chap, references, p, kls,refII,refIII,specref,klsaddparas,sectorskls,sectorschap,insertindices,insertloc,klslist): - #chap is the file string(actually a list), references is the specific references for the file, - #and p is the paras variable(not sure if needed) kls is the KLSadd.tex as a list - chapticker3 = 0 - - #HyperGeoFix(kls, chap) - global check - print(len(check)) - #check = [] +# method to change file string(actually a list right now), returns string to be written to file +# If you write a method that changes something, it is preffered that you call the method in here +def fix_chapter(chap, references, p, kls, kls_list_all, chapticker2): + """ + Removes specific lines stopping the latex file from converting into python, as well as running the + functions responsible for sorting sections and placing the correct additions in the correct places - #SpecialValue(kls,chap) + :param chap: Chapter being processed. + :param references: List containing references (used in reference placer). + :param p: List containing additions to be added. + :param kls: The addendum that contains sections to be added (not processed). + :param kls_list_all: List of all identified keywords, fed into the chapter sorter. + :param chapticker2: Which chapter is being searched (9 or 14). + :return: + """ - #global check - #print(len(check)) - #check = [] - - #LimRelFix(kls, chap) sortloc = 0 - #FixChapterII(kls,chap,klslist) - bob = ['Uniqueness of orthogonality measure'] - for i in klslistIII: - #print(i) - FixChapterIII(kls, chap, i, sortloc) + + for i in kls_list_all: + fix_chapter_sort(kls, chap, i, sortloc) sortloc += 1 - referencePlacer(chap, references, p, kls, refII,refIII,specref,klsaddparas) - #referencePlacerII(chap, references, p, kls, refII, refIII, specref, klsaddparas,sectorskls,sectorschap,insertindices,insertloc) - chap = prepareForPDF(chap) - cms = getCommands(kls) - chap = insertCommands(kls,chap, cms) + for a in range(len(p)-1): + for b in sortmatch_2: + for c in b: + with open("compare.tex", "a") as spook: + spook.write(p[a]) + spook.write("NEXT: ") + print c + print "pause" + print a + print "l0" + print len(c) + print "l1" + print a + print p[a] + p[a] = cutwords(c, p[a]) + print("l2") + + + reference_placer(chap, references, p, chapticker2) + chap = prepare_for_PDF(chap) + cms = get_commands(kls) + chap = insert_commands(kls, chap, cms) commentticker = 0 + # Hard coded command remover for word in chap: word2 = chap[chap.index(word)-1] - if ("\\newcommand{\qhypK}[5]{\,\mbox{}_{#1}\phi_{#2}\!\left(" not in word2): - if ("\\newcommand\\half{\\frac12}" in word): + if "\\newcommand{\qhypK}[5]{\,\mbox{}_{#1}\phi_{#2}\!\left(" not in word2: + if "\\newcommand\\half{\\frac12}" in word: wordtoadd = "%" + word chap[commentticker] = wordtoadd - elif ("\\newcommand{\\hyp}[5]{\\,\\mbox{}_{#1}F_{#2}\\!\\left(" in word): + elif "\\newcommand{\\hyp}[5]{\\,\\mbox{}_{#1}F_{#2}\\!\\left(" in word: wordtoadd = "%" + word chap[commentticker] = wordtoadd - elif ("\\genfrac{}{}{0pt}{}{#3}{#4};#5\\right)}" in word): + elif "\\genfrac{}{}{0pt}{}{#3}{#4};#5\\right)}" in word: wordtoadd = "%" + word chap[commentticker] = wordtoadd - elif ("\\newcommand{\\qhyp}[5]{\\,\\mbox{}_{#1}\\phi_{#2}\\!\\left(" in word): + elif "\\newcommand{\\qhyp}[5]{\\,\\mbox{}_{#1}\\phi_{#2}\\!\\left(" in word: wordtoadd = "%" + word chap[commentticker] = wordtoadd commentticker += 1 + ticker1 = 0 # Formatting to make the Latex file run while ticker1 < len(chap): - if ('\\myciteKLS' in chap[ticker1]): + if '\\myciteKLS' in chap[ticker1]: chap[ticker1] = chap[ticker1].replace('\\myciteKLS', '\\cite') ticker1 += 1 return chap -#open the KLSadd file to do things with -with open("KLSadd(3).tex", "r") as add: - #store the file as a string - addendum = add.readlines() - NewKeywords(addendum) - #Makes sections look like other sections - for word in addendum: - if ("paragraph{" in word): - lenword = len(word) - 1 - temp = word[0:word.find("{") + 1] + "\large\\bf KLSadd: " + word[word.find("{") + 1: lenword] - addendum[addendum.index(word)] = temp - if ("subsubsection*{" in word): - lenword = len(word) - 1 - addendum[addendum.index(word)] = word[0:word.find("{") + 1] + "\large\\bf KLSadd: " + word[word.find("{") + 1: lenword] - index = 0 - indexes = [] - # Designates sections that need stuff added - # get the index - for word in addendum: - index+=1 - if("." in word and "\\subsection*{" in word): - if ("9." in word): - chapNums.append(9) - name = word[word.find("{") + 1: word.find("}") ] - mathPeople.append(name + "#") - specref9.append(index-1) - if("14." in word): - chapNums.append(14) - name = word[word.find("{") + 1: word.find("}") ] - mathPeople.append(name + "#") - if len(specref14) == 0: - specref9.append(index - 1) - specref14.append(index - 1) - indexes.append(index-1) - if ("paragraph{" in word) and (index > 313): - klsaddparas.append(index-1) - if ("subsub" in word) and (index > 313): - klsaddparas.append(index - 1) - if ("\subsection*{" in word) and (index > 313): - klsaddparas.append(index - 1) - klsaddparas.append(2246) - #print(indexes) - #print(specref9) - #print(specref14) - #print(klsaddparas) - #print(mathPeople) - #now indexes holds all of the places there is a section - #using these indexes, get all of the words in between and add that to the paras[] - for i in range(len(indexes)-1): - box = ''.join(addendum[indexes[i]: indexes[i+1]-1]) - paras.append(box) - box2 = ''.join(addendum[indexes[35]: 2245]) - paras.append(box2) - #paras now holds the paragraphs that need to go into the chapter files, but they need to go in the appropriate - #section(like Wilson, Racah, Hahn, etc.) so we use the mathPeople variable - #we can use the section names to place the relevant paragraphs in the right place - - #as of 2/8/16 the paragraphs will go before the References paragraph of the relevant section - #parse both files 9 and 14 as strings - - #chapter 9 - with open("chap09.tex", "r") as ch9: - entire9 = ch9.readlines() #reads in as a list of strings - ch9.close() - - #chapter 14 - with open("chap14.tex", "r") as ch14: - entire14 = ch14.readlines() - ch14.close() - #call the findReferences method to find the index of the References paragraph in the two file strings - #two variables for the references lists one for chapter 9 one for chapter 14 - chapticker = 0 - references9 = findReferences(entire9) - chapticker += 1 - references14 = findReferences(entire14) - ref14III.insert(88, 1217) - ref14III.insert(244, 3053) - ref14III.insert(198, 2554) - #call the fixChapter method to get a list with the addendum paragraphs added in - chapticker2 = 0 - str9 = ''.join(fixChapter(entire9, references9, paras, addendum,ref9II,ref9III,specref9,klsaddparas,sectorskls,sectorschap,insertindices,insertloc,klslist)) - #with open("updated9.tex", "w") as temp9: - # temp9.write(str9) - #print("dignsakfibasjksdjhsfs") - #check4.append("memes") - chapticker2 += 1 - str14 = ''.join(fixChapter(entire14, references14, paras, addendum,ref14II,ref14III,specref14,klsaddparas,sectorskls,sectorschap,insertindices,insertloc,klslist)) - -""" -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -If you are writing something that will make a change to the chapter files, write it BEFORE this line, this part -is where the lists representing the words/strings in the chapter are joined together and updated as a string! -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -""" - - -#write to files -#new output files for safety -with open("updated9.tex","w") as temp9: - temp9.write(str9) - -with open("updated14.tex", "w") as temp14: - temp14.write(str14) -print(klsaddparas) -print(sorterIIIcheck) -print(ref9III) \ No newline at end of file + +def main(): + """ + Runs all of the other functions and outputs their results into an output file as well as putting together the list + of additions that is fed into fix_chapter. + + :return: + """ + # open the KLSadd file to do things with + with open("KLSadd(3).tex", "r") as add: + # store the file as a string + addendum = add.readlines() + kls_list_all = new_keywords(addendum) + # Makes sections look like other sections + + for word in addendum: + if "paragraph{" in word: + lenword = len(word) - 1 + temp = word[0:word.find("{") + 1] + "\large\\bf KLSadd: " + word[word.find("{") + 1: lenword] + addendum[addendum.index(word)] = temp + if "subsubsection*{" in word: + lenword = len(word) - 1 + addendum[addendum.index(word)] = word[0:word.find("{") + 1] + "\large\\bf KLSadd: " + \ + word[word.find("{") + 1: lenword] + index = 0 + indexes = [] + + # Designates sections that need stuff added + # get the index + for word in addendum: + index += 1 + if "." in word and "\\subsection*{" in word: + if "9." in word: + chap_nums.append(9) + name = word[word.find("{") + 1: word.find("}")] + math_people.append(name + "#") + specref9.append(index-1) + if "14." in word: + chap_nums.append(14) + name = word[word.find("{") + 1: word.find("}")] + math_people.append(name + "#") + if len(specref14) == 0: + specref9.append(index - 1) + specref14.append(index - 1) + indexes.append(index-1) + if "paragraph{" in word and index > 313: + klsaddparas.append(index-1) + if "subsub" in word and index > 313: + klsaddparas.append(index - 1) + if "\subsection*{" in word and index > 313: + klsaddparas.append(index - 1) + klsaddparas.append(2246) + print(indexes) + # now indexes holds all of the places there is a section + # using these indexes, get all of the words in between and add that to the paras[] + for i in range(len(indexes)-1): + box = ''.join(addendum[indexes[i]: indexes[i+1]-1]) + paras.append(box) + print(box) + box2 = ''.join(addendum[indexes[35]: 2245]) + paras.append(box2) + + + # paras now holds the paragraphs that need to go into the chapter files, but they need to go in the appropriate + # section(like Wilson, Racah, Hahn, etc.) so we use the mathPeople variable + # we can use the section names to place the relevant paragraphs in the right place + + # as of 2/8/16 the paragraphs will go before the References paragraph of the relevant section + # parse both files 9 and 14 as strings + + # chapter 9 + with open("chap09.tex", "r") as ch9: + entire9 = ch9.readlines() # reads in as a list of strings + + # chapter 14 + with open("chap14.tex", "r") as ch14: + entire14 = ch14.readlines() + + # call the findReferences method to find the index of the References paragraph in the two file strings + # two variables for the references lists one for chapter 9 one for chapter 14 + chapticker = 0 + references9 = find_references(entire9, chapticker) + chapticker += 1 + references14 = find_references(entire14, chapticker) + ref14_3.insert(88, 1217) + ref14_3.insert(244, 3053) + ref14_3.insert(198, 2554) + + # call the fixChapter method to get a list with the addendum paragraphs added in + chapticker2 = 0 + str9 = ''.join(fix_chapter(entire9, references9, paras, addendum, kls_list_all, chapticker2)) + + chapticker2 += 1 + str14 = ''.join(fix_chapter(entire14, references14, paras, addendum, kls_list_all, chapticker2)) + + # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + # If you are writing something that will make a change to the chapter files, write it BEFORE this line, this part + # is where the lists representing the words/strings in the chapter are joined together and updated as a string! + # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + + # write to files + # new output files for safety + with open("updated9.tex", "w") as temp9: + temp9.write(str9) + + with open("updated14.tex", "w") as temp14: + temp14.write(str14) + +if __name__ == '__main__': + main() \ No newline at end of file From 51ff4e71f7c1779b3ddf9fb1c40beaae15c99486 Mon Sep 17 00:00:00 2001 From: Edward Bian Date: Mon, 15 Aug 2016 11:17:43 -0400 Subject: [PATCH 235/402] Refactor directories and decancerify --- KLSadd_insertion/README.md | 105 +++++++ .../{src/KLSadd.pdf => data/KLSaddII.pdf} | Bin 414643 -> 414643 bytes .../{src/KLSadd.tex => data/KLSaddII.tex} | 2 +- KLSadd_insertion/{src => data}/KoLeSw.pdf | Bin KLSadd_insertion/src/__init__.py | 0 KLSadd_insertion/src/linetest.py | 278 +++++++++--------- KLSadd_insertion/src/updateChapters.py | 63 ++-- KLSadd_insertion/test/test_fix_chapter.py | 0 KLSadd_insertion/test/test_new_keywords.py | 6 + KLSadd_insertion/updateChaptersRoadMap.txt | 97 ------ 10 files changed, 288 insertions(+), 263 deletions(-) rename KLSadd_insertion/{src/KLSadd.pdf => data/KLSaddII.pdf} (99%) rename KLSadd_insertion/{src/KLSadd.tex => data/KLSaddII.tex} (99%) rename KLSadd_insertion/{src => data}/KoLeSw.pdf (100%) create mode 100644 KLSadd_insertion/src/__init__.py create mode 100644 KLSadd_insertion/test/test_fix_chapter.py create mode 100644 KLSadd_insertion/test/test_new_keywords.py delete mode 100644 KLSadd_insertion/updateChaptersRoadMap.txt diff --git a/KLSadd_insertion/README.md b/KLSadd_insertion/README.md index 71ad1ef..f08662d 100644 --- a/KLSadd_insertion/README.md +++ b/KLSadd_insertion/README.md @@ -15,5 +15,110 @@ NOTE: when you run updateChapters.py or linetest.py you need the KLSadd.tex file **THIS MEANS YOU SHOULD NOT RUN updateChapters.py OR linetest.py FROM YOUR GIT DIRECTORY. THE CHAPTER FILES ARE NOT PUBLIC DOMAIN, DO NOT COMMIT THEM! COPY THE FILES OVER INTO A SEPERATE DIRECTORY AND THEN COPY THEM BACK OVER WHEN YOU ARE READY TO COMMIT** +This is a roadmap of updateChapters.py. It will help explain every piece of code and exactly how everything works in relation to each other. + + +_______________________________________________________________________________ + + +First: the files + + - KLSadd.tex: This is the addendum we are working with. It has sections that correspond with sections in the chapter files, and it contains paragraphs that must be *inserted* into the chapter files. CAN BE FOUND ONINE IN PDF FORM: https://staff.fnwi.uva.nl/t.h.koornwinder/art/informal/KLSadd.pdf + + - chap09.tex and chap14.tex: these are chapter files. These are LATEX documents that are chapters in a book. This is a book written by smart math people. But they did some things wrong, so another math person wants to fix them. That math person wrote an addendum file called KLSadd.tex and our job is to pull paragraphs out of KLSadd and insert them at the end of the relevant section in the chapter files. Every chapter is made up of sections and every section has subsections + + - updateChapters.py: this is our code. This document will be going over the variables and methods in this program. This program should: take paragraphs from every section in KLSadd and insert them into the relevant chapter file and into the relevant section within that chapter. + + - linetest.py: this abomination of code is the original program. It did the job, and it did it well. However, I realized that it was very very unreadable. This program is the child I never wanted. This program does 100% work and its outputs can be used to check against updateChapters.py's output. + + - newtempchap09.tex and newtempchap14.tex these are output files of linetest.py and this is what the output to updateChapters.py should be like!!!!!!! +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +Next: the specifications + +Toward the end of every section in the chapter file, there is a subsection called "References". It basically just contains a bunch of references to sources and other papers. Every paragraph in every subsection in KLSadd.tex must be put *before* the "References" subsection in the corresponding section in the chapter file. For example: +**These aren't real sections, obviously, just an example** + +in chap09.tex: +```latex +\section{Dogs}\index{Dog polynomials} +\subsection*{Hypergeometric dogs} +blah blah blah +\subsection*{References} +\cite{Dogs101} +``` + +in KLSadd.tex: + +```latex +\subsection*{9.15 Dogs} +\paragraph{Beagles} +The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetric +in $a,b,c,d$. +``` + +So in the chapter 9 file, section 15 "Dogs", before the References subsection, we need to add in the Beagles paragraph. HOWEVER, there are some changes that need to be made before we insert. The easiest change is adding a comment that tells us something was inserted. We just append something like "%RS insertion begin" and "%RS insertion complete" before and after, respectively, the Beagles paragraph when we insert. Another change is adding \large\bf before the paragraph name. This changes the size and format so it looks prettier. So the heading in this example would be: + +```latex +\paragraph{\large\bf Beagles} +``` + +**It is important to note that these changes are currently (3/25/16) unimplemented in updateChapters.py and only exist in linetest.py, if you want to see how the output to updateChapters.py should look, look at newtempchap09.tex** + +So after updateChapters.py is called, the chapter 9 file should look like this: + +```latex +\section{Dogs}\index{Dog polynomials} +\subsection*{Hypergeometric dogs} +blah blah blah +%RS insertion begin +\paragraph{\large\bf Beagles} +The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetric +in $a,b,c,d$. +%RS insertion complete +\subsection*{References} +\cite{Dogs101} +``` + +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +The variables: + +These are the variables in the beginning + +-chapNums: chapNums is a list, denoted by the [], that holds the chapter number (in this case it's either a 9 or a 14) that corresponds to where a section in KLSadd should go + +-paras: holds all of the text in the paragraphs of KLSadd.tex. This is the list that gets read when its time to insert paragraphs into the chapter file. Every element is a loooong string + +-mathPeople: this holds the *name* of every section in KLSadd. It stores names like Wilson, Racah, Dual Hahn, etc. In our example with the Dogs above, it would hold "Dogs". This is useful to finding where to insert the correct paragraph + +-newCommands: this is a list that holds ints representing line numbers in KLSadd.tex that correspond to commands. There are a few special commands in the file that help turn LATEX files into PDF files and they need to be copied over into both chapter files + +-comms: holds the actual strings of the commands found from the line numbers stored in newCommands + +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +The methods: + +-prepareForPDF(chap): this method inserts some packages needed by LATEX files to be turned into pdf files. It takes a list called chap which is a String containing the contents of the chapter fie. It returns the chap String edited with the special packages in place. + +-getCommands(kls): takes the KLSadd.tex as a string as a parameter and stores the special commands in the comms variable mentioned earlier + +-insertCommands(kls, chap, cms): kls is KLSadd.tex, chap is a chapter file string, cms is a list of commands, it is the comms variable. This method returns the chap with the special commands inserted in place + +-findReferences(str): takes a string representation of a chapter file. Returns a list of the line numbers of the "References" subsections. This references variable is used as an index as to where the paragraphs in paras should be inserted + +-fixChapter(chap, references, p, kls): takes a chapter file string, a references list, the paras variable, and the KLSadd file string. This method basically just calls all of the methods above and adds all of the extra stuff like the commands and packages. +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +At this point, the code is like 90% comments. The big chunk at the bottom under the "with open..." can be explained through the comments. + +As a quick rundown: + +The chapters and KLSadd are turned into Strings and passed through the fixChapter methods. + +Then the strings are written into seperate updated chapter files. + +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +However, its broken. Inserted text comes in random places between paragraphs and sometimes stuff isn't added. This shouldn't be that hard to fix but it will require combing through the code a lot of times and changing little things here and there until everything works. As a last ditch effort, we can always resort to using linetext.py and adding stuff in from there. diff --git a/KLSadd_insertion/src/KLSadd.pdf b/KLSadd_insertion/data/KLSaddII.pdf similarity index 99% rename from KLSadd_insertion/src/KLSadd.pdf rename to KLSadd_insertion/data/KLSaddII.pdf index c0562e08dde4eec14d3e67b54f20dbaa5f694ed5..c903e54b27ce6517a9d4e62c3c7f8fa955b90a63 100644 GIT binary patch delta 185 zcmdnITypbr$%Yoj7N!>F7M2#)Eo_apjONpuY}u4~EDQ_{&5X^A40R1mrf;-m6Gs)f zV#^k)VXmMNk{Y4llUbIkU}Ruurm2u!kXVvYoSLXmmReMtnV+X%re~yQv^~&{?K!8j tsjG>hlbNBbse!YL0nltWQ&(3*Am7r>(ag}?*w{|NhLDo&)gEjX+yJx3F%JL$ delta 185 zcmdnITypbr$%Yoj7N!>F7M2#)Eo_apjAqlDY}u4~42_Ho4UJ8W%ybP*r*E`n6Gs)f zV#^k)VWyxFk{Y4llUbIkU}Ruus;Q7&kXVvYoSLXmmReMtnV+X%re~yQxINI0?K!8j vnW>YBlckBFiJ7C3qmzr1iIJ&=o1>Ylg}I5dqm!wnoq`P^CEKe#*etjKwHGl> diff --git a/KLSadd_insertion/src/KLSadd.tex b/KLSadd_insertion/data/KLSaddII.tex similarity index 99% rename from KLSadd_insertion/src/KLSadd.tex rename to KLSadd_insertion/data/KLSaddII.tex index e8de6df..f9f3e63 100644 --- a/KLSadd_insertion/src/KLSadd.tex +++ b/KLSadd_insertion/data/KLSaddII.tex @@ -2634,4 +2634,4 @@ \subsection*{14.29 Discrete $q$-Hermite II} email: }{\tt T.H.Koornwinder@uva.nl} \end{quote} \end{footnotesize} -\end{document} +\end{document} \ No newline at end of file diff --git a/KLSadd_insertion/src/KoLeSw.pdf b/KLSadd_insertion/data/KoLeSw.pdf similarity index 100% rename from KLSadd_insertion/src/KoLeSw.pdf rename to KLSadd_insertion/data/KoLeSw.pdf diff --git a/KLSadd_insertion/src/__init__.py b/KLSadd_insertion/src/__init__.py new file mode 100644 index 0000000..e69de29 diff --git a/KLSadd_insertion/src/linetest.py b/KLSadd_insertion/src/linetest.py index ebdc7d9..19de78a 100644 --- a/KLSadd_insertion/src/linetest.py +++ b/KLSadd_insertion/src/linetest.py @@ -1,10 +1,7 @@ -__author__ = "Rahul Shah" -__status__ = "Development" - """ TODO: Rewrite the whole thing so another human being can read it jeez this is like my handwriting; it's nonsense to everyone but me -This started out as a test file but I just kept writing it +This started out as a test file but I just kept writing it and writing it and writing it and now its the project @@ -17,151 +14,152 @@ sections = [] mathPeople = [] newCommands = [] -# can you tell I like lists? +#can you tell I like lists? name = "" -temp = "" +temp = "" with open("KLSadd.tex", "r") as file: - entireFile = file.readlines() - for word in entireFile: - index += 1 - if("smallskipamount" in word): - newCommands.append(index - 1) - if("mybibitem[1]" in word): - newCommands.append(index) - # 1/27/16 RS added \large\bf KLSadd: to make KLSadd additions appear - # like the other paragraphs - if("paragraph{" in word): - temp = word[0:word.find( - "{") + 1] + "\large\\bf KLSadd: " + word[word.find("{") + 1: word.find("}") + 1] - entireFile[ - entireFile.index(word): entireFile.index(word) + - 1] = temp - if("\\subsection*{" in word and "." in word): - subsections.append(index) - headings.append(word) - name = word[word.find(" ") + 1:word.find("}")] - mathPeople.append(name) - name = "" - # 1/29/16 get all of the new commands - comms = ''.join(entireFile[newCommands[0]:newCommands[1]]) - paras = [] - for i in range(len(subsections) - 1): - str = ''.join(entireFile[subsections[i]: subsections[i + 1] - 1]) - paras.append(str) - sec = "" - for f in headings: - for i in f: - try: - int(i) - sec += i - except: - pass - if("." in i): - sec += i - sections.append(sec) - + entireFile = file.readlines() + for word in entireFile: + index+=1 + if("smallskipamount" in word): + newCommands.append(index-1) + if("mybibitem[1]" in word): + newCommands.append(index) + #1/27/16 RS added \large\bf KLSadd: to make KLSadd additions appear like the other paragraphs + if("paragraph{" in word): + temp = word[0:word.find("{")+ 1] + "\large\\bf KLSadd: " + word[word.find("{")+ 1: word.find("}")+1] + entireFile[entireFile.index(word): entireFile.index(word)+1] = temp + if("\\subsection*{" in word and "." in word): + subsections.append(index) + headings.append(word) + name = word[word.find(" ")+1:word.find("}")] + mathPeople.append(name) + name = "" + #1/29/16 get all of the new commands + comms = ''.join(entireFile[newCommands[0]:newCommands[1]]) + paras = [] + for i in range(len(subsections)-1): + str = ''.join(entireFile[subsections[i]: subsections[i+1]-1]) + paras.append(str) sec = "" - # sections now holds all sections + for f in headings: + for i in f: + try: + int(i) + sec += i + except: + pass + if("." in i): + sec+=i + sections.append(sec) + + sec = "" + #sections now holds all sections - numFile = "" - filenums = [] - for s in sections: numFile = "" - if("." in s[0:2]): - numFile = s[0] - else: - numFile = s[0:2] - filenums.append(int(numFile)) - # filenums now has the file numbers like 1,9,14,etc + filenums = [] + for s in sections: + numFile = "" + if("." in s[0:2]): + numFile = s[0] + else: + numFile = s[0:2] + filenums.append(int(numFile)) + #filenums now has the file numbers like 1,9,14,etc + + #SO python is nothing like java, I have to read in the entire chapter + #file and then change it and THEN put it back into the chapter file + + #chapter 9 + with open("tempchap9.tex", "r") as ch9: + entire9 = ch9.readlines() + ch9.close() + #chapter 14 + with open("tempchap14.tex", "r") as ch14: + entire14 = ch14.readlines() + ch14.close() + + references9 = [] + footmiscIndex = 0 + index = 0 + beginIndex9= -1 + #now I have a list of the names, only store indices + #of the paragraph if its in a section with these names + #TODO make a method to do this + canAdd = False + for word in entire9: + index+=1 + if("begin{document}" in word): + beginIndex9 += index + if("footmisc" in word): + footmiscIndex+= index + #need to check subsection and section because + #KLSadd also fixes some subsections + if("section{" in word or "subsection*{" in word): + w = word[word.find("{")+1:word.find("}")] + if(w in mathPeople): + canAdd = True + if("\\subsection*{References}" in word and canAdd == True): + references9.append(index) + canAdd = False + canAdd = False + references14 = [] + index = 0 + footmiscIndex14 = 0 + beginIndex14 = -1 + for word in entire14: + index+=1 + if("begin{document}" in word): + beginIndex14 += index + if("footmisc" in word): + footmiscIndex14+= index + #need to check subsection and section because + #KLSadd also fixes some subsections + if("section{" in word or "subsection*{" in word): + w = word[word.find("{")+1:word.find("}")] + if(w in mathPeople): + canAdd = True + if("\\subsection*{References}" in word and canAdd == True): + references14.append(index) + canAdd = False + #references9 is now full of indexes to put fixed paragraphs + + #TODO make into a method + with open("newtempchap09.tex", "w") as temp9: + count = 0 - # SO python is nothing like java, I have to read in the entire chapter - # file and then change it and THEN put it back into the chapter file + #1/29/16 add hyperref, xparse, cite packages after the footmisc package + entire9[footmiscIndex] += "\\usepackage[pdftex]{hyperref} \n\\usepackage {xparse} \n\\usepackage{cite} \n" + #1/29/16 add in the newcommands from KLSadd.tex + entire9[beginIndex9-1] += comms + #1/27/16 add \large\bf KLSadd: after every paragraph{ + for i in references9: + entire9[i-3] += "% RS: add begin" + entire9[i-3] += paras[count] + entire9[i-3] += "% RS: add end" + count+=1 + entire9[i-1] += paras[count] - # chapter 9 - with open("tempchap9.tex", "r") as ch9: - entire9 = ch9.readlines() - ch9.close() - # chapter 14 - with open("tempchap14.tex", "r") as ch14: - entire14 = ch14.readlines() - ch14.close() + + str = ''.join(entire9) + temp9.write(str) + - references9 = [] - footmiscIndex = 0 - index = 0 - beginIndex9 = -1 - # now I have a list of the names, only store indices - # of the paragraph if its in a section with these names - # TODO make a method to do this - canAdd = False - for word in entire9: - index += 1 - if("begin{document}" in word): - beginIndex9 += index - if("footmisc" in word): - footmiscIndex += index - # need to check subsection and section because - # KLSadd also fixes some subsections - if("section{" in word or "subsection*{" in word): - w = word[word.find("{") + 1:word.find("}")] - if(w in mathPeople): - canAdd = True - if("\\subsection*{References}" in word and canAdd): - references9.append(index) - canAdd = False - canAdd = False - references14 = [] - index = 0 - footmiscIndex14 = 0 - beginIndex14 = -1 - for word in entire14: - index += 1 - if("begin{document}" in word): - beginIndex14 += index - if("footmisc" in word): - footmiscIndex14 += index - # need to check subsection and section because - # KLSadd also fixes some subsections - if("section{" in word or "subsection*{" in word): - w = word[word.find("{") + 1:word.find("}")] - if(w in mathPeople): - canAdd = True - if("\\subsection*{References}" in word and canAdd): - references14.append(index) - canAdd = False - # references9 is now full of indexes to put fixed paragraphs - # TODO make into a method - with open("newtempchap09.tex", "w") as temp9: - count = 0 + #works, as far as I know, huzzah + #references14 is now full of indexes to put fixed paragraphs + with open("newtempchap14.tex", "w") as temp14: + entire14[footmiscIndex14] += "\\usepackage[pdftex]{hyperref} \n\\usepackage {xparse} \n\\usepackage{cite} \n" + entire14[beginIndex14 -3] += comms + for i in references14: + entire14[i-3] += " RS: add begin" + entire14[i-3] += paras[count] + #entire14[i-3] += "% RS: add end" + count+=1 + str = ''.join(entire14) + temp14.write(str) + - # 1/29/16 add hyperref, xparse, cite packages after the footmisc - # package - entire9[ - footmiscIndex] += "\\usepackage[pdftex]{hyperref} \n\\usepackage {xparse} \n\\usepackage{cite} \n" - # 1/29/16 add in the newcommands from KLSadd.tex - entire9[beginIndex9 - 1] += comms - # 1/27/16 add \large\bf KLSadd: after every paragraph{ - for i in references9: - entire9[i - 3] += "% RS: add begin" - entire9[i - 3] += paras[count] - entire9[i - 3] += "% RS: add end" - count += 1 - entire9[i - 1] += paras[count] + - str = ''.join(entire9) - temp9.write(str) - # works, as far as I know, huzzah - # references14 is now full of indexes to put fixed paragraphs - with open("newtempchap14.tex", "w") as temp14: - entire14[ - footmiscIndex14] += "\\usepackage[pdftex]{hyperref} \n\\usepackage {xparse} \n\\usepackage{cite} \n" - entire14[beginIndex14 - 3] += comms - for i in references14: - entire14[i - 3] += " RS: add begin" - entire14[i - 3] += paras[count] - #entire14[i-3] += "% RS: add end" - count += 1 - str = ''.join(entire14) - temp14.write(str) diff --git a/KLSadd_insertion/src/updateChapters.py b/KLSadd_insertion/src/updateChapters.py index 4dc9f89..a188d3a 100644 --- a/KLSadd_insertion/src/updateChapters.py +++ b/KLSadd_insertion/src/updateChapters.py @@ -1,7 +1,14 @@ +import os + __author__ = "Rahul Shah" __status__ = "Development" __credits__ = ["Rahul Shah", "Edward Bian"] +# Path to data directory +DATA_DIR = os.path.dirname(os.path.realpath(__file__)) + "/../data/" + + + # start out by reading KLSadd.tex to get all of the paragraphs that must be added to the chapter files # also keep track of which chapter each one is in @@ -45,7 +52,8 @@ sortmatch_2 = [] - +test = "memesbananashgvlksrjc%/n" +print test[-2] # 2/18/16 this method addresses the goal of hardcoding in the necessary packages to let the chapter files run as pdf's. # Currently only works with chapter 9, ask Dr. Cohl to help port your chapter 14 output file into a pdf @@ -84,10 +92,6 @@ def new_keywords(kls): add = False if add: kls_list_chap.append(i) - liststr2 = '\n'.join(kls_list_chap) - with open("trunckeywords.tex", "w") as kw: - kw.write(liststr2) - print kls_list_chap return kls_list_chap @@ -272,14 +276,11 @@ def cutwords(word_to_find, word_to_search_in): while checkloop == False: if "\\paragraph{\\large\\bf KLSadd: " in b[b.find(a)-precheck:b.find(a)]: cut = b[:b.find(a) - precheck] + b[b.find(a) + len(a):] - checkloop = True return cut elif "\\subsubsection*{\\large\\bf KLSadd: " in b[b.find(a)-precheck:b.find(a)]: cut = b[:b.find(a) - precheck] + b[b.find(a) + len(a):] - checkloop = True return cut else: - print "ding2" precheck += 1 else: @@ -352,7 +353,7 @@ def insert_commands(kls, chap, cms): # method to find the indices of the reference paragraphs -def find_references(chapter,chapticker): +def find_references(chapter, chapticker): """ This function searches the chapters and locates potential destinations of additions from the addendum. It puts these destinations in a list. @@ -492,20 +493,32 @@ def fix_chapter(chap, references, p, kls, kls_list_all, chapticker2): for a in range(len(p)-1): for b in sortmatch_2: for c in b: - with open("compare.tex", "a") as spook: + with open(DATA_DIR + "compare.tex", "a") as spook: spook.write(p[a]) spook.write("NEXT: ") - print c - print "pause" - print a - print "l0" - print len(c) - print "l1" - print a - print p[a] + if "%" == c[-2]: + c = c[:-3] + #print "memes" + #print c + elif "%" == c[-1]: + c = c[:-2] + #print "extramemes" + #print c + if "Formula (9.8.15) was first obtained by Brafman \myciteKLS{109}." in c: + print "memes" + print c + print "memes" + if "Formula (9.8.15) was first obtained by Brafman \myciteKLS{109}." in p[a]: + print "extramemes" + print p[a] + print "extramemes" + #if "9.1 Wilson" in p[a]: + # print "l2" + # print c + # print "pause" + # print p[a] + # print "l2" p[a] = cutwords(c, p[a]) - print("l2") - reference_placer(chap, references, p, chapticker2) chap = prepare_for_PDF(chap) @@ -548,7 +561,7 @@ def main(): :return: """ # open the KLSadd file to do things with - with open("KLSadd(3).tex", "r") as add: + with open(DATA_DIR + "KLSaddII.tex", "r") as add: # store the file as a string addendum = add.readlines() kls_list_all = new_keywords(addendum) @@ -610,11 +623,11 @@ def main(): # parse both files 9 and 14 as strings # chapter 9 - with open("chap09.tex", "r") as ch9: + with open(DATA_DIR + "chap09.tex", "r") as ch9: entire9 = ch9.readlines() # reads in as a list of strings # chapter 14 - with open("chap14.tex", "r") as ch14: + with open(DATA_DIR + "chap14.tex", "r") as ch14: entire14 = ch14.readlines() # call the findReferences method to find the index of the References paragraph in the two file strings @@ -641,10 +654,10 @@ def main(): # write to files # new output files for safety - with open("updated9.tex", "w") as temp9: + with open(DATA_DIR + "updated9.tex", "w") as temp9: temp9.write(str9) - with open("updated14.tex", "w") as temp14: + with open(DATA_DIR + "updated14.tex", "w") as temp14: temp14.write(str14) if __name__ == '__main__': diff --git a/KLSadd_insertion/test/test_fix_chapter.py b/KLSadd_insertion/test/test_fix_chapter.py new file mode 100644 index 0000000..e69de29 diff --git a/KLSadd_insertion/test/test_new_keywords.py b/KLSadd_insertion/test/test_new_keywords.py new file mode 100644 index 0000000..45e5add --- /dev/null +++ b/KLSadd_insertion/test/test_new_keywords.py @@ -0,0 +1,6 @@ + +__author__ = 'Edward Bian' +__status__ = 'Development' + +from unittest import TestCase +from updateChapters import \ No newline at end of file diff --git a/KLSadd_insertion/updateChaptersRoadMap.txt b/KLSadd_insertion/updateChaptersRoadMap.txt deleted file mode 100644 index e89b864..0000000 --- a/KLSadd_insertion/updateChaptersRoadMap.txt +++ /dev/null @@ -1,97 +0,0 @@ -This is a roadmap of updateChapters.py. It will help explain every piece of code and exactly how everything works in relation to each other. - - -_______________________________________________________________________________ - - -First: the files - ->KLSadd.tex: This is the addendum we are working with. It has sections that correspond with sections in the chapter files, and it contains paragraphs that must be *inserted* into the chapter files. CAN BE FOUND ONINE IN PDF FORM: https://staff.fnwi.uva.nl/t.h.koornwinder/art/informal/KLSadd.pdf - ->chap09.tex and chap14.tex: these are chapter files. These are LATEX documents that are chapters in a book. This is a book written by smart math people. But they did some things wrong, so another math person wants to fix them. That math person wrote an addendum file called KLSadd.tex and our job is to pull paragraphs out of KLSadd and insert them at the end of the relevant section in the chapter files. Every chapter is made up of sections and every section has subsections - ->updateChapters.py: this is our code. This document will be going over the variables and methods in this program. This program should: take paragraphs from every section in KLSadd and insert them into the relevant chapter file and into the relevant section within that chapter. - ->linetest.py: this abomination of code is the original program. It did the job, and it did it well. However, I realized that it was very very unreadable. This program is the child I never wanted. This program does 100% work and its outputs can be used to check against updateChapters.py's output. - ->newtempchap09.tex and newtempchap14.tex these are output files of linetest.py and this is what the output to updateChapters.py should be like!!!!!!! -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - -Next: the specifications - -Toward the end of every section in the chapter file, there is a subsection called "References". It basically just contains a bunch of references to sources and other papers. Every paragraph in every subsection in KLSadd.tex must be put *before* the "References" subsection in the corresponding section in the chapter file. For example: -**These aren't real sections, obviously, just an example** - -in chap09.tex: ->\section{Dogs}\index{Dog polynomials} ->\subsection*{Hypergeometric dogs} ->blah blah blah ->\subsection*{References} ->\cite{Dogs101} - -in KLSadd.tex: ->\subsection*{9.15 Dogs} ->\paragraph{Beagles} ->The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetric ->in $a,b,c,d$. - -So in the chapter 9 file, section 15 "Dogs", before the References subsection, we need to add in the Beagles paragraph. HOWEVER, there are some changes that need to be made before we insert. The easiest change is adding a comment that tells us something was inserted. We just append something like "%RS insertion begin" and "%RS insertion complete" before and after, respectively, the Beagles paragraph when we insert. Another change is adding \large\bf before the paragraph name. This changes the size and format so it looks prettier. So the heading in this example would be: - ->\paragraph{\large\bf Beagles} - -**It is important to note that these changes are currently (3/25/16) unimplemented in updateChapters.py and only exist in linetest.py, if you want to see how the output to updateChapters.py should look, look at newtempchap09.tex** - -So after updateChapters.py is called, the chapter 9 file should look like this: - ->\section{Dogs}\index{Dog polynomials} ->\subsection*{Hypergeometric dogs} ->blah blah blah ->%RS insertion begin ->\paragraph{\large\bf Beagles} ->The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetric ->in $a,b,c,d$. ->%RS insertion complete ->\subsection*{References} ->\cite{Dogs101} - - -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -The variables: - -These are the variables in the beginning - --chapNums: chapNums is a list, denoted by the [], that holds the chapter number (in this case it's either a 9 or a 14) that corresponds to where a section in KLSadd should go - --paras: holds all of the text in the paragraphs of KLSadd.tex. This is the list that gets read when its time to insert paragraphs into the chapter file. Every element is a loooong string - --mathPeople: this holds the *name* of every section in KLSadd. It stores names like Wilson, Racah, Dual Hahn, etc. In our example with the Dogs above, it would hold "Dogs". This is useful to finding where to insert the correct paragraph - --newCommands: this is a list that holds ints representing line numbers in KLSadd.tex that correspond to commands. There are a few special commands in the file that help turn LATEX files into PDF files and they need to be copied over into both chapter files - --comms: holds the actual strings of the commands found from the line numbers stored in newCommands - -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -The methods: - --prepareForPDF(chap): this method inserts some packages needed by LATEX files to be turned into pdf files. It takes a list called chap which is a String containing the contents of the chapter fie. It returns the chap String edited with the special packages in place. - --getCommands(kls): takes the KLSadd.tex as a string as a parameter and stores the special commands in the comms variable mentioned earlier - --insertCommands(kls, chap, cms): kls is KLSadd.tex, chap is a chapter file string, cms is a list of commands, it is the comms variable. This method returns the chap with the special commands inserted in place - --findReferences(str): takes a string representation of a chapter file. Returns a list of the line numbers of the "References" subsections. This references variable is used as an index as to where the paragraphs in paras should be inserted - --fixChapter(chap, references, p, kls): takes a chapter file string, a references list, the paras variable, and the KLSadd file string. This method basically just calls all of the methods above and adds all of the extra stuff like the commands and packages. -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - -At this point, the code is like 90% comments. The big chunk at the bottom under the "with open..." can be explained through the comments. - -As a quick rundown: - -The chapters and KLSadd are turned into Strings and passed through the fixChapter methods. - -Then the strings are written into seperate updated chapter files. - -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - -However, its broken. Inserted text comes in random places between paragraphs and sometimes stuff isn't added. This shouldn't be that hard to fix but it will require combing through the code a lot of times and changing little things here and there until everything works. As a last ditch effort, we can always resort to using linetext.py and adding stuff in from there. From b80bbc03c99373584143c5ffd48154f5dd117d70 Mon Sep 17 00:00:00 2001 From: Edward Bian Date: Mon, 15 Aug 2016 11:52:48 -0400 Subject: [PATCH 236/402] Update .travis.yml with KLSadd_insertion --- .travis.yml | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/.travis.yml b/.travis.yml index dd5c68f..0dac659 100644 --- a/.travis.yml +++ b/.travis.yml @@ -2,6 +2,6 @@ language: python install: - pip install coveralls script: - nosetests --with-coverage tex2Wiki DLMF_preprocessing maple2latex + nosetests --with-coverage tex2Wiki DLMF_preprocessing maple2latex KLSadd_insertion after_success: coveralls From eff1e484e185b82e29b31ebea71dc9ebb4e5ea6f Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Mon, 15 Aug 2016 13:08:59 -0400 Subject: [PATCH 237/402] Add tests for piecewise function --- eCF/src/MathematicaToLaTeX.py | 2 +- eCF/test/test_piecewise.py | 46 +++++++++++++++++++++++++++++++++++ 2 files changed, 47 insertions(+), 1 deletion(-) diff --git a/eCF/src/MathematicaToLaTeX.py b/eCF/src/MathematicaToLaTeX.py index 4bc54e9..7d753a4 100644 --- a/eCF/src/MathematicaToLaTeX.py +++ b/eCF/src/MathematicaToLaTeX.py @@ -762,7 +762,7 @@ def main(): Opens Mathematica file with identities and puts converted lines into newIdentities.tex. """ - test = False + test = True with open(os.path.dirname(os.path.realpath(__file__)) + '/../data/newIdentities.tex', 'w') as latex: diff --git a/eCF/test/test_piecewise.py b/eCF/test/test_piecewise.py index 5775dde..7ee6298 100644 --- a/eCF/test/test_piecewise.py +++ b/eCF/test/test_piecewise.py @@ -5,3 +5,49 @@ from unittest import TestCase from MathematicaToLaTeX import piecewise +class TestPiecewise(TestCase): + + def test_single(self): + self.assertEqual(piecewise( + 'Piecewise[{{var,cond}}]'), + '{\\begin{cases} var & cond \\\\ 0 & \\text{True} \\end{cases}}') + self.assertEqual(piecewise( + '--Piecewise[{{var,cond}}]--'), + '--{\\begin{cases} var & cond \\\\ 0 & \\text{True} \\end{cases}}--') + self.assertEqual(piecewise( + 'Piecewise[{{var,cond}},0]'), + '{\\begin{cases} var & cond \\\\ 0 & \\text{True} \\end{cases}}') + self.assertEqual(piecewise( + '--Piecewise[{{var,cond}},0]--'), + '--{\\begin{cases} var & cond \\\\ 0 & \\text{True} \\end{cases}}--') + self.assertEqual(piecewise( + 'Piecewise[{{var,cond}},num]'), + '{\\begin{cases} var & cond \\\\ num & \\text{True} \\end{cases}}') + self.assertEqual(piecewise( + '--Piecewise[{{var,cond}},num]--'), + '--{\\begin{cases} var & cond \\\\ num & \\text{True} \\end{cases}}--') + + def test_multiple(self): + self.assertEqual(piecewise( + 'Piecewise[{{var1,cond1},{var2,cond2}},0]'), + '{\\begin{cases} var1 & cond1 \\\\ var2 & cond2 \\\\ 0 & \\text{True} \\end{cases}}') + self.assertEqual(piecewise( + '--Piecewise[{{var1,cond1},{var2,cond2}},0]--'), + '--{\\begin{cases} var1 & cond1 \\\\ var2 & cond2 \\\\ 0 & \\text{True} \\end{cases}}--') + self.assertEqual(piecewise( + 'Piecewise[{{var1,cond1},{var2,cond2},{var3,cond3}},0]'), + '{\\begin{cases} var1 & cond1 \\\\ var2 & cond2 \\\\ var3 & cond3 \\\\ 0 & \\text{True} \\end{cases}}') + self.assertEqual(piecewise( + '--Piecewise[{{var1,cond1},{var2,cond2},{var3,cond3}},0]--'), + '--{\\begin{cases} var1 & cond1 \\\\ var2 & cond2 \\\\ var3 & cond3 \\\\ 0 & \\text{True} \\end{cases}}--') + + def test_nested(self): + self.assertEqual(piecewise( + 'Piecewise[{{Piecewise[{{var,Piecewise[{{var,cond}}]}}],Piecewise[{{var,cond}}]}}]'), + '{\\begin{cases} {\\begin{cases} var & {\\begin{cases} var & cond \\\\ 0 & \\text{True} \\end{cases}} \\\\ 0 & \\text{True} \\end{cases}} & {\\begin{cases} var & cond \\\\ 0 & \\text{True} \\end{cases}} \\\\ 0 & \\text{True} \\end{cases}}') + self.assertEqual(piecewise( + '--Piecewise[{{Piecewise[{{var,Piecewise[{{var,cond}}]}}],Piecewise[{{var,cond}}]}}]--'), + '--{\\begin{cases} {\\begin{cases} var & {\\begin{cases} var & cond \\\\ 0 & \\text{True} \\end{cases}} \\\\ 0 & \\text{True} \\end{cases}} & {\\begin{cases} var & cond \\\\ 0 & \\text{True} \\end{cases}} \\\\ 0 & \\text{True} \\end{cases}}--') + + def test_none(self): + self.assertEqual(piecewise('nopiecewise'), 'nopiecewise') From f6d3a29590ed5d8243c1705e41c9a26701493fae Mon Sep 17 00:00:00 2001 From: Joon Bang Date: Mon, 15 Aug 2016 14:48:19 -0400 Subject: [PATCH 238/402] Increase coverage (1) --- main_page/test/add_usage_tests/res.1.txt | 8 ++++++-- main_page/test/fake.Glossary.csv | 4 ++-- 2 files changed, 8 insertions(+), 4 deletions(-) diff --git a/main_page/test/add_usage_tests/res.1.txt b/main_page/test/add_usage_tests/res.1.txt index 2ce81ca..b4d2586 100644 --- a/main_page/test/add_usage_tests/res.1.txt +++ b/main_page/test/add_usage_tests/res.1.txt @@ -7,8 +7,10 @@ The LaTeX DLMF and DRMF macro '''\DefinitionOne''' represents the DefinitionOne. This macro is in the category of real or complex valued functions. -In math mode, this macro can be called in the following way: +In math mode, this macro can be called in the following ways: +:'''\DefinitionOne''' produces {\displaystyle \DefinitionOne}
+:'''\DefinitionOne@{x}''' produces {\displaystyle \DefinitionOne@{x}}
:'''\DefinitionOne@{x}''' produces {\displaystyle \DefinitionOne@{x}}
These are defined by @@ -23,8 +25,10 @@ The LaTeX DLMF and DRMF macro '''\DefinitionTwo''' represents the DefinitionTwo. This macro is in the category of real or complex valued functions. -In math mode, this macro can be called in the following way: +In math mode, this macro can be called in the following ways: +:'''\DefinitionTwo''' produces {\displaystyle \DefinitionTwo}
+:'''\DefinitionTwo@{x}''' produces {\displaystyle \DefinitionTwo@{x}}
:'''\DefinitionTwo@{x}''' produces {\displaystyle \DefinitionTwo@{x}}
These are defined by diff --git a/main_page/test/fake.Glossary.csv b/main_page/test/fake.Glossary.csv index 30db2f0..3e367dd 100644 --- a/main_page/test/fake.Glossary.csv +++ b/main_page/test/fake.Glossary.csv @@ -1,3 +1,3 @@ -\DefinitionOne@{x},sample definition 1,sample-definition-1,F|P,$\mathrm{Def1}{x}$,http://fake-hyperlink.com/Definition1, -\DefinitionTwo@{x},sample definition 2,sample-definition-2,F|P,$\mathrm{Def2}{x}$,http://fake-hyperlink.com/Definition2, +\DefinitionOne@{x},sample definition 1,sample-definition-1,F|F:P:nP,$\mathrm{Def1}{x}$,http://fake-hyperlink.com/Definition1, +\DefinitionTwo@{x},sample definition 2,sample-definition-2,F|FnO:PnO:nPnO,$\mathrm{Def2}{x}$,http://fake-hyperlink.com/Definition2, \DefinitionTwoTwo,sample definition 22,sample-definition-22,C|P,$\mathrm{twenty two}$,http://fake-hyperlink.com/Definition22, \ No newline at end of file From 7e247dac290e2a005dfd26687d8d1b3cb3c6510e Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Mon, 15 Aug 2016 14:49:01 -0400 Subject: [PATCH 239/402] Add tests for replace_vars function --- eCF/src/MathematicaToLaTeX.py | 14 ++++++------- eCF/test/test_replace_vars.py | 37 +++++++++++++++++++++++++++++++++++ 2 files changed, 44 insertions(+), 7 deletions(-) diff --git a/eCF/src/MathematicaToLaTeX.py b/eCF/src/MathematicaToLaTeX.py index 7d753a4..ddf081a 100644 --- a/eCF/src/MathematicaToLaTeX.py +++ b/eCF/src/MathematicaToLaTeX.py @@ -13,7 +13,7 @@ import os -symbols = { +SYMBOLS = { 'Alpha': 'alpha', 'Beta': 'beta', 'Gamma': 'gamma', 'Delta': 'delta', 'Epsilon': 'epsilon', 'Zeta': 'zeta', 'Eta': 'eta', 'Theta': 'theta', 'Iota': 'iota', 'Kappa': 'kappa', 'Lambda': 'lambda', 'Mu': 'mu', @@ -746,13 +746,13 @@ def replace_vars(line): Replaces the easy to convert variables in Mathematica to its equivalent LaTeX code in the dictionary 'symbols'. """ - for word in symbols: - if symbols[word][0] == ' ': - line = line.replace('\\[' + word + ']', symbols[word][1:]) + for word in SYMBOLS: + if SYMBOLS[word][0] == ' ': + line = line.replace('\\[' + word + ']', SYMBOLS[word][1:]) elif word == 'Infinity': - line = line.replace('Infinity', r'\infty') + line = line.replace('Infinity', '\\infty') else: - line = line.replace('[' + word + ']', symbols[word]) + line = line.replace('[' + word + ']', SYMBOLS[word]) return line @@ -762,7 +762,7 @@ def main(): Opens Mathematica file with identities and puts converted lines into newIdentities.tex. """ - test = True + test = False with open(os.path.dirname(os.path.realpath(__file__)) + '/../data/newIdentities.tex', 'w') as latex: diff --git a/eCF/test/test_replace_vars.py b/eCF/test/test_replace_vars.py index 041adc4..9423174 100644 --- a/eCF/test/test_replace_vars.py +++ b/eCF/test/test_replace_vars.py @@ -5,3 +5,40 @@ from unittest import TestCase from MathematicaToLaTeX import replace_vars +SYMBOLS = { + 'Alpha': 'alpha', 'Beta': 'beta', 'Gamma': 'gamma', 'Delta': 'delta', + 'Epsilon': 'epsilon', 'Zeta': 'zeta', 'Eta': 'eta', 'Theta': 'theta', + 'Iota': 'iota', 'Kappa': 'kappa', 'Lambda': 'lambda', 'Mu': 'mu', + 'Nu': 'nu', 'Xi': 'xi', 'Omicron': 'o', 'Pi': 'pi', 'Rho': 'rho', + 'Sigma': 'sigma', 'Tau': 'tau', 'Upsilon': 'upsilon', 'Phi': 'phi', + 'Chi': 'chi', 'Psi': 'phi', 'Omega': 'omega', + + 'CapitalAlpha': ' A', 'CapitalBeta': ' B', 'CapitalGamma': 'Gamma', + 'CapitalDelta': 'Delta', 'CapitalEpsilon': 'E', 'CapitalZeta': ' Z', + 'CapitalEta': ' H', 'CapitalTheta': 'Theta', 'CapitalIota': ' I', + 'CapitalKappa': 'K', 'CapitalLambda': 'Lambda', 'CapitalMu': ' M', + 'CapitalNu': ' N', 'CapitalXi': 'Xi', 'CapitalOmicron': 'O', + 'CapitalPi': 'Pi', 'CapitalRho': ' P', 'CapitalSigma': 'Sigma', + 'CapitalTau': ' T', 'CapitalUpsilon': ' Y', 'CapitalPhi': 'Phi', + 'CapitalChi': ' X', 'CapitalPsi': 'Psi', 'CapitalOmega': 'Omega', + + 'CurlyEpsilon': 'varepsilon', 'CurlyTheta': 'vartheta', + 'CurlyKappa': 'varkappa', 'CurlyPi': 'varpi', 'CurlyRho': 'varrho', + 'FinalSigma': 'varsigma', 'CurlyPhi': 'varphi', + 'CurlyCapitalUpsilon': 'varUpsilon', + + 'Aleph': 'aleph', 'Bet': 'beth', 'Gimel': 'gimel', 'Dalet': 'daleth'} + + +class TestReplaceVars(TestCase): + + def test_replace_symbols(self): + for word in SYMBOLS: + after = '\\' + SYMBOLS[word] + self.assertEquals(replace_vars('\\[' + word + ']'), after.replace('\\ ', '')) + + def test_replace_infinity(self): + self.assertEquals(replace_vars('Infinity'), '\\infty') + + def test_none(self): + self.assertEqual(replace_vars('novariables'), 'novariables') From 6f41a342a6aeeafa17b5d0d85a5e8bc69d9c2415 Mon Sep 17 00:00:00 2001 From: joonbang Date: Mon, 15 Aug 2016 15:11:09 -0400 Subject: [PATCH 240/402] Update README.md --- README.md | 20 +++++++++++++++----- 1 file changed, 15 insertions(+), 5 deletions(-) diff --git a/README.md b/README.md index 508bd50..ad8ae5e 100644 --- a/README.md +++ b/README.md @@ -22,15 +22,25 @@ Parts of this code were developed by Alex Danoff and Azeem Mohammed. To achieve the desired result, one should execute the progams in the following order (using the output of the previous program as input for the next one): -1. remove_excess.py -2. replace_special.py -3. prepare_annotations.py +1. `remove_excess.py` +2. `replace_special.py` +3. `prepare_annotations.py` -## eCF Seeding Project +## BMP Seeding Project + +## CFSF Seeding Project + +This Python code (developed by Joon Bang) translates Maple files from Annie Cuyt's continued fractions package for special functions (CFSF) to LaTeX. + +The project is located in the `maple2latex` folder. ## DLMF Seeding Project -## BMP Seeding Project +## eCF Seeding Project + +This Python code (mostly developed by Kevin Chen) translates Mathematica files to LaTeX. + +The project is currently awaiting approval to be merged into the main branch of the repository. ## KLSadd Insertion Project linetest.py and updateChapters.py written by Rahul Shah and Edward Bian From d6e464afda9357106e8453b17160fd2f3e257bdb Mon Sep 17 00:00:00 2001 From: joonbang Date: Mon, 15 Aug 2016 15:14:06 -0400 Subject: [PATCH 241/402] Update README.md --- README.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/README.md b/README.md index ad8ae5e..4039241 100644 --- a/README.md +++ b/README.md @@ -28,7 +28,7 @@ To achieve the desired result, one should execute the progams in the following o ## BMP Seeding Project -## CFSF Seeding Project +## Maple CFSF Seeding Project This Python code (developed by Joon Bang) translates Maple files from Annie Cuyt's continued fractions package for special functions (CFSF) to LaTeX. @@ -36,7 +36,7 @@ The project is located in the `maple2latex` folder. ## DLMF Seeding Project -## eCF Seeding Project +## Mathematica eCF Seeding Project This Python code (mostly developed by Kevin Chen) translates Mathematica files to LaTeX. From 8152f457dd5b06469c9c2b5f4bba62e8f85607da Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Mon, 15 Aug 2016 16:03:08 -0400 Subject: [PATCH 242/402] Add tests for convert_fraction function --- eCF/src/MathematicaToLaTeX.py | 4 ++-- eCF/test/test_convert_fraction.py | 32 +++++++++++++++++++++++++++++++ 2 files changed, 34 insertions(+), 2 deletions(-) diff --git a/eCF/src/MathematicaToLaTeX.py b/eCF/src/MathematicaToLaTeX.py index ddf081a..f4e4242 100644 --- a/eCF/src/MathematicaToLaTeX.py +++ b/eCF/src/MathematicaToLaTeX.py @@ -527,7 +527,7 @@ def summation(line): def constraint(line): """ Converts Mathematica's 'Element', 'NotElement', and 'Inequality' functions - to LaTeX formatting using \constraint{}. + to LaTeX formatting using \\constraint{}. """ sections = arg_split(line, ',') @@ -601,7 +601,7 @@ def constraint(line): def convert_fraction(line): """ Converts Mathematica fractions, which are only '/', to LaTeX - \frac{}{}-ions. + \\frac{}{}-ions. """ l = list('([{') r = list(')]}') diff --git a/eCF/test/test_convert_fraction.py b/eCF/test/test_convert_fraction.py index 1d6be0f..7ab59d6 100644 --- a/eCF/test/test_convert_fraction.py +++ b/eCF/test/test_convert_fraction.py @@ -5,3 +5,35 @@ from unittest import TestCase from MathematicaToLaTeX import convert_fraction + +class TestConvertFraction(TestCase): + + def test_stop(self): + self.assertEqual(convert_fraction('a*b/c*d'), 'a*\\frac{b}{c}*d') + self.assertEqual(convert_fraction('a+b/c+d'), 'a+\\frac{b}{c}+d') + self.assertEqual(convert_fraction('a-b/c-d'), 'a-\\frac{b}{c}-d') + self.assertEqual(convert_fraction('a=b/c=d'), 'a=\\frac{b}{c}=d') + self.assertEqual(convert_fraction('a,b/c,d'), 'a,\\frac{b}{c},d') + self.assertEqual(convert_fraction('ab/c>d'), 'a>\\frac{b}{c}>d') + self.assertEqual(convert_fraction('a&b/c&d'), 'a&\\frac{b}{c}&d') + + def test_paren(self): + self.assertEqual(convert_fraction('a*(b+c)/(d+e)*f'), 'a*\\frac{b+c}{d+e}*f') + self.assertEqual(convert_fraction('a+(b+c)/(d+e)+f'), 'a+\\frac{b+c}{d+e}+f') + self.assertEqual(convert_fraction('a-(b+c)/(d+e)-f'), 'a-\\frac{b+c}{d+e}-f') + self.assertEqual(convert_fraction('a=(b+c)/(d+e)=f'), 'a=\\frac{b+c}{d+e}=f') + self.assertEqual(convert_fraction('a,(b+c)/(d+e),f'), 'a,\\frac{b+c}{d+e},f') + self.assertEqual(convert_fraction('a<(b+c)/(d+e)(b+c)/(d+e)>f'), 'a>\\frac{b+c}{d+e}>f') + self.assertEqual(convert_fraction('a&(b+c)/(d+e)&f'), 'a&\\frac{b+c}{d+e}&f') + + self.assertEqual(convert_fraction('(a)/(b)'), '\\frac{a}{b}') + self.assertEqual(convert_fraction('a/(b)'), '\\frac{a}{b}') + self.assertEqual(convert_fraction('(a)/b'), '\\frac{a}{b}') + self.assertEqual(convert_fraction('a/b'), '\\frac{a}{b}') + + self.assertEqual(convert_fraction('a(b+c)/d(e+f)'), '\\frac{a(b+c)}{d(e+f)}') + + def test_none(self): + self.assertEqual(convert_fraction('nofraction'), 'nofraction') From 8f04b8bc02366c73217c27f5a47a83d87c77dcab Mon Sep 17 00:00:00 2001 From: Edward Bian Date: Mon, 15 Aug 2016 16:33:42 -0400 Subject: [PATCH 243/402] Better Formatted --- KLSadd_insertion/src/svmono.cls | 1690 ------------------------ KLSadd_insertion/src/updateChapters.py | 276 ++-- 2 files changed, 116 insertions(+), 1850 deletions(-) delete mode 100644 KLSadd_insertion/src/svmono.cls diff --git a/KLSadd_insertion/src/svmono.cls b/KLSadd_insertion/src/svmono.cls deleted file mode 100644 index 2341bee..0000000 --- a/KLSadd_insertion/src/svmono.cls +++ /dev/null @@ -1,1690 +0,0 @@ -% SVMONO DOCUMENT CLASS -- version 3.9 (23-Jul-03) -% Springer Verlag global LaTeX2e support for monographs -%% -%% -%% \CharacterTable -%% {Upper-case \A\B\C\D\E\F\G\H\I\J\K\L\M\N\O\P\Q\R\S\T\U\V\W\X\Y\Z -%% Lower-case \a\b\c\d\e\f\g\h\i\j\k\l\m\n\o\p\q\r\s\t\u\v\w\x\y\z -%% Digits \0\1\2\3\4\5\6\7\8\9 -%% Exclamation \! Double quote \" Hash (number) \# -%% Dollar \$ Percent \% Ampersand \& -%% Acute accent \' Left paren \( Right paren \) -%% Asterisk \* Plus \+ Comma \, -%% Minus \- Point \. Solidus \/ -%% Colon \: Semicolon \; Less than \< -%% Equals \= Greater than \> Question mark \? -%% Commercial at \@ Left bracket \[ Backslash \\ -%% Right bracket \] Circumflex \^ Underscore \_ -%% Grave accent \` Left brace \{ Vertical bar \| -%% Right brace \} Tilde \~} -%% -\NeedsTeXFormat{LaTeX2e}[1995/12/01] -\ProvidesClass{svmono}[2003/07/23 3.9 -^^JSpringer Verlag global LaTeX document class for monographs] - -% Options -% citations -\DeclareOption{natbib}{\ExecuteOptions{oribibl}% -\AtEndOfClass{% Loading package 'NATBIB' -\RequirePackage{natbib} -% Changing some parameters of NATBIB -\setlength{\bibhang}{\parindent} -%\setlength{\bibsep}{0mm} -\let\bibfont=\small -\def\@biblabel#1{#1.} -\newcommand{\etal}{\textit{et al}.} -%\bibpunct[,]{(}{)}{;}{a}{}{,}}} -}} -% Springer environment -\let\if@spthms\iftrue -\DeclareOption{nospthms}{\let\if@spthms\iffalse} -% -\let\envankh\@empty % no anchor for "theorems" -% -\let\if@envcntreset\iffalse % environment counter is not reset -\let\if@envcntresetsect=\iffalse % reset each section? -\DeclareOption{envcountresetchap}{\let\if@envcntreset\iftrue} -\DeclareOption{envcountresetsect}{\let\if@envcntreset\iftrue -\let\if@envcntresetsect=\iftrue} -% -\let\if@envcntsame\iffalse % NOT all environments work like "Theorem", - % each using its own counter -\DeclareOption{envcountsame}{\let\if@envcntsame\iftrue} -% -\let\if@envcntshowhiercnt=\iffalse % do not show hierarchy counter at all -% -% enhance theorem counter -\DeclareOption{envcountchap}{\def\envankh{chapter}% show \thechapter along with theorem number -\let\if@envcntshowhiercnt=\iftrue -\ExecuteOptions{envcountreset}} -% -\DeclareOption{envcountsect}{\def\envankh{section}% show \thesection along with theorem number -\let\if@envcntshowhiercnt=\iftrue -\ExecuteOptions{envcountreset}} -% -% languages -\let\switcht@@therlang\relax -\let\svlanginfo\relax -\def\ds@deutsch{\def\switcht@@therlang{\switcht@deutsch}% -\gdef\svlanginfo{\typeout{Man spricht deutsch.}\global\let\svlanginfo\relax}} -\def\ds@francais{\def\switcht@@therlang{\switcht@francais}% -\gdef\svlanginfo{\typeout{On parle francais.}\global\let\svlanginfo\relax}} -% -\AtBeginDocument{\@ifpackageloaded{babel}{% -\@ifundefined{extrasenglish}{}{\addto\extrasenglish{\switcht@albion}}% -\@ifundefined{extrasUKenglish}{}{\addto\extrasUKenglish{\switcht@albion}}% -\@ifundefined{extrasfrenchb}{}{\addto\extrasfrenchb{\switcht@francais}}% -\@ifundefined{extrasgerman}{}{\addto\extrasgerman{\switcht@deutsch}}% -}{\switcht@@therlang}% -} -% numbering style of floats, equations -\newif\if@numart \@numartfalse -\DeclareOption{numart}{\@numarttrue} -\def\set@numbering{\if@numart\else\num@book\fi} -\AtEndOfClass{\set@numbering} -% style for vectors -\DeclareOption{vecphys}{\def\vec@style{phys}} -\DeclareOption{vecarrow}{\def\vec@style{arrow}} -% running heads -\let\if@runhead\iftrue -\DeclareOption{norunningheads}{\let\if@runhead\iffalse} -% referee option -\let\if@referee\iffalse -\def\makereferee{\def\baselinestretch{2}\selectfont -\newbox\refereebox -\setbox\refereebox=\vbox to\z@{\vskip0.5cm% - \hbox to\textwidth{\normalsize\tt\hrulefill\lower0.5ex - \hbox{\kern5\p@ referee's copy\kern5\p@}\hrulefill}\vss}% -\def\@oddfoot{\copy\refereebox}\let\@evenfoot=\@oddfoot} -\DeclareOption{referee}{\let\if@referee\iftrue -\AtBeginDocument{\makereferee\small\normalsize}} -% modification of thebibliography -\let\if@openbib\iffalse -\DeclareOption{openbib}{\let\if@openbib\iftrue} -% LaTeX standard, sectionwise references -\DeclareOption{oribibl}{\let\oribibl=Y} -\DeclareOption{sectrefs}{\let\secbibl=Y} -% -% footinfo option (provides an informatory line on every page) -\def\SpringerMacroPackageNameA{svmono.cls} -% \thetime, \thedate and \timstamp are macros to include -% time, date (or both) of the TeX run in the document -\def\maketimestamp{\count255=\time -\divide\count255 by 60\relax -\edef\thetime{\the\count255:}% -\multiply\count255 by-60\relax -\advance\count255 by\time -\edef\thetime{\thetime\ifnum\count255<10 0\fi\the\count255} -\edef\thedate{\number\day-\ifcase\month\or Jan\or Feb\or Mar\or - Apr\or May\or Jun\or Jul\or Aug\or Sep\or Oct\or - Nov\or Dec\fi-\number\year} -\def\timstamp{\hbox to\hsize{\tt\hfil\thedate\hfil\thetime\hfil}}} -\maketimestamp -% -% \footinfo generates a info footline on every page containing -% pagenumber, jobname, macroname, and timestamp -\DeclareOption{footinfo}{\AtBeginDocument{\maketimestamp - \def\@oddfoot{\footnotesize\tt Page: \thepage\hfil job: \jobname\hfil - macro: \SpringerMacroPackageNameA\hfil - date/time: \thedate/\thetime}% - \let\@evenfoot=\@oddfoot}} -% -% start new chapter on any page -\newif\if@openright \@openrighttrue -\DeclareOption{openany}{\@openrightfalse} -% -% no size changing allowed -\DeclareOption{11pt}{\OptionNotUsed} -\DeclareOption{12pt}{\OptionNotUsed} -% options for the article class -\def\@rticle@options{10pt,twoside} -% fleqn -\DeclareOption{fleqn}{\def\@rticle@options{10pt,twoside,fleqn}% -\AtEndOfClass{\let\leftlegendglue\relax}% -\AtBeginDocument{\mathindent\parindent}} -% hanging sectioning titles -\let\if@sechang\iffalse -\DeclareOption{sechang}{\let\if@sechang\iftrue} -\def\ClassInfoNoLine#1#2{% - \ClassInfo{#1}{#2\@gobble}% -} -\let\SVMonoOpt\@empty -\DeclareOption*{\InputIfFileExists{sv\CurrentOption.clo}{% -\global\let\SVMonoOpt\CurrentOption}{% -\ClassWarning{Springer-SVMono}{Specified option or subpackage -"\CurrentOption" \MessageBreak not found -passing it to article class \MessageBreak --}\PassOptionsToClass{\CurrentOption}{article}% -}} -\ProcessOptions\relax -\ifx\SVMonoOpt\@empty\relax -\ClassInfoNoLine{Springer-SVMono}{extra/valid Springer sub-package -\MessageBreak not found in option list - using "global" style}{} -\fi -\LoadClass[\@rticle@options]{article} -\raggedbottom - -% various sizes and settings for monographs - -%\setlength{\textwidth}{28pc} % 11.8cm -\setlength{\textwidth}{43pc} % 11.8cm -%\setlength{\textheight}{12pt}\multiply\textheight by 45\relax -%\setlength{\textheight}{540\p@} -\setlength{\textheight}{720\p@} -%\setlength{\topmargin}{0cm} -\setlength{\topmargin}{-2cm} -%\setlength\oddsidemargin {63\p@} -%\setlength\evensidemargin {63\p@} -\setlength\oddsidemargin {-28\p@} -\setlength\evensidemargin {-28\p@} -\setlength\marginparwidth{90\p@} -\setlength\headsep {12\p@} - -\setlength{\parindent}{15\p@} -\setlength{\parskip}{\z@ \@plus \p@} -\setlength{\hfuzz}{2\p@} -\setlength{\arraycolsep}{1.5\p@} - -\frenchspacing - -\tolerance=500 - -\predisplaypenalty=0 -\clubpenalty=10000 -\widowpenalty=10000 - -\setlength\footnotesep{7.7\p@} - -\newdimen\betweenumberspace % dimension for space between -\betweenumberspace=5\p@ % number and text of titles -\newdimen\headlineindent % dimension for space of -\headlineindent=2.5cc % number and gap of running heads - -% fonts, sizes, and the like -\renewcommand\small{% - \@setfontsize\small\@ixpt{11}% - \abovedisplayskip 8.5\p@ \@plus3\p@ \@minus4\p@ - \abovedisplayshortskip \z@ \@plus2\p@ - \belowdisplayshortskip 4\p@ \@plus2\p@ \@minus2\p@ - \def\@listi{\leftmargin\leftmargini - \parsep \z@ \@plus\p@ \@minus\p@ - \topsep 6\p@ \@plus2\p@ \@minus4\p@ - \itemsep\z@}% - \belowdisplayskip \abovedisplayskip -} -% -\let\footnotesize=\small -% -\newenvironment{petit}{\par\addvspace{6\p@}\small}{\par\addvspace{6\p@}} -% - -% modification of automatic positioning of floating objects -\setlength\@fptop{\z@ } -\setlength\@fpsep{12\p@ } -\setlength\@fpbot{\z@ \@plus 1fil } -\def\textfraction{.01} -\def\floatpagefraction{.8} -\setlength{\intextsep}{20\p@ \@plus 2\p@ \@minus 2\p@} -\setcounter{topnumber}{4} -\def\topfraction{.9} -\setcounter{bottomnumber}{2} -\def\bottomfraction{.7} -\setcounter{totalnumber}{6} -% -% size and style of headings -\newcommand{\partsize}{\Large} -\newcommand{\partstyle}{\bfseries\boldmath} -\newcommand{\chapsize}{\Large} -\newcommand{\chapstyle}{\bfseries\boldmath} -\newcommand{\secsize}{\large} -\newcommand{\secstyle}{\bfseries\boldmath} -\newcommand{\subsecsize}{\normalsize} -\newcommand{\subsecstyle}{\bfseries\boldmath} - -\def\newendpage {\par - \ifdim \pagetotal>\topskip - \else\thispagestyle{empty} - \fi - \vfil\penalty -\@M} -\def\clearpage{% - \ifvmode - \ifnum \@dbltopnum =\m@ne - \ifdim \pagetotal <\topskip - \hbox{} - \fi - \fi - \fi - \newendpage - \write\m@ne{} - \vbox{} - \penalty -\@Mi -} -\def\cleardoublepage{\clearpage\if@twoside \ifodd\c@page\else - \hbox{}\newendpage\if@twocolumn\hbox{}\newpage\fi\fi\fi} - -\newcommand{\clearemptydoublepage}{% - \newpage{\pagestyle{empty}\cleardoublepage}} -\newcommand{\startnewpage}{\if@openright\clearemptydoublepage\else\clearpage\fi} - -% redefinition of \part -\renewcommand\part{\clearemptydoublepage - \thispagestyle{empty} - \if@twocolumn - \onecolumn - \@tempswatrue - \else - \@tempswafalse - \fi - \@ifundefined{thispagecropped}{}{\thispagecropped} - \secdef\@part\@spart} - -\def\@part[#1]#2{\ifnum \c@secnumdepth >-2\relax - \refstepcounter{part} - \addcontentsline{toc}{part}{\partname\ - \thepart\thechapterend\hskip\betweenumberspace - #1}\else - \addcontentsline{toc}{part}{#1}\fi - \markboth{}{} - {\raggedleft - \ifnum \c@secnumdepth >-2\relax - \normalfont\partstyle\partsize\vrule height 34pt width 0pt depth 0pt% - \partname\ \thepart\llap{\smash{\lower 5pt\hbox to\textwidth{\hrulefill}}} - \par - \vskip 128.3\p@ \fi - #2\par}\@endpart} -% -% \@endpart finishes the part page -% -\def\@endpart{\vfil\newpage - \if@twoside - \hbox{} - \thispagestyle{empty} - \newpage - \fi - \if@tempswa - \twocolumn - \fi} -% -\def\@spart#1{{\raggedleft - \normalfont\partsize\partstyle - #1\par}\@endpart} -% -% (re)define sectioning -\setcounter{secnumdepth}{2} - -\def\seccounterend{} -\def\seccountergap{\hskip\betweenumberspace} -\def\@seccntformat#1{\csname the#1\endcsname\seccounterend\seccountergap\ignorespaces} -% -\let\firstmark=\botmark -% -\@ifundefined{thechapterend}{\def\thechapterend{}}{} -% -\if@sechang - \def\sec@hangfrom#1{\setbox\@tempboxa\hbox{#1}% - \hangindent\wd\@tempboxa\noindent\box\@tempboxa} -\else - \def\sec@hangfrom#1{\setbox\@tempboxa\hbox{#1}% - \hangindent\z@\noindent\box\@tempboxa} -\fi - -\def\chap@hangfrom#1{\noindent\vrule height 34pt width 0pt depth 0pt -\rlap{\smash{\lower 5pt\hbox to\textwidth{\hrulefill}}}\hbox{#1} -\vskip10pt} -\def\schap@hangfrom{\chap@hangfrom{}} - -\newcounter{chapter} -% -\@addtoreset{section}{chapter} -\@addtoreset{footnote}{chapter} - -\newif\if@mainmatter \@mainmattertrue -\newcommand\frontmatter{\startnewpage - \@mainmatterfalse\pagenumbering{Roman} - \setcounter{page}{5}} -% -\newcommand\mainmatter{\clearemptydoublepage - \@mainmattertrue\pagenumbering{arabic}} -% -\newcommand\backmatter{\clearemptydoublepage\@mainmatterfalse} - -\def\@chapapp{\chaptername} - -\newdimen\chapstarthookwidth -\newcommand\chapstarthook[2][0.66\textwidth]{% -\setlength{\chapstarthookwidth}{#1}% -\gdef\chapst@rthook{#2}} - -\newcommand{\processchapstarthook}{\@ifundefined{chapst@rthook}{}{% - \setbox0=\hbox{\vbox{ - \begin{flushright} - \begin{minipage}{\chapstarthookwidth} - \vrule\@width\z@\@height21\p@\@depth\z@ - \normalfont\small\itshape\chapst@rthook - \end{minipage} - \end{flushright}}}% - \@tempdima=\pagetotal - \advance\@tempdima by\ht0 - \ifdim\@tempdima<106\p@ - \multiply\@tempdima by-1 - \advance\@tempdima by106\p@ - \vskip\@tempdima - \fi - \box0\par - \global\let\chapst@rthook=\undefined}} - -\newcommand\chapter{\startnewpage - \@ifundefined{thispagecropped}{}{\thispagecropped} - \thispagestyle{empty}% - \global\@topnum\z@ - \@afterindentfalse - \secdef\@chapter\@schapter} - -\def\@chapter[#1]#2{\ifnum \c@secnumdepth >\m@ne - \refstepcounter{chapter}% - \if@mainmatter - \typeout{\@chapapp\space\thechapter.}% - \addcontentsline{toc}{chapter}{\protect - \numberline{\thechapter\thechapterend}#1}% - \else - \addcontentsline{toc}{chapter}{#1}% - \fi - \else - \addcontentsline{toc}{chapter}{#1}% - \fi - \chaptermark{#1}% - \addtocontents{lof}{\protect\addvspace{10\p@}}% - \addtocontents{lot}{\protect\addvspace{10\p@}}% - \if@twocolumn - \@topnewpage[\@makechapterhead{#2}]% - \else - \@makechapterhead{#2}% - \@afterheading - \fi} - -\def\@schapter#1{\if@twocolumn - \@topnewpage[\@makeschapterhead{#1}]% - \else - \@makeschapterhead{#1}% - \@afterheading - \fi} - -%%changes position and layout of numbered chapter headings -\def\@makechapterhead#1{{\parindent\z@\raggedright\normalfont - \hyphenpenalty \@M - \interlinepenalty\@M - \chapsize\chapstyle - \chap@hangfrom{\thechapter\thechapterend\hskip\betweenumberspace}%!!! - \ignorespaces#1\par\nobreak - \processchapstarthook - \ifdim\pagetotal>157\p@ - \vskip 11\p@ - \else - \@tempdima=168\p@\advance\@tempdima by-\pagetotal - \vskip\@tempdima - \fi}} - -%%changes position and layout of unnumbered chapter headings -\def\@makeschapterhead#1{{\parindent \z@ \raggedright\normalfont - \hyphenpenalty \@M - \interlinepenalty\@M - \chapsize\chapstyle - \schap@hangfrom - \ignorespaces#1\par\nobreak - \processchapstarthook - \ifdim\pagetotal>157\p@ - \vskip 11\p@ - \else - \@tempdima=168\p@\advance\@tempdima by-\pagetotal - \vskip\@tempdima - \fi}} - -% predefined unnumbered headings -\newcommand{\preface}[1][\prefacename]{\chapter*{#1}\markboth{#1}{#1}} - -% measures and setting of sections -\renewcommand\section{\@startsection{section}{1}{\z@}% - {-24\p@ \@plus -4\p@ \@minus -4\p@}% - {12\p@ \@plus 4\p@ \@minus 4\p@}% - {\normalfont\secsize\secstyle - \rightskip=\z@ \@plus 8em\pretolerance=10000 }} -\renewcommand\subsection{\@startsection{subsection}{2}{\z@}% - {-17\p@ \@plus -4\p@ \@minus -4\p@}% - {10\p@ \@plus 4\p@ \@minus 4\p@}% - {\normalfont\subsecsize\subsecstyle - \rightskip=\z@ \@plus 8em\pretolerance=10000 }} -\renewcommand\subsubsection{\@startsection{subsubsection}{3}{\z@}% - {-17\p@ \@plus -4\p@ \@minus -4\p@}% - {10\p@ \@plus 4\p@ \@minus 4\p@}% - {\normalfont\normalsize\subsecstyle - \rightskip=\z@ \@plus 8em\pretolerance=10000 }} -\renewcommand\paragraph{\@startsection{paragraph}{4}{\z@}% - {-10\p@ \@plus -4\p@ \@minus -4\p@}% - {10\p@ \@plus 4\p@ \@minus 4\p@}% - {\normalfont\normalsize\itshape - \rightskip=\z@ \@plus 8em\pretolerance=10000 }} -\def\subparagraph{\@startsection{subparagraph}{5}{\z@}% - {-5.388\p@ \@plus-4\p@ \@minus-4\p@}{-5\p@}{\normalfont\normalsize\itshape}} - -% Appendix -\renewcommand\appendix{\par - \stepcounter{chapter} - \setcounter{chapter}{0} - \stepcounter{section} - \setcounter{section}{0} - \setcounter{equation}{0} - \setcounter{figure}{0} - \setcounter{table}{0} - \setcounter{footnote}{0} - \def\@chapapp{\appendixname}% - \renewcommand\thechapter{\@Alph\c@chapter}} - -% definition of sections -% \hyphenpenalty and \raggedright added, so that there is no -% hyphenation and the text is set ragged-right in sectioning - -\def\runinsep{} -\def\aftertext{\unskip\runinsep} -% -\def\thesection{\thechapter.\arabic{section}} -\def\thesubsection{\thesection.\arabic{subsection}} -\def\thesubsubsection{\thesubsection.\arabic{subsubsection}} -\def\theparagraph{\thesubsubsection.\arabic{paragraph}} -\def\thesubparagraph{\theparagraph.\arabic{subparagraph}} -\def\chaptermark#1{} -% -\def\@ssect#1#2#3#4#5{% - \@tempskipa #3\relax - \ifdim \@tempskipa>\z@ - \begingroup - #4{% - \@hangfrom{\hskip #1}% - \raggedright - \hyphenpenalty \@M - \interlinepenalty \@M #5\@@par}% - \endgroup - \else - \def\@svsechd{#4{\hskip #1\relax #5}}% - \fi - \@xsect{#3}} -% -\def\@sect#1#2#3#4#5#6[#7]#8{% - \ifnum #2>\c@secnumdepth - \let\@svsec\@empty - \else - \refstepcounter{#1}% - \protected@edef\@svsec{\@seccntformat{#1}\relax}% - \fi - \@tempskipa #5\relax - \ifdim \@tempskipa>\z@ - \begingroup #6\relax - \sec@hangfrom{\hskip #3\relax\@svsec}% - {\raggedright - \hyphenpenalty \@M - \interlinepenalty \@M #8\@@par}% - \endgroup - \csname #1mark\endcsname{#7\seccounterend}% - \addcontentsline{toc}{#1}{\ifnum #2>\c@secnumdepth - \else - \protect\numberline{\csname the#1\endcsname\seccounterend}% - \fi - #7}% - \else - \def\@svsechd{% - #6\hskip #3\relax - \@svsec #8\aftertext\ignorespaces - \csname #1mark\endcsname{#7}% - \addcontentsline{toc}{#1}{% - \ifnum #2>\c@secnumdepth \else - \protect\numberline{\csname the#1\endcsname\seccounterend}% - \fi - #7}}% - \fi - \@xsect{#5}} - -% figures and tables are processed in small print -\def \@floatboxreset {% - \reset@font - \small - \@setnobreak - \@setminipage -} -\def\fps@figure{htbp} -\def\fps@table{htbp} - -% Frame for paste-in figures or tables -\def\mpicplace#1#2{% #1 =width #2 =height -\vbox{\hbox to #1{\vrule\@width \fboxrule \@height #2\hfill}}} - -% labels of enumerate -\renewcommand\labelenumii{\theenumii)} -\renewcommand\theenumii{\@alph\c@enumii} - -% labels of itemize -\renewcommand\labelitemi{\textbullet} -\renewcommand\labelitemii{\textendash} -\let\labelitemiii=\labelitemiv - -% labels of description -\renewcommand*\descriptionlabel[1]{\hspace\labelsep #1\hfil} - -% fixed indentation for standard itemize-environment -\newdimen\svitemindent \setlength{\svitemindent}{\parindent} - - -% make indentations changeable - -\def\setitemindent#1{\settowidth{\labelwidth}{#1}% - \let\setit@m=Y% - \leftmargini\labelwidth - \advance\leftmargini\labelsep - \def\@listi{\leftmargin\leftmargini - \labelwidth\leftmargini\advance\labelwidth by -\labelsep - \parsep=\parskip - \topsep=\medskipamount - \itemsep=\parskip \advance\itemsep by -\parsep}} -\def\setitemitemindent#1{\settowidth{\labelwidth}{#1}% - \let\setit@m=Y% - \leftmarginii\labelwidth - \advance\leftmarginii\labelsep -\def\@listii{\leftmargin\leftmarginii - \labelwidth\leftmarginii\advance\labelwidth by -\labelsep - \parsep=\parskip - \topsep=\z@ - \itemsep=\parskip \advance\itemsep by -\parsep}} -% -% adjusted environment "description" -% if an optional parameter (at the first two levels of lists) -% is present, its width is considered to be the widest mark -% throughout the current list. -\def\description{\@ifnextchar[{\@describe}{\list{}{\labelwidth\z@ - \itemindent-\leftmargin \let\makelabel\descriptionlabel}}} -% -\def\describelabel#1{#1\hfil} -\def\@describe[#1]{\relax\ifnum\@listdepth=0 -\setitemindent{#1}\else\ifnum\@listdepth=1 -\setitemitemindent{#1}\fi\fi -\list{--}{\let\makelabel\describelabel}} -% -\def\itemize{% - \ifnum \@itemdepth >\thr@@\@toodeep\else - \advance\@itemdepth\@ne - \ifx\setit@m\undefined - \ifnum \@itemdepth=1 \leftmargini=\svitemindent - \labelwidth\leftmargini\advance\labelwidth-\labelsep - \leftmarginii=\leftmargini \leftmarginiii=\leftmargini - \fi - \fi - \edef\@itemitem{labelitem\romannumeral\the\@itemdepth}% - \expandafter\list - \csname\@itemitem\endcsname - {\def\makelabel##1{\rlap{##1}\hss}}% - \fi} -% -\newdimen\verbatimindent \verbatimindent\parindent -\def\verbatim{\advance\@totalleftmargin by\verbatimindent -\@verbatim \frenchspacing\@vobeyspaces \@xverbatim} - -% -% special signs and characters -\newcommand{\D}{\mathrm{d}} -\newcommand{\E}{\mathrm{e}} -\let\eul=\E -\newcommand{\I}{{\rm i}} -\let\imag=\I -% -% the definition of uppercase Greek characters -% Springer likes them as italics to depict variables -\DeclareMathSymbol{\Gamma}{\mathalpha}{letters}{"00} -\DeclareMathSymbol{\Delta}{\mathalpha}{letters}{"01} -\DeclareMathSymbol{\Theta}{\mathalpha}{letters}{"02} -\DeclareMathSymbol{\Lambda}{\mathalpha}{letters}{"03} -\DeclareMathSymbol{\Xi}{\mathalpha}{letters}{"04} -\DeclareMathSymbol{\Pi}{\mathalpha}{letters}{"05} -\DeclareMathSymbol{\Sigma}{\mathalpha}{letters}{"06} -\DeclareMathSymbol{\Upsilon}{\mathalpha}{letters}{"07} -\DeclareMathSymbol{\Phi}{\mathalpha}{letters}{"08} -\DeclareMathSymbol{\Psi}{\mathalpha}{letters}{"09} -\DeclareMathSymbol{\Omega}{\mathalpha}{letters}{"0A} -% the upright forms are defined here as \var -\DeclareMathSymbol{\varGamma}{\mathalpha}{operators}{"00} -\DeclareMathSymbol{\varDelta}{\mathalpha}{operators}{"01} -\DeclareMathSymbol{\varTheta}{\mathalpha}{operators}{"02} -\DeclareMathSymbol{\varLambda}{\mathalpha}{operators}{"03} -\DeclareMathSymbol{\varXi}{\mathalpha}{operators}{"04} -\DeclareMathSymbol{\varPi}{\mathalpha}{operators}{"05} -\DeclareMathSymbol{\varSigma}{\mathalpha}{operators}{"06} -\DeclareMathSymbol{\varUpsilon}{\mathalpha}{operators}{"07} -\DeclareMathSymbol{\varPhi}{\mathalpha}{operators}{"08} -\DeclareMathSymbol{\varPsi}{\mathalpha}{operators}{"09} -\DeclareMathSymbol{\varOmega}{\mathalpha}{operators}{"0A} -% Upright Lower Case Greek letters without using a new MathAlphabet -\newcommand{\greeksym}[1]{\usefont{U}{psy}{m}{n}#1} -\newcommand{\greeksymbold}[1]{{\usefont{U}{psy}{b}{n}#1}} -\newcommand{\allmodesymb}[2]{\relax\ifmmode{\mathchoice -{\mbox{\fontsize{\tf@size}{\tf@size}#1{#2}}} -{\mbox{\fontsize{\tf@size}{\tf@size}#1{#2}}} -{\mbox{\fontsize{\sf@size}{\sf@size}#1{#2}}} -{\mbox{\fontsize{\ssf@size}{\ssf@size}#1{#2}}}} -\else -\mbox{#1{#2}}\fi} -% Definition of lower case Greek letters -\newcommand{\ualpha}{\allmodesymb{\greeksym}{a}} -\newcommand{\ubeta}{\allmodesymb{\greeksym}{b}} -\newcommand{\uchi}{\allmodesymb{\greeksym}{c}} -\newcommand{\udelta}{\allmodesymb{\greeksym}{d}} -\newcommand{\ugamma}{\allmodesymb{\greeksym}{g}} -\newcommand{\umu}{\allmodesymb{\greeksym}{m}} -\newcommand{\unu}{\allmodesymb{\greeksym}{n}} -\newcommand{\upi}{\allmodesymb{\greeksym}{p}} -\newcommand{\utau}{\allmodesymb{\greeksym}{t}} -% redefines the \vec accent to a bold character - if desired -\def\fig@type{arrow}% temporarily abused -\ifx\vec@style\fig@type\else -\@ifundefined{vec@style}{% - \def\vec#1{\ensuremath{\mathchoice - {\mbox{\boldmath$\displaystyle\mathbf{#1}$}} - {\mbox{\boldmath$\textstyle\mathbf{#1}$}} - {\mbox{\boldmath$\scriptstyle\mathbf{#1}$}} - {\mbox{\boldmath$\scriptscriptstyle\mathbf{#1}$}}}}% -} -{\def\vec#1{\ensuremath{\mathchoice - {\mbox{\boldmath$\displaystyle#1$}} - {\mbox{\boldmath$\textstyle#1$}} - {\mbox{\boldmath$\scriptstyle#1$}} - {\mbox{\boldmath$\scriptscriptstyle#1$}}}}% -} -\fi -% tensor -\def\tens#1{\relax\ifmmode\mathsf{#1}\else\textsf{#1}\fi} - -% end of proof symbol -\newcommand\qedsymbol{\hbox{\rlap{$\sqcap$}$\sqcup$}} -\newcommand\qed{\relax\ifmmode\else\unskip\quad\fi\qedsymbol} -\newcommand\smartqed{\renewcommand\qed{\relax\ifmmode\qedsymbol\else - {\unskip\nobreak\hfil\penalty50\hskip1em\null\nobreak\hfil\qedsymbol - \parfillskip=\z@\finalhyphendemerits=0\endgraf}\fi}} -% -\def\num@book{% -\renewcommand\thesection{\thechapter.\@arabic\c@section}% -\renewcommand\thesubsection{\thesection.\@arabic\c@subsection}% -\renewcommand\theequation{\thechapter.\@arabic\c@equation}% -\renewcommand\thefigure{\thechapter.\@arabic\c@figure}% -\renewcommand\thetable{\thechapter.\@arabic\c@table}% -\@addtoreset{section}{chapter}% -\@addtoreset{figure}{chapter}% -\@addtoreset{table}{chapter}% -\@addtoreset{equation}{chapter}} -% -% Ragged bottom for the actual page -\def\thisbottomragged{\def\@textbottom{\vskip\z@ \@plus.0001fil -\global\let\@textbottom\relax}} - -% This is texte.tex -% it defines various texts and their translations -% called up with documentstyle options -\def\switcht@albion{% -\def\abstractname{Summary.}% -\def\ackname{Acknowledgement.}% -\def\andname{and}% -\def\bibname{References}% -\def\lastandname{, and}% -\def\appendixname{Appendix}% -\def\chaptername{Chapter}% -\def\claimname{Claim}% -\def\conjecturename{Conjecture}% -\def\contentsname{Contents}% -\def\corollaryname{Corollary}% -\def\definitionname{Definition}% -\def\examplename{Example}% -\def\exercisename{Exercise}% -\def\figurename{Fig.}% -\def\keywordname{{\bf Key words:}}% -\def\indexname{Index}% -\def\lemmaname{Lemma}% -\def\contriblistname{List of Contributors}% -\def\listfigurename{List of Figures}% -\def\listtablename{List of Tables}% -\def\mailname{{\it Correspondence to\/}:}% -\def\noteaddname{Note added in proof}% -\def\notename{Note}% -\def\partname{Part}% -\def\prefacename{Preface}% -\def\problemname{Problem}% -\def\proofname{Proof}% -\def\propertyname{Property}% -\def\propositionname{Proposition}% -\def\questionname{Question}% -\def\refname{References}% -\def\remarkname{Remark}% -\def\seename{see}% -\def\solutionname{Solution}% -\def\subclassname{{\it Subject Classifications\/}:}% -\def\tablename{Table}% -\def\theoremname{Theorem}} -\switcht@albion -% Names of theorem like environments are already defined -% but must be translated if another language is chosen -% -% French section -\def\switcht@francais{\svlanginfo - \def\abstractname{R\'esum\'e.}% - \def\ackname{Remerciements.}% - \def\andname{et}% - \def\lastandname{ et}% - \def\appendixname{Appendice}% - \def\bibname{Bibliographie}% - \def\chaptername{Chapitre}% - \def\claimname{Pr\'etention}% - \def\conjecturename{Hypoth\`ese}% - \def\contentsname{Table des mati\`eres}% - \def\corollaryname{Corollaire}% - \def\definitionname{D\'efinition}% - \def\examplename{Exemple}% - \def\exercisename{Exercice}% - \def\figurename{Fig.}% - \def\keywordname{{\bf Mots-cl\'e:}}% - \def\indexname{Index}% - \def\lemmaname{Lemme}% - \def\contriblistname{Liste des contributeurs}% - \def\listfigurename{Liste des figures}% - \def\listtablename{Liste des tables}% - \def\mailname{{\it Correspondence to\/}:}% - \def\noteaddname{Note ajout\'ee \`a l'\'epreuve}% - \def\notename{Remarque}% - \def\partname{Partie}% - \def\prefacename{Avant-propos}% ou Pr\'eface - \def\problemname{Probl\`eme}% - \def\proofname{Preuve}% - \def\propertyname{Caract\'eristique}% -%\def\propositionname{Proposition}% - \def\questionname{Question}% - \def\refname{Lit\'erature}% - \def\remarkname{Remarque}% - \def\seename{voir}% - \def\solutionname{Solution}% - \def\subclassname{{\it Subject Classifications\/}:}% - \def\tablename{Tableau}% - \def\theoremname{Th\'eor\`eme}% -} -% -% German section -\def\switcht@deutsch{\svlanginfo - \def\abstractname{Zusammenfassung.}% - \def\ackname{Danksagung.}% - \def\andname{und}% - \def\lastandname{ und}% - \def\appendixname{Anhang}% - \def\bibname{Literaturverzeichnis}% - \def\chaptername{Kapitel}% - \def\claimname{Behauptung}% - \def\conjecturename{Hypothese}% - \def\contentsname{Inhaltsverzeichnis}% - \def\corollaryname{Korollar}% -%\def\definitionname{Definition}% - \def\examplename{Beispiel}% - \def\exercisename{\"Ubung}% - \def\figurename{Abb.}% - \def\keywordname{{\bf Schl\"usselw\"orter:}}% - \def\indexname{Sachverzeichnis}% -%\def\lemmaname{Lemma}% - \def\contriblistname{Mitarbeiter}% - \def\listfigurename{Abbildungsverzeichnis}% - \def\listtablename{Tabellenverzeichnis}% - \def\mailname{{\it Correspondence to\/}:}% - \def\noteaddname{Nachtrag}% - \def\notename{Anmerkung}% - \def\partname{Teil}% - \def\prefacename{Vorwort}% -%\def\problemname{Problem}% - \def\proofname{Beweis}% - \def\propertyname{Eigenschaft}% -%\def\propositionname{Proposition}% - \def\questionname{Frage}% - \def\refname{Literaturverzeichnis}% - \def\remarkname{Anmerkung}% - \def\seename{siehe}% - \def\solutionname{L\"osung}% - \def\subclassname{{\it Subject Classifications\/}:}% - \def\tablename{Tabelle}% -%\def\theoremname{Theorem}% -} - -\def\getsto{\mathrel{\mathchoice {\vcenter{\offinterlineskip -\halign{\hfil -$\displaystyle##$\hfil\cr\gets\cr\to\cr}}} -{\vcenter{\offinterlineskip\halign{\hfil$\textstyle##$\hfil\cr\gets -\cr\to\cr}}} -{\vcenter{\offinterlineskip\halign{\hfil$\scriptstyle##$\hfil\cr\gets -\cr\to\cr}}} -{\vcenter{\offinterlineskip\halign{\hfil$\scriptscriptstyle##$\hfil\cr -\gets\cr\to\cr}}}}} -\def\lid{\mathrel{\mathchoice {\vcenter{\offinterlineskip\halign{\hfil -$\displaystyle##$\hfil\cr<\cr\noalign{\vskip1.2\p@}=\cr}}} -{\vcenter{\offinterlineskip\halign{\hfil$\textstyle##$\hfil\cr<\cr -\noalign{\vskip1.2\p@}=\cr}}} -{\vcenter{\offinterlineskip\halign{\hfil$\scriptstyle##$\hfil\cr<\cr -\noalign{\vskip\p@}=\cr}}} -{\vcenter{\offinterlineskip\halign{\hfil$\scriptscriptstyle##$\hfil\cr -<\cr -\noalign{\vskip0.9\p@}=\cr}}}}} -\def\gid{\mathrel{\mathchoice {\vcenter{\offinterlineskip\halign{\hfil -$\displaystyle##$\hfil\cr>\cr\noalign{\vskip1.2\p@}=\cr}}} -{\vcenter{\offinterlineskip\halign{\hfil$\textstyle##$\hfil\cr>\cr -\noalign{\vskip1.2\p@}=\cr}}} -{\vcenter{\offinterlineskip\halign{\hfil$\scriptstyle##$\hfil\cr>\cr -\noalign{\vskip\p@}=\cr}}} -{\vcenter{\offinterlineskip\halign{\hfil$\scriptscriptstyle##$\hfil\cr ->\cr -\noalign{\vskip0.9\p@}=\cr}}}}} -\def\grole{\mathrel{\mathchoice {\vcenter{\offinterlineskip -\halign{\hfil -$\displaystyle##$\hfil\cr>\cr\noalign{\vskip-\p@}<\cr}}} -{\vcenter{\offinterlineskip\halign{\hfil$\textstyle##$\hfil\cr ->\cr\noalign{\vskip-\p@}<\cr}}} -{\vcenter{\offinterlineskip\halign{\hfil$\scriptstyle##$\hfil\cr ->\cr\noalign{\vskip-0.8\p@}<\cr}}} -{\vcenter{\offinterlineskip\halign{\hfil$\scriptscriptstyle##$\hfil\cr ->\cr\noalign{\vskip-0.3\p@}<\cr}}}}} -\def\bbbr{{\rm I\!R}} %reelle Zahlen -\def\bbbm{{\rm I\!M}} -\def\bbbn{{\rm I\!N}} %natuerliche Zahlen -\def\bbbf{{\rm I\!F}} -\def\bbbh{{\rm I\!H}} -\def\bbbk{{\rm I\!K}} -\def\bbbp{{\rm I\!P}} -\def\bbbone{{\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l} -{\rm 1\mskip-4.5mu l} {\rm 1\mskip-5mu l}}} -\def\bbbc{{\mathchoice {\setbox0=\hbox{$\displaystyle\rm C$}\hbox{\hbox -to\z@{\kern0.4\wd0\vrule\@height0.9\ht0\hss}\box0}} -{\setbox0=\hbox{$\textstyle\rm C$}\hbox{\hbox -to\z@{\kern0.4\wd0\vrule\@height0.9\ht0\hss}\box0}} -{\setbox0=\hbox{$\scriptstyle\rm C$}\hbox{\hbox -to\z@{\kern0.4\wd0\vrule\@height0.9\ht0\hss}\box0}} -{\setbox0=\hbox{$\scriptscriptstyle\rm C$}\hbox{\hbox -to\z@{\kern0.4\wd0\vrule\@height0.9\ht0\hss}\box0}}}} -\def\bbbq{{\mathchoice {\setbox0=\hbox{$\displaystyle\rm -Q$}\hbox{\raise -0.15\ht0\hbox to\z@{\kern0.4\wd0\vrule\@height0.8\ht0\hss}\box0}} -{\setbox0=\hbox{$\textstyle\rm Q$}\hbox{\raise -0.15\ht0\hbox to\z@{\kern0.4\wd0\vrule\@height0.8\ht0\hss}\box0}} -{\setbox0=\hbox{$\scriptstyle\rm Q$}\hbox{\raise -0.15\ht0\hbox to\z@{\kern0.4\wd0\vrule\@height0.7\ht0\hss}\box0}} -{\setbox0=\hbox{$\scriptscriptstyle\rm Q$}\hbox{\raise -0.15\ht0\hbox to\z@{\kern0.4\wd0\vrule\@height0.7\ht0\hss}\box0}}}} -\def\bbbt{{\mathchoice {\setbox0=\hbox{$\displaystyle\rm -T$}\hbox{\hbox to\z@{\kern0.3\wd0\vrule\@height0.9\ht0\hss}\box0}} -{\setbox0=\hbox{$\textstyle\rm T$}\hbox{\hbox -to\z@{\kern0.3\wd0\vrule\@height0.9\ht0\hss}\box0}} -{\setbox0=\hbox{$\scriptstyle\rm T$}\hbox{\hbox -to\z@{\kern0.3\wd0\vrule\@height0.9\ht0\hss}\box0}} -{\setbox0=\hbox{$\scriptscriptstyle\rm T$}\hbox{\hbox -to\z@{\kern0.3\wd0\vrule\@height0.9\ht0\hss}\box0}}}} -\def\bbbs{{\mathchoice -{\setbox0=\hbox{$\displaystyle \rm S$}\hbox{\raise0.5\ht0\hbox -to\z@{\kern0.35\wd0\vrule\@height0.45\ht0\hss}\hbox -to\z@{\kern0.55\wd0\vrule\@height0.5\ht0\hss}\box0}} -{\setbox0=\hbox{$\textstyle \rm S$}\hbox{\raise0.5\ht0\hbox -to\z@{\kern0.35\wd0\vrule\@height0.45\ht0\hss}\hbox -to\z@{\kern0.55\wd0\vrule\@height0.5\ht0\hss}\box0}} -{\setbox0=\hbox{$\scriptstyle \rm S$}\hbox{\raise0.5\ht0\hbox -to\z@{\kern0.35\wd0\vrule\@height0.45\ht0\hss}\raise0.05\ht0\hbox -to\z@{\kern0.5\wd0\vrule\@height0.45\ht0\hss}\box0}} -{\setbox0=\hbox{$\scriptscriptstyle\rm S$}\hbox{\raise0.5\ht0\hbox -to\z@{\kern0.4\wd0\vrule\@height0.45\ht0\hss}\raise0.05\ht0\hbox -to\z@{\kern0.55\wd0\vrule\@height0.45\ht0\hss}\box0}}}} -\def\bbbz{{\mathchoice {\hbox{$\textstyle\sf Z\kern-0.4em Z$}} -{\hbox{$\textstyle\sf Z\kern-0.4em Z$}} -{\hbox{$\scriptstyle\sf Z\kern-0.3em Z$}} -{\hbox{$\scriptscriptstyle\sf Z\kern-0.2em Z$}}}} - -\let\ts\, - -\setlength \labelsep {5\p@} -\setlength\leftmargini {17\p@} -\setlength\leftmargin {\leftmargini} -\setlength\leftmarginii {\leftmargini} -\setlength\leftmarginiii {\leftmargini} -\setlength\leftmarginiv {\leftmargini} -\setlength\labelwidth {\leftmargini} -\addtolength\labelwidth{-\labelsep} - -\def\@listI{\leftmargin\leftmargini - \parsep=\parskip - \topsep=\medskipamount - \itemsep=\parskip \advance\itemsep by -\parsep} -\let\@listi\@listI -\@listi - -\def\@listii{\leftmargin\leftmarginii - \labelwidth\leftmarginii - \advance\labelwidth by -\labelsep - \parsep=\parskip - \topsep=\z@ - \itemsep=\parskip - \advance\itemsep by -\parsep} - -\def\@listiii{\leftmargin\leftmarginiii - \labelwidth\leftmarginiii\advance\labelwidth by -\labelsep - \parsep=\parskip - \topsep=\z@ - \itemsep=\parskip - \advance\itemsep by -\parsep - \partopsep=\topsep} - -\setlength\arraycolsep{1.5\p@} -\setlength\tabcolsep{1.5\p@} - -\def\tableofcontents{\@restonecolfalse\if@twocolumn\@restonecoltrue\onecolumn - \fi\chapter*{\contentsname \@mkboth{{\contentsname}}{{\contentsname}}} - \@starttoc{toc}\if@restonecol\twocolumn\fi} - -\setcounter{tocdepth}{2} - -\def\l@part#1#2{\addpenalty{\@secpenalty}% - \addvspace{2em \@plus\p@}% - \begingroup - \parindent \z@ - \rightskip \z@ \@plus 5em - \hrule\vskip5\p@ - \bfseries\boldmath - \leavevmode - #1\par - \vskip5\p@ - \hrule - \vskip\p@ - \nobreak - \endgroup} - -\def\@dotsep{2} - -\def\addnumcontentsmark#1#2#3{% -\addtocontents{#1}{\protect\contentsline{#2}{\protect\numberline - {\thechapter}#3}{\thepage}}} -\def\addcontentsmark#1#2#3{% -\addtocontents{#1}{\protect\contentsline{#2}{#3}{\thepage}}} -\def\addcontentsmarkwop#1#2#3{% -\addtocontents{#1}{\protect\contentsline{#2}{#3}{0}}} - -\def\@adcmk[#1]{\ifcase #1 \or -\def\@gtempa{\addnumcontentsmark}% - \or \def\@gtempa{\addcontentsmark}% - \or \def\@gtempa{\addcontentsmarkwop}% - \fi\@gtempa{toc}{chapter}} -\def\addtocmark{\@ifnextchar[{\@adcmk}{\@adcmk[3]}} - -\def\l@chapter#1#2{\par\addpenalty{-\@highpenalty} - \addvspace{1.0em \@plus \p@} - \@tempdima \tocchpnum \begingroup - \parindent \z@ \rightskip \@tocrmarg - \advance\rightskip by \z@ \@plus 2cm - \parfillskip -\rightskip \pretolerance=10000 - \leavevmode \advance\leftskip\@tempdima \hskip -\leftskip - {\bfseries\boldmath#1}\ifx0#2\hfil\null - \else - \nobreak - \leaders\hbox{$\m@th \mkern \@dotsep mu.\mkern - \@dotsep mu$}\hfill - \nobreak\hbox to\@pnumwidth{\hss #2}% - \fi\par - \penalty\@highpenalty \endgroup} - -\newdimen\tocchpnum -\newdimen\tocsecnum -\newdimen\tocsectotal -\newdimen\tocsubsecnum -\newdimen\tocsubsectotal -\newdimen\tocsubsubsecnum -\newdimen\tocsubsubsectotal -\newdimen\tocparanum -\newdimen\tocparatotal -\newdimen\tocsubparanum -\tocchpnum=20\p@ % chapter {\bf 88.} \@plus 5.3\p@ -\tocsecnum=22.5\p@ % section 88.8. plus 4.722\p@ -\tocsubsecnum=30.5\p@ % subsection 88.8.8 plus 4.944\p@ -\tocsubsubsecnum=38\p@ % subsubsection 88.8.8.8 plus 4.666\p@ -\tocparanum=45\p@ % paragraph 88.8.8.8.8 plus 3.888\p@ -\tocsubparanum=53\p@ % subparagraph 88.8.8.8.8.8 plus 4.11\p@ -\def\calctocindent{% -\tocsectotal=\tocchpnum -\advance\tocsectotal by\tocsecnum -\tocsubsectotal=\tocsectotal -\advance\tocsubsectotal by\tocsubsecnum -\tocsubsubsectotal=\tocsubsectotal -\advance\tocsubsubsectotal by\tocsubsubsecnum -\tocparatotal=\tocsubsubsectotal -\advance\tocparatotal by\tocparanum} -\calctocindent - -\def\@dottedtocline#1#2#3#4#5{% - \ifnum #1>\c@tocdepth \else - \vskip \z@ \@plus.2\p@ - {\leftskip #2\relax \rightskip \@tocrmarg \advance\rightskip by \z@ \@plus 2cm - \parfillskip -\rightskip \pretolerance=10000 - \parindent #2\relax\@afterindenttrue - \interlinepenalty\@M - \leavevmode - \@tempdima #3\relax - \advance\leftskip \@tempdima \null\nobreak\hskip -\leftskip - {#4}\nobreak - \leaders\hbox{$\m@th - \mkern \@dotsep mu\hbox{.}\mkern \@dotsep - mu$}\hfill - \nobreak - \hb@xt@\@pnumwidth{\hfil\normalfont \normalcolor #5}% - \par}% - \fi} -% -\def\l@section{\@dottedtocline{1}{\tocchpnum}{\tocsecnum}} -\def\l@subsection{\@dottedtocline{2}{\tocsectotal}{\tocsubsecnum}} -\def\l@subsubsection{\@dottedtocline{3}{\tocsubsectotal}{\tocsubsubsecnum}} -\def\l@paragraph{\@dottedtocline{4}{\tocsubsubsectotal}{\tocparanum}} -\def\l@subparagraph{\@dottedtocline{5}{\tocparatotal}{\tocsubparanum}} - -\renewcommand\listoffigures{% - \chapter*{\listfigurename - \@mkboth{\listfigurename}{\listfigurename}}% - \@starttoc{lof}% - } - -\renewcommand\listoftables{% - \chapter*{\listtablename - \@mkboth{\listtablename}{\listtablename}}% - \@starttoc{lot}% - } - -\renewcommand\footnoterule{% - \kern-3\p@ - \hrule\@width 50\p@ - \kern2.6\p@} - -\newdimen\foot@parindent -\foot@parindent 10.83\p@ - -\long\def\@makefntext#1{\@setpar{\@@par\@tempdima \hsize - \advance\@tempdima-\foot@parindent\parshape\@ne\foot@parindent - \@tempdima}\par - \parindent \foot@parindent\noindent \hbox to \z@{% - \hss\hss$^{\@thefnmark}$ }#1} - -\if@spthms -% Definition of the "\spnewtheorem" command. -% -% Usage: -% -% \spnewtheorem{env_nam}{caption}[within]{cap_font}{body_font} -% or \spnewtheorem{env_nam}[numbered_like]{caption}{cap_font}{body_font} -% or \spnewtheorem*{env_nam}{caption}{cap_font}{body_font} -% -% New is "cap_font" and "body_font". It stands for -% fontdefinition of the caption and the text itself. -% -% "\spnewtheorem*" gives a theorem without number. -% -% A defined spnewthoerem environment is used as described -% by Lamport. -% -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -\def\@thmcountersep{.} -\def\@thmcounterend{.} -\newcommand\nocaption{\noexpand\@gobble} -\newdimen\spthmsep \spthmsep=3pt - -\def\spnewtheorem{\@ifstar{\@sthm}{\@Sthm}} - -% definition of \spnewtheorem with number - -\def\@spnthm#1#2{% - \@ifnextchar[{\@spxnthm{#1}{#2}}{\@spynthm{#1}{#2}}} -\def\@Sthm#1{\@ifnextchar[{\@spothm{#1}}{\@spnthm{#1}}} - -\def\@spxnthm#1#2[#3]#4#5{\expandafter\@ifdefinable\csname #1\endcsname - {\@definecounter{#1}\@addtoreset{#1}{#3}% - \expandafter\xdef\csname the#1\endcsname{\expandafter\noexpand - \csname the#3\endcsname \noexpand\@thmcountersep \@thmcounter{#1}}% - \expandafter\xdef\csname #1name\endcsname{#2}% - \global\@namedef{#1}{\@spthm{#1}{\csname #1name\endcsname}{#4}{#5}}% - \global\@namedef{end#1}{\@endtheorem}}} - -\def\@spynthm#1#2#3#4{\expandafter\@ifdefinable\csname #1\endcsname - {\@definecounter{#1}% - \expandafter\xdef\csname the#1\endcsname{\@thmcounter{#1}}% - \expandafter\xdef\csname #1name\endcsname{#2}% - \global\@namedef{#1}{\@spthm{#1}{\csname #1name\endcsname}{#3}{#4}}% - \global\@namedef{end#1}{\@endtheorem}}} - -\def\@spothm#1[#2]#3#4#5{% - \@ifundefined{c@#2}{\@latexerr{No theorem environment `#2' defined}\@eha}% - {\expandafter\@ifdefinable\csname #1\endcsname - {\global\@namedef{the#1}{\@nameuse{the#2}}% - \expandafter\xdef\csname #1name\endcsname{#3}% - \global\@namedef{#1}{\@spthm{#2}{\csname #1name\endcsname}{#4}{#5}}% - \global\@namedef{end#1}{\@endtheorem}}}} - -\def\@spthm#1#2#3#4{\topsep 7\p@ \@plus2\p@ \@minus4\p@ -\labelsep=\spthmsep\refstepcounter{#1}% -\@ifnextchar[{\@spythm{#1}{#2}{#3}{#4}}{\@spxthm{#1}{#2}{#3}{#4}}} - -\def\@spxthm#1#2#3#4{\@spbegintheorem{#2}{\csname the#1\endcsname}{#3}{#4}% - \ignorespaces} - -\def\@spythm#1#2#3#4[#5]{\@spopargbegintheorem{#2}{\csname - the#1\endcsname}{#5}{#3}{#4}\ignorespaces} - -\def\normalthmheadings{\def\@spbegintheorem##1##2##3##4{\trivlist - \item[\hskip\labelsep{##3##1\ ##2\@thmcounterend}]##4} -\def\@spopargbegintheorem##1##2##3##4##5{\trivlist - \item[\hskip\labelsep{##4##1\ ##2}]{##4(##3)\@thmcounterend\ }##5}} -\normalthmheadings - -\def\reversethmheadings{\def\@spbegintheorem##1##2##3##4{\trivlist - \item[\hskip\labelsep{##3##2\ ##1\@thmcounterend}]##4} -\def\@spopargbegintheorem##1##2##3##4##5{\trivlist - \item[\hskip\labelsep{##4##2\ ##1}]{##4(##3)\@thmcounterend\ }##5}} - -% definition of \spnewtheorem* without number - -\def\@sthm#1#2{\@Ynthm{#1}{#2}} - -\def\@Ynthm#1#2#3#4{\expandafter\@ifdefinable\csname #1\endcsname - {\global\@namedef{#1}{\@Thm{\csname #1name\endcsname}{#3}{#4}}% - \expandafter\xdef\csname #1name\endcsname{#2}% - \global\@namedef{end#1}{\@endtheorem}}} - -\def\@Thm#1#2#3{\topsep 7\p@ \@plus2\p@ \@minus4\p@ -\@ifnextchar[{\@Ythm{#1}{#2}{#3}}{\@Xthm{#1}{#2}{#3}}} - -\def\@Xthm#1#2#3{\@Begintheorem{#1}{#2}{#3}\ignorespaces} - -\def\@Ythm#1#2#3[#4]{\@Opargbegintheorem{#1} - {#4}{#2}{#3}\ignorespaces} - -\def\@Begintheorem#1#2#3{#3\trivlist - \item[\hskip\labelsep{#2#1\@thmcounterend}]} - -\def\@Opargbegintheorem#1#2#3#4{#4\trivlist - \item[\hskip\labelsep{#3#1}]{#3(#2)\@thmcounterend\ }} - -% initialize theorem environment - -\if@envcntshowhiercnt % show hierarchy counter - \def\@thmcountersep{.} - \spnewtheorem{theorem}{Theorem}[\envankh]{\bfseries}{\itshape} - \@addtoreset{theorem}{chapter} -\else % theorem counter only - \spnewtheorem{theorem}{Theorem}{\bfseries}{\itshape} - \if@envcntreset - \@addtoreset{theorem}{chapter} - \if@envcntresetsect - \@addtoreset{theorem}{section} - \fi - \fi -\fi - -%definition of divers theorem environments -\spnewtheorem*{claim}{Claim}{\itshape}{\rmfamily} -\spnewtheorem*{proof}{Proof}{\itshape}{\rmfamily} -% -\if@envcntsame % all environments like "Theorem" - using its counter - \def\spn@wtheorem#1#2#3#4{\@spothm{#1}[theorem]{#2}{#3}{#4}} -\else % all environments with their own counter - \if@envcntshowhiercnt % show hierarchy counter - \def\spn@wtheorem#1#2#3#4{\@spxnthm{#1}{#2}[\envankh]{#3}{#4}} - \else % environment counter only - \if@envcntreset % environment counter is reset each section - \if@envcntresetsect - \def\spn@wtheorem#1#2#3#4{\@spynthm{#1}{#2}{#3}{#4} - \@addtoreset{#1}{chapter}\@addtoreset{#1}{section}} - \else - \def\spn@wtheorem#1#2#3#4{\@spynthm{#1}{#2}{#3}{#4} - \@addtoreset{#1}{chapter}} - \fi - \else - \let\spn@wtheorem=\@spynthm - \fi - \fi -\fi -% -\let\spdefaulttheorem=\spn@wtheorem -% -\spn@wtheorem{case}{Case}{\itshape}{\rmfamily} -\spn@wtheorem{conjecture}{Conjecture}{\itshape}{\rmfamily} -\spn@wtheorem{corollary}{Corollary}{\bfseries}{\itshape} -\spn@wtheorem{definition}{Definition}{\bfseries}{\itshape} -\spn@wtheorem{example}{Example}{\itshape}{\rmfamily} -\spn@wtheorem{exercise}{Exercise}{\bfseries}{\rmfamily} -\spn@wtheorem{lemma}{Lemma}{\bfseries}{\itshape} -\spn@wtheorem{note}{Note}{\itshape}{\rmfamily} -\spn@wtheorem{problem}{Problem}{\bfseries}{\rmfamily} -\spn@wtheorem{property}{Property}{\itshape}{\rmfamily} -\spn@wtheorem{proposition}{Proposition}{\bfseries}{\itshape} -\spn@wtheorem{question}{Question}{\itshape}{\rmfamily} -\spn@wtheorem{solution}{Solution}{\bfseries}{\rmfamily} -\spn@wtheorem{remark}{Remark}{\itshape}{\rmfamily} -% -\newenvironment{theopargself} - {\def\@spopargbegintheorem##1##2##3##4##5{\trivlist - \item[\hskip\labelsep{##4##1\ ##2}]{##4##3\@thmcounterend\ }##5} - \def\@Opargbegintheorem##1##2##3##4{##4\trivlist - \item[\hskip\labelsep{##3##1}]{##3##2\@thmcounterend\ }}}{} -\newenvironment{theopargself*} - {\def\@spopargbegintheorem##1##2##3##4##5{\trivlist - \item[\hskip\labelsep{##4##1\ ##2}]{\hspace*{-\labelsep}##4##3\@thmcounterend}##5} - \def\@Opargbegintheorem##1##2##3##4{##4\trivlist - \item[\hskip\labelsep{##3##1}]{\hspace*{-\labelsep}##3##2\@thmcounterend}}}{} -\fi - -\def\@takefromreset#1#2{% - \def\@tempa{#1}% - \let\@tempd\@elt - \def\@elt##1{% - \def\@tempb{##1}% - \ifx\@tempa\@tempb\else - \@addtoreset{##1}{#2}% - \fi}% - \expandafter\expandafter\let\expandafter\@tempc\csname cl@#2\endcsname - \expandafter\def\csname cl@#2\endcsname{}% - \@tempc - \let\@elt\@tempd} - -% redefininition of the captions for "figure" and "table" environments -% -\@ifundefined{floatlegendstyle}{\def\floatlegendstyle{\bfseries}}{} -\def\floatcounterend{.\ } -\def\capstrut{\vrule\@width\z@\@height\topskip} -\@ifundefined{captionstyle}{\def\captionstyle{\normalfont\small}}{} -\@ifundefined{instindent}{\newdimen\instindent}{} - -\long\def\@caption#1[#2]#3{\par\addcontentsline{\csname - ext@#1\endcsname}{#1}{\protect\numberline{\csname - the#1\endcsname}{\ignorespaces #2}}\begingroup - \@parboxrestore\if@minipage\@setminipage\fi - \@makecaption{\csname fnum@#1\endcsname}{\ignorespaces #3}\par - \endgroup} - -\def\twocaptionwidth#1#2{\def\first@capwidth{#1}\def\second@capwidth{#2}} -% Default: .46\textwidth -\twocaptionwidth{.46\textwidth}{.46\textwidth} - -\def\leftcaption{\refstepcounter\@captype\@dblarg% - {\@leftcaption\@captype}} - -\def\rightcaption{\refstepcounter\@captype\@dblarg% - {\@rightcaption\@captype}} - -\long\def\@leftcaption#1[#2]#3{\addcontentsline{\csname - ext@#1\endcsname}{#1}{\protect\numberline{\csname - the#1\endcsname}{\ignorespaces #2}}\begingroup - \@parboxrestore - \vskip\figcapgap - \@maketwocaptions{\csname fnum@#1\endcsname}{\ignorespaces #3}% - {\first@capwidth}\ignorespaces\hspace{.073\textwidth}\hfill% - \endgroup} - -\long\def\@rightcaption#1[#2]#3{\addcontentsline{\csname - ext@#1\endcsname}{#1}{\protect\numberline{\csname - the#1\endcsname}{\ignorespaces #2}}\begingroup - \@parboxrestore - \@maketwocaptions{\csname fnum@#1\endcsname}{\ignorespaces #3}% - {\second@capwidth}\par - \endgroup} - -\long\def\@maketwocaptions#1#2#3{% - \parbox[t]{#3}{{\floatlegendstyle #1\floatcounterend}#2}} - -\def\fig@pos{l} -\newcommand{\leftfigure}[2][\fig@pos]{\makebox[.4635\textwidth][#1]{#2}} -\let\rightfigure\leftfigure - -\newdimen\figgap\figgap=0.5cm % hgap between figure and sidecaption -% -\long\def\@makesidecaption#1#2{% - \setbox0=\vbox{\hsize=\@tempdima - \captionstyle{\floatlegendstyle - #1\floatcounterend}#2}% - \ifdim\instindent<\z@ - \ifdim\ht0>-\instindent - \advance\instindent by\ht0 - \typeout{^^JClass-Warning: Legend of \string\sidecaption\space for - \@captype\space\csname the\@captype\endcsname - ^^Jis \the\instindent\space taller than the corresponding float - - ^^Jyou'd better switch the environment. }% - \instindent\z@ - \fi - \else - \ifdim\ht0<\instindent - \advance\instindent by-\ht0 - \advance\instindent by-\dp0\relax - \advance\instindent by\topskip - \advance\instindent by-11\p@ - \else - \advance\instindent by-\ht0 - \instindent=-\instindent - \typeout{^^JClass-Warning: Legend of \string\sidecaption\space for - \@captype\space\csname the\@captype\endcsname - ^^Jis \the\instindent\space taller than the corresponding float - - ^^Jyou'd better switch the environment. }% - \instindent\z@ - \fi - \fi - \parbox[b]{\@tempdima}{\captionstyle{\floatlegendstyle - #1\floatcounterend}#2% - \ifdim\instindent>\z@ \\ - \vrule\@width\z@\@height\instindent - \@depth\z@ - \fi}} -\def\sidecaption{\@ifnextchar[\sidec@ption{\sidec@ption[b]}} -\def\sidec@ption[#1]#2\caption{% -\setbox\@tempboxa=\hbox{\ignorespaces#2\unskip}% -\if@twocolumn - \ifdim\hsize<\textwidth\else - \ifdim\wd\@tempboxa<\columnwidth - \typeout{Double column float fits into single column - - ^^Jyou'd better switch the environment. }% - \fi - \fi -\fi - \instindent=\ht\@tempboxa - \advance\instindent by\dp\@tempboxa -\if t#1 -\else - \instindent=-\instindent -\fi -\@tempdima=\hsize -\advance\@tempdima by-\figgap -\advance\@tempdima by-\wd\@tempboxa -\ifdim\@tempdima<3cm - \ClassWarning{SVMono}{\string\sidecaption: No sufficient room for the legend; - ^^Jusing normal \string\caption}% - \unhbox\@tempboxa - \let\@capcommand=\@caption -\else - \ifdim\@tempdima<4.5cm - \ClassWarning{SVMono}{\string\sidecaption: Room for the legend very narrow; - ^^Jusing \string\raggedright}% - \toks@\expandafter{\captionstyle\sloppy - \rightskip=\z@\@plus6mm\relax}% - \def\captionstyle{\the\toks@}% - \fi - \let\@capcommand=\@sidecaption - \leavevmode - \unhbox\@tempboxa - \hfill -\fi -\refstepcounter\@captype -\@dblarg{\@capcommand\@captype}} -\long\def\@sidecaption#1[#2]#3{\addcontentsline{\csname - ext@#1\endcsname}{#1}{\protect\numberline{\csname - the#1\endcsname}{\ignorespaces #2}}\begingroup - \@parboxrestore - \@makesidecaption{\csname fnum@#1\endcsname}{\ignorespaces #3}\par - \endgroup} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -\def\fig@type{figure} - -\def\leftlegendglue{\hfil} -\newdimen\figcapgap\figcapgap=5\p@ % vgap between figure and caption -\newdimen\tabcapgap\tabcapgap=5.5\p@ % vgap between caption and table - -\long\def\@makecaption#1#2{% - \captionstyle - \ifx\@captype\fig@type - \vskip\figcapgap - \fi - \setbox\@tempboxa\hbox{{\floatlegendstyle #1\floatcounterend}% - \capstrut #2}% - \ifdim \wd\@tempboxa >\hsize - {\floatlegendstyle #1\floatcounterend}\capstrut #2\par - \else - \hbox to\hsize{\leftlegendglue\unhbox\@tempboxa\hfil}% - \fi - \ifx\@captype\fig@type\else - \vskip\tabcapgap - \fi} - -\newcounter{merk} - -\def\endfigure{\resetsubfig\end@float} - -\@namedef{endfigure*}{\resetsubfig\end@dblfloat} - -\def\resetsubfig{\global\let\last@subfig=\undefined} - -\def\r@setsubfig{\xdef\last@subfig{\number\value{figure}}% -\setcounter{figure}{\value{merk}}% -\setcounter{merk}{0}} - -\def\subfigures{\refstepcounter{figure}% - \@tempcnta=\value{merk}% - \setcounter{merk}{\value{figure}}% - \setcounter{figure}{\the\@tempcnta}% - \def\thefigure{\if@numart\else\thechapter.\fi - \@arabic\c@merk\alph{figure}}% - \let\resetsubfig=\r@setsubfig} - -\def\samenumber{\addtocounter{\@captype}{-1}% -\@ifundefined{last@subfig}{}{\setcounter{merk}{\last@subfig}}} - -% redefinition of the "bibliography" environment -% -\def\biblstarthook#1{\gdef\biblst@rthook{#1}} -% -\AtBeginDocument{% -\ifx\secbibl\undefined - \def\bibsection{\chapter*{\refname}\markboth{\refname}{\refname}% - \addcontentsline{toc}{chapter}{\refname}% - \csname biblst@rthook\endcsname} -\else - \def\bibsection{\section*{\refname}\markright{\refname}% - \addcontentsline{toc}{section}{\refname}% - \csname biblst@rthook\endcsname} -\fi} -\ifx\oribibl\undefined % Springer way of life - \renewenvironment{thebibliography}[1]{\bibsection - \global\let\biblst@rthook=\undefined - \def\@biblabel##1{##1.} - \small - \list{\@biblabel{\@arabic\c@enumiv}}% - {\settowidth\labelwidth{\@biblabel{#1}}% - \leftmargin\labelwidth - \advance\leftmargin\labelsep - \if@openbib - \advance\leftmargin\bibindent - \itemindent -\bibindent - \listparindent \itemindent - \parsep \z@ - \fi - \usecounter{enumiv}% - \let\p@enumiv\@empty - \renewcommand\theenumiv{\@arabic\c@enumiv}}% - \if@openbib - \renewcommand\newblock{\par}% - \else - \renewcommand\newblock{\hskip .11em \@plus.33em \@minus.07em}% - \fi - \sloppy\clubpenalty4000\widowpenalty4000% - \sfcode`\.=\@m} - {\def\@noitemerr - {\@latex@warning{Empty `thebibliography' environment}}% - \endlist} - \def\@lbibitem[#1]#2{\item[{[#1]}\hfill]\if@filesw - {\let\protect\noexpand\immediate - \write\@auxout{\string\bibcite{#2}{#1}}}\fi\ignorespaces} -\else % original bibliography is required - \let\bibname=\refname - \renewenvironment{thebibliography}[1] - {\chapter*{\bibname - \@mkboth{\bibname}{\bibname}}% - \list{\@biblabel{\@arabic\c@enumiv}}% - {\settowidth\labelwidth{\@biblabel{#1}}% - \leftmargin\labelwidth - \advance\leftmargin\labelsep - \@openbib@code - \usecounter{enumiv}% - \let\p@enumiv\@empty - \renewcommand\theenumiv{\@arabic\c@enumiv}}% - \sloppy - \clubpenalty4000 - \@clubpenalty \clubpenalty - \widowpenalty4000% - \sfcode`\.\@m} - {\def\@noitemerr - {\@latex@warning{Empty `thebibliography' environment}}% - \endlist} -\fi - -\let\if@threecolind\iffalse -\def\threecolindex{\let\if@threecolind\iftrue} -\def\indexstarthook#1{\gdef\indexst@rthook{#1}} -\renewenvironment{theindex} - {\if@twocolumn - \@restonecolfalse - \else - \@restonecoltrue - \fi - \columnseprule \z@ - \columnsep 1cc - \@nobreaktrue - \if@threecolind - \begin{multicols}{3}[\chapter*{\indexname}% - \else - \begin{multicols}{2}[\chapter*{\indexname}% - \fi - {\csname indexst@rthook\endcsname}]% - \global\let\indexst@rthook=\undefined - \markboth{\indexname}{\indexname}% - \addcontentsline{toc}{chapter}{\indexname}% - \flushbottom - \parindent\z@ - \rightskip\z@ \@plus 40\p@ - \parskip\z@ \@plus .3\p@\relax - \flushbottom - \let\item\@idxitem - \def\,{\relax\ifmmode\mskip\thinmuskip - \else\hskip0.2em\ignorespaces\fi}% - \normalfont\small} - {\end{multicols} - \global\let\if@threecolind\iffalse - \if@restonecol\onecolumn\else\clearpage\fi} - -\def\idxquad{\hskip 10\p@}% space that divides entry from number - -\def\@idxitem{\par\setbox0=\hbox{--\,--\,--\enspace}% - \hangindent\wd0\relax} - -\def\subitem{\par\noindent\setbox0=\hbox{--\enspace}% second order - \kern\wd0\setbox0=\hbox{--\,--\,--\enspace}% - \hangindent\wd0\relax}% indexentry - -\def\subsubitem{\par\noindent\setbox0=\hbox{--\,--\enspace}% third order - \kern\wd0\setbox0=\hbox{--\,--\,--\enspace}% - \hangindent\wd0\relax}% indexentry - -\def\indexspace{\par \vskip 10\p@ \@plus5\p@ \@minus3\p@\relax} - -\def\subtitle#1{\gdef\@subtitle{#1}} -\def\@subtitle{} - -\def\maketitle{\par - \begingroup - \def\thefootnote{\fnsymbol{footnote}}% - \def\@makefnmark{\hbox - to\z@{$\m@th^{\@thefnmark}$\hss}}% - \if@twocolumn - \twocolumn[\@maketitle]% - \else \newpage - \global\@topnum\z@ % Prevents figures from going at top of page. - \@maketitle \fi\thispagestyle{empty}\@thanks - \par\penalty -\@M - \endgroup - \setcounter{footnote}{0}% - \let\maketitle\relax - \let\@maketitle\relax - \gdef\@thanks{}\gdef\@author{}\gdef\@title{}\let\thanks\relax} - -\def\@maketitle{\newpage - \null - \vskip 2em % Vertical space above title. -\begingroup - \def\and{\unskip, } - \parindent=\z@ - \pretolerance=10000 - \rightskip=\z@ \@plus 3cm - {\LARGE % each author set in \LARGE - \lineskip .5em - \@author - \par}% - \vskip 2cm % Vertical space after author. - {\Huge \@title \par}% % Title set in \Huge size. - \vskip 1cm % Vertical space after title. - \if!\@subtitle!\else - {\LARGE\ignorespaces\@subtitle \par} - \vskip 1cm % Vertical space after subtitle. - \fi - \if!\@date!\else - {\large \@date}% % Date set in \large size. - \par - \vskip 1.5em % Vertical space after date. - \fi - \vfill - {\Large Springer\par} - \vskip 5\p@ - \large - Berlin\enspace Heidelberg\enspace New\kern0.1em York\\ - Hong\thinspace Kong\enspace London\\ - Milan\enspace Paris\enspace Tokyo\par -\endgroup} - -% Useful environments -\newenvironment{acknowledgement}{\par\addvspace{17\p@}\small\rm -\trivlist\item[\hskip\labelsep{\it\ackname}]} -{\endtrivlist\addvspace{6\p@}} -% -\newenvironment{noteadd}{\par\addvspace{17\p@}\small\rm -\trivlist\item[\hskip\labelsep{\it\noteaddname}]} -{\endtrivlist\addvspace{6\p@}} -% -\renewenvironment{abstract}{% - \advance\topsep by0.35cm\relax\small - \labelwidth=\z@ - \listparindent=\z@ - \itemindent\listparindent - \trivlist\item[\hskip\labelsep\bfseries\abstractname]% - \if!\abstractname!\hskip-\labelsep\fi - } - {\endtrivlist} - -% define the running headings of a twoside text -\def\runheadsize{\small} -\def\runheadstyle{\rmfamily\upshape} -\def\customizhead{\hspace{\headlineindent}} - -\def\ps@headings{\let\@mkboth\markboth - \let\@oddfoot\@empty\let\@evenfoot\@empty - \def\@evenhead{\runheadsize\runheadstyle\rlap{\thepage}\customizhead - \leftmark\hfil} - \def\@oddhead{\hfil\runheadsize\runheadstyle\rightmark\customizhead - \llap{\thepage}} - \def\chaptermark##1{\markboth{{\ifnum\c@secnumdepth>\m@ne - \thechapter\thechapterend\hskip\betweenumberspace\fi ##1}}{{\ifnum %!!! - \c@secnumdepth>\m@ne\thechapter\thechapterend\hskip\betweenumberspace\fi ##1}}}%!!! - \def\sectionmark##1{\markright{{\ifnum\c@secnumdepth>\z@ - \thesection\seccounterend\hskip\betweenumberspace\fi ##1}}}} - -\def\ps@myheadings{\let\@mkboth\@gobbletwo - \let\@oddfoot\@empty\let\@evenfoot\@empty - \def\@evenhead{\runheadsize\runheadstyle\rlap{\thepage}\customizhead - \leftmark\hfil} - \def\@oddhead{\hfil\runheadsize\runheadstyle\rightmark\customizhead - \llap{\thepage}} - \let\chaptermark\@gobble - \let\sectionmark\@gobble - \let\subsectionmark\@gobble} - - -\ps@headings - -\endinput -%end of file svmono.cls diff --git a/KLSadd_insertion/src/updateChapters.py b/KLSadd_insertion/src/updateChapters.py index a188d3a..a68fb59 100644 --- a/KLSadd_insertion/src/updateChapters.py +++ b/KLSadd_insertion/src/updateChapters.py @@ -7,62 +7,25 @@ # Path to data directory DATA_DIR = os.path.dirname(os.path.realpath(__file__)) + "/../data/" - - -# start out by reading KLSadd.tex to get all of the paragraphs that must be added to the chapter files -# also keep track of which chapter each one is in - - -# chapNums is needed to know which file to open (9 or 14) -# mathPeople is just the name for the sections that are used, like Wilson, Racah, etc. -chap_nums = [] -paras = [] - -# klsaddparas contains the locations of things with "paragraph{" in KLSadd, these are subsections that need to be sorted -klsaddparas = [] -math_people = [] -new_commands = [] # used to hold the indexes of the commands - -# ref9II holds the beginnings and ends of each chapter 9 section, provided and area to search -ref9_2 = [] - -# ref14II holds the beginnings and ends of each chapter 14 section, provided and area to search -ref14_2 = [] - -# ref9III holds all subsections in each chapter 9 section along with the what ref9II holds +# holds all subsections in each chapter section along with the what holds ref9_3 = [] - -# ref14III holds all subsections in each chapter 14 section along with the what ref14II holds ref14_3 = [] -# specref9 is the KLS indexes that go in chapter 9 -specref9 = [] - -# specref14 is the KLS indexes that in chapter 14 -specref14 = [] - -check = [] -check2 = [] -check3 = [] -kls_list = [] -sortmatch = [] +w, h = 2, 1000 +sorter_check = [[0 for _ in range(w)] for __ in range(h)] -w, h = 3, 9999 -sorterIIIcheck = [[0 for x in range(w)] for y in range(h)] -sortmatch_2 = [] - -test = "memesbananashgvlksrjc%/n" -print test[-2] -# 2/18/16 this method addresses the goal of hardcoding in the necessary packages to let the chapter files run as pdf's. -# Currently only works with chapter 9, ask Dr. Cohl to help port your chapter 14 output file into a pdf - -# 5/5/16 this method implements the Hypergeometric paragraphs into relevant sections for practicality -# Later versions should attempt to: fix paragraphs with variations on this name like -# Basic Hypergeometric Representation, q-Hypergeometric Representation +def repeated_item_deleter(list): + item = 0 + while item < len(list): + if "reference" in list[item].lower() or "limit relation" in list[item].lower(): + del list[item] + else: + item += 1 + return list -def new_keywords(kls): +def new_keywords(kls, kls_list): """ This section checks through the Addendum and identifies words that are valid keywords of sections It then takes that list and removes duplicates, as well as printing the output to another file. @@ -72,30 +35,20 @@ def new_keywords(kls): """ kls_list_chap = [] - for i in kls: - if kls.index(i) > 313: - if "paragraph{" in i or "subsubsection*{" in i: - kls_list.append(i[i.find("{") + 1: len(i) - 2]) + for item in kls: + if "paragraph{" in item or "subsubsection*{" in item: + kls_list.append(item[item.find("{") + 1: len(item) - 2]) - i = 0 - while i < len(kls_list): - if "reference" in kls_list[i].lower() or "limit relation" in kls_list[i].lower(): + kls_list = repeated_item_deleter(kls_list) - del kls_list[i] - else: - i += 1 kls_list.append("Limit Relation") - for i in kls_list: - add = True - for j in kls_list_chap: - if i == j: - add = False - if add: - kls_list_chap.append(i) + for item in kls_list: + if item not in kls_list_chap: + kls_list_chap.append(item) return kls_list_chap -def fix_chapter_sort(kls, chap, word, sortloc): +def fix_chapter_sort(kls, chap, word, sortloc, klsaddparas, sortmatch_2): """ This function sorts through the input files and identifies and places sections from the addendum after their respective subsections in the chapter. @@ -120,60 +73,60 @@ def fix_chapter_sort(kls, chap, word, sortloc): k_hyper_sub_chap = [] index = 0 - for i in kls: + for item in kls: index += 1 - line = str(i) + line = str(item) if "\\subsection" in line: temp = line[line.find(" ", 12)+1: line.find("}", 12)] # get just the name (like mathpeople) if sep1 == 0: - if name_chap in line.lower() and kls.index(i) > 313 and ("\\paragraph{" in line or + if name_chap in line.lower() and kls.index(item) > 313 and ("\\paragraph{" in line or "\\subsubsection*{" in line): - for i in klsaddparas: - if index < i: - klsloc = klsaddparas.index(i) + for item in klsaddparas: + if index < item: + klsloc = klsaddparas.index(item) break t = ''.join(kls[index: klsaddparas[klsloc]]) k_hyper_sub_chap.append(t) # append the whole paragraph, pray every paragraph ends with a % comment khyper_header_chap.append(temp) # append the name of subsection elif sep1 == 1: - if name_chap in line.lower() and kls.index(i) > 313 and ("\\paragraph{" in line or + if name_chap in line.lower() and kls.index(item) > 313 and ("\\paragraph{" in line or "\\subsubsection*{" in line) and "orthogonality relation" not in line.lower(): - for i in klsaddparas: - if index < i: - klsloc = klsaddparas.index(i) + for item in klsaddparas: + if index < item: + klsloc = klsaddparas.index(item) break t = ''.join(kls[index: klsaddparas[klsloc]]) k_hyper_sub_chap.append(t) # append the whole paragraph, pray every paragraph ends with a % comment khyper_header_chap.append(temp) # append the name of subsection elif sep1 == 2: - if name_chap in line.lower() and kls.index(i) > 313 and ("\\paragraph{" in line or + if name_chap in line.lower() and kls.index(item) > 313 and ("\\paragraph{" in line or "\\subsubsection*{" in line): - for i in klsaddparas: - if index < i: - klsloc = klsaddparas.index(i) + for item in klsaddparas: + if index < item: + klsloc = klsaddparas.index(item) break t = ''.join(kls[index: klsaddparas[klsloc]]) k_hyper_sub_chap.append(t) # append the whole paragraph, pray every paragraph ends with a % comment khyper_header_chap.append(temp) # append the name of subsection - for i in khyper_header_chap: - if i == "Pseudo Jacobi (or Routh-Romanovski)": - khyper_header_chap[khyper_header_chap.index(i)] = "Pseudo Jacobi" + for item in khyper_header_chap: + if item == "Pseudo Jacobi (or Routh-Romanovski)": + khyper_header_chap[khyper_header_chap.index(item)] = "Pseudo Jacobi" k_hyp_index_iii = 0 offset = 0 - i = 0 - while i < len(khyper_header_chap): + item = 0 + while item < len(khyper_header_chap): try: - if khyper_header_chap[i] == khyper_header_chap[i+1]: - k_hyper_sub_chap[i + 1] = k_hyper_sub_chap[i] + "\paragraph{\\bf KLS Addendum: " + \ - word + "}\n" + k_hyper_sub_chap[i + 1] - khyper_header_chap[i] = "memes" - k_hyper_sub_chap[i] = "memes" + if khyper_header_chap[item] == khyper_header_chap[item+1]: + k_hyper_sub_chap[item + 1] = k_hyper_sub_chap[item] + "\paragraph{\\bf KLS Addendum: " + \ + word + "}\n" + k_hyper_sub_chap[item + 1] + khyper_header_chap[item] = "memes" + k_hyper_sub_chap[item] = "memes" else: - i += 1 + item += 1 except IndexError: - i += 1 + item += 1 a = 0 while a < len(khyper_header_chap): @@ -183,22 +136,22 @@ def fix_chapter_sort(kls, chap, word, sortloc): else: a += 1 - global sorterIIIcheck + global sorter_check chap9 = 0 - if sorterIIIcheck[sortloc][0] == 0: + if sorter_check[sortloc][0] == 0: tempref = ref9_3 - sorterIIIcheck[sortloc][0] += 1 + sorter_check[sortloc][0] += 1 chap9 = 1 else: tempref = ref14_3 - offset = sorterIIIcheck[sortloc][1] + offset = sorter_check[sortloc][1] if sep1 == 1: offset = 8 for d in range(0, len(tempref)): # check every section and subsection line - i = tempref[d] - line = str(chap[i]) + item = tempref[d] + line = str(chap[item]) if "\\section{" in line or "\\subsection{" in line: if "\\subsection{" in line: temp = line[12:line.find("}", 7)] @@ -215,7 +168,7 @@ def fix_chapter_sort(kls, chap, word, sortloc): chap[tempref[d + 1] - 1] += "\paragraph{\\bf KLS Addendum: " + word + "}" chap[tempref[d + 1] - 1] += k_hyper_sub_chap[k_hyp_index_iii + offset] if chap9 == 1: - sorterIIIcheck[sortloc][1] += 1 + sorter_check[sortloc][1] += 1 k_hyp_index_iii += 1 except IndexError: print("Warning! Code has found an error involving section finding for '" + name_chap + "'.") @@ -246,14 +199,12 @@ def fix_chapter_sort(kls, chap, word, sortloc): print("Warning! Code has found an error involving section finding for '" + name_chap + "'.") if len(hyper_headers_chap) != 0: - print "Stuff for checking" - print k_hyper_sub_chap - print khyper_header_chap - print hyper_headers_chap - print word - sortmatch.append(word) + # print "Stuff for checking" + # print k_hyper_sub_chap + # print khyper_header_chap + # print hyper_headers_chap + # print word sortmatch_2.append(k_hyper_sub_chap) - sorterIIIcheck[sortloc][2] = word def cutwords(word_to_find, word_to_search_in): @@ -270,16 +221,12 @@ def cutwords(word_to_find, word_to_search_in): a = word_to_find b = word_to_search_in precheck = 1 - checkloop = False if a in b: if "\paragraph{\large\bf KLSadd: " or "\subsubsection*{\large\bf KLSadd: " in b: - while checkloop == False: - if "\\paragraph{\\large\\bf KLSadd: " in b[b.find(a)-precheck:b.find(a)]: - cut = b[:b.find(a) - precheck] + b[b.find(a) + len(a):] - return cut - elif "\\subsubsection*{\\large\\bf KLSadd: " in b[b.find(a)-precheck:b.find(a)]: - cut = b[:b.find(a) - precheck] + b[b.find(a) + len(a):] - return cut + while True: + if "\\paragraph{\\large\\bf KLSadd: " in b[b.find(a)-precheck:b.find(a)] or \ + "\\subsubsection*{\\large\\bf KLSadd: " in b[b.find(a) - precheck:b.find(a)]: + return b[:b.find(a) - precheck] + b[b.find(a) + len(a):] else: precheck += 1 @@ -307,7 +254,7 @@ def prepare_for_PDF(chap): return chap -def get_commands(kls): +def get_commands(kls, new_commands): """ this method reads in relevant commands that are in KLSadd.tex and returns them as a list @@ -353,7 +300,7 @@ def insert_commands(kls, chap, cms): # method to find the indices of the reference paragraphs -def find_references(chapter, chapticker): +def find_references(chapter, chapticker, math_people): """ This function searches the chapters and locates potential destinations of additions from the addendum. It puts these destinations in a list. @@ -387,20 +334,16 @@ def find_references(chapter, chapticker): ("Pseudo Jacobi" in w and "Pseudo Jacobi (or Routh-Romanovski)" in subunit): canadd = True if chapticker == 0: - ref9_2.append(index) ref9_3.append(index) elif chapticker == 1: - ref14_2.append(index) ref14_3.append(index) specialdetector = 0 if"\\subsection*{References}" in word and canadd: # Appends valid locations references.append(index) if chapticker == 0: - ref9_2.append(index) ref9_3.append(index) elif chapticker == 1: - ref14_2.append(index) ref14_3.append(index) canadd = False if "subsection*{" in word and "References" not in word: @@ -470,59 +413,64 @@ def reference_placer(chap, references, p, chapticker2): # method to change file string(actually a list right now), returns string to be written to file # If you write a method that changes something, it is preffered that you call the method in here -def fix_chapter(chap, references, p, kls, kls_list_all, chapticker2): +def fix_chapter(chap, references, paragraphs_to_be_added, kls, kls_list_all, chapticker2, new_commands, klsaddparas, sortmatch_2): """ Removes specific lines stopping the latex file from converting into python, as well as running the functions responsible for sorting sections and placing the correct additions in the correct places :param chap: Chapter being processed. :param references: List containing references (used in reference placer). - :param p: List containing additions to be added. + :param paragraphs_to_be_added: List containing additions to be added. :param kls: The addendum that contains sections to be added (not processed). :param kls_list_all: List of all identified keywords, fed into the chapter sorter. :param chapticker2: Which chapter is being searched (9 or 14). :return: """ - sortloc = 0 + sort_location = 0 for i in kls_list_all: - fix_chapter_sort(kls, chap, i, sortloc) - sortloc += 1 + fix_chapter_sort(kls, chap, i, sort_location, klsaddparas, sortmatch_2) + sort_location += 1 - for a in range(len(p)-1): + for a in range(len(paragraphs_to_be_added)-1): for b in sortmatch_2: for c in b: with open(DATA_DIR + "compare.tex", "a") as spook: - spook.write(p[a]) + spook.write(paragraphs_to_be_added[a]) spook.write("NEXT: ") + # i = 0 + # while i < len(c) - 1: + # if c[i:i + 1] == '\n': + # c = c[:i] + c[i + 1:] + # else: + # i += 1 + # i = 0 + # shortened_paras = paragraphs_to_be_added[a] + # while i < len(shortened_paras) - 1: + # if shortened_paras[i:i + 1] == '\n': + # shortened_paras = shortened_paras[:i] + shortened_paras[i + 1:] + # else: + # i += 1 + # paragraphs_to_be_added[a] = shortened_paras if "%" == c[-2]: - c = c[:-3] - #print "memes" - #print c + c = c[:-3] + #print "memes" + #print c elif "%" == c[-1]: - c = c[:-2] + c = c[:-2] #print "extramemes" #print c - if "Formula (9.8.15) was first obtained by Brafman \myciteKLS{109}." in c: - print "memes" - print c - print "memes" - if "Formula (9.8.15) was first obtained by Brafman \myciteKLS{109}." in p[a]: - print "extramemes" - print p[a] - print "extramemes" - #if "9.1 Wilson" in p[a]: - # print "l2" - # print c - # print "pause" - # print p[a] - # print "l2" - p[a] = cutwords(c, p[a]) - - reference_placer(chap, references, p, chapticker2) + # if "Formula (9.8.15) was first obtained by Brafman \myciteKLS{109}." in c and \ + # "Formula (9.8.15) was first obtained by Brafman \myciteKLS{109}." in p[a]: + # print "MEMESBEGIN"*20 + # print c + # print "MEMESEND"*24 + paragraphs_to_be_added[a] = cutwords(c, paragraphs_to_be_added[a]) + + reference_placer(chap, references, paragraphs_to_be_added, chapticker2) chap = prepare_for_PDF(chap) - cms = get_commands(kls) + cms = get_commands(kls,new_commands) chap = insert_commands(kls, chap, cms) commentticker = 0 @@ -560,11 +508,23 @@ def main(): :return: """ + chap_nums = [] + new_commands = [] # used to hold the indexes of the commands + + # contains the locations of things with "paragraph{" in KLSadd, these are subsections that need to be sorted + klsaddparas = [] + math_people = [] + + # Stores all keywords + kls_list_full = [] + + sortmatch_2 = [] + # open the KLSadd file to do things with with open(DATA_DIR + "KLSaddII.tex", "r") as add: # store the file as a string addendum = add.readlines() - kls_list_all = new_keywords(addendum) + kls_list_all = new_keywords(addendum, kls_list_full) # Makes sections look like other sections for word in addendum: @@ -588,14 +548,10 @@ def main(): chap_nums.append(9) name = word[word.find("{") + 1: word.find("}")] math_people.append(name + "#") - specref9.append(index-1) if "14." in word: chap_nums.append(14) name = word[word.find("{") + 1: word.find("}")] math_people.append(name + "#") - if len(specref14) == 0: - specref9.append(index - 1) - specref14.append(index - 1) indexes.append(index-1) if "paragraph{" in word and index > 313: klsaddparas.append(index-1) @@ -607,10 +563,10 @@ def main(): print(indexes) # now indexes holds all of the places there is a section # using these indexes, get all of the words in between and add that to the paras[] + paras = [] for i in range(len(indexes)-1): box = ''.join(addendum[indexes[i]: indexes[i+1]-1]) paras.append(box) - print(box) box2 = ''.join(addendum[indexes[35]: 2245]) paras.append(box2) @@ -633,19 +589,19 @@ def main(): # call the findReferences method to find the index of the References paragraph in the two file strings # two variables for the references lists one for chapter 9 one for chapter 14 chapticker = 0 - references9 = find_references(entire9, chapticker) + references9 = find_references(entire9, chapticker, math_people) chapticker += 1 - references14 = find_references(entire14, chapticker) + references14 = find_references(entire14, chapticker, math_people) ref14_3.insert(88, 1217) ref14_3.insert(244, 3053) ref14_3.insert(198, 2554) # call the fixChapter method to get a list with the addendum paragraphs added in chapticker2 = 0 - str9 = ''.join(fix_chapter(entire9, references9, paras, addendum, kls_list_all, chapticker2)) + str9 = ''.join(fix_chapter(entire9, references9, paras, addendum, kls_list_all, chapticker2, new_commands, klsaddparas, sortmatch_2)) chapticker2 += 1 - str14 = ''.join(fix_chapter(entire14, references14, paras, addendum, kls_list_all, chapticker2)) + str14 = ''.join(fix_chapter(entire14, references14, paras, addendum, kls_list_all, chapticker2, new_commands, klsaddparas, sortmatch_2)) # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ # If you are writing something that will make a change to the chapter files, write it BEFORE this line, this part From 921542804ba6a3427c643ac5067183cb4117706e Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Tue, 16 Aug 2016 10:16:36 -0400 Subject: [PATCH 244/402] Add tests for main function --- eCF/src/MathematicaToLaTeX.py | 27 ++++++------- eCF/test/test_main.py | 73 +++++++++++++++++++++++++++++++++++ 2 files changed, 85 insertions(+), 15 deletions(-) create mode 100644 eCF/test/test_main.py diff --git a/eCF/src/MathematicaToLaTeX.py b/eCF/src/MathematicaToLaTeX.py index f4e4242..e14a029 100644 --- a/eCF/src/MathematicaToLaTeX.py +++ b/eCF/src/MathematicaToLaTeX.py @@ -651,22 +651,22 @@ def convert_fraction(line): # ()/() line = '{0}\\frac{{{1}}}{{{2}}}{3}'\ .format(line[:j + 1], line[j + 2:i - 1], - line[i + 2:k], line[k + 1:]) + line[i + 2:k], line[k + 1:]) elif line[j + 1] == '(' and line[i - 1] == ')': # ()/-- line = '{0}\\frac{{{1}}}{{{2}}}{3}'\ .format(line[:j + 1], line[j + 2:i - 1], - line[i + 1:k + 1], line[k + 1:]) + line[i + 1:k + 1], line[k + 1:]) elif line[i + 1] == '(' and line[k] == ')': # --/() line = '{0}\\frac{{{1}}}{{{2}}}{3}'\ .format(line[:j + 1], line[j + 1:i], - line[i + 2:k], line[k + 1:]) + line[i + 2:k], line[k + 1:]) else: # --/-- line = '{0}\\frac{{{1}}}{{{2}}}{3}'\ .format(line[:j + 1], line[j + 1:i], - line[i + 1:k + 1], line[k + 1:]) + line[i + 1:k + 1], line[k + 1:]) i += 1 @@ -757,21 +757,14 @@ def replace_vars(line): return line -def main(): +def main(pathw, pathr, test): """ Opens Mathematica file with identities and puts converted lines into newIdentities.tex. """ - test = False - with open(os.path.dirname(os.path.realpath(__file__)) + - '/../data/newIdentities.tex', 'w') as latex: - if test: - to_open = 'IdentitiesTest.m' - else: - to_open = 'Identities.m' - with open(os.path.dirname(os.path.realpath(__file__)) + '/../data/' + - to_open, 'r') as mathematica: + with open(pathw, 'w') as latex: + with open(pathr, 'r') as mathematica: latex.write('\n\\documentclass{article}\n\n' '\\usepackage{amsmath}\n' @@ -866,4 +859,8 @@ def main(): FUNCTION_CONVERSIONS = tuple(FUNCTION_CONVERSIONS) if __name__ == '__main__': - main() + main( + os.path.dirname(os.path.realpath(__file__)) + + '/../data/newIdentities.tex', + os.path.dirname(os.path.realpath(__file__)) + + '/../data/IdentitiesTest.m', False) diff --git a/eCF/test/test_main.py b/eCF/test/test_main.py new file mode 100644 index 0000000..9c85af9 --- /dev/null +++ b/eCF/test/test_main.py @@ -0,0 +1,73 @@ + +__author__ = 'Kevin Chen' +__status__ = 'Development' + +from unittest import TestCase +from MathematicaToLaTeX import main + +import os + +PATHW = os.path.dirname(os.path.realpath(__file__)) + '/data/test.tex' +PATHR = os.path.dirname(os.path.realpath(__file__)) + '/data/test.m' + + +class TestMain(TestCase): + + def test_test(self): + main(PATHW, PATHR, True) + with open(PATHW, 'r') as l: + latex = l.read() + self.assertEqual( + latex, ('\n' + '\\documentclass{article}\n' + '\n' + '\\usepackage{amsmath}\n' + '\\usepackage{amsthm}\n' + '\\usepackage{amssymb}\n' + '\\usepackage{amsfonts}\n' + '\\usepackage{breqn}\n' + '\\usepackage{DLMFmath}\n' + '\\usepackage{DRMFfcns}\n' + '\\usepackage{DLMFfcns}\n' + '\\usepackage{graphicx}\n' + '\\usepackage[paperwidth=20in, paperheight=20in, margin=0.5in]{geometry}\n' + '\n' + '\\begin{document}\n' + '\n' + '\n' + '% {"description", number}%\n' + '\\begin{equation*}\n' + 'equation\n' + '\\end{equation*}\n' + '\n' + '\n' + '\\end{document}\n')) + + def test_test(self): + main(PATHW, PATHR, False) + with open(PATHW, 'r') as l: + latex = l.read() + self.assertEqual( + latex, ('\n' + '\\documentclass{article}\n' + '\n' + '\\usepackage{amsmath}\n' + '\\usepackage{amsthm}\n' + '\\usepackage{amssymb}\n' + '\\usepackage{amsfonts}\n' + '\\usepackage{breqn}\n' + '\\usepackage{DLMFmath}\n' + '\\usepackage{DRMFfcns}\n' + '\\usepackage{DLMFfcns}\n' + '\\usepackage{graphicx}\n' + '\\usepackage[paperwidth=20in, paperheight=20in, margin=0.5in]{geometry}\n' + '\n' + '\\begin{document}\n' + '\n' + '\n' + '\\begin{equation*} \\tag{description, number}\n' + 'equation\n' + '\\end{equation*}\n' + '\n' + '\n' + '\\end{document}\n')) From b391ffba08cd4614b1a84b92ed769dff7d20b28e Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Tue, 16 Aug 2016 10:27:04 -0400 Subject: [PATCH 245/402] Add test/data directory --- eCF/test/data/test.m | 2 ++ eCF/test/data/test.tex | 23 +++++++++++++++++++++++ 2 files changed, 25 insertions(+) create mode 100644 eCF/test/data/test.m create mode 100644 eCF/test/data/test.tex diff --git a/eCF/test/data/test.m b/eCF/test/data/test.m new file mode 100644 index 0000000..740f5d1 --- /dev/null +++ b/eCF/test/data/test.m @@ -0,0 +1,2 @@ +(* {"description", number}*) +equation diff --git a/eCF/test/data/test.tex b/eCF/test/data/test.tex new file mode 100644 index 0000000..8585591 --- /dev/null +++ b/eCF/test/data/test.tex @@ -0,0 +1,23 @@ + +\documentclass{article} + +\usepackage{amsmath} +\usepackage{amsthm} +\usepackage{amssymb} +\usepackage{amsfonts} +\usepackage{breqn} +\usepackage{DLMFmath} +\usepackage{DRMFfcns} +\usepackage{DLMFfcns} +\usepackage{graphicx} +\usepackage[paperwidth=20in, paperheight=20in, margin=0.5in]{geometry} + +\begin{document} + + +\begin{equation*} \tag{description, number} +equation +\end{equation*} + + +\end{document} From db031eede63d0a4fceaac4fb5e8ed0919a7db6a3 Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Tue, 16 Aug 2016 10:27:37 -0400 Subject: [PATCH 246/402] Move specific gitignore from repo to project --- .gitignore | 11 ----------- eCF/.gitignore | 11 +++++++++++ 2 files changed, 11 insertions(+), 11 deletions(-) diff --git a/.gitignore b/.gitignore index 1d43b2a..5c0f18c 100644 --- a/.gitignore +++ b/.gitignore @@ -251,14 +251,3 @@ sympy-plots-for-*.tex/ *.bak *.sav -# ecf project -*.sty -Identities.* -newIdentities.* -*.csv -macros.txt -test.* -*.log -*.aux -ZE.3.* - diff --git a/eCF/.gitignore b/eCF/.gitignore index e6caac9..26cb15a 100644 --- a/eCF/.gitignore +++ b/eCF/.gitignore @@ -16,3 +16,14 @@ tasks.txt *.sty *.aux *.csv + + +# ecf project +*.sty +Identities.* +newIdentities.* +*.csv +macros.txt +*.log +*.aux +ZE.3.* \ No newline at end of file From 33175adc7ecd9dec622f0c91f8d68d9c0da0de26 Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Tue, 16 Aug 2016 10:33:33 -0400 Subject: [PATCH 247/402] Fix duplicate function names --- eCF/test/test_main.py | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/eCF/test/test_main.py b/eCF/test/test_main.py index 9c85af9..e27319a 100644 --- a/eCF/test/test_main.py +++ b/eCF/test/test_main.py @@ -43,7 +43,7 @@ def test_test(self): '\n' '\\end{document}\n')) - def test_test(self): + def test_nottest(self): main(PATHW, PATHR, False) with open(PATHW, 'r') as l: latex = l.read() From 20a51bcad25b79db0157f565e28a6c67d757390e Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Tue, 16 Aug 2016 10:37:05 -0400 Subject: [PATCH 248/402] Modify main function to use defaults --- eCF/src/MathematicaToLaTeX.py | 11 +++++------ eCF/test/data/test.tex | 3 ++- eCF/test/test_main.py | 4 ++-- 3 files changed, 9 insertions(+), 9 deletions(-) diff --git a/eCF/src/MathematicaToLaTeX.py b/eCF/src/MathematicaToLaTeX.py index e14a029..52f047b 100644 --- a/eCF/src/MathematicaToLaTeX.py +++ b/eCF/src/MathematicaToLaTeX.py @@ -757,7 +757,10 @@ def replace_vars(line): return line -def main(pathw, pathr, test): +def main(pathw=os.path.dirname(os.path.realpath(__file__)) + + '/../data/newIdentities.tex', + pathr=os.path.dirname(os.path.realpath(__file__)) + + '/../data/Identities.m', test=False): """ Opens Mathematica file with identities and puts converted lines into newIdentities.tex. @@ -859,8 +862,4 @@ def main(pathw, pathr, test): FUNCTION_CONVERSIONS = tuple(FUNCTION_CONVERSIONS) if __name__ == '__main__': - main( - os.path.dirname(os.path.realpath(__file__)) + - '/../data/newIdentities.tex', - os.path.dirname(os.path.realpath(__file__)) + - '/../data/IdentitiesTest.m', False) + main() diff --git a/eCF/test/data/test.tex b/eCF/test/data/test.tex index 8585591..c6c4f9e 100644 --- a/eCF/test/data/test.tex +++ b/eCF/test/data/test.tex @@ -15,7 +15,8 @@ \begin{document} -\begin{equation*} \tag{description, number} +% {"description", number}% +\begin{equation*} equation \end{equation*} diff --git a/eCF/test/test_main.py b/eCF/test/test_main.py index e27319a..0945eea 100644 --- a/eCF/test/test_main.py +++ b/eCF/test/test_main.py @@ -14,7 +14,7 @@ class TestMain(TestCase): def test_test(self): - main(PATHW, PATHR, True) + main(pathw=PATHW, pathr=PATHR, test=True) with open(PATHW, 'r') as l: latex = l.read() self.assertEqual( @@ -44,7 +44,7 @@ def test_test(self): '\\end{document}\n')) def test_nottest(self): - main(PATHW, PATHR, False) + main(pathw=PATHW, pathr=PATHR, test=False) with open(PATHW, 'r') as l: latex = l.read() self.assertEqual( From 5181097b114fa6f72d91de1b1819755654e3aa12 Mon Sep 17 00:00:00 2001 From: Edward Bian Date: Tue, 16 Aug 2016 13:05:29 -0400 Subject: [PATCH 249/402] First Unit Tests & Code Cleaning --- KLSadd_insertion/README.md | 13 ++- KLSadd_insertion/src/updateChapters.py | 79 +++++-------------- KLSadd_insertion/test/test_fix_chapter.py | 8 ++ .../test/test_generic_functions.py | 20 +++++ KLSadd_insertion/test/test_new_keywords.py | 6 -- 5 files changed, 55 insertions(+), 71 deletions(-) create mode 100644 KLSadd_insertion/test/test_generic_functions.py delete mode 100644 KLSadd_insertion/test/test_new_keywords.py diff --git a/KLSadd_insertion/README.md b/KLSadd_insertion/README.md index f08662d..9b6f635 100644 --- a/KLSadd_insertion/README.md +++ b/KLSadd_insertion/README.md @@ -13,7 +13,7 @@ NOTE: The KLSadd.tex file only deals with chapters 9 and 14, as stated in the do NOTE: when you run updateChapters.py or linetest.py you need the KLSadd.tex file, tempchap9.tex, and tempchap14.tex in order to run the program. The program *generates* updated9.tex and updated14.tex files. -**THIS MEANS YOU SHOULD NOT RUN updateChapters.py OR linetest.py FROM YOUR GIT DIRECTORY. THE CHAPTER FILES ARE NOT PUBLIC DOMAIN, DO NOT COMMIT THEM! COPY THE FILES OVER INTO A SEPERATE DIRECTORY AND THEN COPY THEM BACK OVER WHEN YOU ARE READY TO COMMIT** +**DO NOT REMOVE INPUT FILES FROM .GITIGNORE, THEY ARE NOT PUBLIC** This is a roadmap of updateChapters.py. It will help explain every piece of code and exactly how everything works in relation to each other. @@ -32,7 +32,6 @@ First: the files - linetest.py: this abomination of code is the original program. It did the job, and it did it well. However, I realized that it was very very unreadable. This program is the child I never wanted. This program does 100% work and its outputs can be used to check against updateChapters.py's output. - newtempchap09.tex and newtempchap14.tex these are output files of linetest.py and this is what the output to updateChapters.py should be like!!!!!!! -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Next: the specifications @@ -62,7 +61,7 @@ So in the chapter 9 file, section 15 "Dogs", before the References subsection, w ```latex \paragraph{\large\bf Beagles} ``` - +--- **It is important to note that these changes are currently (3/25/16) unimplemented in updateChapters.py and only exist in linetest.py, if you want to see how the output to updateChapters.py should look, look at newtempchap09.tex** So after updateChapters.py is called, the chapter 9 file should look like this: @@ -80,7 +79,6 @@ in $a,b,c,d$. \cite{Dogs101} ``` -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The variables: These are the variables in the beginning @@ -95,7 +93,8 @@ These are the variables in the beginning -comms: holds the actual strings of the commands found from the line numbers stored in newCommands -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +--- + The methods: -prepareForPDF(chap): this method inserts some packages needed by LATEX files to be turned into pdf files. It takes a list called chap which is a String containing the contents of the chapter fie. It returns the chap String edited with the special packages in place. @@ -107,7 +106,7 @@ The methods: -findReferences(str): takes a string representation of a chapter file. Returns a list of the line numbers of the "References" subsections. This references variable is used as an index as to where the paragraphs in paras should be inserted -fixChapter(chap, references, p, kls): takes a chapter file string, a references list, the paras variable, and the KLSadd file string. This method basically just calls all of the methods above and adds all of the extra stuff like the commands and packages. -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + At this point, the code is like 90% comments. The big chunk at the bottom under the "with open..." can be explained through the comments. @@ -117,7 +116,7 @@ The chapters and KLSadd are turned into Strings and passed through the fixChapte Then the strings are written into seperate updated chapter files. -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +--- However, its broken. Inserted text comes in random places between paragraphs and sometimes stuff isn't added. This shouldn't be that hard to fix but it will require combing through the code a lot of times and changing little things here and there until everything works. As a last ditch effort, we can always resort to using linetext.py and adding stuff in from there. diff --git a/KLSadd_insertion/src/updateChapters.py b/KLSadd_insertion/src/updateChapters.py index a68fb59..18e3a61 100644 --- a/KLSadd_insertion/src/updateChapters.py +++ b/KLSadd_insertion/src/updateChapters.py @@ -15,7 +15,7 @@ sorter_check = [[0 for _ in range(w)] for __ in range(h)] -def repeated_item_deleter(list): +def extraneous_section_deleter(list): item = 0 while item < len(list): if "reference" in list[item].lower() or "limit relation" in list[item].lower(): @@ -38,8 +38,9 @@ def new_keywords(kls, kls_list): for item in kls: if "paragraph{" in item or "subsubsection*{" in item: kls_list.append(item[item.find("{") + 1: len(item) - 2]) + print item[item.find("{") + 1: len(item) - 2] - kls_list = repeated_item_deleter(kls_list) + kls_list = extraneous_section_deleter(kls_list) kls_list.append("Limit Relation") for item in kls_list: @@ -73,42 +74,25 @@ def fix_chapter_sort(kls, chap, word, sortloc, klsaddparas, sortmatch_2): k_hyper_sub_chap = [] index = 0 + chapterstart = False for item in kls: index += 1 line = str(item) if "\\subsection" in line: temp = line[line.find(" ", 12)+1: line.find("}", 12)] # get just the name (like mathpeople) + if temp.lower() == 'wilson': + chapterstart = True + if sep1 in (0, 2) and name_chap in line.lower() and chapterstart and ("\\paragraph{" in line or "\\subsubsection*{" in line) or \ + (sep1 == 1 and "orthogonality relation" not in line.lower() and name_chap in line.lower() and chapterstart + and ("\\paragraph{" in line or "\\subsubsection*{" in line)): + for item in klsaddparas: + if index < item: + klsloc = klsaddparas.index(item) + break + t = ''.join(kls[index: klsaddparas[klsloc]]) + k_hyper_sub_chap.append(t) # append the whole paragraph, pray every paragraph ends with a % comment + khyper_header_chap.append(temp) # append the name of subsection - if sep1 == 0: - if name_chap in line.lower() and kls.index(item) > 313 and ("\\paragraph{" in line or - "\\subsubsection*{" in line): - for item in klsaddparas: - if index < item: - klsloc = klsaddparas.index(item) - break - t = ''.join(kls[index: klsaddparas[klsloc]]) - k_hyper_sub_chap.append(t) # append the whole paragraph, pray every paragraph ends with a % comment - khyper_header_chap.append(temp) # append the name of subsection - elif sep1 == 1: - if name_chap in line.lower() and kls.index(item) > 313 and ("\\paragraph{" in line or - "\\subsubsection*{" in line) and "orthogonality relation" not in line.lower(): - for item in klsaddparas: - if index < item: - klsloc = klsaddparas.index(item) - break - t = ''.join(kls[index: klsaddparas[klsloc]]) - k_hyper_sub_chap.append(t) # append the whole paragraph, pray every paragraph ends with a % comment - khyper_header_chap.append(temp) # append the name of subsection - elif sep1 == 2: - if name_chap in line.lower() and kls.index(item) > 313 and ("\\paragraph{" in line or - "\\subsubsection*{" in line): - for item in klsaddparas: - if index < item: - klsloc = klsaddparas.index(item) - break - t = ''.join(kls[index: klsaddparas[klsloc]]) - k_hyper_sub_chap.append(t) # append the whole paragraph, pray every paragraph ends with a % comment - khyper_header_chap.append(temp) # append the name of subsection for item in khyper_header_chap: if item == "Pseudo Jacobi (or Routh-Romanovski)": khyper_header_chap[khyper_header_chap.index(item)] = "Pseudo Jacobi" @@ -149,6 +133,9 @@ def fix_chapter_sort(kls, chap, word, sortloc, klsaddparas, sortmatch_2): if sep1 == 1: offset = 8 + if sep1 == 2: + name_chap = 'hypergeometric representation' + for d in range(0, len(tempref)): # check every section and subsection line item = tempref[d] line = str(chap[item]) @@ -158,8 +145,8 @@ def fix_chapter_sort(kls, chap, word, sortloc, klsaddparas, sortmatch_2): else: temp = line[9:line.find("}", 7)] - if sep1 == 0: - if name_chap in line.lower(): + if name_chap in line.lower(): + if sep1 in (0, 2) or sep1 == 1 and "orthogonality relation" not in line: hyper_subs_chap.append([tempref[d + 1]]) # appends the index for the line before following subsection hyper_headers_chap.append(temp) # appends the name of the section the hypergeo subsection is in @@ -172,31 +159,7 @@ def fix_chapter_sort(kls, chap, word, sortloc, klsaddparas, sortmatch_2): k_hyp_index_iii += 1 except IndexError: print("Warning! Code has found an error involving section finding for '" + name_chap + "'.") - elif sep1 == 1: - if name_chap in line.lower() and "orthogonality relation" not in line: - hyper_subs_chap.append([tempref[d + 1]]) # appends the index for the line before following subsection - hyper_headers_chap.append(temp) # appends the name of the section the hypergeo subsection is in - - if temp in khyper_header_chap: - try: - chap[tempref[d + 1] - 1] += "\paragraph{\\bf KLS Addendum: " + word + "}" - chap[tempref[d + 1] - 1] += k_hyper_sub_chap[k_hyp_index_iii + offset] - k_hyp_index_iii += 1 - except IndexError: - print("Warning! Code has found an error involving section finding for '" + name_chap + "'.") - elif sep1 == 2: - if "hypergeometric representation" in line.lower(): - hyper_subs_chap.append([tempref[d + 1]]) # appends the index for the line before following subsection - hyper_headers_chap.append( - temp) # appends the name of the section the hypergeo subsection is in so we can compare - if temp in khyper_header_chap: - try: - chap[tempref[d + 1] - 1] += "\paragraph{\\bf KLS Addendum: Special Value }" - chap[tempref[d + 1] - 1] += k_hyper_sub_chap[k_hyp_index_iii + offset] - k_hyp_index_iii += 1 - except IndexError: - print("Warning! Code has found an error involving section finding for '" + name_chap + "'.") if len(hyper_headers_chap) != 0: # print "Stuff for checking" diff --git a/KLSadd_insertion/test/test_fix_chapter.py b/KLSadd_insertion/test/test_fix_chapter.py index e69de29..ab93c2f 100644 --- a/KLSadd_insertion/test/test_fix_chapter.py +++ b/KLSadd_insertion/test/test_fix_chapter.py @@ -0,0 +1,8 @@ +bob = "memes\nbanana" +i = 0 +while i < len(bob) - 1: + if bob[i:i + 1] == '\n': + bob = bob[:i] + bob[i+1:] + else: + i += 1 +print bob \ No newline at end of file diff --git a/KLSadd_insertion/test/test_generic_functions.py b/KLSadd_insertion/test/test_generic_functions.py new file mode 100644 index 0000000..0a80086 --- /dev/null +++ b/KLSadd_insertion/test/test_generic_functions.py @@ -0,0 +1,20 @@ + +__author__ = 'Edward Bian' +__status__ = 'Development' + +from unittest import TestCase +from updateChapters import new_keywords +from updateChapters import extraneous_section_deleter + + +class TestNewKeywords(TestCase): + + def test_new_keywords(self): + self.assertEquals(new_keywords(['\\subsubsection*{ThisIsATest}\n'],[]), ['ThisIsATest', 'Limit Relation']) + + +class TestExtraneousSectionDeleter(TestCase): + + def testExtraneousSectionDeleter(self): + self.assertEquals(extraneous_section_deleter(['reference', 'bananas', 'limit relation', '458673', + 'limit relations', 'apple']), ['bananas', '458673', 'apple']) \ No newline at end of file diff --git a/KLSadd_insertion/test/test_new_keywords.py b/KLSadd_insertion/test/test_new_keywords.py deleted file mode 100644 index 45e5add..0000000 --- a/KLSadd_insertion/test/test_new_keywords.py +++ /dev/null @@ -1,6 +0,0 @@ - -__author__ = 'Edward Bian' -__status__ = 'Development' - -from unittest import TestCase -from updateChapters import \ No newline at end of file From fb5cd99c594985425d796b24300cd997086ef4ed Mon Sep 17 00:00:00 2001 From: Edward Bian Date: Tue, 16 Aug 2016 13:36:13 -0400 Subject: [PATCH 250/402] More Unit Tests --- KLSadd_insertion/src/updateChapters.py | 16 ++++------------ KLSadd_insertion/test/test_generic_functions.py | 17 ++++++++++++++--- 2 files changed, 18 insertions(+), 15 deletions(-) diff --git a/KLSadd_insertion/src/updateChapters.py b/KLSadd_insertion/src/updateChapters.py index 18e3a61..ea66d42 100644 --- a/KLSadd_insertion/src/updateChapters.py +++ b/KLSadd_insertion/src/updateChapters.py @@ -16,13 +16,7 @@ def extraneous_section_deleter(list): - item = 0 - while item < len(list): - if "reference" in list[item].lower() or "limit relation" in list[item].lower(): - del list[item] - else: - item += 1 - return list + return [item for item in list if "reference" not in item.lower() and "limit relation" not in item.lower()] def new_keywords(kls, kls_list): @@ -38,8 +32,6 @@ def new_keywords(kls, kls_list): for item in kls: if "paragraph{" in item or "subsubsection*{" in item: kls_list.append(item[item.find("{") + 1: len(item) - 2]) - print item[item.find("{") + 1: len(item) - 2] - kls_list = extraneous_section_deleter(kls_list) kls_list.append("Limit Relation") @@ -170,7 +162,7 @@ def fix_chapter_sort(kls, chap, word, sortloc, klsaddparas, sortmatch_2): sortmatch_2.append(k_hyper_sub_chap) -def cutwords(word_to_find, word_to_search_in): +def cut_words(word_to_find, word_to_search_in): """ This function checks through the outputs of later sections and removes duplicates, so that the output file does not have duplicates. @@ -185,7 +177,7 @@ def cutwords(word_to_find, word_to_search_in): b = word_to_search_in precheck = 1 if a in b: - if "\paragraph{\large\bf KLSadd: " or "\subsubsection*{\large\bf KLSadd: " in b: + if "\\paragraph{\\large\\bf KLSadd: " in b or "\\subsubsection*{\\large\\bf KLSadd: " in b: while True: if "\\paragraph{\\large\\bf KLSadd: " in b[b.find(a)-precheck:b.find(a)] or \ "\\subsubsection*{\\large\\bf KLSadd: " in b[b.find(a) - precheck:b.find(a)]: @@ -429,7 +421,7 @@ def fix_chapter(chap, references, paragraphs_to_be_added, kls, kls_list_all, cha # print "MEMESBEGIN"*20 # print c # print "MEMESEND"*24 - paragraphs_to_be_added[a] = cutwords(c, paragraphs_to_be_added[a]) + paragraphs_to_be_added[a] = cut_words(c, paragraphs_to_be_added[a]) reference_placer(chap, references, paragraphs_to_be_added, chapticker2) chap = prepare_for_PDF(chap) diff --git a/KLSadd_insertion/test/test_generic_functions.py b/KLSadd_insertion/test/test_generic_functions.py index 0a80086..c87c788 100644 --- a/KLSadd_insertion/test/test_generic_functions.py +++ b/KLSadd_insertion/test/test_generic_functions.py @@ -5,7 +5,7 @@ from unittest import TestCase from updateChapters import new_keywords from updateChapters import extraneous_section_deleter - +from updateChapters import cut_words class TestNewKeywords(TestCase): @@ -15,6 +15,17 @@ def test_new_keywords(self): class TestExtraneousSectionDeleter(TestCase): - def testExtraneousSectionDeleter(self): + def test_extraneous_section_deleter(self): self.assertEquals(extraneous_section_deleter(['reference', 'bananas', 'limit relation', '458673', - 'limit relations', 'apple']), ['bananas', '458673', 'apple']) \ No newline at end of file + 'limit relations', 'apple']), ['bananas', '458673', 'apple']) + + +class TestCutWords(TestCase): + + def test_cut_words(self): + self.assertEquals(cut_words('MoreWords','WordsAndSomeMoreWords'), 'WordsAndSome') + self.assertEquals(cut_words('SupposedToFail', 'WordsAndSomeMoreWords'), 'WordsAndSomeMoreWords') + self.assertEquals(cut_words('StuffFromAddenum', '''\\paragraph{\\large\\bf KLSadd: TestCase}StuffFromAddenum\\begin{equation}\\end{equation}'''), + '''\\begin{equation}\\end{equation}''') + #self.assertEquals(extraneous_section_deleter(['reference', 'bananas', 'limit relation', '458673', + # 'limit relations', 'apple']), ['bananas', '458673', 'apple']) \ No newline at end of file From 9792d35369108ae814a0c10ce91276c0cda51b7b Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Tue, 16 Aug 2016 13:45:54 -0400 Subject: [PATCH 251/402] Add tests for master_function function --- eCF/src/MathematicaToLaTeX.py | 10 +++-- eCF/test/test_master_function.py | 71 ++++++++++++++++++++++++++++++++ eCF/test/test_replace_vars.py | 4 +- 3 files changed, 79 insertions(+), 6 deletions(-) diff --git a/eCF/src/MathematicaToLaTeX.py b/eCF/src/MathematicaToLaTeX.py index 52f047b..92d9bcc 100644 --- a/eCF/src/MathematicaToLaTeX.py +++ b/eCF/src/MathematicaToLaTeX.py @@ -758,9 +758,9 @@ def replace_vars(line): def main(pathw=os.path.dirname(os.path.realpath(__file__)) + - '/../data/newIdentities.tex', + '/../data/newIdentities.tex', pathr=os.path.dirname(os.path.realpath(__file__)) + - '/../data/Identities.m', test=False): + '/../data/IdentitiesTest.m', test=True): """ Opens Mathematica file with identities and puts converted lines into newIdentities.tex. @@ -806,8 +806,8 @@ def main(pathw=os.path.dirname(os.path.realpath(__file__)) + line = line.replace('EulerGamma', '\\EulerConstant') - for i in FUNCTION_CONVERSIONS: - line = master_function(line, i) + for func in FUNCTION_CONVERSIONS: + line = master_function(line, func) line = carat(line) @@ -861,5 +861,7 @@ def main(pathw=os.path.dirname(os.path.realpath(__file__)) + FUNCTION_CONVERSIONS = tuple(FUNCTION_CONVERSIONS) + if __name__ == '__main__': main() + diff --git a/eCF/test/test_master_function.py b/eCF/test/test_master_function.py index 427bd1d..39434fc 100644 --- a/eCF/test/test_master_function.py +++ b/eCF/test/test_master_function.py @@ -2,6 +2,77 @@ __author__ = 'Kevin Chen' __status__ = 'Development' +import os from unittest import TestCase from MathematicaToLaTeX import master_function +from MathematicaToLaTeX import arg_split + +with open(os.path.dirname(os.path.realpath(__file__)) + '/../data/functions') as functions: + FUNCTION_CONVERSIONS = list(arg_split(line.replace(' ', ''), ',') for line + in functions.read().split('\n') + if (line != '' and '#' not in line)) + + +for index, item in enumerate(FUNCTION_CONVERSIONS): + FUNCTION_CONVERSIONS[index][2] = tuple(arg_split(FUNCTION_CONVERSIONS[index][2][1:-1], ',')) + + if FUNCTION_CONVERSIONS[index][3] == '()': + FUNCTION_CONVERSIONS[index][3] = '' + else: + FUNCTION_CONVERSIONS[index][3] = tuple(FUNCTION_CONVERSIONS[index][3][1:-1].split(',')) + + FUNCTION_CONVERSIONS[index] = tuple(FUNCTION_CONVERSIONS[index]) + +FUNCTION_CONVERSIONS = tuple(FUNCTION_CONVERSIONS) + + +class TestMasterFunction(TestCase): + + def test_conversion(self): + for function in FUNCTION_CONVERSIONS: + for case in function[2]: + args = list('abcdefg'[:case.count('-')]) + before = function[0] + '[' + ','.join(args) + ']' + after = function[1] + case.replace('-', '%s') + + if function[0] == 'GegenbauerC': + args[0], args[1] = args[1], args[0] + if function[0] == 'HarmonicNumber' and case.count('-') == 2: + args[0], args[1] = args[1], args[0] + if function[0] == 'LaguerreL' and case.count('-') == 3: + args[0], args[1] = args[1], args[0] + + after %= tuple(args) + + if function[0] == 'HypergeometricPFQ': + self.assertEqual(master_function('HypergeometricPFQ[{},{},a]', function), + '\\HyperpFq{0}{0}@@{}{}{a}') + self.assertEqual(master_function('--HypergeometricPFQ[{},{},a]--', function), + '--\\HyperpFq{0}{0}@@{}{}{a}--') + self.assertEqual(master_function('HypergeometricPFQ[{a,b,c},{d,e,f},g]', function), + '\\HyperpFq{3}{3}@@{a,b,c}{d,e,f}{g}') + self.assertEqual(master_function('--HypergeometricPFQ[{a,b,c},{d,e,f},g]--', function), + '--\\HyperpFq{3}{3}@@{a,b,c}{d,e,f}{g}--') + elif function[0] == 'QHypergeometricPFQ': + self.assertEqual(master_function('QHypergeometricPFQ[{},{},a,b]', function), + '\\qHyperrphis{0}{0}@@{}{}{a}{b}') + self.assertEqual(master_function('--QHypergeometricPFQ[{},{},a,b]--', function), + '--\\qHyperrphis{0}{0}@@{}{}{a}{b}--') + self.assertEqual(master_function('QHypergeometricPFQ[{a,b,c},{d,e,f},g,h]', function), + '\\qHyperrphis{3}{3}@@{a,b,c}{d,e,f}{g}{h}') + self.assertEqual(master_function('--QHypergeometricPFQ[{a,b,c},{d,e,f},g,h]--', function), + '--\\qHyperrphis{3}{3}@@{a,b,c}{d,e,f}{g}{h}--') + else: + self.assertEqual(master_function(before, function), after) + self.assertEqual(master_function('--{0}--'.format(before), function), '--{0}--'.format(after)) + + def test_exception(self): + for function in FUNCTION_CONVERSIONS: + if function[3] != '': + for exception in function[3]: + self.assertEqual(master_function(exception + '[]', function), exception + '[]') + + def test_none(self): + for func in FUNCTION_CONVERSIONS: + self.assertEqual(master_function('nofunction', func), 'nofunction') diff --git a/eCF/test/test_replace_vars.py b/eCF/test/test_replace_vars.py index 9423174..5f4adb7 100644 --- a/eCF/test/test_replace_vars.py +++ b/eCF/test/test_replace_vars.py @@ -35,10 +35,10 @@ class TestReplaceVars(TestCase): def test_replace_symbols(self): for word in SYMBOLS: after = '\\' + SYMBOLS[word] - self.assertEquals(replace_vars('\\[' + word + ']'), after.replace('\\ ', '')) + self.assertEqual(replace_vars('\\[' + word + ']'), after.replace('\\ ', '')) def test_replace_infinity(self): - self.assertEquals(replace_vars('Infinity'), '\\infty') + self.assertEqual(replace_vars('Infinity'), '\\infty') def test_none(self): self.assertEqual(replace_vars('novariables'), 'novariables') From ca2fa5f1db0573eb1b9a414a69f1741cbba3bc66 Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Tue, 16 Aug 2016 13:51:38 -0400 Subject: [PATCH 252/402] Add one more test case for master_function function --- eCF/test/test_master_function.py | 2 ++ 1 file changed, 2 insertions(+) diff --git a/eCF/test/test_master_function.py b/eCF/test/test_master_function.py index 39434fc..c9f5a32 100644 --- a/eCF/test/test_master_function.py +++ b/eCF/test/test_master_function.py @@ -54,6 +54,8 @@ def test_conversion(self): '\\HyperpFq{3}{3}@@{a,b,c}{d,e,f}{g}') self.assertEqual(master_function('--HypergeometricPFQ[{a,b,c},{d,e,f},g]--', function), '--\\HyperpFq{3}{3}@@{a,b,c}{d,e,f}{g}--') + self.assertEqual(master_function('HypergeometricPFQ[{},{},HypergeometricPFQ[{},{},a]]', function), + '\\HyperpFq{0}{0}@@{}{}{\\HyperpFq{0}{0}@@{}{}{a}}') elif function[0] == 'QHypergeometricPFQ': self.assertEqual(master_function('QHypergeometricPFQ[{},{},a,b]', function), '\\qHyperrphis{0}{0}@@{}{}{a}{b}') From 0dc3acf371b1ddbcb2a7548111387dac05424565 Mon Sep 17 00:00:00 2001 From: Edward Bian Date: Tue, 16 Aug 2016 14:13:03 -0400 Subject: [PATCH 253/402] More Unit Tests --- KLSadd_insertion/test/test_commands.py | 43 ++++++++++++++++++++++++++ 1 file changed, 43 insertions(+) create mode 100644 KLSadd_insertion/test/test_commands.py diff --git a/KLSadd_insertion/test/test_commands.py b/KLSadd_insertion/test/test_commands.py new file mode 100644 index 0000000..30cd77b --- /dev/null +++ b/KLSadd_insertion/test/test_commands.py @@ -0,0 +1,43 @@ + +__author__ = 'Edward Bian' +__status__ = 'Development' + +from unittest import TestCase +from updateChapters import get_commands +from updateChapters import insert_commands +from updateChapters import prepare_for_pdf + + +class TestGetCommands(TestCase): + + def test_get_commands(self): + self.assertEquals(get_commands(['FakeCommands', 'MoreFakeCommands', '\\newcommand\\smallskipamount', 'Words', + '\\newcommandBob', '\\newcommand\\mybibitem[1]', 'KLSAddendumStuff', + '\\begin{document}'],[]), ['\\newcommand\\smallskipamount', 'Words', + '\\newcommandBob', '\\newcommand\\mybibitem[1]']) + + +class TestInsertCommands(TestCase): + + def test_insert_commands(self): + self.assertEquals(insert_commands(['Words','Commands','\\begin{document}','KLSAddendum', + '\subsection*{Introduction}'], + ['Words','Commands','\\begin{document}', + '\subsection*{Hypergeometric representation}'], + ['\\newcommand\\smallskipamount', '\\newcommand\\de\\delta', + '\\newcommand\\la\\lambda' + '\\newcommand\\mybibitem[1]']), + (['Words','Commands','\\newcommand\\smallskipamount', + '\\newcommand\\de\\delta', '\\newcommand\\la\\lambda' + '\\newcommand\\mybibitem[1]', '\\begin{document}', + '\subsection*{Hypergeometric representation}'])) + + +class TestPrepareForPDF(TestCase): + + def test_prepare_for_pdf(self): + self.assertEquals(prepare_for_pdf(['\\newcommand\\smallskipamount', '\usepackage[bottom]{footmisc}', + '\subsection*{Hypergeometric representation}']), + ['\\newcommand\\smallskipamount', '\usepackage[bottom]{footmisc}', + '\\usepackage[pdftex]{hyperref} \n\\usepackage {xparse} \n\\usepackage{cite} \n', + '\subsection*{Hypergeometric representation}']) From acc6916efe2445a671d6c175a14ce5546e371b44 Mon Sep 17 00:00:00 2001 From: Edward Bian Date: Tue, 16 Aug 2016 14:17:13 -0400 Subject: [PATCH 254/402] More Unit Tests --- KLSadd_insertion/src/updateChapters.py | 7 +++++-- 1 file changed, 5 insertions(+), 2 deletions(-) diff --git a/KLSadd_insertion/src/updateChapters.py b/KLSadd_insertion/src/updateChapters.py index ea66d42..3f65f90 100644 --- a/KLSadd_insertion/src/updateChapters.py +++ b/KLSadd_insertion/src/updateChapters.py @@ -191,7 +191,8 @@ def cut_words(word_to_find, word_to_search_in): else: return b -def prepare_for_PDF(chap): + +def prepare_for_pdf(chap): """ Edits the chapter string sent to include hyperref, xparse, and cite packages @@ -223,6 +224,7 @@ def get_commands(kls, new_commands): new_commands.append(index - 1) if "mybibitem[1]" in word: new_commands.append(index) + print new_commands return kls[new_commands[0]:new_commands[1]] @@ -424,8 +426,9 @@ def fix_chapter(chap, references, paragraphs_to_be_added, kls, kls_list_all, cha paragraphs_to_be_added[a] = cut_words(c, paragraphs_to_be_added[a]) reference_placer(chap, references, paragraphs_to_be_added, chapticker2) - chap = prepare_for_PDF(chap) + chap = prepare_for_pdf(chap) cms = get_commands(kls,new_commands) + print cms chap = insert_commands(kls, chap, cms) commentticker = 0 From c30290655bfa336c1d630bdcb310a00fc2c808f8 Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Tue, 16 Aug 2016 14:23:48 -0400 Subject: [PATCH 255/402] Add documentation to function and add extra space after eCF/.gitignore file --- eCF/.gitignore | 2 +- eCF/src/MathematicaToLaTeX.py | 139 ++++++++++++++++++++++++++++++---- 2 files changed, 126 insertions(+), 15 deletions(-) diff --git a/eCF/.gitignore b/eCF/.gitignore index 26cb15a..08e53cf 100644 --- a/eCF/.gitignore +++ b/eCF/.gitignore @@ -26,4 +26,4 @@ newIdentities.* macros.txt *.log *.aux -ZE.3.* \ No newline at end of file +ZE.3.* diff --git a/eCF/src/MathematicaToLaTeX.py b/eCF/src/MathematicaToLaTeX.py index 92d9bcc..848e528 100644 --- a/eCF/src/MathematicaToLaTeX.py +++ b/eCF/src/MathematicaToLaTeX.py @@ -13,6 +13,8 @@ import os +DIR_NAME = os.path.dirname(os.path.realpath(__file__)) + '/../data/' + SYMBOLS = { 'Alpha': 'alpha', 'Beta': 'beta', 'Gamma': 'gamma', 'Delta': 'delta', 'Epsilon': 'epsilon', 'Zeta': 'zeta', 'Eta': 'eta', 'Theta': 'theta', @@ -41,9 +43,21 @@ def find_surrounding(line, function, ex=(), start=0): + # (str, str(, tuple, int)) -> tuple """ Finds the indices of the beginning and end of a function; this is the main function that powers the converter. + + :param line: line to be converted + :param function: the function you're trying to find the surrounding + brackets for + :param ex: exceptions that shouldn't be converted, because conversions do + not see if a function is a part of another function (e.g. the + function "NotEquals" could get converted if a program was told + the original is "Equals" + :param start: index of where to start finding (used if there are multiple + of one function in a line + :returns: converted line """ positions = [0, 0] line = line[start:] @@ -79,9 +93,14 @@ def find_surrounding(line, function, ex=(), start=0): def arg_split(line, sep): + # (str, str) -> list """ - Does the same thing as 'split', but does not split when the separator is + Works very much like 'split', but does not split when the separator is inside parentheses, brackets, or braces. Useful for nested statements. + + :param line: line to be split + :param sep: seperator (character) + :returns: list of segments """ l = list('([{') r = list(')]}') @@ -107,9 +126,16 @@ def arg_split(line, sep): def master_function(line, params): + # (str, tuple) -> str """ A master function, reads in the conversion templates from the 'functions' file and performs the conversion. + + :param line: line to be converted + :param params: tuple containing Mathematica function, equivalent LaTeX + function, the format, using "-" as argument placings, and + exceptions, if any + :returns: converted line """ m, l, sep, ex = params[:5] sep = [i.split('-') for i in sep] @@ -154,7 +180,13 @@ def master_function(line, params): def remove_inactive(line): - """Removes 'Inactive' and its surrounding brackets.""" + # (str) -> str + """ + Removes 'Inactive' and its surrounding brackets. + + :param line: line to be converted + :returns: converted line + """ for _ in range(line.count('Inactive')): pos = find_surrounding(line, 'Inactive') if pos[0] != pos[1]: @@ -164,7 +196,13 @@ def remove_inactive(line): def remove_conditionalexpression(line): - """Removes 'ConditionalExpression' and its surrounding brackets.""" + # (str) -> str + """ + Removes 'ConditionalExpression' and its surrounding brackets. + + :param line: line to be converted + :returns: converted line + """ for _ in range(line.count('ConditionalExpression')): pos = find_surrounding(line, 'ConditionalExpression') if pos[0] != pos[1]: @@ -174,7 +212,13 @@ def remove_conditionalexpression(line): def remove_symbol(line): - """Removes 'Symbol' and its surrounding brackets.""" + # (str) -> str + """ + Removes 'Symbol' and its surrounding brackets. + + :param line: line to be converted + :returns: converted line + """ for _ in range(line.count('Symbol')): pos = find_surrounding(line, 'Symbol') if pos[0] != pos[1]: @@ -184,12 +228,16 @@ def remove_symbol(line): def carat(line): + # (str) -> str """ Converts carats ('^') to ones with braces instead of parentheses. e.g: 'a ^ (b + c)' would only show the first character 'b' as superscript in LaTeX, but converting it to 'a ^ {b + c}' would make it look correct in LaTeX, with 'b + c' as the superscript. + + :param line: line to be converted + :returns: converted line """ l = list('([{') r = list(')]}') @@ -224,13 +272,12 @@ def carat(line): def beta(line): + # (str) -> str """ Converts Mathematica's 'Beta' function to the equivalent LaTeX macro, taking into account the variations for the different number of arguments. - :type line: str :param line: line to be converted - :rtype: str :returns: converted line """ for _ in range(line.count('Beta')): @@ -258,8 +305,12 @@ def beta(line): def cfk(line): + # (str) -> str """ Converts Mathematica's 'ContinuedFractionK' to the equivalent LaTeX macro. + + :param line: line to be converted + :returns: converted line """ for _ in range(line.count('ContinuedFractionK')): try: @@ -289,9 +340,13 @@ def cfk(line): def gamma(line): + # (str) -> str """ Converts Mathematica's 'Gamma' function to the equivalent LaTeX macro, taking into account the variations for the different number of arguments. + + :param line: line to be converted + :returns: converted line """ for _ in range(line.count('Gamma')): try: @@ -329,8 +384,12 @@ def gamma(line): def integrate(line): + # (str) -> str """ Converts Mathematica's 'Integrate' function to the equivalent LaTeX macro. + + :param line: line to be converted + :returns: converted line """ for _ in range(line.count('Integrate')): try: @@ -353,9 +412,13 @@ def integrate(line): def legendrep(line): + # (str) -> str """ Converts Mathematica's 'LegendreP' function to the equivalent LaTeX macro, taking into account the variations for the different number of arguments. + + :param line: line to be converted + :returns: converted line """ for _ in range(line.count('LegendreP')): try: @@ -386,9 +449,13 @@ def legendrep(line): def legendreq(line): + # (str) -> str """ Converts Mathematica's 'LegendreQ' function to the equivalent LaTeX macro, taking into account the variations for the different number of arguments. + + :param line: line to be converted + :returns: converted line """ for _ in range(line.count('LegendreQ')): try: @@ -419,9 +486,13 @@ def legendreq(line): def polyeulergamma(line): + # (str) -> str """ Converts Mathematica's 'Polygamma' function to the equivalent LaTeX macro, taking into account the variations for the different number of arguments. + + :param line: line to be converted + :returns: converted line """ for _ in range(line.count('PolyGamma')): try: @@ -446,8 +517,12 @@ def polyeulergamma(line): def product(line): + # (str) -> str """ Converts Mathematica's product sum function to the equivalent LaTeX macro. + + :param line: line to be converted + :returns: converted line """ for _ in range(line.count('Product')): try: @@ -470,9 +545,13 @@ def product(line): def qpochhammer(line): + # (str) -> str """ Converts Mathematica's 'QPochhammer' function to the equivalent LaTeX macro. + + :param line: line to be converted + :returns: converted line """ for _ in range(line.count('QPochhammer')): try: @@ -501,8 +580,12 @@ def qpochhammer(line): def summation(line): + # (str) -> str """ Converts Mathematica's summation function to the equivalent LaTeX macro. + + :param line: line to be converted + :returns: converted line """ for _ in range(line.count('Sum')): try: @@ -525,9 +608,13 @@ def summation(line): def constraint(line): + # (str) -> str """ Converts Mathematica's 'Element', 'NotElement', and 'Inequality' functions to LaTeX formatting using \\constraint{}. + + :param line: line to be converted + :returns: converted line """ sections = arg_split(line, ',') @@ -599,9 +686,13 @@ def constraint(line): def convert_fraction(line): + # (str) -> str """ Converts Mathematica fractions, which are only '/', to LaTeX \\frac{}{}-ions. + + :param line: line to be converted + :returns: converted line """ l = list('([{') r = list(')]}') @@ -674,7 +765,13 @@ def convert_fraction(line): def piecewise(line): - """Converts Mathematica's piecewise function to LaTeX, using 'cases'.""" + # (str) -> str + """ + Converts Mathematica's piecewise function to LaTeX, using 'cases'. + + :param line: line to be converted + :returns: converted line + """ for _ in range(line.count('Piecewise')): try: pos @@ -703,7 +800,13 @@ def piecewise(line): def replace_operators(line): - """Replaces basic operators.""" + # (str) -> str + """ + Replaces basic operators. + + :param line: line to be converted + :returns: converted line + """ line = line.replace('==', '=') line = line.replace('||', ' \\lor ') line = line.replace('>=', ' \\geq ') @@ -742,9 +845,13 @@ def replace_operators(line): def replace_vars(line): + # (str) -> str """ Replaces the easy to convert variables in Mathematica to its equivalent LaTeX code in the dictionary 'symbols'. + + :param line: line to be converted + :returns: converted line """ for word in SYMBOLS: if SYMBOLS[word][0] == ' ': @@ -757,13 +864,18 @@ def replace_vars(line): return line -def main(pathw=os.path.dirname(os.path.realpath(__file__)) + - '/../data/newIdentities.tex', - pathr=os.path.dirname(os.path.realpath(__file__)) + - '/../data/IdentitiesTest.m', test=True): +def main(pathw=DIR_NAME + 'newIdentities.tex', + pathr=DIR_NAME + 'IdentitiesTest.m', test=False): + # ((str, str, bool)) -> None """ Opens Mathematica file with identities and puts converted lines into newIdentities.tex. + + :param pathw: directory of file to be written to + :param pathr: directory of file to be read from + :param test: if True, replaces "(* *)" with quotes; if False: uses "\\tag" + to mark the functions + :returns: None """ with open(pathw, 'w') as latex: @@ -841,7 +953,7 @@ def main(pathw=os.path.dirname(os.path.realpath(__file__)) + latex.write('\n\n\\end{document}\n') -with open(os.path.dirname(os.path.realpath(__file__)) + '/../data/functions') \ +with open(DIR_NAME + 'functions') \ as functions: FUNCTION_CONVERSIONS = list(arg_split(line.replace(' ', ''), ',') for line in functions.read().split('\n') @@ -864,4 +976,3 @@ def main(pathw=os.path.dirname(os.path.realpath(__file__)) + if __name__ == '__main__': main() - From e413cae570fe1913cb4bbcea6d2167b251fbbe71 Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Tue, 16 Aug 2016 14:47:12 -0400 Subject: [PATCH 256/402] Add comments for sections in code --- eCF/src/MathematicaToLaTeX.py | 25 ++++++++++++++----------- 1 file changed, 14 insertions(+), 11 deletions(-) diff --git a/eCF/src/MathematicaToLaTeX.py b/eCF/src/MathematicaToLaTeX.py index 848e528..6033024 100644 --- a/eCF/src/MathematicaToLaTeX.py +++ b/eCF/src/MathematicaToLaTeX.py @@ -63,20 +63,18 @@ def find_surrounding(line, function, ex=(), start=0): line = line[start:] positions[0] = line.find(function) + # Finds the exceptions (if any) and returns indeces after the exception if ex != '' and len(ex) >= 1: for e in ex: if (line.find(e) != -1 and line.find(e) <= positions[0] and - line.find(e) + len(e) >= positions[0] + len( - function)): + line.find(e) + len(e) >= positions[0] + len(function)): return [line.find(e) + len(e) + start, line.find(e) + len(e) + start] + # Finds the start and end of a function count = 0 for j in range(positions[0] + len(function), len(line) + 1): - if j == len(line) and count == 0: - positions[1] = positions[0] - break if line[j] in list('([{'): count += 1 @@ -153,7 +151,7 @@ def master_function(line, params): if pos[0] != pos[1]: args = arg_split(line[pos[0] + len(m) + 1:pos[1] - 1], ',') - # exceptions: + # Special functions that change the order of arguments: if m == 'GegenbauerC': args[0], args[1] = args[1], args[0] if m == 'HarmonicNumber' and len(args) == 2: @@ -247,6 +245,8 @@ def carat(line): while i != len(line): if line[i] == '^': count = 0 + + # Searches for when a carat ends for k in range(i + 1, len(line)): if line[k] in l: count += 1 @@ -702,7 +702,7 @@ def convert_fraction(line): while i != len(line): if line[i] == '/': - # Searching left. + # Searches left count = 0 for j in range(i - 1, -1, -1): if line[j] in r: @@ -717,7 +717,7 @@ def convert_fraction(line): j -= 1 break - # Searching right. + # Searches right count = 0 for k in range(i + 1, len(line)): if line[k] in l: @@ -822,6 +822,8 @@ def replace_operators(line): line = line.replace('-', ' - ') line = line.replace(',', ', ') + # This is so that things in a constraint, which is denoted by percentage + # signs, don't get operators converted if '%' in line: parts = (line[:line.index('%')], line[line.index('%'):]) line = parts[0] @@ -865,7 +867,7 @@ def replace_vars(line): def main(pathw=DIR_NAME + 'newIdentities.tex', - pathr=DIR_NAME + 'IdentitiesTest.m', test=False): + pathr=DIR_NAME + 'Identities.m', test=False): # ((str, str, bool)) -> None """ Opens Mathematica file with identities and puts converted lines into @@ -898,8 +900,8 @@ def main(pathw=DIR_NAME + 'newIdentities.tex', for line in mathematica: line = line.replace('\n', '') + # If line is a comment, make it a LaTeX comment or "\tag" if '(*' in line and '*)' in line: - if test: line = line.replace('(*', '%').replace('*)', '%') else: @@ -907,7 +909,6 @@ def main(pathw=DIR_NAME + 'newIdentities.tex', line[4:-3].replace('"', '') + '}') latex.write(line + '\n') - else: line = line.replace(' ', '') @@ -953,6 +954,8 @@ def main(pathw=DIR_NAME + 'newIdentities.tex', latex.write('\n\n\\end{document}\n') +# Open data/functions, and process the data into a comprehensible tuple that +# gets fed into "master_function" function with open(DIR_NAME + 'functions') \ as functions: FUNCTION_CONVERSIONS = list(arg_split(line.replace(' ', ''), ',') for line From 6a89c1da3321d2d9dd2d9d38fb9d8489a7bf05a2 Mon Sep 17 00:00:00 2001 From: Edward Bian Date: Tue, 16 Aug 2016 14:54:52 -0400 Subject: [PATCH 257/402] More Unit Tests --- KLSadd_insertion/src/updateChapters.py | 16 +++++++++++++--- KLSadd_insertion/test/test_generic_functions.py | 4 +--- KLSadd_insertion/test/test_reference_finder.py | 12 ++++++++++++ 3 files changed, 26 insertions(+), 6 deletions(-) create mode 100644 KLSadd_insertion/test/test_reference_finder.py diff --git a/KLSadd_insertion/src/updateChapters.py b/KLSadd_insertion/src/updateChapters.py index 3f65f90..4c42612 100644 --- a/KLSadd_insertion/src/updateChapters.py +++ b/KLSadd_insertion/src/updateChapters.py @@ -16,6 +16,12 @@ def extraneous_section_deleter(list): + """ + Removes sections that are irrelevant and will slow or confuse the program + + :param list: + :return: Extraneous name free output + """ return [item for item in list if "reference" not in item.lower() and "limit relation" not in item.lower()] @@ -128,6 +134,8 @@ def fix_chapter_sort(kls, chap, word, sortloc, klsaddparas, sortmatch_2): if sep1 == 2: name_chap = 'hypergeometric representation' + # Uses of numbers like 12, or 9 are for specific lengths of strings (like \\subsection{) where the + # section does not change for d in range(0, len(tempref)): # check every section and subsection line item = tempref[d] line = str(chap[item]) @@ -197,7 +205,7 @@ def prepare_for_pdf(chap): Edits the chapter string sent to include hyperref, xparse, and cite packages :param chap: The chapter (9 or 14) that is being processed - :return: + :return: The processed chapter, ready for additional processing """ foot_misc_index = 0 index = 0 @@ -212,7 +220,7 @@ def prepare_for_pdf(chap): def get_commands(kls, new_commands): """ - this method reads in relevant commands that are in KLSadd.tex and returns them as a list + This method reads in relevant commands that are in KLSadd.tex and returns them as a list :param kls: The addendum is the source of what is being found :return: @@ -315,6 +323,7 @@ def find_references(chapter, chapticker, math_people): besselcheck = 1 bqlegendrecheck = 2 + # Special lines that the program does not normally register for i in ref9_3: if i == 2646: besselcheck = 0 @@ -432,7 +441,7 @@ def fix_chapter(chap, references, paragraphs_to_be_added, kls, kls_list_all, cha chap = insert_commands(kls, chap, cms) commentticker = 0 - # Hard coded command remover + # These commands mess up PDF reading and mus tbe commented out for word in chap: word2 = chap[chap.index(word)-1] if "\\newcommand{\qhypK}[5]{\,\mbox{}_{#1}\phi_{#2}\!\left(" not in word2: @@ -573,6 +582,7 @@ def main(): with open(DATA_DIR + "updated14.tex", "w") as temp14: temp14.write(str14) + print math_people if __name__ == '__main__': main() \ No newline at end of file diff --git a/KLSadd_insertion/test/test_generic_functions.py b/KLSadd_insertion/test/test_generic_functions.py index c87c788..0504cd1 100644 --- a/KLSadd_insertion/test/test_generic_functions.py +++ b/KLSadd_insertion/test/test_generic_functions.py @@ -26,6 +26,4 @@ def test_cut_words(self): self.assertEquals(cut_words('MoreWords','WordsAndSomeMoreWords'), 'WordsAndSome') self.assertEquals(cut_words('SupposedToFail', 'WordsAndSomeMoreWords'), 'WordsAndSomeMoreWords') self.assertEquals(cut_words('StuffFromAddenum', '''\\paragraph{\\large\\bf KLSadd: TestCase}StuffFromAddenum\\begin{equation}\\end{equation}'''), - '''\\begin{equation}\\end{equation}''') - #self.assertEquals(extraneous_section_deleter(['reference', 'bananas', 'limit relation', '458673', - # 'limit relations', 'apple']), ['bananas', '458673', 'apple']) \ No newline at end of file + '''\\begin{equation}\\end{equation}''') \ No newline at end of file diff --git a/KLSadd_insertion/test/test_reference_finder.py b/KLSadd_insertion/test/test_reference_finder.py new file mode 100644 index 0000000..54e6e98 --- /dev/null +++ b/KLSadd_insertion/test/test_reference_finder.py @@ -0,0 +1,12 @@ + +__author__ = 'Edward Bian' +__status__ = 'Development' + +from unittest import TestCase +from updateChapters import find_references + +class TestFindReferences(TestCase): + + def test_find_references(self): + self.assertEquals(find_references(['section{'],0,), ) + From c4d7d01fafe29e0867ff013f9a1d61cc36ccaf7d Mon Sep 17 00:00:00 2001 From: Edward Bian Date: Tue, 16 Aug 2016 16:01:20 -0400 Subject: [PATCH 258/402] More Unit Tests --- KLSadd_insertion/src/updateChapters.py | 8 ++-- .../test/test_reference_finder.py | 42 ++++++++++++++++++- 2 files changed, 45 insertions(+), 5 deletions(-) diff --git a/KLSadd_insertion/src/updateChapters.py b/KLSadd_insertion/src/updateChapters.py index 4c42612..83ed3a7 100644 --- a/KLSadd_insertion/src/updateChapters.py +++ b/KLSadd_insertion/src/updateChapters.py @@ -287,7 +287,7 @@ def find_references(chapter, chapticker, math_people): for word in chapter: index += 1 - specialdetector = 1 + special_detector = 1 # check sections and subsections if("section{" in word or "subsection*{" in word) and ("subsubsection*{" not in word): w = word[word.find("{")+1: word.find("}")] @@ -302,8 +302,8 @@ def find_references(chapter, chapticker, math_people): ref9_3.append(index) elif chapticker == 1: ref14_3.append(index) - specialdetector = 0 - if"\\subsection*{References}" in word and canadd: + special_detector = 0 + if "\\subsection*{References}" in word and canadd: # Appends valid locations references.append(index) if chapticker == 0: @@ -316,7 +316,7 @@ def find_references(chapter, chapticker, math_people): ref9_3.append(index) elif chapticker == 1: ref14_3.append(index) - if "\\section{" in word and specialdetector == 1 and chaptercheck == "14": + if "\\section{" in word and special_detector == 1 and chaptercheck == "14": w2 = word[word.find("{") + 1: word.find("}")] if "Bessel" not in w2: ref14_3.append(index) diff --git a/KLSadd_insertion/test/test_reference_finder.py b/KLSadd_insertion/test/test_reference_finder.py index 54e6e98..2cb49c0 100644 --- a/KLSadd_insertion/test/test_reference_finder.py +++ b/KLSadd_insertion/test/test_reference_finder.py @@ -8,5 +8,45 @@ class TestFindReferences(TestCase): def test_find_references(self): - self.assertEquals(find_references(['section{'],0,), ) + self.assertEquals(find_references(['\\section{Pseudo Jacobi}\\index{Racah polynomials}', + '\\subsection*{Hypergeometric representation}', '\\begin{eqnarray}', + '\\label{DefRacah}', '& &R_n(\lambda(x);\alpha,\beta,\gamma,\delta)\nonumber\\' + , '& &{}=\hyp{4}{3}{-n,n+\alpha+\beta+1,-x,' + 'x+\gamma+\delta+1}{\alpha+1,\beta+\delta+1,\gamma+1}{1},', '\quad n=0,1,2,\ldots,N,' + , '\end{eqnarray}', 'where', '$$\lambda(x)=x(x+\gamma+\delta+1)$$', 'and' + , '$$\alpha+1=-N\quad\textrm{or}\quad\beta+\delta+1=-N\quad\textrm{or}\quad\gamma+1=-N$$' + , 'with $N$ a nonnegative integer.', '\subsection*{References}', + '\cite{Askey89I}, \cite{AskeyWilson79}, \cite{AskeyWilson85}'], + 0, ['9.9 Pseudo Jacobi (or Routh-Romanovski)#']), [13]) + self.assertEquals(find_references(['\\section{Pseudo Jacobi}\\index{Racah polynomials}', + '\\subsection*{Hypergeometric representation}', '\\begin{eqnarray}', + '\\label{DefRacah}', '& &R_n(\lambda(x);\alpha,\beta,\gamma,\delta)\nonumber\\' + , '& &{}=\hyp{4}{3}{-n,n+\alpha+\beta+1,-x,' + 'x+\gamma+\delta+1}{\alpha+1,\beta+\delta+1,\gamma+1}{1},', '\quad n=0,1,2,\ldots,N,' + , '\end{eqnarray}', 'where', '$$\lambda(x)=x(x+\gamma+\delta+1)$$', 'and' + , '$$\alpha+1=-N\quad\textrm{or}\quad\beta+\delta+1=-N\quad\textrm{or}\quad\gamma+1=-N$$' + , 'with $N$ a nonnegative integer.', '\subsection*{References}', + '\cite{Askey89I}, \cite{AskeyWilson79}, \cite{AskeyWilson85}'], + 1, ['9.9 Pseudo Jacobi (or Routh-Romanovski)#']), [13]) + self.assertEquals(find_references(['\\section{Racah}\\index{Racah polynomials}', + '\\subsection*{Hypergeometric representation}', '\\begin{eqnarray}', + '\\label{DefRacah}', '& &R_n(\lambda(x);\alpha,\beta,\gamma,\delta)\nonumber\\' + , '& &{}=\hyp{4}{3}{-n,n+\alpha+\beta+1,-x,' + 'x+\gamma+\delta+1}{\alpha+1,\beta+\delta+1,\gamma+1}{1},', '\quad n=0,1,2,\ldots,N,' + , '\end{eqnarray}', 'where', '$$\lambda(x)=x(x+\gamma+\delta+1)$$', 'and' + , '$$\alpha+1=-N\quad\textrm{or}\quad\beta+\delta+1=-N\quad\textrm{or}\quad\gamma+1=-N$$' + , 'with $N$ a nonnegative integer.', '\subsection*{References}', + '\cite{Askey89I}, \cite{AskeyWilson79}, \cite{AskeyWilson85}'], + 0, ['9.2 Racah#']), [13]) + self.assertEquals(find_references(['\\section{Racah}\\index{Racah polynomials}', + '\\subsection*{Hypergeometric representation}', '\\begin{eqnarray}', + '\\label{DefRacah}', '& &R_n(\lambda(x);\alpha,\beta,\gamma,\delta)\nonumber\\' + , '& &{}=\hyp{4}{3}{-n,n+\alpha+\beta+1,-x,' + 'x+\gamma+\delta+1}{\alpha+1,\beta+\delta+1,\gamma+1}{1},', '\quad n=0,1,2,\ldots,N,' + , '\end{eqnarray}', 'where', '$$\lambda(x)=x(x+\gamma+\delta+1)$$', 'and' + , '$$\alpha+1=-N\quad\textrm{or}\quad\beta+\delta+1=-N\quad\textrm{or}\quad\gamma+1=-N$$' + , 'with $N$ a nonnegative integer.', '\subsection*{References}', + '\cite{Askey89I}, \cite{AskeyWilson79}, \cite{AskeyWilson85}'], + 0, ['9.2 Racah#']), [13]) + From 94936827b91edfca8742d205eb3204fe62c00d8d Mon Sep 17 00:00:00 2001 From: Edward Bian Date: Tue, 16 Aug 2016 16:15:11 -0400 Subject: [PATCH 259/402] More Unit Tests --- KLSadd_insertion/test/test_reference_finder.py | 12 ++++++++++-- 1 file changed, 10 insertions(+), 2 deletions(-) diff --git a/KLSadd_insertion/test/test_reference_finder.py b/KLSadd_insertion/test/test_reference_finder.py index 2cb49c0..b65437f 100644 --- a/KLSadd_insertion/test/test_reference_finder.py +++ b/KLSadd_insertion/test/test_reference_finder.py @@ -48,5 +48,13 @@ def test_find_references(self): , 'with $N$ a nonnegative integer.', '\subsection*{References}', '\cite{Askey89I}, \cite{AskeyWilson79}, \cite{AskeyWilson85}'], 0, ['9.2 Racah#']), [13]) - - + self.assertEquals(find_references(['\\section{Unique}\\index{Racah polynomials}', + '\\subsection*{Hypergeometric representation}', '\\begin{eqnarray}', + '\\label{DefRacah}', '& &R_n(\lambda(x);\alpha,\beta,\gamma,\delta)\nonumber\\' + , '& &{}=\hyp{4}{3}{-n,n+\alpha+\beta+1,-x,' + 'x+\gamma+\delta+1}{\alpha+1,\beta+\delta+1,\gamma+1}{1},', '\quad n=0,1,2,\ldots,N,' + , '\end{eqnarray}', 'where', '$$\lambda(x)=x(x+\gamma+\delta+1)$$', 'and' + , '$$\alpha+1=-N\quad\textrm{or}\quad\beta+\delta+1=-N\quad\textrm{or}\quad\gamma+1=-N$$' + , 'with $N$ a nonnegative integer.', '\subsection*{References}', + '\cite{Askey89I}, \cite{AskeyWilson79}, \cite{AskeyWilson85}'], + 1, ['9.2 Racah#']), []) \ No newline at end of file From 32ed7d8e3f1df6b1e52b3b945f2e4fc2d08b9459 Mon Sep 17 00:00:00 2001 From: Edward Bian Date: Tue, 16 Aug 2016 16:19:20 -0400 Subject: [PATCH 260/402] More Unit Tests --- KLSadd_insertion/src/updateChapters.py | 3 --- 1 file changed, 3 deletions(-) diff --git a/KLSadd_insertion/src/updateChapters.py b/KLSadd_insertion/src/updateChapters.py index 83ed3a7..31810ce 100644 --- a/KLSadd_insertion/src/updateChapters.py +++ b/KLSadd_insertion/src/updateChapters.py @@ -329,9 +329,6 @@ def find_references(chapter, chapticker, math_people): besselcheck = 0 if besselcheck == 1: ref9_3.insert(230, 2646) - for i in ref14_3: - if i == 1217: - bqlegendrecheck -= 1 if bqlegendrecheck > 0: pass From 85d438e689f06f97ea0c6e45c109510c6b21d45b Mon Sep 17 00:00:00 2001 From: notjagan Date: Wed, 17 Aug 2016 09:25:58 -0400 Subject: [PATCH 261/402] Update .travis.yml --- .travis.yml | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/.travis.yml b/.travis.yml index 4d90f83..d749bde 100644 --- a/.travis.yml +++ b/.travis.yml @@ -2,6 +2,6 @@ language: python install: - pip install coveralls script: - nosetests --with-coverage tex2Wiki DLMF_preprocessing maple2latex main_page eCF + nosetests --with-coverage -v tex2Wiki DLMF_preprocessing maple2latex main_page eCF after_success: coveralls From 65c55ef55bbf45e87cc9864a2232b047250254c2 Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Wed, 17 Aug 2016 10:05:30 -0400 Subject: [PATCH 262/402] Add search function and modified carat function to improve the functionality --- eCF/.gitignore | 2 +- eCF/src/MathematicaToLaTeX.py | 113 ++++++++---------- eCF/test/test_carat.py | 12 +- .../test_single_macro_conversion_functions.py | 3 - 4 files changed, 54 insertions(+), 76 deletions(-) diff --git a/eCF/.gitignore b/eCF/.gitignore index 08e53cf..5e5ddb9 100644 --- a/eCF/.gitignore +++ b/eCF/.gitignore @@ -8,7 +8,7 @@ data/References.txt src/Glossary.csv src/new.Glossary.csv src/new.Glossary.updated.csv -src/pythonTest.py +src/MathematicaToLaTeXTest.py IdentitiesTest.txt README_Glossary x.log diff --git a/eCF/src/MathematicaToLaTeX.py b/eCF/src/MathematicaToLaTeX.py index 6033024..539e6ed 100644 --- a/eCF/src/MathematicaToLaTeX.py +++ b/eCF/src/MathematicaToLaTeX.py @@ -41,6 +41,9 @@ 'Infinity': 'infty'} +LEFT_BRACKETS = list('([{') +RIGHT_BRACKETS = list(')]}') + def find_surrounding(line, function, ex=(), start=0): # (str, str(, tuple, int)) -> tuple @@ -67,8 +70,9 @@ def find_surrounding(line, function, ex=(), start=0): if ex != '' and len(ex) >= 1: for e in ex: if (line.find(e) != -1 and - line.find(e) <= positions[0] and - line.find(e) + len(e) >= positions[0] + len(function)): + line.find(e) <= positions[0] and + line.find(e) + len(e) >= positions[0] + len( + function)): return [line.find(e) + len(e) + start, line.find(e) + len(e) + start] @@ -100,17 +104,15 @@ def arg_split(line, sep): :param sep: seperator (character) :returns: list of segments """ - l = list('([{') - r = list(')]}') args = [] count = i = 0 end = len(line) + 1 line = line + sep while i != end: - if line[i] in l: + if line[i] in LEFT_BRACKETS: count += 1 - if line[i] in r: + if line[i] in RIGHT_BRACKETS: count -= 1 if count == 0 and line[i] == sep: args.append(line[:i]) @@ -123,6 +125,30 @@ def arg_split(line, sep): return args +def search(line, i, sign, direction=-1): + j = i + direction + if direction == -1: + end = -1 + else: + end = len(line) + count = 0 + for j in range(i + direction, end, direction): + if line[j] in LEFT_BRACKETS: + count += direction + if line[j] in RIGHT_BRACKETS: + count -= direction + if count == 0 and line[j] in sign: + count -= 1 + if count < 0: + break + if count == 0 and j == end - direction: + j += direction + break + if direction == 1: + j -= 1 + return j + + def master_function(line, params): # (str, tuple) -> str """ @@ -237,31 +263,14 @@ def carat(line): :param line: line to be converted :returns: converted line """ - l = list('([{') - r = list(')]}') - sign = list('*/+-=, ') i = 0 while i != len(line): if line[i] == '^': - count = 0 - - # Searches for when a carat ends - for k in range(i + 1, len(line)): - if line[k] in l: - count += 1 - if line[k] in r: - count -= 1 - if count == 0 and line[k] in sign: - count -= 1 - if count < 0: - break - if count == 0 and k == len(line) - 1: - k += 1 - break - k -= 1 - - if line[i + 1] == '(' and line[-1] == ')': + + k = search(line, i, list('*/+-=, '), 1) + + if line[i + 1] == '(' and line[k] == ')': line = line[:i] + '^{' + line[i + 2:k] + '}' + line[k + 1:] else: line = line[:i] + '^{' + line[i + 1:k + 1] + '}' + line[k + 1:] @@ -694,44 +703,17 @@ def convert_fraction(line): :param line: line to be converted :returns: converted line """ - l = list('([{') - r = list(')]}') - sign = list('*+-=,<>&') i = 0 + l = list("([{") + r = list(")}]") + sign = list('*+-=,<>&') while i != len(line): if line[i] == '/': - # Searches left - count = 0 - for j in range(i - 1, -1, -1): - if line[j] in r: - count += 1 - if line[j] in l: - count -= 1 - if count == 0 and line[j] in sign: - count -= 1 - if count < 0: - break - if count == 0 and j == 0: - j -= 1 - break - - # Searches right - count = 0 - for k in range(i + 1, len(line)): - if line[k] in l: - count += 1 - if line[k] in r: - count -= 1 - if count == 0 and line[k] in sign: - count -= 1 - if count < 0: - break - if count == 0 and k == len(line) - 1: - k += 1 - break - k -= 1 + j = search(line, i, sign) + + k = search(line, i, sign, 1) # Removes extra surrounding parentheses, if there are any. # This won't work if you're doing "( )( )/( )( )", it will @@ -740,22 +722,22 @@ def convert_fraction(line): if (line[j + 1] == '(' and line[i - 1] == ')' and line[i + 1] == '(' and line[k] == ')'): # ()/() - line = '{0}\\frac{{{1}}}{{{2}}}{3}'\ + line = '{0}\\frac{{{1}}}{{{2}}}{3}' \ .format(line[:j + 1], line[j + 2:i - 1], line[i + 2:k], line[k + 1:]) elif line[j + 1] == '(' and line[i - 1] == ')': # ()/-- - line = '{0}\\frac{{{1}}}{{{2}}}{3}'\ + line = '{0}\\frac{{{1}}}{{{2}}}{3}' \ .format(line[:j + 1], line[j + 2:i - 1], line[i + 1:k + 1], line[k + 1:]) elif line[i + 1] == '(' and line[k] == ')': # --/() - line = '{0}\\frac{{{1}}}{{{2}}}{3}'\ + line = '{0}\\frac{{{1}}}{{{2}}}{3}' \ .format(line[:j + 1], line[j + 1:i], line[i + 2:k], line[k + 1:]) else: # --/-- - line = '{0}\\frac{{{1}}}{{{2}}}{3}'\ + line = '{0}\\frac{{{1}}}{{{2}}}{3}' \ .format(line[:j + 1], line[j + 1:i], line[i + 1:k + 1], line[k + 1:]) @@ -976,6 +958,5 @@ def main(pathw=DIR_NAME + 'newIdentities.tex', FUNCTION_CONVERSIONS = tuple(FUNCTION_CONVERSIONS) - if __name__ == '__main__': - main() + main() \ No newline at end of file diff --git a/eCF/test/test_carat.py b/eCF/test/test_carat.py index 6edcfe9..9c4092a 100644 --- a/eCF/test/test_carat.py +++ b/eCF/test/test_carat.py @@ -20,12 +20,12 @@ def test_seperate(self): def test_parentheses(self): self.assertEqual(carat('a^(b+c)'), 'a^{b+c}') - self.assertEqual(carat('a^(b+c)*d'), 'a^{(b+c)}*d') - self.assertEqual(carat('a^(b+c)/d'), 'a^{(b+c)}/d') - self.assertEqual(carat('a^(b+c)+d'), 'a^{(b+c)}+d') - self.assertEqual(carat('a^(b+c)-d'), 'a^{(b+c)}-d') - self.assertEqual(carat('a^(b+c)=d'), 'a^{(b+c)}=d') - self.assertEqual(carat('a^(b+c),d'), 'a^{(b+c)},d') + self.assertEqual(carat('a^(b+c)*d'), 'a^{b+c}*d') + self.assertEqual(carat('a^(b+c)/d'), 'a^{b+c}/d') + self.assertEqual(carat('a^(b+c)+d'), 'a^{b+c}+d') + self.assertEqual(carat('a^(b+c)-d'), 'a^{b+c}-d') + self.assertEqual(carat('a^(b+c)=d'), 'a^{b+c}=d') + self.assertEqual(carat('a^(b+c),d'), 'a^{b+c},d') def test_none(self): self.assertEqual(carat('nocarat'), 'nocarat') diff --git a/eCF/test/test_single_macro_conversion_functions.py b/eCF/test/test_single_macro_conversion_functions.py index 4ea7693..c204d0a 100644 --- a/eCF/test/test_single_macro_conversion_functions.py +++ b/eCF/test/test_single_macro_conversion_functions.py @@ -35,9 +35,6 @@ def test_exceptions(self): self.assertEqual(beta('BetaRegularized[z,a,b]'), 'BetaRegularized[z,a,b]') self.assertEqual(beta('\\[Beta]'), '\\[Beta]') - def test_none(self): - self.assertEqual(beta('none'), 'none') - class TestCFK(TestCase): From 0916099d9ceb496a58af638229794d5829c18920 Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Wed, 17 Aug 2016 10:15:23 -0400 Subject: [PATCH 263/402] Add exception to .coveragerc --- .coveragerc | 2 ++ 1 file changed, 2 insertions(+) diff --git a/.coveragerc b/.coveragerc index 378def2..e907e53 100644 --- a/.coveragerc +++ b/.coveragerc @@ -3,3 +3,5 @@ omit = */python?.?/* */test/* */site-packages/nose/* +exclude_lines = + if __name__ == .__main__.: From 3d3f8022b554d1ec2195f8399c11e69b91c9ba5d Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Wed, 17 Aug 2016 11:06:15 -0400 Subject: [PATCH 264/402] Add test case for cfk function and add documentation for new search function --- eCF/test/test_find_surrounding.py | 1 + eCF/test/test_search.py | 15 +++++++++++++++ .../test_single_macro_conversion_functions.py | 1 + 3 files changed, 17 insertions(+) create mode 100644 eCF/test/test_search.py diff --git a/eCF/test/test_find_surrounding.py b/eCF/test/test_find_surrounding.py index fd545d0..5cbb908 100644 --- a/eCF/test/test_find_surrounding.py +++ b/eCF/test/test_find_surrounding.py @@ -5,3 +5,4 @@ from unittest import TestCase from MathematicaToLaTeX import find_surrounding + diff --git a/eCF/test/test_search.py b/eCF/test/test_search.py new file mode 100644 index 0000000..43ab606 --- /dev/null +++ b/eCF/test/test_search.py @@ -0,0 +1,15 @@ + +__author__ = 'Kevin Chen' +__status__ = 'Development' + +from unittest import TestCase +from MathematicaToLaTeX import search + + +class TestSearch(TestCase): + + def test_end(self): + pass + + def test_parens(self): + pass \ No newline at end of file diff --git a/eCF/test/test_single_macro_conversion_functions.py b/eCF/test/test_single_macro_conversion_functions.py index c204d0a..9fde00d 100644 --- a/eCF/test/test_single_macro_conversion_functions.py +++ b/eCF/test/test_single_macro_conversion_functions.py @@ -45,6 +45,7 @@ def test_single(self): self.assertEqual(cfk('--ContinuedFractionK[g,{i,imin,imax}]--'), '--\\CFK{i}{imin}{imax}@@{1}{g}--') self.assertEqual(cfk('ContinuedFractionK[f,g,{i,imin,imax}]ContinuedFractionK[g,{i,imin,imax}]'), '\\CFK{i}{imin}{imax}@@{f}{g}\\CFK{i}{imin}{imax}@@{1}{g}') self.assertEqual(cfk('--ContinuedFractionK[f,g,{i,imin,imax}]--ContinuedFractionK[g,{i,imin,imax}]--'), '--\\CFK{i}{imin}{imax}@@{f}{g}--\\CFK{i}{imin}{imax}@@{1}{g}--') + self.assertEqual(cfk('ContinuedFractionK[\[Delta] + \[Epsilon], a, {k, 1, Infinity}]'), '\CFK{k}{1}{\infty}@@{\delta + \epsilon}{a}') def test_nested(self): self.assertEqual(cfk('ContinuedFractionK[ContinuedFractionK[f,g,{i,imin,imax}],ContinuedFractionK[f,g,{i,imin,imax}],{i,imin,imax}]'), '\\CFK{i}{imin}{imax}@@{\\CFK{i}{imin}{imax}@@{f}{g}}{\\CFK{i}{imin}{imax}@@{f}{g}}') From 72cb0038457975f2b8b8feee57c2fd927addd637 Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Wed, 17 Aug 2016 11:17:28 -0400 Subject: [PATCH 265/402] Edit test case for cfk, add newline after end of code --- eCF/test/test_search.py | 2 +- eCF/test/test_single_macro_conversion_functions.py | 2 +- 2 files changed, 2 insertions(+), 2 deletions(-) diff --git a/eCF/test/test_search.py b/eCF/test/test_search.py index 43ab606..83c2f7b 100644 --- a/eCF/test/test_search.py +++ b/eCF/test/test_search.py @@ -12,4 +12,4 @@ def test_end(self): pass def test_parens(self): - pass \ No newline at end of file + pass diff --git a/eCF/test/test_single_macro_conversion_functions.py b/eCF/test/test_single_macro_conversion_functions.py index 9fde00d..67e7cdf 100644 --- a/eCF/test/test_single_macro_conversion_functions.py +++ b/eCF/test/test_single_macro_conversion_functions.py @@ -45,7 +45,7 @@ def test_single(self): self.assertEqual(cfk('--ContinuedFractionK[g,{i,imin,imax}]--'), '--\\CFK{i}{imin}{imax}@@{1}{g}--') self.assertEqual(cfk('ContinuedFractionK[f,g,{i,imin,imax}]ContinuedFractionK[g,{i,imin,imax}]'), '\\CFK{i}{imin}{imax}@@{f}{g}\\CFK{i}{imin}{imax}@@{1}{g}') self.assertEqual(cfk('--ContinuedFractionK[f,g,{i,imin,imax}]--ContinuedFractionK[g,{i,imin,imax}]--'), '--\\CFK{i}{imin}{imax}@@{f}{g}--\\CFK{i}{imin}{imax}@@{1}{g}--') - self.assertEqual(cfk('ContinuedFractionK[\[Delta] + \[Epsilon], a, {k, 1, Infinity}]'), '\CFK{k}{1}{\infty}@@{\delta + \epsilon}{a}') + self.assertEqual(cfk('ContinuedFractionK[\\[Delta]+\\[Epsilon],a,{k,1,Infinity}]'), '\CFK{k}{1}{Infinity}@@{\\[Delta]+\\[Epsilon]}{a}') def test_nested(self): self.assertEqual(cfk('ContinuedFractionK[ContinuedFractionK[f,g,{i,imin,imax}],ContinuedFractionK[f,g,{i,imin,imax}],{i,imin,imax}]'), '\\CFK{i}{imin}{imax}@@{\\CFK{i}{imin}{imax}@@{f}{g}}{\\CFK{i}{imin}{imax}@@{f}{g}}') From 6f9cfe4558a495490686d191813d1badfe21123b Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Wed, 17 Aug 2016 11:22:11 -0400 Subject: [PATCH 266/402] Add correct code to MathematicaToLaTeX (was accidentally editing test file before) --- eCF/src/MathematicaToLaTeX.py | 40 +++++++++++++++++++++++------------ 1 file changed, 26 insertions(+), 14 deletions(-) diff --git a/eCF/src/MathematicaToLaTeX.py b/eCF/src/MathematicaToLaTeX.py index 539e6ed..3b1eb4e 100644 --- a/eCF/src/MathematicaToLaTeX.py +++ b/eCF/src/MathematicaToLaTeX.py @@ -51,7 +51,7 @@ def find_surrounding(line, function, ex=(), start=0): Finds the indices of the beginning and end of a function; this is the main function that powers the converter. - :param line: line to be converted + :param line: line with functions that are going to be searched :param function: the function you're trying to find the surrounding brackets for :param ex: exceptions that shouldn't be converted, because conversions do @@ -60,7 +60,7 @@ def find_surrounding(line, function, ex=(), start=0): the original is "Equals" :param start: index of where to start finding (used if there are multiple of one function in a line - :returns: converted line + :returns: positions of opening and ending brackets """ positions = [0, 0] line = line[start:] @@ -70,9 +70,8 @@ def find_surrounding(line, function, ex=(), start=0): if ex != '' and len(ex) >= 1: for e in ex: if (line.find(e) != -1 and - line.find(e) <= positions[0] and - line.find(e) + len(e) >= positions[0] + len( - function)): + line.find(e) <= positions[0] and + line.find(e) + len(e) >= positions[0] + len(function)): return [line.find(e) + len(e) + start, line.find(e) + len(e) + start] @@ -86,7 +85,7 @@ def find_surrounding(line, function, ex=(), start=0): count -= 1 if count == 0: if j == positions[0] + len(function): - positions[1] = positions[0] + positions[1], positions[0] = j, j else: positions[1] = j + 1 break @@ -126,12 +125,23 @@ def arg_split(line, sep): def search(line, i, sign, direction=-1): + # (str, list, str(, int)) -> int + """ + Searches for the ends of fractions or carats; excludes signs in brackets + + :param line: line to be searched + :param i: the starting point, usually a "/" or a "^" + :param sign: list of excluding symbols + :param direction: direction of search, left: -1, right: 1 + :returns: indice of end + """ j = i + direction if direction == -1: end = -1 else: end = len(line) count = 0 + for j in range(i + direction, end, direction): if line[j] in LEFT_BRACKETS: count += direction @@ -144,8 +154,10 @@ def search(line, i, sign, direction=-1): if count == 0 and j == end - direction: j += direction break + if direction == 1: j -= 1 + return j @@ -314,6 +326,8 @@ def beta(line): def cfk(line): + print(line) + print(line) # (str) -> str """ Converts Mathematica's 'ContinuedFractionK' to the equivalent LaTeX macro. @@ -344,6 +358,7 @@ def cfk(line): '\\CFK{{{0}}}{{{1}}}{{{2}}}@@{{1}}{{{3}}}' .format(moreargs[0], moreargs[1], moreargs[2], args[0]) + line[pos[1]:]) + print(line) return line @@ -632,13 +647,13 @@ def constraint(line): constraints = arg_split(sections[-1].replace('&&', '&'), '&') - for i, item, in enumerate(constraints): + for i, element, in enumerate(constraints): if i == 0: constraints[i] = ('\n% \\constraint{$' + - item.replace('&', ' \\land ')) + element.replace('&', ' \\land ')) else: constraints[i] = ('\n% & $' + - item.replace('&', ' \\land ')) + element.replace('&', ' \\land ')) if i == len(constraints) - 1: constraints[i] += '$}' else: @@ -704,15 +719,12 @@ def convert_fraction(line): :returns: converted line """ i = 0 - l = list("([{") - r = list(")}]") sign = list('*+-=,<>&') while i != len(line): if line[i] == '/': j = search(line, i, sign) - k = search(line, i, sign, 1) # Removes extra surrounding parentheses, if there are any. @@ -849,7 +861,7 @@ def replace_vars(line): def main(pathw=DIR_NAME + 'newIdentities.tex', - pathr=DIR_NAME + 'Identities.m', test=False): + pathr=DIR_NAME + 'IdentitiesTest.m', test=True): # ((str, str, bool)) -> None """ Opens Mathematica file with identities and puts converted lines into @@ -959,4 +971,4 @@ def main(pathw=DIR_NAME + 'newIdentities.tex', FUNCTION_CONVERSIONS = tuple(FUNCTION_CONVERSIONS) if __name__ == '__main__': - main() \ No newline at end of file + main() From eaa1761096f6091c149af37a965e3dae44dcc076 Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Wed, 17 Aug 2016 11:23:23 -0400 Subject: [PATCH 267/402] Edit main function --- eCF/src/MathematicaToLaTeX.py | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/eCF/src/MathematicaToLaTeX.py b/eCF/src/MathematicaToLaTeX.py index 3b1eb4e..4f034d3 100644 --- a/eCF/src/MathematicaToLaTeX.py +++ b/eCF/src/MathematicaToLaTeX.py @@ -861,7 +861,7 @@ def replace_vars(line): def main(pathw=DIR_NAME + 'newIdentities.tex', - pathr=DIR_NAME + 'IdentitiesTest.m', test=True): + pathr=DIR_NAME + 'Identities.m', test=False): # ((str, str, bool)) -> None """ Opens Mathematica file with identities and puts converted lines into From eb1127b9fddf54223731a3f44076274fbcaf083d Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Wed, 17 Aug 2016 11:24:18 -0400 Subject: [PATCH 268/402] Remove useless print statements in cfk function --- eCF/src/MathematicaToLaTeX.py | 3 --- 1 file changed, 3 deletions(-) diff --git a/eCF/src/MathematicaToLaTeX.py b/eCF/src/MathematicaToLaTeX.py index 4f034d3..1a984b0 100644 --- a/eCF/src/MathematicaToLaTeX.py +++ b/eCF/src/MathematicaToLaTeX.py @@ -326,8 +326,6 @@ def beta(line): def cfk(line): - print(line) - print(line) # (str) -> str """ Converts Mathematica's 'ContinuedFractionK' to the equivalent LaTeX macro. @@ -358,7 +356,6 @@ def cfk(line): '\\CFK{{{0}}}{{{1}}}{{{2}}}@@{{1}}{{{3}}}' .format(moreargs[0], moreargs[1], moreargs[2], args[0]) + line[pos[1]:]) - print(line) return line From 4d9438be4d4aaf35cf1c08a7fe2a94801c43a09f Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Wed, 17 Aug 2016 11:35:56 -0400 Subject: [PATCH 269/402] Add test in master_function that should finally cover that one last line --- eCF/src/MathematicaToLaTeX.py | 7 +++++-- eCF/test/test_master_function.py | 2 ++ eCF/test/test_single_macro_conversion_functions.py | 1 - 3 files changed, 7 insertions(+), 3 deletions(-) diff --git a/eCF/src/MathematicaToLaTeX.py b/eCF/src/MathematicaToLaTeX.py index 1a984b0..946e8ab 100644 --- a/eCF/src/MathematicaToLaTeX.py +++ b/eCF/src/MathematicaToLaTeX.py @@ -46,6 +46,9 @@ def find_surrounding(line, function, ex=(), start=0): + print(function) + print(line) + print(ex) # (str, str(, tuple, int)) -> tuple """ Finds the indices of the beginning and end of a function; this is the main @@ -85,7 +88,7 @@ def find_surrounding(line, function, ex=(), start=0): count -= 1 if count == 0: if j == positions[0] + len(function): - positions[1], positions[0] = j, j + positions[0] = positions[1] else: positions[1] = j + 1 break @@ -858,7 +861,7 @@ def replace_vars(line): def main(pathw=DIR_NAME + 'newIdentities.tex', - pathr=DIR_NAME + 'Identities.m', test=False): + pathr=DIR_NAME + 'IdentitiesTest.m', test=True): # ((str, str, bool)) -> None """ Opens Mathematica file with identities and puts converted lines into diff --git a/eCF/test/test_master_function.py b/eCF/test/test_master_function.py index c9f5a32..73a6312 100644 --- a/eCF/test/test_master_function.py +++ b/eCF/test/test_master_function.py @@ -68,6 +68,8 @@ def test_conversion(self): else: self.assertEqual(master_function(before, function), after) self.assertEqual(master_function('--{0}--'.format(before), function), '--{0}--'.format(after)) + if function[0] == 'D': + self.assertEqual(master_function('\\[Delta]', function), '\\[Delta]') def test_exception(self): for function in FUNCTION_CONVERSIONS: diff --git a/eCF/test/test_single_macro_conversion_functions.py b/eCF/test/test_single_macro_conversion_functions.py index 67e7cdf..c204d0a 100644 --- a/eCF/test/test_single_macro_conversion_functions.py +++ b/eCF/test/test_single_macro_conversion_functions.py @@ -45,7 +45,6 @@ def test_single(self): self.assertEqual(cfk('--ContinuedFractionK[g,{i,imin,imax}]--'), '--\\CFK{i}{imin}{imax}@@{1}{g}--') self.assertEqual(cfk('ContinuedFractionK[f,g,{i,imin,imax}]ContinuedFractionK[g,{i,imin,imax}]'), '\\CFK{i}{imin}{imax}@@{f}{g}\\CFK{i}{imin}{imax}@@{1}{g}') self.assertEqual(cfk('--ContinuedFractionK[f,g,{i,imin,imax}]--ContinuedFractionK[g,{i,imin,imax}]--'), '--\\CFK{i}{imin}{imax}@@{f}{g}--\\CFK{i}{imin}{imax}@@{1}{g}--') - self.assertEqual(cfk('ContinuedFractionK[\\[Delta]+\\[Epsilon],a,{k,1,Infinity}]'), '\CFK{k}{1}{Infinity}@@{\\[Delta]+\\[Epsilon]}{a}') def test_nested(self): self.assertEqual(cfk('ContinuedFractionK[ContinuedFractionK[f,g,{i,imin,imax}],ContinuedFractionK[f,g,{i,imin,imax}],{i,imin,imax}]'), '\\CFK{i}{imin}{imax}@@{\\CFK{i}{imin}{imax}@@{f}{g}}{\\CFK{i}{imin}{imax}@@{f}{g}}') From 6c8e65e87279c97d3636c7cf36e15532268c7cf6 Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Wed, 17 Aug 2016 11:36:25 -0400 Subject: [PATCH 270/402] Modified main function --- eCF/src/MathematicaToLaTeX.py | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/eCF/src/MathematicaToLaTeX.py b/eCF/src/MathematicaToLaTeX.py index 946e8ab..fe096d3 100644 --- a/eCF/src/MathematicaToLaTeX.py +++ b/eCF/src/MathematicaToLaTeX.py @@ -861,7 +861,7 @@ def replace_vars(line): def main(pathw=DIR_NAME + 'newIdentities.tex', - pathr=DIR_NAME + 'IdentitiesTest.m', test=True): + pathr=DIR_NAME + 'Identities.m', test=False): # ((str, str, bool)) -> None """ Opens Mathematica file with identities and puts converted lines into From 5f5e0ea7959282f1fb320a99ed760e800fd62cf8 Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Wed, 17 Aug 2016 11:45:17 -0400 Subject: [PATCH 271/402] Remove some useless print statements --- eCF/src/MathematicaToLaTeX.py | 3 --- 1 file changed, 3 deletions(-) diff --git a/eCF/src/MathematicaToLaTeX.py b/eCF/src/MathematicaToLaTeX.py index fe096d3..322fdb8 100644 --- a/eCF/src/MathematicaToLaTeX.py +++ b/eCF/src/MathematicaToLaTeX.py @@ -46,9 +46,6 @@ def find_surrounding(line, function, ex=(), start=0): - print(function) - print(line) - print(ex) # (str, str(, tuple, int)) -> tuple """ Finds the indices of the beginning and end of a function; this is the main From 0531f61888fafc3b047b7c9563ccd756549cf8b2 Mon Sep 17 00:00:00 2001 From: notjagan Date: Wed, 17 Aug 2016 14:41:02 -0400 Subject: [PATCH 272/402] Update .coveragerc Tests all Python files in a src/ directory. --- .coveragerc | 4 ++++ 1 file changed, 4 insertions(+) diff --git a/.coveragerc b/.coveragerc index e907e53..2879bc7 100644 --- a/.coveragerc +++ b/.coveragerc @@ -5,3 +5,7 @@ omit = */site-packages/nose/* exclude_lines = if __name__ == .__main__.: + +[run] +include = + */src/*.py From 9fea1c639221bbb5c3ac6143ca7fd05178ee57ca Mon Sep 17 00:00:00 2001 From: notjagan Date: Wed, 17 Aug 2016 14:41:57 -0400 Subject: [PATCH 273/402] Update .travis.yml --- .travis.yml | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/.travis.yml b/.travis.yml index d749bde..a8b9a9b 100644 --- a/.travis.yml +++ b/.travis.yml @@ -2,6 +2,6 @@ language: python install: - pip install coveralls script: - nosetests --with-coverage -v tex2Wiki DLMF_preprocessing maple2latex main_page eCF + nosetests --with-coverage --with-isolation -v tex2Wiki DLMF_preprocessing maple2latex main_page eCF after_success: coveralls From 07fca1fd18a000bbf6f8b5f0d8903fb5350db316 Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Wed, 17 Aug 2016 15:55:45 -0400 Subject: [PATCH 274/402] Start to modify MathematicaToLaTeX for trig functions, add LaTeXConstraintModifier.py --- eCF/.gitignore | 26 +++++++---------- eCF/src/LaTeXConstraintModifier.py | 46 ++++++++++++++++++++++++++++++ 2 files changed, 57 insertions(+), 15 deletions(-) create mode 100644 eCF/src/LaTeXConstraintModifier.py diff --git a/eCF/.gitignore b/eCF/.gitignore index 5e5ddb9..557d2b6 100644 --- a/eCF/.gitignore +++ b/eCF/.gitignore @@ -1,29 +1,25 @@ Divya/ other/ -src/test.txt + data/Identities.* data/newIdentities.* data/IdentitiesTest.m data/References.txt +data/ZE.3.* +data/ZE.4.* + +src/test.txt src/Glossary.csv src/new.Glossary.csv src/new.Glossary.updated.csv -src/MathematicaToLaTeXTest.py -IdentitiesTest.txt -README_Glossary -x.log -tasks.txt +src/pythonTest.py +src/tasks.txt +src/README_Glossary +src/x.log + *.sty *.aux *.csv - - -# ecf project *.sty -Identities.* -newIdentities.* -*.csv -macros.txt +*.out *.log -*.aux -ZE.3.* diff --git a/eCF/src/LaTeXConstraintModifier.py b/eCF/src/LaTeXConstraintModifier.py new file mode 100644 index 0000000..cd24e01 --- /dev/null +++ b/eCF/src/LaTeXConstraintModifier.py @@ -0,0 +1,46 @@ + +__author__ = 'Kevin Chen' +__status__ = 'Development' + +import os +import argparse + +parser = argparse.ArgumentParser( + description='Receive .tex file with constraints and convert' + ' them to lines using flushright and flushright.') +parser.add_argument('PATHR', type=str, + help='path of input .tex file, with the current' + ' directory as the starting point',) +parser.add_argument('PATHW', type=str, + help='path of file to be outputted to, with the current' + ' directory as the starting point') +args = parser.parse_args() + +PATHR = args.PATHR +PATHW = args.PATHW +DIR_NAME = os.path.dirname(os.path.realpath(__file__)) + '/../data/' + + +def combine_percent(lines): + # (list) -> list + """ + Description + + :param lines: + :return: + """ + + pass + + +def main(): + # + """ + + :return: + """ + pass + + +if __name__ == '__main__': + main() From 6a67155cdbd468503df3f8d5b4235f0b0e039fe0 Mon Sep 17 00:00:00 2001 From: Edward Bian Date: Thu, 18 Aug 2016 10:25:25 -0400 Subject: [PATCH 275/402] Minor fixes and unit tests --- KLSadd_insertion/src/updateChapters.py | 22 +++++++++++-------- .../test/test_generic_functions.py | 4 ++-- 2 files changed, 15 insertions(+), 11 deletions(-) diff --git a/KLSadd_insertion/src/updateChapters.py b/KLSadd_insertion/src/updateChapters.py index 31810ce..2330aff 100644 --- a/KLSadd_insertion/src/updateChapters.py +++ b/KLSadd_insertion/src/updateChapters.py @@ -22,7 +22,8 @@ def extraneous_section_deleter(list): :param list: :return: Extraneous name free output """ - return [item for item in list if "reference" not in item.lower() and "limit relation" not in item.lower()] + return [item for item in list if "reference" not in item.lower() and "limit relation" not in item.lower() + and "symmetry" not in item.lower()] def new_keywords(kls, kls_list): @@ -41,6 +42,7 @@ def new_keywords(kls, kls_list): kls_list = extraneous_section_deleter(kls_list) kls_list.append("Limit Relation") + kls_list.append("Symmetry") for item in kls_list: if item not in kls_list_chap: kls_list_chap.append(item) @@ -61,12 +63,14 @@ def fix_chapter_sort(kls, chap, word, sortloc, klsaddparas, sortmatch_2): hyper_headers_chap = [] hyper_subs_chap = [] - sep1 = 0 + specialinput = 0 name_chap = word.lower() if name_chap == "orthogonality": - sep1 = 1 + specialinput = 1 elif name_chap == "special value": - sep1 = 2 + specialinput = 2 + elif name_chap == "symmetry": + specialinput = 3 khyper_header_chap = [] k_hyper_sub_chap = [] @@ -80,8 +84,8 @@ def fix_chapter_sort(kls, chap, word, sortloc, klsaddparas, sortmatch_2): temp = line[line.find(" ", 12)+1: line.find("}", 12)] # get just the name (like mathpeople) if temp.lower() == 'wilson': chapterstart = True - if sep1 in (0, 2) and name_chap in line.lower() and chapterstart and ("\\paragraph{" in line or "\\subsubsection*{" in line) or \ - (sep1 == 1 and "orthogonality relation" not in line.lower() and name_chap in line.lower() and chapterstart + if specialinput in (0,2) and name_chap in line.lower() and chapterstart and ("\\paragraph{" in line or "\\subsubsection*{" in line) or \ + (specialinput in (1, 3) and "orthogonality relation" not in line.lower() and name_chap in line.lower() and chapterstart and ("\\paragraph{" in line or "\\subsubsection*{" in line)): for item in klsaddparas: if index < item: @@ -128,10 +132,10 @@ def fix_chapter_sort(kls, chap, word, sortloc, klsaddparas, sortmatch_2): else: tempref = ref14_3 offset = sorter_check[sortloc][1] - if sep1 == 1: + if specialinput == 1: offset = 8 - if sep1 == 2: + if specialinput in (2, 3): name_chap = 'hypergeometric representation' # Uses of numbers like 12, or 9 are for specific lengths of strings (like \\subsection{) where the @@ -146,7 +150,7 @@ def fix_chapter_sort(kls, chap, word, sortloc, klsaddparas, sortmatch_2): temp = line[9:line.find("}", 7)] if name_chap in line.lower(): - if sep1 in (0, 2) or sep1 == 1 and "orthogonality relation" not in line: + if specialinput in (0, 2, 3) or specialinput == 1 and "orthogonality relation" not in line: hyper_subs_chap.append([tempref[d + 1]]) # appends the index for the line before following subsection hyper_headers_chap.append(temp) # appends the name of the section the hypergeo subsection is in diff --git a/KLSadd_insertion/test/test_generic_functions.py b/KLSadd_insertion/test/test_generic_functions.py index 0504cd1..887e9bc 100644 --- a/KLSadd_insertion/test/test_generic_functions.py +++ b/KLSadd_insertion/test/test_generic_functions.py @@ -10,13 +10,13 @@ class TestNewKeywords(TestCase): def test_new_keywords(self): - self.assertEquals(new_keywords(['\\subsubsection*{ThisIsATest}\n'],[]), ['ThisIsATest', 'Limit Relation']) + self.assertEquals(new_keywords(['\\subsubsection*{ThisIsATest}\n'],[]), ['ThisIsATest', 'Limit Relation', 'Symmetry']) class TestExtraneousSectionDeleter(TestCase): def test_extraneous_section_deleter(self): - self.assertEquals(extraneous_section_deleter(['reference', 'bananas', 'limit relation', '458673', + self.assertEquals(extraneous_section_deleter(['reference', 'bananas', 'limit relation', 'symmetry','458673', 'limit relations', 'apple']), ['bananas', '458673', 'apple']) From 83353e9681bf0750b27af10a3a5bd8e6bcad89ac Mon Sep 17 00:00:00 2001 From: Parth Oza Date: Wed, 20 Jul 2016 14:48:47 -0400 Subject: [PATCH 276/402] Attempt to make tex2Wiki code more readable for future developers --- tex2Wiki/src/DLMFtex2Wiki.py | 418 +++++++++++++++++------------------ tex2Wiki/src/tex2Wiki.py | 409 ++++++++++++++++++---------------- 2 files changed, 423 insertions(+), 404 deletions(-) diff --git a/tex2Wiki/src/DLMFtex2Wiki.py b/tex2Wiki/src/DLMFtex2Wiki.py index f53aa29..7c20de0 100644 --- a/tex2Wiki/src/DLMFtex2Wiki.py +++ b/tex2Wiki/src/DLMFtex2Wiki.py @@ -6,113 +6,102 @@ import csv # imported for using csv format import sys # imported for getting args from shutil import copyfile -from tex2Wiki import append_text, append_revision, getString,\ - getSym, getEq, secLabel, getEqP, unmodLabel, isnumber +from tex2Wiki import append_text, append_revision, get_string_within_, getSym, getEq, secLabel, getEqP, unmodLabel, isnumber -def modLabel(label): - isNumer = False - newlabel = "" +def mod_label(label): + is_numeric = False + new_label = "" num = "" - for i in range(0, len(label)): - if isNumer and not isnumber(label[i]): + for i in range(len(label)): + if is_numeric and not isnumber(label[i]): if len(num) > 1: - newlabel += num + new_label += num num = "" else: - newlabel += "0" + str(num) + new_label += "0" + str(num) num = "" if isnumber(label[i]): - isNumer = True + is_numeric = True num += str(label[i]) else: - isNumer = False - newlabel += label[i] + is_numeric = False + new_label += label[i] if len(num) > 1: - newlabel += num + new_label += num elif len(num) == 1: - newlabel += "0" + num - return newlabel + new_label += "0" + num + return new_label -def DLMF(ofname, mmd, llinks, n): +def dlmf(of_name, mmd, llinks, n): for iterations in range(0, 1): - tex = open(ofname, 'r') + tex = open(of_name, 'r') main_file = open(mmd, "r") - lLinks = open(llinks, 'r') - mainPrepend = "" - mainWrite = open("ZetaFunctions.mmd.new", "w") - lLink = lLinks.readlines() + l_links = open(llinks, 'r') + main_prepend = "" + main_write = open("ZetaFunctions.mmd.new", "w") + l_link_list = l_links.readlines() math = False constraint = False substitution = False symbols = [] lines = tex.readlines() - refLines = [] + ref_lines = [] sections = [] labels = [] eqs = [] - refEqs = [] + ref_eqs = [] parse = False head = False try: - chapRef = [("GA", open("../../data/GA.tex", 'r')), - ("ZE", open("../../data/ZE.3.tex", 'r'))] + chap_ref = [("GA", open("../../data/GA.tex", 'r')), + ("ZE", open("../../data/ZE.3.tex", 'r'))] except IOError: - chapRef = [ - ("GA", open( - "GA.tex", 'r')), ("ZE", open( - "ZE.3.tex", 'r'))] - refLabels = [] - for c in chapRef: - refLines = refLines + (c[1].readlines()) - c[1].close() - for i in range(0, len(refLines)): - line = refLines[i] + chap_ref = [("GA", open("GA.tex", 'r')), ("ZE", open("ZE.3.tex", 'r'))] + ref_labels = [] + for chap in chap_ref: + ref_lines = ref_lines + (chap[1].readlines()) + chap[1].close() + for i in range(len(ref_lines)): + line = ref_lines[i] if "\\begin{equation}" in line: - sLabel = line.find("\\label{") + 7 - eLabel = line.find("}", sLabel) - label = line[sLabel:eLabel] - for l in lLink: - if l.find(label) != -1 and l[len(label) + 1] == "=": - rlabel = l[l.find("=>") + 3:l.find("\\n")] - rlabel = rlabel.replace("/", "") - rlabel = rlabel.replace("#", ":") + s_label = line.find("\\label{") + 7 + e_label = line.find("}", s_label) + label = line[s_label:e_label] + for l_link in l_link_list: + if l_link.find(label) != -1 and l_link[len(label) + 1] == "=": + r_label = l_link[l_link.find("=>") + 3:l_link.find("\\n")] + r_label = r_label.replace("/", "") + r_label = r_label.replace("#", ":") break - label = rlabel - refLabels.append(label) - refEqs.append("") + label = r_label + ref_labels.append(label) + ref_eqs.append("") math = True elif math: - refEqs[-1] += line - if "\\end{equation}" in refLines[ - i + - 1] or "\\constraint" in refLines[ - i + - 1] or "\\substitution" in refLines[ - i + - 1] or "\\drmfn" in refLines[ - i + - 1]: + ref_eqs[-1] += line + if any([item in ref_lines[i + 1] + for item in ["\\end{equation}", "\\constraint", "\\substitution", "\\drmfn"]]): math = False - startFlag = True - for i in range(0, len(lines)): + start_flag = True + for i in range(len(lines)): line = lines[i] if "\\begin{document}" in line: parse = True elif "\\end{document}" in line: - mainPrepend += "
\n" - mainText = mainPrepend - mainText = mainText.replace("drmf_bof\n", "") - mainText = mainText.replace("drmf_eof\n", "") - mainText = mainText.replace( + main_prepend += "
\n" + main_text = main_prepend + main_text = main_text.replace("drmf_bof\n", "") + main_text = main_text.replace("drmf_eof\n", "") + main_text = main_text.replace( "\'\'\'Zeta and Related Functions\'\'\'\n", "") - mainText = mainText.replace("{{#set:Section=0}}\n", "") + main_text = main_text.replace("{{#set:Section=0}}\n", "") append_revision('Zeta and Related Functions') - mainText = "{{#set:Section=0}}\n" + mainText - mainWrite.write(mainText) - mainWrite.close() + main_text = "{{#set:Section=0}}\n" + main_text + main_write.write(main_text) + main_write.close() main_file.close() copyfile(mmd, 'ZetaFunctions.mmd.new') parse = False @@ -128,24 +117,24 @@ def DLMF(ofname, mmd, llinks, n): stringWrite = "\'\'\'" stringWrite += getString(line) + "\'\'\'\n" chapter = getString(line) - if startFlag: - mainPrepend += ( + if start_flag: + main_prepend += ( "\n== Sections in " + chapter + " ==\n\n
\n") - startFlag = False + start_flag = False else: - mainPrepend += ("
\n\n== Sections in " + - chapter + - " ==\n\n
\n") + main_prepend += ("
\n\n== Sections in " + + chapter + + " ==\n\n
\n") head = True elif "\\section" in line: - mainPrepend += ("* [[" + - secLabel(getString(line)) + - "|" + - getString(line) + - "]]\n") + main_prepend += ("* [[" + + secLabel(getString(line)) + + "|" + + getString(line) + + "]]\n") sections.append([getString(line)]) secCounter = 0 @@ -255,29 +244,29 @@ def DLMF(ofname, mmd, llinks, n): if head: append_text("\n") head = False - sLabel = line.find("\\label{") + 7 - eLabel = line.find("}", sLabel) - label = (line[sLabel:eLabel]) + s_label = line.find("\\label{") + 7 + e_label = line.find("}", s_label) + label = (line[s_label:e_label]) eqCounter += 1 - for l in lLink: - if label == l[0:l.find("=") - 1]: - rlabel = l[l.find("=>") + 3:l.find("\\n")] - rlabel = rlabel.replace("/", "") - rlabel = rlabel.replace("#", ":") - rlabel = rlabel.replace("!", ":") + for l_link in l_link_list: + if label == l_link[0:l_link.find("=") - 1]: + r_label = l_link[l_link.find("=>") + 3:l_link.find("\\n")] + r_label = r_label.replace("/", "") + r_label = r_label.replace("#", ":") + r_label = r_label.replace("!", ":") break - label = modLabel(rlabel) - labels.append("Formula:" + rlabel) + label = mod_label(r_label) + labels.append("Formula:" + r_label) eqs.append("") append_text( "\n") math = True elif "\\begin{equation}" in line and not parse: - sLabel = line.find("\\label{") + 7 - eLabel = line.find("}", sLabel) - label = modLabel(line[sLabel:eLabel]) + s_label = line.find("\\label{") + 7 + e_label = line.find("}", s_label) + label = mod_label(line[s_label:e_label]) labels.append("*" + label) # special marker eqs.append("") math = True @@ -306,16 +295,16 @@ def DLMF(ofname, mmd, llinks, n): eqs[-1] += line if not ( - (not ( - "\\end{equation}" in lines[ - i + - 1]) or "\\subsection" in lines[ - i + - 3]) or "\\section" in lines[ - i + - 3]) and not "\\part" in lines[ + (not ( + "\\end{equation}" in lines[ + i + + 1]) or "\\subsection" in lines[ + i + + 3]) or "\\section" in lines[ + i + + 3]) and not "\\part" in lines[ i + - 3]: + 3]: u = i flagM2 = False while flagM: @@ -323,7 +312,7 @@ def DLMF(ofname, mmd, llinks, n): if "\\begin{equation}" in lines[u] in lines[u]: flagM = False if "\\section" in lines[u] or "\\subsection" in lines[ - i] or "\\part" in lines[u] or "\\end{document}" in lines[u]: + i] or "\\part" in lines[u] or "\\end{document}" in lines[u]: flagM = False flagM2 = True if not flagM2: @@ -344,28 +333,28 @@ def DLMF(ofname, mmd, llinks, n): elif math and not parse: eqs[-1] += line if "\\end{equation}" in lines[ - i + - 1] or "\\constraint" in lines[ - i + - 1] or "\\substitution" in lines[ - i + - 1] or "\\drmfn" in lines[ - i + - 1]: + i + + 1] or "\\constraint" in lines[ + i + + 1] or "\\substitution" in lines[ + i + + 1] or "\\drmfn" in lines[ + i + + 1]: math = False if substitution and parse: subLine += line.replace("&", "&
") if "\\end{equation}" in lines[ - i + - 1] or "\\substitution" in lines[ - i + - 1] or "\\constraint" in lines[ - i + - 1] or "\\drmfn" in lines[ - i + - 1] or "\\proof" in lines[ - i + - 1]: + i + + 1] or "\\substitution" in lines[ + i + + 1] or "\\constraint" in lines[ + i + + 1] or "\\drmfn" in lines[ + i + + 1] or "\\proof" in lines[ + i + + 1]: substitution = False append_text( "
Substitution(s): " + @@ -375,16 +364,16 @@ def DLMF(ofname, mmd, llinks, n): if constraint and parse: conLine += line.replace("&", "&
") if "\\end{equation}" in lines[ - i + - 1] or "\\substitution" in lines[ - i + - 1] or "\\constraint" in lines[ - i + - 1] or "\\drmfn" in lines[ - i + - 1] or "\\proof" in lines[ - i + - 1]: + i + + 1] or "\\substitution" in lines[ + i + + 1] or "\\constraint" in lines[ + i + + 1] or "\\drmfn" in lines[ + i + + 1] or "\\proof" in lines[ + i + + 1]: constraint = False append_text( "
Constraint(s): " + @@ -474,7 +463,7 @@ def DLMF(ofname, mmd, llinks, n): "#" + secLabel( labels[eqCounter][ - len("Formula:"):]) + + len("Formula:"):]) + "|formula in " + secLabel( sections[ @@ -542,7 +531,7 @@ def DLMF(ofname, mmd, llinks, n): "#" + secLabel( labels[eqCounter][ - len("Formula:"):]) + + len("Formula:"):]) + "|formula in " + secLabel( sections[ @@ -702,7 +691,7 @@ def DLMF(ofname, mmd, llinks, n): else: ArgCx += 1 if G[0].find(symbol) == 0 and (len(G[0]) == len(symbol) or not G[0][ - len(symbol)].isalpha()): # and (numPar!=0 or numArg!=0): + len(symbol)].isalpha()): # and (numPar!=0 or numArg!=0): checkFlag = True get = True preG = S @@ -719,7 +708,7 @@ def DLMF(ofname, mmd, llinks, n): Q = symbolPar listArgs = [] if len(Q) > len(symbol) and (Q[ - len(symbol)] == "{" or Q[len(symbol)] == "["): + len(symbol)] == "{" or Q[len(symbol)] == "["): ap = "" for o in range(len(symbol), len(Q)): if Q[o] == "{" or z == "[": @@ -734,7 +723,7 @@ def DLMF(ofname, mmd, llinks, n): for t in range(5, len(G)): if G[t] != "": websiteF = websiteF + \ - " [" + G[t] + " " + G[t] + "]" + " [" + G[t] + " " + G[t] + "]" p1 = G[4].strip("$") p1 = "" + p1 + "" # if checkFlag: @@ -791,7 +780,7 @@ def DLMF(ofname, mmd, llinks, n): equation + " Equation (" + equation[ - 1:] + + 1:] + "), Section " + section + "] of [[Bibliography#DLMF|'''DLMF''']].\n\n") @@ -837,7 +826,7 @@ def DLMF(ofname, mmd, llinks, n): "#" + secLabel( labels[eqCounter][ - len("Formula:"):]) + + len("Formula:"):]) + "|formula in " + secLabel( sections[ @@ -885,7 +874,7 @@ def DLMF(ofname, mmd, llinks, n): "_") + "#" + labels[endNum][ - 8:] + + 8:] + "|formula in " + labels[0] + "]]
\n") @@ -922,7 +911,7 @@ def DLMF(ofname, mmd, llinks, n): elif "\\drmfname" in line and parse: math = False comToWrite = "\n== Name ==\n\n
" + \ - getString(line) + "

\n" + comToWrite + getString(line) + "

\n" + comToWrite elif "\\drmfnote" in line and parse: symbols = symbols + getSym(line) @@ -948,37 +937,37 @@ def DLMF(ofname, mmd, llinks, n): pause = True eqR = line[ind:line.find("}", ind) + 1] rLab = getString(eqR) - for l in lLink: - if rLab == l[0:l.find("=") - 1]: - rlabel = l[l.find("=>") + 3:l.find("\\n")] - rlabel = rlabel.replace("/", "") - rlabel = rlabel.replace("#", ":") - rlabel = rlabel.replace("!", ":") + for l_link in l_link_list: + if rLab == l_link[0:l_link.find("=") - 1]: + r_label = l_link[l_link.find("=>") + 3:l_link.find("\\n")] + r_label = r_label.replace("/", "") + r_label = r_label.replace("#", ":") + r_label = r_label.replace("!", ":") break - eInd = refLabels.index("" + rlabel) + eInd = ref_labels.index("" + r_label) z = line[line.find("}", ind + 7) + 1] if z == "." or z == ",": pauseP = True proofLine += ("
\n\n" + - refEqs[eInd] + + ref_eqs[eInd] + "" + z + "
\n") else: if z == "}": proofLine += ("
\n\n" + - refEqs[eInd] + + ref_eqs[eInd] + "
") else: proofLine += ("
\n\n" + - refEqs[eInd] + + ref_eqs[eInd] + "
\n") else: if pause: @@ -1007,30 +996,30 @@ def DLMF(ofname, mmd, llinks, n): pause = True eqR = line[ind:line.find("}", ind) + 1] rLab = getString(eqR) - for l in lLink: - if rLab == l[0:l.find("=") - 1]: - rlabel = l[l.find("=>") + 3:l.find("\\n")] - rlabel = rlabel.replace("/", "") - rlabel = rlabel.replace("#", ":") - rlabel = rlabel.replace("!", ":") + for l_link in l_link_list: + if rLab == l_link[0:l_link.find("=") - 1]: + r_label = l_link[l_link.find("=>") + 3:l_link.find("\\n")] + r_label = r_label.replace("/", "") + r_label = r_label.replace("#", ":") + r_label = r_label.replace("!", ":") break - eInd = refLabels.index("" + rlabel.lstrip("Formula:")) + eInd = ref_labels.index("" + r_label.lstrip("Formula:")) z = line[line.find("}", ind + 7) + 1] if z == "." or z == ",": pauseP = True proofLine += ("
\n\n" + - refEqs[eInd] + + ref_eqs[eInd] + "" + z + "
\n") else: proofLine += ("
\n\n" + - refEqs[eInd] + + ref_eqs[eInd] + "
\n") else: @@ -1056,18 +1045,18 @@ def DLMF(ofname, mmd, llinks, n): elif math: if "\\end{equation}" in lines[ - i + - 1] or "\\constraint" in lines[ - i + - 1] or "\\substitution" in lines[ - i + - 1] or "\\proof" in lines[ - i + - 1] or "\\drmfnote" in lines[ - i + - 1] or "\\drmfname" in lines[ i + - 1]: + 1] or "\\constraint" in lines[ + i + + 1] or "\\substitution" in lines[ + i + + 1] or "\\proof" in lines[ + i + + 1] or "\\drmfnote" in lines[ + i + + 1] or "\\drmfname" in lines[ + i + + 1]: append_text(line.rstrip("\n")) symLine += line.strip("\n") symbols = symbols + getSym(symLine) @@ -1080,26 +1069,31 @@ def DLMF(ofname, mmd, llinks, n): noteLine = noteLine + line symbols = symbols + getSym(line) if "\\end{equation}" in lines[ - i + - 1] or "\\drmfn" in lines[ - i + - 1] or "\\constraint" in lines[ - i + - 1] or "\\substitution" in lines[ - i + - 1] or "\\proof" in lines[ - i + - 1]: + i + + 1] or "\\drmfn" in lines[ + i + + 1] or "\\constraint" in lines[ + i + + 1] or "\\substitution" in lines[ + i + + 1] or "\\proof" in lines[ + i + + 1]: note = False if "\\emph" in noteLine: noteLine = noteLine[ - 0:noteLine.find("\\emph{")] + "\'\'" + noteLine[ - noteLine.find("\\emph{") + len("\\emph{"): noteLine.find( - "}", noteLine.find("\\emph{") + len("\\emph{"))] + "\'\'" + noteLine[ - noteLine.find( - "}", noteLine.find("\\emph{") + len("\\emph{")) + 1:] + 0:noteLine.find("\\emph{")] + "\'\'" + noteLine[ + noteLine.find("\\emph{") + len( + "\\emph{"): noteLine.find( + "}", noteLine.find("\\emph{") + len( + "\\emph{"))] + "\'\'" + noteLine[ + noteLine.find( + "}", + noteLine.find( + "\\emph{") + len( + "\\emph{")) + 1:] comToWrite = comToWrite + "
" + \ - getEq(noteLine) + "

\n" + getEq(noteLine) + "

\n" if constraint and parse: conLine += line.replace("&", "&
") @@ -1107,16 +1101,16 @@ def DLMF(ofname, mmd, llinks, n): symLine += line.strip("\n") # symbols=symbols+getSym(line) if "\\end{equation}" in lines[ - i + - 1] or "\\drmfn" in lines[ - i + - 1] or "\\constraint" in lines[ - i + - 1] or "\\substitution" in lines[ - i + - 1] or "\\proof" in lines[ - i + - 1]: + i + + 1] or "\\drmfn" in lines[ + i + + 1] or "\\constraint" in lines[ + i + + 1] or "\\substitution" in lines[ + i + + 1] or "\\proof" in lines[ + i + + 1]: constraint = False symbols = symbols + getSym(symLine) symLine = "" @@ -1132,16 +1126,16 @@ def DLMF(ofname, mmd, llinks, n): symLine += line.strip("\n") # symbols=symbols+getSym(line) if "\\end{equation}" in lines[ - i + - 1] or "\\drmfn" in lines[ - i + - 1] or "\\substitution" in lines[ - i + - 1] or "\\constraint" in lines[ - i + - 1] or "\\proof" in lines[ - i + - 1]: + i + + 1] or "\\drmfn" in lines[ + i + + 1] or "\\substitution" in lines[ + i + + 1] or "\\constraint" in lines[ + i + + 1] or "\\proof" in lines[ + i + + 1]: substitution = False symbols = symbols + getSym(symLine) symLine = "" @@ -1157,4 +1151,4 @@ def DLMF(ofname, mmd, llinks, n): tex = "../../data/ZE.3.tex" mainFile = "../../data/OrthogonalPolynomials.mmd" lLinks = "../../data/BruceLabelLinks" - DLMF(tex, mainFile, lLinks, 0) + dlmf(tex, mainFile, lLinks, 0) diff --git a/tex2Wiki/src/tex2Wiki.py b/tex2Wiki/src/tex2Wiki.py index c5e70a2..8140140 100644 --- a/tex2Wiki/src/tex2Wiki.py +++ b/tex2Wiki/src/tex2Wiki.py @@ -16,34 +16,41 @@ root = ET.Element('{http://www.mediawiki.org/xml/export-0.10/}mediawiki') -def isnumber(char): # Function to check if char is a number (assuming 1 character) - return char[0] in "0123456789" +def is_number(char): # Function to check if char is a number (assuming 1 character) + return char[0].isdigit() -def getString(line): # Gets all data within curly braces on a line - # -------Initialization------- - stringWrite = "" - getStr = False - pW = "" - # ---------------------------- - for c in line: - if c == "{" or c == "}": # if there is a curly brace in the line - getStr = not getStr # toggle the getStr flag - if c == "}": # no more info needed +def get_string_within_curly_braces(line): + """Gets all data within curly braces on a line""" + # TODO: Possibly have to fix for nested curly braces + + string_within_braces = "" + get_str = False + prev_char = "" + + for char in line: + if char in ["{", "}"]: + get_str = not get_str + if char == "}": break - elif getStr: # if within curly braces - if not ( - pW == c and c == "-"): # if there is no double dash (makes single dash) - if c != "$": # if not a $ sign - stringWrite += c # this character is part of the data - else: # replace $ with '' - stringWrite += "\'\'" # add double ' - pW = c # change last character for finding double dashes - # return the data without newlines and without leading spaces - return stringWrite.rstrip('\n').lstrip() - - -def getG(line): # gets equation for symbols list + # If within curly braces + elif get_str: + # If there is no double dash (makes single dash) + if not (prev_char == char and char == "-"): + if char != "$": + # this character is part of the data + string_within_braces += char + else: + # Add double single quote instead of $ + string_within_braces += "''" + # Change last character for finding double dashes + prev_char = char + # Return the data without newlines and without leading spaces + return string_within_braces.rstrip('\n').lstrip() + + +def get_symbol_list_eq(line): + """Gets equation for symbols list""" start = True final = "" for c in line: @@ -58,56 +65,69 @@ def getG(line): # gets equation for symbols list return final -def getEq(line): # Gets all data within constraints,substitutions - # -------Initialization------- +def get_equation(line, proof=False): + """Gets all data within constraints, substitutions""" per = 1 - stringWrite = "" - fEq = False + string_write = "" + end_of_equation = False count = 0 - # ---------------------------- - for c in line: # read each character + length = 0 + # TODO: figure out what count and per do -_- + + # read each character + for char in line: + if proof and end_of_equation and char not in [" ", "$", "{", "}"] and not char.isalpha(): + length += 1 if count >= 0 and per != 0: per += 1 - if c != " " and c != "%": + if char != " " and char != "%": per = 0 - if c == "{" and per == 0: + if char == "{" and per == 0: if count > 0: - stringWrite += c + string_write += char count += 1 - elif c == "}" and per == 0: + elif char == "}" and per == 0: count -= 1 if count > 0: - stringWrite += c - elif c == "$" and per == 0: # either begin or end equation - fEq = not fEq # toggle fEq flag to know if begin or end equation - if fEq: # if begin - stringWrite += "" - else: # if end - stringWrite += "" - elif c == "\n" and per == 0: # if newline - stringWrite = stringWrite.strip() # remove all leading and trailing whitespace + string_write += char + # either begin or end equation + elif char == "$" and per == 0: + # toggle end_of_equation flag to know if begin or end equation + end_of_equation = not end_of_equation + # if begin + if end_of_equation: + string_write += "" + else: + if proof and length >= 10: + string_write += "
" + else: + string_write += "" + length = 0 + # if newline + elif char == "\n" and per == 0: + string_write = string_write.strip() # should above be # rstrip?<-------------------------------------------------------------------CHECK # THIS - stringWrite += c # add the newline character + string_write += char per += 1 # watch for % signs elif count > 0 and per == 0: # not special character - stringWrite += c # write the character + string_write += char # write the character - return stringWrite.rstrip().lstrip() + return string_write.strip() -def getEqP(line): # Gets all data within proofs +def get_proofs_data(line): + """Gets all data within proofs""" per = 1 stringWrite = "" fEq = False count = 0 length = 0 for c in line: - if fEq and c != " " and c != "$" and c != "{" and c != "}" and not c.isalpha( - ): + if fEq and c != " " and c != "$" and c != "{" and c != "}" and not c.isalpha(): length += 1 if count >= 0 and per != 0: per += 1 @@ -236,14 +256,14 @@ def modLabel(line): newlabel = "" num = "" for i in range(0, len(label)): - if isNumer and not isnumber(label[i]): + if isNumer and not is_number(label[i]): if len(num) > 1: newlabel += num num = "" else: newlabel += "0" + str(num) num = "" - if isnumber(label[i]): + if is_number(label[i]): isNumer = True num += str(label[i]) else: @@ -281,25 +301,6 @@ def writeout(ofname): tree.write(ofname, xml_declaration=True, encoding='utf-8', method='xml') -def main(): - if len(sys.argv) != 6: - fname = "../../data/ZE.3.tex" - ofname = "../../data/ZE.4.xml" - lname = "../../data/BruceLabelLinks" - glossary = "../../data/new.Glossary.csv" - mmd = "../../data/OrthogonalPolynomials.mmd" - - else: - fname = sys.argv[1] - ofname = sys.argv[2] - lname = sys.argv[3] - glossary = sys.argv[4] - mmd = sys.argv[5] - setup_label_links(lname) - readin(fname, glossary, mmd) - writeout(ofname) - - def setup_label_links(ofname): global lLink lLink = open(ofname, "r").readlines() @@ -338,14 +339,14 @@ def readin(ofname, glossary, mmd): elif math: refEqs[-1] += line if "\\end{equation}" in refLines[ - i + - 1] or "\\constraint" in refLines[ - i + - 1] or "\\substitution" in refLines[ - i + - 1] or "\\drmfn" in refLines[ - i + - 1]: + i + + 1] or "\\constraint" in refLines[ + i + + 1] or "\\substitution" in refLines[ + i + + 1] or "\\drmfn" in refLines[ + i + + 1]: math = False for i in range(0, len(lines)): @@ -368,29 +369,29 @@ def readin(ofname, glossary, mmd): parse = False elif "\\title" in line and parse: stringWrite = "\'\'\'" - stringWrite += getString(line) + "\'\'\'\n" + stringWrite += get_string_within_curly_braces(line) + "\'\'\'\n" labels.append("Orthogonal Polynomials") sections.append(["Orthogonal Polynomials", 0]) - chapter = getString(line) + chapter = get_string_within_curly_braces(line) mainPrepend += ( "\n== Sections in " + chapter + " ==\n\n
\n") elif "\\part" in line: - if getString(line) == "BOF": + if get_string_within_curly_braces(line) == "BOF": parse = False - elif getString(line) == "EOF": + elif get_string_within_curly_braces(line) == "EOF": parse = True elif parse: - mainPrepend += ("\n
\n= " + getString(line) + " =\n") + mainPrepend += ("\n
\n= " + get_string_within_curly_braces(line) + " =\n") head = True elif "\\section" in line: mainPrepend += ("* [[" + - secLabel(getString(line)) + + secLabel(get_string_within_curly_braces(line)) + "|" + - getString(line) + + get_string_within_curly_braces(line) + "]]\n") - sections.append([getString(line)]) + sections.append([get_string_within_curly_braces(line)]) secCounter = 0 eqCounter = 0 @@ -401,14 +402,14 @@ def readin(ofname, glossary, mmd): if "\\section" in line: parse = True secCounter += 1 - append_revision(secLabel(getString(line))) + append_revision(secLabel(get_string_within_curly_braces(line))) append_text( "{{DISPLAYTITLE:" + (sections[secCounter][0]) + "}}\n") append_text("{{#set:Chapter=" + chapter + "}}\n") append_text("{{#set:Section=" + str(secCounter) + "}}\n") append_text("{{headSection}}\n") head = True - append_text("== " + getString(line) + " ==\n") + append_text("== " + get_string_within_curly_braces(line) + " ==\n") elif ("\\section" in lines[(i + 1) % len(lines)] or "\\end{document}" in lines[ (i + 1) % len(lines)]) and parse: append_text("{{footSection}}\n") @@ -416,13 +417,13 @@ def readin(ofname, glossary, mmd): eqCounter = 0 elif "\\subsection" in line and parse: - append_text("\n== " + getString(line) + " ==\n") + append_text("\n== " + get_string_within_curly_braces(line) + " ==\n") head = True elif "\\paragraph" in line and parse: - append_text("\n=== " + getString(line) + " ===\n") + append_text("\n=== " + get_string_within_curly_braces(line) + " ===\n") head = True elif "\\subsubsection" in line and parse: - append_text("\n=== " + getString(line) + " ===\n") + append_text("\n=== " + get_string_within_curly_braces(line) + " ===\n") head = True elif "\\begin{equation}" in line and parse: @@ -458,21 +459,21 @@ def readin(ofname, glossary, mmd): if "\\drmfname" in line and parse: append_text( "
This formula has the name: " + - getString(line) + + get_string_within_curly_braces(line) + "

\n") elif math and parse: flagM = True eqs[-1] += line if "\\end{equation}" in lines[ - i + - 1] and "\\subsection" not in lines[ - i + - 3] and "\\section" not in lines[ - i + - 3] and "\\part" not in lines[ - i + - 3]: + i + + 1] and "\\subsection" not in lines[ + i + + 3] and "\\section" not in lines[ + i + + 3] and "\\part" not in lines[ + i + + 3]: u = i flagM2 = False while flagM: @@ -480,7 +481,7 @@ def readin(ofname, glossary, mmd): if "\\begin{equation}" in lines[u] in lines[u]: flagM = False if "\\section" in lines[u] or "\\subsection" in lines[ - i] or "\\part" in lines[u] or "\\end{document}" in lines[u]: + i] or "\\part" in lines[u] or "\\end{document}" in lines[u]: flagM = False flagM2 = True if not flagM2: @@ -501,58 +502,58 @@ def readin(ofname, glossary, mmd): elif math and not parse: eqs[-1] += line if "\\end{equation}" in lines[ - i + - 1] or "\\constraint" in lines[ - i + - 1] or "\\substitution" in lines[ - i + - 1] or "\\drmfn" in lines[ - i + - 1]: + i + + 1] or "\\constraint" in lines[ + i + + 1] or "\\substitution" in lines[ + i + + 1] or "\\drmfn" in lines[ + i + + 1]: math = False if substitution and parse: subLine += line.replace("&", "&
") if "\\end{equation}" in lines[ - i + - 1] or "\\substitution" in lines[ - i + - 1] or "\\constraint" in lines[ - i + - 1] or "\\drmfn" in lines[ - i + - 1] or "\\proof" in lines[ - i + - 1]: + i + + 1] or "\\substitution" in lines[ + i + + 1] or "\\constraint" in lines[ + i + + 1] or "\\drmfn" in lines[ + i + + 1] or "\\proof" in lines[ + i + + 1]: lineR = "" for i in range(0, len(subLine)): if subLine[i] == "&" and not ( - i > subLine.find("\\begin{array}") and i < subLine.find("\\end{array}")): + i > subLine.find("\\begin{array}") and i < subLine.find("\\end{array}")): lineR += "&
" else: lineR += subLine[i] substitution = False append_text( "
Substitution(s): " + - getEq(subLine) + + get_equation(subLine) + "

\n") if constraint and parse: conLine += line.replace("&", "&
") if "\\end{equation}" in lines[ - i + - 1] or "\\substitution" in lines[ - i + - 1] or "\\constraint" in lines[ - i + - 1] or "\\drmfn" in lines[ - i + - 1] or "\\proof" in lines[ - i + - 1]: + i + + 1] or "\\substitution" in lines[ + i + + 1] or "\\constraint" in lines[ + i + + 1] or "\\drmfn" in lines[ + i + + 1] or "\\proof" in lines[ + i + + 1]: constraint = False append_text( "
Constraint(s): " + - getEq(conLine) + + get_equation(conLine) + "

\n") eqCounter = 0 @@ -639,7 +640,7 @@ def readin(ofname, glossary, mmd): "#" + secLabel( labels[eqCounter][ - len("Formula:"):]) + + len("Formula:"):]) + "|formula in " + secLabel( sections[ @@ -723,7 +724,7 @@ def readin(ofname, glossary, mmd): "#" + secLabel( labels[eqCounter][ - len("Formula:"):]) + + len("Formula:"):]) + "|formula in " + secLabel( sections[ @@ -893,7 +894,7 @@ def readin(ofname, glossary, mmd): Q = symbolPar listArgs = [] if len(Q) > len(symbol) and (Q[ - len(symbol)] == "{" or Q[len(symbol)] == "["): + len(symbol)] == "{" or Q[len(symbol)] == "["): ap = "" for o in range(len(symbol), len(Q)): if Q[o] == "}" or z == "]": @@ -906,7 +907,7 @@ def readin(ofname, glossary, mmd): for t in range(5, len(G)): if G[t] != "": websiteF = websiteF + \ - " [" + G[t] + " " + G[t] + "]" + " [" + G[t] + " " + G[t] + "]" p1 = G[4].strip("$") p1 = "" + p1 + "" # if checkFlag: @@ -1001,7 +1002,7 @@ def readin(ofname, glossary, mmd): "#" + secLabel( labels[eqCounter][ - len("Formula:"):]) + + len("Formula:"):]) + "|formula in " + secLabel( sections[ @@ -1064,7 +1065,7 @@ def readin(ofname, glossary, mmd): "_") + "#" + labels[endNum][ - 8:] + + 8:] + "|formula in " + labels[0] + "]]
\n") @@ -1102,7 +1103,7 @@ def readin(ofname, glossary, mmd): elif "\\drmfname" in line and parse: math = False comToWrite = "\n== Name ==\n\n
" + \ - getString(line) + "

\n" + comToWrite + get_string_within_curly_braces(line) + "

\n" + comToWrite elif "\\drmfnote" in line and parse: symbols = symbols + getSym(line) @@ -1167,7 +1168,7 @@ def readin(ofname, glossary, mmd): proof = False append_text( comToWrite + - getEqP(proofLine) + + get_proofs_data(proofLine) + "
\n
\n") comToWrite = "" symbols = symbols + getSym(symLine) @@ -1215,7 +1216,7 @@ def readin(ofname, glossary, mmd): proof = False append_text( comToWrite + - getEqP(proofLine).rstrip("\n") + + get_proofs_data(proofLine).rstrip("\n") + "
\n
\n") comToWrite = "" symbols = symbols + getSym(symLine) @@ -1223,18 +1224,18 @@ def readin(ofname, glossary, mmd): elif math: if "\\end{equation}" in lines[ - i + - 1] or "\\constraint" in lines[ - i + - 1] or "\\substitution" in lines[ - i + - 1] or "\\proof" in lines[ - i + - 1] or "\\drmfnote" in lines[ - i + - 1] or "\\drmfname" in lines[ i + - 1]: + 1] or "\\constraint" in lines[ + i + + 1] or "\\substitution" in lines[ + i + + 1] or "\\proof" in lines[ + i + + 1] or "\\drmfnote" in lines[ + i + + 1] or "\\drmfname" in lines[ + i + + 1]: append_text(line.rstrip("\n")) symLine += line.strip("\n") symbols = symbols + getSym(symLine) @@ -1247,26 +1248,31 @@ def readin(ofname, glossary, mmd): noteLine += line symbols = symbols + getSym(line) if "\\end{equation}" in lines[ - i + - 1] or "\\drmfn" in lines[ - i + - 1] or "\\constraint" in lines[ - i + - 1] or "\\substitution" in lines[ - i + - 1] or "\\proof" in lines[ - i + - 1]: + i + + 1] or "\\drmfn" in lines[ + i + + 1] or "\\constraint" in lines[ + i + + 1] or "\\substitution" in lines[ + i + + 1] or "\\proof" in lines[ + i + + 1]: note = False if "\\emph" in noteLine: noteLine = noteLine[ - 0:noteLine.find("\\emph{")] + "\'\'" + noteLine[ - noteLine.find("\\emph{") + len("\\emph{"):noteLine.find( - "}", noteLine.find("\\emph{") + len("\\emph{"))] + "\'\'" + noteLine[ - noteLine.find( - "}", noteLine.find("\\emph{") + len("\\emph{")) + 1:] + 0:noteLine.find("\\emph{")] + "\'\'" + noteLine[ + noteLine.find("\\emph{") + len( + "\\emph{"):noteLine.find( + "}", noteLine.find("\\emph{") + len( + "\\emph{"))] + "\'\'" + noteLine[ + noteLine.find( + "}", + noteLine.find( + "\\emph{") + len( + "\\emph{")) + 1:] comToWrite = comToWrite + "
" + \ - getEq(noteLine) + "

\n" + get_equation(noteLine) + "

\n" if constraint and parse: conLine += line.replace("&", "&
") @@ -1274,23 +1280,23 @@ def readin(ofname, glossary, mmd): symLine += line.strip("\n") # symbols=symbols+getSym(line) if "\\end{equation}" in lines[ - i + - 1] or "\\drmfn" in lines[ - i + - 1] or "\\constraint" in lines[ - i + - 1] or "\\substitution" in lines[ - i + - 1] or "\\proof" in lines[ - i + - 1]: + i + + 1] or "\\drmfn" in lines[ + i + + 1] or "\\constraint" in lines[ + i + + 1] or "\\substitution" in lines[ + i + + 1] or "\\proof" in lines[ + i + + 1]: constraint = False symbols = symbols + getSym(symLine) symLine = "" append_text( comToWrite + "
" + - getEq(conLine) + + get_equation(conLine) + "

\n") comToWrite = "" if substitution and parse: @@ -1299,33 +1305,52 @@ def readin(ofname, glossary, mmd): symLine += line.strip("\n") if "\\end{equation}" in lines[ - i + - 1] or "\\drmfn" in lines[ - i + - 1] or "\\substitution" in lines[ - i + - 1] or "\\constraint" in lines[ - i + - 1] or "\\proof" in lines[ - i + - 1]: + i + + 1] or "\\drmfn" in lines[ + i + + 1] or "\\substitution" in lines[ + i + + 1] or "\\constraint" in lines[ + i + + 1] or "\\proof" in lines[ + i + + 1]: substitution = False symbols = symbols + getSym(symLine) symLine = "" lineR = "" for i in range(0, len(subLine)): if subLine[i] == "&" and not ( - i > subLine.find("\\begin{array}") and i < subLine.find("\\end{array}")): + i > subLine.find("\\begin{array}") and i < subLine.find("\\end{array}")): lineR += "&
" else: lineR += subLine[i] append_text( comToWrite + "
" + - getEq(subLine) + + get_equation(subLine) + "

\n") comToWrite = "" +def main(): + if len(sys.argv) != 6: + fname = "../../data/ZE.3.tex" + ofname = "../../data/ZE.4.xml" + lname = "../../data/BruceLabelLinks" + glossary = "../../data/new.Glossary.csv" + mmd = "../../data/OrthogonalPolynomials.mmd" + + else: + fname = sys.argv[1] + ofname = sys.argv[2] + lname = sys.argv[3] + glossary = sys.argv[4] + mmd = sys.argv[5] + setup_label_links(lname) + readin(fname, glossary, mmd) + writeout(ofname) + + if __name__ == "__main__": main() From e2435ce590e3dca923a589eb545f579e8af1242f Mon Sep 17 00:00:00 2001 From: Joon Bang Date: Tue, 16 Aug 2016 14:28:05 -0400 Subject: [PATCH 277/402] Start rewrite of tex2Wiki --- tex2Wiki/README.md | 2 +- tex2Wiki/experiments/xml_output.py | 20 - tex2Wiki/src/DLMFtex2Wiki.py | 1154 ---------- tex2Wiki/src/OrthogonalPolynomials.mmd | 65 + tex2Wiki/src/OrthogonalPolynomials.mmd.new | 65 + tex2Wiki/src/__init__.py | 0 tex2Wiki/src/testData.txt | 0 tex2Wiki/src/tex2Wiki.py | 2436 +++++++++----------- tex2Wiki/src/tex2Wiki2.py | 136 ++ tex2Wiki/test/llinks | 1 - tex2Wiki/test/test_getEq.py | 9 - tex2Wiki/test/test_getString.py | 11 - tex2Wiki/test/test_isnumber.py | 11 - tex2Wiki/test/test_modlabel.py | 19 - tex2Wiki/test/test_secLabel.py | 6 - tex2Wiki/test/test_unmodLabel.py | 8 - 16 files changed, 1347 insertions(+), 2596 deletions(-) delete mode 100644 tex2Wiki/experiments/xml_output.py delete mode 100644 tex2Wiki/src/DLMFtex2Wiki.py create mode 100644 tex2Wiki/src/OrthogonalPolynomials.mmd create mode 100644 tex2Wiki/src/OrthogonalPolynomials.mmd.new create mode 100644 tex2Wiki/src/__init__.py create mode 100644 tex2Wiki/src/testData.txt create mode 100644 tex2Wiki/src/tex2Wiki2.py delete mode 100644 tex2Wiki/test/llinks delete mode 100644 tex2Wiki/test/test_getEq.py delete mode 100644 tex2Wiki/test/test_getString.py delete mode 100644 tex2Wiki/test/test_isnumber.py delete mode 100644 tex2Wiki/test/test_modlabel.py delete mode 100644 tex2Wiki/test/test_secLabel.py delete mode 100644 tex2Wiki/test/test_unmodLabel.py diff --git a/tex2Wiki/README.md b/tex2Wiki/README.md index 760607b..3ae558d 100644 --- a/tex2Wiki/README.md +++ b/tex2Wiki/README.md @@ -1,6 +1,6 @@ # tex2Wiki -## Azeem Mohammed +## Azeem Mohammed, Joon Bang WIP code diff --git a/tex2Wiki/experiments/xml_output.py b/tex2Wiki/experiments/xml_output.py deleted file mode 100644 index f91670d..0000000 --- a/tex2Wiki/experiments/xml_output.py +++ /dev/null @@ -1,20 +0,0 @@ -import xml.etree.ElementTree as ET - - -def main(): - ET.register_namespace('', 'http://www.mediawiki.org/xml/export-0.10/') - root = ET.Element('{http://www.mediawiki.org/xml/export-0.10/}mediawiki') - page = ET.SubElement(root, 'page') - title = ET.SubElement(page, 'title') - title.text = 'the Title & stuff' - ET.dump(root) - tree = ET.ElementTree(root) - tree.write( - "page.xhtml", - xml_declaration=True, - encoding='utf-8', - method='xml') - - -if __name__ == "__main__": - main() diff --git a/tex2Wiki/src/DLMFtex2Wiki.py b/tex2Wiki/src/DLMFtex2Wiki.py deleted file mode 100644 index 7c20de0..0000000 --- a/tex2Wiki/src/DLMFtex2Wiki.py +++ /dev/null @@ -1,1154 +0,0 @@ -__author__ = "Azeem Mohammed" -__status__ = "Development" - -# Version 15: Minor Fixes -# Convert tex to wikiText -import csv # imported for using csv format -import sys # imported for getting args -from shutil import copyfile -from tex2Wiki import append_text, append_revision, get_string_within_, getSym, getEq, secLabel, getEqP, unmodLabel, isnumber - - -def mod_label(label): - is_numeric = False - new_label = "" - num = "" - for i in range(len(label)): - if is_numeric and not isnumber(label[i]): - if len(num) > 1: - new_label += num - num = "" - else: - new_label += "0" + str(num) - num = "" - if isnumber(label[i]): - is_numeric = True - num += str(label[i]) - else: - is_numeric = False - new_label += label[i] - if len(num) > 1: - new_label += num - elif len(num) == 1: - new_label += "0" + num - return new_label - - -def dlmf(of_name, mmd, llinks, n): - for iterations in range(0, 1): - tex = open(of_name, 'r') - main_file = open(mmd, "r") - l_links = open(llinks, 'r') - main_prepend = "" - main_write = open("ZetaFunctions.mmd.new", "w") - l_link_list = l_links.readlines() - math = False - constraint = False - substitution = False - symbols = [] - lines = tex.readlines() - ref_lines = [] - sections = [] - labels = [] - eqs = [] - ref_eqs = [] - parse = False - head = False - try: - chap_ref = [("GA", open("../../data/GA.tex", 'r')), - ("ZE", open("../../data/ZE.3.tex", 'r'))] - except IOError: - chap_ref = [("GA", open("GA.tex", 'r')), ("ZE", open("ZE.3.tex", 'r'))] - ref_labels = [] - for chap in chap_ref: - ref_lines = ref_lines + (chap[1].readlines()) - chap[1].close() - for i in range(len(ref_lines)): - line = ref_lines[i] - if "\\begin{equation}" in line: - s_label = line.find("\\label{") + 7 - e_label = line.find("}", s_label) - label = line[s_label:e_label] - for l_link in l_link_list: - if l_link.find(label) != -1 and l_link[len(label) + 1] == "=": - r_label = l_link[l_link.find("=>") + 3:l_link.find("\\n")] - r_label = r_label.replace("/", "") - r_label = r_label.replace("#", ":") - break - label = r_label - ref_labels.append(label) - ref_eqs.append("") - math = True - elif math: - ref_eqs[-1] += line - if any([item in ref_lines[i + 1] - for item in ["\\end{equation}", "\\constraint", "\\substitution", "\\drmfn"]]): - math = False - - start_flag = True - for i in range(len(lines)): - line = lines[i] - if "\\begin{document}" in line: - parse = True - elif "\\end{document}" in line: - main_prepend += "
\n" - main_text = main_prepend - main_text = main_text.replace("drmf_bof\n", "") - main_text = main_text.replace("drmf_eof\n", "") - main_text = main_text.replace( - "\'\'\'Zeta and Related Functions\'\'\'\n", "") - main_text = main_text.replace("{{#set:Section=0}}\n", "") - append_revision('Zeta and Related Functions') - main_text = "{{#set:Section=0}}\n" + main_text - main_write.write(main_text) - main_write.close() - main_file.close() - copyfile(mmd, 'ZetaFunctions.mmd.new') - parse = False - elif "\\title" in line and parse: - labels.append("Zeta and Related Functions") - sections.append(["Zeta and Related Functions", 0]) - elif "\\part" in line: - if getString(line) == "BOF": - parse = False - elif getString(line) == "EOF": - parse = True - elif parse: - stringWrite = "\'\'\'" - stringWrite += getString(line) + "\'\'\'\n" - chapter = getString(line) - if start_flag: - main_prepend += ( - "\n== Sections in " + - chapter + - " ==\n\n
\n") - start_flag = False - else: - main_prepend += ("
\n\n== Sections in " + - chapter + - " ==\n\n
\n") - head = True - elif "\\section" in line: - main_prepend += ("* [[" + - secLabel(getString(line)) + - "|" + - getString(line) + - "]]\n") - sections.append([getString(line)]) - - secCounter = 0 - eqCounter = 0 - for i in range(0, len(lines)): - line = lines[i] - if "\\section" in line: - parse = True - secCounter += 1 - append_revision(secLabel(getString(line))) - append_text( - "{{DISPLAYTITLE:" + (sections[secCounter][0]) + "}}\n") - append_text("
\n") - append_text( - "
<< [[" + - secLabel( - sections[ - secCounter - - 1][0]) + - "|" + - secLabel( - sections[ - secCounter - - 1][0]) + - "]]
\n") - append_text( - "
[[Zeta_and_Related_Functions#" + - "Sections_in_" + - chapter.replace( - " ", - "_") + - "|" + - secLabel( - sections[secCounter][0]) + - "]]
\n") - append_text( - "
[[" + - secLabel( - sections[ - (secCounter + - 1) % - len(sections)][0]) + - "|" + - secLabel( - sections[ - (secCounter + - 1) % - len(sections)][0]) + - "]] >>
\n
\n\n") - head = True - append_text("== " + getString(line) + " ==\n") - elif ("\\section" in lines[(i + 1) % len(lines)] or "\\end{document}" in lines[ - (i + 1) % len(lines)]) and parse: - append_text("
\n") - append_text( - "
<< [[" + - secLabel( - sections[ - secCounter - - 1][0]) + - "|" + - secLabel( - sections[ - secCounter - - 1][0]) + - "]]
\n") - append_text( - "
[[Zeta_and_Related_Functions#" + - "Sections_in_" - "" + - chapter.replace( - " ", - "_") + - "|" + - secLabel( - sections[secCounter][0]) + - "]]
\n") - append_text( - "
[[" + - secLabel( - sections[ - (secCounter + - 1) % - len(sections)][0]) + - "|" + - secLabel( - sections[ - (secCounter + - 1) % - len(sections)][0]) + - "]] >>
\n
\n\n") - append_text("drmf_eof\n") - sections[secCounter].append(eqCounter) - eqCounter = 0 - - elif "\\subsection" in line and parse: - append_text("\n== " + getString(line) + " ==\n") - head = True - elif "\\paragraph" in line and parse: - append_text("\n=== " + getString(line) + " ===\n") - head = True - elif "\\subsubsection" in line and parse: - append_text("\n=== " + getString(line) + " ===\n") - head = True - - elif "\\begin{equation}" in line and parse: - if head: - append_text("\n") - head = False - s_label = line.find("\\label{") + 7 - e_label = line.find("}", s_label) - label = (line[s_label:e_label]) - eqCounter += 1 - for l_link in l_link_list: - if label == l_link[0:l_link.find("=") - 1]: - r_label = l_link[l_link.find("=>") + 3:l_link.find("\\n")] - r_label = r_label.replace("/", "") - r_label = r_label.replace("#", ":") - r_label = r_label.replace("!", ":") - break - label = mod_label(r_label) - labels.append("Formula:" + r_label) - eqs.append("") - append_text( - "\n") - math = True - elif "\\begin{equation}" in line and not parse: - s_label = line.find("\\label{") + 7 - e_label = line.find("}", s_label) - label = mod_label(line[s_label:e_label]) - labels.append("*" + label) # special marker - eqs.append("") - math = True - elif "\\end{equation}" in line: - - math = False - elif "\\constraint" in line and parse: - constraint = True - math = False - conLine = "" - elif "\\substitution" in line and parse: - substitution = True - math = False - subLine = "" - elif "\\proof" in line and parse: - math = False - elif "\\drmfn" in line and parse: - math = False - if "\\drmfname" in line and parse: - append_text( - "
This formula has the name: " + - getString(line) + - "

\n") - elif math and parse: - flagM = True - eqs[-1] += line - - if not ( - (not ( - "\\end{equation}" in lines[ - i + - 1]) or "\\subsection" in lines[ - i + - 3]) or "\\section" in lines[ - i + - 3]) and not "\\part" in lines[ - i + - 3]: - u = i - flagM2 = False - while flagM: - u += 1 - if "\\begin{equation}" in lines[u] in lines[u]: - flagM = False - if "\\section" in lines[u] or "\\subsection" in lines[ - i] or "\\part" in lines[u] or "\\end{document}" in lines[u]: - flagM = False - flagM2 = True - if not flagM2: - - append_text(line.rstrip("\n")) - append_text("\n
\n") - else: - append_text(line.rstrip("\n")) - append_text("\n\n") - elif "\\end{equation}" in lines[i + 1]: - append_text(line.rstrip("\n")) - append_text("\n\n") - elif "\\constraint" in lines[i + 1] or "\\substitution" in lines[i + 1] or "\\drmfn" in lines[i + 1]: - append_text(line.rstrip("\n")) - append_text("\n\n") - else: - append_text(line) - elif math and not parse: - eqs[-1] += line - if "\\end{equation}" in lines[ - i + - 1] or "\\constraint" in lines[ - i + - 1] or "\\substitution" in lines[ - i + - 1] or "\\drmfn" in lines[ - i + - 1]: - math = False - if substitution and parse: - subLine += line.replace("&", "&
") - if "\\end{equation}" in lines[ - i + - 1] or "\\substitution" in lines[ - i + - 1] or "\\constraint" in lines[ - i + - 1] or "\\drmfn" in lines[ - i + - 1] or "\\proof" in lines[ - i + - 1]: - substitution = False - append_text( - "
Substitution(s): " + - getEq(subLine) + - "

\n") - - if constraint and parse: - conLine += line.replace("&", "&
") - if "\\end{equation}" in lines[ - i + - 1] or "\\substitution" in lines[ - i + - 1] or "\\constraint" in lines[ - i + - 1] or "\\drmfn" in lines[ - i + - 1] or "\\proof" in lines[ - i + - 1]: - constraint = False - append_text( - "
Constraint(s): " + - getEq(conLine) + - "

\n") - - eqCounter = n - endNum = len(labels) - 1 - parse = False - constraint = False - substitution = False - note = False - hCon = True - hSub = True - hNote = True - hProof = True - proof = False - comToWrite = "" - secCount = -1 - newSec = False - for i in range(0, len(lines)): - line = lines[i] - - if "\\section" in line: - secCount += 1 - newSec = True - eqS = 0 - - if "\\begin{equation}" in line: - symLine = line.strip("\n") - eqS += 1 - constraint = False - substitution = False - note = False - comToWrite = "" - hCon = True - hSub = True - hNote = True - hProof = True - proof = False - parse = True - symbols = [] - eqCounter += 1 - label = labels[eqCounter] - append_revision(secLabel(label)) - append_text("{{DISPLAYTITLE:" + (labels[eqCounter]) + "}}\n") - if eqCounter == len(labels) - 1: - break - if eqCounter < endNum: # FOR ANYTHING THAT IS NOT THE EXTRA EQUATIONS - append_text("
\n") - if newSec: - append_text( - "
" - "<< [[" + - secLabel( - sections[secCount][0]).replace( - " ", - "_") + - "|" + - secLabel( - sections[secCount][0]) + - "]]
\n") - else: - append_text( - "
" - "<< [[" + - secLabel( - labels[ - eqCounter - - 1]).replace( - " ", - "_") + - "|" + - secLabel( - labels[ - eqCounter - - 1]) + - "]]
\n") - append_text( - "
[[" + - secLabel( - sections[ - secCount + - 1][0]).replace( - " ", - "_") + - "#" + - secLabel( - labels[eqCounter][ - len("Formula:"):]) + - "|formula in " + - secLabel( - sections[ - secCount + - 1][0]) + - "]]
\n") - if True: - append_text( - "
[[" + - secLabel( - labels[ - (eqCounter + - 1) % - (endNum + - 1)]).replace( - " ", - "_") + - "|" + - secLabel( - labels[ - (eqCounter + - 1) % - (endNum + - 1)]) + - "]] >>
\n") - append_text("
\n\n") - elif eqCounter == endNum: - append_text("
\n") - if newSec: - newSec = False - append_text( - "
<< [[" + - secLabel( - sections[secCount][0]).replace( - " ", - "_") + - "|" + - secLabel( - sections[secCount][0]) + - "]]
\n") - else: - append_text( - "
" - "<< [[" + - secLabel( - labels[ - eqCounter - - 1]).replace( - " ", - "_") + - "|" + - secLabel( - labels[ - eqCounter - - 1]) + - "]]
\n") - append_text( - "
[[" + - secLabel( - sections[ - secCount + - 1][0]).replace( - " ", - "_") + - "#" + - secLabel( - labels[eqCounter][ - len("Formula:"):]) + - "|formula in " + - secLabel( - sections[ - secCount + - 1][0]) + - "]]
\n") - append_text( - "
[[" + - secLabel( - labels[ - (eqCounter + - 1) % - (endNum + - 1)].replace( - " ", - "_")) + - "|" + - secLabel( - labels[ - (eqCounter + - 1) % - (endNum + - 1)]) + - "]]
\n") - append_text("
\n\n") - - append_text("
\n") - math = True - elif "\\end{equation}" in line: - append_text(comToWrite) - parse = False - math = False - if hProof: - append_text("\n== Proof ==\n\nWe ask users to provide proof(s), \ - reference(s) to proof(s), or further clarification on the proof(s) in this space.\n") - - append_text("\n== Symbols List ==\n\n") - newSym = [] - # if "09.07:04" in label: - for x in symbols: - flagA = True - # if x not in newSym: - cN = 0 - cC = 0 - flag = True - ArgCx = 0 - for z in x: - if z.isalpha() and flag or z == "&": - cN += 1 - else: - flag = False - if z == "{" or z == "[": - cC += 1 - if z == "}" or z == "]": - cC -= 1 - if cC == 0: - ArgCx += 1 - noA = x[:cN] - for y in newSym: - cN = 0 - cC = 0 - ArgC = 0 - flag = True - for z in y: - if z.isalpha() and flag or z == "&": - cN += 1 - else: - flag = False - if z == "{" or z == "[": - cC += 1 - if z == "}" or z == "]": - cC -= 1 - if cC == 0: - ArgC += 1 - - if y[:cN] == noA: # and ArgC==ArgCx: - flagA = False - break - if flagA: - newSym.append(x) - newSym.reverse() - ampFlag = False - finSym = [] - for s in range(len(newSym) - 1, -1, -1): - symbolPar = "\\" + newSym[s] - ArgCx = 0 - parCx = 0 - parFlag = False - cC = 0 - for z in symbolPar: - if z == "@": - parFlag = True - elif z.isalpha() or z == "&": - cN += 1 - else: - if z == "{" or z == "[": - cC += 1 - if z == "}" or z == "]": - cC -= 1 - if cC == 0: - if parFlag: - parCx += 1 - else: - ArgCx += 1 - - if symbolPar.find("{") != -1 or symbolPar.find("[") != -1: - if symbolPar.find("[") == -1: - symbol = symbolPar[0:symbolPar.find("{")] - elif symbolPar.find("{") == -1 or symbolPar.find("[") < symbolPar.find("{"): - symbol = symbolPar[0:symbolPar.find("[")] - else: - symbol = symbolPar[0:symbolPar.find("{")] - else: - symbol = symbolPar - gFlag = False - checkFlag = False - get = False - try: - gCSV = csv.reader( - open( - sys.argv[3], - 'rb'), - delimiter=',', - quotechar='\"') - except (IOError, IndexError): - gCSV = csv.reader( - open( - '../../data/new.Glossary.csv', - 'rb'), - delimiter=',', - quotechar='\"') - preG = "" - if symbol == "\\&": - ampFlag = True - for S in gCSV: - G = S - ArgCx = 0 - parCx = 0 - parFlag = False - cC = 0 - ind = G[0].find("@") - if ind == -1: - ind = len(G[0]) - 1 - for z in G[0]: - if z == "@": - parFlag = True - elif z.isalpha(): - cN += 1 - else: - if z == "{" or z == "[": - cC += 1 - if z == "}" or z == "]": - cC -= 1 - if cC == 0: - if parFlag: - parCx += 1 - else: - ArgCx += 1 - if G[0].find(symbol) == 0 and (len(G[0]) == len(symbol) or not G[0][ - len(symbol)].isalpha()): # and (numPar!=0 or numArg!=0): - checkFlag = True - get = True - preG = S - elif checkFlag: - get = True - checkFlag = False - if get: - if get: - G = preG - if True: - if symbolPar.find("@") != -1: - Q = symbolPar[:symbolPar.find("@")] - else: - Q = symbolPar - listArgs = [] - if len(Q) > len(symbol) and (Q[ - len(symbol)] == "{" or Q[len(symbol)] == "["): - ap = "" - for o in range(len(symbol), len(Q)): - if Q[o] == "{" or z == "[": - pass - elif Q[o] == "}" or z == "]": - listArgs.append(ap) - ap = "" - else: - ap += Q[o] - websiteF = "" - web1 = G[5] - for t in range(5, len(G)): - if G[t] != "": - websiteF = websiteF + \ - " [" + G[t] + " " + G[t] + "]" - p1 = G[4].strip("$") - p1 = "" + p1 + "" - # if checkFlag: - new2 = "" - pause = False - mathF = True - p2 = G[1] - for k in range(0, len(p2)): - if p2[k] == "$": - if mathF: - new2 += " " - else: - new2 += "" - mathF = not mathF - else: - new2 += p2[k] - p2 = new2 - finSym.append( - web1 + " " + p1 + "] : " + p2 + " :" + websiteF) - break - if not gFlag: - del newSym[s] - - gFlag = True - if ampFlag: - append_text("& : logical and") - gFlag = False - for y in finSym: - if y == "& : logical and": - pass - elif gFlag: - gFlag = False - append_text("[" + y) - else: - append_text("
\n[" + y) - - append_text("\n
\n") - - # should there be a space between bibliography and ==? - append_text("\n== Bibliography==\n\n") - r = unmodLabel(labels[eqCounter]) - q = r.find("DLMF:") + 5 - p = r.find(":", q) - section = r[q:p] - equation = r[p + 1:] - if equation.find(":") != -1: - equation = equation[0:equation.find(":")] - if isnumber(section) == False: - return eqCounter - append_text( - "[HTTP://DLMF.NIST.GOV/" + - section + - "#" + - equation + - " Equation (" + - equation[ - 1:] + - "), Section " + - section + - "] of [[Bibliography#DLMF|'''DLMF''']].\n\n") - append_text( - "== URL links ==\n\nWe ask users to provide relevant URL links in this space.\n\n") - if eqCounter < endNum: - append_text("
\n") - if newSec: - newSec = False - append_text( - "
<< [[" + - secLabel( - sections[secCount][0]).replace( - " ", - "_") + - "|" + - secLabel( - sections[secCount][0]) + - "]]
\n") - else: - append_text( - "
<< [[" + - secLabel( - labels[ - eqCounter - - 1]).replace( - " ", - "_") + - "|" + - secLabel( - labels[ - eqCounter - - 1]) + - "]]
\n") - append_text( - "
[[" + - secLabel( - sections[ - secCount + - 1][0]).replace( - " ", - "_") + - "#" + - secLabel( - labels[eqCounter][ - len("Formula:"):]) + - "|formula in " + - secLabel( - sections[ - secCount + - 1][0]) + - "]]
\n") - if True: - append_text( - "
[[" + - secLabel( - labels[ - (eqCounter + - 1) % - (endNum + - 1)]).replace( - " ", - "_") + - "|" + - secLabel( - labels[ - (eqCounter + - 1) % - (endNum + - 1)]) + - "]] >>
\n") - append_text("
\n\ndrmf_eof\n") - else: # FOR EXTRA EQUATIONS - append_text("
\n") - append_text( - "
<< [[" + - labels[ - endNum - - 1].replace( - " ", - "_") + - "|" + - labels[ - endNum - - 1] + - "]]
\n") - append_text( - "
[[" + - labels[0].replace( - " ", - "_") + - "#" + - labels[endNum][ - 8:] + - "|formula in " + - labels[0] + - "]]
\n") - append_text( - "
[[" + - labels[ - (0) % - endNum].replace( - " ", - "_") + - "|" + - labels[ - 0 % - endNum] + - "]]
\n") - append_text("
\n\ndrmf_eof\n") - - elif "\\constraint" in line and parse: - symLine = line.strip("\n") - if hCon: - comToWrite += "\n== Constraint(s) ==\n\n" - hCon = False - constraint = True - math = False - conLine = "" - elif "\\substitution" in line and parse: - symLine = line.strip("\n") - if hSub: - comToWrite += "\n== Substitution(s) ==\n\n" - hSub = False - substitution = True - math = False - subLine = "" - elif "\\drmfname" in line and parse: - math = False - comToWrite = "\n== Name ==\n\n
" + \ - getString(line) + "

\n" + comToWrite - - elif "\\drmfnote" in line and parse: - symbols = symbols + getSym(line) - if hNote: - comToWrite += "\n== Note(s) ==\n\n" - hNote = False - note = True - math = False - noteLine = "" - - elif "\\proof" in line and parse: - symLine = line.strip("\n") - if hProof: - hProof = False - comToWrite += "\n== Proof ==\n\nWe ask users to provide proof(s), reference(s) to proof(s), " \ - "or further clarification on the proof(s) in this space. \n

\n
" - proof = True - proofLine = "" - pause = False - pauseP = False - for ind in range(0, len(line)): - if line[ind:ind + 7] == "\\eqref{": - pause = True - eqR = line[ind:line.find("}", ind) + 1] - rLab = getString(eqR) - for l_link in l_link_list: - if rLab == l_link[0:l_link.find("=") - 1]: - r_label = l_link[l_link.find("=>") + 3:l_link.find("\\n")] - r_label = r_label.replace("/", "") - r_label = r_label.replace("#", ":") - r_label = r_label.replace("!", ":") - break - - eInd = ref_labels.index("" + r_label) - z = line[line.find("}", ind + 7) + 1] - if z == "." or z == ",": - pauseP = True - proofLine += ("
\n\n" + - ref_eqs[eInd] + - "" + - z + - "
\n") - else: - if z == "}": - proofLine += ("
\n\n" + - ref_eqs[eInd] + - "
") - else: - proofLine += ("
\n\n" + - ref_eqs[eInd] + - "
\n") - else: - if pause: - if line[ind] == "}": - pause = False - elif pauseP: - pauseP = False - elif line[ind] == "\n" and "\\end{equation}" in lines[i + 1]: - pass - else: - proofLine += (line[ind]) - if "\\end{equation}" in lines[i + 1]: - proof = False - append_text( - comToWrite + - getEqP(proofLine) + - "
\n
\n") - comToWrite = "" - symbols = symbols + getSym(symLine) - symLine = "" - elif proof: - symLine += line.strip("\n") - pauseP = False - for ind in range(0, len(line)): - if line[ind:ind + 7] == "\\eqref{": - pause = True - eqR = line[ind:line.find("}", ind) + 1] - rLab = getString(eqR) - for l_link in l_link_list: - if rLab == l_link[0:l_link.find("=") - 1]: - r_label = l_link[l_link.find("=>") + 3:l_link.find("\\n")] - r_label = r_label.replace("/", "") - r_label = r_label.replace("#", ":") - r_label = r_label.replace("!", ":") - break - - eInd = ref_labels.index("" + r_label.lstrip("Formula:")) - z = line[line.find("}", ind + 7) + 1] - if z == "." or z == ",": - pauseP = True - proofLine += ("
\n\n" + - ref_eqs[eInd] + - "" + - z + - "
\n") - else: - proofLine += ("
\n\n" + - ref_eqs[eInd] + - "
\n") - - else: - if pause: - if line[ind] == "}": - pause = False - elif pauseP: - pauseP = False - elif line[ind] == "\n" and "\\end{equation}" in lines[i + 1]: - pass - - else: - proofLine += (line[ind]) - if "\\end{equation}" in lines[i + 1]: - proof = False - append_text( - comToWrite + - getEqP(proofLine).rstrip("\n") + - "
\n
\n") - comToWrite = "" - symbols = symbols + getSym(symLine) - symLine = "" - - elif math: - if "\\end{equation}" in lines[ - i + - 1] or "\\constraint" in lines[ - i + - 1] or "\\substitution" in lines[ - i + - 1] or "\\proof" in lines[ - i + - 1] or "\\drmfnote" in lines[ - i + - 1] or "\\drmfname" in lines[ - i + - 1]: - append_text(line.rstrip("\n")) - symLine += line.strip("\n") - symbols = symbols + getSym(symLine) - symLine = "" - append_text("\n
\n") - else: - symLine += line.strip("\n") - append_text(line) - if note and parse: - noteLine = noteLine + line - symbols = symbols + getSym(line) - if "\\end{equation}" in lines[ - i + - 1] or "\\drmfn" in lines[ - i + - 1] or "\\constraint" in lines[ - i + - 1] or "\\substitution" in lines[ - i + - 1] or "\\proof" in lines[ - i + - 1]: - note = False - if "\\emph" in noteLine: - noteLine = noteLine[ - 0:noteLine.find("\\emph{")] + "\'\'" + noteLine[ - noteLine.find("\\emph{") + len( - "\\emph{"): noteLine.find( - "}", noteLine.find("\\emph{") + len( - "\\emph{"))] + "\'\'" + noteLine[ - noteLine.find( - "}", - noteLine.find( - "\\emph{") + len( - "\\emph{")) + 1:] - comToWrite = comToWrite + "
" + \ - getEq(noteLine) + "

\n" - - if constraint and parse: - conLine += line.replace("&", "&
") - - symLine += line.strip("\n") - # symbols=symbols+getSym(line) - if "\\end{equation}" in lines[ - i + - 1] or "\\drmfn" in lines[ - i + - 1] or "\\constraint" in lines[ - i + - 1] or "\\substitution" in lines[ - i + - 1] or "\\proof" in lines[ - i + - 1]: - constraint = False - symbols = symbols + getSym(symLine) - symLine = "" - append_text( - comToWrite + - "
" + - getEq(conLine) + - "

\n") - comToWrite = "" - if substitution and parse: - subLine = subLine + line.replace("&", "&
") - - symLine += line.strip("\n") - # symbols=symbols+getSym(line) - if "\\end{equation}" in lines[ - i + - 1] or "\\drmfn" in lines[ - i + - 1] or "\\substitution" in lines[ - i + - 1] or "\\constraint" in lines[ - i + - 1] or "\\proof" in lines[ - i + - 1]: - substitution = False - symbols = symbols + getSym(symLine) - symLine = "" - append_text( - comToWrite + - "
" + - getEq(subLine) + - "

\n") - comToWrite = "" - - -if __name__ == "__main__": - tex = "../../data/ZE.3.tex" - mainFile = "../../data/OrthogonalPolynomials.mmd" - lLinks = "../../data/BruceLabelLinks" - dlmf(tex, mainFile, lLinks, 0) diff --git a/tex2Wiki/src/OrthogonalPolynomials.mmd b/tex2Wiki/src/OrthogonalPolynomials.mmd new file mode 100644 index 0000000..cab1d14 --- /dev/null +++ b/tex2Wiki/src/OrthogonalPolynomials.mmd @@ -0,0 +1,65 @@ +drmf_bof +'''Orthogonal Polynomials''' +{{#set:Section=0}} +== Sections in KLS Chapter 1 == + +
+* [[Orthogonal polynomials|Orthogonal polynomials]] +* [[The gamma and beta function|The gamma and beta function]] +* [[The shifted factorial and binomial coefficients|The shifted factorial and binomial coefficients]] +* [[Hypergeometric functions|Hypergeometric functions]] +* [[The binomial theorem and other summation formulas|The binomial theorem and other summation formulas]] +* [[Some integrals|Some integrals]] +* [[Transformation formulas|Transformation formulas]] +* [[The q-shifted factorial|The ''q''-shifted factorial]] +* [[The q-gamma function and q-binomial coefficients|The ''q''-gamma function and ''q''-binomial coefficients]] +* [[Basic hypergeometric functions|Basic hypergeometric functions]] +* [[The q-binomial theorem and other summation formulas|The ''q''-binomial theorem and other summation formulas]] +* [[More integrals|More integrals]] +* [[Transformation formulas|Transformation formulas]] +* [[Some q-analogues of special functions|Some ''q''-analogues of special functions]] +* [[The q-derivative and q-integral|The ''q''-derivative and ''q''-integral]] +* [[Shift operators and Rodrigues-type formulas|Shift operators and Rodrigues-type formulas]] +
+== Sections in KLS Chapter 1 == + +
+* [[Orthogonal polynomials|Orthogonal polynomials]] +* [[The gamma and beta function|The gamma and beta function]] +* [[The shifted factorial and binomial coefficients|The shifted factorial and binomial coefficients]] +* [[Hypergeometric functions|Hypergeometric functions]] +* [[The binomial theorem and other summation formulas|The binomial theorem and other summation formulas]] +* [[Some integrals|Some integrals]] +* [[Transformation formulas|Transformation formulas]] +* [[The q-shifted factorial|The ''q''-shifted factorial]] +* [[The q-gamma function and q-binomial coefficients|The ''q''-gamma function and ''q''-binomial coefficients]] +* [[Basic hypergeometric functions|Basic hypergeometric functions]] +* [[The q-binomial theorem and other summation formulas|The ''q''-binomial theorem and other summation formulas]] +* [[More integrals|More integrals]] +* [[Transformation formulas|Transformation formulas]] +* [[Some q-analogues of special functions|Some ''q''-analogues of special functions]] +* [[The q-derivative and q-integral|The ''q''-derivative and ''q''-integral]] +* [[Shift operators and Rodrigues-type formulas|Shift operators and Rodrigues-type formulas]] +
+== Sections in KLS Chapter 1 == + +
+* [[Orthogonal polynomials|Orthogonal polynomials]] +* [[The gamma and beta function|The gamma and beta function]] +* [[The shifted factorial and binomial coefficients|The shifted factorial and binomial coefficients]] +* [[Hypergeometric functions|Hypergeometric functions]] +* [[The binomial theorem and other summation formulas|The binomial theorem and other summation formulas]] +* [[Some integrals|Some integrals]] +* [[Transformation formulas|Transformation formulas]] +* [[The q-shifted factorial|The ''q''-shifted factorial]] +* [[The q-gamma function and q-binomial coefficients|The ''q''-gamma function and ''q''-binomial coefficients]] +* [[Basic hypergeometric functions|Basic hypergeometric functions]] +* [[The q-binomial theorem and other summation formulas|The ''q''-binomial theorem and other summation formulas]] +* [[More integrals|More integrals]] +* [[Transformation formulas|Transformation formulas]] +* [[Some q-analogues of special functions|Some ''q''-analogues of special functions]] +* [[The q-derivative and q-integral|The ''q''-derivative and ''q''-integral]] +* [[Shift operators and Rodrigues-type formulas|Shift operators and Rodrigues-type formulas]] +
+ +drmf_eof diff --git a/tex2Wiki/src/OrthogonalPolynomials.mmd.new b/tex2Wiki/src/OrthogonalPolynomials.mmd.new new file mode 100644 index 0000000..cab1d14 --- /dev/null +++ b/tex2Wiki/src/OrthogonalPolynomials.mmd.new @@ -0,0 +1,65 @@ +drmf_bof +'''Orthogonal Polynomials''' +{{#set:Section=0}} +== Sections in KLS Chapter 1 == + +
+* [[Orthogonal polynomials|Orthogonal polynomials]] +* [[The gamma and beta function|The gamma and beta function]] +* [[The shifted factorial and binomial coefficients|The shifted factorial and binomial coefficients]] +* [[Hypergeometric functions|Hypergeometric functions]] +* [[The binomial theorem and other summation formulas|The binomial theorem and other summation formulas]] +* [[Some integrals|Some integrals]] +* [[Transformation formulas|Transformation formulas]] +* [[The q-shifted factorial|The ''q''-shifted factorial]] +* [[The q-gamma function and q-binomial coefficients|The ''q''-gamma function and ''q''-binomial coefficients]] +* [[Basic hypergeometric functions|Basic hypergeometric functions]] +* [[The q-binomial theorem and other summation formulas|The ''q''-binomial theorem and other summation formulas]] +* [[More integrals|More integrals]] +* [[Transformation formulas|Transformation formulas]] +* [[Some q-analogues of special functions|Some ''q''-analogues of special functions]] +* [[The q-derivative and q-integral|The ''q''-derivative and ''q''-integral]] +* [[Shift operators and Rodrigues-type formulas|Shift operators and Rodrigues-type formulas]] +
+== Sections in KLS Chapter 1 == + +
+* [[Orthogonal polynomials|Orthogonal polynomials]] +* [[The gamma and beta function|The gamma and beta function]] +* [[The shifted factorial and binomial coefficients|The shifted factorial and binomial coefficients]] +* [[Hypergeometric functions|Hypergeometric functions]] +* [[The binomial theorem and other summation formulas|The binomial theorem and other summation formulas]] +* [[Some integrals|Some integrals]] +* [[Transformation formulas|Transformation formulas]] +* [[The q-shifted factorial|The ''q''-shifted factorial]] +* [[The q-gamma function and q-binomial coefficients|The ''q''-gamma function and ''q''-binomial coefficients]] +* [[Basic hypergeometric functions|Basic hypergeometric functions]] +* [[The q-binomial theorem and other summation formulas|The ''q''-binomial theorem and other summation formulas]] +* [[More integrals|More integrals]] +* [[Transformation formulas|Transformation formulas]] +* [[Some q-analogues of special functions|Some ''q''-analogues of special functions]] +* [[The q-derivative and q-integral|The ''q''-derivative and ''q''-integral]] +* [[Shift operators and Rodrigues-type formulas|Shift operators and Rodrigues-type formulas]] +
+== Sections in KLS Chapter 1 == + +
+* [[Orthogonal polynomials|Orthogonal polynomials]] +* [[The gamma and beta function|The gamma and beta function]] +* [[The shifted factorial and binomial coefficients|The shifted factorial and binomial coefficients]] +* [[Hypergeometric functions|Hypergeometric functions]] +* [[The binomial theorem and other summation formulas|The binomial theorem and other summation formulas]] +* [[Some integrals|Some integrals]] +* [[Transformation formulas|Transformation formulas]] +* [[The q-shifted factorial|The ''q''-shifted factorial]] +* [[The q-gamma function and q-binomial coefficients|The ''q''-gamma function and ''q''-binomial coefficients]] +* [[Basic hypergeometric functions|Basic hypergeometric functions]] +* [[The q-binomial theorem and other summation formulas|The ''q''-binomial theorem and other summation formulas]] +* [[More integrals|More integrals]] +* [[Transformation formulas|Transformation formulas]] +* [[Some q-analogues of special functions|Some ''q''-analogues of special functions]] +* [[The q-derivative and q-integral|The ''q''-derivative and ''q''-integral]] +* [[Shift operators and Rodrigues-type formulas|Shift operators and Rodrigues-type formulas]] +
+ +drmf_eof diff --git a/tex2Wiki/src/__init__.py b/tex2Wiki/src/__init__.py new file mode 100644 index 0000000..e69de29 diff --git a/tex2Wiki/src/testData.txt b/tex2Wiki/src/testData.txt new file mode 100644 index 0000000..e69de29 diff --git a/tex2Wiki/src/tex2Wiki.py b/tex2Wiki/src/tex2Wiki.py index 8140140..9d137bb 100644 --- a/tex2Wiki/src/tex2Wiki.py +++ b/tex2Wiki/src/tex2Wiki.py @@ -1,1356 +1,1080 @@ -__author__ = "Azeem Mohammed" -__status__ = "Development" - -# Version 15: Minor Fixes -# Convert tex to wikiText -import csv # imported for using csv format -import sys # imported for getting args -from shutil import copyfile -import xml.etree.ElementTree as ET -import datetime - -wiki = '' -next_formula_number = 0 -lLink = '' -ET.register_namespace('', 'http://www.mediawiki.org/xml/export-0.10/') -root = ET.Element('{http://www.mediawiki.org/xml/export-0.10/}mediawiki') - - -def is_number(char): # Function to check if char is a number (assuming 1 character) - return char[0].isdigit() - - -def get_string_within_curly_braces(line): - """Gets all data within curly braces on a line""" - # TODO: Possibly have to fix for nested curly braces - - string_within_braces = "" - get_str = False - prev_char = "" - - for char in line: - if char in ["{", "}"]: - get_str = not get_str - if char == "}": - break - # If within curly braces - elif get_str: - # If there is no double dash (makes single dash) - if not (prev_char == char and char == "-"): - if char != "$": - # this character is part of the data - string_within_braces += char - else: - # Add double single quote instead of $ - string_within_braces += "''" - # Change last character for finding double dashes - prev_char = char - # Return the data without newlines and without leading spaces - return string_within_braces.rstrip('\n').lstrip() - - -def get_symbol_list_eq(line): - """Gets equation for symbols list""" - start = True - final = "" - for c in line: - if c == "$" and start: - final += " " - start = False - elif c == "$": - final += "" - start = True - else: - final += c - return final - - -def get_equation(line, proof=False): - """Gets all data within constraints, substitutions""" - per = 1 - string_write = "" - end_of_equation = False - count = 0 - length = 0 - # TODO: figure out what count and per do -_- - - # read each character - for char in line: - if proof and end_of_equation and char not in [" ", "$", "{", "}"] and not char.isalpha(): - length += 1 - if count >= 0 and per != 0: - per += 1 - if char != " " and char != "%": - per = 0 - if char == "{" and per == 0: - if count > 0: - string_write += char - count += 1 - elif char == "}" and per == 0: - count -= 1 - if count > 0: - string_write += char - # either begin or end equation - elif char == "$" and per == 0: - # toggle end_of_equation flag to know if begin or end equation - end_of_equation = not end_of_equation - # if begin - if end_of_equation: - string_write += "" - else: - if proof and length >= 10: - string_write += "
" - else: - string_write += "" - length = 0 - # if newline - elif char == "\n" and per == 0: - string_write = string_write.strip() - - # should above be - # rstrip?<-------------------------------------------------------------------CHECK - # THIS - - string_write += char - per += 1 # watch for % signs - elif count > 0 and per == 0: # not special character - string_write += char # write the character - - return string_write.strip() - - -def get_proofs_data(line): - """Gets all data within proofs""" - per = 1 - stringWrite = "" - fEq = False - count = 0 - length = 0 - for c in line: - if fEq and c != " " and c != "$" and c != "{" and c != "}" and not c.isalpha(): - length += 1 - if count >= 0 and per != 0: - per += 1 - if c != " " and c != "%": - # if c!="%": - per = 0 - if c == "{" and per == 0: - if count > 0: - stringWrite += c - count += 1 - elif c == "}" and per == 0: - count -= 1 - if count > 0: - stringWrite += c - elif c == "$" and per == 0: - fEq = not fEq - if fEq: - stringWrite += " " - else: - if length < 10: - stringWrite += "" - else: - stringWrite += "" + "
" - length = 0 - elif c == "\n" and per == 0: - stringWrite = stringWrite.strip() - stringWrite += c - per += 1 - elif count > 0 and per == 0: - stringWrite += c - - return stringWrite.lstrip() - - -def getSym(line): # Gets all symbols on a line for symbols list - symList = [] - if line == "": - return symList - symbol = "" - symFlag = False - argFlag = False - cC = 0 - for i in range(0, len(line)): - # if - # "MeixnerPollaczek{\\lambda}{n}@{x}{\\phi}=\frac{\\pochhammer{2\\lambda}{n}}" - # in line: - c = line[i] - if symFlag: - if c == "{" or c == "[": - cC += 1 - argFlag = True - if c != "}" and c != "]": - if argFlag or c.isalpha(): - symbol += c - else: - symFlag = False - argFlag = False - symList.append(symbol) - symList += (getSym(symbol)) - symbol = "" - else: - cC -= 1 - symbol += c - if i + 1 == len(line): - p = "" - else: - p = line[i + 1] - if cC <= 0 and p != "{" and p != "[" and p != "@": - symFlag = False - # if - # "monicAskeyWilson{n+1}@{x}{a}{b}{c}{d}{q}+\\frac{1}{2}" - # in line: - symList.append(symbol) - symList += (getSym(symbol)) - argFlag = False - symbol = "" - - elif c == "\\": - symFlag = True - elif c == "&" and not (i > line.find("\\begin{array}") and i < line.find("\\end{array}")): - symList.append("&") - - # if - # "MeixnerPollaczek{\\lambda}{n}@{x}{\\phi}=\frac{\\pochhammer{2\\lambda}{n}}" - # in line: - symList.append(symbol) - symList += getSym(symbol) - return symList - - -def unmodLabel(label): - label = label.replace(".0", ".") - label = label.replace(":0", ":") - return label - - -def secLabel(label): - return label.replace("\'\'", "") - - -def modLabel(line): - global lLink - start_label = line.find("\\formula{") - if start_label > 0: - start_label += len("\\formula{") - else: - start_label = line.find("\\label{") - if start_label > 0: - start_label += len("\\label{") - else: - global next_formula_number - next_formula_number += 1 - return 'auto-number-' + str(next_formula_number) - end_label = line.find("}", start_label) - label = line[start_label:end_label] - rlabel = label - for l in lLink: - if l.find(label) != -1 and l[len(label) + 1] == "=": - rlabel = l[l.find("=>") + 3:l.find("\\n")] - rlabel = rlabel.replace("/", "") - rlabel = rlabel.replace("#", ":") - break - label = rlabel - label = label.replace('eq:', 'Formula:') - isNumer = False - newlabel = "" - num = "" - for i in range(0, len(label)): - if isNumer and not is_number(label[i]): - if len(num) > 1: - newlabel += num - num = "" - else: - newlabel += "0" + str(num) - num = "" - if is_number(label[i]): - isNumer = True - num += str(label[i]) - else: - isNumer = False - newlabel += label[i] - if len(num) > 1: - newlabel += num - elif len(num) == 1: - newlabel += "0" + num - return newlabel - - -def append_text(text): - global wiki - wiki.text = wiki.text + text - - -def append_revision(param): - global wiki - page = ET.SubElement(root, 'page') - title = ET.SubElement(page, 'title') - title.text = param - revision = ET.SubElement(page, 'revision') - timestamp = ET.SubElement(revision, 'timestamp') - timestamp.text = str(datetime.datetime.utcnow()) - contributor = ET.SubElement(revision, 'contributor') - username = ET.SubElement(contributor, 'username') - username.text = 'SeedBot' - wiki = ET.SubElement(revision, 'text') - wiki.text = '' - - -def writeout(ofname): - tree = ET.ElementTree(root) - tree.write(ofname, xml_declaration=True, encoding='utf-8', method='xml') - - -def setup_label_links(ofname): - global lLink - lLink = open(ofname, "r").readlines() - - -def readin(ofname, glossary, mmd): - # try: - for iterations in range(0, 1): - tex = open(ofname, 'r') - main_file = open(mmd, "r") - mainText = main_file.read() - mainPrepend = "" - mainWrite = open("OrthogonalPolynomials.mmd.new", "w") - glossary = open('new.Glossary.csv', 'rb') - math = False - constraint = False - substitution = False - symbols = [] - lines = tex.readlines() - refLines = [] - sections = [] - labels = [] - eqs = [] - refEqs = [] - parse = False - head = False - refLabels = [] - chapter = '' - for i in range(0, len(refLines)): - line = refLines[i] - if "\\begin{equation}" in line: - label = modLabel(line) - refLabels.append(label) - refEqs.append("") - math = True - elif math: - refEqs[-1] += line - if "\\end{equation}" in refLines[ - i + - 1] or "\\constraint" in refLines[ - i + - 1] or "\\substitution" in refLines[ - i + - 1] or "\\drmfn" in refLines[ - i + - 1]: - math = False - - for i in range(0, len(lines)): - line = lines[i] - if "\\begin{document}" in line: - parse = True - elif "\\end{document}" in line and parse: - mainPrepend += "
\n" - mainText = mainPrepend + mainText - mainText = mainText.replace( - "\'\'\'Orthogonal Polynomials\'\'\'\n", "") - mainText = mainText.replace("{{#set:Section=0}}\n", "") - mainText = mainText[0:mainText.rfind("== Sections ")] - append_revision('Orthogonal Polynomials') - mainText = "{{#set:Section=0}}\n" + mainText - mainWrite.write(mainText) - mainWrite.close() - main_file.close() - copyfile(mmd, 'OrthogonalPolynomials.mmd.new') - parse = False - elif "\\title" in line and parse: - stringWrite = "\'\'\'" - stringWrite += get_string_within_curly_braces(line) + "\'\'\'\n" - labels.append("Orthogonal Polynomials") - sections.append(["Orthogonal Polynomials", 0]) - chapter = get_string_within_curly_braces(line) - mainPrepend += ( - "\n== Sections in " + - chapter + - " ==\n\n
\n") - elif "\\part" in line: - if get_string_within_curly_braces(line) == "BOF": - parse = False - elif get_string_within_curly_braces(line) == "EOF": - parse = True - elif parse: - mainPrepend += ("\n
\n= " + get_string_within_curly_braces(line) + " =\n") - head = True - elif "\\section" in line: - mainPrepend += ("* [[" + - secLabel(get_string_within_curly_braces(line)) + - "|" + - get_string_within_curly_braces(line) + - "]]\n") - sections.append([get_string_within_curly_braces(line)]) - - secCounter = 0 - eqCounter = 0 - subLine = '' - conLine = '' - for i in range(0, len(lines)): - line = lines[i] - if "\\section" in line: - parse = True - secCounter += 1 - append_revision(secLabel(get_string_within_curly_braces(line))) - append_text( - "{{DISPLAYTITLE:" + (sections[secCounter][0]) + "}}\n") - append_text("{{#set:Chapter=" + chapter + "}}\n") - append_text("{{#set:Section=" + str(secCounter) + "}}\n") - append_text("{{headSection}}\n") - head = True - append_text("== " + get_string_within_curly_braces(line) + " ==\n") - elif ("\\section" in lines[(i + 1) % len(lines)] or "\\end{document}" in lines[ - (i + 1) % len(lines)]) and parse: - append_text("{{footSection}}\n") - sections[secCounter].append(eqCounter) - eqCounter = 0 - - elif "\\subsection" in line and parse: - append_text("\n== " + get_string_within_curly_braces(line) + " ==\n") - head = True - elif "\\paragraph" in line and parse: - append_text("\n=== " + get_string_within_curly_braces(line) + " ===\n") - head = True - elif "\\subsubsection" in line and parse: - append_text("\n=== " + get_string_within_curly_braces(line) + " ===\n") - head = True - - elif "\\begin{equation}" in line and parse: - if head: - append_text("\n") - head = False - label = modLabel(line) - eqCounter += 1 - labels.append(label) - eqs.append("") - append_text("") - math = True - elif "\\begin{equation}" in line and not parse: - label = modLabel(line) - labels.append("*" + label) # special marker - eqs.append("") - math = True - elif "\\end{equation}" in line: - - math = False - elif "\\constraint" in line and parse: - constraint = True - math = False - conLine = "" - elif "\\substitution" in line and parse: - substitution = True - math = False - subLine = "" - elif "\\proof" in line and parse: - math = False - elif "\\drmfn" in line and parse: - math = False - if "\\drmfname" in line and parse: - append_text( - "
This formula has the name: " + - get_string_within_curly_braces(line) + - "

\n") - elif math and parse: - flagM = True - eqs[-1] += line - - if "\\end{equation}" in lines[ - i + - 1] and "\\subsection" not in lines[ - i + - 3] and "\\section" not in lines[ - i + - 3] and "\\part" not in lines[ - i + - 3]: - u = i - flagM2 = False - while flagM: - u += 1 - if "\\begin{equation}" in lines[u] in lines[u]: - flagM = False - if "\\section" in lines[u] or "\\subsection" in lines[ - i] or "\\part" in lines[u] or "\\end{document}" in lines[u]: - flagM = False - flagM2 = True - if not flagM2: - - append_text(line.rstrip("\n")) - append_text("\n
\n") - else: - append_text(line.rstrip("\n")) - append_text("\n\n") - elif "\\end{equation}" in lines[i + 1]: - append_text(line.rstrip("\n")) - append_text("\n\n") - elif "\\constraint" in lines[i + 1] or "\\substitution" in lines[i + 1] or "\\drmfn" in lines[i + 1]: - append_text(line.rstrip("\n")) - append_text("\n\n") - else: - append_text(line) - elif math and not parse: - eqs[-1] += line - if "\\end{equation}" in lines[ - i + - 1] or "\\constraint" in lines[ - i + - 1] or "\\substitution" in lines[ - i + - 1] or "\\drmfn" in lines[ - i + - 1]: - math = False - if substitution and parse: - subLine += line.replace("&", "&
") - if "\\end{equation}" in lines[ - i + - 1] or "\\substitution" in lines[ - i + - 1] or "\\constraint" in lines[ - i + - 1] or "\\drmfn" in lines[ - i + - 1] or "\\proof" in lines[ - i + - 1]: - lineR = "" - for i in range(0, len(subLine)): - if subLine[i] == "&" and not ( - i > subLine.find("\\begin{array}") and i < subLine.find("\\end{array}")): - lineR += "&
" - else: - lineR += subLine[i] - substitution = False - append_text( - "
Substitution(s): " + - get_equation(subLine) + - "

\n") - - if constraint and parse: - conLine += line.replace("&", "&
") - if "\\end{equation}" in lines[ - i + - 1] or "\\substitution" in lines[ - i + - 1] or "\\constraint" in lines[ - i + - 1] or "\\drmfn" in lines[ - i + - 1] or "\\proof" in lines[ - i + - 1]: - constraint = False - append_text( - "
Constraint(s): " + - get_equation(conLine) + - "

\n") - - eqCounter = 0 - endNum = len(labels) - 1 - parse = False - constraint = False - substitution = False - note = False - hCon = True - hSub = True - hNote = True - hProof = True - proof = False - comToWrite = "" - secCount = -1 - newSec = False - symLine = '' - eqS = '' - noteLine = '' - proofLine = '' - pause = False - for i in range(0, len(lines)): - line = lines[i] - - if "\\section" in line: - secCount += 1 - newSec = True - eqS = 0 - - if "\\begin{equation}" in line: - symLine = line.strip("\n") - eqS += 1 - constraint = False - substitution = False - note = False - comToWrite = "" - hCon = True - hSub = True - hNote = True - hProof = True - proof = False - parse = True - symbols = [] - eqCounter += 1 - label = labels[eqCounter] - append_revision(secLabel(label)) - append_text("{{DISPLAYTITLE:" + (labels[eqCounter]) + "}}\n") - if eqCounter < endNum: # FOR ANYTHING THAT IS NOT THE EXTRA EQUATIONS - append_text("
\n") - if newSec: - append_text( - "
<< [[" + - secLabel( - sections[secCount][0]).replace( - " ", - "_") + - "|" + - secLabel( - sections[secCount][0]) + - "]]
\n") - else: - append_text( - "
<< [[" + - secLabel( - labels[ - eqCounter - - 1]).replace( - " ", - "_") + - "|" + - secLabel( - labels[ - eqCounter - - 1]) + - "]]
\n") - append_text( - "
[[" + - secLabel( - sections[ - secCount + - 1][0]).replace( - " ", - "_") + - "#" + - secLabel( - labels[eqCounter][ - len("Formula:"):]) + - "|formula in " + - secLabel( - sections[ - secCount + - 1][0]) + - "]]
\n") - if eqS == sections[secCount][1]: - append_text( - "
[[" + - secLabel( - sections[ - (secCount + - 1) % - len(sections)][0]).replace( - " ", - "_") + - "|" + - secLabel( - sections[ - (secCount + - 1) % - len(sections)][0]) + - "]] >>
\n") - else: - append_text( - "
[[" + - secLabel( - labels[ - (eqCounter + - 1) % - (endNum + - 1)]).replace( - " ", - "_") + - "|" + - secLabel( - labels[ - (eqCounter + - 1) % - (endNum + - 1)]) + - "]] >>
\n") - append_text("
\n\n") - elif eqCounter == endNum: - append_text("
\n") - if newSec: - newSec = False - append_text( - "
<< [[" + - secLabel( - sections[secCount][0]).replace( - " ", - "_") + - "|" + - secLabel( - sections[secCount][0]) + - "]]
\n") - else: - append_text( - "
<< [[" + - secLabel( - labels[ - eqCounter - - 1]).replace( - " ", - "_") + - "|" + - secLabel( - labels[ - eqCounter - - 1]) + - "]]
\n") - append_text( - "
[[" + - secLabel( - sections[ - secCount + - 1][0]).replace( - " ", - "_") + - "#" + - secLabel( - labels[eqCounter][ - len("Formula:"):]) + - "|formula in " + - secLabel( - sections[ - secCount + - 1][0]) + - "]]
\n") - append_text( - "
[[" + - secLabel( - labels[ - (eqCounter + - 1) % - (endNum + - 1)].replace( - " ", - "_")) + - "|" + - secLabel( - labels[ - (eqCounter + - 1) % - (endNum + - 1)]) + - "]]
\n") - append_text("
\n\n") - - append_text("
\n") - math = True - elif "\\end{equation}" in line: - append_text(comToWrite) - parse = False - math = False - if hProof: - append_text( - "\n== Proof ==\n\nWe ask users to provide proof(s), reference(s) to proof(s), or" - " further clarification on the proof(s) in this space.\n") - append_text("\n== Symbols List ==\n\n") - newSym = [] - # if "09.07:04" in label: - for x in symbols: - flagA = True - # if x not in newSym: - cN = 0 - cC = 0 - flag = True - ArgCx = 0 - for z in x: - if z.isalpha() and flag or z == "&": - cN += 1 - else: - flag = False - if z == "{" or z == "[": - cC += 1 - if z == "}" or z == "]": - cC -= 1 - if cC == 0: - ArgCx += 1 - noA = x[:cN] - for y in newSym: - cN = 0 - cC = 0 - ArgC = 0 - flag = True - for z in y: - if z.isalpha() and flag or z == "&": - cN += 1 - else: - flag = False - if z == "{" or z == "[": - cC += 1 - if z == "}" or z == "]": - cC -= 1 - if cC == 0: - ArgC += 1 - - if y[:cN] == noA: # and ArgC==ArgCx: - flagA = False - break - if flagA: - newSym.append(x) - newSym.reverse() - ampFlag = False - finSym = [] - for s in range(len(newSym) - 1, -1, -1): - symbolPar = "\\" + newSym[s] - ArgCx = 0 - parCx = 0 - parFlag = False - cC = 0 - cN = 0 - for z in symbolPar: - if z == "@": - parFlag = True - elif z.isalpha() or z == "&": - cN += 1 - else: - if z == "{" or z == "[": - cC += 1 - if z == "}" or z == "]": - cC -= 1 - if cC == 0: - if parFlag: - parCx += 1 - else: - ArgCx += 1 - - if symbolPar.find("{") != -1 or symbolPar.find("[") != -1: - if symbolPar.find("[") == -1: - symbol = symbolPar[0:symbolPar.find("{")] - elif symbolPar.find("{") == -1 or symbolPar.find("[") < symbolPar.find("{"): - symbol = symbolPar[0:symbolPar.find("[")] - else: - symbol = symbolPar[0:symbolPar.find("{")] - else: - symbol = symbolPar - gFlag = False - checkFlag = False - get = False - gCSV = csv.reader( - open( - glossary, - 'rb'), - delimiter=',', - quotechar='\"') - preG = "" - if symbol == "\\&": - ampFlag = True - for S in gCSV: - G = S - ArgCx = 0 - parCx = 0 - parFlag = False - cC = 0 - ind = G[0].find("@") - if ind == -1: - ind = len(G[0]) - 1 - for z in G[0]: - if z == "@": - parFlag = True - elif z.isalpha(): - cN += 1 - else: - if z == "{" or z == "[": - cC += 1 - if z == "}" or z == "]": - cC -= 1 - if cC == 0: - if parFlag: - parCx += 1 - else: - ArgCx += 1 - if G[0].find(symbol) == 0 and (len(G[0]) == len( - symbol) or not G[0][len(symbol)].isalpha()): - checkFlag = True - get = True - preG = S - elif checkFlag: - get = True - checkFlag = False - if get: - if get: - G = preG - if True: - if symbolPar.find("@") != -1: - Q = symbolPar[:symbolPar.find("@")] - else: - Q = symbolPar - listArgs = [] - if len(Q) > len(symbol) and (Q[ - len(symbol)] == "{" or Q[len(symbol)] == "["): - ap = "" - for o in range(len(symbol), len(Q)): - if Q[o] == "}" or z == "]": - listArgs.append(ap) - ap = "" - else: - ap += Q[o] - websiteF = "" - web1 = G[5] - for t in range(5, len(G)): - if G[t] != "": - websiteF = websiteF + \ - " [" + G[t] + " " + G[t] + "]" - p1 = G[4].strip("$") - p1 = "" + p1 + "" - # if checkFlag: - new2 = "" - pause = False - mathF = True - p2 = G[1] - for k in range(0, len(p2)): - if p2[k] == "$": - if mathF: - new2 += " " - else: - new2 += "" - mathF = not mathF - else: - new2 += p2[k] - p2 = new2 - finSym.append( - web1 + " " + p1 + "] : " + p2 + " :" + websiteF) - break - if not gFlag: - del newSym[s] - - gFlag = True - if ampFlag: - append_text("& : logical and") - gFlag = False - for y in finSym: - if y == "& : logical and": - pass - elif gFlag: - gFlag = False - append_text("[" + y) - else: - append_text("
\n[" + y) - - append_text("\n
\n") - - # should there be a space between bibliography and ==? - append_text("\n== Bibliography==\n\n") - r = unmodLabel(labels[eqCounter]) - q = r.find("KLS:") + 4 - p = r.find(":", q) - section = r[q:p] - equation = r[p + 1:] - if equation.find(":") != -1: - equation = equation[0:equation.find(":")] - append_text( - "[http://homepage.tudelft.nl/11r49/askey/contents.html " - "Equation in Section " + - section + - "] of [[Bibliography#KLS|'''KLS''']].\n\n") - append_text( - "== URL links ==\n\nWe ask users to provide relevant URL links in this space.\n\n") - if eqCounter < endNum: - append_text("
\n") - if newSec: - newSec = False - append_text( - "
<< [[" + - secLabel( - sections[secCount][0]).replace( - " ", - "_") + - "|" + - secLabel( - sections[secCount][0]) + - "]]
\n") - else: - append_text( - "
<< [[" + - secLabel( - labels[ - eqCounter - - 1]).replace( - " ", - "_") + - "|" + - secLabel( - labels[ - eqCounter - - 1]) + - "]]
\n") - append_text( - "
[[" + - secLabel( - sections[ - secCount + - 1][0]).replace( - " ", - "_") + - "#" + - secLabel( - labels[eqCounter][ - len("Formula:"):]) + - "|formula in " + - secLabel( - sections[ - secCount + - 1][0]) + - "]]
\n") - if eqS == sections[secCount][1]: - append_text( - "
[[" + - sections[ - (secCount + - 1) % - len(sections)][0].replace( - " ", - "_") + - "|" + - sections[ - (secCount + - 1) % - len(sections)][0] + - "]] >>
\n") - else: - append_text( - "
[[" + - secLabel( - labels[ - (eqCounter + - 1) % - (endNum + - 1)]).replace( - " ", - "_") + - "|" + - secLabel( - labels[ - (eqCounter + - 1) % - (endNum + - 1)]) + - "]] >>
\n") - append_text("
\n") - else: # FOR EXTRA EQUATIONS - append_text("
\n") - append_text( - "
<< [[" + - labels[ - endNum - - 1].replace( - " ", - "_") + - "|" + - labels[ - endNum - - 1] + - "]]
\n") - append_text( - "
[[" + - labels[0].replace( - " ", - "_") + - "#" + - labels[endNum][ - 8:] + - "|formula in " + - labels[0] + - "]]
\n") - append_text( - "
[[" + - labels[ - 0 % - endNum].replace( - " ", - "_") + - "|" + - labels[ - 0 % - endNum] + - "]]
\n") - append_text("
\n") - elif "\\constraint" in line and parse: - # symbols=symbols+getSym(line) - symLine = line.strip("\n") - if hCon: - comToWrite += "\n== Constraint(s) ==\n\n" - hCon = False - constraint = True - math = False - conLine = "" - elif "\\substitution" in line and parse: - # symbols=symbols+getSym(line) - symLine = line.strip("\n") - if hSub: - comToWrite += "\n== Substitution(s) ==\n\n" - hSub = False - substitution = True - math = False - subLine = "" - elif "\\drmfname" in line and parse: - math = False - comToWrite = "\n== Name ==\n\n
" + \ - get_string_within_curly_braces(line) + "

\n" + comToWrite - - elif "\\drmfnote" in line and parse: - symbols = symbols + getSym(line) - if hNote: - comToWrite += "\n== Note(s) ==\n\n" - hNote = False - note = True - math = False - noteLine = "" - - elif "\\proof" in line and parse: - # symbols=symbols+getSym(line) - symLine = line.strip("\n") - if hProof: - hProof = False - comToWrite += "\n== Proof ==\n\nWe ask users to provide proof(s), reference(s) to proof(s), or " \ - "further clarification on the proof(s) in this space. \n

\n" \ - "
" - proof = True - proofLine = "" - pause = False - pauseP = False - for ind in range(0, len(line)): - if line[ind:ind + 7] == "\\eqref{": - # TODO: figure out how eqR is defined - # rLab = getString(eqR) - pause = True - eInd = refLabels.index( - "" + label) # This should be rLab - z = line[line.find("}", ind + 7) + 1] - if z == "." or z == ",": - pauseP = True - proofLine += ("
\n" + - refEqs[eInd] + - "" + - z + - "
\n") - else: - if z == "}": - proofLine += ( - "
\n" + refEqs[ - eInd] + "
") - else: - proofLine += ("
\n \n" + - refEqs[eInd] + - "
\n") - else: - if pause: - if line[ind] == "}": - pause = False - elif pauseP: - pauseP = False - elif line[ind] == "\n" and "\\end{equation}" in lines[i + 1]: - pass - else: - proofLine += (line[ind]) - if "\\end{equation}" in lines[i + 1]: - proof = False - append_text( - comToWrite + - get_proofs_data(proofLine) + - "
\n
\n") - comToWrite = "" - symbols = symbols + getSym(symLine) - symLine = "" - - elif proof: - symLine += line.strip("\n") - pauseP = False - for ind in range(0, len(line)): - if line[ind:ind + 7] == "\\eqref{": - pause = True - # TODO: Figure out how this is used - # eqR = line[ind:line.find("}", ind) + 1] - # rLab = getString(eqR) - eInd = refLabels.index("" + label) - z = line[line.find("}", ind + 7) + 1] - if z == "." or z == ",": - pauseP = True - proofLine += ("
\n" + - refEqs[eInd] + - "" + - z + - "
\n") - else: - proofLine += ("
\n" + - refEqs[eInd] + - "
\n") - - else: - if pause: - if line[ind] == "}": - pause = False - elif pauseP: - pauseP = False - elif line[ind] == "\n" and "\\end{equation}" in lines[i + 1]: - pass - - else: - proofLine += (line[ind]) - if "\\end{equation}" in lines[i + 1]: - proof = False - append_text( - comToWrite + - get_proofs_data(proofLine).rstrip("\n") + - "
\n
\n") - comToWrite = "" - symbols = symbols + getSym(symLine) - symLine = "" - - elif math: - if "\\end{equation}" in lines[ - i + - 1] or "\\constraint" in lines[ - i + - 1] or "\\substitution" in lines[ - i + - 1] or "\\proof" in lines[ - i + - 1] or "\\drmfnote" in lines[ - i + - 1] or "\\drmfname" in lines[ - i + - 1]: - append_text(line.rstrip("\n")) - symLine += line.strip("\n") - symbols = symbols + getSym(symLine) - symLine = "" - append_text("\n
\n") - else: - symLine += line.strip("\n") - append_text(line) - if note and parse: - noteLine += line - symbols = symbols + getSym(line) - if "\\end{equation}" in lines[ - i + - 1] or "\\drmfn" in lines[ - i + - 1] or "\\constraint" in lines[ - i + - 1] or "\\substitution" in lines[ - i + - 1] or "\\proof" in lines[ - i + - 1]: - note = False - if "\\emph" in noteLine: - noteLine = noteLine[ - 0:noteLine.find("\\emph{")] + "\'\'" + noteLine[ - noteLine.find("\\emph{") + len( - "\\emph{"):noteLine.find( - "}", noteLine.find("\\emph{") + len( - "\\emph{"))] + "\'\'" + noteLine[ - noteLine.find( - "}", - noteLine.find( - "\\emph{") + len( - "\\emph{")) + 1:] - comToWrite = comToWrite + "
" + \ - get_equation(noteLine) + "

\n" - - if constraint and parse: - conLine += line.replace("&", "&
") - - symLine += line.strip("\n") - # symbols=symbols+getSym(line) - if "\\end{equation}" in lines[ - i + - 1] or "\\drmfn" in lines[ - i + - 1] or "\\constraint" in lines[ - i + - 1] or "\\substitution" in lines[ - i + - 1] or "\\proof" in lines[ - i + - 1]: - constraint = False - symbols = symbols + getSym(symLine) - symLine = "" - append_text( - comToWrite + - "
" + - get_equation(conLine) + - "

\n") - comToWrite = "" - if substitution and parse: - # TODO: Figure out if .replace is needed - subLine += line.replace("&", "&
") - - symLine += line.strip("\n") - if "\\end{equation}" in lines[ - i + - 1] or "\\drmfn" in lines[ - i + - 1] or "\\substitution" in lines[ - i + - 1] or "\\constraint" in lines[ - i + - 1] or "\\proof" in lines[ - i + - 1]: - substitution = False - symbols = symbols + getSym(symLine) - symLine = "" - lineR = "" - for i in range(0, len(subLine)): - if subLine[i] == "&" and not ( - i > subLine.find("\\begin{array}") and i < subLine.find("\\end{array}")): - lineR += "&
" - else: - lineR += subLine[i] - append_text( - comToWrite + - "
" + - get_equation(subLine) + - "

\n") - comToWrite = "" - - -def main(): - if len(sys.argv) != 6: - fname = "../../data/ZE.3.tex" - ofname = "../../data/ZE.4.xml" - lname = "../../data/BruceLabelLinks" - glossary = "../../data/new.Glossary.csv" - mmd = "../../data/OrthogonalPolynomials.mmd" - - else: - fname = sys.argv[1] - ofname = sys.argv[2] - lname = sys.argv[3] - glossary = sys.argv[4] - mmd = sys.argv[5] - setup_label_links(lname) - readin(fname, glossary, mmd) - writeout(ofname) - - -if __name__ == "__main__": - main() +__author__ = "Joon Bang" +__status__ = "Development" +__credits__ = ["Joon Bang", "Azeem Mohammed"] + +# Version 0.2.0: Rewrite +# Convert tex to wikiText +import csv +import os +import sys + + +def get_data_str(string): + # (str) -> str + """Gets all characters from input string that are in between curly braces.""" + + left = string.find("{") + 1 + right = string.find("}", left) + text = string[left:right].replace("$", "''").lstrip().rstrip("\n") + + return text + + +def getG(line): # gets equation for symbols list + start = True + final = "" + for ch in line: + if ch == "$" and start: + final += "{\\displaystyle " + start = False + elif ch == "$": + final += "}" + start = True + else: + final += ch + + return final + + +def getEq(line): # Gets all data within constraints,substitutions + per = 1 + stringWrite = "" + fEq = False + count = 0 + + for ch in line: # read each character + if count >= 0 and per != 0: + per += 1 + if ch != " " and ch != "%": + per = 0 + + if ch == "{" and per == 0: + if count > 0: + stringWrite += ch + count += 1 + + elif ch == "}" and per == 0: + count -= 1 + if count > 0: + stringWrite += ch + + elif ch == "$" and per == 0: # either begin or end equation + fEq = not (fEq) # toggle fEq flag to know if begin or end equation + if fEq: # if begin + stringWrite += "{\\displaystyle " + else: # if end + stringWrite += "}" + + elif ch == "\n" and per == 0: # if newline + stringWrite = stringWrite.strip() + "\n" + per += 1 # watch for % signs + + elif count > 0 and per == 0: # not special character + stringWrite += ch # write the character + + return stringWrite.strip() + + +def getEqP(line, Flag): # Gets all data within proofs + if Flag: + a = 1 + else: + a = 0 + per = 1 + stringWrite = "" + fEq = False + count = 0 + length = 0 + for c in line: + if fEq and c != " " and c != "$" and c != "{" and c != "}" and not c.isalpha(): + length += 1 + if count >= 0 and per != 0: + per += 1 + if c != " " and c != "%": + # if c!="%": + per = 0 + if c == "{" and per == 0: + if count > 0: + stringWrite += c + count += 1 + elif c == "}" and per == 0: + count -= 1 + if count > 0: + stringWrite += c + elif c == "$" and per == 0: + fEq = not (fEq) + if fEq: + stringWrite += "{\\displaystyle " + else: + if length < 10: + stringWrite += "}" + else: + stringWrite += "}" + "
" + length = 0 + elif c == "\n" and per == 0: + stringWrite = stringWrite.strip() + stringWrite += c + per += 1 + elif count > 0 and per == 0: + stringWrite += c + + return (stringWrite.lstrip()) + + +def getSym(line): # Gets all symbols on a line for symbols list + symList = [] + if line == "": + return symList + symbol = "" + symFlag = False + argFlag = False + cC = 0 + for i in range(0, len(line)): + # if "MeixnerPollaczek{\\lambda}{n}@{x}{\\phi}=\frac{\\pochhammer{2\\lambda}{n}}" in line: + c = line[i] + if symFlag: + if c == "{" or c == "[": + cC += 1 + argFlag = True + if c != "}" and c != "]": + if argFlag or c.isalpha(): + symbol += c + else: + symFlag = False + argFlag = False + symList.append(symbol) + symList += (getSym(symbol)) + symbol = "" + else: + cC -= 1 + symbol += c + if i + 1 == len(line): + p = "" + else: + p = line[i + 1] + if cC <= 0 and p != "{" and p != "[" and p != "@": + symFlag = False + # if "monicAskeyWilson{n+1}@{x}{a}{b}{c}{d}{q}+\\frac{1}{2}" in line: + symList.append(symbol) + symList += (getSym(symbol)) + argFlag = False + symbol = "" + + elif c == "\\": + symFlag = True + elif c == "&" and not (i > line.find("\\begin{array}") and i < line.find("\\end{array}")): + symList.append("&") + + # if "MeixnerPollaczek{\\lambda}{n}@{x}{\\phi}=\frac{\\pochhammer{2\\lambda}{n}}" in line: + symList.append(symbol) + symList += getSym(symbol) + return (symList) + + +def unmodLabel(label): + label = label.replace(".0", ".") + label = label.replace(":0", ":") + return (label) + + +def secLabel(label): + return (label.replace("\'\'", "")) + + +def modLabel(label): + # label.replace("Formula:KLS:","KLS;") + isNumer = False + newlabel = "" + num = "" + for i in range(0, len(label)): + if isNumer and not label[i][0].isdigit(): + if len(num) > 1: + newlabel += num + isNumer = False + num = "" + else: + newlabel += "0" + str(num) + num = "" + isNumer = False + if label[i][0].isdigit(): + isNumer = True + num += str(label[i]) + else: + isNumer = False + newlabel += label[i] + if len(num) > 1: + newlabel += num + elif len(num) == 1: + newlabel += "0" + num + return (newlabel) + + +def main(): + wiki = open(sys.argv[2], 'w') + with open(sys.argv[1], 'r') as tex: + lines = tex.readlines() + + with open("OrthogonalPolynomials.mmd", "r") as main: + mainText = main.read() + + mainPrepend = "" + mainWrite = open("OrthogonalPolynomials.mmd.new", "w") + math = False + constraint = False + substitution = False + parse = False + head = False + symbols = [] + refLines = [] + sections = [] + labels = [] + eqs = [] + refEqs = [] + refLabels = [] + + for i in xrange(len(refLines)): + line = refLines[i] + if "\\begin{equation}" in line: + sLabel = line.find("\\formula{") + 9 + eLabel = line.find("}", sLabel) + label = modLabel(line[sLabel:eLabel]) + '''for l in lLink: + if l.find(label)!=-1 and l[len(label)+1]=="=": + rlabel=l[l.find("=>")+3:l.find("\\n")] + rlabel=rlabel.replace("/","") + rlabel=rlabel.replace("#",":") + break''' + refLabels.append(label) + refEqs.append("") + math = True + elif math: + refEqs[len(refEqs) - 1] += line + if "\\end{equation}" in refLines[i + 1] or "\\constraint" in refLines[i + 1] or "\\substitution" in \ + refLines[i + 1] or "\\drmfn" in refLines[i + 1]: + math = False + + for i in range(0, len(lines)): + line = lines[i] + if "\\begin{document}" in line: + # wiki.write("drmf_bof\n") + parse = True + elif "\\end{document}" in line and parse: + # wiki.write("
\n") + mainPrepend += "
\n" + mainText = mainPrepend + mainText + mainText = mainText.replace("drmf_bof\n", "") + mainText = mainText.replace("drmf_eof\n", "") + mainText = mainText.replace("\'\'\'Orthogonal Polynomials\'\'\'\n", "") + mainText = mainText.replace("{{#set:Section=0}}\n", "") + mainText = mainText[0:mainText.rfind("== Sections ")] + mainText = "drmf_bof\n\'\'\'Orthogonal Polynomials\'\'\'\n{{#set:Section=0}}" + mainText + "\ndrmf_eof\n" + mainWrite.write(mainText) + mainWrite.close() + os.system("cp -f OrthogonalPolynomials.mmd.new OrthogonalPolynomials.mmd") + # wiki.write("\ndrmf_eof\n") + parse = False + elif "\\title" in line and parse: + stringWrite = "\'\'\'" + stringWrite += get_data_str(line) + "\'\'\'\n" + labels.append("Orthogonal Polynomials") + sections.append(["Orthogonal Polynomials", 0]) + # wiki.write(stringWrite) + # mainPrepend+=stringWrite + chapter = get_data_str(line) + mainPrepend += ( + "\n== Sections in " + chapter + " ==\n\n
\n") + elif "\\part" in line: + if get_data_str(line) == "BOF": + parse = False + elif get_data_str(line) == "EOF": + parse = True + elif parse: + mainPrepend += ("\n
\n= " + get_data_str(line) + " =\n") + head = True + elif "\\section" in line: + mainPrepend += ("* [[" + secLabel(get_data_str(line)) + "|" + get_data_str(line) + "]]\n") + sections.append([get_data_str(line)]) + + secCounter = 0 + eqCounter = 0 + for i in range(0, len(lines)): + line = lines[i] + # line.replace("\\begin{equation}","$") + # line.replace("\\end{equation},"$") + '''if "\\begin{document}" in line: + wiki.write("drmf_bof\n") + parse=True + elif "\\end{document}" in line and parse: + wiki.write("\ndrmf_eof\n") + parse=False + elif "\\title" in line and parse: + stringWrite="\'\'\'" + stringWrite+=getString(line)+"\'\'\'\n" + labels.append("Orthogonal Polynomials") + wiki.write(stringWrite\n) + elif "\\part" in line: + if getString(line)=="BOF": + parse=False + elif getString(line)=="EOF": + parse=True + elif parse: + wiki.write("\n
\n= "+getString(line)+" =\n") + head=True + ''' + if "\\section" in line: + parse = True + secCounter += 1 + wiki.write("drmf_bof\n") + wiki.write("\'\'\'" + secLabel(get_data_str(line)) + "\'\'\'\n") + wiki.write("{{DISPLAYTITLE:" + (sections[secCounter][0]) + "}}\n") + wiki.write("
\n") + wiki.write("
<< [[" + secLabel(sections[secCounter - 1][0]) + "|" + secLabel( + sections[secCounter - 1][0]) + "]]
\n") + # wiki.write("
[[Orthogonal_Polynomials#"+secLabel(sections[secCounter][0])+"|"+secLabel(sections[secCounter][0])+"]]
\n") + wiki.write("
[[Orthogonal_Polynomials#" + "Sections_in_" + chapter.replace(" ", + "_") + "|" + secLabel( + sections[secCounter][0]) + "]]
\n") + wiki.write( + "
[[" + secLabel(sections[(secCounter + 1) % len(sections)][0]) + "|" + secLabel( + sections[(secCounter + 1) % len(sections)][0]) + "]] >>
\n
\n\n") + # wiki.write("{{head|pre="+secLabel(sections[secCounter-1][0])+"|cur="+secLabel(sections[secCounter][0])+"|next="+secLabel(sections[(secCounter+1)%len(sections)][0])+"}}\n") + # wiki.write("{{#set:Chapter="+chapter+"}}\n") + # wiki.write("{{#set:Section="+str(secCounter)+"}}\n") + # wiki.write("{{head}}\n") + head = True + wiki.write("== " + get_data_str(line) + " ==\n") + elif ("\\section" in lines[(i + 1) % len(lines)] or "\\end{document}" in lines[(i + 1) % len(lines)]) and parse: + wiki.write("
\n") + wiki.write("
<< [[" + secLabel(sections[secCounter - 1][0]) + "|" + secLabel( + sections[secCounter - 1][0]) + "]]
\n") + # wiki.write("
[[Orthogonal_Polynomials#"+secLabel(sections[secCounter][0])+"|"+secLabel(sections[secCounter][0])+"]]
\n") + wiki.write("
[[Orthogonal_Polynomials#" + "Sections_in_" + chapter.replace(" ", + "_") + "|" + secLabel( + sections[secCounter][0]) + "]]
\n") + wiki.write( + "
[[" + secLabel(sections[(secCounter + 1) % len(sections)][0]) + "|" + secLabel( + sections[(secCounter + 1) % len(sections)][0]) + "]] >>
\n
\n\n") + # wiki.write("{{foot|pre="+secLabel(sections[secCounter-1][0])+"|cur="+secLabel(sections[secCounter][0])+"|next="+secLabel(sections[(secCounter+1)%len(sections)][0])+"}}\n") + # wiki.write("{{foot}}\n") + wiki.write("drmf_eof\n") + sections[secCounter].append(eqCounter) + eqCounter = 0 + + elif "\\subsection" in line and parse: + wiki.write("\n== " + get_data_str(line) + " ==\n") + head = True + elif "\\paragraph" in line and parse: + wiki.write("\n=== " + get_data_str(line) + " ===\n") + head = True + elif "\\subsubsection" in line and parse: + wiki.write("\n=== " + get_data_str(line) + " ===\n") + head = True + + elif "\\begin{equation}" in line and parse: + # symLine="" + if head: + wiki.write("\n") + head = False + sLabel = line.find("\\formula{") + 9 + eLabel = line.find("}", sLabel) + label = modLabel(line[sLabel:eLabel]) + eqCounter += 1 + '''for l in lLink: + if label==l[0:l.find("=")-1]: + rlabel=l[l.find("=>")+3:l.find("\\n")] + rlabel=rlabel.replace("/","") + rlabel=rlabel.replace("#",":") + rlabel=rlabel.replace("!",":") + break''' + labels.append(label) + eqs.append("") + # wiki.write("\n\n") + wiki.write("{\displaystyle\n") + math = True + elif "\\begin{equation}" in line and not parse: + sLabel = line.find("\\formula{") + 9 + eLabel = line.find("}", sLabel) + label = modLabel(line[sLabel:eLabel]) + '''for l in lLink: + if l.find(label)!=-1: + rlabel=l[l.find("=>")+3:l.find("\\n")] + rlabel=rlabel.replace("/","") + rlabel=rlabel.replace("#",":") + break''' + labels.append("*" + label) # special marker + eqs.append("") + math = True + elif "\\end{equation}" in line: + + math = False + elif "\\constraint" in line and parse: + constraint = True + math = False + conLine = "" + elif "\\substitution" in line and parse: + substitution = True + math = False + subLine = "" + # wiki.write("
Substitution(s): "+getEq(line)+"

\n") + elif "\\proof" in line and parse: + math = False + elif "\\drmfn" in line and parse: + math = False + if "\\drmfname" in line and parse: + wiki.write("
This formula has the name: " + get_data_str(line) + "

\n") + elif math and parse: + flagM = True + eqs[len(eqs) - 1] += line + + if "\\end{equation}" in lines[i + 1] and not "\\subsection" in lines[i + 3] and not "\\section" in lines[ + i + 3] and not "\\part" in lines[i + 3]: + u = i + flagM2 = False + while flagM: + u += 1 + if "\\begin{equation}" in lines[u] in lines[u]: + flagM = False + if "\\section" in lines[u] or "\\subsection" in lines[i] or "\\part" in lines[ + u] or "\\end{document}" in lines[u]: + flagM = False + flagM2 = True + if not (flagM2): + + wiki.write(line.rstrip("\n")) + wiki.write("\n}
\n") + else: + wiki.write(line.rstrip("\n")) + wiki.write("\n}\n") + elif "\\end{equation}" in lines[i + 1]: + wiki.write(line.rstrip("\n")) + wiki.write("\n}\n") + elif "\\constraint" in lines[i + 1] or "\\substitution" in lines[i + 1] or "\\drmfn" in lines[i + 1]: + wiki.write(line.rstrip("\n")) + wiki.write("\n}\n") + else: + wiki.write(line) + elif math and not parse: + eqs[len(eqs) - 1] += line + if "\\end{equation}" in lines[i + 1] or "\\constraint" in lines[i + 1] or "\\substitution" in lines[ + i + 1] or "\\drmfn" in lines[i + 1]: + math = False + if substitution and parse: + subLine = subLine + line # .replace("&","&
") + if "\\end{equation}" in lines[i + 1] or "\\substitution" in lines[i + 1] or "\\constraint" in lines[ + i + 1] or "\\drmfn" in lines[i + 1] or "\\proof" in lines[i + 1]: + lineR = "" + for i in range(0, len(subLine)): + if subLine[i] == "&" and not ( + i > subLine.find("\\begin{array}") and i < subLine.find("\\end{array}")): + lineR += "&
" + else: + lineR += subLine[i] + substitution = False + wiki.write("
Substitution(s): " + getEq(lineR) + "

\n") + + if constraint and parse: + conLine = conLine + line.replace("&", "&
") + if "\\end{equation}" in lines[i + 1] or "\\substitution" in lines[i + 1] or "\\constraint" in lines[ + i + 1] or "\\drmfn" in lines[i + 1] or "\\proof" in lines[i + 1]: + constraint = False + wiki.write("
Constraint(s): " + getEq(conLine) + "

\n") + + eqCounter = 0 + endNum = len(labels) - 1 + '''for n in range(1,len(labels)): + if n+1!=len(labels) and labels[n+1][0]=="*" and labels[n][0]!="*": + labels[n]=""+labels[n] + endNum=n + elif labels[n][0]=="*": + labels[n]=""+labels[n][1:] + else: + labels[n]=""+labels[n]''' + '''for n in range(0,len(refLabels)): + refLabels[n]=""+refLabels[n]''' + parse = False + constraint = False + substitution = False + note = False + hCon = True + hSub = True + hNote = True + hProof = True + proof = False + comToWrite = "" + secCount = -1 + newSec = False + for i in range(0, len(lines)): + line = lines[i] + + if "\\section" in line: + secCount += 1 + newSec = True + eqS = 0 + + if "\\begin{equation}" in line: + symLine = line.strip("\n") + eqS += 1 + constraint = False + substitution = False + note = False + comToWrite = "" + hCon = True + hSub = True + hNote = True + hProof = True + proof = False + parse = True + symbols = [] + eqCounter += 1 + wiki.write("drmf_bof\n") + label = labels[eqCounter] + wiki.write("\'\'\'" + secLabel(label) + "\'\'\'\n") + wiki.write("{{DISPLAYTITLE:" + (labels[eqCounter]) + "}}\n") + if eqCounter < endNum: # FOR ANYTHING THAT IS NOT THE EXTRA EQUATIONS + wiki.write("
\n") + if newSec: + wiki.write("
<< [[" + secLabel(sections[secCount][0]).replace(" ", + "_") + "|" + secLabel( + sections[secCount][0]) + "]]
\n") + else: + wiki.write("
<< [[" + secLabel(labels[eqCounter - 1]).replace(" ", + "_") + "|" + secLabel( + labels[eqCounter - 1]) + "]]
\n") + wiki.write("
[[" + secLabel(sections[secCount + 1][0]).replace(" ", + "_") + "#" + secLabel( + labels[eqCounter][len("Formula:"):]) + "|formula in " + secLabel( + sections[secCount + 1][0]) + "]]
\n") + # if eqS==sections[secCount][1]: + # wiki.write("
[["+secLabel(sections[(secCount+1)%len(sections)][0]).replace(" ","_")+"|"+secLabel(sections[(secCount+1)%len(sections)][0])+"]] >>
\n") + # else: + if True: + wiki.write( + "
[[" + secLabel(labels[(eqCounter + 1) % (endNum + 1)]).replace(" ", + "_") + "|" + secLabel( + labels[(eqCounter + 1) % (endNum + 1)]) + "]] >>
\n") + wiki.write("
\n\n") + elif eqCounter == endNum: + wiki.write("
\n") + if newSec: + newSec = False + wiki.write("
<< [[" + secLabel(sections[secCount][0]).replace(" ", + "_") + "|" + secLabel( + sections[secCount][0]) + "]]
\n") + else: + wiki.write("
<< [[" + secLabel(labels[eqCounter - 1]).replace(" ", + "_") + "|" + secLabel( + labels[eqCounter - 1]) + "]]
\n") + wiki.write("
[[" + secLabel(sections[secCount + 1][0]).replace(" ", + "_") + "#" + secLabel( + labels[eqCounter][len("Formula:"):]) + "|formula in " + secLabel( + sections[secCount + 1][0]) + "]]
\n") + wiki.write("
[[" + secLabel( + labels[(eqCounter + 1) % (endNum + 1)].replace(" ", "_")) + "|" + secLabel( + labels[(eqCounter + 1) % (endNum + 1)]) + "]]
\n") + wiki.write("
\n\n") + + wiki.write("
{\displaystyle\n") + math = True + elif "\\end{equation}" in line: + wiki.write(comToWrite) + parse = False + math = False + if hProof: + wiki.write( + "\n== Proof ==\n\nWe ask users to provide proof(s), reference(s) to proof(s), or further clarification on the proof(s) in this space.\n") + + wiki.write("\n== Symbols List ==\n\n") + newSym = [] + # if "09.07:04" in label: + for x in symbols: + flagA = True + # if x not in newSym: + cN = 0 + cC = 0 + flag = True + ArgCx = 0 + for z in x: + if z.isalpha() and flag or z == "&": + cN += 1 + else: + flag = False + if z == "{" or z == "[": + cC += 1 + if z == "}" or z == "]": + cC -= 1 + if cC == 0: + ArgCx += 1 + noA = x[:cN] + for y in newSym: + cN = 0 + cC = 0 + ArgC = 0 + flag = True + for z in y: + if z.isalpha() and flag or z == "&": + cN += 1 + else: + flag = False + if z == "{" or z == "[": + cC += 1 + if z == "}" or z == "]": + cC -= 1 + if cC == 0: + ArgC += 1 + + if y[:cN] == noA: # and ArgC==ArgCx: + flagA = False + break + if flagA: + newSym.append(x) + newSym.reverse() + symF = False + ampFlag = False + finSym = [] + for s in range(len(newSym) - 1, -1, -1): + symbolPar = "\\" + newSym[s] + ArgCx = 0 + parCx = 0 + parFlag = False + cC = 0 + for z in symbolPar: + if z == "@": + parFlag = True + elif z.isalpha() or z == "&": + cN += 1 + else: + if z == "{" or z == "[": + cC += 1 + if z == "}" or z == "]": + cC -= 1 + if cC == 0: + if parFlag: + parCx += 1 + else: + ArgCx += 1 + + if symbolPar.find("{") != -1 or symbolPar.find("[") != -1: + if symbolPar.find("[") == -1: + symbol = symbolPar[0:symbolPar.find("{")] + elif symbolPar.find("{") == -1 or symbolPar.find("[") < symbolPar.find("{"): + symbol = symbolPar[0:symbolPar.find("[")] + else: + symbol = symbolPar[0:symbolPar.find("{")] + else: + symbol = symbolPar + numArg = parCx + numPar = ArgCx + gFlag = False + checkFlag = False + get = False + gCSV = csv.reader(open('new.Glossary.csv', 'rb'), delimiter=',', quotechar='\"') + preG = "" + if symbol == "\\&": + ampFlag = True + for S in gCSV: + G = S + ArgCx = 0 + parCx = 0 + parFlag = False + cC = 0 + ind = G[0].find("@") + if ind == -1: + ind = len(G[0]) - 1 + for z in G[0]: + if z == "@": + parFlag = True + elif z.isalpha(): + cN += 1 + else: + if z == "{" or z == "[": + cC += 1 + if z == "}" or z == "]": + cC -= 1 + if cC == 0: + if parFlag: + parCx += 1 + else: + ArgCx += 1 + if G[0].find(symbol) == 0 and (len(G[0]) == len(symbol) or not G[0][ + len(symbol)].isalpha()): # and (numPar!=0 or numArg!=0): + checkFlag = True + get = True + preG = S + + elif checkFlag: + get = True + checkFlag = False + + if (get): + if get: + G = preG + get = False + checkFlag = False + if True: + if symbolPar.find("@") != -1: + Q = symbolPar[:symbolPar.find("@")] + else: + Q = symbolPar + listArgs = [] + if len(Q) > len(symbol) and (Q[len(symbol)] == "{" or Q[len(symbol)] == "["): + ap = "" + for o in range(len(symbol), len(Q)): + if Q[o] == "{" or z == "[": + argFlag = True + elif Q[o] == "}" or z == "]": + argFlag = False + listArgs.append(ap) + ap = "" + else: + ap += Q[o] + '''websiteU=g[g.find("http://"):].strip("\n") + k=0 + websites=[] + for r in range(0,len(websiteU)): + if websiteU[r]==",": + if websiteU[k:r].find("http://")!=-1: + websites.append(websiteU[k:r+1].strip(" ")) + k=r + + + if websiteU[k:r].find("http://")!=-1: + websites.append(websiteU[k:r+1].strip(" ")) + websiteF="" + for d in websites: + websiteF=websiteF+" ["+d+" "+d+"]"''' + # websiteF=G[4].strip("\n") + websiteF = "" + web1 = G[5] + for t in range(5, len(G)): + if G[t] != "": + websiteF = websiteF + " [" + G[t] + " " + G[t] + "]" + + # p1=Q + # if Q.find("@")!=-1: + # p1=Q[:Q.find("@")] + p1 = G[4].strip("$") + p1 = "{\\displaystyle " + p1 + "}" + # if checkFlag: + new1 = "" + new2 = "" + pause = False + mathF = True + '''for k in range(0,len(p1)): + if p1[k]=="$": + if mathF: + new1+="{\\displaystyle " + else: + new1+="}" + mathF=not mathF + + elif p1[k]=="#" and p1[k+1].isdigit(): + pause=True + elif pause: + num=int(p1[k]) + #letter=chr(num+96) + letter=listArgs[num-1] + new1+=letter + pause=False + + else: + new1+=p1[k]''' + p2 = G[1] + for k in range(0, len(p2)): + if p2[k] == "$": + if mathF: + new2 += "{\\displaystyle " + else: + new2 += "}" + mathF = not mathF + else: + new2 += p2[k] + # p1=new1 + p2 = new2 + finSym.append(web1 + " " + p1 + "] : " + p2 + " :" + websiteF) + break + # gFlag=True + # if not symF: + # symF=True + # wiki.write("[") + # else: + # wiki.write("
\n") + # wiki.write("[") + # wiki.write(g[g.find("http://"):].strip("\n")) + # wiki.write(" ") + # wiki.write(getEq(g[g.find("{$"):]).strip("\n").replace("\\\\","\\")) + # wiki.write("] : ") + # wiki.write(g[g.find(" {")+2:g.find("} ",g.find(" {")+1)]) + # wiki.write(" : [") + # wiki.write(g[g.find("http://"):].strip("\n")) + # wiki.write(" ") + # wiki.write(g[g.find("http://"):].strip("\n")) + # wiki.write("] ")''' + + # preG=S + if not gFlag: + del newSym[s] + + gFlag = True + # finSym.reverse() + if ampFlag: + wiki.write("& : logical and") + gFlag = False + for y in finSym: + if y == "& : logical and": + pass + elif gFlag: + gFlag = False + wiki.write("[" + y) + else: + wiki.write("
\n[" + y) + + wiki.write("\n
\n") + + wiki.write("\n== Bibliography==\n\n") # should there be a space between bibliography and ==? + r = unmodLabel(labels[eqCounter]) + q = r.find("KLS:") + 4 + p = r.find(":", q) + section = r[q:p] + equation = r[p + 1:] + if equation.find(":") != -1: + equation = equation[0:equation.find(":")] + + wiki.write( + "[http://homepage.tudelft.nl/11r49/askey/contents.html Equation in Section " + section + "] of [[Bibliography#KLS|'''KLS''']].\n\n") # Where should it link to? + wiki.write("== URL links ==\n\nWe ask users to provide relevant URL links in this space.\n\n") + if eqCounter < endNum: + wiki.write("
\n") + if newSec: + newSec = False + wiki.write("
<< [[" + secLabel(sections[secCount][0]).replace(" ", + "_") + "|" + secLabel( + sections[secCount][0]) + "]]
\n") + else: + wiki.write("
<< [[" + secLabel(labels[eqCounter - 1]).replace(" ", + "_") + "|" + secLabel( + labels[eqCounter - 1]) + "]]
\n") + wiki.write("
[[" + secLabel(sections[secCount + 1][0]).replace(" ", + "_") + "#" + secLabel( + labels[eqCounter][len("Formula:"):]) + "|formula in " + secLabel( + sections[secCount + 1][0]) + "]]
\n") + # if eqS==sections[secCount][1]: + # wiki.write("
[["+sections[(secCount+1)%len(sections)][0].replace(" ","_")+"|"+sections[(secCount+1)%len(sections)][0]+"]] >>
\n") + # else: + if True: + wiki.write( + "
[[" + secLabel(labels[(eqCounter + 1) % (endNum + 1)]).replace(" ", + "_") + "|" + secLabel( + labels[(eqCounter + 1) % (endNum + 1)]) + "]] >>
\n") + wiki.write("
\n\ndrmf_eof\n") + else: # FOR EXTRA EQUATIONS + wiki.write("
\n") + wiki.write("
<< [[" + labels[endNum - 1].replace(" ", "_") + "|" + labels[ + endNum - 1] + "]]
\n") + wiki.write("
[[" + labels[0].replace(" ", "_") + "#" + labels[endNum][ + 8:] + "|formula in " + + labels[0] + "]]
\n") + wiki.write("
[[" + labels[(0) % endNum].replace(" ", "_") + "|" + labels[ + 0 % endNum] + "]]
\n") + wiki.write("
\n\ndrmf_eof\n") + + + + elif "\\constraint" in line and parse: + # symbols=symbols+getSym(line) + symLine = line.strip("\n") + if hCon: + comToWrite = comToWrite + "\n== Constraint(s) ==\n\n" + hCon = False + constraint = True + math = False + conLine = "" + # wiki.write("
"+getEq(line)+"

\n") + elif "\\substitution" in line and parse: + # symbols=symbols+getSym(line) + symLine = line.strip("\n") + if hSub: + comToWrite = comToWrite + "\n== Substitution(s) ==\n\n" + hSub = False + # wiki.write("
"+getEq(line)+"

\n") + substitution = True + math = False + subLine = "" + elif "\\drmfname" in line and parse: + math = False + comToWrite = "\n== Name ==\n\n
" + get_data_str(line) + "

\n" + comToWrite + + elif "\\drmfnote" in line and parse: + symbols = symbols + getSym(line) + if hNote: + comToWrite = comToWrite + "\n== Note(s) ==\n\n" + hNote = False + note = True + math = False + noteLine = "" + + elif "\\proof" in line and parse: + # symbols=symbols+getSym(line) + symLine = line.strip("\n") + if hProof: + hProof = False + comToWrite = comToWrite + "\n== Proof ==\n\nWe ask users to provide proof(s), reference(s) to proof(s), or further clarification on the proof(s) in this space. \n

\n
" + proof = True + proofLine = "" + pause = False + pauseP = False + for ind in range(0, len(line)): + if line[ind:ind + 7] == "\\eqref{": + pause = True + eqR = line[ind:line.find("}", ind) + 1] + rLab = get_data_str(eqR) + '''for l in lLink: + if rLab==l[0:l.find("=")-1]: + rlabel=l[l.find("=>")+3:l.find("\\n")] + rlabel=rlabel.replace("/","") + rlabel=rlabel.replace("#",":") + rlabel=rlabel.replace("!",":") + break''' + + eInd = refLabels.index("" + rLab) + z = line[line.find("}", ind + 7) + 1] + if z == "." or z == ",": + pauseP = True + proofLine += ("
\n{\displaystyle\n" + refEqs[ + eInd] + "}" + z + "
\n") + else: + if z == "}": + proofLine += ( + "
\n{\displaystyle\n" + refEqs[eInd] + "}
") + else: + proofLine += ( + "
\n{\displaystyle\n" + refEqs[eInd] + "}
\n") + + + else: + if pause: + if line[ind] == "}": + pause = False + elif pauseP: + pauseP = False + elif line[ind] == "\n" and "\\end{equation}" in lines[i + 1]: + pass + else: + proofLine += (line[ind]) + if "\\end{equation}" in lines[i + 1]: + proof = False + # symLine+=line.strip("\n") + wiki.write(comToWrite + getEqP(proofLine, False) + "
\n
\n") + comToWrite = "" + symbols = symbols + getSym(symLine) + symLine = "" + + # wiki.write(line) + + elif proof: + symLine += line.strip("\n") + pauseP = False + # symbols=symbols+getSym(line) + for ind in range(0, len(line)): + if line[ind:ind + 7] == "\\eqref{": + pause = True + eqR = line[ind:line.find("}", ind) + 1] + rLab = get_data_str(eqR) + '''for l in lLink: + if rLab==l[0:l.find("=")-1]: + rlabel=l[l.find("=>")+3:l.find("\\n")] + rlabel=rlabel.replace("/","") + rlabel=rlabel.replace("#",":") + rlabel=rlabel.replace("!",":") + break''' + + eInd = refLabels.index("" + label) + z = line[line.find("}", ind + 7) + 1] + if z == "." or z == ",": + pauseP = True + proofLine += ("
\n{\displaystyle\n" + refEqs[ + eInd] + "}" + z + "
\n") + else: + proofLine += ( + "
\n{\displaystyle\n" + refEqs[eInd] + "}
\n") + + else: + if pause: + if line[ind] == "}": + pause = False + elif pauseP: + pauseP = False + elif line[ind] == "\n" and "\\end{equation}" in lines[i + 1]: + pass + + else: + proofLine += (line[ind]) + if "\\end{equation}" in lines[i + 1]: + proof = False + # symLine+=line.strip("\n") + wiki.write(comToWrite + getEqP(proofLine, False).rstrip("\n") + "
\n
\n") + comToWrite = "" + symbols = symbols + getSym(symLine) + symLine = "" + + elif math: + if "\\end{equation}" in lines[i + 1] or "\\constraint" in lines[i + 1] or "\\substitution" in lines[ + i + 1] or "\\proof" in lines[i + 1] or "\\drmfnote" in lines[i + 1] or "\\drmfname" in lines[ + i + 1]: + wiki.write(line.rstrip("\n")) + symLine += line.strip("\n") + symbols = symbols + getSym(symLine) + symLine = "" + wiki.write("\n}
\n") + else: + symLine += line.strip("\n") + wiki.write(line) + if note and parse: + noteLine = noteLine + line + symbols = symbols + getSym(line) + if "\\end{equation}" in lines[i + 1] or "\\drmfn" in lines[i + 1] or "\\constraint" in lines[ + i + 1] or "\\substitution" in lines[i + 1] or "\\proof" in lines[i + 1]: + note = False + if "\\emph" in noteLine: + noteLine = noteLine[0:noteLine.find("\\emph{")] + "\'\'" + noteLine[noteLine.find("\\emph{") + len( + "\\emph{"):noteLine.find("}", noteLine.find("\\emph{") + len("\\emph{"))] + "\'\'" + noteLine[ + noteLine.find( + "}", + noteLine.find( + "\\emph{") + len( + "\\emph{")) + 1:] + comToWrite = comToWrite + "
" + getEq(noteLine) + "

\n" + + if constraint and parse: + conLine = conLine + line.replace("&", "&
") + + symLine += line.strip("\n") + # symbols=symbols+getSym(line) + if "\\end{equation}" in lines[i + 1] or "\\drmfn" in lines[i + 1] or "\\constraint" in lines[ + i + 1] or "\\substitution" in lines[i + 1] or "\\proof" in lines[i + 1]: + constraint = False + symbols = symbols + getSym(symLine) + symLine = "" + wiki.write(comToWrite + "
" + getEq(conLine) + "

\n") + comToWrite = "" + if substitution and parse: + subLine = subLine + line # .replace("&","&
") + + symLine += line.strip("\n") + # symbols=symbols+getSym(line) + if "\\end{equation}" in lines[i + 1] or "\\drmfn" in lines[i + 1] or "\\substitution" in lines[ + i + 1] or "\\constraint" in lines[i + 1] or "\\proof" in lines[i + 1]: + substitution = False + symbols = symbols + getSym(symLine) + symLine = "" + lineR = "" + for i in range(0, len(subLine)): + if subLine[i] == "&" and not ( + i > subLine.find("\\begin{array}") and i < subLine.find("\\end{array}")): + lineR += "&
" + else: + lineR += subLine[i] + wiki.write(comToWrite + "
" + getEq(lineR) + "

\n") + comToWrite = "" + +if __name__ == "__main__": + main() diff --git a/tex2Wiki/src/tex2Wiki2.py b/tex2Wiki/src/tex2Wiki2.py new file mode 100644 index 0000000..4e15939 --- /dev/null +++ b/tex2Wiki/src/tex2Wiki2.py @@ -0,0 +1,136 @@ +__author__ = "Joon Bang" +__status__ = "Prototype" + + +class LatexEquation(object): + def __init__(self, label, equation, metadata): + self.label = label + self.equation = equation + self.metadata = metadata + + def __str__(self): + return self.label + "\n" + self.equation + "\n" + str(self.metadata) + + +def generate_html(tag_name, options, text, spacing=True): + # (str, str, str(, bool)) -> str + """Generates an html tag, with optional html parameters.""" + result = "<" + tag_name + if options != "": + result += " " + options + result += ">\n" + text + "\n\n" + + if not spacing: + return result.replace("\n", " ")[:-1] + "\n" + + return result + + +def multi_split(s, seps): + """Copy pasted from internet!""" + res = [s] + for sep in seps: + s, res = res, [] + for seq in s: + res += seq.split(sep) + return res + + +def convert_dollar_signs(string): + count = 0 + result = "" + for ch in string: + if ch == "$" and count % 2 == 0: + result += "{\\displaystyle " + count += 1 + elif ch == "$": + result += "}" + count += 1 + else: + result += ch + + return result + + +def translate(data): + for section_data in data: + print section_data[0] + equations = section_data[1] + for i, equation in enumerate(equations): + # formula stuff + try: + formula = get_data_str(equation, latex="\\formula") + formula = formula.split("Formula:", 1)[1] + except IndexError: # there is no formula. + break + + # get metadata + raw_metadata = list() + metadata_types = {"substitution": "Substitution(s)", "constraint": "Constraint(s)"} + + equation = equation.split("\n")[1:] + + j = 0 + while j < len(equation): + if "%" in equation[j]: + raw_metadata.append(equation.pop(j)) + else: + j += 1 + + raw_metadata = ''.join([line[1:].strip() for line in raw_metadata]) + + metadata = dict() + for data_type in metadata_types: + metadata[data_type] = convert_dollar_signs(get_data_str(raw_metadata, latex="\\"+data_type)) + + equations[i] = LatexEquation(formula, ''.join(equation), metadata) + + for equation in equations: + print equation + + +def find_end(text, left_delimiter, right_delimiter, start=0): + net = 0 # left delimiters encountered - right delimiters encountered + for i, ch in enumerate(text[start:]): + if ch == left_delimiter: + net += 1 + elif ch == right_delimiter: + if net == 1: + return i + start + else: + net -= 1 + + return -1 + + +def get_data_str(text, latex=""): + if latex + "{" not in text: + return "" + + start = text.find(latex + "{") + return text[start + len(latex + "{"):find_end(text, "{", "}", start)] + + +def main(): + with open("tex2Wiki/data/01outb.tex") as input_file: + text = input_file.read() + + text = text.split("\\begin{document}", 1)[1] + text = text.split("\\section") + + title = get_data_str(text[0]) + print title + + data = list() + for section in text[1:]: + section_name = get_data_str(section).replace("$", "''") + equations = section.split("\\end{equation}") + for i, equation in enumerate(equations[:-1]): + equations[i] = equation.split("\\begin{equation}", 1)[1].strip() + + data.append([section_name, equations]) + + translate(data) + +if __name__ == '__main__': + main() diff --git a/tex2Wiki/test/llinks b/tex2Wiki/test/llinks deleted file mode 100644 index 72ed132..0000000 --- a/tex2Wiki/test/llinks +++ /dev/null @@ -1 +0,0 @@ -eq:EF.EX.TM => DLMF:/4.9#E4 \ No newline at end of file diff --git a/tex2Wiki/test/test_getEq.py b/tex2Wiki/test/test_getEq.py deleted file mode 100644 index 6dcdbf4..0000000 --- a/tex2Wiki/test/test_getEq.py +++ /dev/null @@ -1,9 +0,0 @@ -from unittest import TestCase -from tex2Wiki import getEq -from tex2Wiki import setup_label_links -class TestGetEq(TestCase): - - def test_getEq(self): - self.assertEqual('\realpart{s} > 1', getEq('% \constraint{$\realpart{s} > 1$}')) - self.assertEqual('\realpart{s} > -1 &
s\neq 1', getEq('% \constraint{$\realpart{s} > -1$ &
$s \neq 1$}\n')) - self.assertEqual('\realpart{s} > -1 &
s\neq 1', getEq('% \constraint{$\realpart{s} > -1$ &
$s \neq 1$}')) \ No newline at end of file diff --git a/tex2Wiki/test/test_getString.py b/tex2Wiki/test/test_getString.py deleted file mode 100644 index 9414d8e..0000000 --- a/tex2Wiki/test/test_getString.py +++ /dev/null @@ -1,11 +0,0 @@ - -from unittest import TestCase -from tex2Wiki import getString -from tex2Wiki import setup_label_links -class TestGetString(TestCase): - - def test_getString(self): - self.assertEqual('Mathematical Applications', getString('\\section{Mathematical Applications}\\label{sec:ZE.APPL}%ZE.16\n')) - self.assertEqual('eq:ZE.INT.EL1',getString('\\eqref{eq:ZE.INT.EL1}')) - self.assertEqual('eq:ZE.INT.EL3',getString('\\eqref{eq:ZE.INT.EL3}\n')) - self.assertEqual('eq:ZE.INT.EL5', getString(' \n \\eqref{eq:ZE.INT.EL5}\n ')) \ No newline at end of file diff --git a/tex2Wiki/test/test_isnumber.py b/tex2Wiki/test/test_isnumber.py deleted file mode 100644 index 42ed552..0000000 --- a/tex2Wiki/test/test_isnumber.py +++ /dev/null @@ -1,11 +0,0 @@ -__author__ = "Azeem Mohammed" -__status__ = "Development" - -from unittest import TestCase -from tex2Wiki import isnumber - - -class TestIsnumber(TestCase): - def test_isnumber(self): - self.assertEqual(True, isnumber("1")) - self.assertEqual(False, isnumber("a")) diff --git a/tex2Wiki/test/test_modlabel.py b/tex2Wiki/test/test_modlabel.py deleted file mode 100644 index 1e4f369..0000000 --- a/tex2Wiki/test/test_modlabel.py +++ /dev/null @@ -1,19 +0,0 @@ -__author__ = "Azeem Mohammed" -__status__ = "Development" - -from unittest import TestCase -from tex2Wiki import modLabel -from tex2Wiki import setup_label_links - - -class TestModLabel(TestCase): - def test_modLabel(self): - self.assertEqual('Formula:EF.EX.TM', modLabel('\\begin{equation}\label{eq:EF.EX.TM}')) - self.assertEqual('Formula:EF.EX.TM', modLabel('\\begin{equation}\\formula{eq:EF.EX.TM}')) - self.assertEqual('auto-number-1', modLabel('\\begin{equation}')) - self.assertEqual('auto-number-2', modLabel('\\begin{equation}')) - - def test_modLabelsWithList(self): - setup_label_links('testdata/llinks') - self.assertEqual('DLMF:04.09:E', modLabel('\\begin{equation}\label{eq:EF.EX.TM}')) - self.assertEqual('DLMF:04.09:E', modLabel('\\begin{equation}\\formula{eq:EF.EX.TM}')) diff --git a/tex2Wiki/test/test_secLabel.py b/tex2Wiki/test/test_secLabel.py deleted file mode 100644 index f0f7603..0000000 --- a/tex2Wiki/test/test_secLabel.py +++ /dev/null @@ -1,6 +0,0 @@ -from unittest import TestCase -from tex2Wiki import secLabel -from tex2Wiki import setup_label_links -class TestSecLabel(TestCase): - def test_secLabel(self): - self.assertEqual('Formula:DLMF:25.5:E12',secLabel("Formula:DLMF:25.5:E12''")) \ No newline at end of file diff --git a/tex2Wiki/test/test_unmodLabel.py b/tex2Wiki/test/test_unmodLabel.py deleted file mode 100644 index fb89e25..0000000 --- a/tex2Wiki/test/test_unmodLabel.py +++ /dev/null @@ -1,8 +0,0 @@ -from unittest import TestCase -from tex2Wiki import unmodLabel -from tex2Wiki import setup_label_links -class TestModLabel(TestCase): - - def test_unmodLabel(self): - self.assertEqual('Formula:DLMF:15.2:E1', unmodLabel('Formula:DLMF:15.02:E1')) #remove 0's before decimal point - self.assertEqual('Formula:DLMF:5.2:E1', unmodLabel('Formula:DLMF:05.02:E1')) #remove 0's before both decimal points and after colon From b5a32ed0b2164ebf756f7bb5085ec11e51ede352 Mon Sep 17 00:00:00 2001 From: Joon Bang Date: Tue, 16 Aug 2016 14:56:28 -0400 Subject: [PATCH 278/402] Add mostly functional generation of first part of output file --- tex2Wiki/src/tex2Wiki2.py | 62 ++++++++++++++++++++++++++++++++------- 1 file changed, 52 insertions(+), 10 deletions(-) diff --git a/tex2Wiki/src/tex2Wiki2.py b/tex2Wiki/src/tex2Wiki2.py index 4e15939..bd3eff4 100644 --- a/tex2Wiki/src/tex2Wiki2.py +++ b/tex2Wiki/src/tex2Wiki2.py @@ -1,6 +1,8 @@ __author__ = "Joon Bang" __status__ = "Prototype" +METADATA_TYPES = {"substitution": "Substitution(s)", "constraint": "Constraint(s)"} + class LatexEquation(object): def __init__(self, label, equation, metadata): @@ -18,7 +20,18 @@ def generate_html(tag_name, options, text, spacing=True): result = "<" + tag_name if options != "": result += " " + options - result += ">\n" + text + "\n\n" + + result += ">" + + if tag_name == "math": + result += "{\\displaystyle" + + result += "\n" + text + "\n" + + if tag_name == "math": + result += "}" + + result += "\n" if not spacing: return result.replace("\n", " ")[:-1] + "\n" @@ -53,9 +66,9 @@ def convert_dollar_signs(string): def translate(data): + result = list() for section_data in data: - print section_data[0] - equations = section_data[1] + equations = section_data[1][:-1] for i, equation in enumerate(equations): # formula stuff try: @@ -66,7 +79,6 @@ def translate(data): # get metadata raw_metadata = list() - metadata_types = {"substitution": "Substitution(s)", "constraint": "Constraint(s)"} equation = equation.split("\n")[1:] @@ -80,13 +92,42 @@ def translate(data): raw_metadata = ''.join([line[1:].strip() for line in raw_metadata]) metadata = dict() - for data_type in metadata_types: - metadata[data_type] = convert_dollar_signs(get_data_str(raw_metadata, latex="\\"+data_type)) + for data_type in METADATA_TYPES: + metadata[data_type] = convert_dollar_signs(get_data_str(raw_metadata, latex="\\"+data_type)).strip() + + equations[i] = LatexEquation(formula, ' '.join(equation), metadata) + + result.append([section_data[0], equations]) + + return result + + +def create_general_pages(data): + ret = "" + + for section_data in data: + section_name = section_data[0] + result = "drmf_bof\n'''" + section_name + "'''\n{{DISPLAYTITLE:" + section_name + "}}\n" + + # TODO: headers code (recycle from main_page code?) + + result += "\n== " + section_name + " ==\n\n" + + text = "" + for equation in section_data[1]: + # equation is of type LatexEquation + text += generate_html("math", "id=\"" + equation.label + "\"", equation.equation) + + for data_type, info in equation.metadata.iteritems(): + if info != "": + text += generate_html("div", "align=\"right\"", METADATA_TYPES[data_type] + ": " + info, + spacing=False)[:-1] + "
\n" + + result += text + "drmf_eof\n" - equations[i] = LatexEquation(formula, ''.join(equation), metadata) + ret += result - for equation in equations: - print equation + return ret def find_end(text, left_delimiter, right_delimiter, start=0): @@ -130,7 +171,8 @@ def main(): data.append([section_name, equations]) - translate(data) + data = translate(data) + print create_general_pages(data) if __name__ == '__main__': main() From 9c3d3fd16ed217af84f97ab9c2c8fd5a0671e81f Mon Sep 17 00:00:00 2001 From: Joon Bang Date: Wed, 17 Aug 2016 10:07:12 -0400 Subject: [PATCH 279/402] Update .gitignore --- main_page/.gitignore | 4 ++++ maple2latex/.gitignore | 5 +++++ tex2Wiki/.gitignore | 2 ++ 3 files changed, 11 insertions(+) create mode 100644 main_page/.gitignore create mode 100644 maple2latex/.gitignore create mode 100644 tex2Wiki/.gitignore diff --git a/main_page/.gitignore b/main_page/.gitignore new file mode 100644 index 0000000..3caccc4 --- /dev/null +++ b/main_page/.gitignore @@ -0,0 +1,4 @@ +.DS_Store +backups/ +*Glossary.csv* +!test/fake.Glossary.csv diff --git a/maple2latex/.gitignore b/maple2latex/.gitignore new file mode 100644 index 0000000..1beb704 --- /dev/null +++ b/maple2latex/.gitignore @@ -0,0 +1,5 @@ +functions/ +out/ +notes.txt +resources/ +.DS_Store diff --git a/tex2Wiki/.gitignore b/tex2Wiki/.gitignore new file mode 100644 index 0000000..00a7b41 --- /dev/null +++ b/tex2Wiki/.gitignore @@ -0,0 +1,2 @@ +AzeemOldCode/ +*Glossary.csv* \ No newline at end of file From f9aac473abd66eecaeb027bfbe66ee47b8c1b035 Mon Sep 17 00:00:00 2001 From: Joon Bang Date: Wed, 17 Aug 2016 13:30:45 -0400 Subject: [PATCH 280/402] Correct generation of first part of output file --- tex2Wiki/src/testData.txt | 0 tex2Wiki/src/tex2Wiki2.py | 180 ++++++++++++++++++++++++++++---------- 2 files changed, 134 insertions(+), 46 deletions(-) delete mode 100644 tex2Wiki/src/testData.txt diff --git a/tex2Wiki/src/testData.txt b/tex2Wiki/src/testData.txt deleted file mode 100644 index e69de29..0000000 diff --git a/tex2Wiki/src/tex2Wiki2.py b/tex2Wiki/src/tex2Wiki2.py index bd3eff4..1839c5f 100644 --- a/tex2Wiki/src/tex2Wiki2.py +++ b/tex2Wiki/src/tex2Wiki2.py @@ -1,6 +1,8 @@ __author__ = "Joon Bang" __status__ = "Prototype" +INPUT_FILE = "tex2Wiki/data/01outb.tex" +OUTPUT_FILE = "tex2Wiki/data/01outb.mmd" METADATA_TYPES = {"substitution": "Substitution(s)", "constraint": "Constraint(s)"} @@ -10,33 +12,50 @@ def __init__(self, label, equation, metadata): self.equation = equation self.metadata = metadata - def __str__(self): - return self.label + "\n" + self.equation + "\n" + str(self.metadata) +def generate_html(tag_name, text, options=dict(), spacing=2): + # (str, str(, dict, bool)) -> str + """ + Generates an html tag, with optional html parameters. + When spacing = 0, there should be no spacing. + When spacing = 1, the tag, text, and end tag are padded with spaces. + When spacing = 2, the tag, text, and end tag are padded with newlines. + """ + option_text = [key + "=\"" + value + "\"" for key, value in options.iteritems()] + result = "<" + tag_name + " " * (option_text != []) + ", ".join(option_text) + ">" + + if spacing == 2: + result += "\n" + text + "\n\n" + elif spacing == 1: + result += " " + text + " " + else: + result += text + "" -def generate_html(tag_name, options, text, spacing=True): - # (str, str, str(, bool)) -> str - """Generates an html tag, with optional html parameters.""" - result = "<" + tag_name - if options != "": - result += " " + options + return result - result += ">" - if tag_name == "math": - result += "{\\displaystyle" +def generate_math_html(text, options=dict(), spacing=True): + """ + Special case of generate_html, where the tag is "math". + """ + option_text = [key + "=\"" + value + "\"" for key, value in options.iteritems()] + result = "" - result += "\n" + text + "\n" + if spacing: + result += "{\\displaystyle\n" + text + "\n}\n" + else: + result += "{\\displaystyle " + text + "}\n" - if tag_name == "math": - result += "}" + return result - result += "\n" - if not spacing: - return result.replace("\n", " ")[:-1] + "\n" +def generate_link(left, right=""): + """Generates a link thingie.""" - return result + if right == "": + return "[[" + left + "|" + left + "]]" + + return "[[" + left + "|" + right + "]]" def multi_split(s, seps): @@ -52,7 +71,7 @@ def multi_split(s, seps): def convert_dollar_signs(string): count = 0 result = "" - for ch in string: + for i, ch in enumerate(string): if ch == "$" and count % 2 == 0: result += "{\\displaystyle " count += 1 @@ -65,65 +84,111 @@ def convert_dollar_signs(string): return result -def translate(data): +def format_formula(formula): + formula = multi_split(formula.split("Formula:", 1)[1], [".", ":"]) + + for j in [-1, -2, -3]: + formula[j] = formula[j].zfill(2) + + if len(formula) == 3: + formula = formula[0] + "." + formula[1] + ":" + formula[2] + else: + formula = ":".join(formula[:-3]) + ":" + formula[-3] + "." + formula[-2] + ":" + formula[-1] + + return formula + + +def format_metadata(string): + if string == "": + return "" + + # modified convert_dollar_sign algorithm + dollar_count = 0 + result = "" + for i, ch in enumerate(string): + if ch == "$": + if dollar_count % 2 == 1: + result += "}" + elif dollar_count > 1: + result = result[:-1] + "
\n{\\displaystyle " + else: + result += "{\\displaystyle " + dollar_count += 1 + else: + result += ch + + return result.strip() + + +def extract_data(data): result = list() for section_data in data: equations = section_data[1][:-1] for i, equation in enumerate(equations): # formula stuff try: - formula = get_data_str(equation, latex="\\formula") - formula = formula.split("Formula:", 1)[1] + formula = format_formula(get_data_str(equation, latex="\\formula")) except IndexError: # there is no formula. break - # get metadata - raw_metadata = list() - equation = equation.split("\n")[1:] + # get metadata + percent_list = list() + raw_metadata = "" + j = 0 while j < len(equation): if "%" in equation[j]: - raw_metadata.append(equation.pop(j)) + raw_metadata += equation.pop(j)[1:].strip().strip("\n") + "\n" else: j += 1 - raw_metadata = ''.join([line[1:].strip() for line in raw_metadata]) - metadata = dict() for data_type in METADATA_TYPES: - metadata[data_type] = convert_dollar_signs(get_data_str(raw_metadata, latex="\\"+data_type)).strip() + metadata[data_type] = format_metadata(get_data_str(raw_metadata, latex="\\"+data_type)) - equations[i] = LatexEquation(formula, ' '.join(equation), metadata) + # print metadata + + equations[i] = LatexEquation(formula, '\n'.join(equation), metadata) result.append([section_data[0], equations]) return result -def create_general_pages(data): +def create_general_pages(data, title): ret = "" - for section_data in data: + print [d[0] for d in data] + + section_names = [d[0] for d in data] # list of section names + + for i, section_data in enumerate(data): section_name = section_data[0] - result = "drmf_bof\n'''" + section_name + "'''\n{{DISPLAYTITLE:" + section_name + "}}\n" + result = "drmf_bof\n'''" + section_name.replace("''", "") + "'''\n{{DISPLAYTITLE:" + section_name + "}}\n" - # TODO: headers code (recycle from main_page code?) + # header/footer code + header, footer = generate_nav_bar(title, section_names, i) - result += "\n== " + section_name + " ==\n\n" + result += header + "\n== " + section_name + " ==\n\n" text = "" - for equation in section_data[1]: + for eq in section_data[1]: # equation is of type LatexEquation - text += generate_html("math", "id=\"" + equation.label + "\"", equation.equation) + text += generate_math_html(eq.equation, options={"id": eq.label}) - for data_type, info in equation.metadata.iteritems(): - if info != "": - text += generate_html("div", "align=\"right\"", METADATA_TYPES[data_type] + ": " + info, - spacing=False)[:-1] + "
\n" + metadata_exists = False + for data_type in sorted(eq.metadata.keys()): + if eq.metadata[data_type] != "": + text += generate_html("div", METADATA_TYPES[data_type] + ": " + eq.metadata[data_type], + options={"align": "right"}, spacing=False) + "
\n" + metadata_exists = True - result += text + "drmf_eof\n" + if not metadata_exists: + text = text[:-1] + "
\n" + + result += text + footer + "\n" + "drmf_eof\n" ret += result @@ -152,15 +217,35 @@ def get_data_str(text, latex=""): return text[start + len(latex + "{"):find_end(text, "{", "}", start)] +def generate_nav_bar(title, sections, i): + main_title = "Orthogonal Polynomials" + sections = [main_title] + sections + [main_title] + + center_text = main_title.replace(" ", "_") + "#Sections_in_" + title.replace(" ", "_") + + sections = [s.replace("''", "") for s in sections] + + nav_section = generate_html("div", "<< " + generate_link(sections[i]), options={"id": "alignleft"}, + spacing=1) + "\n" + \ + generate_html("div", generate_link(center_text, sections[i + 1]), options={"id": "aligncenter"}, + spacing=1) + "\n" + \ + generate_html("div", generate_link(sections[i + 2]) + " >>", options={"id": "alignright"}, + spacing=1) + + header = generate_html("div", nav_section, options={"id": "drmf_head"}) + footer = generate_html("div", nav_section, options={"id": "drmf_foot"}) + + return header, footer + + def main(): - with open("tex2Wiki/data/01outb.tex") as input_file: + with open(INPUT_FILE) as input_file: text = input_file.read() text = text.split("\\begin{document}", 1)[1] text = text.split("\\section") title = get_data_str(text[0]) - print title data = list() for section in text[1:]: @@ -171,8 +256,11 @@ def main(): data.append([section_name, equations]) - data = translate(data) - print create_general_pages(data) + data = extract_data(data) + output_1 = create_general_pages(data, title) + + with open(OUTPUT_FILE, "w") as output_file: + output_file.write(output_1) if __name__ == '__main__': main() From 0d10d10f104f40f31bd375af44d8b3da4eaa8dfe Mon Sep 17 00:00:00 2001 From: Joon Bang Date: Wed, 17 Aug 2016 14:40:54 -0400 Subject: [PATCH 281/402] Add mostly functional generation for .mmd --- tex2Wiki/src/tex2Wiki2.py | 195 +++++++++++++++++++++++++++++++------- 1 file changed, 160 insertions(+), 35 deletions(-) diff --git a/tex2Wiki/src/tex2Wiki2.py b/tex2Wiki/src/tex2Wiki2.py index 1839c5f..cf4af57 100644 --- a/tex2Wiki/src/tex2Wiki2.py +++ b/tex2Wiki/src/tex2Wiki2.py @@ -3,8 +3,11 @@ INPUT_FILE = "tex2Wiki/data/01outb.tex" OUTPUT_FILE = "tex2Wiki/data/01outb.mmd" +GLOSSARY_LOCATION = "tex2Wiki/src/new.Glossary.csv" METADATA_TYPES = {"substitution": "Substitution(s)", "constraint": "Constraint(s)"} +import csv + class LatexEquation(object): def __init__(self, label, equation, metadata): @@ -157,11 +160,121 @@ def extract_data(data): return result +def find_end(text, left_delimiter, right_delimiter, start=0): + net = 0 # left delimiters encountered - right delimiters encountered + for i, ch in enumerate(text[start:]): + if ch == left_delimiter: + net += 1 + elif ch == right_delimiter: + if net == 1: + return i + start + else: + net -= 1 + + return -1 + + +def get_data_str(text, latex=""): + if latex + "{" not in text: + return "" + + start = text.find(latex + "{") + return text[start + len(latex + "{"):find_end(text, "{", "}", start)] + + +def generate_nav_bar(main_title, hash_code, sections, i, middle=""): + # TODO: Rewrite this function. + + sections = ["Orthogonal Polynomials"] + sections + ["Orthogonal Polynomials"] + + center_text = (main_title + "#" + hash_code).replace(" ", "_") + + sections = [s.replace("''", "") for s in sections] + + if middle == "": + middle = sections[i + 1] + + nav_section = generate_html("div", "<< " + generate_link(sections[i]), options={"id": "alignleft"}, + spacing=1) + "\n" + \ + generate_html("div", generate_link(center_text, middle), options={"id": "aligncenter"}, + spacing=1) + "\n" + \ + generate_html("div", generate_link(sections[i + 2]) + " >>", options={"id": "alignright"}, + spacing=1) + + header = generate_html("div", nav_section, options={"id": "drmf_head"}) + footer = generate_html("div", nav_section, options={"id": "drmf_foot"}) + + return header, footer + + +def get_macro_name(macro): + # (str) -> str + """Obtains the macro name.""" + macro_name = "" + for ch in macro: + if ch.isalpha() or ch == "\\": + macro_name += ch + elif ch in ["@", "{", "["]: + break + + return macro_name + + +def find_all(pattern, string): + # (str, str) -> generator + """Finds all instances of pattern in string.""" + + i = string.find(pattern) + while i != -1: + yield i + i = string.find(pattern, i + 1) + + +def get_symbols(text, glossary): + # (str, dict) -> str + """Generates span text based on symbols present in text.""" + symbols = set() + + for keyword in glossary: + for index in find_all(keyword, text): + # if the macro is present in the text + if index != -1: + index += len(keyword) # now index of next character + if index >= len(text) or not text[index].isalpha(): + symbols.add(keyword) + + span_text = "" + for symbol in sorted(symbols, key=str.lower): + links = list() + for cell in glossary[symbol]: + if "http://" in cell or "https://" in cell: + links.append(cell) + + id_link = links[0] + links = ["[" + link + " " + link + "]" for link in links] + + meaning = list(glossary[symbol][1]) + + count = 0 + for i, ch in enumerate(meaning): + if ch == "$" and count % 2 == 0: + meaning[i] = "{\\displaystyle " + count += 1 + elif ch == "$": + meaning[i] = "}" + count += 1 + + appearance = glossary[symbol][4].strip("$") + + span_text += "[" + id_link + " {\\displaystyle " + appearance + \ + "}] : " + ''.join(meaning) + " : " + " ".join(links) + "
\n" + + return span_text[:-7] # slice off the extra br and endline + + def create_general_pages(data, title): ret = "" - print [d[0] for d in data] - section_names = [d[0] for d in data] # list of section names for i, section_data in enumerate(data): @@ -169,7 +282,7 @@ def create_general_pages(data, title): result = "drmf_bof\n'''" + section_name.replace("''", "") + "'''\n{{DISPLAYTITLE:" + section_name + "}}\n" # header/footer code - header, footer = generate_nav_bar(title, section_names, i) + header, footer = generate_nav_bar("Orthogonal Polynomials", "Sections in " + title, section_names, i) result += header + "\n== " + section_name + " ==\n\n" @@ -195,47 +308,59 @@ def create_general_pages(data, title): return ret -def find_end(text, left_delimiter, right_delimiter, start=0): - net = 0 # left delimiters encountered - right delimiters encountered - for i, ch in enumerate(text[start:]): - if ch == left_delimiter: - net += 1 - elif ch == right_delimiter: - if net == 1: - return i + start - else: - net -= 1 +def create_specific_pages(data): + pages = list() - return -1 + formulae = list() + for section_data in data: + for equations in section_data[1:]: + for eq in equations: + formulae.append("Formula:" + eq.label) + formulae.append(section_data[0]) + print formulae -def get_data_str(text, latex=""): - if latex + "{" not in text: - return "" + glossary = dict() + with open(GLOSSARY_LOCATION, "rb") as csv_file: + glossary_file = csv.reader(csv_file, delimiter=',', quotechar='\"') + for row in glossary_file: + glossary[get_macro_name(row[0])] = row - start = text.find(latex + "{") - return text[start + len(latex + "{"):find_end(text, "{", "}", start)] + i = 0 + for j, section_data in enumerate(data): + section_name = section_data[0] + for eq in section_data[1]: + header, footer = generate_nav_bar(section_name, eq.label, formulae, i, middle="formula in " + section_name) + result = "drmf_bof\n'''Formula:" + eq.label + "'''\n{{DISPLAYTITLE:Formula:" + eq.label + "}}\n" + header \ + + "\n
" -def generate_nav_bar(title, sections, i): - main_title = "Orthogonal Polynomials" - sections = [main_title] + sections + [main_title] + result += generate_html("div", generate_math_html(eq.equation)[:-1], options={"align": "center"}, + spacing=0) + "\n\n" - center_text = main_title.replace(" ", "_") + "#Sections_in_" + title.replace(" ", "_") + for data_type, info in eq.metadata.iteritems(): + if info != "": + result += "== " + METADATA_TYPES[data_type] + " ==\n\n" + result += generate_html("div", info, options={"align": "left"}, spacing=0) + "
\n\n" - sections = [s.replace("''", "") for s in sections] + result += "== Proof ==\n\nWe ask users to provide proof(s), reference(s) to proof(s), or further " \ + "clarification on the proof(s) in this space.\n\n== Symbols List ==\n\n" + result += get_symbols(result, glossary) + "\n
\n\n" - nav_section = generate_html("div", "<< " + generate_link(sections[i]), options={"id": "alignleft"}, - spacing=1) + "\n" + \ - generate_html("div", generate_link(center_text, sections[i + 1]), options={"id": "aligncenter"}, - spacing=1) + "\n" + \ - generate_html("div", generate_link(sections[i + 2]) + " >>", options={"id": "alignright"}, - spacing=1) + result += "== Bibliography ==\n\n" + result += "[http://homepage.tudelft.nl/11r49/askey/contents.html " \ + "Equation in Section 1." + str(j + 1) + "] of [[Bibliography#KLS|'''KLS''']]." - header = generate_html("div", nav_section, options={"id": "drmf_head"}) - footer = generate_html("div", nav_section, options={"id": "drmf_foot"}) + result += "\n\n== URL links ==\n\nWe ask users to provide relevant URL links in this space.\n\n" - return header, footer + result += "
" + footer + "\ndrmf_eof" + pages.append(result) + + i += 1 + + i += 1 + + return "\n".join(pages) + "\n" def main(): @@ -257,10 +382,10 @@ def main(): data.append([section_name, equations]) data = extract_data(data) - output_1 = create_general_pages(data, title) + output = create_general_pages(data, title) + create_specific_pages(data) with open(OUTPUT_FILE, "w") as output_file: - output_file.write(output_1) + output_file.write(output) if __name__ == '__main__': main() From c59da8da2b73fa2edc9b9492c3ff22bf840d4267 Mon Sep 17 00:00:00 2001 From: Joon Bang Date: Thu, 18 Aug 2016 10:54:45 -0400 Subject: [PATCH 282/402] Rewrite generate_nav_bar function; add some comments --- main_page/src/.functions.py.swp | Bin 0 -> 16384 bytes tex2Wiki/src/OrthogonalPolynomials.mmd | 65 ------------ tex2Wiki/src/OrthogonalPolynomials.mmd.new | 65 ------------ {tex2Wiki => tex2wiki}/.gitignore | 0 {tex2Wiki => tex2wiki}/README.md | 0 {tex2Wiki => tex2wiki}/src/__init__.py | 0 {tex2Wiki => tex2wiki}/src/tex2Wiki.py | 0 .../tex2Wiki2.py => tex2wiki/src/tex2wiki2.py | 96 ++++++++++-------- 8 files changed, 56 insertions(+), 170 deletions(-) create mode 100644 main_page/src/.functions.py.swp delete mode 100644 tex2Wiki/src/OrthogonalPolynomials.mmd delete mode 100644 tex2Wiki/src/OrthogonalPolynomials.mmd.new rename {tex2Wiki => tex2wiki}/.gitignore (100%) rename {tex2Wiki => tex2wiki}/README.md (100%) rename {tex2Wiki => tex2wiki}/src/__init__.py (100%) rename {tex2Wiki => tex2wiki}/src/tex2Wiki.py (100%) rename tex2Wiki/src/tex2Wiki2.py => tex2wiki/src/tex2wiki2.py (81%) diff --git a/main_page/src/.functions.py.swp b/main_page/src/.functions.py.swp new file mode 100644 index 0000000000000000000000000000000000000000..876aecd1e0a7fac875906508349952435f2fd69e GIT binary patch literal 16384 zcmeHOYm6jS6~2ngOP6P2)C9cLnoKv_(=)S>Xxiy@U+%*!%w~5HvAff$?z%l)OLbLK zRXwxg!Y+#@VuFt#e~1P#iV~wl5E2Y15c#Lkm>57!5KWN8#Ngu>V#M#Yz=bm$`r#Juc1K05n)~{u_E@3S6AHDH&^X{e{`x!gt_dZ0$XS<^D z#Hsh~QQdc=xV{{$Y|!8&37p$=_`1EbM}%-(e|h#$7#tJ*cy467es^pyi@7N5&kZJi zKXwB@s*l$;kX06!43rGKRR%V*Yql-SQ>!i3Rs8bHZg{H#%K9Y(B?Bb`B?Bb`B?Bb` zB?Bb`B?JG%48-Ge*eQ(v?OF%=o36L1ujKm;eF;v8pTnStOy!KAU{s?>nxEuHk-~-nK*8&{4 z0C@dE#(obx3;YE50q_X$FmMlW8dwBofPY=U*w29<13v;D1Wp16f%gD^KcBI`0Z#z; z11Es}z(v4j;Fa?rA9w^f0el454qOWS`I~aQr_!1BSp9F3Kb^%uaui{|lJHQ>lY2Z@e1UK#e!z190q6 zNA!$-H^k|8L#l%Q5m9|Ka_W#G$F_x6qByMb+3g(P3`61*DZMKR=*m%%byhKG*+QGR z;s^3Uf4IhN^jh(RUy}i4mWd9g0z~BD%qt+=T7n$6=v8iZuD@YJfyY zqRFuQL;jN0;Cu^w6t(51Eer&0L3(snWu^pUq%BA$z5GK<^mXV6eJ*XYI_r+VHM`J( zVQ(;er6bju>WNf1vj${0XICLRA8I=C6F_uHt)x;~Cos3ZI%cXj&Rq&9C}T=(kZ{j-AUQ$lrwR9o;*OX4=mdTr|1u{F)}7u-qeofgTMff>D9F zFp>SC^c9`VbO|pK`N<=(d}|YxnQ1CKG?s$%RZU-eZAeLKo4V3ObeApNCCws=FxP5H zD0#9=cmTC$8L5c|l4f#h;7@x*<<+E3^FbJl!0JK!sQ3hr#0W&Y{j-yN=0$C|CJJ28 znO(tSJBBmzYk05~#+)d=<_Zs+v%Fs&x}HFVtD1avL20>x7#*)c$-RjFB55zKk9g1T zlfp#2>@Fp3!#z>QYJuA5DVQWPswCr8N)$InY1&=i9*Mj)^+I9aoG2%4Kuui>yBwD3 z3HoT6^$DKjl}alZk+i=wBA}o_8%Mi6WgSwAqAepZF%gip)T4!_f+SgmlLiH9a)A%6 zs#KWXOwBy&ut)iml!sh#y15`)gYF#BVuN-(75y{PA6il<6pigZoB*U#q4rW(RMvKt zCa6V5NyFkC#i_BqEN6g3Q5D&&&=ehR)e9h!&(W7!4-uJ}rAbU&qP3WnEL(uLeL9|p zq4}I*dg{BzSKWA+t}a=QbqGdj+bysvgE09~sd0+Kss*028H3TRX2x2RaW9%Hikh^{ zrAh8PxOC*m&cio#4=(NAdGx^2^;mXR`rCM4tk(B=K@{2HTD>2wSd6)&F@idd8^*d_ zESY!`b-UDTk6002Fs9_e!XtYU4}(xW`d9#`yvz2NEk^nOuaWN^K(0vnzj=TE9CG_- z0LuOE1Uz67xEy!^dHj>WL%XhB*bOWI^T0omr#}aL4Y&^&0~LVs_opvF zP7J&oI3M^Oa`G2}?*ZQh?gg5_mB7o$$sYi22TlQ9U>k55@GSE2hk*Nmj{cD_@i0lS2MrIL9mzs34g8xcC zcwVqdaXn}xg{_D({Ysved^gyKj;3NEU&}JJD7eGqiDevutk~WJy`}&Bl~rOoNGIj< z@fnrSWrQnD-^>t5n~|_7X<<@?{y=!46Feg`AN}KD%{0_zZSpNXi`Xa;mOUWTqFhm& zO6pWjC54SDOp~{X6ska1=2?1BgZ?*2nwV^)KcJFou8xZ=M4qHP9U02-OCKFkdYbA= z!&m21n7?HmfuxU335I|`1TmuRh_Dji{IX&mb-e(3<0z-bXTshL4wxK-(<69^usKtk zsqtGTK`i=18@Fw1CS^_$BD32>kE}y59%kIBhZNk8D{>8}C6Kfr<;MlbQ-^)kxKGs= zl8V!IR&3Vx&t}n5Y9f)krmJ{@D*LU<3^D5t-=z3#Rz=b=t^bLX#ayex8865IH{@XQ zOI&tj$ZRz{F^FXXfaXPOqV{ys*^)>3`6FfZfLb@?(~jN{PFPi=9V6A~(r*HaqXpB{ zUXPL(#sD&>QSxDayHRy$-kFq)kJR`RrTj*6PJ#xRBk`mWHO{*qOo2~_3ntm68Yn1^ zIeUtZ{8f(bfsPdHE|QkWhXh+XY4wk!SrQPh_qQf zJ|2TV%^~R|JFX z_L8kiUNl-8^#U)#xxq4aM&ra%lgZY%Fy}{9eSQ$HIA8u_P7o0c#ds%mztMZDDFu7lTEvZFx|#BFOk?0!CRs}ypj z1dzOGUrD8f9vGFG3!2Ja)l0slE0|$n-B5!JCm3yV6h@Dxdp?uig%KH@r>9No%cujo zFxYjbR=&KT-qP(adEtO}RyMET-3?-=5HXE91aP>AcPU}YU0$*L$)VOmvXs17F_RB> jXIa&{+UC@JFgaBHK}WByUpVwkPhO;Kgl$rsUI70F0MhV3 literal 0 HcmV?d00001 diff --git a/tex2Wiki/src/OrthogonalPolynomials.mmd b/tex2Wiki/src/OrthogonalPolynomials.mmd deleted file mode 100644 index cab1d14..0000000 --- a/tex2Wiki/src/OrthogonalPolynomials.mmd +++ /dev/null @@ -1,65 +0,0 @@ -drmf_bof -'''Orthogonal Polynomials''' -{{#set:Section=0}} -== Sections in KLS Chapter 1 == - -
-* [[Orthogonal polynomials|Orthogonal polynomials]] -* [[The gamma and beta function|The gamma and beta function]] -* [[The shifted factorial and binomial coefficients|The shifted factorial and binomial coefficients]] -* [[Hypergeometric functions|Hypergeometric functions]] -* [[The binomial theorem and other summation formulas|The binomial theorem and other summation formulas]] -* [[Some integrals|Some integrals]] -* [[Transformation formulas|Transformation formulas]] -* [[The q-shifted factorial|The ''q''-shifted factorial]] -* [[The q-gamma function and q-binomial coefficients|The ''q''-gamma function and ''q''-binomial coefficients]] -* [[Basic hypergeometric functions|Basic hypergeometric functions]] -* [[The q-binomial theorem and other summation formulas|The ''q''-binomial theorem and other summation formulas]] -* [[More integrals|More integrals]] -* [[Transformation formulas|Transformation formulas]] -* [[Some q-analogues of special functions|Some ''q''-analogues of special functions]] -* [[The q-derivative and q-integral|The ''q''-derivative and ''q''-integral]] -* [[Shift operators and Rodrigues-type formulas|Shift operators and Rodrigues-type formulas]] -
-== Sections in KLS Chapter 1 == - -
-* [[Orthogonal polynomials|Orthogonal polynomials]] -* [[The gamma and beta function|The gamma and beta function]] -* [[The shifted factorial and binomial coefficients|The shifted factorial and binomial coefficients]] -* [[Hypergeometric functions|Hypergeometric functions]] -* [[The binomial theorem and other summation formulas|The binomial theorem and other summation formulas]] -* [[Some integrals|Some integrals]] -* [[Transformation formulas|Transformation formulas]] -* [[The q-shifted factorial|The ''q''-shifted factorial]] -* [[The q-gamma function and q-binomial coefficients|The ''q''-gamma function and ''q''-binomial coefficients]] -* [[Basic hypergeometric functions|Basic hypergeometric functions]] -* [[The q-binomial theorem and other summation formulas|The ''q''-binomial theorem and other summation formulas]] -* [[More integrals|More integrals]] -* [[Transformation formulas|Transformation formulas]] -* [[Some q-analogues of special functions|Some ''q''-analogues of special functions]] -* [[The q-derivative and q-integral|The ''q''-derivative and ''q''-integral]] -* [[Shift operators and Rodrigues-type formulas|Shift operators and Rodrigues-type formulas]] -
-== Sections in KLS Chapter 1 == - -
-* [[Orthogonal polynomials|Orthogonal polynomials]] -* [[The gamma and beta function|The gamma and beta function]] -* [[The shifted factorial and binomial coefficients|The shifted factorial and binomial coefficients]] -* [[Hypergeometric functions|Hypergeometric functions]] -* [[The binomial theorem and other summation formulas|The binomial theorem and other summation formulas]] -* [[Some integrals|Some integrals]] -* [[Transformation formulas|Transformation formulas]] -* [[The q-shifted factorial|The ''q''-shifted factorial]] -* [[The q-gamma function and q-binomial coefficients|The ''q''-gamma function and ''q''-binomial coefficients]] -* [[Basic hypergeometric functions|Basic hypergeometric functions]] -* [[The q-binomial theorem and other summation formulas|The ''q''-binomial theorem and other summation formulas]] -* [[More integrals|More integrals]] -* [[Transformation formulas|Transformation formulas]] -* [[Some q-analogues of special functions|Some ''q''-analogues of special functions]] -* [[The q-derivative and q-integral|The ''q''-derivative and ''q''-integral]] -* [[Shift operators and Rodrigues-type formulas|Shift operators and Rodrigues-type formulas]] -
- -drmf_eof diff --git a/tex2Wiki/src/OrthogonalPolynomials.mmd.new b/tex2Wiki/src/OrthogonalPolynomials.mmd.new deleted file mode 100644 index cab1d14..0000000 --- a/tex2Wiki/src/OrthogonalPolynomials.mmd.new +++ /dev/null @@ -1,65 +0,0 @@ -drmf_bof -'''Orthogonal Polynomials''' -{{#set:Section=0}} -== Sections in KLS Chapter 1 == - -
-* [[Orthogonal polynomials|Orthogonal polynomials]] -* [[The gamma and beta function|The gamma and beta function]] -* [[The shifted factorial and binomial coefficients|The shifted factorial and binomial coefficients]] -* [[Hypergeometric functions|Hypergeometric functions]] -* [[The binomial theorem and other summation formulas|The binomial theorem and other summation formulas]] -* [[Some integrals|Some integrals]] -* [[Transformation formulas|Transformation formulas]] -* [[The q-shifted factorial|The ''q''-shifted factorial]] -* [[The q-gamma function and q-binomial coefficients|The ''q''-gamma function and ''q''-binomial coefficients]] -* [[Basic hypergeometric functions|Basic hypergeometric functions]] -* [[The q-binomial theorem and other summation formulas|The ''q''-binomial theorem and other summation formulas]] -* [[More integrals|More integrals]] -* [[Transformation formulas|Transformation formulas]] -* [[Some q-analogues of special functions|Some ''q''-analogues of special functions]] -* [[The q-derivative and q-integral|The ''q''-derivative and ''q''-integral]] -* [[Shift operators and Rodrigues-type formulas|Shift operators and Rodrigues-type formulas]] -
-== Sections in KLS Chapter 1 == - -
-* [[Orthogonal polynomials|Orthogonal polynomials]] -* [[The gamma and beta function|The gamma and beta function]] -* [[The shifted factorial and binomial coefficients|The shifted factorial and binomial coefficients]] -* [[Hypergeometric functions|Hypergeometric functions]] -* [[The binomial theorem and other summation formulas|The binomial theorem and other summation formulas]] -* [[Some integrals|Some integrals]] -* [[Transformation formulas|Transformation formulas]] -* [[The q-shifted factorial|The ''q''-shifted factorial]] -* [[The q-gamma function and q-binomial coefficients|The ''q''-gamma function and ''q''-binomial coefficients]] -* [[Basic hypergeometric functions|Basic hypergeometric functions]] -* [[The q-binomial theorem and other summation formulas|The ''q''-binomial theorem and other summation formulas]] -* [[More integrals|More integrals]] -* [[Transformation formulas|Transformation formulas]] -* [[Some q-analogues of special functions|Some ''q''-analogues of special functions]] -* [[The q-derivative and q-integral|The ''q''-derivative and ''q''-integral]] -* [[Shift operators and Rodrigues-type formulas|Shift operators and Rodrigues-type formulas]] -
-== Sections in KLS Chapter 1 == - -
-* [[Orthogonal polynomials|Orthogonal polynomials]] -* [[The gamma and beta function|The gamma and beta function]] -* [[The shifted factorial and binomial coefficients|The shifted factorial and binomial coefficients]] -* [[Hypergeometric functions|Hypergeometric functions]] -* [[The binomial theorem and other summation formulas|The binomial theorem and other summation formulas]] -* [[Some integrals|Some integrals]] -* [[Transformation formulas|Transformation formulas]] -* [[The q-shifted factorial|The ''q''-shifted factorial]] -* [[The q-gamma function and q-binomial coefficients|The ''q''-gamma function and ''q''-binomial coefficients]] -* [[Basic hypergeometric functions|Basic hypergeometric functions]] -* [[The q-binomial theorem and other summation formulas|The ''q''-binomial theorem and other summation formulas]] -* [[More integrals|More integrals]] -* [[Transformation formulas|Transformation formulas]] -* [[Some q-analogues of special functions|Some ''q''-analogues of special functions]] -* [[The q-derivative and q-integral|The ''q''-derivative and ''q''-integral]] -* [[Shift operators and Rodrigues-type formulas|Shift operators and Rodrigues-type formulas]] -
- -drmf_eof diff --git a/tex2Wiki/.gitignore b/tex2wiki/.gitignore similarity index 100% rename from tex2Wiki/.gitignore rename to tex2wiki/.gitignore diff --git a/tex2Wiki/README.md b/tex2wiki/README.md similarity index 100% rename from tex2Wiki/README.md rename to tex2wiki/README.md diff --git a/tex2Wiki/src/__init__.py b/tex2wiki/src/__init__.py similarity index 100% rename from tex2Wiki/src/__init__.py rename to tex2wiki/src/__init__.py diff --git a/tex2Wiki/src/tex2Wiki.py b/tex2wiki/src/tex2Wiki.py similarity index 100% rename from tex2Wiki/src/tex2Wiki.py rename to tex2wiki/src/tex2Wiki.py diff --git a/tex2Wiki/src/tex2Wiki2.py b/tex2wiki/src/tex2wiki2.py similarity index 81% rename from tex2Wiki/src/tex2Wiki2.py rename to tex2wiki/src/tex2wiki2.py index cf4af57..0173020 100644 --- a/tex2Wiki/src/tex2Wiki2.py +++ b/tex2wiki/src/tex2wiki2.py @@ -1,10 +1,11 @@ __author__ = "Joon Bang" __status__ = "Prototype" -INPUT_FILE = "tex2Wiki/data/01outb.tex" -OUTPUT_FILE = "tex2Wiki/data/01outb.mmd" -GLOSSARY_LOCATION = "tex2Wiki/src/new.Glossary.csv" -METADATA_TYPES = {"substitution": "Substitution(s)", "constraint": "Constraint(s)"} +INPUT_FILE = "tex2wiki/data/09outb.tex" +OUTPUT_FILE = "tex2wiki/data/09outb.mmd" +GLOSSARY_LOCATION = "tex2wiki/src/new.Glossary.csv" +METADATA_TYPES = ["substitution", "constraint"] +METADATA_MEANING = {"substitution": "Substitution(s)", "constraint": "Constraint(s)"} import csv @@ -88,6 +89,9 @@ def convert_dollar_signs(string): def format_formula(formula): + if formula[0] == ":": + return formula[1:].zfill(2) + formula = multi_split(formula.split("Formula:", 1)[1], [".", ":"]) for j in [-1, -2, -3]: @@ -182,24 +186,16 @@ def get_data_str(text, latex=""): return text[start + len(latex + "{"):find_end(text, "{", "}", start)] -def generate_nav_bar(main_title, hash_code, sections, i, middle=""): - # TODO: Rewrite this function. - - sections = ["Orthogonal Polynomials"] + sections + ["Orthogonal Polynomials"] - - center_text = (main_title + "#" + hash_code).replace(" ", "_") +def generate_nav_bar(info): + links = list() + for link, text in info: + link = link.replace("''", "") + text = text.replace("''", "") + links.append(generate_link(link, text)) - sections = [s.replace("''", "") for s in sections] - - if middle == "": - middle = sections[i + 1] - - nav_section = generate_html("div", "<< " + generate_link(sections[i]), options={"id": "alignleft"}, - spacing=1) + "\n" + \ - generate_html("div", generate_link(center_text, middle), options={"id": "aligncenter"}, - spacing=1) + "\n" + \ - generate_html("div", generate_link(sections[i + 2]) + " >>", options={"id": "alignright"}, - spacing=1) + nav_section = generate_html("div", "<< " + links[0], options={"id": "alignleft"}, spacing=1) + "\n" + \ + generate_html("div", links[1], options={"id": "aligncenter"}, spacing=1) + "\n" + \ + generate_html("div", links[2] + " >>", options={"id": "alignright"}, spacing=1) header = generate_html("div", nav_section, options={"id": "drmf_head"}) footer = generate_html("div", nav_section, options={"id": "drmf_foot"}) @@ -275,26 +271,33 @@ def get_symbols(text, glossary): def create_general_pages(data, title): ret = "" - section_names = [d[0] for d in data] # list of section names + # get list of section names + section_names = [d[0] for d in data] + section_names = ["Orthogonal Polynomials"] + section_names + ["Orthogonal Polynomials"] for i, section_data in enumerate(data): section_name = section_data[0] result = "drmf_bof\n'''" + section_name.replace("''", "") + "'''\n{{DISPLAYTITLE:" + section_name + "}}\n" - # header/footer code - header, footer = generate_nav_bar("Orthogonal Polynomials", "Sections in " + title, section_names, i) + # get header and footer + center_text = ("Orthogonal Polynomials" + "#Sections in " + title).replace(" ", "_") + link_info = [[section_names[i], section_names[i]], [center_text, section_names[i + 1]], + [section_names[i + 2], section_names[i + 2]]] + header, footer = generate_nav_bar(link_info) result += header + "\n== " + section_name + " ==\n\n" text = "" for eq in section_data[1]: # equation is of type LatexEquation + print eq + print text += generate_math_html(eq.equation, options={"id": eq.label}) metadata_exists = False for data_type in sorted(eq.metadata.keys()): if eq.metadata[data_type] != "": - text += generate_html("div", METADATA_TYPES[data_type] + ": " + eq.metadata[data_type], + text += generate_html("div", METADATA_MEANING[data_type] + ": " + eq.metadata[data_type], options={"align": "right"}, spacing=False) + "
\n" metadata_exists = True @@ -308,9 +311,7 @@ def create_general_pages(data, title): return ret -def create_specific_pages(data): - pages = list() - +def create_specific_pages(data, glossary): formulae = list() for section_data in data: for equations in section_data[1:]: @@ -318,41 +319,50 @@ def create_specific_pages(data): formulae.append("Formula:" + eq.label) formulae.append(section_data[0]) - print formulae - - glossary = dict() - with open(GLOSSARY_LOCATION, "rb") as csv_file: - glossary_file = csv.reader(csv_file, delimiter=',', quotechar='\"') - for row in glossary_file: - glossary[get_macro_name(row[0])] = row + formulae = ["Orthogonal Polynomials"] + formulae[:-1] + ["Orthogonal Polynomials"] i = 0 + pages = list() for j, section_data in enumerate(data): section_name = section_data[0] for eq in section_data[1]: - header, footer = generate_nav_bar(section_name, eq.label, formulae, i, middle="formula in " + section_name) - + # get header and footer + center_text = (section_name + "#" + eq.label).replace(" ", "_") + middle = "formula in " + section_name + link_info = [[formulae[i].replace(" ", "_"), formulae[i]], [center_text, middle], + [formulae[i + 2].replace(" ", "_"), formulae[i + 2]]] + header, footer = generate_nav_bar(link_info) + + # add title of page, navigation headers result = "drmf_bof\n'''Formula:" + eq.label + "'''\n{{DISPLAYTITLE:Formula:" + eq.label + "}}\n" + header \ + "\n
" + # add formula result += generate_html("div", generate_math_html(eq.equation)[:-1], options={"align": "center"}, spacing=0) + "\n\n" + # add metadata for data_type, info in eq.metadata.iteritems(): if info != "": - result += "== " + METADATA_TYPES[data_type] + " ==\n\n" + result += "== " + METADATA_MEANING[data_type] + " ==\n\n" result += generate_html("div", info, options={"align": "left"}, spacing=0) + "
\n\n" + # proof section result += "== Proof ==\n\nWe ask users to provide proof(s), reference(s) to proof(s), or further " \ - "clarification on the proof(s) in this space.\n\n== Symbols List ==\n\n" + "clarification on the proof(s) in this space.\n\n" + + # symbols list section + result += "== Symbols List ==\n\n" result += get_symbols(result, glossary) + "\n
\n\n" + # bibliography section result += "== Bibliography ==\n\n" result += "[http://homepage.tudelft.nl/11r49/askey/contents.html " \ "Equation in Section 1." + str(j + 1) + "] of [[Bibliography#KLS|'''KLS''']]." result += "\n\n== URL links ==\n\nWe ask users to provide relevant URL links in this space.\n\n" + # end of page result += "
" + footer + "\ndrmf_eof" pages.append(result) @@ -367,6 +377,12 @@ def main(): with open(INPUT_FILE) as input_file: text = input_file.read() + glossary = dict() + with open(GLOSSARY_LOCATION, "rb") as csv_file: + glossary_file = csv.reader(csv_file, delimiter=',', quotechar='\"') + for row in glossary_file: + glossary[get_macro_name(row[0])] = row + text = text.split("\\begin{document}", 1)[1] text = text.split("\\section") @@ -382,7 +398,7 @@ def main(): data.append([section_name, equations]) data = extract_data(data) - output = create_general_pages(data, title) + create_specific_pages(data) + output = create_general_pages(data, title) + create_specific_pages(data, glossary) with open(OUTPUT_FILE, "w") as output_file: output_file.write(output) From 93f4c9be8b090fe60d04137d5c71d99528832f2d Mon Sep 17 00:00:00 2001 From: Joon Bang Date: Thu, 18 Aug 2016 11:09:37 -0400 Subject: [PATCH 283/402] Update .gitignore --- tex2wiki/.gitignore | 5 ++++- 1 file changed, 4 insertions(+), 1 deletion(-) diff --git a/tex2wiki/.gitignore b/tex2wiki/.gitignore index 00a7b41..1fa607a 100644 --- a/tex2wiki/.gitignore +++ b/tex2wiki/.gitignore @@ -1,2 +1,5 @@ AzeemOldCode/ -*Glossary.csv* \ No newline at end of file +*Glossary.csv* +*.mmd +*.bak +*.tex \ No newline at end of file From 6340bc665775f781ad440eed3192ad1402af9653 Mon Sep 17 00:00:00 2001 From: Joon Bang Date: Thu, 18 Aug 2016 11:12:43 -0400 Subject: [PATCH 284/402] Remove old tex2wiki.py --- main_page/src/.functions.py.swp | Bin 16384 -> 0 bytes tex2wiki/src/tex2Wiki.py | 1080 -------------------- tex2wiki/src/{tex2wiki2.py => tex2wiki.py} | 6 +- 3 files changed, 3 insertions(+), 1083 deletions(-) delete mode 100644 main_page/src/.functions.py.swp delete mode 100644 tex2wiki/src/tex2Wiki.py rename tex2wiki/src/{tex2wiki2.py => tex2wiki.py} (98%) diff --git a/main_page/src/.functions.py.swp b/main_page/src/.functions.py.swp deleted file mode 100644 index 876aecd1e0a7fac875906508349952435f2fd69e..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 16384 zcmeHOYm6jS6~2ngOP6P2)C9cLnoKv_(=)S>Xxiy@U+%*!%w~5HvAff$?z%l)OLbLK zRXwxg!Y+#@VuFt#e~1P#iV~wl5E2Y15c#Lkm>57!5KWN8#Ngu>V#M#Yz=bm$`r#Juc1K05n)~{u_E@3S6AHDH&^X{e{`x!gt_dZ0$XS<^D z#Hsh~QQdc=xV{{$Y|!8&37p$=_`1EbM}%-(e|h#$7#tJ*cy467es^pyi@7N5&kZJi zKXwB@s*l$;kX06!43rGKRR%V*Yql-SQ>!i3Rs8bHZg{H#%K9Y(B?Bb`B?Bb`B?Bb` zB?Bb`B?JG%48-Ge*eQ(v?OF%=o36L1ujKm;eF;v8pTnStOy!KAU{s?>nxEuHk-~-nK*8&{4 z0C@dE#(obx3;YE50q_X$FmMlW8dwBofPY=U*w29<13v;D1Wp16f%gD^KcBI`0Z#z; 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" - length = 0 - elif c == "\n" and per == 0: - stringWrite = stringWrite.strip() - stringWrite += c - per += 1 - elif count > 0 and per == 0: - stringWrite += c - - return (stringWrite.lstrip()) - - -def getSym(line): # Gets all symbols on a line for symbols list - symList = [] - if line == "": - return symList - symbol = "" - symFlag = False - argFlag = False - cC = 0 - for i in range(0, len(line)): - # if "MeixnerPollaczek{\\lambda}{n}@{x}{\\phi}=\frac{\\pochhammer{2\\lambda}{n}}" in line: - c = line[i] - if symFlag: - if c == "{" or c == "[": - cC += 1 - argFlag = True - if c != "}" and c != "]": - if argFlag or c.isalpha(): - symbol += c - else: - symFlag = False - argFlag = False - symList.append(symbol) - symList += (getSym(symbol)) - symbol = "" - else: - cC -= 1 - symbol += c - if i + 1 == len(line): - p = "" - else: - p = line[i + 1] - if cC <= 0 and p != "{" and p != "[" and p != "@": - symFlag = False - # if "monicAskeyWilson{n+1}@{x}{a}{b}{c}{d}{q}+\\frac{1}{2}" in line: - symList.append(symbol) - symList += (getSym(symbol)) - argFlag = False - symbol = "" - - elif c == "\\": - symFlag = True - elif c == "&" and not (i > line.find("\\begin{array}") and i < line.find("\\end{array}")): - symList.append("&") - - # if "MeixnerPollaczek{\\lambda}{n}@{x}{\\phi}=\frac{\\pochhammer{2\\lambda}{n}}" in line: - symList.append(symbol) - symList += getSym(symbol) - return (symList) - - -def unmodLabel(label): - label = label.replace(".0", ".") - label = label.replace(":0", ":") - return (label) - - -def secLabel(label): - return (label.replace("\'\'", "")) - - -def modLabel(label): - # label.replace("Formula:KLS:","KLS;") - isNumer = False - newlabel = "" - num = "" - for i in range(0, len(label)): - if isNumer and not label[i][0].isdigit(): - if len(num) > 1: - newlabel += num - isNumer = False - num = "" - else: - newlabel += "0" + str(num) - num = "" - isNumer = False - if label[i][0].isdigit(): - isNumer = True - num += str(label[i]) - else: - isNumer = False - newlabel += label[i] - if len(num) > 1: - newlabel += num - elif len(num) == 1: - newlabel += "0" + num - return (newlabel) - - -def main(): - wiki = open(sys.argv[2], 'w') - with open(sys.argv[1], 'r') as tex: - lines = tex.readlines() - - with open("OrthogonalPolynomials.mmd", "r") as main: - mainText = main.read() - - mainPrepend = "" - mainWrite = open("OrthogonalPolynomials.mmd.new", "w") - math = False - constraint = False - substitution = False - parse = False - head = False - symbols = [] - refLines = [] - sections = [] - labels = [] - eqs = [] - refEqs = [] - refLabels = [] - - for i in xrange(len(refLines)): - line = refLines[i] - if "\\begin{equation}" in line: - sLabel = line.find("\\formula{") + 9 - eLabel = line.find("}", sLabel) - label = modLabel(line[sLabel:eLabel]) - '''for l in lLink: - if l.find(label)!=-1 and l[len(label)+1]=="=": - rlabel=l[l.find("=>")+3:l.find("\\n")] - rlabel=rlabel.replace("/","") - rlabel=rlabel.replace("#",":") - break''' - refLabels.append(label) - refEqs.append("") - math = True - elif math: - refEqs[len(refEqs) - 1] += line - if "\\end{equation}" in refLines[i + 1] or "\\constraint" in refLines[i + 1] or "\\substitution" in \ - refLines[i + 1] or "\\drmfn" in refLines[i + 1]: - math = False - - for i in range(0, len(lines)): - line = lines[i] - if "\\begin{document}" in line: - # wiki.write("drmf_bof\n") - parse = True - elif "\\end{document}" in line and parse: - # wiki.write("
\n") - mainPrepend += "
\n" - mainText = mainPrepend + mainText - mainText = mainText.replace("drmf_bof\n", "") - mainText = mainText.replace("drmf_eof\n", "") - mainText = mainText.replace("\'\'\'Orthogonal Polynomials\'\'\'\n", "") - mainText = mainText.replace("{{#set:Section=0}}\n", "") - mainText = mainText[0:mainText.rfind("== Sections ")] - mainText = "drmf_bof\n\'\'\'Orthogonal Polynomials\'\'\'\n{{#set:Section=0}}" + mainText + "\ndrmf_eof\n" - mainWrite.write(mainText) - mainWrite.close() - os.system("cp -f OrthogonalPolynomials.mmd.new OrthogonalPolynomials.mmd") - # wiki.write("\ndrmf_eof\n") - parse = False - elif "\\title" in line and parse: - stringWrite = "\'\'\'" - stringWrite += get_data_str(line) + "\'\'\'\n" - labels.append("Orthogonal Polynomials") - sections.append(["Orthogonal Polynomials", 0]) - # wiki.write(stringWrite) - # mainPrepend+=stringWrite - chapter = get_data_str(line) - mainPrepend += ( - "\n== Sections in " + chapter + " ==\n\n
\n") - elif "\\part" in line: - if get_data_str(line) == "BOF": - parse = False - elif get_data_str(line) == "EOF": - parse = True - elif parse: - mainPrepend += ("\n
\n= " + get_data_str(line) + " =\n") - head = True - elif "\\section" in line: - mainPrepend += ("* [[" + secLabel(get_data_str(line)) + "|" + get_data_str(line) + "]]\n") - sections.append([get_data_str(line)]) - - secCounter = 0 - eqCounter = 0 - for i in range(0, len(lines)): - line = lines[i] - # line.replace("\\begin{equation}","$") - # line.replace("\\end{equation},"$") - '''if "\\begin{document}" in line: - wiki.write("drmf_bof\n") - parse=True - elif "\\end{document}" in line and parse: - wiki.write("\ndrmf_eof\n") - parse=False - elif "\\title" in line and parse: - stringWrite="\'\'\'" - stringWrite+=getString(line)+"\'\'\'\n" - labels.append("Orthogonal Polynomials") - wiki.write(stringWrite\n) - elif "\\part" in line: - if getString(line)=="BOF": - parse=False - elif getString(line)=="EOF": - parse=True - elif parse: - wiki.write("\n
\n= "+getString(line)+" =\n") - head=True - ''' - if "\\section" in line: - parse = True - secCounter += 1 - wiki.write("drmf_bof\n") - wiki.write("\'\'\'" + secLabel(get_data_str(line)) + "\'\'\'\n") - wiki.write("{{DISPLAYTITLE:" + (sections[secCounter][0]) + "}}\n") - wiki.write("
\n") - wiki.write("
<< [[" + secLabel(sections[secCounter - 1][0]) + "|" + secLabel( - sections[secCounter - 1][0]) + "]]
\n") - # wiki.write("
[[Orthogonal_Polynomials#"+secLabel(sections[secCounter][0])+"|"+secLabel(sections[secCounter][0])+"]]
\n") - wiki.write("
[[Orthogonal_Polynomials#" + "Sections_in_" + chapter.replace(" ", - "_") + "|" + secLabel( - sections[secCounter][0]) + "]]
\n") - wiki.write( - "
[[" + secLabel(sections[(secCounter + 1) % len(sections)][0]) + "|" + secLabel( - sections[(secCounter + 1) % len(sections)][0]) + "]] >>
\n
\n\n") - # wiki.write("{{head|pre="+secLabel(sections[secCounter-1][0])+"|cur="+secLabel(sections[secCounter][0])+"|next="+secLabel(sections[(secCounter+1)%len(sections)][0])+"}}\n") - # wiki.write("{{#set:Chapter="+chapter+"}}\n") - # wiki.write("{{#set:Section="+str(secCounter)+"}}\n") - # wiki.write("{{head}}\n") - head = True - wiki.write("== " + get_data_str(line) + " ==\n") - elif ("\\section" in lines[(i + 1) % len(lines)] or "\\end{document}" in lines[(i + 1) % len(lines)]) and parse: - wiki.write("
\n") - wiki.write("
<< [[" + secLabel(sections[secCounter - 1][0]) + "|" + secLabel( - sections[secCounter - 1][0]) + "]]
\n") - # wiki.write("
[[Orthogonal_Polynomials#"+secLabel(sections[secCounter][0])+"|"+secLabel(sections[secCounter][0])+"]]
\n") - wiki.write("
[[Orthogonal_Polynomials#" + "Sections_in_" + chapter.replace(" ", - "_") + "|" + secLabel( - sections[secCounter][0]) + "]]
\n") - wiki.write( - "
[[" + secLabel(sections[(secCounter + 1) % len(sections)][0]) + "|" + secLabel( - sections[(secCounter + 1) % len(sections)][0]) + "]] >>
\n
\n\n") - # wiki.write("{{foot|pre="+secLabel(sections[secCounter-1][0])+"|cur="+secLabel(sections[secCounter][0])+"|next="+secLabel(sections[(secCounter+1)%len(sections)][0])+"}}\n") - # wiki.write("{{foot}}\n") - wiki.write("drmf_eof\n") - sections[secCounter].append(eqCounter) - eqCounter = 0 - - elif "\\subsection" in line and parse: - wiki.write("\n== " + get_data_str(line) + " ==\n") - head = True - elif "\\paragraph" in line and parse: - wiki.write("\n=== " + get_data_str(line) + " ===\n") - head = True - elif "\\subsubsection" in line and parse: - wiki.write("\n=== " + get_data_str(line) + " ===\n") - head = True - - elif "\\begin{equation}" in line and parse: - # symLine="" - if head: - wiki.write("\n") - head = False - sLabel = line.find("\\formula{") + 9 - eLabel = line.find("}", sLabel) - label = modLabel(line[sLabel:eLabel]) - eqCounter += 1 - '''for l in lLink: - if label==l[0:l.find("=")-1]: - rlabel=l[l.find("=>")+3:l.find("\\n")] - rlabel=rlabel.replace("/","") - rlabel=rlabel.replace("#",":") - rlabel=rlabel.replace("!",":") - break''' - labels.append(label) - eqs.append("") - # wiki.write("\n\n") - wiki.write("{\displaystyle\n") - math = True - elif "\\begin{equation}" in line and not parse: - sLabel = line.find("\\formula{") + 9 - eLabel = line.find("}", sLabel) - label = modLabel(line[sLabel:eLabel]) - '''for l in lLink: - if l.find(label)!=-1: - rlabel=l[l.find("=>")+3:l.find("\\n")] - rlabel=rlabel.replace("/","") - rlabel=rlabel.replace("#",":") - break''' - labels.append("*" + label) # special marker - eqs.append("") - math = True - elif "\\end{equation}" in line: - - math = False - elif "\\constraint" in line and parse: - constraint = True - math = False - conLine = "" - elif "\\substitution" in line and parse: - substitution = True - math = False - subLine = "" - # wiki.write("
Substitution(s): "+getEq(line)+"

\n") - elif "\\proof" in line and parse: - math = False - elif "\\drmfn" in line and parse: - math = False - if "\\drmfname" in line and parse: - wiki.write("
This formula has the name: " + get_data_str(line) + "

\n") - elif math and parse: - flagM = True - eqs[len(eqs) - 1] += line - - if "\\end{equation}" in lines[i + 1] and not "\\subsection" in lines[i + 3] and not "\\section" in lines[ - i + 3] and not "\\part" in lines[i + 3]: - u = i - flagM2 = False - while flagM: - u += 1 - if "\\begin{equation}" in lines[u] in lines[u]: - flagM = False - if "\\section" in lines[u] or "\\subsection" in lines[i] or "\\part" in lines[ - u] or "\\end{document}" in lines[u]: - flagM = False - flagM2 = True - if not (flagM2): - - wiki.write(line.rstrip("\n")) - wiki.write("\n}
\n") - else: - wiki.write(line.rstrip("\n")) - wiki.write("\n}\n") - elif "\\end{equation}" in lines[i + 1]: - wiki.write(line.rstrip("\n")) - wiki.write("\n}\n") - elif "\\constraint" in lines[i + 1] or "\\substitution" in lines[i + 1] or "\\drmfn" in lines[i + 1]: - wiki.write(line.rstrip("\n")) - wiki.write("\n}\n") - else: - wiki.write(line) - elif math and not parse: - eqs[len(eqs) - 1] += line - if "\\end{equation}" in lines[i + 1] or "\\constraint" in lines[i + 1] or "\\substitution" in lines[ - i + 1] or "\\drmfn" in lines[i + 1]: - math = False - if substitution and parse: - subLine = subLine + line # .replace("&","&
") - if "\\end{equation}" in lines[i + 1] or "\\substitution" in lines[i + 1] or "\\constraint" in lines[ - i + 1] or "\\drmfn" in lines[i + 1] or "\\proof" in lines[i + 1]: - lineR = "" - for i in range(0, len(subLine)): - if subLine[i] == "&" and not ( - i > subLine.find("\\begin{array}") and i < subLine.find("\\end{array}")): - lineR += "&
" - else: - lineR += subLine[i] - substitution = False - wiki.write("
Substitution(s): " + getEq(lineR) + "

\n") - - if constraint and parse: - conLine = conLine + line.replace("&", "&
") - if "\\end{equation}" in lines[i + 1] or "\\substitution" in lines[i + 1] or "\\constraint" in lines[ - i + 1] or "\\drmfn" in lines[i + 1] or "\\proof" in lines[i + 1]: - constraint = False - wiki.write("
Constraint(s): " + getEq(conLine) + "

\n") - - eqCounter = 0 - endNum = len(labels) - 1 - '''for n in range(1,len(labels)): - if n+1!=len(labels) and labels[n+1][0]=="*" and labels[n][0]!="*": - labels[n]=""+labels[n] - endNum=n - elif labels[n][0]=="*": - labels[n]=""+labels[n][1:] - else: - labels[n]=""+labels[n]''' - '''for n in range(0,len(refLabels)): - refLabels[n]=""+refLabels[n]''' - parse = False - constraint = False - substitution = False - note = False - hCon = True - hSub = True - hNote = True - hProof = True - proof = False - comToWrite = "" - secCount = -1 - newSec = False - for i in range(0, len(lines)): - line = lines[i] - - if "\\section" in line: - secCount += 1 - newSec = True - eqS = 0 - - if "\\begin{equation}" in line: - symLine = line.strip("\n") - eqS += 1 - constraint = False - substitution = False - note = False - comToWrite = "" - hCon = True - hSub = True - hNote = True - hProof = True - proof = False - parse = True - symbols = [] - eqCounter += 1 - wiki.write("drmf_bof\n") - label = labels[eqCounter] - wiki.write("\'\'\'" + secLabel(label) + "\'\'\'\n") - wiki.write("{{DISPLAYTITLE:" + (labels[eqCounter]) + "}}\n") - if eqCounter < endNum: # FOR ANYTHING THAT IS NOT THE EXTRA EQUATIONS - wiki.write("
\n") - if newSec: - wiki.write("
<< [[" + secLabel(sections[secCount][0]).replace(" ", - "_") + "|" + secLabel( - sections[secCount][0]) + "]]
\n") - else: - wiki.write("
<< [[" + secLabel(labels[eqCounter - 1]).replace(" ", - "_") + "|" + secLabel( - labels[eqCounter - 1]) + "]]
\n") - wiki.write("
[[" + secLabel(sections[secCount + 1][0]).replace(" ", - "_") + "#" + secLabel( - labels[eqCounter][len("Formula:"):]) + "|formula in " + secLabel( - sections[secCount + 1][0]) + "]]
\n") - # if eqS==sections[secCount][1]: - # wiki.write("
[["+secLabel(sections[(secCount+1)%len(sections)][0]).replace(" ","_")+"|"+secLabel(sections[(secCount+1)%len(sections)][0])+"]] >>
\n") - # else: - if True: - wiki.write( - "
[[" + secLabel(labels[(eqCounter + 1) % (endNum + 1)]).replace(" ", - "_") + "|" + secLabel( - labels[(eqCounter + 1) % (endNum + 1)]) + "]] >>
\n") - wiki.write("
\n\n") - elif eqCounter == endNum: - wiki.write("
\n") - if newSec: - newSec = False - wiki.write("
<< [[" + secLabel(sections[secCount][0]).replace(" ", - "_") + "|" + secLabel( - sections[secCount][0]) + "]]
\n") - else: - wiki.write("
<< [[" + secLabel(labels[eqCounter - 1]).replace(" ", - "_") + "|" + secLabel( - labels[eqCounter - 1]) + "]]
\n") - wiki.write("
[[" + secLabel(sections[secCount + 1][0]).replace(" ", - "_") + "#" + secLabel( - labels[eqCounter][len("Formula:"):]) + "|formula in " + secLabel( - sections[secCount + 1][0]) + "]]
\n") - wiki.write("
[[" + secLabel( - labels[(eqCounter + 1) % (endNum + 1)].replace(" ", "_")) + "|" + secLabel( - labels[(eqCounter + 1) % (endNum + 1)]) + "]]
\n") - wiki.write("
\n\n") - - wiki.write("
{\displaystyle\n") - math = True - elif "\\end{equation}" in line: - wiki.write(comToWrite) - parse = False - math = False - if hProof: - wiki.write( - "\n== Proof ==\n\nWe ask users to provide proof(s), reference(s) to proof(s), or further clarification on the proof(s) in this space.\n") - - wiki.write("\n== Symbols List ==\n\n") - newSym = [] - # if "09.07:04" in label: - for x in symbols: - flagA = True - # if x not in newSym: - cN = 0 - cC = 0 - flag = True - ArgCx = 0 - for z in x: - if z.isalpha() and flag or z == "&": - cN += 1 - else: - flag = False - if z == "{" or z == "[": - cC += 1 - if z == "}" or z == "]": - cC -= 1 - if cC == 0: - ArgCx += 1 - noA = x[:cN] - for y in newSym: - cN = 0 - cC = 0 - ArgC = 0 - flag = True - for z in y: - if z.isalpha() and flag or z == "&": - cN += 1 - else: - flag = False - if z == "{" or z == "[": - cC += 1 - if z == "}" or z == "]": - cC -= 1 - if cC == 0: - ArgC += 1 - - if y[:cN] == noA: # and ArgC==ArgCx: - flagA = False - break - if flagA: - newSym.append(x) - newSym.reverse() - symF = False - ampFlag = False - finSym = [] - for s in range(len(newSym) - 1, -1, -1): - symbolPar = "\\" + newSym[s] - ArgCx = 0 - parCx = 0 - parFlag = False - cC = 0 - for z in symbolPar: - if z == "@": - parFlag = True - elif z.isalpha() or z == "&": - cN += 1 - else: - if z == "{" or z == "[": - cC += 1 - if z == "}" or z == "]": - cC -= 1 - if cC == 0: - if parFlag: - parCx += 1 - else: - ArgCx += 1 - - if symbolPar.find("{") != -1 or symbolPar.find("[") != -1: - if symbolPar.find("[") == -1: - symbol = symbolPar[0:symbolPar.find("{")] - elif symbolPar.find("{") == -1 or symbolPar.find("[") < symbolPar.find("{"): - symbol = symbolPar[0:symbolPar.find("[")] - else: - symbol = symbolPar[0:symbolPar.find("{")] - else: - symbol = symbolPar - numArg = parCx - numPar = ArgCx - gFlag = False - checkFlag = False - get = False - gCSV = csv.reader(open('new.Glossary.csv', 'rb'), delimiter=',', quotechar='\"') - preG = "" - if symbol == "\\&": - ampFlag = True - for S in gCSV: - G = S - ArgCx = 0 - parCx = 0 - parFlag = False - cC = 0 - ind = G[0].find("@") - if ind == -1: - ind = len(G[0]) - 1 - for z in G[0]: - if z == "@": - parFlag = True - elif z.isalpha(): - cN += 1 - else: - if z == "{" or z == "[": - cC += 1 - if z == "}" or z == "]": - cC -= 1 - if cC == 0: - if parFlag: - parCx += 1 - else: - ArgCx += 1 - if G[0].find(symbol) == 0 and (len(G[0]) == len(symbol) or not G[0][ - len(symbol)].isalpha()): # and (numPar!=0 or numArg!=0): - checkFlag = True - get = True - preG = S - - elif checkFlag: - get = True - checkFlag = False - - if (get): - if get: - G = preG - get = False - checkFlag = False - if True: - if symbolPar.find("@") != -1: - Q = symbolPar[:symbolPar.find("@")] - else: - Q = symbolPar - listArgs = [] - if len(Q) > len(symbol) and (Q[len(symbol)] == "{" or Q[len(symbol)] == "["): - ap = "" - for o in range(len(symbol), len(Q)): - if Q[o] == "{" or z == "[": - argFlag = True - elif Q[o] == "}" or z == "]": - argFlag = False - listArgs.append(ap) - ap = "" - else: - ap += Q[o] - '''websiteU=g[g.find("http://"):].strip("\n") - k=0 - websites=[] - for r in range(0,len(websiteU)): - if websiteU[r]==",": - if websiteU[k:r].find("http://")!=-1: - websites.append(websiteU[k:r+1].strip(" ")) - k=r - - - if websiteU[k:r].find("http://")!=-1: - websites.append(websiteU[k:r+1].strip(" ")) - websiteF="" - for d in websites: - websiteF=websiteF+" ["+d+" "+d+"]"''' - # websiteF=G[4].strip("\n") - websiteF = "" - web1 = G[5] - for t in range(5, len(G)): - if G[t] != "": - websiteF = websiteF + " [" + G[t] + " " + G[t] + "]" - - # p1=Q - # if Q.find("@")!=-1: - # p1=Q[:Q.find("@")] - p1 = G[4].strip("$") - p1 = "{\\displaystyle " + p1 + "}" - # if checkFlag: - new1 = "" - new2 = "" - pause = False - mathF = True - '''for k in range(0,len(p1)): - if p1[k]=="$": - if mathF: - new1+="{\\displaystyle " - else: - new1+="}" - mathF=not mathF - - elif p1[k]=="#" and p1[k+1].isdigit(): - pause=True - elif pause: - num=int(p1[k]) - #letter=chr(num+96) - letter=listArgs[num-1] - new1+=letter - pause=False - - else: - new1+=p1[k]''' - p2 = G[1] - for k in range(0, len(p2)): - if p2[k] == "$": - if mathF: - new2 += "{\\displaystyle " - else: - new2 += "}" - mathF = not mathF - else: - new2 += p2[k] - # p1=new1 - p2 = new2 - finSym.append(web1 + " " + p1 + "] : " + p2 + " :" + websiteF) - break - # gFlag=True - # if not symF: - # symF=True - # wiki.write("[") - # else: - # wiki.write("
\n") - # wiki.write("[") - # wiki.write(g[g.find("http://"):].strip("\n")) - # wiki.write(" ") - # wiki.write(getEq(g[g.find("{$"):]).strip("\n").replace("\\\\","\\")) - # wiki.write("] : ") - # wiki.write(g[g.find(" {")+2:g.find("} ",g.find(" {")+1)]) - # wiki.write(" : [") - # wiki.write(g[g.find("http://"):].strip("\n")) - # wiki.write(" ") - # wiki.write(g[g.find("http://"):].strip("\n")) - # wiki.write("] ")''' - - # preG=S - if not gFlag: - del newSym[s] - - gFlag = True - # finSym.reverse() - if ampFlag: - wiki.write("& : logical and") - gFlag = False - for y in finSym: - if y == "& : logical and": - pass - elif gFlag: - gFlag = False - wiki.write("[" + y) - else: - wiki.write("
\n[" + y) - - wiki.write("\n
\n") - - wiki.write("\n== Bibliography==\n\n") # should there be a space between bibliography and ==? - r = unmodLabel(labels[eqCounter]) - q = r.find("KLS:") + 4 - p = r.find(":", q) - section = r[q:p] - equation = r[p + 1:] - if equation.find(":") != -1: - equation = equation[0:equation.find(":")] - - wiki.write( - "[http://homepage.tudelft.nl/11r49/askey/contents.html Equation in Section " + section + "] of [[Bibliography#KLS|'''KLS''']].\n\n") # Where should it link to? - wiki.write("== URL links ==\n\nWe ask users to provide relevant URL links in this space.\n\n") - if eqCounter < endNum: - wiki.write("
\n") - if newSec: - newSec = False - wiki.write("
<< [[" + secLabel(sections[secCount][0]).replace(" ", - "_") + "|" + secLabel( - sections[secCount][0]) + "]]
\n") - else: - wiki.write("
<< [[" + secLabel(labels[eqCounter - 1]).replace(" ", - "_") + "|" + secLabel( - labels[eqCounter - 1]) + "]]
\n") - wiki.write("
[[" + secLabel(sections[secCount + 1][0]).replace(" ", - "_") + "#" + secLabel( - labels[eqCounter][len("Formula:"):]) + "|formula in " + secLabel( - sections[secCount + 1][0]) + "]]
\n") - # if eqS==sections[secCount][1]: - # wiki.write("
[["+sections[(secCount+1)%len(sections)][0].replace(" ","_")+"|"+sections[(secCount+1)%len(sections)][0]+"]] >>
\n") - # else: - if True: - wiki.write( - "
[[" + secLabel(labels[(eqCounter + 1) % (endNum + 1)]).replace(" ", - "_") + "|" + secLabel( - labels[(eqCounter + 1) % (endNum + 1)]) + "]] >>
\n") - wiki.write("
\n\ndrmf_eof\n") - else: # FOR EXTRA EQUATIONS - wiki.write("
\n") - wiki.write("
<< [[" + labels[endNum - 1].replace(" ", "_") + "|" + labels[ - endNum - 1] + "]]
\n") - wiki.write("
[[" + labels[0].replace(" ", "_") + "#" + labels[endNum][ - 8:] + "|formula in " + - labels[0] + "]]
\n") - wiki.write("
[[" + labels[(0) % endNum].replace(" ", "_") + "|" + labels[ - 0 % endNum] + "]]
\n") - wiki.write("
\n\ndrmf_eof\n") - - - - elif "\\constraint" in line and parse: - # symbols=symbols+getSym(line) - symLine = line.strip("\n") - if hCon: - comToWrite = comToWrite + "\n== Constraint(s) ==\n\n" - hCon = False - constraint = True - math = False - conLine = "" - # wiki.write("
"+getEq(line)+"

\n") - elif "\\substitution" in line and parse: - # symbols=symbols+getSym(line) - symLine = line.strip("\n") - if hSub: - comToWrite = comToWrite + "\n== Substitution(s) ==\n\n" - hSub = False - # wiki.write("
"+getEq(line)+"

\n") - substitution = True - math = False - subLine = "" - elif "\\drmfname" in line and parse: - math = False - comToWrite = "\n== Name ==\n\n
" + get_data_str(line) + "

\n" + comToWrite - - elif "\\drmfnote" in line and parse: - symbols = symbols + getSym(line) - if hNote: - comToWrite = comToWrite + "\n== Note(s) ==\n\n" - hNote = False - note = True - math = False - noteLine = "" - - elif "\\proof" in line and parse: - # symbols=symbols+getSym(line) - symLine = line.strip("\n") - if hProof: - hProof = False - comToWrite = comToWrite + "\n== Proof ==\n\nWe ask users to provide proof(s), reference(s) to proof(s), or further clarification on the proof(s) in this space. \n

\n
" - proof = True - proofLine = "" - pause = False - pauseP = False - for ind in range(0, len(line)): - if line[ind:ind + 7] == "\\eqref{": - pause = True - eqR = line[ind:line.find("}", ind) + 1] - rLab = get_data_str(eqR) - '''for l in lLink: - if rLab==l[0:l.find("=")-1]: - rlabel=l[l.find("=>")+3:l.find("\\n")] - rlabel=rlabel.replace("/","") - rlabel=rlabel.replace("#",":") - rlabel=rlabel.replace("!",":") - break''' - - eInd = refLabels.index("" + rLab) - z = line[line.find("}", ind + 7) + 1] - if z == "." or z == ",": - pauseP = True - proofLine += ("
\n{\displaystyle\n" + refEqs[ - eInd] + "}" + z + "
\n") - else: - if z == "}": - proofLine += ( - "
\n{\displaystyle\n" + refEqs[eInd] + "}
") - else: - proofLine += ( - "
\n{\displaystyle\n" + refEqs[eInd] + "}
\n") - - - else: - if pause: - if line[ind] == "}": - pause = False - elif pauseP: - pauseP = False - elif line[ind] == "\n" and "\\end{equation}" in lines[i + 1]: - pass - else: - proofLine += (line[ind]) - if "\\end{equation}" in lines[i + 1]: - proof = False - # symLine+=line.strip("\n") - wiki.write(comToWrite + getEqP(proofLine, False) + "
\n
\n") - comToWrite = "" - symbols = symbols + getSym(symLine) - symLine = "" - - # wiki.write(line) - - elif proof: - symLine += line.strip("\n") - pauseP = False - # symbols=symbols+getSym(line) - for ind in range(0, len(line)): - if line[ind:ind + 7] == "\\eqref{": - pause = True - eqR = line[ind:line.find("}", ind) + 1] - rLab = get_data_str(eqR) - '''for l in lLink: - if rLab==l[0:l.find("=")-1]: - rlabel=l[l.find("=>")+3:l.find("\\n")] - rlabel=rlabel.replace("/","") - rlabel=rlabel.replace("#",":") - rlabel=rlabel.replace("!",":") - break''' - - eInd = refLabels.index("" + label) - z = line[line.find("}", ind + 7) + 1] - if z == "." or z == ",": - pauseP = True - proofLine += ("
\n{\displaystyle\n" + refEqs[ - eInd] + "}" + z + "
\n") - else: - proofLine += ( - "
\n{\displaystyle\n" + refEqs[eInd] + "}
\n") - - else: - if pause: - if line[ind] == "}": - pause = False - elif pauseP: - pauseP = False - elif line[ind] == "\n" and "\\end{equation}" in lines[i + 1]: - pass - - else: - proofLine += (line[ind]) - if "\\end{equation}" in lines[i + 1]: - proof = False - # symLine+=line.strip("\n") - wiki.write(comToWrite + getEqP(proofLine, False).rstrip("\n") + "
\n
\n") - comToWrite = "" - symbols = symbols + getSym(symLine) - symLine = "" - - elif math: - if "\\end{equation}" in lines[i + 1] or "\\constraint" in lines[i + 1] or "\\substitution" in lines[ - i + 1] or "\\proof" in lines[i + 1] or "\\drmfnote" in lines[i + 1] or "\\drmfname" in lines[ - i + 1]: - wiki.write(line.rstrip("\n")) - symLine += line.strip("\n") - symbols = symbols + getSym(symLine) - symLine = "" - wiki.write("\n}
\n") - else: - symLine += line.strip("\n") - wiki.write(line) - if note and parse: - noteLine = noteLine + line - symbols = symbols + getSym(line) - if "\\end{equation}" in lines[i + 1] or "\\drmfn" in lines[i + 1] or "\\constraint" in lines[ - i + 1] or "\\substitution" in lines[i + 1] or "\\proof" in lines[i + 1]: - note = False - if "\\emph" in noteLine: - noteLine = noteLine[0:noteLine.find("\\emph{")] + "\'\'" + noteLine[noteLine.find("\\emph{") + len( - "\\emph{"):noteLine.find("}", noteLine.find("\\emph{") + len("\\emph{"))] + "\'\'" + noteLine[ - noteLine.find( - "}", - noteLine.find( - "\\emph{") + len( - "\\emph{")) + 1:] - comToWrite = comToWrite + "
" + getEq(noteLine) + "

\n" - - if constraint and parse: - conLine = conLine + line.replace("&", "&
") - - symLine += line.strip("\n") - # symbols=symbols+getSym(line) - if "\\end{equation}" in lines[i + 1] or "\\drmfn" in lines[i + 1] or "\\constraint" in lines[ - i + 1] or "\\substitution" in lines[i + 1] or "\\proof" in lines[i + 1]: - constraint = False - symbols = symbols + getSym(symLine) - symLine = "" - wiki.write(comToWrite + "
" + getEq(conLine) + "

\n") - comToWrite = "" - if substitution and parse: - subLine = subLine + line # .replace("&","&
") - - symLine += line.strip("\n") - # symbols=symbols+getSym(line) - if "\\end{equation}" in lines[i + 1] or "\\drmfn" in lines[i + 1] or "\\substitution" in lines[ - i + 1] or "\\constraint" in lines[i + 1] or "\\proof" in lines[i + 1]: - substitution = False - symbols = symbols + getSym(symLine) - symLine = "" - lineR = "" - for i in range(0, len(subLine)): - if subLine[i] == "&" and not ( - i > subLine.find("\\begin{array}") and i < subLine.find("\\end{array}")): - lineR += "&
" - else: - lineR += subLine[i] - wiki.write(comToWrite + "
" + getEq(lineR) + "

\n") - comToWrite = "" - -if __name__ == "__main__": - main() diff --git a/tex2wiki/src/tex2wiki2.py b/tex2wiki/src/tex2wiki.py similarity index 98% rename from tex2wiki/src/tex2wiki2.py rename to tex2wiki/src/tex2wiki.py index 0173020..f5d31e2 100644 --- a/tex2wiki/src/tex2wiki2.py +++ b/tex2wiki/src/tex2wiki.py @@ -1,9 +1,9 @@ __author__ = "Joon Bang" __status__ = "Prototype" -INPUT_FILE = "tex2wiki/data/09outb.tex" -OUTPUT_FILE = "tex2wiki/data/09outb.mmd" -GLOSSARY_LOCATION = "tex2wiki/src/new.Glossary.csv" +INPUT_FILE = "tex2wiki/data/01outb.tex" +OUTPUT_FILE = "tex2wiki/data/01outb.mmd" +GLOSSARY_LOCATION = "tex2wiki/data/new.Glossary.csv" METADATA_TYPES = ["substitution", "constraint"] METADATA_MEANING = {"substitution": "Substitution(s)", "constraint": "Constraint(s)"} From 2c43edc30338fe2ee6817fb70e929ddaaa5b1cbe Mon Sep 17 00:00:00 2001 From: Joon Bang Date: Thu, 18 Aug 2016 11:21:49 -0400 Subject: [PATCH 285/402] Fix QuantifiedCode & TravisCI issues --- .travis.yml | 2 +- tex2wiki/src/tex2wiki.py | 10 ++++++++-- 2 files changed, 9 insertions(+), 3 deletions(-) diff --git a/.travis.yml b/.travis.yml index d749bde..5106bef 100644 --- a/.travis.yml +++ b/.travis.yml @@ -2,6 +2,6 @@ language: python install: - pip install coveralls script: - nosetests --with-coverage -v tex2Wiki DLMF_preprocessing maple2latex main_page eCF + nosetests --with-coverage -v DLMF_preprocessing maple2latex main_page eCF after_success: coveralls diff --git a/tex2wiki/src/tex2wiki.py b/tex2wiki/src/tex2wiki.py index f5d31e2..8aa4c5e 100644 --- a/tex2wiki/src/tex2wiki.py +++ b/tex2wiki/src/tex2wiki.py @@ -17,7 +17,7 @@ def __init__(self, label, equation, metadata): self.metadata = metadata -def generate_html(tag_name, text, options=dict(), spacing=2): +def generate_html(tag_name, text, options=None, spacing=2): # (str, str(, dict, bool)) -> str """ Generates an html tag, with optional html parameters. @@ -25,6 +25,9 @@ def generate_html(tag_name, text, options=dict(), spacing=2): When spacing = 1, the tag, text, and end tag are padded with spaces. When spacing = 2, the tag, text, and end tag are padded with newlines. """ + if options is None: + options = {} + option_text = [key + "=\"" + value + "\"" for key, value in options.iteritems()] result = "<" + tag_name + " " * (option_text != []) + ", ".join(option_text) + ">" @@ -38,10 +41,13 @@ def generate_html(tag_name, text, options=dict(), spacing=2): return result -def generate_math_html(text, options=dict(), spacing=True): +def generate_math_html(text, options=None, spacing=True): """ Special case of generate_html, where the tag is "math". """ + if options is None: + options = {} + option_text = [key + "=\"" + value + "\"" for key, value in options.iteritems()] result = "" From 932cf5a53d3d5f5efc811b68b4d454de488a5ffc Mon Sep 17 00:00:00 2001 From: Edward Bian Date: Thu, 18 Aug 2016 13:37:34 -0400 Subject: [PATCH 286/402] Minor Fixes --- KLSadd_insertion/src/updateChapters.py | 21 +++++++++++---------- 1 file changed, 11 insertions(+), 10 deletions(-) diff --git a/KLSadd_insertion/src/updateChapters.py b/KLSadd_insertion/src/updateChapters.py index 2330aff..28775bc 100644 --- a/KLSadd_insertion/src/updateChapters.py +++ b/KLSadd_insertion/src/updateChapters.py @@ -42,10 +42,11 @@ def new_keywords(kls, kls_list): kls_list = extraneous_section_deleter(kls_list) kls_list.append("Limit Relation") - kls_list.append("Symmetry") for item in kls_list: if item not in kls_list_chap: kls_list_chap.append(item) + kls_list_chap[kls_list_chap.index("Special value")] = "Symmetry" + kls_list_chap.append("Special value") return kls_list_chap @@ -63,14 +64,14 @@ def fix_chapter_sort(kls, chap, word, sortloc, klsaddparas, sortmatch_2): hyper_headers_chap = [] hyper_subs_chap = [] - specialinput = 0 + special_input = 0 name_chap = word.lower() if name_chap == "orthogonality": - specialinput = 1 + special_input = 1 elif name_chap == "special value": - specialinput = 2 + special_input = 2 elif name_chap == "symmetry": - specialinput = 3 + special_input = 3 khyper_header_chap = [] k_hyper_sub_chap = [] @@ -84,8 +85,8 @@ def fix_chapter_sort(kls, chap, word, sortloc, klsaddparas, sortmatch_2): temp = line[line.find(" ", 12)+1: line.find("}", 12)] # get just the name (like mathpeople) if temp.lower() == 'wilson': chapterstart = True - if specialinput in (0,2) and name_chap in line.lower() and chapterstart and ("\\paragraph{" in line or "\\subsubsection*{" in line) or \ - (specialinput in (1, 3) and "orthogonality relation" not in line.lower() and name_chap in line.lower() and chapterstart + if special_input in (0, 2) and name_chap in line.lower() and chapterstart and ("\\paragraph{" in line or "\\subsubsection*{" in line) or \ + (special_input in (1, 3) and "orthogonality relation" not in line.lower() and name_chap in line.lower() and chapterstart and ("\\paragraph{" in line or "\\subsubsection*{" in line)): for item in klsaddparas: if index < item: @@ -132,10 +133,10 @@ def fix_chapter_sort(kls, chap, word, sortloc, klsaddparas, sortmatch_2): else: tempref = ref14_3 offset = sorter_check[sortloc][1] - if specialinput == 1: + if special_input == 1: offset = 8 - if specialinput in (2, 3): + if special_input in (2, 3): name_chap = 'hypergeometric representation' # Uses of numbers like 12, or 9 are for specific lengths of strings (like \\subsection{) where the @@ -150,7 +151,7 @@ def fix_chapter_sort(kls, chap, word, sortloc, klsaddparas, sortmatch_2): temp = line[9:line.find("}", 7)] if name_chap in line.lower(): - if specialinput in (0, 2, 3) or specialinput == 1 and "orthogonality relation" not in line: + if special_input in (0, 2, 3) or special_input == 1 and "orthogonality relation" not in line: hyper_subs_chap.append([tempref[d + 1]]) # appends the index for the line before following subsection hyper_headers_chap.append(temp) # appends the name of the section the hypergeo subsection is in From 133e68ac7da82807b6df2139f446522dc02762dc Mon Sep 17 00:00:00 2001 From: Edward Bian Date: Thu, 18 Aug 2016 13:40:26 -0400 Subject: [PATCH 287/402] Travis Fixes --- .travis.yml | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/.travis.yml b/.travis.yml index f747800..f5826d4 100644 --- a/.travis.yml +++ b/.travis.yml @@ -2,6 +2,6 @@ language: python install: - pip install coveralls script: - nosetests --with-coverage --with-isolation -v tex2Wiki DLMF_preprocessing maple2latex main_page eCF KLSadd_insertion + nosetests --with-coverage -v tex2Wiki DLMF_preprocessing maple2latex main_page eCF KLSadd_insertion after_success: coveralls From ec643666407acf083be369fe14212a8552671c4b Mon Sep 17 00:00:00 2001 From: Edward Bian Date: Thu, 18 Aug 2016 13:53:52 -0400 Subject: [PATCH 288/402] Unit Test Fixes --- KLSadd_insertion/src/updateChapters.py | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/KLSadd_insertion/src/updateChapters.py b/KLSadd_insertion/src/updateChapters.py index 28775bc..43c2719 100644 --- a/KLSadd_insertion/src/updateChapters.py +++ b/KLSadd_insertion/src/updateChapters.py @@ -157,8 +157,8 @@ def fix_chapter_sort(kls, chap, word, sortloc, klsaddparas, sortmatch_2): if temp in khyper_header_chap: try: - chap[tempref[d + 1] - 1] += "\paragraph{\\bf KLS Addendum: " + word + "}" - chap[tempref[d + 1] - 1] += k_hyper_sub_chap[k_hyp_index_iii + offset] + chap[tempref[d + 1] - 1] += "\paragraph{\\bf KLS Addendum: " + word + "}" + \ + k_hyper_sub_chap[k_hyp_index_iii + offset] if chap9 == 1: sorter_check[sortloc][1] += 1 k_hyp_index_iii += 1 From d18fd5d28e50da6f1bc6069d189fe5fa31ca4684 Mon Sep 17 00:00:00 2001 From: Edward Bian Date: Thu, 18 Aug 2016 13:58:36 -0400 Subject: [PATCH 289/402] Unit Test Fixes --- KLSadd_insertion/test/test_generic_functions.py | 3 ++- 1 file changed, 2 insertions(+), 1 deletion(-) diff --git a/KLSadd_insertion/test/test_generic_functions.py b/KLSadd_insertion/test/test_generic_functions.py index 887e9bc..fc6fa2f 100644 --- a/KLSadd_insertion/test/test_generic_functions.py +++ b/KLSadd_insertion/test/test_generic_functions.py @@ -10,7 +10,8 @@ class TestNewKeywords(TestCase): def test_new_keywords(self): - self.assertEquals(new_keywords(['\\subsubsection*{ThisIsATest}\n'],[]), ['ThisIsATest', 'Limit Relation', 'Symmetry']) + self.assertEquals(new_keywords(['\\subsubsection*{ThisIsATest}\n', '\\subsubsection*{Special value}\n'] + ,[]), ['ThisIsATest', 'Symmetry', 'Limit Relation', 'Special value']) class TestExtraneousSectionDeleter(TestCase): From b88d60d756174620487c396b795113071564060a Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Thu, 18 Aug 2016 15:16:24 -0400 Subject: [PATCH 290/402] Add mostly finished latex_constraint_modifier, modify MathematicaToLaTeX.py and will modify tests for that --- eCF/.gitignore | 2 + eCF/src/LaTeXConstraintModifier.py | 46 ------- eCF/src/MathematicaToLaTeX.py | 22 ++-- eCF/src/latex_constraint_modifier.py | 172 +++++++++++++++++++++++++++ 4 files changed, 182 insertions(+), 60 deletions(-) delete mode 100644 eCF/src/LaTeXConstraintModifier.py create mode 100644 eCF/src/latex_constraint_modifier.py diff --git a/eCF/.gitignore b/eCF/.gitignore index 557d2b6..278d977 100644 --- a/eCF/.gitignore +++ b/eCF/.gitignore @@ -3,9 +3,11 @@ other/ data/Identities.* data/newIdentities.* +data/newIdentitiesnew.* data/IdentitiesTest.m data/References.txt data/ZE.3.* +data/ZE.3new.* data/ZE.4.* src/test.txt diff --git a/eCF/src/LaTeXConstraintModifier.py b/eCF/src/LaTeXConstraintModifier.py deleted file mode 100644 index cd24e01..0000000 --- a/eCF/src/LaTeXConstraintModifier.py +++ /dev/null @@ -1,46 +0,0 @@ - -__author__ = 'Kevin Chen' -__status__ = 'Development' - -import os -import argparse - -parser = argparse.ArgumentParser( - description='Receive .tex file with constraints and convert' - ' them to lines using flushright and flushright.') -parser.add_argument('PATHR', type=str, - help='path of input .tex file, with the current' - ' directory as the starting point',) -parser.add_argument('PATHW', type=str, - help='path of file to be outputted to, with the current' - ' directory as the starting point') -args = parser.parse_args() - -PATHR = args.PATHR -PATHW = args.PATHW -DIR_NAME = os.path.dirname(os.path.realpath(__file__)) + '/../data/' - - -def combine_percent(lines): - # (list) -> list - """ - Description - - :param lines: - :return: - """ - - pass - - -def main(): - # - """ - - :return: - """ - pass - - -if __name__ == '__main__': - main() diff --git a/eCF/src/MathematicaToLaTeX.py b/eCF/src/MathematicaToLaTeX.py index 322fdb8..ea9ece7 100644 --- a/eCF/src/MathematicaToLaTeX.py +++ b/eCF/src/MathematicaToLaTeX.py @@ -858,7 +858,7 @@ def replace_vars(line): def main(pathw=DIR_NAME + 'newIdentities.tex', - pathr=DIR_NAME + 'Identities.m', test=False): + pathr=DIR_NAME + 'Identities.m'): # ((str, str, bool)) -> None """ Opens Mathematica file with identities and puts converted lines into @@ -884,7 +884,7 @@ def main(pathw=DIR_NAME + 'newIdentities.tex', '\\usepackage{DRMFfcns}\n' '\\usepackage{DLMFfcns}\n' '\\usepackage{graphicx}\n' - '\\usepackage[paperwidth=20in, paperheight=20in, ' + '\\usepackage[paperwidth=15in, paperheight=20in, ' 'margin=0.5in]{geometry}\n\n' '\\begin{document}\n\n\n') @@ -893,12 +893,9 @@ def main(pathw=DIR_NAME + 'newIdentities.tex', # If line is a comment, make it a LaTeX comment or "\tag" if '(*' in line and '*)' in line: - if test: - line = line.replace('(*', '%').replace('*)', '%') - else: - line = ('\\begin{equation*} \\tag{' + - line[4:-3].replace('"', '') + '}') - + mtt = line[4:-3].replace('"', '') + #print(mtt) + line = ('\\begin{equation}') latex.write(line + '\n') else: @@ -932,13 +929,10 @@ def main(pathw=DIR_NAME + 'newIdentities.tex', line = replace_operators(line) line = replace_vars(line) - print(line) - - if test and line != '': - line = '\\begin{equation*}\n' + line - if line != '': - line += '\n\\end{equation*}' + print(mtt) + line += '\n% \\mathematicatag{$\\tt{' + mtt + '}$}' + line += '\n\\end{equation}' latex.write(line + '\n') diff --git a/eCF/src/latex_constraint_modifier.py b/eCF/src/latex_constraint_modifier.py new file mode 100644 index 0000000..ec3ee1d --- /dev/null +++ b/eCF/src/latex_constraint_modifier.py @@ -0,0 +1,172 @@ + +__author__ = 'Kevin Chen' +__status__ = 'Development' + +import argparse + +parser = argparse.ArgumentParser( + description='Receive .tex file with constraints and convert' + ' them viewable metadata.') +parser.add_argument('PATHR', type=str, + help='path of input .tex file, with the current' + ' directory as the starting point',) +parser.add_argument('PATHW', type=str, + help='path of file to be outputted to, with the current' + ' directory as the starting point') + +args = parser.parse_args() +PATHR = args.PATHR +PATHW = args.PATHW + +CONVERSIONS = {'\\constraint': '{\\bf C}:~', + '\\substitution': '{\\bf S}:~', + '\\drmfnote': '{\\bf NOTE}:~', + '\\drmfname': '{\\bf NAME}:~', + '\\proof': '{\\bf PROOF}:~', + '\\mathematicatag': '{\\bf MtT}:~', + '\\mapletag': '{\\bf MpT}:~'} + + +def find_surrounding(line, function): + # (str, str(, tuple, int)) -> tuple + """ + Finds the indices of the beginning and end of a function; this is the main + function that powers the converter. + + :param line: line with functions that are going to be searched + :param function: the function you're trying to find the surrounding + brackets for + :returns: positions of opening and ending brackets + """ + positions = [0, 0] + positions[0] = line.find(function) + + # Finds the start and end of a function + count = 0 + for j in range(positions[0] + len(function), len(line) + 1): + + if line[j] in list('([{'): + count += 1 + if line[j] in list(')]}'): + count -= 1 + if count == 0: + if j == positions[0] + len(function): + positions[0] = positions[1] + else: + positions[1] = j + 1 + break + + return positions[0], positions[1] + + +def combine_percent(lines): + # (list) -> list + """ + Combines + + :param lines: + :return: + """ + index = 0 + while index < len(lines): + if len(lines[index]) >= 3 and lines[index][:3] == '% ' and \ + index > lines.index('\\begin{document}'): + if len(lines[index + 1]) != '' and lines[index + 1][0] == '%': + print(lines[index]) + add = lines.pop(index + 1) + lines[index] += '\n' + add.replace('%', ' ') + else: + lines[index], lines[index + 1] = lines[index + 1], lines[index] + index += 2 + else: + index += 1 + + return lines + + +def replace(line): + # (str) -> str + """ + Replaces constraints with viewable metadata. + + :param line: line to be converted + :returns: converted line + """ + for word in CONVERSIONS: + for _ in range(line.count(word)): + pos = find_surrounding(line, word) + if pos[0] != pos[1]: + arg = line[pos[0] + len(word) + 1:pos[1] - 1]\ + .replace('\n', '').replace(' &', ' &') + arg = ('\\\\[0.2cm]\n ' + CONVERSIONS[word])\ + .join(arg.split('&')) + + if (max([line.find(item) for item in CONVERSIONS]) > pos[1] or + max([line.find(CONVERSIONS[item]) for + item in CONVERSIONS]) > pos[1]): + line = line[:pos[0]] + CONVERSIONS[word] + arg + \ + ' \\\\[0.2cm]' + line[pos[1]:] + else: + line = line[:pos[0]] + CONVERSIONS[word] + arg + \ + line[pos[1]:] + + return line + + +def dollarsign(line): + # (str) -> str + """ + Description. + + :param line: line to be converted + :returns: converted line + """ + count = [] + for index in range(len(line) - 1, 0, -1): + if line[index] == '$': + count.insert(0, index) + if len(count) == 2: + # Checks to see if there is already a "displaystyle", so it doesn't + # add a second useless one. + if line[count[0]:count[0] + 15] == '${\\displaystyle': + pass + else: + begin = count[0] + end = count[1] + line = line[:begin] + '${\\displaystyle ' + \ + line[begin + 1: end] + '}' + line[end:] + count = [] + + return line + + +def main(): + # (None) -> None + """ + Main function that reads data and calls other functions to process data. + + :return: None + """ + with open(PATHR, 'r') as b: + before = b.read().split('\n') + + print(before) + + before = combine_percent(before) + + print(before) + + with open(PATHW, 'w') as after: + for line in before: + if len(line) >= 3 and line[:3] == '% ' and \ + before.index(line) > before.index('\\begin{document}'): + line = replace(line) + line = dollarsign(line) + line = '\\begin{flushright}\n' + line.replace('%', ' ') + \ + '\n\\end{flushright}' + + after.write(line + '\n') + + +if __name__ == '__main__': + main() From 53925547696e919a3e008ace409312bbcdc77d39 Mon Sep 17 00:00:00 2001 From: Edward Bian Date: Thu, 18 Aug 2016 15:27:05 -0400 Subject: [PATCH 291/402] Reference Formatter and Unit Tests --- KLSadd_insertion/src/updateChapters.py | 47 +++++++++++++++++++ .../test/test_generic_functions.py | 22 ++++++++- 2 files changed, 68 insertions(+), 1 deletion(-) diff --git a/KLSadd_insertion/src/updateChapters.py b/KLSadd_insertion/src/updateChapters.py index 43c2719..1cc1545 100644 --- a/KLSadd_insertion/src/updateChapters.py +++ b/KLSadd_insertion/src/updateChapters.py @@ -14,6 +14,51 @@ w, h = 2, 1000 sorter_check = [[0 for _ in range(w)] for __ in range(h)] +def reference_formatter(kls): + """ + This function formats all in addendum references in order to distinguish them from non-addendum references + :param kls: + :return: + """ + for item in kls: + ref_count = item.count('\eqref{') + if ref_count == 1: + kls[kls.index(item)] = item[:item.find('\eqref{')+7] + "KLSadd: " + item[item.find('\eqref{')+7:] + elif ref_count == 2: + subitem = item[:item.find('\eqref{')+7] + "KLSadd: " + item[item.find('\eqref{')+7:item.find('\eqref{')+10] + subitem2 = item[len(subitem)-8:] + subitem2b = subitem2[:subitem2.find('\eqref{')+7] + "KLSadd: " + subitem2[subitem2.find('\eqref{')+7:] + kls[kls.index(item)] = subitem + subitem2b + elif ref_count == 3: + subitem = item[:item.find('\eqref{') + 7] + "KLSadd: " + item[item.find('\eqref{') + 7:item.find('\eqref{') + 10] + subitem2 = item[len(subitem) - 8:] + subitem2b = subitem2[:subitem2.find('\eqref{') + 7] + "KLSadd: " + subitem2[subitem2.find('\eqref{') + 7:subitem2.find('\eqref{') + 10] + subitem3 = item[len(subitem)+len(subitem2b) - 16:] + subitem3b = subitem3[:subitem3.find('\eqref{') + 7] + "KLSadd: " + subitem3[subitem3.find('\eqref{') + 7:] + kls[kls.index(item)] = subitem + subitem2b + subitem3b + elif ref_count == 4: + subitem = item[:item.find('\eqref{') + 7] + "KLSadd: " + item[item.find('\eqref{') + 7:item.find('\eqref{') + 10] + subitem2 = item[len(subitem) - 8:] + subitem2b = subitem2[:subitem2.find('\eqref{') + 7] + "KLSadd: " + subitem2[subitem2.find('\eqref{') + 7:subitem2.find('\eqref{') + 10] + subitem3 = item[len(subitem) + len(subitem2b) - 16:] + subitem3b = subitem3[:subitem3.find('\eqref{') + 7] + "KLSadd: " + subitem3[subitem3.find('\eqref{') + 7:subitem3.find('\eqref{') + 10] + subitem4 = item[len(subitem) + len(subitem2b) +len(subitem3b) - 24:] + subitem4b = subitem4[:subitem4.find('\eqref{') + 7] + "KLSadd: " + subitem4[subitem4.find('\eqref{') + 7:] + kls[kls.index(item)] = subitem + subitem2b + subitem3b + subitem4b + elif ref_count == 5: + subitem = item[:item.find('\eqref{') + 7] + "KLSadd: " + item[item.find('\eqref{') + 7:item.find('\eqref{') + 10] + subitem2 = item[len(subitem) - 8:] + subitem2b = subitem2[:subitem2.find('\eqref{') + 7] + "KLSadd: " + subitem2[subitem2.find('\eqref{') + 7:subitem2.find('\eqref{') + 10] + subitem3 = item[len(subitem) + len(subitem2b) - 16:] + subitem3b = subitem3[:subitem3.find('\eqref{') + 7] + "KLSadd: " + subitem3[subitem3.find('\eqref{') + 7:subitem3.find('\eqref{') + 10] + subitem4 = item[len(subitem) + len(subitem2b) + len(subitem3b) - 24:] + subitem4b = subitem4[:subitem4.find('\eqref{') + 7] + "KLSadd: " + subitem4[subitem4.find('\eqref{') + 7:subitem4.find('\eqref{') + 10] + subitem5 = item[len(subitem) + len(subitem2b) + len(subitem3b) + len(subitem4b) - 32:] + subitem5b = subitem5[:subitem5.find('\eqref{') + 7] + "KLSadd: " + subitem5[subitem5.find('\eqref{') + 7:] + kls[kls.index(item)] = subitem + subitem2b + subitem3b + subitem4b + subitem5b + + return kls + def extraneous_section_deleter(list): """ @@ -557,6 +602,8 @@ def main(): # call the findReferences method to find the index of the References paragraph in the two file strings # two variables for the references lists one for chapter 9 one for chapter 14 + addendum = reference_formatter(addendum) + chapticker = 0 references9 = find_references(entire9, chapticker, math_people) chapticker += 1 diff --git a/KLSadd_insertion/test/test_generic_functions.py b/KLSadd_insertion/test/test_generic_functions.py index fc6fa2f..df5a5d6 100644 --- a/KLSadd_insertion/test/test_generic_functions.py +++ b/KLSadd_insertion/test/test_generic_functions.py @@ -6,6 +6,7 @@ from updateChapters import new_keywords from updateChapters import extraneous_section_deleter from updateChapters import cut_words +from updateChapters import reference_formatter class TestNewKeywords(TestCase): @@ -27,4 +28,23 @@ def test_cut_words(self): self.assertEquals(cut_words('MoreWords','WordsAndSomeMoreWords'), 'WordsAndSome') self.assertEquals(cut_words('SupposedToFail', 'WordsAndSomeMoreWords'), 'WordsAndSomeMoreWords') self.assertEquals(cut_words('StuffFromAddenum', '''\\paragraph{\\large\\bf KLSadd: TestCase}StuffFromAddenum\\begin{equation}\\end{equation}'''), - '''\\begin{equation}\\end{equation}''') \ No newline at end of file + '''\\begin{equation}\\end{equation}''') + +class TestReferenceFormatter(TestCase): + + def test_reference_formatter(self): + self.assertEquals(reference_formatter(['by \eqref{1337}, This should work', 'WordsThatMightBeFoundInTheChapter']) + , ['by \eqref{KLSadd: 1337}, This should work', + 'WordsThatMightBeFoundInTheChapter']) + self.assertEquals(reference_formatter(['by \eqref{1337} and \eqref{360} This should work', 'WordsThatMightBeFoundInTheChapter']) + , ['by \eqref{KLSadd: 1337} and \eqref{KLSadd: 360} This should work', + 'WordsThatMightBeFoundInTheChapter']) + self.assertEquals(reference_formatter(['by \eqref{1337}, \eqref{720} and \eqref{360} This should work', 'WordsThatMightBeFoundInTheChapter']) + , ['by \eqref{KLSadd: 1337}, \eqref{KLSadd: 720} and \eqref{KLSadd: 360} This should work', + 'WordsThatMightBeFoundInTheChapter']) + self.assertEquals(reference_formatter(['by \eqref{1337}, \eqref{720}, \eqref{500} and \eqref{360} This should work', 'WordsThatMightBeFoundInTheChapter']) + , ['by \eqref{KLSadd: 1337}, \eqref{KLSadd: 720}, \eqref{KLSadd: 500} and \eqref{KLSadd: 360} This should work', + 'WordsThatMightBeFoundInTheChapter']) + self.assertEquals(reference_formatter(['by \eqref{1337}, \eqref{720}, \eqref{500}, \eqref{404} and \eqref{360} This should work', 'WordsThatMightBeFoundInTheChapter']) + , ['by \eqref{KLSadd: 1337}, \eqref{KLSadd: 720}, \eqref{KLSadd: 500}, \eqref{KLSadd: 404} and \eqref{KLSadd: 360} This should work', + 'WordsThatMightBeFoundInTheChapter']) \ No newline at end of file From 5d0b063eedc7285915539feeacf9711a33bba3ae Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Thu, 18 Aug 2016 15:38:58 -0400 Subject: [PATCH 292/402] Rename MathematicaToLaTeX to mathematica_to_latex --- eCF/.gitignore | 1 + ...ticaToLaTeX.py => mathematica_to_latex.py} | 6 +-- eCF/test/data/test.tex | 8 ++-- eCF/test/test_arg_split.py | 2 +- eCF/test/test_carat.py | 2 +- eCF/test/test_constraint.py | 2 +- eCF/test/test_convert_fraction.py | 2 +- eCF/test/test_find_surrounding.py | 2 +- eCF/test/test_main.py | 43 +++---------------- eCF/test/test_master_function.py | 4 +- eCF/test/test_piecewise.py | 2 +- eCF/test/test_removal_functions.py | 6 +-- eCF/test/test_replace_operators.py | 4 +- eCF/test/test_replace_vars.py | 2 +- eCF/test/test_search.py | 2 +- .../test_single_macro_conversion_functions.py | 20 ++++----- 16 files changed, 40 insertions(+), 68 deletions(-) rename eCF/src/{MathematicaToLaTeX.py => mathematica_to_latex.py} (99%) diff --git a/eCF/.gitignore b/eCF/.gitignore index 278d977..b5ffbf6 100644 --- a/eCF/.gitignore +++ b/eCF/.gitignore @@ -9,6 +9,7 @@ data/References.txt data/ZE.3.* data/ZE.3new.* data/ZE.4.* +data/questions.txt src/test.txt src/Glossary.csv diff --git a/eCF/src/MathematicaToLaTeX.py b/eCF/src/mathematica_to_latex.py similarity index 99% rename from eCF/src/MathematicaToLaTeX.py rename to eCF/src/mathematica_to_latex.py index ea9ece7..7e4fc2e 100644 --- a/eCF/src/MathematicaToLaTeX.py +++ b/eCF/src/mathematica_to_latex.py @@ -799,6 +799,7 @@ def replace_operators(line): :returns: converted line """ line = line.replace('==', '=') + line = line.replace('!=', ' \\ne ') line = line.replace('||', ' \\lor ') line = line.replace('>=', ' \\geq ') line = line.replace('<=', ' \\leq ') @@ -831,8 +832,7 @@ def replace_operators(line): line = line.replace('Catalan', '\\CatalansConstant') line = line.replace('GoldenRatio', '\\GoldenRatio') line = line.replace('Pi', '\\pi') - line = line.replace('CalculateData`Private`nu', - '\\text{CalculateData`Private`nu}') + line = line.replace('CalculateData`Private`nu', '\\nu') return line @@ -858,7 +858,7 @@ def replace_vars(line): def main(pathw=DIR_NAME + 'newIdentities.tex', - pathr=DIR_NAME + 'Identities.m'): + pathr=DIR_NAME + 'IdentitiesTest.m'): # ((str, str, bool)) -> None """ Opens Mathematica file with identities and puts converted lines into diff --git a/eCF/test/data/test.tex b/eCF/test/data/test.tex index c6c4f9e..a315ca9 100644 --- a/eCF/test/data/test.tex +++ b/eCF/test/data/test.tex @@ -10,15 +10,15 @@ \usepackage{DRMFfcns} \usepackage{DLMFfcns} \usepackage{graphicx} -\usepackage[paperwidth=20in, paperheight=20in, margin=0.5in]{geometry} +\usepackage[paperwidth=15in, paperheight=20in, margin=0.5in]{geometry} \begin{document} -% {"description", number}% -\begin{equation*} +\begin{equation} equation -\end{equation*} +% \mathematicatag{$\tt{description, number}$} +\end{equation} \end{document} diff --git a/eCF/test/test_arg_split.py b/eCF/test/test_arg_split.py index 51b704f..44946a8 100644 --- a/eCF/test/test_arg_split.py +++ b/eCF/test/test_arg_split.py @@ -3,7 +3,7 @@ __status__ = 'Development' from unittest import TestCase -from MathematicaToLaTeX import arg_split +from mathematica_to_latex import arg_split class TestArgumentSplit(TestCase): diff --git a/eCF/test/test_carat.py b/eCF/test/test_carat.py index 9c4092a..f7c6b9f 100644 --- a/eCF/test/test_carat.py +++ b/eCF/test/test_carat.py @@ -3,7 +3,7 @@ __status__ = 'Development' from unittest import TestCase -from MathematicaToLaTeX import carat +from mathematica_to_latex import carat class TestCarat(TestCase): diff --git a/eCF/test/test_constraint.py b/eCF/test/test_constraint.py index c881a08..0f469f5 100644 --- a/eCF/test/test_constraint.py +++ b/eCF/test/test_constraint.py @@ -3,7 +3,7 @@ __status__ = 'Development' from unittest import TestCase -from MathematicaToLaTeX import constraint +from mathematica_to_latex import constraint class TestConstraint(TestCase): diff --git a/eCF/test/test_convert_fraction.py b/eCF/test/test_convert_fraction.py index 7ab59d6..2fe9dd5 100644 --- a/eCF/test/test_convert_fraction.py +++ b/eCF/test/test_convert_fraction.py @@ -3,7 +3,7 @@ __status__ = 'Development' from unittest import TestCase -from MathematicaToLaTeX import convert_fraction +from mathematica_to_latex import convert_fraction class TestConvertFraction(TestCase): diff --git a/eCF/test/test_find_surrounding.py b/eCF/test/test_find_surrounding.py index 5cbb908..af509f3 100644 --- a/eCF/test/test_find_surrounding.py +++ b/eCF/test/test_find_surrounding.py @@ -3,6 +3,6 @@ __status__ = 'Development' from unittest import TestCase -from MathematicaToLaTeX import find_surrounding +from mathematica_to_latex import find_surrounding diff --git a/eCF/test/test_main.py b/eCF/test/test_main.py index 0945eea..9e1c70e 100644 --- a/eCF/test/test_main.py +++ b/eCF/test/test_main.py @@ -3,7 +3,7 @@ __status__ = 'Development' from unittest import TestCase -from MathematicaToLaTeX import main +from mathematica_to_latex import main import os @@ -13,8 +13,8 @@ class TestMain(TestCase): - def test_test(self): - main(pathw=PATHW, pathr=PATHR, test=True) + def test_gen(self): + main(pathw=PATHW, pathr=PATHR) with open(PATHW, 'r') as l: latex = l.read() self.assertEqual( @@ -30,44 +30,15 @@ def test_test(self): '\\usepackage{DRMFfcns}\n' '\\usepackage{DLMFfcns}\n' '\\usepackage{graphicx}\n' - '\\usepackage[paperwidth=20in, paperheight=20in, margin=0.5in]{geometry}\n' + '\\usepackage[paperwidth=15in, paperheight=20in, margin=0.5in]{geometry}\n' '\n' '\\begin{document}\n' '\n' '\n' - '% {"description", number}%\n' - '\\begin{equation*}\n' + '\\begin{equation}\n' 'equation\n' - '\\end{equation*}\n' - '\n' - '\n' - '\\end{document}\n')) - - def test_nottest(self): - main(pathw=PATHW, pathr=PATHR, test=False) - with open(PATHW, 'r') as l: - latex = l.read() - self.assertEqual( - latex, ('\n' - '\\documentclass{article}\n' - '\n' - '\\usepackage{amsmath}\n' - '\\usepackage{amsthm}\n' - '\\usepackage{amssymb}\n' - '\\usepackage{amsfonts}\n' - '\\usepackage{breqn}\n' - '\\usepackage{DLMFmath}\n' - '\\usepackage{DRMFfcns}\n' - '\\usepackage{DLMFfcns}\n' - '\\usepackage{graphicx}\n' - '\\usepackage[paperwidth=20in, paperheight=20in, margin=0.5in]{geometry}\n' - '\n' - '\\begin{document}\n' - '\n' - '\n' - '\\begin{equation*} \\tag{description, number}\n' - 'equation\n' - '\\end{equation*}\n' + '% \\mathematicatag{$\\tt{description, number}$}\n' + '\\end{equation}\n' '\n' '\n' '\\end{document}\n')) diff --git a/eCF/test/test_master_function.py b/eCF/test/test_master_function.py index 73a6312..cde731d 100644 --- a/eCF/test/test_master_function.py +++ b/eCF/test/test_master_function.py @@ -4,8 +4,8 @@ import os from unittest import TestCase -from MathematicaToLaTeX import master_function -from MathematicaToLaTeX import arg_split +from mathematica_to_latex import master_function +from mathematica_to_latex import arg_split with open(os.path.dirname(os.path.realpath(__file__)) + '/../data/functions') as functions: diff --git a/eCF/test/test_piecewise.py b/eCF/test/test_piecewise.py index 7ee6298..2a01df9 100644 --- a/eCF/test/test_piecewise.py +++ b/eCF/test/test_piecewise.py @@ -3,7 +3,7 @@ __status__ = 'Development' from unittest import TestCase -from MathematicaToLaTeX import piecewise +from mathematica_to_latex import piecewise class TestPiecewise(TestCase): diff --git a/eCF/test/test_removal_functions.py b/eCF/test/test_removal_functions.py index eefbd21..7af5f7b 100644 --- a/eCF/test/test_removal_functions.py +++ b/eCF/test/test_removal_functions.py @@ -3,9 +3,9 @@ __status__ = 'Development' from unittest import TestCase -from MathematicaToLaTeX import remove_inactive -from MathematicaToLaTeX import remove_conditionalexpression -from MathematicaToLaTeX import remove_symbol +from mathematica_to_latex import remove_inactive +from mathematica_to_latex import remove_conditionalexpression +from mathematica_to_latex import remove_symbol BEFORE1 = ('Inactive[test]', diff --git a/eCF/test/test_replace_operators.py b/eCF/test/test_replace_operators.py index d8eaf3e..ff6041e 100644 --- a/eCF/test/test_replace_operators.py +++ b/eCF/test/test_replace_operators.py @@ -3,7 +3,7 @@ __status__ = 'Development' from unittest import TestCase -from MathematicaToLaTeX import replace_operators +from mathematica_to_latex import replace_operators class TestReplaceOperators(TestCase): @@ -32,7 +32,7 @@ def test_constants(self): self.assertEqual(replace_operators('Catalan'), '\\CatalansConstant') self.assertEqual(replace_operators('GoldenRatio'), '\\GoldenRatio') self.assertEqual(replace_operators('Pi'), '\\pi') - self.assertEqual(replace_operators('CalculateData`Private`nu'), '\\text{CalculateData`Private`nu}') + self.assertEqual(replace_operators('CalculateData`Private`nu'), '\\nu') def test_with_percent(self): self.assertEqual(replace_operators('(%('), '\\left( %(') diff --git a/eCF/test/test_replace_vars.py b/eCF/test/test_replace_vars.py index 5f4adb7..68c796b 100644 --- a/eCF/test/test_replace_vars.py +++ b/eCF/test/test_replace_vars.py @@ -3,7 +3,7 @@ __status__ = 'Development' from unittest import TestCase -from MathematicaToLaTeX import replace_vars +from mathematica_to_latex import replace_vars SYMBOLS = { 'Alpha': 'alpha', 'Beta': 'beta', 'Gamma': 'gamma', 'Delta': 'delta', diff --git a/eCF/test/test_search.py b/eCF/test/test_search.py index 83c2f7b..cfd50e5 100644 --- a/eCF/test/test_search.py +++ b/eCF/test/test_search.py @@ -3,7 +3,7 @@ __status__ = 'Development' from unittest import TestCase -from MathematicaToLaTeX import search +from mathematica_to_latex import search class TestSearch(TestCase): diff --git a/eCF/test/test_single_macro_conversion_functions.py b/eCF/test/test_single_macro_conversion_functions.py index c204d0a..0c05961 100644 --- a/eCF/test/test_single_macro_conversion_functions.py +++ b/eCF/test/test_single_macro_conversion_functions.py @@ -3,16 +3,16 @@ __status__ = 'Development' from unittest import TestCase -from MathematicaToLaTeX import beta -from MathematicaToLaTeX import cfk -from MathematicaToLaTeX import gamma -from MathematicaToLaTeX import integrate -from MathematicaToLaTeX import legendrep -from MathematicaToLaTeX import legendreq -from MathematicaToLaTeX import polyeulergamma -from MathematicaToLaTeX import product -from MathematicaToLaTeX import qpochhammer -from MathematicaToLaTeX import summation +from mathematica_to_latex import beta +from mathematica_to_latex import cfk +from mathematica_to_latex import gamma +from mathematica_to_latex import integrate +from mathematica_to_latex import legendrep +from mathematica_to_latex import legendreq +from mathematica_to_latex import polyeulergamma +from mathematica_to_latex import product +from mathematica_to_latex import qpochhammer +from mathematica_to_latex import summation class TestBeta(TestCase): From 4aa3469de4fcbf34ee4cd68c8617a08c0c114833 Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Thu, 18 Aug 2016 15:42:52 -0400 Subject: [PATCH 293/402] Add documentation to latex_constraint_modifier.py --- eCF/src/latex_constraint_modifier.py | 9 +++++---- 1 file changed, 5 insertions(+), 4 deletions(-) diff --git a/eCF/src/latex_constraint_modifier.py b/eCF/src/latex_constraint_modifier.py index ec3ee1d..0ff1945 100644 --- a/eCF/src/latex_constraint_modifier.py +++ b/eCF/src/latex_constraint_modifier.py @@ -62,10 +62,11 @@ def find_surrounding(line, function): def combine_percent(lines): # (list) -> list """ - Combines + Combines terms with constraints (ones with percent signs) so that replacing + is easier. - :param lines: - :return: + :param lines: list of all lines + :return: same list, but groups of constraint terms are combined """ index = 0 while index < len(lines): @@ -116,7 +117,7 @@ def replace(line): def dollarsign(line): # (str) -> str """ - Description. + Changes "$ $" to "${\\displaystyle }$". :param line: line to be converted :returns: converted line From 959a713102220992e948e4d8f35e5e1bcf6b6e1d Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Thu, 18 Aug 2016 15:47:18 -0400 Subject: [PATCH 294/402] Remove useless print statements --- eCF/src/latex_constraint_modifier.py | 4 ---- 1 file changed, 4 deletions(-) diff --git a/eCF/src/latex_constraint_modifier.py b/eCF/src/latex_constraint_modifier.py index 0ff1945..c8c8604 100644 --- a/eCF/src/latex_constraint_modifier.py +++ b/eCF/src/latex_constraint_modifier.py @@ -151,12 +151,8 @@ def main(): with open(PATHR, 'r') as b: before = b.read().split('\n') - print(before) - before = combine_percent(before) - print(before) - with open(PATHW, 'w') as after: for line in before: if len(line) >= 3 and line[:3] == '% ' and \ From 901e820b7aaa4a7f55ab85bf952f86706e2dbb66 Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Thu, 18 Aug 2016 15:51:00 -0400 Subject: [PATCH 295/402] Modify docstrings --- eCF/src/latex_constraint_modifier.py | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/eCF/src/latex_constraint_modifier.py b/eCF/src/latex_constraint_modifier.py index c8c8604..b3b090d 100644 --- a/eCF/src/latex_constraint_modifier.py +++ b/eCF/src/latex_constraint_modifier.py @@ -28,7 +28,7 @@ def find_surrounding(line, function): - # (str, str(, tuple, int)) -> tuple + # (str, str) -> tuple """ Finds the indices of the beginning and end of a function; this is the main function that powers the converter. From e4aa55ec56ca4648e5ec207dac56e36fcd3e831c Mon Sep 17 00:00:00 2001 From: Kevin Chen Date: Thu, 18 Aug 2016 16:05:29 -0400 Subject: [PATCH 296/402] Fix typo in README.md --- README.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/README.md b/README.md index 4039241..98ef0ef 100644 --- a/README.md +++ b/README.md @@ -52,4 +52,4 @@ TODO in updateChapters.py: FINISHED- implemented smart fix for "hypergeometric representation" paragraphs. Now the content in these paragraphs are appended directly from the KLS section into the chapter's hypergeometric representation paragraphs Needs to be done: --implemenent the same smart fix for "Limit relations" paragraphs. Current code for this implementation does not work and has spotty logic that does not encompass all variations of limit relations paragraphs (such as the one subsubsection in chapter 9) +-implement the same smart fix for "Limit relations" paragraphs. Current code for this implementation does not work and has spotty logic that does not encompass all variations of limit relations paragraphs (such as the one subsubsection in chapter 9) From ea6b8bcf42188683ae60afa197d078d59e0b808e Mon Sep 17 00:00:00 2001 From: Kevin Chen Date: Thu, 18 Aug 2016 16:07:57 -0400 Subject: [PATCH 297/402] Update README.md --- README.md | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/README.md b/README.md index 98ef0ef..095c02d 100644 --- a/README.md +++ b/README.md @@ -43,11 +43,11 @@ This Python code (mostly developed by Kevin Chen) translates Mathematica files t The project is currently awaiting approval to be merged into the main branch of the repository. ## KLSadd Insertion Project -linetest.py and updateChapters.py written by Rahul Shah and Edward Bian +`linetest.py` and `updateChapters.py` written by Rahul Shah and Edward Bian -Edits the DRMF chapter files to include the relevant KLS addendum additions. Additions are (currently) only being added right before the "References" paragraphs in each section. The linetest.py file is the first working model of the code, but it is very messy and esoteric. Comments have been made to aid interpretation. The new project file, updateChapters.py is a more readable version. +Edits the DRMF chapter files to include the relevant KLS addendum additions. Additions are (currently) only being added right before the "References" paragraphs in each section. The `linetest.py` file is the first working model of the code, but it is very messy and esoteric. Comments have been made to aid interpretation. The new project file, `updateChapters.py` is a more readable version. -TODO in updateChapters.py: +TODO in `updateChapters.py`: FINISHED- implemented smart fix for "hypergeometric representation" paragraphs. Now the content in these paragraphs are appended directly from the KLS section into the chapter's hypergeometric representation paragraphs From 9cf71af72361289d70165bb09707cf85de77789e Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Thu, 18 Aug 2016 16:15:26 -0400 Subject: [PATCH 298/402] Add test case for replace_operators function --- eCF/src/mathematica_to_latex.py | 2 +- eCF/test/test_replace_operators.py | 1 + 2 files changed, 2 insertions(+), 1 deletion(-) diff --git a/eCF/src/mathematica_to_latex.py b/eCF/src/mathematica_to_latex.py index 7e4fc2e..184db3f 100644 --- a/eCF/src/mathematica_to_latex.py +++ b/eCF/src/mathematica_to_latex.py @@ -858,7 +858,7 @@ def replace_vars(line): def main(pathw=DIR_NAME + 'newIdentities.tex', - pathr=DIR_NAME + 'IdentitiesTest.m'): + pathr=DIR_NAME + 'Identities.m'): # ((str, str, bool)) -> None """ Opens Mathematica file with identities and puts converted lines into diff --git a/eCF/test/test_replace_operators.py b/eCF/test/test_replace_operators.py index ff6041e..d129ce2 100644 --- a/eCF/test/test_replace_operators.py +++ b/eCF/test/test_replace_operators.py @@ -10,6 +10,7 @@ class TestReplaceOperators(TestCase): def test_without_percent(self): self.assertEqual(replace_operators('=='), ' = ') + self.assertEqual(replace_operators('!='), ' \\ne ') self.assertEqual(replace_operators('||'), ' \\lor ') self.assertEqual(replace_operators('>='), ' \\geq ') self.assertEqual(replace_operators('<='), ' \\leq ') From f9ab318c5605d487f6e9b986ea2b3f15c4ddc1ec Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Thu, 18 Aug 2016 16:17:30 -0400 Subject: [PATCH 299/402] Modify docstring --- eCF/src/mathematica_to_latex.py | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/eCF/src/mathematica_to_latex.py b/eCF/src/mathematica_to_latex.py index 184db3f..b5e1a58 100644 --- a/eCF/src/mathematica_to_latex.py +++ b/eCF/src/mathematica_to_latex.py @@ -859,7 +859,7 @@ def replace_vars(line): def main(pathw=DIR_NAME + 'newIdentities.tex', pathr=DIR_NAME + 'Identities.m'): - # ((str, str, bool)) -> None + # ((str, str)) -> None """ Opens Mathematica file with identities and puts converted lines into newIdentities.tex. From 783c90ae5ad9735a26f44bbacc8dcf0e9095253a Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Thu, 18 Aug 2016 16:17:46 -0400 Subject: [PATCH 300/402] Modify comments --- eCF/src/mathematica_to_latex.py | 2 -- 1 file changed, 2 deletions(-) diff --git a/eCF/src/mathematica_to_latex.py b/eCF/src/mathematica_to_latex.py index b5e1a58..9345289 100644 --- a/eCF/src/mathematica_to_latex.py +++ b/eCF/src/mathematica_to_latex.py @@ -891,10 +891,8 @@ def main(pathw=DIR_NAME + 'newIdentities.tex', for line in mathematica: line = line.replace('\n', '') - # If line is a comment, make it a LaTeX comment or "\tag" if '(*' in line and '*)' in line: mtt = line[4:-3].replace('"', '') - #print(mtt) line = ('\\begin{equation}') latex.write(line + '\n') else: From f1e82ff81f0f87a8f4501ed70323a78d8e573207 Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Thu, 18 Aug 2016 16:18:02 -0400 Subject: [PATCH 301/402] Remove redundant parentheses --- eCF/src/mathematica_to_latex.py | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/eCF/src/mathematica_to_latex.py b/eCF/src/mathematica_to_latex.py index 9345289..873cf52 100644 --- a/eCF/src/mathematica_to_latex.py +++ b/eCF/src/mathematica_to_latex.py @@ -893,7 +893,7 @@ def main(pathw=DIR_NAME + 'newIdentities.tex', if '(*' in line and '*)' in line: mtt = line[4:-3].replace('"', '') - line = ('\\begin{equation}') + line = '\\begin{equation}' latex.write(line + '\n') else: From 4e29fccab44309dff91b70f394a15ee74be6ec25 Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Thu, 18 Aug 2016 16:18:47 -0400 Subject: [PATCH 302/402] Remove useless print statement, add useful print statement --- eCF/src/mathematica_to_latex.py | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/eCF/src/mathematica_to_latex.py b/eCF/src/mathematica_to_latex.py index 873cf52..72d4a1d 100644 --- a/eCF/src/mathematica_to_latex.py +++ b/eCF/src/mathematica_to_latex.py @@ -928,10 +928,10 @@ def main(pathw=DIR_NAME + 'newIdentities.tex', line = replace_vars(line) if line != '': - print(mtt) line += '\n% \\mathematicatag{$\\tt{' + mtt + '}$}' line += '\n\\end{equation}' + print line latex.write(line + '\n') latex.write('\n\n\\end{document}\n') From 4f8a4b9a82bb564f2958b8e1d4e2e6977f7575c5 Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Thu, 18 Aug 2016 16:24:44 -0400 Subject: [PATCH 303/402] Remove useless newline --- eCF/src/mathematica_to_latex.py | 1 - 1 file changed, 1 deletion(-) diff --git a/eCF/src/mathematica_to_latex.py b/eCF/src/mathematica_to_latex.py index 72d4a1d..1b74b6f 100644 --- a/eCF/src/mathematica_to_latex.py +++ b/eCF/src/mathematica_to_latex.py @@ -896,7 +896,6 @@ def main(pathw=DIR_NAME + 'newIdentities.tex', line = '\\begin{equation}' latex.write(line + '\n') else: - line = line.replace(' ', '') line = remove_inactive(line) From 7fabc1272ae990f98d57bc366fbc8e71f18925f2 Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Thu, 18 Aug 2016 16:26:21 -0400 Subject: [PATCH 304/402] Add metadata conversion case --- eCF/src/latex_constraint_modifier.py | 1 + 1 file changed, 1 insertion(+) diff --git a/eCF/src/latex_constraint_modifier.py b/eCF/src/latex_constraint_modifier.py index b3b090d..21f0097 100644 --- a/eCF/src/latex_constraint_modifier.py +++ b/eCF/src/latex_constraint_modifier.py @@ -24,6 +24,7 @@ '\\drmfname': '{\\bf NAME}:~', '\\proof': '{\\bf PROOF}:~', '\\mathematicatag': '{\\bf MtT}:~', + '\\mathematicareference': '{\\bf MtR}:~', '\\mapletag': '{\\bf MpT}:~'} From a5c9d6bcb2090deeda2e43c0332b76a748b0c34a Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Thu, 18 Aug 2016 16:28:18 -0400 Subject: [PATCH 305/402] Fix line length formatting --- eCF/src/latex_constraint_modifier.py | 3 ++- 1 file changed, 2 insertions(+), 1 deletion(-) diff --git a/eCF/src/latex_constraint_modifier.py b/eCF/src/latex_constraint_modifier.py index 21f0097..0319c98 100644 --- a/eCF/src/latex_constraint_modifier.py +++ b/eCF/src/latex_constraint_modifier.py @@ -157,7 +157,8 @@ def main(): with open(PATHW, 'w') as after: for line in before: if len(line) >= 3 and line[:3] == '% ' and \ - before.index(line) > before.index('\\begin{document}'): + before.index(line) > \ + before.index('\\begin{document}'): line = replace(line) line = dollarsign(line) line = '\\begin{flushright}\n' + line.replace('%', ' ') + \ From 3097f9a374ab40045f4d28a0ce0afcc50501afd6 Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Thu, 18 Aug 2016 16:28:42 -0400 Subject: [PATCH 306/402] Fix line length formatting --- eCF/src/latex_constraint_modifier.py | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/eCF/src/latex_constraint_modifier.py b/eCF/src/latex_constraint_modifier.py index 0319c98..adac470 100644 --- a/eCF/src/latex_constraint_modifier.py +++ b/eCF/src/latex_constraint_modifier.py @@ -128,8 +128,8 @@ def dollarsign(line): if line[index] == '$': count.insert(0, index) if len(count) == 2: - # Checks to see if there is already a "displaystyle", so it doesn't - # add a second useless one. + # This checks to see if there is already a "displaystyle", so it + # doesn't add a second useless one. if line[count[0]:count[0] + 15] == '${\\displaystyle': pass else: From 5c255b39fd85894b0c5f5ea60dfaa2fedbb5649f Mon Sep 17 00:00:00 2001 From: Edward Bian Date: Thu, 18 Aug 2016 16:39:11 -0400 Subject: [PATCH 307/402] Small Fixes and updates --- KLSadd_insertion/src/updateChapters.py | 48 +------------------ .../test/test_generic_functions.py | 20 +------- 2 files changed, 3 insertions(+), 65 deletions(-) diff --git a/KLSadd_insertion/src/updateChapters.py b/KLSadd_insertion/src/updateChapters.py index 1cc1545..727183f 100644 --- a/KLSadd_insertion/src/updateChapters.py +++ b/KLSadd_insertion/src/updateChapters.py @@ -14,51 +14,6 @@ w, h = 2, 1000 sorter_check = [[0 for _ in range(w)] for __ in range(h)] -def reference_formatter(kls): - """ - This function formats all in addendum references in order to distinguish them from non-addendum references - :param kls: - :return: - """ - for item in kls: - ref_count = item.count('\eqref{') - if ref_count == 1: - kls[kls.index(item)] = item[:item.find('\eqref{')+7] + "KLSadd: " + item[item.find('\eqref{')+7:] - elif ref_count == 2: - subitem = item[:item.find('\eqref{')+7] + "KLSadd: " + item[item.find('\eqref{')+7:item.find('\eqref{')+10] - subitem2 = item[len(subitem)-8:] - subitem2b = subitem2[:subitem2.find('\eqref{')+7] + "KLSadd: " + subitem2[subitem2.find('\eqref{')+7:] - kls[kls.index(item)] = subitem + subitem2b - elif ref_count == 3: - subitem = item[:item.find('\eqref{') + 7] + "KLSadd: " + item[item.find('\eqref{') + 7:item.find('\eqref{') + 10] - subitem2 = item[len(subitem) - 8:] - subitem2b = subitem2[:subitem2.find('\eqref{') + 7] + "KLSadd: " + subitem2[subitem2.find('\eqref{') + 7:subitem2.find('\eqref{') + 10] - subitem3 = item[len(subitem)+len(subitem2b) - 16:] - subitem3b = subitem3[:subitem3.find('\eqref{') + 7] + "KLSadd: " + subitem3[subitem3.find('\eqref{') + 7:] - kls[kls.index(item)] = subitem + subitem2b + subitem3b - elif ref_count == 4: - subitem = item[:item.find('\eqref{') + 7] + "KLSadd: " + item[item.find('\eqref{') + 7:item.find('\eqref{') + 10] - subitem2 = item[len(subitem) - 8:] - subitem2b = subitem2[:subitem2.find('\eqref{') + 7] + "KLSadd: " + subitem2[subitem2.find('\eqref{') + 7:subitem2.find('\eqref{') + 10] - subitem3 = item[len(subitem) + len(subitem2b) - 16:] - subitem3b = subitem3[:subitem3.find('\eqref{') + 7] + "KLSadd: " + subitem3[subitem3.find('\eqref{') + 7:subitem3.find('\eqref{') + 10] - subitem4 = item[len(subitem) + len(subitem2b) +len(subitem3b) - 24:] - subitem4b = subitem4[:subitem4.find('\eqref{') + 7] + "KLSadd: " + subitem4[subitem4.find('\eqref{') + 7:] - kls[kls.index(item)] = subitem + subitem2b + subitem3b + subitem4b - elif ref_count == 5: - subitem = item[:item.find('\eqref{') + 7] + "KLSadd: " + item[item.find('\eqref{') + 7:item.find('\eqref{') + 10] - subitem2 = item[len(subitem) - 8:] - subitem2b = subitem2[:subitem2.find('\eqref{') + 7] + "KLSadd: " + subitem2[subitem2.find('\eqref{') + 7:subitem2.find('\eqref{') + 10] - subitem3 = item[len(subitem) + len(subitem2b) - 16:] - subitem3b = subitem3[:subitem3.find('\eqref{') + 7] + "KLSadd: " + subitem3[subitem3.find('\eqref{') + 7:subitem3.find('\eqref{') + 10] - subitem4 = item[len(subitem) + len(subitem2b) + len(subitem3b) - 24:] - subitem4b = subitem4[:subitem4.find('\eqref{') + 7] + "KLSadd: " + subitem4[subitem4.find('\eqref{') + 7:subitem4.find('\eqref{') + 10] - subitem5 = item[len(subitem) + len(subitem2b) + len(subitem3b) + len(subitem4b) - 32:] - subitem5b = subitem5[:subitem5.find('\eqref{') + 7] + "KLSadd: " + subitem5[subitem5.find('\eqref{') + 7:] - kls[kls.index(item)] = subitem + subitem2b + subitem3b + subitem4b + subitem5b - - return kls - def extraneous_section_deleter(list): """ @@ -602,7 +557,8 @@ def main(): # call the findReferences method to find the index of the References paragraph in the two file strings # two variables for the references lists one for chapter 9 one for chapter 14 - addendum = reference_formatter(addendum) + for item in addendum: + addendum[addendum.index(item)] = item.replace("\\eqref{", "\\eqref{KLSadd: ") chapticker = 0 references9 = find_references(entire9, chapticker, math_people) diff --git a/KLSadd_insertion/test/test_generic_functions.py b/KLSadd_insertion/test/test_generic_functions.py index df5a5d6..7aa85f9 100644 --- a/KLSadd_insertion/test/test_generic_functions.py +++ b/KLSadd_insertion/test/test_generic_functions.py @@ -6,7 +6,6 @@ from updateChapters import new_keywords from updateChapters import extraneous_section_deleter from updateChapters import cut_words -from updateChapters import reference_formatter class TestNewKeywords(TestCase): @@ -30,21 +29,4 @@ def test_cut_words(self): self.assertEquals(cut_words('StuffFromAddenum', '''\\paragraph{\\large\\bf KLSadd: TestCase}StuffFromAddenum\\begin{equation}\\end{equation}'''), '''\\begin{equation}\\end{equation}''') -class TestReferenceFormatter(TestCase): - - def test_reference_formatter(self): - self.assertEquals(reference_formatter(['by \eqref{1337}, This should work', 'WordsThatMightBeFoundInTheChapter']) - , ['by \eqref{KLSadd: 1337}, This should work', - 'WordsThatMightBeFoundInTheChapter']) - self.assertEquals(reference_formatter(['by \eqref{1337} and \eqref{360} This should work', 'WordsThatMightBeFoundInTheChapter']) - , ['by \eqref{KLSadd: 1337} and \eqref{KLSadd: 360} This should work', - 'WordsThatMightBeFoundInTheChapter']) - self.assertEquals(reference_formatter(['by \eqref{1337}, \eqref{720} and \eqref{360} This should work', 'WordsThatMightBeFoundInTheChapter']) - , ['by \eqref{KLSadd: 1337}, \eqref{KLSadd: 720} and \eqref{KLSadd: 360} This should work', - 'WordsThatMightBeFoundInTheChapter']) - self.assertEquals(reference_formatter(['by \eqref{1337}, \eqref{720}, \eqref{500} and \eqref{360} This should work', 'WordsThatMightBeFoundInTheChapter']) - , ['by \eqref{KLSadd: 1337}, \eqref{KLSadd: 720}, \eqref{KLSadd: 500} and \eqref{KLSadd: 360} This should work', - 'WordsThatMightBeFoundInTheChapter']) - self.assertEquals(reference_formatter(['by \eqref{1337}, \eqref{720}, \eqref{500}, \eqref{404} and \eqref{360} This should work', 'WordsThatMightBeFoundInTheChapter']) - , ['by \eqref{KLSadd: 1337}, \eqref{KLSadd: 720}, \eqref{KLSadd: 500}, \eqref{KLSadd: 404} and \eqref{KLSadd: 360} This should work', - 'WordsThatMightBeFoundInTheChapter']) \ No newline at end of file + From dd8ff82e55513ac43fd982f46c3cec779baa4a92 Mon Sep 17 00:00:00 2001 From: Edward Bian Date: Fri, 19 Aug 2016 10:41:23 -0400 Subject: [PATCH 308/402] README updates --- KLSadd_insertion/test/test_chap1_placer.py | 60 ++++++++++++++++++++++ 1 file changed, 60 insertions(+) create mode 100644 KLSadd_insertion/test/test_chap1_placer.py diff --git a/KLSadd_insertion/test/test_chap1_placer.py b/KLSadd_insertion/test/test_chap1_placer.py new file mode 100644 index 0000000..b65437f --- /dev/null +++ b/KLSadd_insertion/test/test_chap1_placer.py @@ -0,0 +1,60 @@ + +__author__ = 'Edward Bian' +__status__ = 'Development' + +from unittest import TestCase +from updateChapters import find_references + +class TestFindReferences(TestCase): + + def test_find_references(self): + self.assertEquals(find_references(['\\section{Pseudo Jacobi}\\index{Racah polynomials}', + '\\subsection*{Hypergeometric representation}', '\\begin{eqnarray}', + '\\label{DefRacah}', '& &R_n(\lambda(x);\alpha,\beta,\gamma,\delta)\nonumber\\' + , '& &{}=\hyp{4}{3}{-n,n+\alpha+\beta+1,-x,' + 'x+\gamma+\delta+1}{\alpha+1,\beta+\delta+1,\gamma+1}{1},', '\quad n=0,1,2,\ldots,N,' + , '\end{eqnarray}', 'where', '$$\lambda(x)=x(x+\gamma+\delta+1)$$', 'and' + , '$$\alpha+1=-N\quad\textrm{or}\quad\beta+\delta+1=-N\quad\textrm{or}\quad\gamma+1=-N$$' + , 'with $N$ a nonnegative integer.', '\subsection*{References}', + '\cite{Askey89I}, \cite{AskeyWilson79}, \cite{AskeyWilson85}'], + 0, ['9.9 Pseudo Jacobi (or Routh-Romanovski)#']), [13]) + self.assertEquals(find_references(['\\section{Pseudo Jacobi}\\index{Racah polynomials}', + '\\subsection*{Hypergeometric representation}', '\\begin{eqnarray}', + '\\label{DefRacah}', '& &R_n(\lambda(x);\alpha,\beta,\gamma,\delta)\nonumber\\' + , '& &{}=\hyp{4}{3}{-n,n+\alpha+\beta+1,-x,' + 'x+\gamma+\delta+1}{\alpha+1,\beta+\delta+1,\gamma+1}{1},', '\quad n=0,1,2,\ldots,N,' + , '\end{eqnarray}', 'where', '$$\lambda(x)=x(x+\gamma+\delta+1)$$', 'and' + , '$$\alpha+1=-N\quad\textrm{or}\quad\beta+\delta+1=-N\quad\textrm{or}\quad\gamma+1=-N$$' + , 'with $N$ a nonnegative integer.', '\subsection*{References}', + '\cite{Askey89I}, \cite{AskeyWilson79}, \cite{AskeyWilson85}'], + 1, ['9.9 Pseudo Jacobi (or Routh-Romanovski)#']), [13]) + self.assertEquals(find_references(['\\section{Racah}\\index{Racah polynomials}', + '\\subsection*{Hypergeometric representation}', '\\begin{eqnarray}', + '\\label{DefRacah}', '& &R_n(\lambda(x);\alpha,\beta,\gamma,\delta)\nonumber\\' + , '& &{}=\hyp{4}{3}{-n,n+\alpha+\beta+1,-x,' + 'x+\gamma+\delta+1}{\alpha+1,\beta+\delta+1,\gamma+1}{1},', '\quad n=0,1,2,\ldots,N,' + , '\end{eqnarray}', 'where', '$$\lambda(x)=x(x+\gamma+\delta+1)$$', 'and' + , '$$\alpha+1=-N\quad\textrm{or}\quad\beta+\delta+1=-N\quad\textrm{or}\quad\gamma+1=-N$$' + , 'with $N$ a nonnegative integer.', '\subsection*{References}', + '\cite{Askey89I}, \cite{AskeyWilson79}, \cite{AskeyWilson85}'], + 0, ['9.2 Racah#']), [13]) + self.assertEquals(find_references(['\\section{Racah}\\index{Racah polynomials}', + '\\subsection*{Hypergeometric representation}', '\\begin{eqnarray}', + '\\label{DefRacah}', '& &R_n(\lambda(x);\alpha,\beta,\gamma,\delta)\nonumber\\' + , '& &{}=\hyp{4}{3}{-n,n+\alpha+\beta+1,-x,' + 'x+\gamma+\delta+1}{\alpha+1,\beta+\delta+1,\gamma+1}{1},', '\quad n=0,1,2,\ldots,N,' + , '\end{eqnarray}', 'where', '$$\lambda(x)=x(x+\gamma+\delta+1)$$', 'and' + , '$$\alpha+1=-N\quad\textrm{or}\quad\beta+\delta+1=-N\quad\textrm{or}\quad\gamma+1=-N$$' + , 'with $N$ a nonnegative integer.', '\subsection*{References}', + '\cite{Askey89I}, \cite{AskeyWilson79}, \cite{AskeyWilson85}'], + 0, ['9.2 Racah#']), [13]) + self.assertEquals(find_references(['\\section{Unique}\\index{Racah polynomials}', + '\\subsection*{Hypergeometric representation}', '\\begin{eqnarray}', + '\\label{DefRacah}', '& &R_n(\lambda(x);\alpha,\beta,\gamma,\delta)\nonumber\\' + , '& &{}=\hyp{4}{3}{-n,n+\alpha+\beta+1,-x,' + 'x+\gamma+\delta+1}{\alpha+1,\beta+\delta+1,\gamma+1}{1},', '\quad n=0,1,2,\ldots,N,' + , '\end{eqnarray}', 'where', '$$\lambda(x)=x(x+\gamma+\delta+1)$$', 'and' + , '$$\alpha+1=-N\quad\textrm{or}\quad\beta+\delta+1=-N\quad\textrm{or}\quad\gamma+1=-N$$' + , 'with $N$ a nonnegative integer.', '\subsection*{References}', + '\cite{Askey89I}, \cite{AskeyWilson79}, \cite{AskeyWilson85}'], + 1, ['9.2 Racah#']), []) \ No newline at end of file From d6fecffb90b3f0dd93e6a374b562d1acf6a7154d Mon Sep 17 00:00:00 2001 From: Edward Bian Date: Fri, 19 Aug 2016 10:46:53 -0400 Subject: [PATCH 309/402] README updates and More Functionality --- KLSadd_insertion/src/updateChapters.py | 39 ++++++++++++++ KLSadd_insertion/test/test_chap1_placer.py | 62 ++++------------------ 2 files changed, 48 insertions(+), 53 deletions(-) diff --git a/KLSadd_insertion/src/updateChapters.py b/KLSadd_insertion/src/updateChapters.py index 727183f..17ddcd2 100644 --- a/KLSadd_insertion/src/updateChapters.py +++ b/KLSadd_insertion/src/updateChapters.py @@ -15,6 +15,36 @@ sorter_check = [[0 for _ in range(w)] for __ in range(h)] +def chap1placer(chap1, kls, klsaddparas): + chapterstart = True + index = 0 + kls_sub_chap1 = [] + kls_header_chap1 = [] + for item in kls: + index += 1 + line = str(item) + if "\\subsection" in line: + temp = line[line.find(" ", 12) + 1: line.find("}", 12)+1] # get just the name (like mathpeople) + print temp + if 'wilson' in temp.lower(): + chapterstart = False + print("ding") + if chapterstart and ("\\paragraph{" in line or "\\subsubsection*{" in line): + for item in klsaddparas: + if index < item: + klsloc = klsaddparas.index(item) + break + t = ''.join(kls[index-1: klsaddparas[klsloc]]) + kls_sub_chap1.append(t) # append the whole paragraph, pray every paragraph ends with a % comment + kls_header_chap1.append(temp) # append the name of subsection + break + intro_to_add = ''.join(kls_sub_chap1) + chap1 = chap1[:len(chap1)-1] + chap1 = ''.join(chap1) + total_chap1 = chap1 + "\\paragraph{\\large\\bf KLS Addendum: Generalities}" + intro_to_add + "\\end{document}" + return total_chap1 + + def extraneous_section_deleter(list): """ Removes sections that are irrelevant and will slow or confuse the program @@ -547,6 +577,14 @@ def main(): # as of 2/8/16 the paragraphs will go before the References paragraph of the relevant section # parse both files 9 and 14 as strings + with open(DATA_DIR + "chap01.tex", "r") as ch1: + entire1 = ch1.readlines() # reads in as a list of strings + + chap1placer(entire1, addendum, klsaddparas) + + with open(DATA_DIR + "updated1.tex", "w") as temp1: + temp1.write(chap1placer(entire1, addendum, klsaddparas)) + # chapter 9 with open(DATA_DIR + "chap09.tex", "r") as ch9: entire9 = ch9.readlines() # reads in as a list of strings @@ -588,6 +626,7 @@ def main(): with open(DATA_DIR + "updated14.tex", "w") as temp14: temp14.write(str14) print math_people + print klsaddparas if __name__ == '__main__': main() \ No newline at end of file diff --git a/KLSadd_insertion/test/test_chap1_placer.py b/KLSadd_insertion/test/test_chap1_placer.py index b65437f..0cde1ab 100644 --- a/KLSadd_insertion/test/test_chap1_placer.py +++ b/KLSadd_insertion/test/test_chap1_placer.py @@ -3,58 +3,14 @@ __status__ = 'Development' from unittest import TestCase -from updateChapters import find_references +from updateChapters import chap1placer -class TestFindReferences(TestCase): - def test_find_references(self): - self.assertEquals(find_references(['\\section{Pseudo Jacobi}\\index{Racah polynomials}', - '\\subsection*{Hypergeometric representation}', '\\begin{eqnarray}', - '\\label{DefRacah}', '& &R_n(\lambda(x);\alpha,\beta,\gamma,\delta)\nonumber\\' - , '& &{}=\hyp{4}{3}{-n,n+\alpha+\beta+1,-x,' - 'x+\gamma+\delta+1}{\alpha+1,\beta+\delta+1,\gamma+1}{1},', '\quad n=0,1,2,\ldots,N,' - , '\end{eqnarray}', 'where', '$$\lambda(x)=x(x+\gamma+\delta+1)$$', 'and' - , '$$\alpha+1=-N\quad\textrm{or}\quad\beta+\delta+1=-N\quad\textrm{or}\quad\gamma+1=-N$$' - , 'with $N$ a nonnegative integer.', '\subsection*{References}', - '\cite{Askey89I}, \cite{AskeyWilson79}, \cite{AskeyWilson85}'], - 0, ['9.9 Pseudo Jacobi (or Routh-Romanovski)#']), [13]) - self.assertEquals(find_references(['\\section{Pseudo Jacobi}\\index{Racah polynomials}', - '\\subsection*{Hypergeometric representation}', '\\begin{eqnarray}', - '\\label{DefRacah}', '& &R_n(\lambda(x);\alpha,\beta,\gamma,\delta)\nonumber\\' - , '& &{}=\hyp{4}{3}{-n,n+\alpha+\beta+1,-x,' - 'x+\gamma+\delta+1}{\alpha+1,\beta+\delta+1,\gamma+1}{1},', '\quad n=0,1,2,\ldots,N,' - , '\end{eqnarray}', 'where', '$$\lambda(x)=x(x+\gamma+\delta+1)$$', 'and' - , '$$\alpha+1=-N\quad\textrm{or}\quad\beta+\delta+1=-N\quad\textrm{or}\quad\gamma+1=-N$$' - , 'with $N$ a nonnegative integer.', '\subsection*{References}', - '\cite{Askey89I}, \cite{AskeyWilson79}, \cite{AskeyWilson85}'], - 1, ['9.9 Pseudo Jacobi (or Routh-Romanovski)#']), [13]) - self.assertEquals(find_references(['\\section{Racah}\\index{Racah polynomials}', - '\\subsection*{Hypergeometric representation}', '\\begin{eqnarray}', - '\\label{DefRacah}', '& &R_n(\lambda(x);\alpha,\beta,\gamma,\delta)\nonumber\\' - , '& &{}=\hyp{4}{3}{-n,n+\alpha+\beta+1,-x,' - 'x+\gamma+\delta+1}{\alpha+1,\beta+\delta+1,\gamma+1}{1},', '\quad n=0,1,2,\ldots,N,' - , '\end{eqnarray}', 'where', '$$\lambda(x)=x(x+\gamma+\delta+1)$$', 'and' - , '$$\alpha+1=-N\quad\textrm{or}\quad\beta+\delta+1=-N\quad\textrm{or}\quad\gamma+1=-N$$' - , 'with $N$ a nonnegative integer.', '\subsection*{References}', - '\cite{Askey89I}, \cite{AskeyWilson79}, \cite{AskeyWilson85}'], - 0, ['9.2 Racah#']), [13]) - self.assertEquals(find_references(['\\section{Racah}\\index{Racah polynomials}', - '\\subsection*{Hypergeometric representation}', '\\begin{eqnarray}', - '\\label{DefRacah}', '& &R_n(\lambda(x);\alpha,\beta,\gamma,\delta)\nonumber\\' - , '& &{}=\hyp{4}{3}{-n,n+\alpha+\beta+1,-x,' - 'x+\gamma+\delta+1}{\alpha+1,\beta+\delta+1,\gamma+1}{1},', '\quad n=0,1,2,\ldots,N,' - , '\end{eqnarray}', 'where', '$$\lambda(x)=x(x+\gamma+\delta+1)$$', 'and' - , '$$\alpha+1=-N\quad\textrm{or}\quad\beta+\delta+1=-N\quad\textrm{or}\quad\gamma+1=-N$$' - , 'with $N$ a nonnegative integer.', '\subsection*{References}', - '\cite{Askey89I}, \cite{AskeyWilson79}, \cite{AskeyWilson85}'], - 0, ['9.2 Racah#']), [13]) - self.assertEquals(find_references(['\\section{Unique}\\index{Racah polynomials}', - '\\subsection*{Hypergeometric representation}', '\\begin{eqnarray}', - '\\label{DefRacah}', '& &R_n(\lambda(x);\alpha,\beta,\gamma,\delta)\nonumber\\' - , '& &{}=\hyp{4}{3}{-n,n+\alpha+\beta+1,-x,' - 'x+\gamma+\delta+1}{\alpha+1,\beta+\delta+1,\gamma+1}{1},', '\quad n=0,1,2,\ldots,N,' - , '\end{eqnarray}', 'where', '$$\lambda(x)=x(x+\gamma+\delta+1)$$', 'and' - , '$$\alpha+1=-N\quad\textrm{or}\quad\beta+\delta+1=-N\quad\textrm{or}\quad\gamma+1=-N$$' - , 'with $N$ a nonnegative integer.', '\subsection*{References}', - '\cite{Askey89I}, \cite{AskeyWilson79}, \cite{AskeyWilson85}'], - 1, ['9.2 Racah#']), []) \ No newline at end of file +class TestChap1Placer(TestCase): + + def test_chap1_placer(self): + self.assertEquals(chap1placer(['WordsAndSomeMoreWords','SampleEquation', '\\end{document}'] + , ['\\subsection*{Generalities}', '\\paragraph{MathFunction}', 'MathEquations', 'WordsAndStuff', + '\\subsection*{9.1 Wilson}', '\\paragraph{Symmetry}', 'WordsAndStuff'], [0,1,4,5]) + , 'WordsAndSomeMoreWordsSampleEquation\\paragraph{\\large\\bf KLS Addendum: Generalities}\\paragraph{MathFunction}MathEquationsWordsAndStuff\\end{document}') + From f97ab8f28073d799bc308a0d71fcec664780a45d Mon Sep 17 00:00:00 2001 From: Edward Bian Date: Fri, 19 Aug 2016 10:53:40 -0400 Subject: [PATCH 310/402] More Functionality --- KLSadd_insertion/src/updateChapters.py | 4 ---- 1 file changed, 4 deletions(-) diff --git a/KLSadd_insertion/src/updateChapters.py b/KLSadd_insertion/src/updateChapters.py index 17ddcd2..43fb71f 100644 --- a/KLSadd_insertion/src/updateChapters.py +++ b/KLSadd_insertion/src/updateChapters.py @@ -25,10 +25,6 @@ def chap1placer(chap1, kls, klsaddparas): line = str(item) if "\\subsection" in line: temp = line[line.find(" ", 12) + 1: line.find("}", 12)+1] # get just the name (like mathpeople) - print temp - if 'wilson' in temp.lower(): - chapterstart = False - print("ding") if chapterstart and ("\\paragraph{" in line or "\\subsubsection*{" in line): for item in klsaddparas: if index < item: From 22504c8a92cf73468b039b438c419b1f974590ca Mon Sep 17 00:00:00 2001 From: Edward Bian Date: Fri, 19 Aug 2016 11:15:26 -0400 Subject: [PATCH 311/402] README --- KLSadd_insertion/README.md | 9 ++++----- 1 file changed, 4 insertions(+), 5 deletions(-) diff --git a/KLSadd_insertion/README.md b/KLSadd_insertion/README.md index 9b6f635..104f5d4 100644 --- a/KLSadd_insertion/README.md +++ b/KLSadd_insertion/README.md @@ -1,8 +1,8 @@ ##KLS Addendum Insertion Project -This project uses python and string manipulation to insert sections of the KLSadd.tex file into the appropriate DRMF chapter sections. The updateChapters.py program is the most recently updated and cleanest version, however it is not completed. Should be run with a simple call to +This project uses Python and string manipulation to insert sections of the KLSadd.tex file into the appropriate DRMF chapter sections. The updateChapters.py program is the most recently updated and cleanest version, however it is not completed. Should be run with a simple call to ``` -python updateChapters.py +Python updateChapters.py ``` The program must have a tempchap9.tex and a tempchap14.tex as well as the KLSadd.tex file in the same directory! The linetest.py program is the original program file fully updated. It is much harder to read and should only be used as a reference to update the updateChapters.py program. @@ -62,7 +62,6 @@ So in the chapter 9 file, section 15 "Dogs", before the References subsection, w \paragraph{\large\bf Beagles} ``` --- -**It is important to note that these changes are currently (3/25/16) unimplemented in updateChapters.py and only exist in linetest.py, if you want to see how the output to updateChapters.py should look, look at newtempchap09.tex** So after updateChapters.py is called, the chapter 9 file should look like this: @@ -108,7 +107,7 @@ The methods: -fixChapter(chap, references, p, kls): takes a chapter file string, a references list, the paras variable, and the KLSadd file string. This method basically just calls all of the methods above and adds all of the extra stuff like the commands and packages. -At this point, the code is like 90% comments. The big chunk at the bottom under the "with open..." can be explained through the comments. + As a quick rundown: @@ -118,6 +117,6 @@ Then the strings are written into seperate updated chapter files. --- -However, its broken. Inserted text comes in random places between paragraphs and sometimes stuff isn't added. This shouldn't be that hard to fix but it will require combing through the code a lot of times and changing little things here and there until everything works. As a last ditch effort, we can always resort to using linetext.py and adding stuff in from there. +At this point in time updateChapters.py is mostly functional (there are still a few areas that do not work correctly). Currently, unit tests and additional functions are being added. From 438050bc8ce417c9fb08dfff1e7b5cf637904682 Mon Sep 17 00:00:00 2001 From: Edward Bian Date: Fri, 19 Aug 2016 16:19:35 -0400 Subject: [PATCH 312/402] Unit Tests --- KLSadd_insertion/src/updateChapters.py | 3 +- .../test/test_reference_placer.py | 52 +++++++++++++++++++ 2 files changed, 54 insertions(+), 1 deletion(-) create mode 100644 KLSadd_insertion/test/test_reference_placer.py diff --git a/KLSadd_insertion/src/updateChapters.py b/KLSadd_insertion/src/updateChapters.py index 43fb71f..880b375 100644 --- a/KLSadd_insertion/src/updateChapters.py +++ b/KLSadd_insertion/src/updateChapters.py @@ -403,6 +403,7 @@ def reference_placer(chap, references, p, chapticker2): count += 1 else: count += 1 + return chap # method to change file string(actually a list right now), returns string to be written to file @@ -622,7 +623,7 @@ def main(): with open(DATA_DIR + "updated14.tex", "w") as temp14: temp14.write(str14) print math_people - print klsaddparas + print paras if __name__ == '__main__': main() \ No newline at end of file diff --git a/KLSadd_insertion/test/test_reference_placer.py b/KLSadd_insertion/test/test_reference_placer.py new file mode 100644 index 0000000..dd92a6d --- /dev/null +++ b/KLSadd_insertion/test/test_reference_placer.py @@ -0,0 +1,52 @@ + +__author__ = 'Edward Bian' +__status__ = 'Development' + +from unittest import TestCase +from updateChapters import reference_placer + + +class TestReferencePlacer(TestCase): + + def test_reference_placer(self): + self.assertEquals(reference_placer(['\\section{Racah}\\index{Racah polynomials}', + '\\subsection*{Hypergeometric representation}', '\\begin{eqnarray}', + '\\label{DefRacah}', '& &R_n(\lambda(x);\alpha,\beta,\gamma,\delta)\nonumber\\' + , '& &{}=\hyp{4}{3}{-n,n+\alpha+\beta+1,-x,', '\end{eqnarray}' + , 'with $N$ a nonnegative integer.', '\\subsection*{References}', + '\cite{Askey89I}, \cite{AskeyWilson79}, \cite{AskeyWilson85}'],[9], + ['\\subsection*{9.1 Racah}\\label{sec9.1}\\paragraph{\\large\\bf KLSadd: Hypergeometric representation}In addition to (9.1.1) we have'], + 0), ['\\section{Racah}\\index{Racah polynomials}', '\\subsection*{Hypergeometric representation}' + , '\\begin{eqnarray}' + , '\\label{DefRacah}', '& &R_n(\lambda(x);\alpha,\beta,\gamma,\delta)\nonumber\\' + , '& &{}=\hyp{4}{3}{-n,n+\alpha+\beta+1,-x,', '\end{eqnarray}' + , 'with $N$ a nonnegative integer.%Begin KLSadd additions\\subsection*{9.1 Racah}\\label{sec9.1}\\paragraph{\\large\\bf KLSadd: Hypergeometric representation}In addition to (9.1.1) we have%End of KLSadd additions', '\\subsection*{References}', + '\cite{Askey89I}, \cite{AskeyWilson79}, \cite{AskeyWilson85}']) + + self.assertEquals(reference_placer(['\\section{Racah}\\index{Racah polynomials}', + '\\subsection*{Hypergeometric representation}', '\\begin{eqnarray}', + '\\label{DefRacah}', '& &R_n(\lambda(x);\alpha,\beta,\gamma,\delta)\nonumber\\' + , '& &{}=\hyp{4}{3}{-n,n+\alpha+\beta+1,-x,', '\end{eqnarray}' + , 'with $N$ a nonnegative integer.', '\\subsection*{References}', + '\cite{Askey89I}, \cite{AskeyWilson79}, \cite{AskeyWilson85}'], [9], + ['\\subsection*{14.1 Racah}\\label{sec9.1}\\paragraph{\\large\\bf KLSadd: Hypergeometric representation}In addition to (9.1.1) we have'], + 1), ['\\section{Racah}\\index{Racah polynomials}', '\\subsection*{Hypergeometric representation}' + , '\\begin{eqnarray}' + , '\\label{DefRacah}', '& &R_n(\lambda(x);\alpha,\beta,\gamma,\delta)\nonumber\\' + , '& &{}=\hyp{4}{3}{-n,n+\alpha+\beta+1,-x,', '\end{eqnarray}' + , 'with $N$ a nonnegative integer.%Begin KLSadd additions\\subsection*{14.1 Racah}\\label{sec9.1}\\paragraph{\\large\\bf KLSadd: Hypergeometric representation}In addition to (9.1.1) we have%End of KLSadd additions', '\\subsection*{References}', + '\cite{Askey89I}, \cite{AskeyWilson79}, \cite{AskeyWilson85}']) + + self.assertEquals(reference_placer(['\\section{Racah}\\index{Racah polynomials}', + '\\subsection*{Hypergeometric representation}', '\\begin{eqnarray}', + '\\label{DefRacah}', '& &R_n(\lambda(x);\alpha,\beta,\gamma,\delta)\nonumber\\' + , '& &{}=\hyp{4}{3}{-n,n+\alpha+\beta+1,-x,', '\end{eqnarray}' + , 'with $N$ a nonnegative integer.', '\\subsection*{References}', + '\cite{Askey89I}, \cite{AskeyWilson79}, \cite{AskeyWilson85}'], [9], + ['MiscWords', '\\subsection*{9.1 Racah}\\label{sec9.1}\\paragraph{\\large\\bf KLSadd: Hypergeometric representation}In addition to (9.1.1) we have'], + 0), ['\\section{Racah}\\index{Racah polynomials}', '\\subsection*{Hypergeometric representation}' + , '\\begin{eqnarray}' + , '\\label{DefRacah}', '& &R_n(\lambda(x);\alpha,\beta,\gamma,\delta)\nonumber\\' + , '& &{}=\hyp{4}{3}{-n,n+\alpha+\beta+1,-x,', '\end{eqnarray}' + , 'with $N$ a nonnegative integer.%Begin KLSadd additions\\subsection*{9.1 Racah}\\label{sec9.1}\\paragraph{\\large\\bf KLSadd: Hypergeometric representation}In addition to (9.1.1) we have%End of KLSadd additions', '\\subsection*{References}', + '\cite{Askey89I}, \cite{AskeyWilson79}, \cite{AskeyWilson85}']) \ No newline at end of file From b6e7e1470828ed8647af7f25c2235af447e5ab49 Mon Sep 17 00:00:00 2001 From: Oksana Tkach Date: Mon, 22 Aug 2016 11:32:32 -0400 Subject: [PATCH 313/402] Fix \\iunit stuff --- DLMF_preprocessing/src/math_function.py | 71 +++++---- DLMF_preprocessing/src/remove_excess.py | 172 ++++++++++------------ DLMF_preprocessing/src/replace_special.py | 4 +- 3 files changed, 121 insertions(+), 126 deletions(-) diff --git a/DLMF_preprocessing/src/math_function.py b/DLMF_preprocessing/src/math_function.py index 79b198e..6a248ae 100644 --- a/DLMF_preprocessing/src/math_function.py +++ b/DLMF_preprocessing/src/math_function.py @@ -6,35 +6,42 @@ def math_string(in_file): # Takes input file or string and returns list of strings when in math mode try: # test if file or string - string = open(in_file).read() + math_str = open(in_file).read() except: - string = in_file + math_str = in_file output = [] - ranges = math_mode.find_math_ranges(string) + ranges = math_mode.find_math_ranges(math_str) for i in ranges: - new = string[i[0]:i[1]] + new = math_str[i[0]:i[1]] output.append(new) return output -def change_original(o_file, changed_math_string): +def change_original(o_file, changed_math_string_list): # Places changed string from math mode back into place in the original function try: o_string = open(o_file).read() except: o_string = o_file - ranges = math_mode.find_math_ranges(o_string) + math_ranges = math_mode.find_math_ranges(o_string) num = 0 edited = o_string - for i in ranges: - edited = string.replace(edited, o_string[i[0]:i[1]], changed_math_string[num]) + # print math_ranges + for i in math_ranges[::-1]: + edited = edited[:i[0]] + changed_math_string_list[::-1][num] + edited[i[1]:] + # edited = edited.replace(o_string[i[0]:i[1]], changed_math_string_list[num]) + # print edited num += 1 + # re check NM.2new.tex isn't changing all of the i chars to \iunit + + check = open('whatevs.txt', 'w') + check.write(edited) + check.close() return edited def formatting(file_str): # Proper spacing for the indexes and gets rid of all non begin{eq end{eq text - commands = r'\\[a-zA-Z]+' updated = [] @@ -48,70 +55,82 @@ def formatting(file_str): in_eq = False been_in_eq = False - section = r'\\[a-zA-Z]*section' re.compile(commands) re.compile(section) past_last = 0 - for line in lines: - past_last += len(line) + unnecessary = ['\\begin{align', '\\end{align', '\\index', '\\paragraph', '\\eqref'] + + for lnum, line in enumerate(lines): + + lnum += 1 + + not_added = not any([element in line for element in unnecessary]) + past_last += len(line) + 1 # + 1 to compensate for \n if '\\begin{equation}' in line: + # Make a blank line between equations + if '\\end{equation}' in updated[-1]: + updated.append('') in_eq = True been_in_eq = True if '\\end{equation}' in line: - updated.append(line) + updated.append(line.lstrip()) in_eq = False continue if in_eq: - updated.append(line) + if line != '': + updated.append(line.lstrip()) else: # not been in eq if not been_in_eq: # if text line != command - if re.match(commands, line): - if '\\paragraph' not in line and '\\index' not in line: + if re.match(commands, line.lstrip()): + if not_added: updated.append(line) + elif line == '': + updated.append(line) else: # not in eq but already been in eq - if re.match(section,line) or line == '': + if re.match(section, line.lstrip()) or line == '': updated.append(line) - if past_last > ranges[-1][1]: + if past_last >= ranges[-1][1]: # In section after last eq - if re.match(commands,line) or line == '': - if '\\paragraph' not in line: + if re.match(commands, line.lstrip()) or line == '': + if not_added: updated.append(line) # if this line is an index start storing it,or write it if we're done with the indexes if '\\index{' in line: in_ind = True ind_str += line + "\n" - continue + # if next line also index, continue storing + if '\\index{' in lines[lnum]: + continue - elif in_ind: + if in_ind: in_ind = False # add a preceding newline if one is not already present - if previous.strip() != "": + if previous.strip() != '': ind_str = "\n" + ind_str fullsplit = ind_str.split("\n") updated.extend(fullsplit) ind_str = "" - previous = line + wrote = "\n".join(updated) # remove consecutive blank lines and blank lines between \index groups spaces_pat = re.compile(r'\n{2,}[ ]?\n+') wrote = spaces_pat.sub('\n\n', wrote) - wrote = re.sub(r'\\index{(.*?)}\n\n\\index{(.*?)}', r'\\index{\1}\n\\index{\2}', wrote) + wrote = re.sub(r'\\index{(.*?)}\n{2,}(?=\\index{(.*?)})', r'\\index{\1}\n', wrote) return wrote - diff --git a/DLMF_preprocessing/src/remove_excess.py b/DLMF_preprocessing/src/remove_excess.py index 02740e9..13c5e6e 100644 --- a/DLMF_preprocessing/src/remove_excess.py +++ b/DLMF_preprocessing/src/remove_excess.py @@ -1,18 +1,13 @@ -""" -This program begins with an unprocessed LaTeX file and removes parts that are -unnecessary for the DLMF. -""" - -__author__ = "Alex Danoff" -__status__ = "Development" +"""This program begins with an unprocessed LaTeX file and removes parts that are unnecessary for the DLMF.""" import re import sys import parentheses -from utilities import writeout, readin, get_line_lengths, find_line +from utilities import (writeout, readin, get_line_lengths, + find_line) -EQ_START = r'\begin{equation}' +EQ_START = r'\\begin{equation}' EQ_END = r'\end{equation}' CASES_START = r'\begin{cases}' @@ -21,7 +16,7 @@ EQMIX_START = r'\begin{equationmix}' EQMIX_END = r'\end{equationmix}' -STD_REGEX = r'.*?###open_(\d+)###.*?###close_\1###' +STD_REGEX = r'.*?###open_(\d+)###.*?###close_0###' def main(): @@ -85,6 +80,18 @@ def remove_excess(content): """Removes the excess pieces from the given content and returns the updated version as a string. :param content: """ + start_eq = re.compile(EQ_START) + end_eq = re.compile(EQ_END) + starts = [m.end() for m in start_eq.finditer(content)] + ends = [m.start() for m in end_eq.finditer(content)] + print 'starts', starts + print 'ends', ends + if len(starts) != len(ends): + print 'UNEVEN DISTRIBUTION OF BEGIN END EQ' + sys.exit(-1) + + for i in enumerate(start): + print 'i', i oldest = content @@ -96,12 +103,8 @@ def remove_excess(content): pattern = re.compile(r'\\acknowledgements{.*?}\n\n', re.DOTALL) content = re.sub(pattern, r'', content) content = re.sub(r'\\maketitle', r' ', content) - content = re.sub( - r'\\bibliography{\.\./bib/DLMF}', - r'\\bibliographystyle{plain}' + - "\n" + - r'\\bibliography{/home/hcohl/DRMF/DLMF/DLMF.bib}', - content) + content = re.sub(r'\\bibliography{\.\./bib/DLMF}', + r'\\bibliographystyle{plain}' + "\n" + r'\\bibliography{/home/hcohl/DRMF/DLMF/DLMF.bib}', content) content = re.sub(r'\\acknowledgements{.*?}', r' ', content) content = re.sub(r'\{math}', r'{equation}', content) content = re.sub(r'\\galleryitem{.*?}{.*?}', r' ', content) @@ -121,58 +124,41 @@ def remove_excess(content): content = re.sub(r'\\author\[.*?\]{.*?}', r' ', content) # modify labels, etc - content = re.sub( - r'\\begin\{equation\}\[.*?\]+', - r'\\begin{equation}', - content, - re.DOTALL) + content = re.sub(r'\\begin\{equation\}\[.*?\]+', r'\\begin{equation}', content, re.DOTALL) print(len(old) - len(content)) old = content - # remove from something to the start of the next section/part/etc - content = remove_section( - r'\\section{Special Notation}', - r'\\part', - content) - content = remove_section( - r'\\part{Computation}', - r'\\bibliographystyle{plain}', - content) - content = remove_section( - r'\\part{References}', - r'\\bibliographystyle{plain}', - content) + + content = remove_section(r'\\section{Special Notation}', r'\\part', content) + content = remove_section(r'\\part{Computation}', r'\\bibliographystyle{plain}', content) + content = remove_section(r'\\part{References}', r'\\bibliographystyle{plain}', content) content = remove_section(r'\\section{Graphics}', r'\\section', content) content = remove_section(r'\\section{Graphics}', r'\\section', content) - content = remove_section( - r'\\subsection{Graphics}', - r'\\subsection', - content) + content = remove_section(r'\\subsection{Graphics}', r'\\subsection', content) content = remove_section(r'\\section{Integrals}', r'\\section', content) - content = remove_section( - r'\\subsection{Integrals}', - r'\\subsection', - content) - content = remove_section( - r'\\section{Physical Applications}', - r'\\bibliographystyle{plain}', - content) + content = remove_section(r'\\subsection{Integrals}', r'\\subsection', content) + content = remove_section(r'\\section{Physical Applications}', r'\\bibliographystyle{plain}', content) print(len(old) - len(content)) to_join = [] - + in_eq = False # takes out comment lines first for line in content.split("\n"): - # if line only consists of % replace with nothing, otherwise don't add - # back + if EQ_START in line: + in_eq = True + if EQ_END in line: + in_eq = False + + # if line only consists of % replace with nothing, otherwise don't add back if line.lstrip().startswith("%"): if line.rstrip().endswith("%"): line = "" else: - continue + if not in_eq: + continue to_join.append(line) @@ -188,21 +174,24 @@ def remove_excess(content): content = remove_begin_end("table", content) content = remove_begin_end("table", content) + content = remove_begin_end("%", content) content = remove_begin_end("sidebar", content) content = parentheses.remove(content, curly=True) + + to_remove = [ r'\\citet', r'\\lxDeclare\[.*?\]', r'\\note', r'\\origref', r'\\lxRefDeclaration', - r'\\MarkDefn.*?###open_(\d+)###.*?###close_\1###', + r'\\MarkDefn', r'\\indexdefn', - r'\\MarkNotation.*?###open_(\d+)###.*?###close_\1###', - r'\\affiliation', + r'\\MarkNotation', + r'\\affiliation.*?###open_(\d+)###.*?###close_0###', ] lengths = get_line_lengths(content) @@ -210,68 +199,70 @@ def remove_excess(content): # go through each group and remove it for name in to_remove: - str_regex = name + STD_REGEX + # for each improper name, put it in ###open### and ###closes### + str_regex = name # + STD_REGEX not really necessary to add STD_REGEX, edit if wrong -oksana pattern = re.compile(str_regex, re.DOTALL) + lines = content.split("\n") # go through every match, finding start and end for match in pattern.finditer(content): + + # for each match of compiled improper name in file + start = find_line(match.start(), lengths) end = find_line(match.end(), lengths) # print("{0}: {1} - {2}".format(group, start, end)) - skip_lines.update(range(start, end + 1)) - # define items that cannot be at the beginning of any line or be - # contined in any line - illegal_starts = [ - r'\documentclass{DLMF}', - r'\thischapter', - r'\part{Notation}', - r'\begin{equationgroup', - r'\end{equationgroup', - r'\begin{onecolumn', - r'\end{onecolumn'] + # define items that cannot be at the beginning of any line or be contined in any line + illegal_starts = [r'\documentclass{DLMF}', r'\thischapter', r'\part{Notation}', r'\begin{equationgroup', + r'\end{equationgroup', r'\begin{onecolumn', r'\end{onecolumn'] illegal_elements = [r'TwoToOneRule', r'OneToTwoRule', r'\citet'] lines = content.split("\n") - # various flags that will be useful when going through the lines - in_eq = False + # various flags that will be useful when going through the lines, in_eq above for comment control in_const = False in_cases = False eqmix_label = "" const_str = "" - # remove trailing % and whitespace and add to updated - also remove lines - # that should be skipped + + + # remove trailing % and whitespace and add to updated - also remove lines that should be skipped for lnum, line in enumerate(lines): lnum += 1 # don't add line back if it should be skipped + + if lnum in skip_lines: + + # print lnum, line continue + # replaces long ###open and close phrases with curly brackets line = parentheses.insert(line, curly=True) - line_checks = [line.lstrip().startswith(start) for start in illegal_starts] + \ - [element in line for element in illegal_elements] + line_checks = [line.lstrip().startswith(start) for start in illegal_starts] + [element in line for element in + illegal_elements] # skip current line if it starts with or contains an illegal element if any(line_checks): + continue + # no more ugly % characters after cleaned cleaned = line.rstrip().rstrip("%").rstrip() - # line marks the start of an equationmix, set the flag and remove the - # line + # line marks the start of an equationmix, set the flag and remove the line if EQMIX_START in cleaned: eqmix_label = cleaned[cleaned.index(r'\label'):] continue - # line marks the end of an equationmix, set the flag and remove the - # line + # line marks the end of an equationmix, set the flag and remove the line if EQMIX_END in cleaned: eqmix_label = "" continue @@ -298,9 +289,9 @@ def remove_excess(content): if CASES_END in cleaned: in_cases = False - # remove commas, periods, colons, and semi-colons from the end of - # equations + # remove commas, periods, colons, and semi-colons from the end of equations if in_eq and not in_const: + to_strip = ":;," # don't take off trailing commas when in a cases block @@ -317,8 +308,7 @@ def remove_excess(content): # we're done with the constraint, make substitutions if cleaned.rstrip().rstrip(".,").endswith("}"): - const_str = re.sub( - r'\$[;,](\s*(?:\$|or))', r'$ &\1', const_str) + const_str = re.sub(r'\$[;,](\s*(?:\$|or))', r'$ &\1', const_str) in_const = False # constraint ends with two }, put one on next line @@ -371,27 +361,15 @@ def remove_excess(content): # returns the preamble as a list of it's lines def _get_preamble(): - preamble = [ - '\\documentclass{article}', - '\\usepackage{amsmath}', - '\\usepackage{amsfonts}', - '\\usepackage{DLMFbreqn}', - '\\usepackage{DLMFmath}', - '\\usepackage{DRMFfcns}', - '', - '\\oddsidemargin -0.7cm', - '\\textwidth 18.3cm', - '\\textheight 26.0cm', - '\\topmargin -2.0cm', - '', - '% \constraint{', - '% \substitution{', - '% \drmfnote{', - '% \drmfname{', - ''] + preamble = ['\\documentclass{article}', '\\usepackage{amsmath}', '\\usepackage{amsfonts}', + '\\usepackage{breqn}', + '\\usepackage{DLMFmath}', '\\usepackage{DRMFfcns}', '', '\\oddsidemargin -0.7cm', '\\textwidth 18.3cm', + '\\textheight 26.0cm', '\\topmargin -2.0cm', '', '% \constraint{', '% \substitution{', + '% \drmfnote{', '% \drmfname{', ''] return preamble if __name__ == "__main__": main() + diff --git a/DLMF_preprocessing/src/replace_special.py b/DLMF_preprocessing/src/replace_special.py index 070584a..c001215 100644 --- a/DLMF_preprocessing/src/replace_special.py +++ b/DLMF_preprocessing/src/replace_special.py @@ -77,8 +77,6 @@ def remove_special(content): "proof": [r'\proof{', False] } - should_replace = False - in_ind = False in_eq = True pi_pat = re.compile(r'(\s*)\\pi(\s*\b|[aeiou])') @@ -176,7 +174,7 @@ def remove_special(content): content[counter] = function counter += 1 - return "\n".join(content) + return content # replaces "i"s as necessary in words def _replace_i(words): From 1039231737eb5911a0febf6ba4ee98208e5cfc57 Mon Sep 17 00:00:00 2001 From: Edward Bian Date: Mon, 22 Aug 2016 14:27:45 -0400 Subject: [PATCH 314/402] Unit Tests --- KLSadd_insertion/src/updateChapters.py | 69 ++++++++----------- KLSadd_insertion/test/test_fix_chapter.py | 8 --- .../test/test_fix_chapter_sort.py | 36 ++++++++++ 3 files changed, 63 insertions(+), 50 deletions(-) delete mode 100644 KLSadd_insertion/test/test_fix_chapter.py create mode 100644 KLSadd_insertion/test/test_fix_chapter_sort.py diff --git a/KLSadd_insertion/src/updateChapters.py b/KLSadd_insertion/src/updateChapters.py index 880b375..bafe41f 100644 --- a/KLSadd_insertion/src/updateChapters.py +++ b/KLSadd_insertion/src/updateChapters.py @@ -8,8 +8,6 @@ DATA_DIR = os.path.dirname(os.path.realpath(__file__)) + "/../data/" # holds all subsections in each chapter section along with the what holds -ref9_3 = [] -ref14_3 = [] w, h = 2, 1000 sorter_check = [[0 for _ in range(w)] for __ in range(h)] @@ -76,7 +74,7 @@ def new_keywords(kls, kls_list): return kls_list_chap -def fix_chapter_sort(kls, chap, word, sortloc, klsaddparas, sortmatch_2): +def fix_chapter_sort(kls, chap, word, sortloc, klsaddparas, sortmatch_2, tempref): """ This function sorts through the input files and identifies and places sections from the addendum after their respective subsections in the chapter. @@ -134,6 +132,9 @@ def fix_chapter_sort(kls, chap, word, sortloc, klsaddparas, sortmatch_2): if khyper_header_chap[item] == khyper_header_chap[item+1]: k_hyper_sub_chap[item + 1] = k_hyper_sub_chap[item] + "\paragraph{\\bf KLS Addendum: " + \ word + "}\n" + k_hyper_sub_chap[item + 1] + print khyper_header_chap[item] + print "ding" + print k_hyper_sub_chap[item + 1] khyper_header_chap[item] = "memes" k_hyper_sub_chap[item] = "memes" else: @@ -153,11 +154,9 @@ def fix_chapter_sort(kls, chap, word, sortloc, klsaddparas, sortmatch_2): chap9 = 0 if sorter_check[sortloc][0] == 0: - tempref = ref9_3 sorter_check[sortloc][0] += 1 chap9 = 1 else: - tempref = ref14_3 offset = sorter_check[sortloc][1] if special_input == 1: offset = 8 @@ -199,7 +198,7 @@ def fix_chapter_sort(kls, chap, word, sortloc, klsaddparas, sortmatch_2): # print hyper_headers_chap # print word sortmatch_2.append(k_hyper_sub_chap) - + return chap def cut_words(word_to_find, word_to_search_in): """ @@ -305,6 +304,8 @@ def find_references(chapter, chapticker, math_people): :param chapticker: Which chapter is being searched (9 or 14). :return: List of references. """ + ref9_3 = [] + ref14_3 = [] references = [] index = -1 # chaptercheck designates which chapter is being searched for references @@ -363,7 +364,7 @@ def find_references(chapter, chapticker, math_people): if bqlegendrecheck > 0: pass - return references + return references, ref9_3, ref14_3 def reference_placer(chap, references, p, chapticker2): @@ -408,7 +409,7 @@ def reference_placer(chap, references, p, chapticker2): # method to change file string(actually a list right now), returns string to be written to file # If you write a method that changes something, it is preffered that you call the method in here -def fix_chapter(chap, references, paragraphs_to_be_added, kls, kls_list_all, chapticker2, new_commands, klsaddparas, sortmatch_2): +def fix_chapter(chap, references, paragraphs_to_be_added, kls, kls_list_all, chapticker2, new_commands, klsaddparas, sortmatch_2, tempref): """ Removes specific lines stopping the latex file from converting into python, as well as running the functions responsible for sorting sections and placing the correct additions in the correct places @@ -425,7 +426,7 @@ def fix_chapter(chap, references, paragraphs_to_be_added, kls, kls_list_all, cha sort_location = 0 for i in kls_list_all: - fix_chapter_sort(kls, chap, i, sort_location, klsaddparas, sortmatch_2) + fix_chapter_sort(kls, chap, i, sort_location, klsaddparas, sortmatch_2, tempref) sort_location += 1 for a in range(len(paragraphs_to_be_added)-1): @@ -434,33 +435,12 @@ def fix_chapter(chap, references, paragraphs_to_be_added, kls, kls_list_all, cha with open(DATA_DIR + "compare.tex", "a") as spook: spook.write(paragraphs_to_be_added[a]) spook.write("NEXT: ") - # i = 0 - # while i < len(c) - 1: - # if c[i:i + 1] == '\n': - # c = c[:i] + c[i + 1:] - # else: - # i += 1 - # i = 0 - # shortened_paras = paragraphs_to_be_added[a] - # while i < len(shortened_paras) - 1: - # if shortened_paras[i:i + 1] == '\n': - # shortened_paras = shortened_paras[:i] + shortened_paras[i + 1:] - # else: - # i += 1 - # paragraphs_to_be_added[a] = shortened_paras if "%" == c[-2]: c = c[:-3] #print "memes" #print c elif "%" == c[-1]: c = c[:-2] - #print "extramemes" - #print c - # if "Formula (9.8.15) was first obtained by Brafman \myciteKLS{109}." in c and \ - # "Formula (9.8.15) was first obtained by Brafman \myciteKLS{109}." in p[a]: - # print "MEMESBEGIN"*20 - # print c - # print "MEMESEND"*24 paragraphs_to_be_added[a] = cut_words(c, paragraphs_to_be_added[a]) reference_placer(chap, references, paragraphs_to_be_added, chapticker2) @@ -523,6 +503,9 @@ def main(): kls_list_all = new_keywords(addendum, kls_list_full) # Makes sections look like other sections + for item in addendum: + addendum[addendum.index(item)] = item.replace("\\eqref{","\\eqref{KLSadd: ") + for word in addendum: if "paragraph{" in word: lenword = len(word) - 1 @@ -537,6 +520,7 @@ def main(): # Designates sections that need stuff added # get the index + startindex = 9999 for word in addendum: index += 1 if "." in word and "\\subsection*{" in word: @@ -548,12 +532,15 @@ def main(): chap_nums.append(14) name = word[word.find("{") + 1: word.find("}")] math_people.append(name + "#") + if name == "9.1 Wilson": + startindex = addendum.index(word) + print startindex indexes.append(index-1) - if "paragraph{" in word and index > 313: + if "paragraph{" in word and index > startindex-1: klsaddparas.append(index-1) - if "subsub" in word and index > 313: + if "subsub" in word and index > startindex-1: klsaddparas.append(index - 1) - if "\subsection*{" in word and index > 313: + if "\subsection*{" in word and index > startindex-1: klsaddparas.append(index - 1) klsaddparas.append(2246) print(indexes) @@ -577,11 +564,8 @@ def main(): with open(DATA_DIR + "chap01.tex", "r") as ch1: entire1 = ch1.readlines() # reads in as a list of strings - chap1placer(entire1, addendum, klsaddparas) - with open(DATA_DIR + "updated1.tex", "w") as temp1: temp1.write(chap1placer(entire1, addendum, klsaddparas)) - # chapter 9 with open(DATA_DIR + "chap09.tex", "r") as ch9: entire9 = ch9.readlines() # reads in as a list of strings @@ -592,23 +576,26 @@ def main(): # call the findReferences method to find the index of the References paragraph in the two file strings # two variables for the references lists one for chapter 9 one for chapter 14 - for item in addendum: - addendum[addendum.index(item)] = item.replace("\\eqref{", "\\eqref{KLSadd: ") chapticker = 0 references9 = find_references(entire9, chapticker, math_people) + ref9_3 = references9[1] + references9 = references9[0] chapticker += 1 references14 = find_references(entire14, chapticker, math_people) + ref14_3 = references14[2] + references14 = references14[0] + ref14_3.insert(88, 1217) ref14_3.insert(244, 3053) ref14_3.insert(198, 2554) # call the fixChapter method to get a list with the addendum paragraphs added in chapticker2 = 0 - str9 = ''.join(fix_chapter(entire9, references9, paras, addendum, kls_list_all, chapticker2, new_commands, klsaddparas, sortmatch_2)) + str9 = ''.join(fix_chapter(entire9, references9, paras, addendum, kls_list_all, chapticker2, new_commands, klsaddparas, sortmatch_2, ref9_3)) chapticker2 += 1 - str14 = ''.join(fix_chapter(entire14, references14, paras, addendum, kls_list_all, chapticker2, new_commands, klsaddparas, sortmatch_2)) + str14 = ''.join(fix_chapter(entire14, references14, paras, addendum, kls_list_all, chapticker2, new_commands, klsaddparas, sortmatch_2, ref14_3)) # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ # If you are writing something that will make a change to the chapter files, write it BEFORE this line, this part @@ -622,8 +609,6 @@ def main(): with open(DATA_DIR + "updated14.tex", "w") as temp14: temp14.write(str14) - print math_people - print paras if __name__ == '__main__': main() \ No newline at end of file diff --git a/KLSadd_insertion/test/test_fix_chapter.py b/KLSadd_insertion/test/test_fix_chapter.py deleted file mode 100644 index ab93c2f..0000000 --- a/KLSadd_insertion/test/test_fix_chapter.py +++ /dev/null @@ -1,8 +0,0 @@ -bob = "memes\nbanana" -i = 0 -while i < len(bob) - 1: - if bob[i:i + 1] == '\n': - bob = bob[:i] + bob[i+1:] - else: - i += 1 -print bob \ No newline at end of file diff --git a/KLSadd_insertion/test/test_fix_chapter_sort.py b/KLSadd_insertion/test/test_fix_chapter_sort.py new file mode 100644 index 0000000..01aee0e --- /dev/null +++ b/KLSadd_insertion/test/test_fix_chapter_sort.py @@ -0,0 +1,36 @@ + +__author__ = 'Edward Bian' +__status__ = 'Development' + +from unittest import TestCase +from updateChapters import fix_chapter_sort + + +class TestFixChapterSort(TestCase): + def test_fix_chapter_sort(self): + self.assertEquals(fix_chapter_sort(['%' + , '\subsection*{9.1 Wilson}' + , '\paragraph{Hypergeometric Representation}' + , 'The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetric' + , 'in $a,b,c,d$.' + , '\renewcommand{\refname}{Standard references}'] + ,['\section{Wilson}\index{Wilson polynomials}' + ,'\subsection*{Hypergeometric representation}' + ,'\begin{eqnarray}' + ,'& &\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\nonumber\\' + ,'\end{eqnarray}' + ,'\subsection*{Orthogonality relation}' + ,'\begin{eqnarray}' + ,'\label{OrthIWilson}' + ,'& &\frac{1}{2\pi}\int_0^{\infty}' + ,'\end{eqnarray}'],'Hypergeometric Representation',0,[1,2,5],[],[0,1,5]) + ,['\\section{Wilson}\\index{Wilson polynomials}' + , '\\subsection*{Hypergeometric representation}' + , '\begin{eqnarray}' + , '& &\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\nonumber\\' + , '\\end{eqnarray}\\paragraph{\\bf KLS Addendum: Hypergeometric Representation}The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetricin $a,b,c,d$.' + , '\\subsection*{Orthogonality relation}' + , '\begin{eqnarray}' + , '\\label{OrthIWilson}' + , '& &\frac{1}{2\\pi}\\int_0^{\\infty}' + , '\\end{eqnarray}']) \ No newline at end of file From 0226ef50f34d6385b751d4741bdcebdc8bf3617e Mon Sep 17 00:00:00 2001 From: Edward Bian Date: Mon, 22 Aug 2016 14:35:24 -0400 Subject: [PATCH 315/402] Unit Tests --- .../test/test_reference_finder.py | 35 ++++++------------- 1 file changed, 10 insertions(+), 25 deletions(-) diff --git a/KLSadd_insertion/test/test_reference_finder.py b/KLSadd_insertion/test/test_reference_finder.py index b65437f..64ff297 100644 --- a/KLSadd_insertion/test/test_reference_finder.py +++ b/KLSadd_insertion/test/test_reference_finder.py @@ -11,50 +11,35 @@ def test_find_references(self): self.assertEquals(find_references(['\\section{Pseudo Jacobi}\\index{Racah polynomials}', '\\subsection*{Hypergeometric representation}', '\\begin{eqnarray}', '\\label{DefRacah}', '& &R_n(\lambda(x);\alpha,\beta,\gamma,\delta)\nonumber\\' - , '& &{}=\hyp{4}{3}{-n,n+\alpha+\beta+1,-x,' - 'x+\gamma+\delta+1}{\alpha+1,\beta+\delta+1,\gamma+1}{1},', '\quad n=0,1,2,\ldots,N,' - , '\end{eqnarray}', 'where', '$$\lambda(x)=x(x+\gamma+\delta+1)$$', 'and' - , '$$\alpha+1=-N\quad\textrm{or}\quad\beta+\delta+1=-N\quad\textrm{or}\quad\gamma+1=-N$$' + , '& &{}=\hyp{4}{3}{-n,n+\alpha+\beta+1,-x,', '\end{eqnarray}' , 'with $N$ a nonnegative integer.', '\subsection*{References}', '\cite{Askey89I}, \cite{AskeyWilson79}, \cite{AskeyWilson85}'], - 0, ['9.9 Pseudo Jacobi (or Routh-Romanovski)#']), [13]) + 0, ['9.9 Pseudo Jacobi (or Routh-Romanovski)#']), ([8], [0, 1, 8, 2646], [])) self.assertEquals(find_references(['\\section{Pseudo Jacobi}\\index{Racah polynomials}', '\\subsection*{Hypergeometric representation}', '\\begin{eqnarray}', '\\label{DefRacah}', '& &R_n(\lambda(x);\alpha,\beta,\gamma,\delta)\nonumber\\' - , '& &{}=\hyp{4}{3}{-n,n+\alpha+\beta+1,-x,' - 'x+\gamma+\delta+1}{\alpha+1,\beta+\delta+1,\gamma+1}{1},', '\quad n=0,1,2,\ldots,N,' - , '\end{eqnarray}', 'where', '$$\lambda(x)=x(x+\gamma+\delta+1)$$', 'and' - , '$$\alpha+1=-N\quad\textrm{or}\quad\beta+\delta+1=-N\quad\textrm{or}\quad\gamma+1=-N$$' + , '& &{}=\hyp{4}{3}{-n,n+\alpha+\beta+1,-x,', '\end{eqnarray}' , 'with $N$ a nonnegative integer.', '\subsection*{References}', '\cite{Askey89I}, \cite{AskeyWilson79}, \cite{AskeyWilson85}'], - 1, ['9.9 Pseudo Jacobi (or Routh-Romanovski)#']), [13]) + 1, ['9.9 Pseudo Jacobi (or Routh-Romanovski)#']), ([8], [2646], [0, 1, 8])) self.assertEquals(find_references(['\\section{Racah}\\index{Racah polynomials}', '\\subsection*{Hypergeometric representation}', '\\begin{eqnarray}', '\\label{DefRacah}', '& &R_n(\lambda(x);\alpha,\beta,\gamma,\delta)\nonumber\\' - , '& &{}=\hyp{4}{3}{-n,n+\alpha+\beta+1,-x,' - 'x+\gamma+\delta+1}{\alpha+1,\beta+\delta+1,\gamma+1}{1},', '\quad n=0,1,2,\ldots,N,' - , '\end{eqnarray}', 'where', '$$\lambda(x)=x(x+\gamma+\delta+1)$$', 'and' - , '$$\alpha+1=-N\quad\textrm{or}\quad\beta+\delta+1=-N\quad\textrm{or}\quad\gamma+1=-N$$' + , '& &{}=\hyp{4}{3}{-n,n+\alpha+\beta+1,-x,', '\end{eqnarray}' , 'with $N$ a nonnegative integer.', '\subsection*{References}', '\cite{Askey89I}, \cite{AskeyWilson79}, \cite{AskeyWilson85}'], - 0, ['9.2 Racah#']), [13]) + 0, ['9.2 Racah#']), ([8], [0, 1, 8, 2646], [])) self.assertEquals(find_references(['\\section{Racah}\\index{Racah polynomials}', '\\subsection*{Hypergeometric representation}', '\\begin{eqnarray}', '\\label{DefRacah}', '& &R_n(\lambda(x);\alpha,\beta,\gamma,\delta)\nonumber\\' - , '& &{}=\hyp{4}{3}{-n,n+\alpha+\beta+1,-x,' - 'x+\gamma+\delta+1}{\alpha+1,\beta+\delta+1,\gamma+1}{1},', '\quad n=0,1,2,\ldots,N,' - , '\end{eqnarray}', 'where', '$$\lambda(x)=x(x+\gamma+\delta+1)$$', 'and' - , '$$\alpha+1=-N\quad\textrm{or}\quad\beta+\delta+1=-N\quad\textrm{or}\quad\gamma+1=-N$$' + , '& &{}=\hyp{4}{3}{-n,n+\alpha+\beta+1,-x,', '\end{eqnarray}' , 'with $N$ a nonnegative integer.', '\subsection*{References}', '\cite{Askey89I}, \cite{AskeyWilson79}, \cite{AskeyWilson85}'], - 0, ['9.2 Racah#']), [13]) + 0, ['9.2 Racah#']), ([8], [0, 1, 8, 2646], [])) self.assertEquals(find_references(['\\section{Unique}\\index{Racah polynomials}', '\\subsection*{Hypergeometric representation}', '\\begin{eqnarray}', '\\label{DefRacah}', '& &R_n(\lambda(x);\alpha,\beta,\gamma,\delta)\nonumber\\' - , '& &{}=\hyp{4}{3}{-n,n+\alpha+\beta+1,-x,' - 'x+\gamma+\delta+1}{\alpha+1,\beta+\delta+1,\gamma+1}{1},', '\quad n=0,1,2,\ldots,N,' - , '\end{eqnarray}', 'where', '$$\lambda(x)=x(x+\gamma+\delta+1)$$', 'and' - , '$$\alpha+1=-N\quad\textrm{or}\quad\beta+\delta+1=-N\quad\textrm{or}\quad\gamma+1=-N$$' + , '& &{}=\hyp{4}{3}{-n,n+\alpha+\beta+1,-x,', '\end{eqnarray}' , 'with $N$ a nonnegative integer.', '\subsection*{References}', '\cite{Askey89I}, \cite{AskeyWilson79}, \cite{AskeyWilson85}'], - 1, ['9.2 Racah#']), []) \ No newline at end of file + 1, ['9.2 Racah#']), ([], [2646], [0, 1])) \ No newline at end of file From 1e1e700edce524f6484b04e3044496de872e5daa Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Mon, 22 Aug 2016 15:04:46 -0400 Subject: [PATCH 316/402] Add mathematica references and tests for that --- eCF/src/latex_constraint_modifier.py | 20 ++++++++++---- eCF/src/mathematica_to_latex.py | 40 +++++++++++++++++++++++----- eCF/test/data/test.tex | 1 + eCF/test/data/testref.txt | 3 +++ eCF/test/test_main.py | 6 +++-- 5 files changed, 57 insertions(+), 13 deletions(-) create mode 100644 eCF/test/data/testref.txt diff --git a/eCF/src/latex_constraint_modifier.py b/eCF/src/latex_constraint_modifier.py index adac470..9d7f955 100644 --- a/eCF/src/latex_constraint_modifier.py +++ b/eCF/src/latex_constraint_modifier.py @@ -25,7 +25,10 @@ '\\proof': '{\\bf PROOF}:~', '\\mathematicatag': '{\\bf MtT}:~', '\\mathematicareference': '{\\bf MtR}:~', - '\\mapletag': '{\\bf MpT}:~'} + '\\mapletag': '{\\bf MpT}:~', + '\\category': '{\\bf CAT}:~'} +TT = ('\\mathematicatag', '\\mapletag') +TEXT = ('\\mathematicareference', '\\category') def find_surrounding(line, function): @@ -72,7 +75,7 @@ def combine_percent(lines): index = 0 while index < len(lines): if len(lines[index]) >= 3 and lines[index][:3] == '% ' and \ - index > lines.index('\\begin{document}'): + index > lines.index('\\begin{equation}'): if len(lines[index + 1]) != '' and lines[index + 1][0] == '%': print(lines[index]) add = lines.pop(index + 1) @@ -100,8 +103,15 @@ def replace(line): if pos[0] != pos[1]: arg = line[pos[0] + len(word) + 1:pos[1] - 1]\ .replace('\n', '').replace(' &', ' &') - arg = ('\\\\[0.2cm]\n ' + CONVERSIONS[word])\ - .join(arg.split('&')) + if word in TT: + arg = ('}$ \\\\[0.2cm]\n ' + CONVERSIONS[word] + + '$\\tt{').join(arg.split('&')) + if word in TEXT: + arg = ('}$ \\\\[0.2cm]\n ' + CONVERSIONS[word] + + '$\\text{').join(arg.split('&')) + else: + arg = ('\\\\[0.2cm]\n ' + CONVERSIONS[word])\ + .join(arg.split('&')) if (max([line.find(item) for item in CONVERSIONS]) > pos[1] or max([line.find(CONVERSIONS[item]) for @@ -158,7 +168,7 @@ def main(): for line in before: if len(line) >= 3 and line[:3] == '% ' and \ before.index(line) > \ - before.index('\\begin{document}'): + before.index('\\begin{equation}'): line = replace(line) line = dollarsign(line) line = '\\begin{flushright}\n' + line.replace('%', ' ') + \ diff --git a/eCF/src/mathematica_to_latex.py b/eCF/src/mathematica_to_latex.py index 1b74b6f..1f6867e 100644 --- a/eCF/src/mathematica_to_latex.py +++ b/eCF/src/mathematica_to_latex.py @@ -161,6 +161,28 @@ def search(line, i, sign, direction=-1): return j +def process_references(pathr): + # (str) -> dict + """ + Opens references file and process it into a dictionary + + :param pathr: directory of file to be read from + :return: dictionary of processed references + """ + with open(pathr) as refs: + references = list(line.split('\n') for line in + refs.read().split('\n\n')) + + key = [] + value = [] + print(references) + for pair in references: + key.append(pair[0][3:-1].replace('"', '')) + value.append('&'.join(pair[1:])) + + return dict(zip(key, value)) + + def master_function(line, params): # (str, tuple) -> str """ @@ -858,7 +880,8 @@ def replace_vars(line): def main(pathw=DIR_NAME + 'newIdentities.tex', - pathr=DIR_NAME + 'Identities.m'): + pathr=DIR_NAME + 'Identities.m', + pathref=DIR_NAME + 'References.txt'): # ((str, str)) -> None """ Opens Mathematica file with identities and puts converted lines into @@ -866,10 +889,10 @@ def main(pathw=DIR_NAME + 'newIdentities.tex', :param pathw: directory of file to be written to :param pathr: directory of file to be read from - :param test: if True, replaces "(* *)" with quotes; if False: uses "\\tag" - to mark the functions + :param pathref: directory of file with references to be inserted :returns: None """ + references = process_references(pathref) with open(pathw, 'w') as latex: with open(pathr, 'r') as mathematica: @@ -928,9 +951,14 @@ def main(pathw=DIR_NAME + 'newIdentities.tex', if line != '': line += '\n% \\mathematicatag{$\\tt{' + mtt + '}$}' + try: + line += '\n% \\mathematicareference{$\\text{' + \ + references[mtt] + '}$}' + except KeyError: + print(mtt) line += '\n\\end{equation}' - print line + # print line latex.write(line + '\n') latex.write('\n\n\\end{document}\n') @@ -938,8 +966,7 @@ def main(pathw=DIR_NAME + 'newIdentities.tex', # Open data/functions, and process the data into a comprehensible tuple that # gets fed into "master_function" function -with open(DIR_NAME + 'functions') \ - as functions: +with open(DIR_NAME + 'functions') as functions: FUNCTION_CONVERSIONS = list(arg_split(line.replace(' ', ''), ',') for line in functions.read().split('\n') if (line != '' and '#' not in line)) @@ -958,5 +985,6 @@ def main(pathw=DIR_NAME + 'newIdentities.tex', FUNCTION_CONVERSIONS = tuple(FUNCTION_CONVERSIONS) + if __name__ == '__main__': main() diff --git a/eCF/test/data/test.tex b/eCF/test/data/test.tex index a315ca9..ceaa441 100644 --- a/eCF/test/data/test.tex +++ b/eCF/test/data/test.tex @@ -18,6 +18,7 @@ \begin{equation} equation % \mathematicatag{$\tt{description, number}$} +% \mathematicareference{$\text{test reference line 1&test reference line 2}$} \end{equation} diff --git a/eCF/test/data/testref.txt b/eCF/test/data/testref.txt new file mode 100644 index 0000000..b61ac73 --- /dev/null +++ b/eCF/test/data/testref.txt @@ -0,0 +1,3 @@ +% {"description", number} +test reference line 1 +test reference line 2 \ No newline at end of file diff --git a/eCF/test/test_main.py b/eCF/test/test_main.py index 9e1c70e..1be6ff8 100644 --- a/eCF/test/test_main.py +++ b/eCF/test/test_main.py @@ -9,12 +9,13 @@ PATHW = os.path.dirname(os.path.realpath(__file__)) + '/data/test.tex' PATHR = os.path.dirname(os.path.realpath(__file__)) + '/data/test.m' +PATHREF = os.path.dirname(os.path.realpath(__file__)) + '/data/testref.txt' class TestMain(TestCase): def test_gen(self): - main(pathw=PATHW, pathr=PATHR) + main(pathw=PATHW, pathr=PATHR, pathref=PATHREF) with open(PATHW, 'r') as l: latex = l.read() self.assertEqual( @@ -38,7 +39,8 @@ def test_gen(self): '\\begin{equation}\n' 'equation\n' '% \\mathematicatag{$\\tt{description, number}$}\n' + '% \\mathematicareference{$\\text{test reference line 1&test reference line 2}$}\n' '\\end{equation}\n' '\n' '\n' - '\\end{document}\n')) + '\\end{document}\n')) \ No newline at end of file From 278a774801549d5e9746693110f5729b74a976ac Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Mon, 22 Aug 2016 15:28:41 -0400 Subject: [PATCH 317/402] Add another test for main function --- eCF/src/mathematica_to_latex.py | 2 +- eCF/test/data/test.m | 3 +++ eCF/test/data/test.tex | 5 +++++ eCF/test/test_main.py | 5 +++++ 4 files changed, 14 insertions(+), 1 deletion(-) diff --git a/eCF/src/mathematica_to_latex.py b/eCF/src/mathematica_to_latex.py index 1f6867e..f7464a8 100644 --- a/eCF/src/mathematica_to_latex.py +++ b/eCF/src/mathematica_to_latex.py @@ -955,7 +955,7 @@ def main(pathw=DIR_NAME + 'newIdentities.tex', line += '\n% \\mathematicareference{$\\text{' + \ references[mtt] + '}$}' except KeyError: - print(mtt) + pass line += '\n\\end{equation}' # print line diff --git a/eCF/test/data/test.m b/eCF/test/data/test.m index 740f5d1..1864fd5 100644 --- a/eCF/test/data/test.m +++ b/eCF/test/data/test.m @@ -1,2 +1,5 @@ (* {"description", number}*) equation + +(* {"nodescription", number}*) +equation \ No newline at end of file diff --git a/eCF/test/data/test.tex b/eCF/test/data/test.tex index ceaa441..fbcacdb 100644 --- a/eCF/test/data/test.tex +++ b/eCF/test/data/test.tex @@ -21,5 +21,10 @@ % \mathematicareference{$\text{test reference line 1&test reference line 2}$} \end{equation} +\begin{equation} +equation +% \mathematicatag{$\tt{nodescription, number}$} +\end{equation} + \end{document} diff --git a/eCF/test/test_main.py b/eCF/test/test_main.py index 1be6ff8..fde72f5 100644 --- a/eCF/test/test_main.py +++ b/eCF/test/test_main.py @@ -42,5 +42,10 @@ def test_gen(self): '% \\mathematicareference{$\\text{test reference line 1&test reference line 2}$}\n' '\\end{equation}\n' '\n' + '\\begin{equation}\n' + 'equation\n' + '% \\mathematicatag{$\\tt{nodescription, number}$}\n' + '\\end{equation}\n' + '\n' '\n' '\\end{document}\n')) \ No newline at end of file From 34d0cb3872efe254f70d85169ffc5631ece02697 Mon Sep 17 00:00:00 2001 From: Oksana Tkach Date: Tue, 23 Aug 2016 10:30:35 -0400 Subject: [PATCH 318/402] Remove in-line comments after equation labels --- DLMF_preprocessing/src/math_function.py | 8 -- DLMF_preprocessing/src/remove_excess.py | 132 +++++++++++++++------- DLMF_preprocessing/src/replace_special.py | 2 - 3 files changed, 93 insertions(+), 49 deletions(-) diff --git a/DLMF_preprocessing/src/math_function.py b/DLMF_preprocessing/src/math_function.py index 6a248ae..d362e77 100644 --- a/DLMF_preprocessing/src/math_function.py +++ b/DLMF_preprocessing/src/math_function.py @@ -1,5 +1,4 @@ import math_mode -import string import re @@ -29,14 +28,7 @@ def change_original(o_file, changed_math_string_list): # print math_ranges for i in math_ranges[::-1]: edited = edited[:i[0]] + changed_math_string_list[::-1][num] + edited[i[1]:] - # edited = edited.replace(o_string[i[0]:i[1]], changed_math_string_list[num]) - # print edited num += 1 - # re check NM.2new.tex isn't changing all of the i chars to \iunit - - check = open('whatevs.txt', 'w') - check.write(edited) - check.close() return edited diff --git a/DLMF_preprocessing/src/remove_excess.py b/DLMF_preprocessing/src/remove_excess.py index 13c5e6e..d5519c8 100644 --- a/DLMF_preprocessing/src/remove_excess.py +++ b/DLMF_preprocessing/src/remove_excess.py @@ -8,7 +8,7 @@ find_line) EQ_START = r'\\begin{equation}' -EQ_END = r'\end{equation}' +EQ_END = r'\\end{equation}' CASES_START = r'\begin{cases}' CASES_END = r'\end{cases}' @@ -80,24 +80,15 @@ def remove_excess(content): """Removes the excess pieces from the given content and returns the updated version as a string. :param content: """ + oldest = content + start_eq = re.compile(EQ_START) end_eq = re.compile(EQ_END) - starts = [m.end() for m in start_eq.finditer(content)] - ends = [m.start() for m in end_eq.finditer(content)] - print 'starts', starts - print 'ends', ends - if len(starts) != len(ends): - print 'UNEVEN DISTRIBUTION OF BEGIN END EQ' - sys.exit(-1) - - for i in enumerate(start): - print 'i', i - - oldest = content updated = _get_preamble() old = content content = remove_macro(r"\\lxID", content) + # edit single lines # removed unecessary information that caused an error with pdflatex pattern = re.compile(r'\\acknowledgements{.*?}\n\n', re.DOTALL) @@ -115,7 +106,7 @@ def remove_excess(content): pattern = re.compile(r'\\onlyprint{.*?}', re.DOTALL) content = re.sub(pattern, r' ', content) pattern = re.compile(r'\\origref\[.*?\]{.*?}', re.DOTALL) - content = re.sub(pattern, r' ', content) + content = re.sub(pattern, r'', content) content = re.sub(r'\\begin\{electroniconly}', r' ', content) content = re.sub(r'\\end\{electroniconly}', r' ', content) content = re.sub(r'\\end\{printonly}', r' ', content) @@ -129,7 +120,6 @@ def remove_excess(content): print(len(old) - len(content)) old = content - content = remove_section(r'\\section{Special Notation}', r'\\part', content) content = remove_section(r'\\part{Computation}', r'\\bibliographystyle{plain}', content) content = remove_section(r'\\part{References}', r'\\bibliographystyle{plain}', content) @@ -144,12 +134,13 @@ def remove_excess(content): to_join = [] in_eq = False + # takes out comment lines first for line in content.split("\n"): - if EQ_START in line: + if re.match(EQ_START, line.lstrip()): in_eq = True - if EQ_END in line: + if re.match(EQ_END, line.lstrip()): in_eq = False # if line only consists of % replace with nothing, otherwise don't add back @@ -174,14 +165,11 @@ def remove_excess(content): content = remove_begin_end("table", content) content = remove_begin_end("table", content) - content = remove_begin_end("%", content) content = remove_begin_end("sidebar", content) content = parentheses.remove(content, curly=True) - - to_remove = [ r'\\citet', r'\\lxDeclare\[.*?\]', @@ -195,14 +183,31 @@ def remove_excess(content): ] lengths = get_line_lengths(content) + lines = content.split('\n') + + entering = re.compile(r'\\begin###open_\d###equation###close_\d###') + exiting = re.compile(r'\\end###open_\d###equation###close_\d###') + + # Find the lines on which equations start and end + eq_starts = [m.end() for m in entering.finditer(content)] + eq_ends = [m.start() for m in exiting.finditer(content)] + start_end_eq = [] + + for i in range(len(eq_starts)): + eq_starts[i] = find_line(eq_starts[i], lengths) + eq_ends[i] = find_line(eq_ends[i], lengths) + start_end_eq.append((eq_starts[i], eq_ends[i])) + + if len(eq_starts) != len(eq_ends): + print 'UNEVEN DISTRIBUTION OF \\begin{equation} and \\end{equation}' + sys.exit(-1) # go through each group and remove it for name in to_remove: # for each improper name, put it in ###open### and ###closes### - str_regex = name # + STD_REGEX not really necessary to add STD_REGEX, edit if wrong -oksana + str_regex = name # + STD_REGEX not really necessary to add STD_REGEX, edit if wrong -oksana pattern = re.compile(str_regex, re.DOTALL) - lines = content.split("\n") # go through every match, finding start and end for match in pattern.finditer(content): @@ -211,17 +216,27 @@ def remove_excess(content): start = find_line(match.start(), lengths) end = find_line(match.end(), lengths) + dont_skip = False + + # check that element isn't a comment within an equation + for eq in start_end_eq: - # print("{0}: {1} - {2}".format(group, start, end)) - skip_lines.update(range(start, end + 1)) + if eq[0] <= start <= eq[1]: + if lines[start - 1].lstrip().startswith('%'): + dont_skip = True + + if lines[start - 1].count('$') % 2 != 0: + # don't skip if only 1 $ on that line so as to avoid causing wrong math_mode ranges + dont_skip = True + + if not dont_skip: + skip_lines.update(range(start, end + 1)) # define items that cannot be at the beginning of any line or be contined in any line illegal_starts = [r'\documentclass{DLMF}', r'\thischapter', r'\part{Notation}', r'\begin{equationgroup', r'\end{equationgroup', r'\begin{onecolumn', r'\end{onecolumn'] illegal_elements = [r'TwoToOneRule', r'OneToTwoRule', r'\citet'] - lines = content.split("\n") - # various flags that will be useful when going through the lines, in_eq above for comment control in_const = False in_cases = False @@ -229,7 +244,20 @@ def remove_excess(content): eqmix_label = "" const_str = "" + lengths = get_line_lengths(content) + + eq_starts = [m.end() for m in start_eq.finditer(content)] + eq_ends = [m.start() for m in end_eq.finditer(content)] + start_end_eq = [] + + for i in range(len(eq_starts)): + eq_starts[i] = find_line(eq_starts[i], lengths) + eq_ends[i] = find_line(eq_ends[i], lengths) + start_end_eq.append((eq_starts[i], eq_ends[i])) + + eq_counter = None + in_eqmix = False # remove trailing % and whitespace and add to updated - also remove lines that should be skipped for lnum, line in enumerate(lines): @@ -237,43 +265,71 @@ def remove_excess(content): lnum += 1 # don't add line back if it should be skipped - - if lnum in skip_lines: - - # print lnum, line continue + if re.match(EQ_START, line.lstrip()): + in_eq = True + if re.match(EQ_END, line.lstrip()): + in_eq = False + # replaces long ###open and close phrases with curly brackets line = parentheses.insert(line, curly=True) line_checks = [line.lstrip().startswith(start) for start in illegal_starts] + [element in line for element in - illegal_elements] + illegal_elements] # skip current line if it starts with or contains an illegal element if any(line_checks): + # don't remove comments in equations + if in_eq: + if not line.lstrip().startswith('%'): + continue - continue + if not in_eq: + # don't remove if 1 $ in line (causes missing $ error) + if line.count('$') % 2 == 0: + continue - # no more ugly % characters after cleaned + # no more trailing % characters after cleaned cleaned = line.rstrip().rstrip("%").rstrip() # line marks the start of an equationmix, set the flag and remove the line if EQMIX_START in cleaned: - eqmix_label = cleaned[cleaned.index(r'\label'):] + in_eqmix = True + eq_counter = 0 + if re.search('%', cleaned): + cleaned = cleaned[:cleaned.index('%')] + eqmix_label = cleaned[cleaned.index(r'\label'):-1] + ".SE" continue # line marks the end of an equationmix, set the flag and remove the line if EQMIX_END in cleaned: + in_eqmix = False + eq_counter = None eqmix_label = "" continue # if this line marks the start of an equation, set the flag - if EQ_START in cleaned: + if re.match(EQ_START, cleaned.lstrip()): + if in_eqmix: + if '\\end{equation}' in cleaned: + # if in eqmix and \end{equation} in line, space the commands, label, and the actual equation + if isinstance(eq_counter, int): + eq_counter += 1 + cleaned = cleaned.lstrip()[:16] + eqmix_label + str(eq_counter) + '}\n' + \ + cleaned[18:cleaned.index('equation}')] + '\n\\end{equation}' + cleaned = re.sub(r'\n{2,}', '\n', cleaned) + else: + # just in eqmix, add the command and label only + eq_counter += 1 + cleaned = cleaned.lstrip()[:16] + eqmix_label + str(eq_counter) + '}' + else: + # just in eq, add command and label only: make sure no in line comments get added + cleaned = cleaned.lstrip()[:16] + cleaned[16: 17 + cleaned[16:].index('}')] + eqmix_label in_eq = True - cleaned = cleaned + eqmix_label # if this line marks the end of an equation, set the flag - if EQ_END in cleaned: + if re.match(EQ_END, cleaned.lstrip()): in_eq = False # comment out constraints and replace commas with double commas; we need to build the entire constraint @@ -291,7 +347,6 @@ def remove_excess(content): # remove commas, periods, colons, and semi-colons from the end of equations if in_eq and not in_const: - to_strip = ":;," # don't take off trailing commas when in a cases block @@ -372,4 +427,3 @@ def _get_preamble(): if __name__ == "__main__": main() - diff --git a/DLMF_preprocessing/src/replace_special.py b/DLMF_preprocessing/src/replace_special.py index c001215..09d39c3 100644 --- a/DLMF_preprocessing/src/replace_special.py +++ b/DLMF_preprocessing/src/replace_special.py @@ -130,8 +130,6 @@ def remove_special(content): comment_str = comment_str[:-1] - # print([x for x in inside if inside[x][SEEN]]) - # reset special block flags for flag in inside: inside[flag][SEEN] = False From c118fdb758f5aebbcaaf41b53d64ea187fa74f87 Mon Sep 17 00:00:00 2001 From: Oksana Tkach Date: Tue, 23 Aug 2016 12:37:43 -0400 Subject: [PATCH 319/402] Match the intended formatting of replace_special.py --- DLMF_preprocessing/test/test_replace_i.py | 7 ++++--- 1 file changed, 4 insertions(+), 3 deletions(-) diff --git a/DLMF_preprocessing/test/test_replace_i.py b/DLMF_preprocessing/test/test_replace_i.py index d58d0aa..210f667 100644 --- a/DLMF_preprocessing/test/test_replace_i.py +++ b/DLMF_preprocessing/test/test_replace_i.py @@ -20,14 +20,15 @@ % \constraint{$s = 0,1,2,\dots$.} """ + class TestReplaceSpecial(TestCase): def test__replace_special(self): tex = "z^a = \\underbrace{z \\cdot z \\cdots z}_{n \n \\text{ times}} = 1 / z^{-a}" result = remove_special(tex) - self.assertEqual(tex, result) + self.assertEqual(tex.split('\n'), result) def test_dollar_loc(self): actual = remove_special(mismatched) - self.assertEqual(mismatched, actual) + self.assertEqual(mismatched.split('\n'), actual) def test_matching(self): result = remove_special(matched) - self.assertEqual(matched, result) \ No newline at end of file + self.assertEqual(matched.split('\n'), result) \ No newline at end of file From 690032f750dfb0b7397a3e16e62a9d0a8fb5314a Mon Sep 17 00:00:00 2001 From: Oksana Tkach Date: Tue, 23 Aug 2016 12:58:17 -0400 Subject: [PATCH 320/402] Check for current formatting rules in the program. --- DLMF_preprocessing/test/test_math_function.py | 14 +++++++++----- 1 file changed, 9 insertions(+), 5 deletions(-) diff --git a/DLMF_preprocessing/test/test_math_function.py b/DLMF_preprocessing/test/test_math_function.py index 9a3c2de..9d724cf 100644 --- a/DLMF_preprocessing/test/test_math_function.py +++ b/DLMF_preprocessing/test/test_math_function.py @@ -15,7 +15,8 @@ where $\\ell$ = last term of the series = $a + (n-1)d$. \\paragraph{Geometric Progression} -\\index{geometric progression (or series)}""" +\\index{geometric progression (or series)} +""" just_math = [""" a + (a + d) + (a + 2d) + \\dots + (a + (n-1)d) @@ -23,15 +24,18 @@ = \\tfrac{1}{2} n (a + \\ell)\n""", "\\ell", "a + (n-1)d"] no_text = """ + \\index{arithmetic progression} \\begin{equation}\\label{eq:AL.ES.AR} - a + (a + d) + (a + 2d) + \\dots + (a + (n-1)d) - = na + \\tfrac{1}{2} n(n-1) d - = \\tfrac{1}{2} n (a + \\ell) +a + (a + d) + (a + 2d) + \\dots + (a + (n-1)d) += na + \\tfrac{1}{2} n(n-1) d += \\tfrac{1}{2} n (a + \\ell) \\end{equation} -\\index{geometric progression (or series)}""" +\\index{geometric progression (or series)} + +""" From 6b3c1f4a0e9dbbfee0c153c8cd9b2971f01aecd1 Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Tue, 23 Aug 2016 13:55:57 -0400 Subject: [PATCH 321/402] Add differences for single variable and multiple variables in a trig function, with @ and @@ --- eCF/data/functions | 52 ++++++++++++++-------------- eCF/src/latex_constraint_modifier.py | 10 +++--- eCF/src/mathematica_to_latex.py | 12 +++++-- eCF/test/test_master_function.py | 12 +++++++ 4 files changed, 53 insertions(+), 33 deletions(-) diff --git a/eCF/data/functions b/eCF/data/functions index 360ce2c..5f63087 100644 --- a/eCF/data/functions +++ b/eCF/data/functions @@ -3,18 +3,18 @@ AiryAiPrime, \AiryAi', (@{-}), () AiryBi, \AiryBi, (@{-}), (AiryBiPrime) AiryBiPrime, \AiryBi', (@{-}), () - ArcCos, \acos, (@{-}), (ArcCosh) - ArcCosh, \acosh, (@{-}), () - ArcCot, \acot, (@{-}), (ArcCoth) - ArcCoth, \acoth, (@{-}), () - ArcCsc, \acsc, (@{-}), (ArcCsch) - ArcCsch, \acsch, (@{-}), () - ArcSec, \asec, (@{-}), (ArcSech) - ArcSech, \asech, (@{-}), () - ArcSin, \asin, (@{-}), (ArcSinh) - ArcSinh, \asinh, (@{-}), () - ArcTan, \atan, (@{-}), (ArcTanh) - ArcTanh, \atanh, (@{-}), () + ArcCos, \acos, (@@{-}), (ArcCosh) + ArcCosh, \acosh, (@@{-}), () + ArcCot, \acot, (@@{-}), (ArcCoth) + ArcCoth, \acoth, (@@{-}), () + ArcCsc, \acsc, (@@{-}), (ArcCsch) + ArcCsch, \acsch, (@@{-}), () + ArcSec, \asec, (@@{-}), (ArcSech) + ArcSech, \asech, (@@{-}), () + ArcSin, \asin, (@@{-}), (ArcSinh) + ArcSinh, \asinh, (@@{-}), () + ArcTan, \atan, (@@{-}), (ArcTanh) + ArcTanh, \atanh, (@@{-}), () Arg, \ph, (@{-}), () BetaRegularized, \IncI, ({-}@{-}{-}), () Binomial, \binom, ({-}{-}), () @@ -25,14 +25,14 @@ BesselY, \BesselY, ({-}@{-}), (SphericalBesselY) ChebyshevT, \ChebyT, ({-}@{-}), () ChebyshevU, \ChebyU, ({-}@{-}), () - Cos, \cos, (@{-}), (ArcCos, Cosh, CosIntegral) - Cosh, \cosh, (@{-}), (ArcCosh, CoshIntegral) + Cos, \cos, (@@{-}), (ArcCos, Cosh, CosIntegral) + Cosh, \cosh, (@@{-}), (ArcCosh, CoshIntegral) CoshIntegral, \CoshInt, (@{-}), () CosIntegral, \CosInt, (@{-}), () - Cot, \cot, (@{-}), (ArcCot, Coth) - Coth, \coth, (@{-}), (ArcCoth) - Csc, \csc, (@{-}), (ArcCsc, Csch) - Csch, \csch, (@{-}), (ArcCsch) + Cot, \cot, (@@{-}), (ArcCot, Coth) + Coth, \coth, (@@{-}), (ArcCoth) + Csc, \csc, (@@{-}), (ArcCsc, Csch) + Csch, \csch, (@@{-}), (ArcCsch) DawsonF, \DawsonsInt, (@{-}), () D, \deriv, ({-}{-}), (ParabolicCylinderD) EllipticE, \CompEllIntE, (@{-}), () @@ -90,12 +90,12 @@ RamanujanTauTheta, \RamanujanTauTheta, (@{-}), () Re, \realpart, ({-}), (Regularized, Real) RiemannSiegelTheta, \RiemannSiegelTheta, (@{-}), () - Sec, \sec, (@{-}), (ArcSec, Sech) - Sech, \sech, (@{-}), (ArcSech) - Sinc, \sinc, (@{-}), () - Sign, \sign, (@{-}), () - Sin, \sin, (@{-}), (ArcSin, Sinh, SinIntegral, Sinc) - Sinh, \sinh, (@{-}), (ArcSinh, SinhIntegral) + Sec, \sec, (@@{-}), (ArcSec, Sech) + Sech, \sech, (@@{-}), (ArcSech) + Sinc, \sinc, (@@{-}), () + Sign, \sign, (@@{-}), () + Sin, \sin, (@@{-}), (ArcSin, Sinh, SinIntegral, Sinc) + Sinh, \sinh, (@@{-}), (ArcSinh, SinhIntegral) SinhIntegral, \SinhInt, (@{-}), () SinIntegral, \SinInt, (@{-}), () SphericalBesselJ, \SphBesselJ, ({-}@{-}), () @@ -108,8 +108,8 @@ StruveH, \StruveH, ({-}@{-}), () StruveL, \StruveL, ({-}@{-}), () Subscript, \text, ({-}_{-}), () - Tan, \tan, (@{-}), (ArcTan, Tanh) - Tanh, \tanh, (@{-}), (ArcTanh) + Tan, \tan, (@@{-}), (ArcTan, Tanh) + Tanh, \tanh, (@@{-}), (ArcTanh) UnitStep, \HeavisideH, (@{-}), () WhittakerM, \WhitM, ({-}{-}@{-}), () WhittakerW, \WhitW, ({-}{-}@{-}), () diff --git a/eCF/src/latex_constraint_modifier.py b/eCF/src/latex_constraint_modifier.py index 9d7f955..e843981 100644 --- a/eCF/src/latex_constraint_modifier.py +++ b/eCF/src/latex_constraint_modifier.py @@ -5,8 +5,8 @@ import argparse parser = argparse.ArgumentParser( - description='Receive .tex file with constraints and convert' - ' them viewable metadata.') + description='Receive .tex file with constraints and convert' + ' them viewable metadata.') parser.add_argument('PATHR', type=str, help='path of input .tex file, with the current' ' directory as the starting point',) @@ -103,6 +103,8 @@ def replace(line): if pos[0] != pos[1]: arg = line[pos[0] + len(word) + 1:pos[1] - 1]\ .replace('\n', '').replace(' &', ' &') + + # Splits lines if there are any "&". if word in TT: arg = ('}$ \\\\[0.2cm]\n ' + CONVERSIONS[word] + '$\\tt{').join(arg.split('&')) @@ -140,9 +142,7 @@ def dollarsign(line): if len(count) == 2: # This checks to see if there is already a "displaystyle", so it # doesn't add a second useless one. - if line[count[0]:count[0] + 15] == '${\\displaystyle': - pass - else: + if line[count[0]:count[0] + 15] != '${\\displaystyle': begin = count[0] end = count[1] line = line[:begin] + '${\\displaystyle ' + \ diff --git a/eCF/src/mathematica_to_latex.py b/eCF/src/mathematica_to_latex.py index f7464a8..6da5118 100644 --- a/eCF/src/mathematica_to_latex.py +++ b/eCF/src/mathematica_to_latex.py @@ -43,6 +43,10 @@ LEFT_BRACKETS = list('([{') RIGHT_BRACKETS = list(')]}') +TRIG = ('ArcCos', 'ArcCosh', 'ArcCot', 'ArcCoth', 'ArcCsc', 'ArcCsch', + 'ArcSec', 'ArcSech', 'ArcSin', 'ArcSinh', 'ArcTan', 'ArcTanh', + 'Cos', 'Cosh', 'Cot', 'Coth', 'Csc', 'Csch', 'Sec', 'Sech', 'Sinc', + 'Sin', 'Sinh', 'Tan', 'Tanh') def find_surrounding(line, function, ex=(), start=0): @@ -175,7 +179,6 @@ def process_references(pathr): key = [] value = [] - print(references) for pair in references: key.append(pair[0][3:-1].replace('"', '')) value.append('&'.join(pair[1:])) @@ -197,6 +200,7 @@ def master_function(line, params): """ m, l, sep, ex = params[:5] sep = [i.split('-') for i in sep] + multi = list('+-*/') for _ in range(line.count(m)): try: @@ -227,6 +231,9 @@ def master_function(line, params): args.insert(0, 0) else: args.insert(0, len(arg_split(args[1][1:-1], ','))) + if m in TRIG and \ + sum([args[0].count(element) for element in multi]) != 0: + sep[0][0] = sep[0][0].replace('@@', '@') line = (line[:pos[0]] + l + '%s'.join(sep[[len(y) for y in sep] .index(len(args) + 1)]) + @@ -958,7 +965,8 @@ def main(pathw=DIR_NAME + 'newIdentities.tex', pass line += '\n\\end{equation}' - # print line + print(line) + latex.write(line + '\n') latex.write('\n\n\\end{document}\n') diff --git a/eCF/test/test_master_function.py b/eCF/test/test_master_function.py index cde731d..b089b08 100644 --- a/eCF/test/test_master_function.py +++ b/eCF/test/test_master_function.py @@ -25,6 +25,11 @@ FUNCTION_CONVERSIONS[index] = tuple(FUNCTION_CONVERSIONS[index]) FUNCTION_CONVERSIONS = tuple(FUNCTION_CONVERSIONS) +TRIG = ('ArcCos', 'ArcCosh', 'ArcCot', 'ArcCoth', 'ArcCsc', 'ArcCsch', + 'ArcSec', 'ArcSech', 'ArcSin', 'ArcSinh', 'ArcTan', 'ArcTanh', + 'Cos', 'Cosh', 'Cot', 'Coth', 'Csc', 'Csch', 'Sec', 'Sech', 'Sinc', + 'Sin', 'Sinh', 'Tan', 'Tanh') +MULTI = list('+-*/') class TestMasterFunction(TestCase): @@ -68,6 +73,13 @@ def test_conversion(self): else: self.assertEqual(master_function(before, function), after) self.assertEqual(master_function('--{0}--'.format(before), function), '--{0}--'.format(after)) + if function[0] in TRIG: + for sep in MULTI: + before = before[:-1] + sep + 'b]' + after = after.replace('@@', '@')[:-1] + sep + 'b}' + self.assertEqual(master_function(before, function), after) + self.assertEqual(master_function('--{0}--'.format(before), function), '--{0}--'.format(after)) + if function[0] == 'D': self.assertEqual(master_function('\\[Delta]', function), '\\[Delta]') From b59483b0af14cf95696180de212171bd15892aa6 Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Tue, 23 Aug 2016 14:10:08 -0400 Subject: [PATCH 322/402] Add documentation to code --- eCF/src/mathematica_to_latex.py | 3 +++ 1 file changed, 3 insertions(+) diff --git a/eCF/src/mathematica_to_latex.py b/eCF/src/mathematica_to_latex.py index 6da5118..5f33f10 100644 --- a/eCF/src/mathematica_to_latex.py +++ b/eCF/src/mathematica_to_latex.py @@ -231,6 +231,9 @@ def master_function(line, params): args.insert(0, 0) else: args.insert(0, len(arg_split(args[1][1:-1], ','))) + + # If the arguments in a trig function are more than one variable, + # then instead of "@@" make it "@" if m in TRIG and \ sum([args[0].count(element) for element in multi]) != 0: sep[0][0] = sep[0][0].replace('@@', '@') From ee81ec3239e5526e937b94305e1a393f7538e818 Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Tue, 23 Aug 2016 14:10:33 -0400 Subject: [PATCH 323/402] Modify documentation --- eCF/src/mathematica_to_latex.py | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/eCF/src/mathematica_to_latex.py b/eCF/src/mathematica_to_latex.py index 5f33f10..ec8db3c 100644 --- a/eCF/src/mathematica_to_latex.py +++ b/eCF/src/mathematica_to_latex.py @@ -976,7 +976,7 @@ def main(pathw=DIR_NAME + 'newIdentities.tex', # Open data/functions, and process the data into a comprehensible tuple that -# gets fed into "master_function" function +# gets fed into the "master_function" function with open(DIR_NAME + 'functions') as functions: FUNCTION_CONVERSIONS = list(arg_split(line.replace(' ', ''), ',') for line in functions.read().split('\n') From 97a4986581983767b5b8ade3719510db0096cd49 Mon Sep 17 00:00:00 2001 From: Oksana Tkach Date: Tue, 23 Aug 2016 14:13:29 -0400 Subject: [PATCH 324/402] Not assume all equations have labels --- DLMF_preprocessing/src/remove_excess.py | 5 ++++- 1 file changed, 4 insertions(+), 1 deletion(-) diff --git a/DLMF_preprocessing/src/remove_excess.py b/DLMF_preprocessing/src/remove_excess.py index d5519c8..f1e2d4d 100644 --- a/DLMF_preprocessing/src/remove_excess.py +++ b/DLMF_preprocessing/src/remove_excess.py @@ -325,7 +325,10 @@ def remove_excess(content): cleaned = cleaned.lstrip()[:16] + eqmix_label + str(eq_counter) + '}' else: # just in eq, add command and label only: make sure no in line comments get added - cleaned = cleaned.lstrip()[:16] + cleaned[16: 17 + cleaned[16:].index('}')] + eqmix_label + if '\\label' not in cleaned: + cleaned = cleaned.lstrip() + eqmix_label + else: + cleaned = cleaned.lstrip()[:16] + cleaned[16: 17 + cleaned[16:].index('}')] + eqmix_label in_eq = True # if this line marks the end of an equation, set the flag From 8bac6649b8f961ed98bcebe18238868d4ed7e270 Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Tue, 23 Aug 2016 14:21:04 -0400 Subject: [PATCH 325/402] Change spacing --- eCF/src/mathematica_to_latex.py | 4 +--- 1 file changed, 1 insertion(+), 3 deletions(-) diff --git a/eCF/src/mathematica_to_latex.py b/eCF/src/mathematica_to_latex.py index ec8db3c..b879e89 100644 --- a/eCF/src/mathematica_to_latex.py +++ b/eCF/src/mathematica_to_latex.py @@ -311,7 +311,6 @@ def carat(line): while i != len(line): if line[i] == '^': - k = search(line, i, list('*/+-=, '), 1) if line[i + 1] == '(' and line[k] == ')': @@ -757,7 +756,7 @@ def convert_fraction(line): k = search(line, i, sign, 1) # Removes extra surrounding parentheses, if there are any. - # This won't work if you're doing "( )( )/( )( )", it will + # This won't work if you're doing "( )( )/( )( )": it will # incorrectly change it to " )( / )( ", but there are no cases of # this happening yet, so I have not gone to fixing this yet. if (line[j + 1] == '(' and line[i - 1] == ')' and @@ -781,7 +780,6 @@ def convert_fraction(line): line = '{0}\\frac{{{1}}}{{{2}}}{3}' \ .format(line[:j + 1], line[j + 1:i], line[i + 1:k + 1], line[k + 1:]) - i += 1 return line From 8a296a95712f0a3f625f4dcf5ffa9d55021354d2 Mon Sep 17 00:00:00 2001 From: Edward Bian Date: Tue, 23 Aug 2016 14:38:33 -0400 Subject: [PATCH 326/402] Unit Tests --- KLSadd_insertion/src/updateChapters.py | 30 +++++++++---------- .../test/test_reference_finder.py | 10 +++---- 2 files changed, 20 insertions(+), 20 deletions(-) diff --git a/KLSadd_insertion/src/updateChapters.py b/KLSadd_insertion/src/updateChapters.py index bafe41f..0127c03 100644 --- a/KLSadd_insertion/src/updateChapters.py +++ b/KLSadd_insertion/src/updateChapters.py @@ -9,7 +9,7 @@ # holds all subsections in each chapter section along with the what holds -w, h = 2, 1000 +w, h = 2, 500 sorter_check = [[0 for _ in range(w)] for __ in range(h)] @@ -47,7 +47,8 @@ def extraneous_section_deleter(list): :return: Extraneous name free output """ return [item for item in list if "reference" not in item.lower() and "limit relation" not in item.lower() - and "symmetry" not in item.lower()] + and "symmetry" not in item.lower() and " hypergeometric representation" not in item.lower() + and "hypergeometric representation " not in item.lower()] def new_keywords(kls, kls_list): @@ -198,7 +199,7 @@ def fix_chapter_sort(kls, chap, word, sortloc, klsaddparas, sortmatch_2, tempref # print hyper_headers_chap # print word sortmatch_2.append(k_hyper_sub_chap) - return chap + return chap, sorter_check def cut_words(word_to_find, word_to_search_in): """ @@ -324,6 +325,10 @@ def find_references(chapter, chapticker, math_people): if("section{" in word or "subsection*{" in word) and ("subsubsection*{" not in word): w = word[word.find("{")+1: word.find("}")] ws = word[word.find("{")+1: word.find("~")] + if "bessel" in word.lower() and chapticker == 0: + ref9_3.append(index) + if ("big $q$-legendre" in word.lower() or "little $q$-legendre" in word.lower() or "continuous $q$-legendre" in word.lower()) and chapticker == 1: + ref14_3.append(index) for unit in math_people: subunit = unit[unit.find(" ")+1: unit.find("#")] # System of checks that verifies if section is in chapter @@ -352,15 +357,10 @@ def find_references(chapter, chapticker, math_people): w2 = word[word.find("{") + 1: word.find("}")] if "Bessel" not in w2: ref14_3.append(index) - besselcheck = 1 + bqlegendrecheck = 2 # Special lines that the program does not normally register - for i in ref9_3: - if i == 2646: - besselcheck = 0 - if besselcheck == 1: - ref9_3.insert(230, 2646) if bqlegendrecheck > 0: pass @@ -542,7 +542,9 @@ def main(): klsaddparas.append(index - 1) if "\subsection*{" in word and index > startindex-1: klsaddparas.append(index - 1) - klsaddparas.append(2246) + if "\\renewcommand{\\refname}{Standard references}" in word: + klsaddparas.append(index-1) + indexes.append(index - 1) print(indexes) # now indexes holds all of the places there is a section # using these indexes, get all of the words in between and add that to the paras[] @@ -550,8 +552,6 @@ def main(): for i in range(len(indexes)-1): box = ''.join(addendum[indexes[i]: indexes[i+1]-1]) paras.append(box) - box2 = ''.join(addendum[indexes[35]: 2245]) - paras.append(box2) # paras now holds the paragraphs that need to go into the chapter files, but they need to go in the appropriate @@ -586,10 +586,8 @@ def main(): ref14_3 = references14[2] references14 = references14[0] - ref14_3.insert(88, 1217) - ref14_3.insert(244, 3053) - ref14_3.insert(198, 2554) + print ref14_3 # call the fixChapter method to get a list with the addendum paragraphs added in chapticker2 = 0 str9 = ''.join(fix_chapter(entire9, references9, paras, addendum, kls_list_all, chapticker2, new_commands, klsaddparas, sortmatch_2, ref9_3)) @@ -610,5 +608,7 @@ def main(): with open(DATA_DIR + "updated14.tex", "w") as temp14: temp14.write(str14) + print sorter_check + if __name__ == '__main__': main() \ No newline at end of file diff --git a/KLSadd_insertion/test/test_reference_finder.py b/KLSadd_insertion/test/test_reference_finder.py index 64ff297..8426f44 100644 --- a/KLSadd_insertion/test/test_reference_finder.py +++ b/KLSadd_insertion/test/test_reference_finder.py @@ -14,32 +14,32 @@ def test_find_references(self): , '& &{}=\hyp{4}{3}{-n,n+\alpha+\beta+1,-x,', '\end{eqnarray}' , 'with $N$ a nonnegative integer.', '\subsection*{References}', '\cite{Askey89I}, \cite{AskeyWilson79}, \cite{AskeyWilson85}'], - 0, ['9.9 Pseudo Jacobi (or Routh-Romanovski)#']), ([8], [0, 1, 8, 2646], [])) + 0, ['9.9 Pseudo Jacobi (or Routh-Romanovski)#']), ([8], [0, 1, 8], [])) self.assertEquals(find_references(['\\section{Pseudo Jacobi}\\index{Racah polynomials}', '\\subsection*{Hypergeometric representation}', '\\begin{eqnarray}', '\\label{DefRacah}', '& &R_n(\lambda(x);\alpha,\beta,\gamma,\delta)\nonumber\\' , '& &{}=\hyp{4}{3}{-n,n+\alpha+\beta+1,-x,', '\end{eqnarray}' , 'with $N$ a nonnegative integer.', '\subsection*{References}', '\cite{Askey89I}, \cite{AskeyWilson79}, \cite{AskeyWilson85}'], - 1, ['9.9 Pseudo Jacobi (or Routh-Romanovski)#']), ([8], [2646], [0, 1, 8])) + 1, ['9.9 Pseudo Jacobi (or Routh-Romanovski)#']), ([8], [], [0, 1, 8])) self.assertEquals(find_references(['\\section{Racah}\\index{Racah polynomials}', '\\subsection*{Hypergeometric representation}', '\\begin{eqnarray}', '\\label{DefRacah}', '& &R_n(\lambda(x);\alpha,\beta,\gamma,\delta)\nonumber\\' , '& &{}=\hyp{4}{3}{-n,n+\alpha+\beta+1,-x,', '\end{eqnarray}' , 'with $N$ a nonnegative integer.', '\subsection*{References}', '\cite{Askey89I}, \cite{AskeyWilson79}, \cite{AskeyWilson85}'], - 0, ['9.2 Racah#']), ([8], [0, 1, 8, 2646], [])) + 0, ['9.2 Racah#']), ([8], [0, 1, 8], [])) self.assertEquals(find_references(['\\section{Racah}\\index{Racah polynomials}', '\\subsection*{Hypergeometric representation}', '\\begin{eqnarray}', '\\label{DefRacah}', '& &R_n(\lambda(x);\alpha,\beta,\gamma,\delta)\nonumber\\' , '& &{}=\hyp{4}{3}{-n,n+\alpha+\beta+1,-x,', '\end{eqnarray}' , 'with $N$ a nonnegative integer.', '\subsection*{References}', '\cite{Askey89I}, \cite{AskeyWilson79}, \cite{AskeyWilson85}'], - 0, ['9.2 Racah#']), ([8], [0, 1, 8, 2646], [])) + 0, ['9.2 Racah#']), ([8], [0, 1, 8], [])) self.assertEquals(find_references(['\\section{Unique}\\index{Racah polynomials}', '\\subsection*{Hypergeometric representation}', '\\begin{eqnarray}', '\\label{DefRacah}', '& &R_n(\lambda(x);\alpha,\beta,\gamma,\delta)\nonumber\\' , '& &{}=\hyp{4}{3}{-n,n+\alpha+\beta+1,-x,', '\end{eqnarray}' , 'with $N$ a nonnegative integer.', '\subsection*{References}', '\cite{Askey89I}, \cite{AskeyWilson79}, \cite{AskeyWilson85}'], - 1, ['9.2 Racah#']), ([], [2646], [0, 1])) \ No newline at end of file + 1, ['9.2 Racah#']), ([], [], [0, 1])) \ No newline at end of file From b23e386b5d64534eebd06e0fb1cbf1285ec67f96 Mon Sep 17 00:00:00 2001 From: Edward Bian Date: Tue, 23 Aug 2016 14:42:13 -0400 Subject: [PATCH 327/402] Unit Tests --- .../test/test_fix_chapter_sort.py | 60 +++++++++++++------ 1 file changed, 43 insertions(+), 17 deletions(-) diff --git a/KLSadd_insertion/test/test_fix_chapter_sort.py b/KLSadd_insertion/test/test_fix_chapter_sort.py index 01aee0e..66e80bf 100644 --- a/KLSadd_insertion/test/test_fix_chapter_sort.py +++ b/KLSadd_insertion/test/test_fix_chapter_sort.py @@ -9,28 +9,54 @@ class TestFixChapterSort(TestCase): def test_fix_chapter_sort(self): self.assertEquals(fix_chapter_sort(['%' - , '\subsection*{9.1 Wilson}' - , '\paragraph{Hypergeometric Representation}' + , '\\subsection*{9.1 Wilson}' + , '\\paragraph{Hypergeometric Representation}' , 'The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetric' , 'in $a,b,c,d$.' - , '\renewcommand{\refname}{Standard references}'] - ,['\section{Wilson}\index{Wilson polynomials}' - ,'\subsection*{Hypergeometric representation}' - ,'\begin{eqnarray}' - ,'& &\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\nonumber\\' - ,'\end{eqnarray}' - ,'\subsection*{Orthogonality relation}' - ,'\begin{eqnarray}' - ,'\label{OrthIWilson}' - ,'& &\frac{1}{2\pi}\int_0^{\infty}' - ,'\end{eqnarray}'],'Hypergeometric Representation',0,[1,2,5],[],[0,1,5]) + , '\\renewcommand{\\refname}{Standard references}'] + ,['\\section{Wilson}\\index{Wilson polynomials}' + ,'\\subsection*{Hypergeometric representation}' + ,'\\begin{eqnarray}' + ,'& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' + ,'\\end{eqnarray}' + ,'\\subsection*{Orthogonality relation}' + ,'\\begin{eqnarray}' + ,'\\label{OrthIWilson}' + ,'& &\\frac{1}{2\\pi}\\int_0^{\\infty}' + ,'\\end{eqnarray}'],'Hypergeometric Representation',0,[1,2,5],[],[0, 1, 5, 9]) ,['\\section{Wilson}\\index{Wilson polynomials}' , '\\subsection*{Hypergeometric representation}' - , '\begin{eqnarray}' - , '& &\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\nonumber\\' + , '\\begin{eqnarray}' + , '& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' , '\\end{eqnarray}\\paragraph{\\bf KLS Addendum: Hypergeometric Representation}The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetricin $a,b,c,d$.' , '\\subsection*{Orthogonality relation}' - , '\begin{eqnarray}' + , '\\begin{eqnarray}' + , '\\label{OrthIWilson}' + , '& &\\frac{1}{2\\pi}\\int_0^{\\infty}' + , '\\end{eqnarray}']) + self.assertEquals(fix_chapter_sort(['%' + , '\\subsection*{9.1 Wilson}' + , '\\paragraph{Orthogonality}' + , 'The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetric' + , 'in $a,b,c,d$.' + , '\\renewcommand{\\refname}{Standard references}'] + , ['\\section{Wilson}\\index{Wilson polynomials}' + , '\\subsection*{Hypergeometric representation}' + , '\\begin{eqnarray}' + , '& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' + , '\\end{eqnarray}' + , '\\subsection*{Orthogonality}' + , '\\begin{eqnarray}' + , '\\label{OrthIWilson}' + , '& &\\frac{1}{2\\pi}\\int_0^{\\infty}' + , '\\end{eqnarray}'], 'Memes', 0, [1, 2, 5], [], [0, 1, 5, 9]) + , ['\\section{Wilson}\\index{Wilson polynomials}' + , '\\subsection*{Hypergeometric representation}' + , '\\begin{eqnarray}' + , '& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' + , '\\end{eqnarray}' + , '\\subsection*{Orthogonality}' + , '\\begin{eqnarray}' , '\\label{OrthIWilson}' - , '& &\frac{1}{2\\pi}\\int_0^{\\infty}' + , '& &\\frac{1}{2\\pi}\\int_0^{\\infty}' , '\\end{eqnarray}']) \ No newline at end of file From 1382aeccb57fcfe730208f7b964e8d3b58603bd6 Mon Sep 17 00:00:00 2001 From: Edward Bian Date: Tue, 23 Aug 2016 14:47:34 -0400 Subject: [PATCH 328/402] Unit Tests --- KLSadd_insertion/test/test_fix_chapter_sort.py | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/KLSadd_insertion/test/test_fix_chapter_sort.py b/KLSadd_insertion/test/test_fix_chapter_sort.py index 66e80bf..6dd705d 100644 --- a/KLSadd_insertion/test/test_fix_chapter_sort.py +++ b/KLSadd_insertion/test/test_fix_chapter_sort.py @@ -33,7 +33,7 @@ def test_fix_chapter_sort(self): , '\\begin{eqnarray}' , '\\label{OrthIWilson}' , '& &\\frac{1}{2\\pi}\\int_0^{\\infty}' - , '\\end{eqnarray}']) + , '\\end{eqnarray}'],[]) self.assertEquals(fix_chapter_sort(['%' , '\\subsection*{9.1 Wilson}' , '\\paragraph{Orthogonality}' @@ -59,4 +59,4 @@ def test_fix_chapter_sort(self): , '\\begin{eqnarray}' , '\\label{OrthIWilson}' , '& &\\frac{1}{2\\pi}\\int_0^{\\infty}' - , '\\end{eqnarray}']) \ No newline at end of file + , '\\end{eqnarray}'],[]) \ No newline at end of file From 8d6f93cf3954123f2b05abeb9cd1479637e8d341 Mon Sep 17 00:00:00 2001 From: Edward Bian Date: Tue, 23 Aug 2016 15:11:52 -0400 Subject: [PATCH 329/402] Unit Tests --- KLSadd_insertion/src/updateChapters.py | 4 ++-- KLSadd_insertion/test/test_fix_chapter_sort.py | 4 ++-- 2 files changed, 4 insertions(+), 4 deletions(-) diff --git a/KLSadd_insertion/src/updateChapters.py b/KLSadd_insertion/src/updateChapters.py index 0127c03..b56e4af 100644 --- a/KLSadd_insertion/src/updateChapters.py +++ b/KLSadd_insertion/src/updateChapters.py @@ -9,7 +9,7 @@ # holds all subsections in each chapter section along with the what holds -w, h = 2, 500 +w, h = 2, 100 sorter_check = [[0 for _ in range(w)] for __ in range(h)] @@ -199,7 +199,7 @@ def fix_chapter_sort(kls, chap, word, sortloc, klsaddparas, sortmatch_2, tempref # print hyper_headers_chap # print word sortmatch_2.append(k_hyper_sub_chap) - return chap, sorter_check + return chap def cut_words(word_to_find, word_to_search_in): """ diff --git a/KLSadd_insertion/test/test_fix_chapter_sort.py b/KLSadd_insertion/test/test_fix_chapter_sort.py index 6dd705d..66e80bf 100644 --- a/KLSadd_insertion/test/test_fix_chapter_sort.py +++ b/KLSadd_insertion/test/test_fix_chapter_sort.py @@ -33,7 +33,7 @@ def test_fix_chapter_sort(self): , '\\begin{eqnarray}' , '\\label{OrthIWilson}' , '& &\\frac{1}{2\\pi}\\int_0^{\\infty}' - , '\\end{eqnarray}'],[]) + , '\\end{eqnarray}']) self.assertEquals(fix_chapter_sort(['%' , '\\subsection*{9.1 Wilson}' , '\\paragraph{Orthogonality}' @@ -59,4 +59,4 @@ def test_fix_chapter_sort(self): , '\\begin{eqnarray}' , '\\label{OrthIWilson}' , '& &\\frac{1}{2\\pi}\\int_0^{\\infty}' - , '\\end{eqnarray}'],[]) \ No newline at end of file + , '\\end{eqnarray}']) \ No newline at end of file From 55ec08a9ea5f6114b29ba65043c8e44bae362a9e Mon Sep 17 00:00:00 2001 From: Edward Bian Date: Tue, 23 Aug 2016 16:24:40 -0400 Subject: [PATCH 330/402] Unit Tests --- KLSadd_insertion/test/test_reference_finder.py | 18 +++++++++++++++++- 1 file changed, 17 insertions(+), 1 deletion(-) diff --git a/KLSadd_insertion/test/test_reference_finder.py b/KLSadd_insertion/test/test_reference_finder.py index 8426f44..b06028d 100644 --- a/KLSadd_insertion/test/test_reference_finder.py +++ b/KLSadd_insertion/test/test_reference_finder.py @@ -42,4 +42,20 @@ def test_find_references(self): , '& &{}=\hyp{4}{3}{-n,n+\alpha+\beta+1,-x,', '\end{eqnarray}' , 'with $N$ a nonnegative integer.', '\subsection*{References}', '\cite{Askey89I}, \cite{AskeyWilson79}, \cite{AskeyWilson85}'], - 1, ['9.2 Racah#']), ([], [], [0, 1])) \ No newline at end of file + 1, ['9.2 Racah#']), ([], [], [0, 1])) + self.assertEquals(find_references(['\\section{Unique}\\index{Racah polynomials}', + '\\subsection*{Bessel}', '\\begin{eqnarray}', + '\\label{DefRacah}', + '& &R_n(\lambda(x);\alpha,\beta,\gamma,\delta)\nonumber\\' + , '& &{}=\hyp{4}{3}{-n,n+\alpha+\beta+1,-x,', '\end{eqnarray}' + , 'with $N$ a nonnegative integer.', '\subsection*{References}', + '\cite{Askey89I}, \cite{AskeyWilson79}, \cite{AskeyWilson85}'], + 0, ['9.2 Racah#']), ([], [1, 1], [])) + self.assertEquals(find_references(['\\section{Unique}\\index{Racah polynomials}', + '\\subsection*{Big $q$-Legendre}', '\\begin{eqnarray}', + '\\label{DefRacah}', + '& &R_n(\lambda(x);\alpha,\beta,\gamma,\delta)\nonumber\\' + , '& &{}=\hyp{4}{3}{-n,n+\alpha+\beta+1,-x,', '\end{eqnarray}' + , 'with $N$ a nonnegative integer.', '\subsection*{References}', + '\cite{Askey89I}, \cite{AskeyWilson79}, \cite{AskeyWilson85}'], + 1, ['9.2 Racah#']), ([], [], [0, 1, 1])) \ No newline at end of file From 84aacdee106844fcd590d17926247097b7923ac0 Mon Sep 17 00:00:00 2001 From: Edward Bian Date: Wed, 24 Aug 2016 14:27:59 -0400 Subject: [PATCH 331/402] Unit Tests --- KLSadd_insertion/src/updateChapters.py | 32 +++++++++++-------- .../test/test_generic_functions.py | 2 +- 2 files changed, 20 insertions(+), 14 deletions(-) diff --git a/KLSadd_insertion/src/updateChapters.py b/KLSadd_insertion/src/updateChapters.py index b56e4af..ea703e2 100644 --- a/KLSadd_insertion/src/updateChapters.py +++ b/KLSadd_insertion/src/updateChapters.py @@ -9,8 +9,8 @@ # holds all subsections in each chapter section along with the what holds -w, h = 2, 100 -sorter_check = [[0 for _ in range(w)] for __ in range(h)] +# w, h = 2, 100 +# sorter_check = [[0 for _ in range(w)] for __ in range(h)] def chap1placer(chap1, kls, klsaddparas): @@ -72,10 +72,13 @@ def new_keywords(kls, kls_list): kls_list_chap.append(item) kls_list_chap[kls_list_chap.index("Special value")] = "Symmetry" kls_list_chap.append("Special value") - return kls_list_chap + w = 2 + h = len(kls_list_chap)+1 + sorter_check = [[0 for _ in range(w)] for __ in range(h)] + return kls_list_chap, sorter_check -def fix_chapter_sort(kls, chap, word, sortloc, klsaddparas, sortmatch_2, tempref): +def fix_chapter_sort(kls, chap, word, sortloc, klsaddparas, sortmatch_2, tempref, sorter_check): """ This function sorts through the input files and identifies and places sections from the addendum after their respective subsections in the chapter. @@ -133,9 +136,6 @@ def fix_chapter_sort(kls, chap, word, sortloc, klsaddparas, sortmatch_2, tempref if khyper_header_chap[item] == khyper_header_chap[item+1]: k_hyper_sub_chap[item + 1] = k_hyper_sub_chap[item] + "\paragraph{\\bf KLS Addendum: " + \ word + "}\n" + k_hyper_sub_chap[item + 1] - print khyper_header_chap[item] - print "ding" - print k_hyper_sub_chap[item + 1] khyper_header_chap[item] = "memes" k_hyper_sub_chap[item] = "memes" else: @@ -151,7 +151,7 @@ def fix_chapter_sort(kls, chap, word, sortloc, klsaddparas, sortmatch_2, tempref else: a += 1 - global sorter_check + #global sorter_check chap9 = 0 if sorter_check[sortloc][0] == 0: @@ -409,7 +409,7 @@ def reference_placer(chap, references, p, chapticker2): # method to change file string(actually a list right now), returns string to be written to file # If you write a method that changes something, it is preffered that you call the method in here -def fix_chapter(chap, references, paragraphs_to_be_added, kls, kls_list_all, chapticker2, new_commands, klsaddparas, sortmatch_2, tempref): +def fix_chapter(chap, references, paragraphs_to_be_added, kls, kls_list_all, chapticker2, new_commands, klsaddparas, sortmatch_2, tempref, sorter_check): """ Removes specific lines stopping the latex file from converting into python, as well as running the functions responsible for sorting sections and placing the correct additions in the correct places @@ -426,8 +426,9 @@ def fix_chapter(chap, references, paragraphs_to_be_added, kls, kls_list_all, cha sort_location = 0 for i in kls_list_all: - fix_chapter_sort(kls, chap, i, sort_location, klsaddparas, sortmatch_2, tempref) + fix_chapter_sort(kls, chap, i, sort_location, klsaddparas, sortmatch_2, tempref, sorter_check) sort_location += 1 + print sorter_check for a in range(len(paragraphs_to_be_added)-1): for b in sortmatch_2: @@ -500,8 +501,9 @@ def main(): with open(DATA_DIR + "KLSaddII.tex", "r") as add: # store the file as a string addendum = add.readlines() - kls_list_all = new_keywords(addendum, kls_list_full) + kls_list_all = new_keywords(addendum, kls_list_full)[0] # Makes sections look like other sections + sorter_check = new_keywords(addendum, kls_list_full)[1] for item in addendum: addendum[addendum.index(item)] = item.replace("\\eqref{","\\eqref{KLSadd: ") @@ -590,10 +592,14 @@ def main(): print ref14_3 # call the fixChapter method to get a list with the addendum paragraphs added in chapticker2 = 0 - str9 = ''.join(fix_chapter(entire9, references9, paras, addendum, kls_list_all, chapticker2, new_commands, klsaddparas, sortmatch_2, ref9_3)) + str9 = ''.join(fix_chapter(entire9, references9, paras, addendum, kls_list_all, chapticker2, new_commands, klsaddparas, sortmatch_2, ref9_3, sorter_check)) + print sorter_check + print "DING" chapticker2 += 1 - str14 = ''.join(fix_chapter(entire14, references14, paras, addendum, kls_list_all, chapticker2, new_commands, klsaddparas, sortmatch_2, ref14_3)) + str14 = ''.join(fix_chapter(entire14, references14, paras, addendum, kls_list_all, chapticker2, new_commands, klsaddparas, sortmatch_2, ref14_3, sorter_check)) + + print "DING" # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ # If you are writing something that will make a change to the chapter files, write it BEFORE this line, this part diff --git a/KLSadd_insertion/test/test_generic_functions.py b/KLSadd_insertion/test/test_generic_functions.py index 7aa85f9..5ff7c81 100644 --- a/KLSadd_insertion/test/test_generic_functions.py +++ b/KLSadd_insertion/test/test_generic_functions.py @@ -11,7 +11,7 @@ class TestNewKeywords(TestCase): def test_new_keywords(self): self.assertEquals(new_keywords(['\\subsubsection*{ThisIsATest}\n', '\\subsubsection*{Special value}\n'] - ,[]), ['ThisIsATest', 'Symmetry', 'Limit Relation', 'Special value']) + ,[]), (['ThisIsATest', 'Symmetry', 'Limit Relation', 'Special value'],[[0,0],[0,0],[0,0],[0,0], [0,0]])) class TestExtraneousSectionDeleter(TestCase): From 7742d7e37b07b464b9e9b18535f04a7e007301b9 Mon Sep 17 00:00:00 2001 From: Edward Bian Date: Wed, 24 Aug 2016 14:32:12 -0400 Subject: [PATCH 332/402] Testing Something --- .../test/test_fix_chapter_sort.py | 124 +++++++++--------- 1 file changed, 62 insertions(+), 62 deletions(-) diff --git a/KLSadd_insertion/test/test_fix_chapter_sort.py b/KLSadd_insertion/test/test_fix_chapter_sort.py index 66e80bf..af24eb0 100644 --- a/KLSadd_insertion/test/test_fix_chapter_sort.py +++ b/KLSadd_insertion/test/test_fix_chapter_sort.py @@ -1,62 +1,62 @@ - -__author__ = 'Edward Bian' -__status__ = 'Development' - -from unittest import TestCase -from updateChapters import fix_chapter_sort - - -class TestFixChapterSort(TestCase): - def test_fix_chapter_sort(self): - self.assertEquals(fix_chapter_sort(['%' - , '\\subsection*{9.1 Wilson}' - , '\\paragraph{Hypergeometric Representation}' - , 'The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetric' - , 'in $a,b,c,d$.' - , '\\renewcommand{\\refname}{Standard references}'] - ,['\\section{Wilson}\\index{Wilson polynomials}' - ,'\\subsection*{Hypergeometric representation}' - ,'\\begin{eqnarray}' - ,'& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' - ,'\\end{eqnarray}' - ,'\\subsection*{Orthogonality relation}' - ,'\\begin{eqnarray}' - ,'\\label{OrthIWilson}' - ,'& &\\frac{1}{2\\pi}\\int_0^{\\infty}' - ,'\\end{eqnarray}'],'Hypergeometric Representation',0,[1,2,5],[],[0, 1, 5, 9]) - ,['\\section{Wilson}\\index{Wilson polynomials}' - , '\\subsection*{Hypergeometric representation}' - , '\\begin{eqnarray}' - , '& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' - , '\\end{eqnarray}\\paragraph{\\bf KLS Addendum: Hypergeometric Representation}The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetricin $a,b,c,d$.' - , '\\subsection*{Orthogonality relation}' - , '\\begin{eqnarray}' - , '\\label{OrthIWilson}' - , '& &\\frac{1}{2\\pi}\\int_0^{\\infty}' - , '\\end{eqnarray}']) - self.assertEquals(fix_chapter_sort(['%' - , '\\subsection*{9.1 Wilson}' - , '\\paragraph{Orthogonality}' - , 'The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetric' - , 'in $a,b,c,d$.' - , '\\renewcommand{\\refname}{Standard references}'] - , ['\\section{Wilson}\\index{Wilson polynomials}' - , '\\subsection*{Hypergeometric representation}' - , '\\begin{eqnarray}' - , '& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' - , '\\end{eqnarray}' - , '\\subsection*{Orthogonality}' - , '\\begin{eqnarray}' - , '\\label{OrthIWilson}' - , '& &\\frac{1}{2\\pi}\\int_0^{\\infty}' - , '\\end{eqnarray}'], 'Memes', 0, [1, 2, 5], [], [0, 1, 5, 9]) - , ['\\section{Wilson}\\index{Wilson polynomials}' - , '\\subsection*{Hypergeometric representation}' - , '\\begin{eqnarray}' - , '& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' - , '\\end{eqnarray}' - , '\\subsection*{Orthogonality}' - , '\\begin{eqnarray}' - , '\\label{OrthIWilson}' - , '& &\\frac{1}{2\\pi}\\int_0^{\\infty}' - , '\\end{eqnarray}']) \ No newline at end of file +# +# __author__ = 'Edward Bian' +# __status__ = 'Development' +# +# from unittest import TestCase +# from updateChapters import fix_chapter_sort +# +# +# class TestFixChapterSort(TestCase): +# def test_fix_chapter_sort(self): +# self.assertEquals(fix_chapter_sort(['%' +# , '\\subsection*{9.1 Wilson}' +# , '\\paragraph{Hypergeometric Representation}' +# , 'The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetric' +# , 'in $a,b,c,d$.' +# , '\\renewcommand{\\refname}{Standard references}'] +# ,['\\section{Wilson}\\index{Wilson polynomials}' +# ,'\\subsection*{Hypergeometric representation}' +# ,'\\begin{eqnarray}' +# ,'& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' +# ,'\\end{eqnarray}' +# ,'\\subsection*{Orthogonality relation}' +# ,'\\begin{eqnarray}' +# ,'\\label{OrthIWilson}' +# ,'& &\\frac{1}{2\\pi}\\int_0^{\\infty}' +# ,'\\end{eqnarray}'],'Hypergeometric Representation',0,[1,2,5],[],[0, 1, 5, 9]) +# ,['\\section{Wilson}\\index{Wilson polynomials}' +# , '\\subsection*{Hypergeometric representation}' +# , '\\begin{eqnarray}' +# , '& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' +# , '\\end{eqnarray}\\paragraph{\\bf KLS Addendum: Hypergeometric Representation}The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetricin $a,b,c,d$.' +# , '\\subsection*{Orthogonality relation}' +# , '\\begin{eqnarray}' +# , '\\label{OrthIWilson}' +# , '& &\\frac{1}{2\\pi}\\int_0^{\\infty}' +# , '\\end{eqnarray}']) +# self.assertEquals(fix_chapter_sort(['%' +# , '\\subsection*{9.1 Wilson}' +# , '\\paragraph{Orthogonality}' +# , 'The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetric' +# , 'in $a,b,c,d$.' +# , '\\renewcommand{\\refname}{Standard references}'] +# , ['\\section{Wilson}\\index{Wilson polynomials}' +# , '\\subsection*{Hypergeometric representation}' +# , '\\begin{eqnarray}' +# , '& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' +# , '\\end{eqnarray}' +# , '\\subsection*{Orthogonality}' +# , '\\begin{eqnarray}' +# , '\\label{OrthIWilson}' +# , '& &\\frac{1}{2\\pi}\\int_0^{\\infty}' +# , '\\end{eqnarray}'], 'Memes', 0, [1, 2, 5], [], [0, 1, 5, 9]) +# , ['\\section{Wilson}\\index{Wilson polynomials}' +# , '\\subsection*{Hypergeometric representation}' +# , '\\begin{eqnarray}' +# , '& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' +# , '\\end{eqnarray}' +# , '\\subsection*{Orthogonality}' +# , '\\begin{eqnarray}' +# , '\\label{OrthIWilson}' +# , '& &\\frac{1}{2\\pi}\\int_0^{\\infty}' +# , '\\end{eqnarray}']) \ No newline at end of file From 30e9e2d912c3f9713148067129bde88074ed56da Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Wed, 24 Aug 2016 14:44:54 -0400 Subject: [PATCH 333/402] Add code to add parentheses around ambiguous funcions (trig) --- eCF/src/latex_constraint_modifier.py | 1 - eCF/src/mathematica_to_latex.py | 17 +++++++++++------ eCF/test/test_master_function.py | 13 +++++++++---- 3 files changed, 20 insertions(+), 11 deletions(-) diff --git a/eCF/src/latex_constraint_modifier.py b/eCF/src/latex_constraint_modifier.py index e843981..1a329e0 100644 --- a/eCF/src/latex_constraint_modifier.py +++ b/eCF/src/latex_constraint_modifier.py @@ -77,7 +77,6 @@ def combine_percent(lines): if len(lines[index]) >= 3 and lines[index][:3] == '% ' and \ index > lines.index('\\begin{equation}'): if len(lines[index + 1]) != '' and lines[index + 1][0] == '%': - print(lines[index]) add = lines.pop(index + 1) lines[index] += '\n' + add.replace('%', ' ') else: diff --git a/eCF/src/mathematica_to_latex.py b/eCF/src/mathematica_to_latex.py index b879e89..9d838c5 100644 --- a/eCF/src/mathematica_to_latex.py +++ b/eCF/src/mathematica_to_latex.py @@ -238,9 +238,14 @@ def master_function(line, params): sum([args[0].count(element) for element in multi]) != 0: sep[0][0] = sep[0][0].replace('@@', '@') - line = (line[:pos[0]] + l + '%s'.join(sep[[len(y) for y in sep] - .index(len(args) + 1)]) + - line[pos[1]:]) + if m in TRIG and len(line) != pos[1] and line[pos[1]] == '^': + line = line[:pos[0]] + '(' + l + \ + '%s'.join(sep[[len(y) for y in sep] + .index(len(args) + 1)]) + ')' + line[pos[1]:] + else: + line = line[:pos[0]] + l + \ + '%s'.join(sep[[len(y) for y in sep]. + index(len(args) + 1)]) + line[pos[1]:] line %= tuple(args) @@ -311,8 +316,8 @@ def carat(line): while i != len(line): if line[i] == '^': + print(line) k = search(line, i, list('*/+-=, '), 1) - if line[i + 1] == '(' and line[k] == ')': line = line[:i] + '^{' + line[i + 2:k] + '}' + line[k + 1:] else: @@ -329,7 +334,7 @@ def beta(line): Converts Mathematica's 'Beta' function to the equivalent LaTeX macro, taking into account the variations for the different number of arguments. - :param line: line to be converted + :param line: line to be convertedmathematica_to_latex.py:324 :returns: converted line """ for _ in range(line.count('Beta')): @@ -966,7 +971,7 @@ def main(pathw=DIR_NAME + 'newIdentities.tex', pass line += '\n\\end{equation}' - print(line) + # print(line) latex.write(line + '\n') diff --git a/eCF/test/test_master_function.py b/eCF/test/test_master_function.py index b089b08..56f20f8 100644 --- a/eCF/test/test_master_function.py +++ b/eCF/test/test_master_function.py @@ -75,10 +75,15 @@ def test_conversion(self): self.assertEqual(master_function('--{0}--'.format(before), function), '--{0}--'.format(after)) if function[0] in TRIG: for sep in MULTI: - before = before[:-1] + sep + 'b]' - after = after.replace('@@', '@')[:-1] + sep + 'b}' - self.assertEqual(master_function(before, function), after) - self.assertEqual(master_function('--{0}--'.format(before), function), '--{0}--'.format(after)) + before2 = before[:-1] + sep + 'b]' + after2 = after.replace('@@', '@')[:-1] + sep + 'b}' + self.assertEqual(master_function(before2, function), after2) + self.assertEqual(master_function('--{0}--'.format(before2), function), '--{0}--'.format(after2)) + + before2 += '^' + after2 = '(' + after2 + ')^' + self.assertEqual(master_function(before2, function), after2) + self.assertEqual(master_function('--{0}--'.format(before2), function), '--{0}--'.format(after2)) if function[0] == 'D': self.assertEqual(master_function('\\[Delta]', function), '\\[Delta]') From 76fec70e5db70f7eb353a66fcf9b8fae59f69d03 Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Wed, 24 Aug 2016 15:02:31 -0400 Subject: [PATCH 334/402] Add comments --- eCF/src/mathematica_to_latex.py | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/eCF/src/mathematica_to_latex.py b/eCF/src/mathematica_to_latex.py index 9d838c5..c87392c 100644 --- a/eCF/src/mathematica_to_latex.py +++ b/eCF/src/mathematica_to_latex.py @@ -238,6 +238,7 @@ def master_function(line, params): sum([args[0].count(element) for element in multi]) != 0: sep[0][0] = sep[0][0].replace('@@', '@') + # Add parens around ambiguous functions (trig functions) if m in TRIG and len(line) != pos[1] and line[pos[1]] == '^': line = line[:pos[0]] + '(' + l + \ '%s'.join(sep[[len(y) for y in sep] @@ -246,7 +247,6 @@ def master_function(line, params): line = line[:pos[0]] + l + \ '%s'.join(sep[[len(y) for y in sep]. index(len(args) + 1)]) + line[pos[1]:] - line %= tuple(args) return line From 7a0ddfabc57f1f9401eaf58ab4b2e59e57950096 Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Wed, 24 Aug 2016 15:25:21 -0400 Subject: [PATCH 335/402] Move latex_constraint_modifier.py to DLMF_preprocessing/src, added __init__.py files to eCF/src and eCF/test --- .../src/latex_constraint_modifier.py | 37 +++++++++---------- eCF/src/__init__.py | 0 eCF/test/__init__.py | 0 3 files changed, 18 insertions(+), 19 deletions(-) rename {eCF => DLMF_preprocessing}/src/latex_constraint_modifier.py (87%) create mode 100644 eCF/src/__init__.py create mode 100644 eCF/test/__init__.py diff --git a/eCF/src/latex_constraint_modifier.py b/DLMF_preprocessing/src/latex_constraint_modifier.py similarity index 87% rename from eCF/src/latex_constraint_modifier.py rename to DLMF_preprocessing/src/latex_constraint_modifier.py index 1a329e0..fb21194 100644 --- a/eCF/src/latex_constraint_modifier.py +++ b/DLMF_preprocessing/src/latex_constraint_modifier.py @@ -4,20 +4,6 @@ import argparse -parser = argparse.ArgumentParser( - description='Receive .tex file with constraints and convert' - ' them viewable metadata.') -parser.add_argument('PATHR', type=str, - help='path of input .tex file, with the current' - ' directory as the starting point',) -parser.add_argument('PATHW', type=str, - help='path of file to be outputted to, with the current' - ' directory as the starting point') - -args = parser.parse_args() -PATHR = args.PATHR -PATHW = args.PATHW - CONVERSIONS = {'\\constraint': '{\\bf C}:~', '\\substitution': '{\\bf S}:~', '\\drmfnote': '{\\bf NOTE}:~', @@ -151,19 +137,21 @@ def dollarsign(line): return line -def main(): - # (None) -> None +def main(pathr, pathw): + # ((str, str)) -> None """ Main function that reads data and calls other functions to process data. + :param pathr: + :param pathw: :return: None """ - with open(PATHR, 'r') as b: + with open(pathr, 'r') as b: before = b.read().split('\n') before = combine_percent(before) - with open(PATHW, 'w') as after: + with open(pathw, 'w') as after: for line in before: if len(line) >= 3 and line[:3] == '% ' and \ before.index(line) > \ @@ -177,4 +165,15 @@ def main(): if __name__ == '__main__': - main() + parser = argparse.ArgumentParser( + description='Receive .tex file with constraints and convert' + ' them viewable metadata.') + parser.add_argument('PATHR', type=str, + help='path of input .tex file, with the current' + ' directory as the starting point', ) + parser.add_argument('PATHW', type=str, + help='path of file to be outputted to, with the current' + ' directory as the starting point') + args = parser.parse_args() + + main(args.PATHR, args.PATHW) diff --git a/eCF/src/__init__.py b/eCF/src/__init__.py new file mode 100644 index 0000000..e69de29 diff --git a/eCF/test/__init__.py b/eCF/test/__init__.py new file mode 100644 index 0000000..e69de29 From 8ca407a1489b3aa300ece845c9387732521b24f7 Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Wed, 24 Aug 2016 15:31:23 -0400 Subject: [PATCH 336/402] Start test cases for latex_constraint_modifier.py, add DLMF_preprocessing/test/data directory, modify documentatoin in latex_constraint_modifier.py --- .../src/latex_constraint_modifier.py | 6 +++--- DLMF_preprocessing/test/data/in.tex | 0 DLMF_preprocessing/test/data/out.tex | 0 .../test/test_latex_constraint_modifier.py | 17 +++++++++++++++++ 4 files changed, 20 insertions(+), 3 deletions(-) create mode 100644 DLMF_preprocessing/test/data/in.tex create mode 100644 DLMF_preprocessing/test/data/out.tex create mode 100644 DLMF_preprocessing/test/test_latex_constraint_modifier.py diff --git a/DLMF_preprocessing/src/latex_constraint_modifier.py b/DLMF_preprocessing/src/latex_constraint_modifier.py index fb21194..72a0a90 100644 --- a/DLMF_preprocessing/src/latex_constraint_modifier.py +++ b/DLMF_preprocessing/src/latex_constraint_modifier.py @@ -138,12 +138,12 @@ def dollarsign(line): def main(pathr, pathw): - # ((str, str)) -> None + # (str, str) -> None """ Main function that reads data and calls other functions to process data. - :param pathr: - :param pathw: + :param pathr: directory of file to be read from + :param pathw: directory of file to be written to :return: None """ with open(pathr, 'r') as b: diff --git a/DLMF_preprocessing/test/data/in.tex b/DLMF_preprocessing/test/data/in.tex new file mode 100644 index 0000000..e69de29 diff --git a/DLMF_preprocessing/test/data/out.tex b/DLMF_preprocessing/test/data/out.tex new file mode 100644 index 0000000..e69de29 diff --git a/DLMF_preprocessing/test/test_latex_constraint_modifier.py b/DLMF_preprocessing/test/test_latex_constraint_modifier.py new file mode 100644 index 0000000..3dfc003 --- /dev/null +++ b/DLMF_preprocessing/test/test_latex_constraint_modifier.py @@ -0,0 +1,17 @@ + +__author__ = 'Kevin Chen' +__status__ = 'Development' + +from unittest import TestCase +from latex_constraint_modifier import * + + +class TestCombinePercent(TestCase): + + +class TestReplace(TestCase): + + +class TestDollarsign(TestCase): + + From 9bc8df2797557bc20ded16353c1248f2c8fad6fa Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Wed, 24 Aug 2016 15:33:39 -0400 Subject: [PATCH 337/402] Remove a few useless lines of code --- DLMF_preprocessing/src/latex_constraint_modifier.py | 5 +---- DLMF_preprocessing/test/test_latex_constraint_modifier.py | 1 + 2 files changed, 2 insertions(+), 4 deletions(-) diff --git a/DLMF_preprocessing/src/latex_constraint_modifier.py b/DLMF_preprocessing/src/latex_constraint_modifier.py index 72a0a90..08ddcd7 100644 --- a/DLMF_preprocessing/src/latex_constraint_modifier.py +++ b/DLMF_preprocessing/src/latex_constraint_modifier.py @@ -40,10 +40,7 @@ def find_surrounding(line, function): if line[j] in list(')]}'): count -= 1 if count == 0: - if j == positions[0] + len(function): - positions[0] = positions[1] - else: - positions[1] = j + 1 + positions[1] = j + 1 break return positions[0], positions[1] diff --git a/DLMF_preprocessing/test/test_latex_constraint_modifier.py b/DLMF_preprocessing/test/test_latex_constraint_modifier.py index 3dfc003..ede2571 100644 --- a/DLMF_preprocessing/test/test_latex_constraint_modifier.py +++ b/DLMF_preprocessing/test/test_latex_constraint_modifier.py @@ -15,3 +15,4 @@ class TestReplace(TestCase): class TestDollarsign(TestCase): +class TestMain(TestCase): \ No newline at end of file From 38979015a88336d18074b6511018e66eb734f410 Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Wed, 24 Aug 2016 15:36:05 -0400 Subject: [PATCH 338/402] Add spacing --- .../test/test_latex_constraint_modifier.py | 16 ++++++++++++---- eCF/test/test_main.py | 2 +- 2 files changed, 13 insertions(+), 5 deletions(-) diff --git a/DLMF_preprocessing/test/test_latex_constraint_modifier.py b/DLMF_preprocessing/test/test_latex_constraint_modifier.py index ede2571..f901bf3 100644 --- a/DLMF_preprocessing/test/test_latex_constraint_modifier.py +++ b/DLMF_preprocessing/test/test_latex_constraint_modifier.py @@ -5,14 +5,22 @@ from unittest import TestCase from latex_constraint_modifier import * +import os +PATHR = os.path.dirname(os.path.realpath(__file__)) + '/data/in.tex' +PATHW = os.path.dirname(os.path.realpath(__file__)) + '/data/out.tex' -class TestCombinePercent(TestCase): +#class TestFindSurrounding(TestCase): -class TestReplace(TestCase): +#class TestCombinePercent(TestCase): -class TestDollarsign(TestCase): +#class TestReplace(TestCase): + + +#class TestDollarsign(TestCase): + + +class TestMain(TestCase): -class TestMain(TestCase): \ No newline at end of file diff --git a/eCF/test/test_main.py b/eCF/test/test_main.py index fde72f5..9ec83a4 100644 --- a/eCF/test/test_main.py +++ b/eCF/test/test_main.py @@ -48,4 +48,4 @@ def test_gen(self): '\\end{equation}\n' '\n' '\n' - '\\end{document}\n')) \ No newline at end of file + '\\end{document}\n')) From 5cfadac6fb7f202a9cc92286b02ef2919f798888 Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Wed, 24 Aug 2016 16:09:05 -0400 Subject: [PATCH 339/402] Add unittests to latex_constraint_modifier.py --- DLMF_preprocessing/test/data/correctout.tex | 84 +++++++++++++++++++ DLMF_preprocessing/test/data/in.tex | 70 ++++++++++++++++ DLMF_preprocessing/test/data/out.tex | 0 DLMF_preprocessing/test/data/testout.tex | 84 +++++++++++++++++++ .../test/test_latex_constraint_modifier.py | 23 +++-- eCF/src/mathematica_to_latex.py | 5 +- eCF/test/data/test.tex | 4 +- eCF/test/test_main.py | 6 +- 8 files changed, 255 insertions(+), 21 deletions(-) create mode 100644 DLMF_preprocessing/test/data/correctout.tex delete mode 100644 DLMF_preprocessing/test/data/out.tex create mode 100644 DLMF_preprocessing/test/data/testout.tex diff --git a/DLMF_preprocessing/test/data/correctout.tex b/DLMF_preprocessing/test/data/correctout.tex new file mode 100644 index 0000000..8614bab --- /dev/null +++ b/DLMF_preprocessing/test/data/correctout.tex @@ -0,0 +1,84 @@ + +\documentclass{article} + +\usepackage{amsmath} +\usepackage{amsthm} +\usepackage{amssymb} +\usepackage{amsfonts} +\usepackage{breqn} +\usepackage{DLMFmath} +\usepackage{DRMFfcns} +\usepackage{DLMFfcns} +\usepackage{graphicx} +\usepackage[paperwidth=15in, paperheight=20in, margin=0.5in]{geometry} + +\begin{document} + + +% \constraint{ +% \substitution{ +% \drmfnote{ +% \drmfname{ +% \proof{ +% \mathematicatag{ +% \mathematicareference{ +% \mapletag +% \category + + +\begin{equation} + tests for each metadata (single) +\end{equation} +\begin{flushright} + {\bf C}:~text \\[0.2cm] + {\bf S}:~text \\[0.2cm] + {\bf NOTE}:~text \\[0.2cm] + {\bf NAME}:~text \\[0.2cm] + {\bf PROOF}:~text \\[0.2cm] + {\bf MtT}:~text \\[0.2cm] + {\bf MtR}:~text \\[0.2cm] + {\bf MpT}:~text \\[0.2cm] + {\bf CAT}:~text +\end{flushright} + +\begin{equation} + tests for each metadata (multiple) +\end{equation} +\begin{flushright} + {\bf C}:~a \\[0.2cm] + {\bf C}:~ b \\[0.2cm] + {\bf S}:~a \\[0.2cm] + {\bf S}:~ b \\[0.2cm] + {\bf NOTE}:~a \\[0.2cm] + {\bf NOTE}:~ b \\[0.2cm] + {\bf NAME}:~a \\[0.2cm] + {\bf NAME}:~ b \\[0.2cm] + {\bf PROOF}:~a \\[0.2cm] + {\bf PROOF}:~ b \\[0.2cm] + {\bf MtT}:~${\displaystyle \tt{a }}$ \\[0.2cm] + {\bf MtT}:~${\displaystyle \tt{ b}}$ \\[0.2cm] + {\bf MtR}:~${\displaystyle \text{a }}$ \\[0.2cm] + {\bf MtR}:~${\displaystyle \text{ b}}$ \\[0.2cm] + {\bf MpT}:~${\displaystyle \tt{a }}$ \\[0.2cm] + {\bf MpT}:~${\displaystyle \tt{ b}}$ \\[0.2cm] + {\bf CAT}:~${\displaystyle \text{a }}$ \\[0.2cm] + {\bf CAT}:~${\displaystyle \text{ b}}$ +\end{flushright} + +\begin{equation} + tests for percent combination +\end{equation} +\begin{flushright} + {\bf C}:~ a \\[0.2cm] + {\bf NOTE}:~this is a multiline note. +\end{flushright} + +\begin{equation} + tests for dollarsign conversion +\end{equation} +\begin{flushright} + {\bf C}:~${\displaystyle a}$ ${\displaystyle alreadyconverted}$ +\end{flushright} + + +\end{document} diff --git a/DLMF_preprocessing/test/data/in.tex b/DLMF_preprocessing/test/data/in.tex index e69de29..3a40265 100644 --- a/DLMF_preprocessing/test/data/in.tex +++ b/DLMF_preprocessing/test/data/in.tex @@ -0,0 +1,70 @@ + +\documentclass{article} + +\usepackage{amsmath} +\usepackage{amsthm} +\usepackage{amssymb} +\usepackage{amsfonts} +\usepackage{breqn} +\usepackage{DLMFmath} +\usepackage{DRMFfcns} +\usepackage{DLMFfcns} +\usepackage{graphicx} +\usepackage[paperwidth=15in, paperheight=20in, margin=0.5in]{geometry} + +\begin{document} + + +% \constraint{ +% \substitution{ +% \drmfnote{ +% \drmfname{ +% \proof{ +% \mathematicatag{ +% \mathematicareference{ +% \mapletag +% \category + + +\begin{equation} + tests for each metadata (single) +% \constraint{text} +% \substitution{text} +% \drmfnote{text} +% \drmfname{text} +% \proof{text} +% \mathematicatag{text} +% \mathematicareference{text} +% \mapletag{text} +% \category{text} +\end{equation} + +\begin{equation} + tests for each metadata (multiple) +% \constraint{a & b} +% \substitution{a & b} +% \drmfnote{a & b} +% \drmfname{a & b} +% \proof{a & b} +% \mathematicatag{$\tt{a & b}$} +% \mathematicareference{$\text{a & b}$} +% \mapletag{$\tt{a & b}$} +% \category{$\text{a & b}$} +\end{equation} + +\begin{equation} + tests for percent combination +% \constraint{ +% a} +% \drmfnote{this +% is a +% multiline note.} +\end{equation} + +\begin{equation} + tests for dollarsign conversion +% \constraint{$a$ ${\displaystyle alreadyconverted}$} +\end{equation} + + +\end{document} \ No newline at end of file diff --git a/DLMF_preprocessing/test/data/out.tex b/DLMF_preprocessing/test/data/out.tex deleted file mode 100644 index e69de29..0000000 diff --git a/DLMF_preprocessing/test/data/testout.tex b/DLMF_preprocessing/test/data/testout.tex new file mode 100644 index 0000000..8614bab --- /dev/null +++ b/DLMF_preprocessing/test/data/testout.tex @@ -0,0 +1,84 @@ + +\documentclass{article} + +\usepackage{amsmath} +\usepackage{amsthm} +\usepackage{amssymb} +\usepackage{amsfonts} +\usepackage{breqn} +\usepackage{DLMFmath} +\usepackage{DRMFfcns} +\usepackage{DLMFfcns} +\usepackage{graphicx} +\usepackage[paperwidth=15in, paperheight=20in, margin=0.5in]{geometry} + +\begin{document} + + +% \constraint{ +% \substitution{ +% \drmfnote{ +% \drmfname{ +% \proof{ +% \mathematicatag{ +% \mathematicareference{ +% \mapletag +% \category + + +\begin{equation} + tests for each metadata (single) +\end{equation} +\begin{flushright} + {\bf C}:~text \\[0.2cm] + {\bf S}:~text \\[0.2cm] + {\bf NOTE}:~text \\[0.2cm] + {\bf NAME}:~text \\[0.2cm] + {\bf PROOF}:~text \\[0.2cm] + {\bf MtT}:~text \\[0.2cm] + {\bf MtR}:~text \\[0.2cm] + {\bf MpT}:~text \\[0.2cm] + {\bf CAT}:~text +\end{flushright} + +\begin{equation} + tests for each metadata (multiple) +\end{equation} +\begin{flushright} + {\bf C}:~a \\[0.2cm] + {\bf C}:~ b \\[0.2cm] + {\bf S}:~a \\[0.2cm] + {\bf S}:~ b \\[0.2cm] + {\bf NOTE}:~a \\[0.2cm] + {\bf NOTE}:~ b \\[0.2cm] + {\bf NAME}:~a \\[0.2cm] + {\bf NAME}:~ b \\[0.2cm] + {\bf PROOF}:~a \\[0.2cm] + {\bf PROOF}:~ b \\[0.2cm] + {\bf MtT}:~${\displaystyle \tt{a }}$ \\[0.2cm] + {\bf MtT}:~${\displaystyle \tt{ b}}$ \\[0.2cm] + {\bf MtR}:~${\displaystyle \text{a }}$ \\[0.2cm] + {\bf MtR}:~${\displaystyle \text{ b}}$ \\[0.2cm] + {\bf MpT}:~${\displaystyle \tt{a }}$ \\[0.2cm] + {\bf MpT}:~${\displaystyle \tt{ b}}$ \\[0.2cm] + {\bf CAT}:~${\displaystyle \text{a }}$ \\[0.2cm] + {\bf CAT}:~${\displaystyle \text{ b}}$ +\end{flushright} + +\begin{equation} + tests for percent combination +\end{equation} +\begin{flushright} + {\bf C}:~ a \\[0.2cm] + {\bf NOTE}:~this is a multiline note. +\end{flushright} + +\begin{equation} + tests for dollarsign conversion +\end{equation} +\begin{flushright} + {\bf C}:~${\displaystyle a}$ ${\displaystyle alreadyconverted}$ +\end{flushright} + + +\end{document} diff --git a/DLMF_preprocessing/test/test_latex_constraint_modifier.py b/DLMF_preprocessing/test/test_latex_constraint_modifier.py index f901bf3..ede45e8 100644 --- a/DLMF_preprocessing/test/test_latex_constraint_modifier.py +++ b/DLMF_preprocessing/test/test_latex_constraint_modifier.py @@ -7,20 +7,17 @@ import os PATHR = os.path.dirname(os.path.realpath(__file__)) + '/data/in.tex' -PATHW = os.path.dirname(os.path.realpath(__file__)) + '/data/out.tex' - - -#class TestFindSurrounding(TestCase): - - -#class TestCombinePercent(TestCase): - - -#class TestReplace(TestCase): - - -#class TestDollarsign(TestCase): +PATHW = os.path.dirname(os.path.realpath(__file__)) + '/data/testout.tex' +PATHC = os.path.dirname(os.path.realpath(__file__)) + '/data/correctout.tex' class TestMain(TestCase): + def test_generation(self): + main(PATHR, PATHW) + with open(PATHW, 'r') as r: + out = r.read() + with open(PATHC, 'r') as c: + correct = c.read() + + self.assertEqual(out, correct) diff --git a/eCF/src/mathematica_to_latex.py b/eCF/src/mathematica_to_latex.py index c87392c..154be73 100644 --- a/eCF/src/mathematica_to_latex.py +++ b/eCF/src/mathematica_to_latex.py @@ -969,10 +969,9 @@ def main(pathw=DIR_NAME + 'newIdentities.tex', references[mtt] + '}$}' except KeyError: pass - line += '\n\\end{equation}' - - # print(line) + line = ' ' + line + '\n\\end{equation}' + print line latex.write(line + '\n') latex.write('\n\n\\end{document}\n') diff --git a/eCF/test/data/test.tex b/eCF/test/data/test.tex index fbcacdb..7ff4f9a 100644 --- a/eCF/test/data/test.tex +++ b/eCF/test/data/test.tex @@ -16,13 +16,13 @@ \begin{equation} -equation + equation % \mathematicatag{$\tt{description, number}$} % \mathematicareference{$\text{test reference line 1&test reference line 2}$} \end{equation} \begin{equation} -equation + equation % \mathematicatag{$\tt{nodescription, number}$} \end{equation} diff --git a/eCF/test/test_main.py b/eCF/test/test_main.py index 9ec83a4..4ccaa9b 100644 --- a/eCF/test/test_main.py +++ b/eCF/test/test_main.py @@ -14,7 +14,7 @@ class TestMain(TestCase): - def test_gen(self): + def test_generation(self): main(pathw=PATHW, pathr=PATHR, pathref=PATHREF) with open(PATHW, 'r') as l: latex = l.read() @@ -37,13 +37,13 @@ def test_gen(self): '\n' '\n' '\\begin{equation}\n' - 'equation\n' + ' equation\n' '% \\mathematicatag{$\\tt{description, number}$}\n' '% \\mathematicareference{$\\text{test reference line 1&test reference line 2}$}\n' '\\end{equation}\n' '\n' '\\begin{equation}\n' - 'equation\n' + ' equation\n' '% \\mathematicatag{$\\tt{nodescription, number}$}\n' '\\end{equation}\n' '\n' From a8ef285412a7015aa7bf9082ab316814f4d6fd6d Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Wed, 24 Aug 2016 16:11:13 -0400 Subject: [PATCH 340/402] Remove import * --- DLMF_preprocessing/test/test_latex_constraint_modifier.py | 3 ++- 1 file changed, 2 insertions(+), 1 deletion(-) diff --git a/DLMF_preprocessing/test/test_latex_constraint_modifier.py b/DLMF_preprocessing/test/test_latex_constraint_modifier.py index ede45e8..77ea343 100644 --- a/DLMF_preprocessing/test/test_latex_constraint_modifier.py +++ b/DLMF_preprocessing/test/test_latex_constraint_modifier.py @@ -3,9 +3,10 @@ __status__ = 'Development' from unittest import TestCase -from latex_constraint_modifier import * +from latex_constraint_modifier import main import os + PATHR = os.path.dirname(os.path.realpath(__file__)) + '/data/in.tex' PATHW = os.path.dirname(os.path.realpath(__file__)) + '/data/testout.tex' PATHC = os.path.dirname(os.path.realpath(__file__)) + '/data/correctout.tex' From 3513c44b17d4e4bc49f31883633cb79e59db4970 Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Wed, 24 Aug 2016 16:23:34 -0400 Subject: [PATCH 341/402] Fix minor issues --- DLMF_preprocessing/src/latex_constraint_modifier.py | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/DLMF_preprocessing/src/latex_constraint_modifier.py b/DLMF_preprocessing/src/latex_constraint_modifier.py index 08ddcd7..56721f4 100644 --- a/DLMF_preprocessing/src/latex_constraint_modifier.py +++ b/DLMF_preprocessing/src/latex_constraint_modifier.py @@ -18,7 +18,7 @@ def find_surrounding(line, function): - # (str, str) -> tuple + # (str, str) -> list """ Finds the indices of the beginning and end of a function; this is the main function that powers the converter. @@ -43,7 +43,7 @@ def find_surrounding(line, function): positions[1] = j + 1 break - return positions[0], positions[1] + return positions def combine_percent(lines): @@ -169,8 +169,8 @@ def main(pathr, pathw): help='path of input .tex file, with the current' ' directory as the starting point', ) parser.add_argument('PATHW', type=str, - help='path of file to be outputted to, with the current' - ' directory as the starting point') + help='path of file to be outputted to, with the' + ' current directory as the starting point') args = parser.parse_args() main(args.PATHR, args.PATHW) From 26fcd49de3df72a0e4c4fea6614cb13c85aaf749 Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Wed, 24 Aug 2016 16:33:47 -0400 Subject: [PATCH 342/402] Change display of log to ln --- eCF/data/functions | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/eCF/data/functions b/eCF/data/functions index 5f63087..5a51260 100644 --- a/eCF/data/functions +++ b/eCF/data/functions @@ -75,8 +75,8 @@ KroneckerDelta, \Kronecker, ({-}{0}, {-}{-}), () LaguerreL, \Laguerre, ({-}@{-}, [-]{-}@{-}), () LerchPhi, \LerchPhi, (@{-}{-}{-}), (HurwitzLerchPhi) - Log, \log, (@{-}), (PolyLog, LogInt, LogGamma, ProductLog) - LogGamma, \log@@, ({\EulerGamma@{-}}), () + Log, \ln, (@{-}), (PolyLog, LogInt, LogGamma, ProductLog) + LogGamma, \ln, (@@{\EulerGamma@{-}}), () LogIntegral, \LogInt, (@{-}), () LucasL, \LucasL, ({-}, {-}@{-}), () Max, \max, ((-,-), (-,-,-)), () From 1f2546f058eb757c640e52a6837fcedf99fbc430 Mon Sep 17 00:00:00 2001 From: Kevin Chen Date: Wed, 24 Aug 2016 16:38:39 -0400 Subject: [PATCH 343/402] Update README.md --- README.md | 2 -- 1 file changed, 2 deletions(-) diff --git a/README.md b/README.md index 095c02d..8b5f07f 100644 --- a/README.md +++ b/README.md @@ -40,8 +40,6 @@ The project is located in the `maple2latex` folder. This Python code (mostly developed by Kevin Chen) translates Mathematica files to LaTeX. -The project is currently awaiting approval to be merged into the main branch of the repository. - ## KLSadd Insertion Project `linetest.py` and `updateChapters.py` written by Rahul Shah and Edward Bian From e22364ce9bbbc1f11d66055f185cc8a9bd4cf2d4 Mon Sep 17 00:00:00 2001 From: Edward Bian Date: Thu, 25 Aug 2016 15:31:51 -0400 Subject: [PATCH 344/402] Unit Tests --- KLSadd_insertion/src/updateChapters.py | 2 + .../test/test_fix_chapter_sort.py | 124 +++++++++--------- 2 files changed, 64 insertions(+), 62 deletions(-) diff --git a/KLSadd_insertion/src/updateChapters.py b/KLSadd_insertion/src/updateChapters.py index ea703e2..e715944 100644 --- a/KLSadd_insertion/src/updateChapters.py +++ b/KLSadd_insertion/src/updateChapters.py @@ -87,6 +87,8 @@ def fix_chapter_sort(kls, chap, word, sortloc, klsaddparas, sortmatch_2, tempref :param chap: The destination of inserted sections. :param word: The keyword that is being processed. :param sortloc: The location in the list of words, of the word being sorted. + :param klsaddparas: + :param sortmatch_2: :return: This function outputs the processed chapter into a larger function for further processing. """ hyper_headers_chap = [] diff --git a/KLSadd_insertion/test/test_fix_chapter_sort.py b/KLSadd_insertion/test/test_fix_chapter_sort.py index af24eb0..205bf32 100644 --- a/KLSadd_insertion/test/test_fix_chapter_sort.py +++ b/KLSadd_insertion/test/test_fix_chapter_sort.py @@ -1,62 +1,62 @@ -# -# __author__ = 'Edward Bian' -# __status__ = 'Development' -# -# from unittest import TestCase -# from updateChapters import fix_chapter_sort -# -# -# class TestFixChapterSort(TestCase): -# def test_fix_chapter_sort(self): -# self.assertEquals(fix_chapter_sort(['%' -# , '\\subsection*{9.1 Wilson}' -# , '\\paragraph{Hypergeometric Representation}' -# , 'The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetric' -# , 'in $a,b,c,d$.' -# , '\\renewcommand{\\refname}{Standard references}'] -# ,['\\section{Wilson}\\index{Wilson polynomials}' -# ,'\\subsection*{Hypergeometric representation}' -# ,'\\begin{eqnarray}' -# ,'& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' -# ,'\\end{eqnarray}' -# ,'\\subsection*{Orthogonality relation}' -# ,'\\begin{eqnarray}' -# ,'\\label{OrthIWilson}' -# ,'& &\\frac{1}{2\\pi}\\int_0^{\\infty}' -# ,'\\end{eqnarray}'],'Hypergeometric Representation',0,[1,2,5],[],[0, 1, 5, 9]) -# ,['\\section{Wilson}\\index{Wilson polynomials}' -# , '\\subsection*{Hypergeometric representation}' -# , '\\begin{eqnarray}' -# , '& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' -# , '\\end{eqnarray}\\paragraph{\\bf KLS Addendum: Hypergeometric Representation}The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetricin $a,b,c,d$.' -# , '\\subsection*{Orthogonality relation}' -# , '\\begin{eqnarray}' -# , '\\label{OrthIWilson}' -# , '& &\\frac{1}{2\\pi}\\int_0^{\\infty}' -# , '\\end{eqnarray}']) -# self.assertEquals(fix_chapter_sort(['%' -# , '\\subsection*{9.1 Wilson}' -# , '\\paragraph{Orthogonality}' -# , 'The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetric' -# , 'in $a,b,c,d$.' -# , '\\renewcommand{\\refname}{Standard references}'] -# , ['\\section{Wilson}\\index{Wilson polynomials}' -# , '\\subsection*{Hypergeometric representation}' -# , '\\begin{eqnarray}' -# , '& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' -# , '\\end{eqnarray}' -# , '\\subsection*{Orthogonality}' -# , '\\begin{eqnarray}' -# , '\\label{OrthIWilson}' -# , '& &\\frac{1}{2\\pi}\\int_0^{\\infty}' -# , '\\end{eqnarray}'], 'Memes', 0, [1, 2, 5], [], [0, 1, 5, 9]) -# , ['\\section{Wilson}\\index{Wilson polynomials}' -# , '\\subsection*{Hypergeometric representation}' -# , '\\begin{eqnarray}' -# , '& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' -# , '\\end{eqnarray}' -# , '\\subsection*{Orthogonality}' -# , '\\begin{eqnarray}' -# , '\\label{OrthIWilson}' -# , '& &\\frac{1}{2\\pi}\\int_0^{\\infty}' -# , '\\end{eqnarray}']) \ No newline at end of file + +__author__ = 'Edward Bian' +__status__ = 'Development' + +from unittest import TestCase +from updateChapters import fix_chapter_sort + + +class TestFixChapterSort(TestCase): + def test_fix_chapter_sort(self): + self.assertEquals(fix_chapter_sort(['%' + , '\\subsection*{9.1 Wilson}' + , '\\paragraph{Hypergeometric Representation}' + , 'The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetric' + , 'in $a,b,c,d$.' + , '\\renewcommand{\\refname}{Standard references}'] + ,['\\section{Wilson}\\index{Wilson polynomials}' + ,'\\subsection*{Hypergeometric representation}' + ,'\\begin{eqnarray}' + ,'& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' + ,'\\end{eqnarray}' + ,'\\subsection*{Orthogonality relation}' + ,'\\begin{eqnarray}' + ,'\\label{OrthIWilson}' + ,'& &\\frac{1}{2\\pi}\\int_0^{\\infty}' + ,'\\end{eqnarray}'],'Hypergeometric Representation',0,[1,2,5],[],[0, 1, 5, 9],[[0, 0], [0, 0]]) + ,['\\section{Wilson}\\index{Wilson polynomials}' + , '\\subsection*{Hypergeometric representation}' + , '\\begin{eqnarray}' + , '& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' + , '\\end{eqnarray}\\paragraph{\\bf KLS Addendum: Hypergeometric Representation}The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetricin $a,b,c,d$.' + , '\\subsection*{Orthogonality relation}' + , '\\begin{eqnarray}' + , '\\label{OrthIWilson}' + , '& &\\frac{1}{2\\pi}\\int_0^{\\infty}' + , '\\end{eqnarray}']) + self.assertEquals(fix_chapter_sort(['%' + , '\\subsection*{9.1 Wilson}' + , '\\paragraph{Orthogonality}' + , 'The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetric' + , 'in $a,b,c,d$.' + , '\\renewcommand{\\refname}{Standard references}'] + , ['\\section{Wilson}\\index{Wilson polynomials}' + , '\\subsection*{Hypergeometric representation}' + , '\\begin{eqnarray}' + , '& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' + , '\\end{eqnarray}' + , '\\subsection*{Orthogonality}' + , '\\begin{eqnarray}' + , '\\label{OrthIWilson}' + , '& &\\frac{1}{2\\pi}\\int_0^{\\infty}' + , '\\end{eqnarray}'], 'Memes', 0, [1, 2, 5], [], [0, 1, 5, 9],[[0, 0], [0, 0]]) + , ['\\section{Wilson}\\index{Wilson polynomials}' + , '\\subsection*{Hypergeometric representation}' + , '\\begin{eqnarray}' + , '& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' + , '\\end{eqnarray}' + , '\\subsection*{Orthogonality}' + , '\\begin{eqnarray}' + , '\\label{OrthIWilson}' + , '& &\\frac{1}{2\\pi}\\int_0^{\\infty}' + , '\\end{eqnarray}']) \ No newline at end of file From 0cac986c138e4cf910ab4c3a3b4fc08d5afa0f3b Mon Sep 17 00:00:00 2001 From: Edward Bian Date: Thu, 25 Aug 2016 16:27:03 -0400 Subject: [PATCH 345/402] Unit Tests --- .../test/test_fix_chapter_sort.py | 54 ++++++++++++++++++- 1 file changed, 53 insertions(+), 1 deletion(-) diff --git a/KLSadd_insertion/test/test_fix_chapter_sort.py b/KLSadd_insertion/test/test_fix_chapter_sort.py index 205bf32..8f1e6aa 100644 --- a/KLSadd_insertion/test/test_fix_chapter_sort.py +++ b/KLSadd_insertion/test/test_fix_chapter_sort.py @@ -49,7 +49,7 @@ def test_fix_chapter_sort(self): , '\\begin{eqnarray}' , '\\label{OrthIWilson}' , '& &\\frac{1}{2\\pi}\\int_0^{\\infty}' - , '\\end{eqnarray}'], 'Memes', 0, [1, 2, 5], [], [0, 1, 5, 9],[[0, 0], [0, 0]]) + , '\\end{eqnarray}'], 'Orthogonality', 0, [1, 2, 5], [], [0, 1, 5, 9],[[0, 0], [0, 0]]) , ['\\section{Wilson}\\index{Wilson polynomials}' , '\\subsection*{Hypergeometric representation}' , '\\begin{eqnarray}' @@ -58,5 +58,57 @@ def test_fix_chapter_sort(self): , '\\subsection*{Orthogonality}' , '\\begin{eqnarray}' , '\\label{OrthIWilson}' + , '& &\\frac{1}{2\\pi}\\int_0^{\\infty}\\paragraph{\\bf KLS Addendum: Orthogonality}The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetricin $a,b,c,d$.' + , '\\end{eqnarray}']) + self.assertEquals(fix_chapter_sort(['%' + , '\\subsection*{9.1 Wilson}' + , '\\paragraph{Special Value}' + , 'The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetric' + , 'in $a,b,c,d$.' + , '\\renewcommand{\\refname}{Standard references}'] + , ['\\section{Wilson}\\index{Wilson polynomials}' + , '\\subsection*{Hypergeometric representation}' + , '\\begin{eqnarray}' + , '& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' + , '\\end{eqnarray}' + , '\\subsection*{Orthogonality}' + , '\\begin{eqnarray}' + , '\\label{OrthIWilson}' + , '& &\\frac{1}{2\\pi}\\int_0^{\\infty}' + , '\\end{eqnarray}'], 'Special value', 0, [1, 2, 5], [], [0, 1, 5, 9], [[0, 0], [0, 0]]) + , ['\\section{Wilson}\\index{Wilson polynomials}' + , '\\subsection*{Hypergeometric representation}' + , '\\begin{eqnarray}' + , '& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' + , '\\end{eqnarray}\\paragraph{\\bf KLS Addendum: Special value}The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetricin $a,b,c,d$.' + , '\\subsection*{Orthogonality}' + , '\\begin{eqnarray}' + , '\\label{OrthIWilson}' + , '& &\\frac{1}{2\\pi}\\int_0^{\\infty}' + , '\\end{eqnarray}']) + self.assertEquals(fix_chapter_sort(['%' + , '\\subsection*{9.1 Wilson}' + , '\\paragraph{Symmetry}' + , 'The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetric' + , 'in $a,b,c,d$.' + , '\\renewcommand{\\refname}{Standard references}'] + , ['\\section{Wilson}\\index{Wilson polynomials}' + , '\\subsection*{Hypergeometric representation}' + , '\\begin{eqnarray}' + , '& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' + , '\\end{eqnarray}' + , '\\subsection*{Orthogonality}' + , '\\begin{eqnarray}' + , '\\label{OrthIWilson}' + , '& &\\frac{1}{2\\pi}\\int_0^{\\infty}' + , '\\end{eqnarray}'], 'Symmetry', 0, [1, 2, 5], [], [0, 1, 5, 9], [[0, 0], [0, 0]]) + , ['\\section{Wilson}\\index{Wilson polynomials}' + , '\\subsection*{Hypergeometric representation}' + , '\\begin{eqnarray}' + , '& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' + , '\\end{eqnarray}\\paragraph{\\bf KLS Addendum: Symmetry}The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetricin $a,b,c,d$.' + , '\\subsection*{Orthogonality}' + , '\\begin{eqnarray}' + , '\\label{OrthIWilson}' , '& &\\frac{1}{2\\pi}\\int_0^{\\infty}' , '\\end{eqnarray}']) \ No newline at end of file From 042010c92ba4384570f1afde5157112eed2ddad2 Mon Sep 17 00:00:00 2001 From: Edward Bian Date: Thu, 25 Aug 2016 16:53:26 -0400 Subject: [PATCH 346/402] Unit Tests --- KLSadd_insertion/src/updateChapters.py | 5 +++-- KLSadd_insertion/test/test_fix_chapter_sort.py | 16 +++++++++------- 2 files changed, 12 insertions(+), 9 deletions(-) diff --git a/KLSadd_insertion/src/updateChapters.py b/KLSadd_insertion/src/updateChapters.py index e715944..1909da2 100644 --- a/KLSadd_insertion/src/updateChapters.py +++ b/KLSadd_insertion/src/updateChapters.py @@ -161,8 +161,8 @@ def fix_chapter_sort(kls, chap, word, sortloc, klsaddparas, sortmatch_2, tempref chap9 = 1 else: offset = sorter_check[sortloc][1] - if special_input == 1: - offset = 8 + # if special_input == 1: + # offset = 8 if special_input in (2, 3): name_chap = 'hypergeometric representation' @@ -201,6 +201,7 @@ def fix_chapter_sort(kls, chap, word, sortloc, klsaddparas, sortmatch_2, tempref # print hyper_headers_chap # print word sortmatch_2.append(k_hyper_sub_chap) + print khyper_header_chap return chap def cut_words(word_to_find, word_to_search_in): diff --git a/KLSadd_insertion/test/test_fix_chapter_sort.py b/KLSadd_insertion/test/test_fix_chapter_sort.py index 8f1e6aa..c618ea1 100644 --- a/KLSadd_insertion/test/test_fix_chapter_sort.py +++ b/KLSadd_insertion/test/test_fix_chapter_sort.py @@ -36,11 +36,12 @@ def test_fix_chapter_sort(self): , '\\end{eqnarray}']) self.assertEquals(fix_chapter_sort(['%' , '\\subsection*{9.1 Wilson}' + , '\\subsection*{14.1 Askey}' , '\\paragraph{Orthogonality}' , 'The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetric' , 'in $a,b,c,d$.' , '\\renewcommand{\\refname}{Standard references}'] - , ['\\section{Wilson}\\index{Wilson polynomials}' + , ['\\section{Askey}\\index{Wilson polynomials}' , '\\subsection*{Hypergeometric representation}' , '\\begin{eqnarray}' , '& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' @@ -49,8 +50,8 @@ def test_fix_chapter_sort(self): , '\\begin{eqnarray}' , '\\label{OrthIWilson}' , '& &\\frac{1}{2\\pi}\\int_0^{\\infty}' - , '\\end{eqnarray}'], 'Orthogonality', 0, [1, 2, 5], [], [0, 1, 5, 9],[[0, 0], [0, 0]]) - , ['\\section{Wilson}\\index{Wilson polynomials}' + , '\\end{eqnarray}'], 'Orthogonality', 0, [1, 2, 3, 6], [], [0, 1, 5, 9],[[1, 0], [1, 0]]) + , ['\\section{Askey}\\index{Wilson polynomials}' , '\\subsection*{Hypergeometric representation}' , '\\begin{eqnarray}' , '& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' @@ -87,12 +88,13 @@ def test_fix_chapter_sort(self): , '& &\\frac{1}{2\\pi}\\int_0^{\\infty}' , '\\end{eqnarray}']) self.assertEquals(fix_chapter_sort(['%' - , '\\subsection*{9.1 Wilson}' + , '\\subsection*{9.1 Wilson)' + , '\\subsection*{9.2 Pseudo Jacobi (or Routh-Romanovski)}' , '\\paragraph{Symmetry}' , 'The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetric' , 'in $a,b,c,d$.' , '\\renewcommand{\\refname}{Standard references}'] - , ['\\section{Wilson}\\index{Wilson polynomials}' + , ['\\section{Pseudo Jacobi}\\index{Wilson polynomials}' , '\\subsection*{Hypergeometric representation}' , '\\begin{eqnarray}' , '& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' @@ -101,8 +103,8 @@ def test_fix_chapter_sort(self): , '\\begin{eqnarray}' , '\\label{OrthIWilson}' , '& &\\frac{1}{2\\pi}\\int_0^{\\infty}' - , '\\end{eqnarray}'], 'Symmetry', 0, [1, 2, 5], [], [0, 1, 5, 9], [[0, 0], [0, 0]]) - , ['\\section{Wilson}\\index{Wilson polynomials}' + , '\\end{eqnarray}'], 'Symmetry', 0, [1, 2, 3, 6], [], [0, 1, 5, 9], [[0, 0], [0, 0]]) + , ['\\section{Pseudo Jacobi}\\index{Wilson polynomials}' , '\\subsection*{Hypergeometric representation}' , '\\begin{eqnarray}' , '& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' From 3190529428bce5072298f0b1b96c096e10646a89 Mon Sep 17 00:00:00 2001 From: Edward Bian Date: Fri, 26 Aug 2016 09:52:37 -0400 Subject: [PATCH 347/402] Testing Something --- KLSadd_insertion/test/test_fix_chapter_sort.py | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/KLSadd_insertion/test/test_fix_chapter_sort.py b/KLSadd_insertion/test/test_fix_chapter_sort.py index c618ea1..01a8d38 100644 --- a/KLSadd_insertion/test/test_fix_chapter_sort.py +++ b/KLSadd_insertion/test/test_fix_chapter_sort.py @@ -72,7 +72,7 @@ def test_fix_chapter_sort(self): , '\\begin{eqnarray}' , '& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' , '\\end{eqnarray}' - , '\\subsection*{Orthogonality}' + , '\\section*{Hypergeometric representation}' , '\\begin{eqnarray}' , '\\label{OrthIWilson}' , '& &\\frac{1}{2\\pi}\\int_0^{\\infty}' @@ -82,7 +82,7 @@ def test_fix_chapter_sort(self): , '\\begin{eqnarray}' , '& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' , '\\end{eqnarray}\\paragraph{\\bf KLS Addendum: Special value}The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetricin $a,b,c,d$.' - , '\\subsection*{Orthogonality}' + , '\\section*{Hypergeometric representation}' , '\\begin{eqnarray}' , '\\label{OrthIWilson}' , '& &\\frac{1}{2\\pi}\\int_0^{\\infty}' From 84cbb61d7b27d24a69bf3b81303e9adf945180ff Mon Sep 17 00:00:00 2001 From: Edward Bian Date: Fri, 26 Aug 2016 09:56:05 -0400 Subject: [PATCH 348/402] Testing Something --- KLSadd_insertion/test/test_fix_chapter_sort.py | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/KLSadd_insertion/test/test_fix_chapter_sort.py b/KLSadd_insertion/test/test_fix_chapter_sort.py index 01a8d38..f7d9c73 100644 --- a/KLSadd_insertion/test/test_fix_chapter_sort.py +++ b/KLSadd_insertion/test/test_fix_chapter_sort.py @@ -72,7 +72,7 @@ def test_fix_chapter_sort(self): , '\\begin{eqnarray}' , '& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' , '\\end{eqnarray}' - , '\\section*{Hypergeometric representation}' + , '\\subsection*{Hypergeometric representation}' , '\\begin{eqnarray}' , '\\label{OrthIWilson}' , '& &\\frac{1}{2\\pi}\\int_0^{\\infty}' @@ -82,7 +82,7 @@ def test_fix_chapter_sort(self): , '\\begin{eqnarray}' , '& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' , '\\end{eqnarray}\\paragraph{\\bf KLS Addendum: Special value}The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetricin $a,b,c,d$.' - , '\\section*{Hypergeometric representation}' + , '\\subsection*{Hypergeometric representation}' , '\\begin{eqnarray}' , '\\label{OrthIWilson}' , '& &\\frac{1}{2\\pi}\\int_0^{\\infty}' From d7fee73c35e65c9d3e3e4fe962d61a7dfc789829 Mon Sep 17 00:00:00 2001 From: Edward Bian Date: Fri, 26 Aug 2016 09:59:11 -0400 Subject: [PATCH 349/402] Testing Something --- KLSadd_insertion/test/test_fix_chapter_sort.py | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/KLSadd_insertion/test/test_fix_chapter_sort.py b/KLSadd_insertion/test/test_fix_chapter_sort.py index f7d9c73..790d87f 100644 --- a/KLSadd_insertion/test/test_fix_chapter_sort.py +++ b/KLSadd_insertion/test/test_fix_chapter_sort.py @@ -72,7 +72,7 @@ def test_fix_chapter_sort(self): , '\\begin{eqnarray}' , '& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' , '\\end{eqnarray}' - , '\\subsection*{Hypergeometric representation}' + , '\\subsection{Hypergeometric representation}' , '\\begin{eqnarray}' , '\\label{OrthIWilson}' , '& &\\frac{1}{2\\pi}\\int_0^{\\infty}' @@ -82,7 +82,7 @@ def test_fix_chapter_sort(self): , '\\begin{eqnarray}' , '& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' , '\\end{eqnarray}\\paragraph{\\bf KLS Addendum: Special value}The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetricin $a,b,c,d$.' - , '\\subsection*{Hypergeometric representation}' + , '\\subsection{Hypergeometric representation}' , '\\begin{eqnarray}' , '\\label{OrthIWilson}' , '& &\\frac{1}{2\\pi}\\int_0^{\\infty}' From 74b0756bbbf65094943c67854f5459713f3ab5d0 Mon Sep 17 00:00:00 2001 From: Edward Bian Date: Fri, 26 Aug 2016 11:01:19 -0400 Subject: [PATCH 350/402] Testing Something --- .../test/test_fix_chapter_sort.py | 26 +++++++++++++++++++ 1 file changed, 26 insertions(+) diff --git a/KLSadd_insertion/test/test_fix_chapter_sort.py b/KLSadd_insertion/test/test_fix_chapter_sort.py index 790d87f..af7f0bf 100644 --- a/KLSadd_insertion/test/test_fix_chapter_sort.py +++ b/KLSadd_insertion/test/test_fix_chapter_sort.py @@ -113,4 +113,30 @@ def test_fix_chapter_sort(self): , '\\begin{eqnarray}' , '\\label{OrthIWilson}' , '& &\\frac{1}{2\\pi}\\int_0^{\\infty}' + , '\\end{eqnarray}']) + self.assertEquals(fix_chapter_sort(['%' + , '\\subsection*{9.1 Wilson}' + , '\\paragraph{Hypergeometric Representation}' + , 'The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetric' + , 'in $a,b,c,d$.' + , '\\renewcommand{\\refname}{Standard references}'] + , ['\\section{Wilson}\\index{Wilson polynomials}' + , '\\subsection*{Hypergeometric representation}' + , '\\begin{eqnarray}' + , '& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' + , '\\end{eqnarray}' + , '\\subsection*{Basic Hypergeometric representation}' + , '\\begin{eqnarray}' + , '\\label{OrthIWilson}' + , '& &\\frac{1}{2\\pi}\\int_0^{\\infty}' + , '\\end{eqnarray}'], 'Hypergeometric Representation', 0, [1, 2, 5], [], [0, 1, 5, 9], [[0, 0], [0, 0]]) + , ['\\section{Wilson}\\index{Wilson polynomials}' + , '\\subsection*{Hypergeometric representation}' + , '\\begin{eqnarray}' + , '& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' + , '\\end{eqnarray}\\paragraph{\\bf KLS Addendum: Hypergeometric Representation}The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetricin $a,b,c,d$.' + , '\\subsection*{Basic Hypergeometric representation}' + , '\\begin{eqnarray}' + , '\\label{OrthIWilson}' + , '& &\\frac{1}{2\\pi}\\int_0^{\\infty}' , '\\end{eqnarray}']) \ No newline at end of file From 174da2236f219a746a1a731f128eb60b7ca28a9f Mon Sep 17 00:00:00 2001 From: Edward Bian Date: Fri, 26 Aug 2016 11:29:54 -0400 Subject: [PATCH 351/402] Testing Something --- KLSadd_insertion/test/test_fix_chapter_sort.py | 8 +++++--- 1 file changed, 5 insertions(+), 3 deletions(-) diff --git a/KLSadd_insertion/test/test_fix_chapter_sort.py b/KLSadd_insertion/test/test_fix_chapter_sort.py index af7f0bf..d0b6372 100644 --- a/KLSadd_insertion/test/test_fix_chapter_sort.py +++ b/KLSadd_insertion/test/test_fix_chapter_sort.py @@ -119,23 +119,25 @@ def test_fix_chapter_sort(self): , '\\paragraph{Hypergeometric Representation}' , 'The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetric' , 'in $a,b,c,d$.' + , '\\paragraph{Orthogonality relation}' + , 'Words and Stuff' , '\\renewcommand{\\refname}{Standard references}'] , ['\\section{Wilson}\\index{Wilson polynomials}' , '\\subsection*{Hypergeometric representation}' , '\\begin{eqnarray}' , '& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' , '\\end{eqnarray}' - , '\\subsection*{Basic Hypergeometric representation}' + , '\\subsection*{Orthogonality relation}' , '\\begin{eqnarray}' , '\\label{OrthIWilson}' , '& &\\frac{1}{2\\pi}\\int_0^{\\infty}' - , '\\end{eqnarray}'], 'Hypergeometric Representation', 0, [1, 2, 5], [], [0, 1, 5, 9], [[0, 0], [0, 0]]) + , '\\end{eqnarray}'], 'Hypergeometric Representation', 0, [1, 2, 5, 7], [], [0, 1, 5, 9], [[0, 0], [0, 0]]) , ['\\section{Wilson}\\index{Wilson polynomials}' , '\\subsection*{Hypergeometric representation}' , '\\begin{eqnarray}' , '& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' , '\\end{eqnarray}\\paragraph{\\bf KLS Addendum: Hypergeometric Representation}The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetricin $a,b,c,d$.' - , '\\subsection*{Basic Hypergeometric representation}' + , '\\subsection*{Orthogonality relation}' , '\\begin{eqnarray}' , '\\label{OrthIWilson}' , '& &\\frac{1}{2\\pi}\\int_0^{\\infty}' From cb607a37db63a3586e0c060965624f9e9bbc3053 Mon Sep 17 00:00:00 2001 From: Edward Bian Date: Fri, 26 Aug 2016 11:32:47 -0400 Subject: [PATCH 352/402] Testing Something --- KLSadd_insertion/test/test_fix_chapter_sort.py | 8 +++----- 1 file changed, 3 insertions(+), 5 deletions(-) diff --git a/KLSadd_insertion/test/test_fix_chapter_sort.py b/KLSadd_insertion/test/test_fix_chapter_sort.py index d0b6372..af7f0bf 100644 --- a/KLSadd_insertion/test/test_fix_chapter_sort.py +++ b/KLSadd_insertion/test/test_fix_chapter_sort.py @@ -119,25 +119,23 @@ def test_fix_chapter_sort(self): , '\\paragraph{Hypergeometric Representation}' , 'The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetric' , 'in $a,b,c,d$.' - , '\\paragraph{Orthogonality relation}' - , 'Words and Stuff' , '\\renewcommand{\\refname}{Standard references}'] , ['\\section{Wilson}\\index{Wilson polynomials}' , '\\subsection*{Hypergeometric representation}' , '\\begin{eqnarray}' , '& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' , '\\end{eqnarray}' - , '\\subsection*{Orthogonality relation}' + , '\\subsection*{Basic Hypergeometric representation}' , '\\begin{eqnarray}' , '\\label{OrthIWilson}' , '& &\\frac{1}{2\\pi}\\int_0^{\\infty}' - , '\\end{eqnarray}'], 'Hypergeometric Representation', 0, [1, 2, 5, 7], [], [0, 1, 5, 9], [[0, 0], [0, 0]]) + , '\\end{eqnarray}'], 'Hypergeometric Representation', 0, [1, 2, 5], [], [0, 1, 5, 9], [[0, 0], [0, 0]]) , ['\\section{Wilson}\\index{Wilson polynomials}' , '\\subsection*{Hypergeometric representation}' , '\\begin{eqnarray}' , '& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' , '\\end{eqnarray}\\paragraph{\\bf KLS Addendum: Hypergeometric Representation}The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetricin $a,b,c,d$.' - , '\\subsection*{Orthogonality relation}' + , '\\subsection*{Basic Hypergeometric representation}' , '\\begin{eqnarray}' , '\\label{OrthIWilson}' , '& &\\frac{1}{2\\pi}\\int_0^{\\infty}' From b398a63f57d9d76c12f1042942448cdb8cbfbc84 Mon Sep 17 00:00:00 2001 From: Edward Bian Date: Fri, 26 Aug 2016 14:20:07 -0400 Subject: [PATCH 353/402] Testing Something --- .../test/test_fix_chapter_sort.py | 22 ++++++++++++++++++- 1 file changed, 21 insertions(+), 1 deletion(-) diff --git a/KLSadd_insertion/test/test_fix_chapter_sort.py b/KLSadd_insertion/test/test_fix_chapter_sort.py index af7f0bf..8a71fd4 100644 --- a/KLSadd_insertion/test/test_fix_chapter_sort.py +++ b/KLSadd_insertion/test/test_fix_chapter_sort.py @@ -139,4 +139,24 @@ def test_fix_chapter_sort(self): , '\\begin{eqnarray}' , '\\label{OrthIWilson}' , '& &\\frac{1}{2\\pi}\\int_0^{\\infty}' - , '\\end{eqnarray}']) \ No newline at end of file + , '\\end{eqnarray}']) + self.assertEquals(fix_chapter_sort(['%' + , '\\subsection*{9.1 Wilson}' + , '\\paragraph{Symmetry}' + , 'The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetric' + ,'\\paragraph{Trivial Symmetry}' + , 'Words' + , '\\renewcommand{\\refname}{Standard references}'] + , ['\\section{Wilson}\\index{Wilson polynomials}' + , '\\subsection*{Hypergeometric representation}' + , '\\begin{eqnarray}' + , '& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' + , '\\end{eqnarray}' + ,'%'] + , 'Symmetry', 0, [1, 2, 4, 6], [], [0, 1, 5], [[0, 0], [0, 0]]) + , ['\\section{Wilson}\\index{Wilson polynomials}' + , '\\subsection*{Hypergeometric representation}' + , '\\begin{eqnarray}' + , '& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' + , '\\end{eqnarray}\\paragraph{\\bf KLS Addendum: Symmetry}The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetric\\paragraph{\\bf KLS Addendum: Symmetry}\nWords' + , '%']) \ No newline at end of file From 46e8cd22b1a709f0974863a849b72d2575d7f02f Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Fri, 26 Aug 2016 16:06:25 -0400 Subject: [PATCH 354/402] Add conversion for E to \expe and add conversion for trig and hyperbolic functions with powers following it --- DLMF_preprocessing/test/data/correctout.tex | 6 +-- DLMF_preprocessing/test/data/in.tex | 6 +-- DLMF_preprocessing/test/data/testout.tex | 6 +-- eCF/src/mathematica_to_latex.py | 52 ++++++++++++++------- eCF/test/data/test.tex | 6 +-- eCF/test/test_main.py | 6 +-- eCF/test/test_master_function.py | 31 ++++++++---- eCF/test/test_replace_operators.py | 2 + 8 files changed, 63 insertions(+), 52 deletions(-) diff --git a/DLMF_preprocessing/test/data/correctout.tex b/DLMF_preprocessing/test/data/correctout.tex index 8614bab..7b049bf 100644 --- a/DLMF_preprocessing/test/data/correctout.tex +++ b/DLMF_preprocessing/test/data/correctout.tex @@ -2,14 +2,10 @@ \documentclass{article} \usepackage{amsmath} -\usepackage{amsthm} -\usepackage{amssymb} \usepackage{amsfonts} -\usepackage{breqn} +\usepackage{amssymb} \usepackage{DLMFmath} \usepackage{DRMFfcns} -\usepackage{DLMFfcns} -\usepackage{graphicx} \usepackage[paperwidth=15in, paperheight=20in, margin=0.5in]{geometry} \begin{document} diff --git a/DLMF_preprocessing/test/data/in.tex b/DLMF_preprocessing/test/data/in.tex index 3a40265..b37dee5 100644 --- a/DLMF_preprocessing/test/data/in.tex +++ b/DLMF_preprocessing/test/data/in.tex @@ -2,14 +2,10 @@ \documentclass{article} \usepackage{amsmath} -\usepackage{amsthm} -\usepackage{amssymb} \usepackage{amsfonts} -\usepackage{breqn} +\usepackage{amssymb} \usepackage{DLMFmath} \usepackage{DRMFfcns} -\usepackage{DLMFfcns} -\usepackage{graphicx} \usepackage[paperwidth=15in, paperheight=20in, margin=0.5in]{geometry} \begin{document} diff --git a/DLMF_preprocessing/test/data/testout.tex b/DLMF_preprocessing/test/data/testout.tex index 8614bab..7b049bf 100644 --- a/DLMF_preprocessing/test/data/testout.tex +++ b/DLMF_preprocessing/test/data/testout.tex @@ -2,14 +2,10 @@ \documentclass{article} \usepackage{amsmath} -\usepackage{amsthm} -\usepackage{amssymb} \usepackage{amsfonts} -\usepackage{breqn} +\usepackage{amssymb} \usepackage{DLMFmath} \usepackage{DRMFfcns} -\usepackage{DLMFfcns} -\usepackage{graphicx} \usepackage[paperwidth=15in, paperheight=20in, margin=0.5in]{geometry} \begin{document} diff --git a/eCF/src/mathematica_to_latex.py b/eCF/src/mathematica_to_latex.py index 154be73..755e82a 100644 --- a/eCF/src/mathematica_to_latex.py +++ b/eCF/src/mathematica_to_latex.py @@ -43,10 +43,13 @@ LEFT_BRACKETS = list('([{') RIGHT_BRACKETS = list(')]}') -TRIG = ('ArcCos', 'ArcCosh', 'ArcCot', 'ArcCoth', 'ArcCsc', 'ArcCsch', - 'ArcSec', 'ArcSech', 'ArcSin', 'ArcSinh', 'ArcTan', 'ArcTanh', - 'Cos', 'Cosh', 'Cot', 'Coth', 'Csc', 'Csch', 'Sec', 'Sech', 'Sinc', - 'Sin', 'Sinh', 'Tan', 'Tanh') +TRIG_OUTER = ('ArcCos', 'ArcCosh', 'ArcCot', 'ArcCoth', 'ArcCsc', 'ArcCsch', + 'ArcSec', 'ArcSech', 'ArcSin', 'ArcSinh', 'ArcTan', 'ArcTanh', + 'Sinc') +TRIG_INNER = ('Cos', 'Cot', 'Csc', 'Sec', 'Sin', 'Tan', + 'Cosh', 'Coth', 'Csch', 'Sech', 'Sinh', 'Tanh') +E_EXCEPT = ('EulerGamma', 'Epsilon', 'EulerConstant', 'EulerBeta', + 'ExpIntn', 'ExpInti', 'CompEllIntE', 'CompEllIntK') def find_surrounding(line, function, ex=(), start=0): @@ -201,7 +204,6 @@ def master_function(line, params): m, l, sep, ex = params[:5] sep = [i.split('-') for i in sep] multi = list('+-*/') - for _ in range(line.count(m)): try: pos @@ -234,15 +236,27 @@ def master_function(line, params): # If the arguments in a trig function are more than one variable, # then instead of "@@" make it "@" - if m in TRIG and \ + if (m in TRIG_OUTER or m in TRIG_INNER) and \ sum([args[0].count(element) for element in multi]) != 0: sep[0][0] = sep[0][0].replace('@@', '@') # Add parens around ambiguous functions (trig functions) - if m in TRIG and len(line) != pos[1] and line[pos[1]] == '^': + if m in TRIG_OUTER and len(line) != pos[1] and line[pos[1]] == '^': line = line[:pos[0]] + '(' + l + \ '%s'.join(sep[[len(y) for y in sep] .index(len(args) + 1)]) + ')' + line[pos[1]:] + # Add the inner square, like \cos^2{x} + elif m in TRIG_INNER and len(line) != pos[1] and \ + line[pos[1]] == '^': + if line[pos[1] + 1] == '{' and line[pos[1] + 3] == '}': + line = line[:pos[0]] + l + line[pos[1]:pos[1] + 4] + \ + '%s'.join(sep[[len(y) for y in sep]. + index(len(args) + 1)]) + line[pos[1] + 4:] + else: + line = line[:pos[0]] + '(' + l + \ + '%s'.join(sep[[len(y) for y in sep]. + index(len(args) + 1)]) + ')' + \ + line[pos[1]:] else: line = line[:pos[0]] + l + \ '%s'.join(sep[[len(y) for y in sep]. @@ -316,7 +330,6 @@ def carat(line): while i != len(line): if line[i] == '^': - print(line) k = search(line, i, list('*/+-=, '), 1) if line[i + 1] == '(' and line[k] == ')': line = line[:i] + '^{' + line[i + 2:k] + '}' + line[k + 1:] @@ -868,6 +881,15 @@ def replace_operators(line): line = line.replace('GoldenRatio', '\\GoldenRatio') line = line.replace('Pi', '\\pi') line = line.replace('CalculateData`Private`nu', '\\nu') + line = line.replace('\\Jacobisn@{t}{m ^ {2}} ^ {2}', + '\\Jacobisn^{2}@{t}{m ^ {2}}') + + # Replaces "E" constant with "\\expe" + for word in E_EXCEPT: + line = line.replace(word, word.replace('E', 'A')) + line = line.replace('E', '\\expe') + for word in E_EXCEPT: + line = line.replace(word.replace('E', 'A'), word) return line @@ -893,7 +915,7 @@ def replace_vars(line): def main(pathw=DIR_NAME + 'newIdentities.tex', - pathr=DIR_NAME + 'Identities.m', + pathr=DIR_NAME + 'IdentitiesTest.m', pathref=DIR_NAME + 'References.txt'): # ((str, str)) -> None """ @@ -912,14 +934,10 @@ def main(pathw=DIR_NAME + 'newIdentities.tex', latex.write('\n\\documentclass{article}\n\n' '\\usepackage{amsmath}\n' - '\\usepackage{amsthm}\n' - '\\usepackage{amssymb}\n' '\\usepackage{amsfonts}\n' - '\\usepackage{breqn}\n' + '\\usepackage{amssymb}\n' '\\usepackage{DLMFmath}\n' '\\usepackage{DRMFfcns}\n' - '\\usepackage{DLMFfcns}\n' - '\\usepackage{graphicx}\n' '\\usepackage[paperwidth=15in, paperheight=20in, ' 'margin=0.5in]{geometry}\n\n' '\\begin{document}\n\n\n') @@ -940,11 +958,11 @@ def main(pathw=DIR_NAME + 'newIdentities.tex', line = line.replace('EulerGamma', '\\EulerConstant') + line = carat(line) + for func in FUNCTION_CONVERSIONS: line = master_function(line, func) - line = carat(line) - line = beta(line) line = cfk(line) line = gamma(line) @@ -971,7 +989,7 @@ def main(pathw=DIR_NAME + 'newIdentities.tex', pass line = ' ' + line + '\n\\end{equation}' - print line + # print line latex.write(line + '\n') latex.write('\n\n\\end{document}\n') diff --git a/eCF/test/data/test.tex b/eCF/test/data/test.tex index 7ff4f9a..1e19a81 100644 --- a/eCF/test/data/test.tex +++ b/eCF/test/data/test.tex @@ -2,14 +2,10 @@ \documentclass{article} \usepackage{amsmath} -\usepackage{amsthm} -\usepackage{amssymb} \usepackage{amsfonts} -\usepackage{breqn} +\usepackage{amssymb} \usepackage{DLMFmath} \usepackage{DRMFfcns} -\usepackage{DLMFfcns} -\usepackage{graphicx} \usepackage[paperwidth=15in, paperheight=20in, margin=0.5in]{geometry} \begin{document} diff --git a/eCF/test/test_main.py b/eCF/test/test_main.py index 4ccaa9b..8c38723 100644 --- a/eCF/test/test_main.py +++ b/eCF/test/test_main.py @@ -23,14 +23,10 @@ def test_generation(self): '\\documentclass{article}\n' '\n' '\\usepackage{amsmath}\n' - '\\usepackage{amsthm}\n' - '\\usepackage{amssymb}\n' '\\usepackage{amsfonts}\n' - '\\usepackage{breqn}\n' + '\\usepackage{amssymb}\n' '\\usepackage{DLMFmath}\n' '\\usepackage{DRMFfcns}\n' - '\\usepackage{DLMFfcns}\n' - '\\usepackage{graphicx}\n' '\\usepackage[paperwidth=15in, paperheight=20in, margin=0.5in]{geometry}\n' '\n' '\\begin{document}\n' diff --git a/eCF/test/test_master_function.py b/eCF/test/test_master_function.py index 56f20f8..53d0025 100644 --- a/eCF/test/test_master_function.py +++ b/eCF/test/test_master_function.py @@ -25,10 +25,11 @@ FUNCTION_CONVERSIONS[index] = tuple(FUNCTION_CONVERSIONS[index]) FUNCTION_CONVERSIONS = tuple(FUNCTION_CONVERSIONS) -TRIG = ('ArcCos', 'ArcCosh', 'ArcCot', 'ArcCoth', 'ArcCsc', 'ArcCsch', - 'ArcSec', 'ArcSech', 'ArcSin', 'ArcSinh', 'ArcTan', 'ArcTanh', - 'Cos', 'Cosh', 'Cot', 'Coth', 'Csc', 'Csch', 'Sec', 'Sech', 'Sinc', - 'Sin', 'Sinh', 'Tan', 'Tanh') +TRIG_OUTER = ('ArcCos', 'ArcCosh', 'ArcCot', 'ArcCoth', 'ArcCsc', 'ArcCsch', + 'ArcSec', 'ArcSech', 'ArcSin', 'ArcSinh', 'ArcTan', 'ArcTanh', + 'Sinc') +TRIG_INNER = ('Cos', 'Cot', 'Csc', 'Sec', 'Sin', 'Tan', + 'Cosh', 'Coth', 'Csch', 'Sech', 'Sinh', 'Tanh') MULTI = list('+-*/') @@ -73,18 +74,28 @@ def test_conversion(self): else: self.assertEqual(master_function(before, function), after) self.assertEqual(master_function('--{0}--'.format(before), function), '--{0}--'.format(after)) - if function[0] in TRIG: + + # Test single and double "@" signs + if function[0] in TRIG_OUTER or function[0] in TRIG_INNER: for sep in MULTI: before2 = before[:-1] + sep + 'b]' after2 = after.replace('@@', '@')[:-1] + sep + 'b}' self.assertEqual(master_function(before2, function), after2) self.assertEqual(master_function('--{0}--'.format(before2), function), '--{0}--'.format(after2)) - before2 += '^' - after2 = '(' + after2 + ')^' - self.assertEqual(master_function(before2, function), after2) - self.assertEqual(master_function('--{0}--'.format(before2), function), '--{0}--'.format(after2)) - + # Test parentheses placement for powers in functions + if function[0] in TRIG_OUTER: + before2 = before + '^' + after2 = '(' + after + ')^' + self.assertEqual(master_function(before2, function), after2) + self.assertEqual(master_function('--{0}--'.format(before2), function), '--{0}--'.format(after2)) + if function[0] in TRIG_INNER: + before2 = before + '^{b}' + after2 = after[:-5] + '^{b}@@{a}' + self.assertEqual(master_function(before2, function), after2) + self.assertEqual(master_function('--{0}--'.format(before2), function), '--{0}--'.format(after2)) + + # Test an exception if function[0] == 'D': self.assertEqual(master_function('\\[Delta]', function), '\\[Delta]') diff --git a/eCF/test/test_replace_operators.py b/eCF/test/test_replace_operators.py index d129ce2..e88fac5 100644 --- a/eCF/test/test_replace_operators.py +++ b/eCF/test/test_replace_operators.py @@ -34,6 +34,8 @@ def test_constants(self): self.assertEqual(replace_operators('GoldenRatio'), '\\GoldenRatio') self.assertEqual(replace_operators('Pi'), '\\pi') self.assertEqual(replace_operators('CalculateData`Private`nu'), '\\nu') + self.assertEqual(replace_operators('E EulerGamma Epsilon EulerConstant EulerBeta ExpIntn ExpInti CompEllIntE CompEllIntK'), + '\\expe EulerGamma Epsilon EulerConstant EulerBeta ExpIntn ExpInti CompEllIntE CompEllIntK') def test_with_percent(self): self.assertEqual(replace_operators('(%('), '\\left( %(') From e9e1bbdd5b21ad43b7be7a347e1d2dab62d5d441 Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Fri, 26 Aug 2016 16:21:57 -0400 Subject: [PATCH 355/402] Fix test cases for more coverage --- eCF/test/test_master_function.py | 12 ++++++------ 1 file changed, 6 insertions(+), 6 deletions(-) diff --git a/eCF/test/test_master_function.py b/eCF/test/test_master_function.py index 53d0025..41ef177 100644 --- a/eCF/test/test_master_function.py +++ b/eCF/test/test_master_function.py @@ -83,12 +83,12 @@ def test_conversion(self): self.assertEqual(master_function(before2, function), after2) self.assertEqual(master_function('--{0}--'.format(before2), function), '--{0}--'.format(after2)) - # Test parentheses placement for powers in functions - if function[0] in TRIG_OUTER: - before2 = before + '^' - after2 = '(' + after + ')^' - self.assertEqual(master_function(before2, function), after2) - self.assertEqual(master_function('--{0}--'.format(before2), function), '--{0}--'.format(after2)) + before2 = before + '^{b+c}' + after2 = '(' + after + ')^{b+c}' + self.assertEqual(master_function(before2, function), after2) + self.assertEqual(master_function('--{0}--'.format(before2), function), '--{0}--'.format(after2)) + + # Test powers for trig and hyperbolic functions if function[0] in TRIG_INNER: before2 = before + '^{b}' after2 = after[:-5] + '^{b}@@{a}' From 015f7dda8d409f140a208dcfe841e1e57e002307 Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Fri, 26 Aug 2016 16:40:53 -0400 Subject: [PATCH 356/402] Add line to generation --- DLMF_preprocessing/test/data/correctout.tex | 1 + DLMF_preprocessing/test/data/in.tex | 1 + DLMF_preprocessing/test/data/testout.tex | 1 + eCF/src/mathematica_to_latex.py | 5 +++-- eCF/test/data/test.tex | 1 + eCF/test/test_main.py | 1 + 6 files changed, 8 insertions(+), 2 deletions(-) diff --git a/DLMF_preprocessing/test/data/correctout.tex b/DLMF_preprocessing/test/data/correctout.tex index 7b049bf..f6d7a9f 100644 --- a/DLMF_preprocessing/test/data/correctout.tex +++ b/DLMF_preprocessing/test/data/correctout.tex @@ -4,6 +4,7 @@ \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} +\usepackage{breqn} \usepackage{DLMFmath} \usepackage{DRMFfcns} \usepackage[paperwidth=15in, paperheight=20in, margin=0.5in]{geometry} diff --git a/DLMF_preprocessing/test/data/in.tex b/DLMF_preprocessing/test/data/in.tex index b37dee5..cdd1670 100644 --- a/DLMF_preprocessing/test/data/in.tex +++ b/DLMF_preprocessing/test/data/in.tex @@ -4,6 +4,7 @@ \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} +\usepackage{breqn} \usepackage{DLMFmath} \usepackage{DRMFfcns} \usepackage[paperwidth=15in, paperheight=20in, margin=0.5in]{geometry} diff --git a/DLMF_preprocessing/test/data/testout.tex b/DLMF_preprocessing/test/data/testout.tex index 7b049bf..f6d7a9f 100644 --- a/DLMF_preprocessing/test/data/testout.tex +++ b/DLMF_preprocessing/test/data/testout.tex @@ -4,6 +4,7 @@ \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} +\usepackage{breqn} \usepackage{DLMFmath} \usepackage{DRMFfcns} \usepackage[paperwidth=15in, paperheight=20in, margin=0.5in]{geometry} diff --git a/eCF/src/mathematica_to_latex.py b/eCF/src/mathematica_to_latex.py index 755e82a..f97771d 100644 --- a/eCF/src/mathematica_to_latex.py +++ b/eCF/src/mathematica_to_latex.py @@ -915,7 +915,7 @@ def replace_vars(line): def main(pathw=DIR_NAME + 'newIdentities.tex', - pathr=DIR_NAME + 'IdentitiesTest.m', + pathr=DIR_NAME + 'Identities.m', pathref=DIR_NAME + 'References.txt'): # ((str, str)) -> None """ @@ -936,6 +936,7 @@ def main(pathw=DIR_NAME + 'newIdentities.tex', '\\usepackage{amsmath}\n' '\\usepackage{amsfonts}\n' '\\usepackage{amssymb}\n' + '\\usepackage{breqn}\n' '\\usepackage{DLMFmath}\n' '\\usepackage{DRMFfcns}\n' '\\usepackage[paperwidth=15in, paperheight=20in, ' @@ -989,7 +990,7 @@ def main(pathw=DIR_NAME + 'newIdentities.tex', pass line = ' ' + line + '\n\\end{equation}' - # print line + print line latex.write(line + '\n') latex.write('\n\n\\end{document}\n') diff --git a/eCF/test/data/test.tex b/eCF/test/data/test.tex index 1e19a81..bef7adc 100644 --- a/eCF/test/data/test.tex +++ b/eCF/test/data/test.tex @@ -4,6 +4,7 @@ \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} +\usepackage{breqn} \usepackage{DLMFmath} \usepackage{DRMFfcns} \usepackage[paperwidth=15in, paperheight=20in, margin=0.5in]{geometry} diff --git a/eCF/test/test_main.py b/eCF/test/test_main.py index 8c38723..f102f6d 100644 --- a/eCF/test/test_main.py +++ b/eCF/test/test_main.py @@ -25,6 +25,7 @@ def test_generation(self): '\\usepackage{amsmath}\n' '\\usepackage{amsfonts}\n' '\\usepackage{amssymb}\n' + '\\usepackage{breqn}\n' '\\usepackage{DLMFmath}\n' '\\usepackage{DRMFfcns}\n' '\\usepackage[paperwidth=15in, paperheight=20in, margin=0.5in]{geometry}\n' From 260e0db69f60371f38f6716928125668687a8bbd Mon Sep 17 00:00:00 2001 From: KChen1250 Date: Tue, 6 Sep 2016 17:07:48 -0400 Subject: [PATCH 357/402] Add fix for spacing before constraing --- DLMF_preprocessing/src/latex_constraint_modifier.py | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/DLMF_preprocessing/src/latex_constraint_modifier.py b/DLMF_preprocessing/src/latex_constraint_modifier.py index 56721f4..f58c117 100644 --- a/DLMF_preprocessing/src/latex_constraint_modifier.py +++ b/DLMF_preprocessing/src/latex_constraint_modifier.py @@ -144,7 +144,7 @@ def main(pathr, pathw): :return: None """ with open(pathr, 'r') as b: - before = b.read().split('\n') + before = [c.replace(' %', '%') for c in b.read().split('\n')] before = combine_percent(before) From 769f962c64ee07bf7388c954dba4ba139046c4af Mon Sep 17 00:00:00 2001 From: Edward Bian Date: Fri, 9 Sep 2016 16:48:45 -0400 Subject: [PATCH 358/402] Unit test fixes --- KLSadd_insertion/src/updateChapters.py | 57 +++--- KLSadd_insertion/test/test_chap1_placer.py | 2 +- KLSadd_insertion/test/test_fix_chapter.py | 162 ++++++++++++++++++ .../test/test_fix_chapter_sort.py | 2 +- .../test/test_generic_functions.py | 2 +- .../test/test_reference_placer.py | 6 +- 6 files changed, 203 insertions(+), 28 deletions(-) create mode 100644 KLSadd_insertion/test/test_fix_chapter.py diff --git a/KLSadd_insertion/src/updateChapters.py b/KLSadd_insertion/src/updateChapters.py index 1909da2..d0e162c 100644 --- a/KLSadd_insertion/src/updateChapters.py +++ b/KLSadd_insertion/src/updateChapters.py @@ -35,7 +35,7 @@ def chap1placer(chap1, kls, klsaddparas): intro_to_add = ''.join(kls_sub_chap1) chap1 = chap1[:len(chap1)-1] chap1 = ''.join(chap1) - total_chap1 = chap1 + "\\paragraph{\\large\\bf KLS Addendum: Generalities}" + intro_to_add + "\\end{document}" + total_chap1 = chap1 + "\\paragraph{\\bf KLS Addendum: Generalities}" + intro_to_add + "\\end{document}" return total_chap1 @@ -96,12 +96,15 @@ def fix_chapter_sort(kls, chap, word, sortloc, klsaddparas, sortmatch_2, tempref special_input = 0 name_chap = word.lower() + print name_chap if name_chap == "orthogonality": special_input = 1 elif name_chap == "special value": special_input = 2 elif name_chap == "symmetry": special_input = 3 + elif name_chap == "bilateral generating function": + special_input = 4 khyper_header_chap = [] k_hyper_sub_chap = [] @@ -115,7 +118,7 @@ def fix_chapter_sort(kls, chap, word, sortloc, klsaddparas, sortmatch_2, tempref temp = line[line.find(" ", 12)+1: line.find("}", 12)] # get just the name (like mathpeople) if temp.lower() == 'wilson': chapterstart = True - if special_input in (0, 2) and name_chap in line.lower() and chapterstart and ("\\paragraph{" in line or "\\subsubsection*{" in line) or \ + if special_input in (0, 2, 4) and name_chap in line.lower() and chapterstart and ("\\paragraph{" in line or "\\subsubsection*{" in line) or \ (special_input in (1, 3) and "orthogonality relation" not in line.lower() and name_chap in line.lower() and chapterstart and ("\\paragraph{" in line or "\\subsubsection*{" in line)): for item in klsaddparas: @@ -135,9 +138,13 @@ def fix_chapter_sort(kls, chap, word, sortloc, klsaddparas, sortmatch_2, tempref item = 0 while item < len(khyper_header_chap): try: - if khyper_header_chap[item] == khyper_header_chap[item+1]: + if khyper_header_chap[item] == khyper_header_chap[item + 1] and "generating functions" in name_chap: + k_hyper_sub_chap[item + 1] = k_hyper_sub_chap[item] + "\paragraph{\\bf KLS Addendum: Bilateral generating functions}" + k_hyper_sub_chap[item + 1] + khyper_header_chap[item] = "memes" + k_hyper_sub_chap[item] = "memes" + elif khyper_header_chap[item] == khyper_header_chap[item+1]: k_hyper_sub_chap[item + 1] = k_hyper_sub_chap[item] + "\paragraph{\\bf KLS Addendum: " + \ - word + "}\n" + k_hyper_sub_chap[item + 1] + word + "}" + k_hyper_sub_chap[item + 1] khyper_header_chap[item] = "memes" k_hyper_sub_chap[item] = "memes" else: @@ -179,7 +186,7 @@ def fix_chapter_sort(kls, chap, word, sortloc, klsaddparas, sortmatch_2, tempref temp = line[9:line.find("}", 7)] if name_chap in line.lower(): - if special_input in (0, 2, 3) or special_input == 1 and "orthogonality relation" not in line: + if special_input in (0, 2, 3, 4) or special_input == 1 and "orthogonality relation" not in line: hyper_subs_chap.append([tempref[d + 1]]) # appends the index for the line before following subsection hyper_headers_chap.append(temp) # appends the name of the section the hypergeo subsection is in @@ -201,9 +208,14 @@ def fix_chapter_sort(kls, chap, word, sortloc, klsaddparas, sortmatch_2, tempref # print hyper_headers_chap # print word sortmatch_2.append(k_hyper_sub_chap) + if "generating function" in name_chap: + print "Stuff for checking" + print k_hyper_sub_chap print khyper_header_chap + print hyper_headers_chap return chap + def cut_words(word_to_find, word_to_search_in): """ This function checks through the outputs of later sections and removes duplicates, so that the output file does @@ -219,10 +231,10 @@ def cut_words(word_to_find, word_to_search_in): b = word_to_search_in precheck = 1 if a in b: - if "\\paragraph{\\large\\bf KLSadd: " in b or "\\subsubsection*{\\large\\bf KLSadd: " in b: + if "\\paragraph{\\bf KLS Addendum: " in b or "\\subsubsection*{\\bf KLS Addendum: " in b: while True: - if "\\paragraph{\\large\\bf KLSadd: " in b[b.find(a)-precheck:b.find(a)] or \ - "\\subsubsection*{\\large\\bf KLSadd: " in b[b.find(a) - precheck:b.find(a)]: + if "\\paragraph{\\bf KLS Addendum: " in b[b.find(a)-precheck:b.find(a)] or \ + "\\subsubsection*{\\bf KLS Addendum: " in b[b.find(a) - precheck:b.find(a)]: return b[:b.find(a) - precheck] + b[b.find(a) + len(a):] else: precheck += 1 @@ -254,7 +266,7 @@ def prepare_for_pdf(chap): def get_commands(kls, new_commands): """ - This method reads in relevant commands that are in KLSadd.tex and returns them as a list + This method reads in relevant commands that are in KLS Addendum.tex and returns them as a list :param kls: The addendum is the source of what is being found :return: @@ -283,7 +295,7 @@ def insert_commands(kls, chap, cms): """ # reads in the newCommands[] and puts them in chap begin_index = -1 # the index of the "begin document" keyphrase, this is where the new commands need to be inserted. - # find index of begin document in KLSadd + # find index of begin document in KLS Addendum index = 0 for word in kls: @@ -393,17 +405,17 @@ def reference_placer(chap, references, p, chapticker2): # Place before References paragraph word1 = str(p[count]) if designator in word1[word1.find("\\subsection*{") + 1: word1.find("}")]: - chap[i - 2] += "%Begin KLSadd additions" + chap[i - 2] += "%Begin KLS Addendum additions" chap[i - 2] += p[count] - chap[i - 2] += "%End of KLSadd additions" + chap[i - 2] += "%End of KLS Addendum additions" count += 1 else: while designator not in word1[word1.find("\\subsection*{") + 1: word1.find("}")]: word1 = str(p[count]) if designator in word1[word1.find("\\subsection*{") + 1: word1.find("}")]: - chap[i - 2] += "%Begin KLSadd additions" + chap[i - 2] += "%Begin KLS Addendum additions" chap[i - 2] += p[count] - chap[i - 2] += "%End of KLSadd additions" + chap[i - 2] += "%End of KLS Addendum additions" count += 1 else: count += 1 @@ -431,7 +443,6 @@ def fix_chapter(chap, references, paragraphs_to_be_added, kls, kls_list_all, cha for i in kls_list_all: fix_chapter_sort(kls, chap, i, sort_location, klsaddparas, sortmatch_2, tempref, sorter_check) sort_location += 1 - print sorter_check for a in range(len(paragraphs_to_be_added)-1): for b in sortmatch_2: @@ -445,6 +456,11 @@ def fix_chapter(chap, references, paragraphs_to_be_added, kls, kls_list_all, cha #print c elif "%" == c[-1]: c = c[:-2] + # if "Formula (9.8.15) was first obtained by Brafman" in c and "Formula (9.8.15) was first obtained by Brafman"in paragraphs_to_be_added[a]: + # print c + # print "DING" + # print paragraphs_to_be_added[a] + paragraphs_to_be_added[a] = cut_words(c, paragraphs_to_be_added[a]) reference_placer(chap, references, paragraphs_to_be_added, chapticker2) @@ -478,6 +494,7 @@ def fix_chapter(chap, references, paragraphs_to_be_added, kls, kls_list_all, cha if '\\myciteKLS' in chap[ticker1]: chap[ticker1] = chap[ticker1].replace('\\myciteKLS', '\\cite') ticker1 += 1 + print chap return chap @@ -509,16 +526,16 @@ def main(): sorter_check = new_keywords(addendum, kls_list_full)[1] for item in addendum: - addendum[addendum.index(item)] = item.replace("\\eqref{","\\eqref{KLSadd: ") + addendum[addendum.index(item)] = item.replace("\\eqref{","\\eqref{KLS Addendum: ") for word in addendum: if "paragraph{" in word: lenword = len(word) - 1 - temp = word[0:word.find("{") + 1] + "\large\\bf KLSadd: " + word[word.find("{") + 1: lenword] + temp = word[0:word.find("{") + 1] + "\\bf KLS Addendum: " + word[word.find("{") + 1: lenword] addendum[addendum.index(word)] = temp if "subsubsection*{" in word: lenword = len(word) - 1 - addendum[addendum.index(word)] = word[0:word.find("{") + 1] + "\large\\bf KLSadd: " + \ + addendum[addendum.index(word)] = word[0:word.find("{") + 1] + "\\bf KLS Addendum: " + \ word[word.find("{") + 1: lenword] index = 0 indexes = [] @@ -596,14 +613,10 @@ def main(): # call the fixChapter method to get a list with the addendum paragraphs added in chapticker2 = 0 str9 = ''.join(fix_chapter(entire9, references9, paras, addendum, kls_list_all, chapticker2, new_commands, klsaddparas, sortmatch_2, ref9_3, sorter_check)) - print sorter_check - print "DING" chapticker2 += 1 str14 = ''.join(fix_chapter(entire14, references14, paras, addendum, kls_list_all, chapticker2, new_commands, klsaddparas, sortmatch_2, ref14_3, sorter_check)) - print "DING" - # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ # If you are writing something that will make a change to the chapter files, write it BEFORE this line, this part # is where the lists representing the words/strings in the chapter are joined together and updated as a string! diff --git a/KLSadd_insertion/test/test_chap1_placer.py b/KLSadd_insertion/test/test_chap1_placer.py index 0cde1ab..121e02e 100644 --- a/KLSadd_insertion/test/test_chap1_placer.py +++ b/KLSadd_insertion/test/test_chap1_placer.py @@ -12,5 +12,5 @@ def test_chap1_placer(self): self.assertEquals(chap1placer(['WordsAndSomeMoreWords','SampleEquation', '\\end{document}'] , ['\\subsection*{Generalities}', '\\paragraph{MathFunction}', 'MathEquations', 'WordsAndStuff', '\\subsection*{9.1 Wilson}', '\\paragraph{Symmetry}', 'WordsAndStuff'], [0,1,4,5]) - , 'WordsAndSomeMoreWordsSampleEquation\\paragraph{\\large\\bf KLS Addendum: Generalities}\\paragraph{MathFunction}MathEquationsWordsAndStuff\\end{document}') + , 'WordsAndSomeMoreWordsSampleEquation\\paragraph{\\bf KLS Addendum: Generalities}\\paragraph{MathFunction}MathEquationsWordsAndStuff\\end{document}') diff --git a/KLSadd_insertion/test/test_fix_chapter.py b/KLSadd_insertion/test/test_fix_chapter.py new file mode 100644 index 0000000..8a71fd4 --- /dev/null +++ b/KLSadd_insertion/test/test_fix_chapter.py @@ -0,0 +1,162 @@ + +__author__ = 'Edward Bian' +__status__ = 'Development' + +from unittest import TestCase +from updateChapters import fix_chapter_sort + + +class TestFixChapterSort(TestCase): + def test_fix_chapter_sort(self): + self.assertEquals(fix_chapter_sort(['%' + , '\\subsection*{9.1 Wilson}' + , '\\paragraph{Hypergeometric Representation}' + , 'The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetric' + , 'in $a,b,c,d$.' + , '\\renewcommand{\\refname}{Standard references}'] + ,['\\section{Wilson}\\index{Wilson polynomials}' + ,'\\subsection*{Hypergeometric representation}' + ,'\\begin{eqnarray}' + ,'& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' + ,'\\end{eqnarray}' + ,'\\subsection*{Orthogonality relation}' + ,'\\begin{eqnarray}' + ,'\\label{OrthIWilson}' + ,'& &\\frac{1}{2\\pi}\\int_0^{\\infty}' + ,'\\end{eqnarray}'],'Hypergeometric Representation',0,[1,2,5],[],[0, 1, 5, 9],[[0, 0], [0, 0]]) + ,['\\section{Wilson}\\index{Wilson polynomials}' + , '\\subsection*{Hypergeometric representation}' + , '\\begin{eqnarray}' + , '& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' + , '\\end{eqnarray}\\paragraph{\\bf KLS Addendum: Hypergeometric Representation}The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetricin $a,b,c,d$.' + , '\\subsection*{Orthogonality relation}' + , '\\begin{eqnarray}' + , '\\label{OrthIWilson}' + , '& &\\frac{1}{2\\pi}\\int_0^{\\infty}' + , '\\end{eqnarray}']) + self.assertEquals(fix_chapter_sort(['%' + , '\\subsection*{9.1 Wilson}' + , '\\subsection*{14.1 Askey}' + , '\\paragraph{Orthogonality}' + , 'The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetric' + , 'in $a,b,c,d$.' + , '\\renewcommand{\\refname}{Standard references}'] + , ['\\section{Askey}\\index{Wilson polynomials}' + , '\\subsection*{Hypergeometric representation}' + , '\\begin{eqnarray}' + , '& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' + , '\\end{eqnarray}' + , '\\subsection*{Orthogonality}' + , '\\begin{eqnarray}' + , '\\label{OrthIWilson}' + , '& &\\frac{1}{2\\pi}\\int_0^{\\infty}' + , '\\end{eqnarray}'], 'Orthogonality', 0, [1, 2, 3, 6], [], [0, 1, 5, 9],[[1, 0], [1, 0]]) + , ['\\section{Askey}\\index{Wilson polynomials}' + , '\\subsection*{Hypergeometric representation}' + , '\\begin{eqnarray}' + , '& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' + , '\\end{eqnarray}' + , '\\subsection*{Orthogonality}' + , '\\begin{eqnarray}' + , '\\label{OrthIWilson}' + , '& &\\frac{1}{2\\pi}\\int_0^{\\infty}\\paragraph{\\bf KLS Addendum: Orthogonality}The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetricin $a,b,c,d$.' + , '\\end{eqnarray}']) + self.assertEquals(fix_chapter_sort(['%' + , '\\subsection*{9.1 Wilson}' + , '\\paragraph{Special Value}' + , 'The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetric' + , 'in $a,b,c,d$.' + , '\\renewcommand{\\refname}{Standard references}'] + , ['\\section{Wilson}\\index{Wilson polynomials}' + , '\\subsection*{Hypergeometric representation}' + , '\\begin{eqnarray}' + , '& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' + , '\\end{eqnarray}' + , '\\subsection{Hypergeometric representation}' + , '\\begin{eqnarray}' + , '\\label{OrthIWilson}' + , '& &\\frac{1}{2\\pi}\\int_0^{\\infty}' + , '\\end{eqnarray}'], 'Special value', 0, [1, 2, 5], [], [0, 1, 5, 9], [[0, 0], [0, 0]]) + , ['\\section{Wilson}\\index{Wilson polynomials}' + , '\\subsection*{Hypergeometric representation}' + , '\\begin{eqnarray}' + , '& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' + , '\\end{eqnarray}\\paragraph{\\bf KLS Addendum: Special value}The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetricin $a,b,c,d$.' + , '\\subsection{Hypergeometric representation}' + , '\\begin{eqnarray}' + , '\\label{OrthIWilson}' + , '& &\\frac{1}{2\\pi}\\int_0^{\\infty}' + , '\\end{eqnarray}']) + self.assertEquals(fix_chapter_sort(['%' + , '\\subsection*{9.1 Wilson)' + , '\\subsection*{9.2 Pseudo Jacobi (or Routh-Romanovski)}' + , '\\paragraph{Symmetry}' + , 'The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetric' + , 'in $a,b,c,d$.' + , '\\renewcommand{\\refname}{Standard references}'] + , ['\\section{Pseudo Jacobi}\\index{Wilson polynomials}' + , '\\subsection*{Hypergeometric representation}' + , '\\begin{eqnarray}' + , '& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' + , '\\end{eqnarray}' + , '\\subsection*{Orthogonality}' + , '\\begin{eqnarray}' + , '\\label{OrthIWilson}' + , '& &\\frac{1}{2\\pi}\\int_0^{\\infty}' + , '\\end{eqnarray}'], 'Symmetry', 0, [1, 2, 3, 6], [], [0, 1, 5, 9], [[0, 0], [0, 0]]) + , ['\\section{Pseudo Jacobi}\\index{Wilson polynomials}' + , '\\subsection*{Hypergeometric representation}' + , '\\begin{eqnarray}' + , '& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' + , '\\end{eqnarray}\\paragraph{\\bf KLS Addendum: Symmetry}The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetricin $a,b,c,d$.' + , '\\subsection*{Orthogonality}' + , '\\begin{eqnarray}' + , '\\label{OrthIWilson}' + , '& &\\frac{1}{2\\pi}\\int_0^{\\infty}' + , '\\end{eqnarray}']) + self.assertEquals(fix_chapter_sort(['%' + , '\\subsection*{9.1 Wilson}' + , '\\paragraph{Hypergeometric Representation}' + , 'The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetric' + , 'in $a,b,c,d$.' + , '\\renewcommand{\\refname}{Standard references}'] + , ['\\section{Wilson}\\index{Wilson polynomials}' + , '\\subsection*{Hypergeometric representation}' + , '\\begin{eqnarray}' + , '& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' + , '\\end{eqnarray}' + , '\\subsection*{Basic Hypergeometric representation}' + , '\\begin{eqnarray}' + , '\\label{OrthIWilson}' + , '& &\\frac{1}{2\\pi}\\int_0^{\\infty}' + , '\\end{eqnarray}'], 'Hypergeometric Representation', 0, [1, 2, 5], [], [0, 1, 5, 9], [[0, 0], [0, 0]]) + , ['\\section{Wilson}\\index{Wilson polynomials}' + , '\\subsection*{Hypergeometric representation}' + , '\\begin{eqnarray}' + , '& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' + , '\\end{eqnarray}\\paragraph{\\bf KLS Addendum: Hypergeometric Representation}The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetricin $a,b,c,d$.' + , '\\subsection*{Basic Hypergeometric representation}' + , '\\begin{eqnarray}' + , '\\label{OrthIWilson}' + , '& &\\frac{1}{2\\pi}\\int_0^{\\infty}' + , '\\end{eqnarray}']) + self.assertEquals(fix_chapter_sort(['%' + , '\\subsection*{9.1 Wilson}' + , '\\paragraph{Symmetry}' + , 'The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetric' + ,'\\paragraph{Trivial Symmetry}' + , 'Words' + , '\\renewcommand{\\refname}{Standard references}'] + , ['\\section{Wilson}\\index{Wilson polynomials}' + , '\\subsection*{Hypergeometric representation}' + , '\\begin{eqnarray}' + , '& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' + , '\\end{eqnarray}' + ,'%'] + , 'Symmetry', 0, [1, 2, 4, 6], [], [0, 1, 5], [[0, 0], [0, 0]]) + , ['\\section{Wilson}\\index{Wilson polynomials}' + , '\\subsection*{Hypergeometric representation}' + , '\\begin{eqnarray}' + , '& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' + , '\\end{eqnarray}\\paragraph{\\bf KLS Addendum: Symmetry}The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetric\\paragraph{\\bf KLS Addendum: Symmetry}\nWords' + , '%']) \ No newline at end of file diff --git a/KLSadd_insertion/test/test_fix_chapter_sort.py b/KLSadd_insertion/test/test_fix_chapter_sort.py index 8a71fd4..8acda86 100644 --- a/KLSadd_insertion/test/test_fix_chapter_sort.py +++ b/KLSadd_insertion/test/test_fix_chapter_sort.py @@ -158,5 +158,5 @@ def test_fix_chapter_sort(self): , '\\subsection*{Hypergeometric representation}' , '\\begin{eqnarray}' , '& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' - , '\\end{eqnarray}\\paragraph{\\bf KLS Addendum: Symmetry}The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetric\\paragraph{\\bf KLS Addendum: Symmetry}\nWords' + , '\\end{eqnarray}\\paragraph{\\bf KLS Addendum: Symmetry}The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetric\\paragraph{\\bf KLS Addendum: Symmetry}Words' , '%']) \ No newline at end of file diff --git a/KLSadd_insertion/test/test_generic_functions.py b/KLSadd_insertion/test/test_generic_functions.py index 5ff7c81..665dfdc 100644 --- a/KLSadd_insertion/test/test_generic_functions.py +++ b/KLSadd_insertion/test/test_generic_functions.py @@ -26,7 +26,7 @@ class TestCutWords(TestCase): def test_cut_words(self): self.assertEquals(cut_words('MoreWords','WordsAndSomeMoreWords'), 'WordsAndSome') self.assertEquals(cut_words('SupposedToFail', 'WordsAndSomeMoreWords'), 'WordsAndSomeMoreWords') - self.assertEquals(cut_words('StuffFromAddenum', '''\\paragraph{\\large\\bf KLSadd: TestCase}StuffFromAddenum\\begin{equation}\\end{equation}'''), + self.assertEquals(cut_words('StuffFromAddenum', '''StuffFromAddenum\\begin{equation}\\end{equation}'''), '''\\begin{equation}\\end{equation}''') diff --git a/KLSadd_insertion/test/test_reference_placer.py b/KLSadd_insertion/test/test_reference_placer.py index dd92a6d..f17989c 100644 --- a/KLSadd_insertion/test/test_reference_placer.py +++ b/KLSadd_insertion/test/test_reference_placer.py @@ -20,7 +20,7 @@ def test_reference_placer(self): , '\\begin{eqnarray}' , '\\label{DefRacah}', '& &R_n(\lambda(x);\alpha,\beta,\gamma,\delta)\nonumber\\' , '& &{}=\hyp{4}{3}{-n,n+\alpha+\beta+1,-x,', '\end{eqnarray}' - , 'with $N$ a nonnegative integer.%Begin KLSadd additions\\subsection*{9.1 Racah}\\label{sec9.1}\\paragraph{\\large\\bf KLSadd: Hypergeometric representation}In addition to (9.1.1) we have%End of KLSadd additions', '\\subsection*{References}', + , 'with $N$ a nonnegative integer.%Begin KLS Addendum additions\\subsection*{9.1 Racah}\\label{sec9.1}\\paragraph{\\large\\bf KLSadd: Hypergeometric representation}In addition to (9.1.1) we have%End of KLS Addendum additions', '\\subsection*{References}', '\cite{Askey89I}, \cite{AskeyWilson79}, \cite{AskeyWilson85}']) self.assertEquals(reference_placer(['\\section{Racah}\\index{Racah polynomials}', @@ -34,7 +34,7 @@ def test_reference_placer(self): , '\\begin{eqnarray}' , '\\label{DefRacah}', '& &R_n(\lambda(x);\alpha,\beta,\gamma,\delta)\nonumber\\' , '& &{}=\hyp{4}{3}{-n,n+\alpha+\beta+1,-x,', '\end{eqnarray}' - , 'with $N$ a nonnegative integer.%Begin KLSadd additions\\subsection*{14.1 Racah}\\label{sec9.1}\\paragraph{\\large\\bf KLSadd: Hypergeometric representation}In addition to (9.1.1) we have%End of KLSadd additions', '\\subsection*{References}', + , 'with $N$ a nonnegative integer.%Begin KLS Addendum additions\\subsection*{14.1 Racah}\\label{sec9.1}\\paragraph{\\large\\bf KLSadd: Hypergeometric representation}In addition to (9.1.1) we have%End of KLS Addendum additions', '\\subsection*{References}', '\cite{Askey89I}, \cite{AskeyWilson79}, \cite{AskeyWilson85}']) self.assertEquals(reference_placer(['\\section{Racah}\\index{Racah polynomials}', @@ -48,5 +48,5 @@ def test_reference_placer(self): , '\\begin{eqnarray}' , '\\label{DefRacah}', '& &R_n(\lambda(x);\alpha,\beta,\gamma,\delta)\nonumber\\' , '& &{}=\hyp{4}{3}{-n,n+\alpha+\beta+1,-x,', '\end{eqnarray}' - , 'with $N$ a nonnegative integer.%Begin KLSadd additions\\subsection*{9.1 Racah}\\label{sec9.1}\\paragraph{\\large\\bf KLSadd: Hypergeometric representation}In addition to (9.1.1) we have%End of KLSadd additions', '\\subsection*{References}', + , 'with $N$ a nonnegative integer.%Begin KLS Addendum additions\\subsection*{9.1 Racah}\\label{sec9.1}\\paragraph{\\large\\bf KLSadd: Hypergeometric representation}In addition to (9.1.1) we have%End of KLS Addendum additions', '\\subsection*{References}', '\cite{Askey89I}, \cite{AskeyWilson79}, \cite{AskeyWilson85}']) \ No newline at end of file From 95ec36204f1d81114e0d160d319f4e82bf18d86d Mon Sep 17 00:00:00 2001 From: Edward Bian Date: Fri, 9 Sep 2016 16:55:33 -0400 Subject: [PATCH 359/402] Unit test fixes --- KLSadd_insertion/test/test_fix_chapter.py | 170 ++-------------------- 1 file changed, 15 insertions(+), 155 deletions(-) diff --git a/KLSadd_insertion/test/test_fix_chapter.py b/KLSadd_insertion/test/test_fix_chapter.py index 8a71fd4..8b4635e 100644 --- a/KLSadd_insertion/test/test_fix_chapter.py +++ b/KLSadd_insertion/test/test_fix_chapter.py @@ -3,160 +3,20 @@ __status__ = 'Development' from unittest import TestCase -from updateChapters import fix_chapter_sort +from updateChapters import fix_chapter -class TestFixChapterSort(TestCase): - def test_fix_chapter_sort(self): - self.assertEquals(fix_chapter_sort(['%' - , '\\subsection*{9.1 Wilson}' - , '\\paragraph{Hypergeometric Representation}' - , 'The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetric' - , 'in $a,b,c,d$.' - , '\\renewcommand{\\refname}{Standard references}'] - ,['\\section{Wilson}\\index{Wilson polynomials}' - ,'\\subsection*{Hypergeometric representation}' - ,'\\begin{eqnarray}' - ,'& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' - ,'\\end{eqnarray}' - ,'\\subsection*{Orthogonality relation}' - ,'\\begin{eqnarray}' - ,'\\label{OrthIWilson}' - ,'& &\\frac{1}{2\\pi}\\int_0^{\\infty}' - ,'\\end{eqnarray}'],'Hypergeometric Representation',0,[1,2,5],[],[0, 1, 5, 9],[[0, 0], [0, 0]]) - ,['\\section{Wilson}\\index{Wilson polynomials}' - , '\\subsection*{Hypergeometric representation}' - , '\\begin{eqnarray}' - , '& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' - , '\\end{eqnarray}\\paragraph{\\bf KLS Addendum: Hypergeometric Representation}The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetricin $a,b,c,d$.' - , '\\subsection*{Orthogonality relation}' - , '\\begin{eqnarray}' - , '\\label{OrthIWilson}' - , '& &\\frac{1}{2\\pi}\\int_0^{\\infty}' - , '\\end{eqnarray}']) - self.assertEquals(fix_chapter_sort(['%' - , '\\subsection*{9.1 Wilson}' - , '\\subsection*{14.1 Askey}' - , '\\paragraph{Orthogonality}' - , 'The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetric' - , 'in $a,b,c,d$.' - , '\\renewcommand{\\refname}{Standard references}'] - , ['\\section{Askey}\\index{Wilson polynomials}' - , '\\subsection*{Hypergeometric representation}' - , '\\begin{eqnarray}' - , '& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' - , '\\end{eqnarray}' - , '\\subsection*{Orthogonality}' - , '\\begin{eqnarray}' - , '\\label{OrthIWilson}' - , '& &\\frac{1}{2\\pi}\\int_0^{\\infty}' - , '\\end{eqnarray}'], 'Orthogonality', 0, [1, 2, 3, 6], [], [0, 1, 5, 9],[[1, 0], [1, 0]]) - , ['\\section{Askey}\\index{Wilson polynomials}' - , '\\subsection*{Hypergeometric representation}' - , '\\begin{eqnarray}' - , '& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' - , '\\end{eqnarray}' - , '\\subsection*{Orthogonality}' - , '\\begin{eqnarray}' - , '\\label{OrthIWilson}' - , '& &\\frac{1}{2\\pi}\\int_0^{\\infty}\\paragraph{\\bf KLS Addendum: Orthogonality}The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetricin $a,b,c,d$.' - , '\\end{eqnarray}']) - self.assertEquals(fix_chapter_sort(['%' - , '\\subsection*{9.1 Wilson}' - , '\\paragraph{Special Value}' - , 'The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetric' - , 'in $a,b,c,d$.' - , '\\renewcommand{\\refname}{Standard references}'] - , ['\\section{Wilson}\\index{Wilson polynomials}' - , '\\subsection*{Hypergeometric representation}' - , '\\begin{eqnarray}' - , '& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' - , '\\end{eqnarray}' - , '\\subsection{Hypergeometric representation}' - , '\\begin{eqnarray}' - , '\\label{OrthIWilson}' - , '& &\\frac{1}{2\\pi}\\int_0^{\\infty}' - , '\\end{eqnarray}'], 'Special value', 0, [1, 2, 5], [], [0, 1, 5, 9], [[0, 0], [0, 0]]) - , ['\\section{Wilson}\\index{Wilson polynomials}' - , '\\subsection*{Hypergeometric representation}' - , '\\begin{eqnarray}' - , '& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' - , '\\end{eqnarray}\\paragraph{\\bf KLS Addendum: Special value}The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetricin $a,b,c,d$.' - , '\\subsection{Hypergeometric representation}' - , '\\begin{eqnarray}' - , '\\label{OrthIWilson}' - , '& &\\frac{1}{2\\pi}\\int_0^{\\infty}' - , '\\end{eqnarray}']) - self.assertEquals(fix_chapter_sort(['%' - , '\\subsection*{9.1 Wilson)' - , '\\subsection*{9.2 Pseudo Jacobi (or Routh-Romanovski)}' - , '\\paragraph{Symmetry}' - , 'The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetric' - , 'in $a,b,c,d$.' - , '\\renewcommand{\\refname}{Standard references}'] - , ['\\section{Pseudo Jacobi}\\index{Wilson polynomials}' - , '\\subsection*{Hypergeometric representation}' - , '\\begin{eqnarray}' - , '& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' - , '\\end{eqnarray}' - , '\\subsection*{Orthogonality}' - , '\\begin{eqnarray}' - , '\\label{OrthIWilson}' - , '& &\\frac{1}{2\\pi}\\int_0^{\\infty}' - , '\\end{eqnarray}'], 'Symmetry', 0, [1, 2, 3, 6], [], [0, 1, 5, 9], [[0, 0], [0, 0]]) - , ['\\section{Pseudo Jacobi}\\index{Wilson polynomials}' - , '\\subsection*{Hypergeometric representation}' - , '\\begin{eqnarray}' - , '& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' - , '\\end{eqnarray}\\paragraph{\\bf KLS Addendum: Symmetry}The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetricin $a,b,c,d$.' - , '\\subsection*{Orthogonality}' - , '\\begin{eqnarray}' - , '\\label{OrthIWilson}' - , '& &\\frac{1}{2\\pi}\\int_0^{\\infty}' - , '\\end{eqnarray}']) - self.assertEquals(fix_chapter_sort(['%' - , '\\subsection*{9.1 Wilson}' - , '\\paragraph{Hypergeometric Representation}' - , 'The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetric' - , 'in $a,b,c,d$.' - , '\\renewcommand{\\refname}{Standard references}'] - , ['\\section{Wilson}\\index{Wilson polynomials}' - , '\\subsection*{Hypergeometric representation}' - , '\\begin{eqnarray}' - , '& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' - , '\\end{eqnarray}' - , '\\subsection*{Basic Hypergeometric representation}' - , '\\begin{eqnarray}' - , '\\label{OrthIWilson}' - , '& &\\frac{1}{2\\pi}\\int_0^{\\infty}' - , '\\end{eqnarray}'], 'Hypergeometric Representation', 0, [1, 2, 5], [], [0, 1, 5, 9], [[0, 0], [0, 0]]) - , ['\\section{Wilson}\\index{Wilson polynomials}' - , '\\subsection*{Hypergeometric representation}' - , '\\begin{eqnarray}' - , '& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' - , '\\end{eqnarray}\\paragraph{\\bf KLS Addendum: Hypergeometric Representation}The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetricin $a,b,c,d$.' - , '\\subsection*{Basic Hypergeometric representation}' - , '\\begin{eqnarray}' - , '\\label{OrthIWilson}' - , '& &\\frac{1}{2\\pi}\\int_0^{\\infty}' - , '\\end{eqnarray}']) - self.assertEquals(fix_chapter_sort(['%' - , '\\subsection*{9.1 Wilson}' - , '\\paragraph{Symmetry}' - , 'The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetric' - ,'\\paragraph{Trivial Symmetry}' - , 'Words' - , '\\renewcommand{\\refname}{Standard references}'] - , ['\\section{Wilson}\\index{Wilson polynomials}' - , '\\subsection*{Hypergeometric representation}' - , '\\begin{eqnarray}' - , '& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' - , '\\end{eqnarray}' - ,'%'] - , 'Symmetry', 0, [1, 2, 4, 6], [], [0, 1, 5], [[0, 0], [0, 0]]) - , ['\\section{Wilson}\\index{Wilson polynomials}' - , '\\subsection*{Hypergeometric representation}' - , '\\begin{eqnarray}' - , '& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' - , '\\end{eqnarray}\\paragraph{\\bf KLS Addendum: Symmetry}The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetric\\paragraph{\\bf KLS Addendum: Symmetry}\nWords' - , '%']) \ No newline at end of file +# class TestFixChapter(TestCase): +# def test_fix_chapter(self): +# self.assertEquals(fix_chapter(['\\newcommand{\qhypK}[5]{\,\mbox{}_{#1}\phi_{#2}\!\left(' +# , '\\newcommand\\half{\\frac12}' +# , '\\newcommand{\\hyp}[5]{\\,\\mbox{}_{#1}F_{#2}\\!\\left(' +# , '\\genfrac{}{}{0pt}{}{#3}{#4};#5\\right)}' +# , '\\newcommand{\\qhyp}[5]{\\,\\mbox{}_{#1}\\phi_{#2}\\!\\left(' +# , '\\subsection*{9.1 Wilson}' +# , '\\paragraph{Hypergeometric Representation}' +# , 'The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetric' +# ,'\\paragraph{Something}' +# , 'Words' +# , '\\renewcommand{\\refname}{Standard references}' +# , '\\myciteKLS{SampleReference}' \ No newline at end of file From e982d172f1e4450d57652b01ef6bdf6e7eafbeff Mon Sep 17 00:00:00 2001 From: Edward Bian Date: Tue, 11 Oct 2016 15:48:56 -0400 Subject: [PATCH 360/402] Main unit test --- KLSadd_insertion/test/test_fix_chapter.py | 111 +++++++++++++++++++--- 1 file changed, 97 insertions(+), 14 deletions(-) diff --git a/KLSadd_insertion/test/test_fix_chapter.py b/KLSadd_insertion/test/test_fix_chapter.py index 8b4635e..9c1e7e1 100644 --- a/KLSadd_insertion/test/test_fix_chapter.py +++ b/KLSadd_insertion/test/test_fix_chapter.py @@ -6,17 +6,100 @@ from updateChapters import fix_chapter -# class TestFixChapter(TestCase): -# def test_fix_chapter(self): -# self.assertEquals(fix_chapter(['\\newcommand{\qhypK}[5]{\,\mbox{}_{#1}\phi_{#2}\!\left(' -# , '\\newcommand\\half{\\frac12}' -# , '\\newcommand{\\hyp}[5]{\\,\\mbox{}_{#1}F_{#2}\\!\\left(' -# , '\\genfrac{}{}{0pt}{}{#3}{#4};#5\\right)}' -# , '\\newcommand{\\qhyp}[5]{\\,\\mbox{}_{#1}\\phi_{#2}\\!\\left(' -# , '\\subsection*{9.1 Wilson}' -# , '\\paragraph{Hypergeometric Representation}' -# , 'The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetric' -# ,'\\paragraph{Something}' -# , 'Words' -# , '\\renewcommand{\\refname}{Standard references}' -# , '\\myciteKLS{SampleReference}' \ No newline at end of file +class TestFixChapter(TestCase): + def test_fix_chapter(self): + ''' + self.assertEquals(fix_chapter(['\section{Wilson}\index{Wilson polynomials}' + ,'\par' + ,'\subsection*{Hypergeometric representation}' + ,'\\begin{eqnarray}' + ,'\label{DefWilson}' + ,'& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' + ,'& &{}=\hyp{4}{3}{-n,n+a+b+c+d-1,a+ix,a-ix}{a+b,a+c,a+d}{1}.' + ,'\end{eqnarray}' + ,'\\newpage' + ,'\subsection*{Orthogonality relation}' + ,'If $Re(a,b,c,d)>0$ and non-real parameters occur in conjugate pairs, then' + ,'\\begin{eqnarray}' + ,'\label{OrthIWilson}' + ,'\end{eqnarray}' + ,'\subsection*{References}'], + [14],['\\subsection*{9.1 Wilson}\\n\\label{sec9.1}\\n%\\n\\n%\\n\\n%\\n\\n%\\n\\n'], + [#'\\newcommand{\hyp}[5]{\,\mbox{}_{#1}F_{#2}\!\left(' + #,' \genfrac{}{}{0pt}{}{#3}{#4};#5\\right)}' + #,'\\newcommand{\qhyp}[5]{\,\mbox{}_{#1}\phi_{#2}\!\left(,' + #,' \genfrac{}{}{0pt}{}{#3}{#4};#5\\right)}' + '%' + ,'\subsection*{9.1 Wilson}' + ,'\label{sec9.1}' + ,'%' + ,'\paragraph{Symmetry}' + ,'The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetric in $a,b,c,d$.' + ,'\\' + ,'This follows from the orthogonality relation (9.1.2)' + ,'together with the value of its coefficient of $y^n$ given in (9.1.5b).' + ,'Alternatively, combine (9.1.1) with \mycite{AAR}{Theorem 3.1.1}.\\' + ,'As a consequence, (9.1.12). Then the generating' + ,'functions (9.1.13), (9.1.14) will follow by symmetry in the parameters.' + ,'\\renewcommand{\\refname}{Standard references}'] + ,['Symmetry', 'Limit Relation', 'Special value'], 0, [],[1,4],[],[0,2,9,14] + , [[0,0],[0,0],[0,0],[0,0]]),[]) + ''' +#fix_chapter(chap, references , paragraphs_to_be_added , kls, kls_list_all (basically word), chapticker2, new_commands, klsaddparas, sortmatch_2, tempref, sorter_check): +#fix_chapter_sort(chap, [5] , [\\subsection*{9.1 Wilson}\n\\label{sec9.1}\n%\n, kls, word, sortloc(should be 0) , 0, , blank?, , klsaddparas, sortmatch_2, tempref, sorter_check): +# fix_chapter_sort( kls, chap, word, sortloc , klsaddparas, sortmatch_2, tempref, sorter_check): +# fix_chapter( chap, references, paragraphs_to_be_added, kls, kls_list_all, chapticker2, new_commands , klsaddparas, sortmatch_2, tempref, sorter_check): + self.assertEquals(fix_chapter(['\\newcommand{\qhypK}[5]{\,\mbox{}_{#1}\phi_{#2}\!\left(' + , '\\newcommand\\half{\\frac12}' + , '\\newcommand{\\hyp}[5]{\\,\\mbox{}_{#1}F_{#2}\\!\\left(' + , '\\genfrac{}{}{0pt}{}{#3}{#4};#5\\right)}' + , '\\newcommand{\\qhyp}[5]{\\,\\mbox{}_{#1}\\phi_{#2}\\!\\left(' + , '\\section{Wilson}\\index{Wilson polynomials}' + , '\\subsection*{Hypergeometric representation}' + , '\\begin{eqnarray}' + , '& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' + , '\\end{eqnarray}' + , '\\subsection*{Orthogonality relation}' + , '\\begin{eqnarray}' + , '\\label{OrthIWilson}' + , '& &\\frac{1}{2\\pi}\\int_0^{\\infty}' + , '\\end{eqnarray}'] + + + ,[5],['\\subsection*{9.1 Wilson}\\n\\label{sec9.1}\\n%\\n'] + + , ['\\newcommand\\smallskipamount' + , '\\newcommand\\mybibitem[1]' + ,'%' + , '\\subsection*{9.1 Wilson}' + , '\\paragraph{Hypergeometric Representation}' + , 'The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetric' + , 'in $a,b,c,d$.' + , '\\renewcommand{\\refname}{Standard references}'] + + + + , ['Hypergeometric Representation'] + , 0, [], [3, 4, 7], [], [5, 6, 11, 14], [[0, 0], [0, 0]]) + , ['\\newcommand\\mybibitem[1]' + , '\\usepackage[pdftex]{hyperref} \n\\usepackage {xparse} \n\\usepackage{cite} \n' + , '\\newcommand{\\qhypK}[5]{\\,\\mbox{}_{#1}\\phi_{#2}\\!\\left(' + , '\\newcommand\\half{\\frac12}' + , '%\\newcommand{\\hyp}[5]{\\,\\mbox{}_{#1}F_{#2}\\!\\left(' + , '%\\genfrac{}{}{0pt}{}{#3}{#4};#5\\right)}%Begin KLS Addendum additions\\subsection*{9.1 Wilson}\\n\\label{sec9.1}\\n%\\n%End of KLS Addendum additions' + , '%\\newcommand{\\qhyp}[5]{\\,\\mbox{}_{#1}\\phi_{#2}\\!\\left(' + , '\\section{Wilson}\\index{Wilson polynomials}' + , '\\subsection*{Hypergeometric representation}' + , '\\begin{eqnarray}' + , '& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' + , '\\end{eqnarray}' + , '\\subsection*{Orthogonality relation}\\paragraph{\\bf KLS Addendum: Hypergeometric Representation}The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetricin $a,b,c,d$.' + , '\\begin{eqnarray}' + , '\\label{OrthIWilson}' + , '& &\\frac{1}{2\\pi}\\int_0^{\\infty}' + , '\\newcommand\\smallskipamount' + , '\\end{eqnarray}']) + + + + From d4fa9cfae627fd303f84267ba6af9da4fd5c3b11 Mon Sep 17 00:00:00 2001 From: Moritz Schubotz Date: Thu, 13 Oct 2016 16:54:03 +0200 Subject: [PATCH 361/402] Update README.md Add link to CSFS project --- README.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/README.md b/README.md index 8b5f07f..b5b6afb 100644 --- a/README.md +++ b/README.md @@ -30,7 +30,7 @@ To achieve the desired result, one should execute the progams in the following o ## Maple CFSF Seeding Project -This Python code (developed by Joon Bang) translates Maple files from Annie Cuyt's continued fractions package for special functions (CFSF) to LaTeX. +This Python code (developed by Joon Bang) translates Maple files from Annie Cuyt's continued fractions package for special functions ([CFSF](http://www.swmath.org/software/13927)) to LaTeX. The project is located in the `maple2latex` folder. From 92702be77aa181db8122fb1102ca50bd17874159 Mon Sep 17 00:00:00 2001 From: Edward Bian Date: Fri, 14 Oct 2016 17:57:37 -0400 Subject: [PATCH 362/402] Filling in holes left by last commit --- KLSadd_insertion/test/test_fix_chapter.py | 109 ++++++++++-------- .../test/test_updateChapters_main.py | 105 +++++++++++++++++ 2 files changed, 166 insertions(+), 48 deletions(-) create mode 100644 KLSadd_insertion/test/test_updateChapters_main.py diff --git a/KLSadd_insertion/test/test_fix_chapter.py b/KLSadd_insertion/test/test_fix_chapter.py index 9c1e7e1..e50a4c4 100644 --- a/KLSadd_insertion/test/test_fix_chapter.py +++ b/KLSadd_insertion/test/test_fix_chapter.py @@ -8,49 +8,14 @@ class TestFixChapter(TestCase): def test_fix_chapter(self): - ''' - self.assertEquals(fix_chapter(['\section{Wilson}\index{Wilson polynomials}' - ,'\par' - ,'\subsection*{Hypergeometric representation}' - ,'\\begin{eqnarray}' - ,'\label{DefWilson}' - ,'& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' - ,'& &{}=\hyp{4}{3}{-n,n+a+b+c+d-1,a+ix,a-ix}{a+b,a+c,a+d}{1}.' - ,'\end{eqnarray}' - ,'\\newpage' - ,'\subsection*{Orthogonality relation}' - ,'If $Re(a,b,c,d)>0$ and non-real parameters occur in conjugate pairs, then' - ,'\\begin{eqnarray}' - ,'\label{OrthIWilson}' - ,'\end{eqnarray}' - ,'\subsection*{References}'], - [14],['\\subsection*{9.1 Wilson}\\n\\label{sec9.1}\\n%\\n\\n%\\n\\n%\\n\\n%\\n\\n'], - [#'\\newcommand{\hyp}[5]{\,\mbox{}_{#1}F_{#2}\!\left(' - #,' \genfrac{}{}{0pt}{}{#3}{#4};#5\\right)}' - #,'\\newcommand{\qhyp}[5]{\,\mbox{}_{#1}\phi_{#2}\!\left(,' - #,' \genfrac{}{}{0pt}{}{#3}{#4};#5\\right)}' - '%' - ,'\subsection*{9.1 Wilson}' - ,'\label{sec9.1}' - ,'%' - ,'\paragraph{Symmetry}' - ,'The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetric in $a,b,c,d$.' - ,'\\' - ,'This follows from the orthogonality relation (9.1.2)' - ,'together with the value of its coefficient of $y^n$ given in (9.1.5b).' - ,'Alternatively, combine (9.1.1) with \mycite{AAR}{Theorem 3.1.1}.\\' - ,'As a consequence, (9.1.12). Then the generating' - ,'functions (9.1.13), (9.1.14) will follow by symmetry in the parameters.' - ,'\\renewcommand{\\refname}{Standard references}'] - ,['Symmetry', 'Limit Relation', 'Special value'], 0, [],[1,4],[],[0,2,9,14] - , [[0,0],[0,0],[0,0],[0,0]]),[]) - ''' + #fix_chapter(chap, references , paragraphs_to_be_added , kls, kls_list_all (basically word), chapticker2, new_commands, klsaddparas, sortmatch_2, tempref, sorter_check): #fix_chapter_sort(chap, [5] , [\\subsection*{9.1 Wilson}\n\\label{sec9.1}\n%\n, kls, word, sortloc(should be 0) , 0, , blank?, , klsaddparas, sortmatch_2, tempref, sorter_check): # fix_chapter_sort( kls, chap, word, sortloc , klsaddparas, sortmatch_2, tempref, sorter_check): # fix_chapter( chap, references, paragraphs_to_be_added, kls, kls_list_all, chapticker2, new_commands , klsaddparas, sortmatch_2, tempref, sorter_check): self.assertEquals(fix_chapter(['\\newcommand{\qhypK}[5]{\,\mbox{}_{#1}\phi_{#2}\!\left(' - , '\\newcommand\\half{\\frac12}' + , '\\newcommand\half{\\frac50}' + , '\\newcommand\half{\\frac12}' , '\\newcommand{\\hyp}[5]{\\,\\mbox{}_{#1}F_{#2}\\!\\left(' , '\\genfrac{}{}{0pt}{}{#3}{#4};#5\\right)}' , '\\newcommand{\\qhyp}[5]{\\,\\mbox{}_{#1}\\phi_{#2}\\!\\left(' @@ -62,12 +27,58 @@ def test_fix_chapter(self): , '\\subsection*{Orthogonality relation}' , '\\begin{eqnarray}' , '\\label{OrthIWilson}' - , '& &\\frac{1}{2\\pi}\\int_0^{\\infty}' + , '\\myciteKLS' , '\\end{eqnarray}'] + ,[5],['\\subsection*{9.1 Wilson}\\n\\label{sec9.1}\\n%\\n'] + , ['\\newcommand\\smallskipamount' + , '\\newcommand\\mybibitem[1]' + ,'%' + , '\\subsection*{9.1 Wilson}' + , '\\paragraph{Hypergeometric Representation}' + , 'The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetric' + , 'in $a,b,c,d$.' + , '\\renewcommand{\\refname}{Standard references}'] + , ['Hypergeometric Representation'] + , 0, [], [3, 4, 7], [['%ab']], [6, 7, 12, 15], [[0, 0], [0, 0]]) + # Expected output: + , ['\\newcommand\\mybibitem[1]' + , '\\usepackage[pdftex]{hyperref} \n\\usepackage {xparse} \n\\usepackage{cite} \n' + , '\\newcommand{\\qhypK}[5]{\\,\\mbox{}_{#1}\\phi_{#2}\\!\\left(' + , '\\newcommand\half{\\frac50}' + , '%\\newcommand\\half{\\frac12}' + , '%\\newcommand{\\hyp}[5]{\\,\\mbox{}_{#1}F_{#2}\\!\\left(%Begin KLS Addendum additions\\subsection*{9.1 Wilson}\\n\\label{sec9.1}\\n%\\n%End of KLS Addendum additions' + , '%\\genfrac{}{}{0pt}{}{#3}{#4};#5\\right)}' + , '%\\newcommand{\\qhyp}[5]{\\,\\mbox{}_{#1}\\phi_{#2}\\!\\left(' + , '\\section{Wilson}\\index{Wilson polynomials}' + , '\\subsection*{Hypergeometric representation}' + , '\\begin{eqnarray}' + , '& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' + , '\\end{eqnarray}' + , '\\subsection*{Orthogonality relation}\\paragraph{\\bf KLS Addendum: Hypergeometric Representation}The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetricin $a,b,c,d$.' + , '\\begin{eqnarray}' + , '\\label{OrthIWilson}' + , '\\cite' + , '\\newcommand\\smallskipamount' + , '\\end{eqnarray}']) + self.assertEquals(fix_chapter(['\\newcommand{\qhypK}[5]{\,\mbox{}_{#1}\phi_{#2}\!\left(' + , '\\newcommand\half{\\frac50}' + , '\\newcommand\half{\\frac12}' + , '\\newcommand{\\hyp}[5]{\\,\\mbox{}_{#1}F_{#2}\\!\\left(' + , '\\genfrac{}{}{0pt}{}{#3}{#4};#5\\right)}' + , '\\newcommand{\\qhyp}[5]{\\,\\mbox{}_{#1}\\phi_{#2}\\!\\left(' + , '\\section{Wilson}\\index{Wilson polynomials}' + , '\\subsection*{Hypergeometric representation}' + , '\\begin{eqnarray}' + , '& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' + , '\\end{eqnarray}' + , '\\subsection*{Orthogonality relation}' + , '\\begin{eqnarray}' + , '\\label{OrthIWilson}' + , '\\myciteKLS' + , '\\end{eqnarray}'] ,[5],['\\subsection*{9.1 Wilson}\\n\\label{sec9.1}\\n%\\n'] - , ['\\newcommand\\smallskipamount' , '\\newcommand\\mybibitem[1]' ,'%' @@ -76,17 +87,18 @@ def test_fix_chapter(self): , 'The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetric' , 'in $a,b,c,d$.' , '\\renewcommand{\\refname}{Standard references}'] + , ['Hypergeometric Representation'] + , 0, [], [3, 4, 7], [['%a']], [6, 7, 12, 15], [[0, 0], [0, 0]]) + # Expected output: - - , ['Hypergeometric Representation'] - , 0, [], [3, 4, 7], [], [5, 6, 11, 14], [[0, 0], [0, 0]]) , ['\\newcommand\\mybibitem[1]' , '\\usepackage[pdftex]{hyperref} \n\\usepackage {xparse} \n\\usepackage{cite} \n' , '\\newcommand{\\qhypK}[5]{\\,\\mbox{}_{#1}\\phi_{#2}\\!\\left(' - , '\\newcommand\\half{\\frac12}' - , '%\\newcommand{\\hyp}[5]{\\,\\mbox{}_{#1}F_{#2}\\!\\left(' - , '%\\genfrac{}{}{0pt}{}{#3}{#4};#5\\right)}%Begin KLS Addendum additions\\subsection*{9.1 Wilson}\\n\\label{sec9.1}\\n%\\n%End of KLS Addendum additions' + , '\\newcommand\half{\\frac50}' + , '%\\newcommand\\half{\\frac12}' + , '%\\newcommand{\\hyp}[5]{\\,\\mbox{}_{#1}F_{#2}\\!\\left(%Begin KLS Addendum additions\\subsection*{9.1 Wilson}\\n\\label{sec9.1}\\n%\\n%End of KLS Addendum additions' + , '%\\genfrac{}{}{0pt}{}{#3}{#4};#5\\right)}' , '%\\newcommand{\\qhyp}[5]{\\,\\mbox{}_{#1}\\phi_{#2}\\!\\left(' , '\\section{Wilson}\\index{Wilson polynomials}' , '\\subsection*{Hypergeometric representation}' @@ -96,10 +108,11 @@ def test_fix_chapter(self): , '\\subsection*{Orthogonality relation}\\paragraph{\\bf KLS Addendum: Hypergeometric Representation}The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetricin $a,b,c,d$.' , '\\begin{eqnarray}' , '\\label{OrthIWilson}' - , '& &\\frac{1}{2\\pi}\\int_0^{\\infty}' + , '\\cite' , '\\newcommand\\smallskipamount' , '\\end{eqnarray}']) - +['\\newcommand\\mybibitem[1]', '\\usepackage[pdftex]{hyperref} \n\\usepackage {xparse} \n\\usepackage{cite} \n', '\\newcommand{\\qhypK}[5]{\\,\\mbox{}_{#1}\\phi_{#2}\\!\\left(', '\newcommand\\half{\x0crac12}', '%\\newcommand{\\hyp}[5]{\\,\\mbox{}_{#1}F_{#2}\\!\\left(', '%\\genfrac{}{}{0pt}{}{#3}{#4};#5\\right)}%Begin KLS Addendum additions\\subsection*{9.1 Wilson}\\n\\label{sec9.1}\\n%\\n%End of KLS Addendum additions', '%\\newcommand{\\qhyp}[5]{\\,\\mbox{}_{#1}\\phi_{#2}\\!\\left(', '\\section{Wilson}\\index{Wilson polynomials}', '\\subsection*{Hypergeometric representation}', '\\begin{eqnarray}', '& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\', '\\end{eqnarray}', '\\subsection*{Orthogonality relation}\\paragraph{\\bf KLS Addendum: Hypergeometric Representation}The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetricin $a,b,c,d$.', '\\begin{eqnarray}', '\\label{OrthIWilson}', '& &\\frac{1}{2\\pi}\\int_0^{\\infty}', '\\newcommand\\smallskipamount', '\\end{eqnarray}'] +['\\newcommand\\mybibitem[1]', '\\usepackage[pdftex]{hyperref} \n\\usepackage {xparse} \n\\usepackage{cite} \n', '\\newcommand{\\qhypK}[5]{\\,\\mbox{}_{#1}\\phi_{#2}\\!\\left(', '\\newcommand\\half{\\frac12}', '%\\newcommand{\\hyp}[5]{\\,\\mbox{}_{#1}F_{#2}\\!\\left(', '%\\genfrac{}{}{0pt}{}{#3}{#4};#5\\right)}%Begin KLS Addendum additions\\subsection*{9.1 Wilson}\\n\\label{sec9.1}\\n%\\n%End of KLS Addendum additions', '%\\newcommand{\\qhyp}[5]{\\,\\mbox{}_{#1}\\phi_{#2}\\!\\left(', '\\section{Wilson}\\index{Wilson polynomials}', '\\subsection*{Hypergeometric representation}', '\\begin{eqnarray}', '& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\', '\\end{eqnarray}', '\\subsection*{Orthogonality relation}\\paragraph{\\bf KLS Addendum: Hypergeometric Representation}The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetricin $a,b,c,d$.', '\\begin{eqnarray}', '\\label{OrthIWilson}', '& &\\frac{1}{2\\pi}\\int_0^{\\infty}', '\\newcommand\\smallskipamount', '\\end{eqnarray}'] diff --git a/KLSadd_insertion/test/test_updateChapters_main.py b/KLSadd_insertion/test/test_updateChapters_main.py new file mode 100644 index 0000000..9c1e7e1 --- /dev/null +++ b/KLSadd_insertion/test/test_updateChapters_main.py @@ -0,0 +1,105 @@ + +__author__ = 'Edward Bian' +__status__ = 'Development' + +from unittest import TestCase +from updateChapters import fix_chapter + + +class TestFixChapter(TestCase): + def test_fix_chapter(self): + ''' + self.assertEquals(fix_chapter(['\section{Wilson}\index{Wilson polynomials}' + ,'\par' + ,'\subsection*{Hypergeometric representation}' + ,'\\begin{eqnarray}' + ,'\label{DefWilson}' + ,'& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' + ,'& &{}=\hyp{4}{3}{-n,n+a+b+c+d-1,a+ix,a-ix}{a+b,a+c,a+d}{1}.' + ,'\end{eqnarray}' + ,'\\newpage' + ,'\subsection*{Orthogonality relation}' + ,'If $Re(a,b,c,d)>0$ and non-real parameters occur in conjugate pairs, then' + ,'\\begin{eqnarray}' + ,'\label{OrthIWilson}' + ,'\end{eqnarray}' + ,'\subsection*{References}'], + [14],['\\subsection*{9.1 Wilson}\\n\\label{sec9.1}\\n%\\n\\n%\\n\\n%\\n\\n%\\n\\n'], + [#'\\newcommand{\hyp}[5]{\,\mbox{}_{#1}F_{#2}\!\left(' + #,' \genfrac{}{}{0pt}{}{#3}{#4};#5\\right)}' + #,'\\newcommand{\qhyp}[5]{\,\mbox{}_{#1}\phi_{#2}\!\left(,' + #,' \genfrac{}{}{0pt}{}{#3}{#4};#5\\right)}' + '%' + ,'\subsection*{9.1 Wilson}' + ,'\label{sec9.1}' + ,'%' + ,'\paragraph{Symmetry}' + ,'The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetric in $a,b,c,d$.' + ,'\\' + ,'This follows from the orthogonality relation (9.1.2)' + ,'together with the value of its coefficient of $y^n$ given in (9.1.5b).' + ,'Alternatively, combine (9.1.1) with \mycite{AAR}{Theorem 3.1.1}.\\' + ,'As a consequence, (9.1.12). Then the generating' + ,'functions (9.1.13), (9.1.14) will follow by symmetry in the parameters.' + ,'\\renewcommand{\\refname}{Standard references}'] + ,['Symmetry', 'Limit Relation', 'Special value'], 0, [],[1,4],[],[0,2,9,14] + , [[0,0],[0,0],[0,0],[0,0]]),[]) + ''' +#fix_chapter(chap, references , paragraphs_to_be_added , kls, kls_list_all (basically word), chapticker2, new_commands, klsaddparas, sortmatch_2, tempref, sorter_check): +#fix_chapter_sort(chap, [5] , [\\subsection*{9.1 Wilson}\n\\label{sec9.1}\n%\n, kls, word, sortloc(should be 0) , 0, , blank?, , klsaddparas, sortmatch_2, tempref, sorter_check): +# fix_chapter_sort( kls, chap, word, sortloc , klsaddparas, sortmatch_2, tempref, sorter_check): +# fix_chapter( chap, references, paragraphs_to_be_added, kls, kls_list_all, chapticker2, new_commands , klsaddparas, sortmatch_2, tempref, sorter_check): + self.assertEquals(fix_chapter(['\\newcommand{\qhypK}[5]{\,\mbox{}_{#1}\phi_{#2}\!\left(' + , '\\newcommand\\half{\\frac12}' + , '\\newcommand{\\hyp}[5]{\\,\\mbox{}_{#1}F_{#2}\\!\\left(' + , '\\genfrac{}{}{0pt}{}{#3}{#4};#5\\right)}' + , '\\newcommand{\\qhyp}[5]{\\,\\mbox{}_{#1}\\phi_{#2}\\!\\left(' + , '\\section{Wilson}\\index{Wilson polynomials}' + , '\\subsection*{Hypergeometric representation}' + , '\\begin{eqnarray}' + , '& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' + , '\\end{eqnarray}' + , '\\subsection*{Orthogonality relation}' + , '\\begin{eqnarray}' + , '\\label{OrthIWilson}' + , '& &\\frac{1}{2\\pi}\\int_0^{\\infty}' + , '\\end{eqnarray}'] + + + ,[5],['\\subsection*{9.1 Wilson}\\n\\label{sec9.1}\\n%\\n'] + + , ['\\newcommand\\smallskipamount' + , '\\newcommand\\mybibitem[1]' + ,'%' + , '\\subsection*{9.1 Wilson}' + , '\\paragraph{Hypergeometric Representation}' + , 'The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetric' + , 'in $a,b,c,d$.' + , '\\renewcommand{\\refname}{Standard references}'] + + + + , ['Hypergeometric Representation'] + , 0, [], [3, 4, 7], [], [5, 6, 11, 14], [[0, 0], [0, 0]]) + , ['\\newcommand\\mybibitem[1]' + , '\\usepackage[pdftex]{hyperref} \n\\usepackage {xparse} \n\\usepackage{cite} \n' + , '\\newcommand{\\qhypK}[5]{\\,\\mbox{}_{#1}\\phi_{#2}\\!\\left(' + , '\\newcommand\\half{\\frac12}' + , '%\\newcommand{\\hyp}[5]{\\,\\mbox{}_{#1}F_{#2}\\!\\left(' + , '%\\genfrac{}{}{0pt}{}{#3}{#4};#5\\right)}%Begin KLS Addendum additions\\subsection*{9.1 Wilson}\\n\\label{sec9.1}\\n%\\n%End of KLS Addendum additions' + , '%\\newcommand{\\qhyp}[5]{\\,\\mbox{}_{#1}\\phi_{#2}\\!\\left(' + , '\\section{Wilson}\\index{Wilson polynomials}' + , '\\subsection*{Hypergeometric representation}' + , '\\begin{eqnarray}' + , '& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' + , '\\end{eqnarray}' + , '\\subsection*{Orthogonality relation}\\paragraph{\\bf KLS Addendum: Hypergeometric Representation}The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetricin $a,b,c,d$.' + , '\\begin{eqnarray}' + , '\\label{OrthIWilson}' + , '& &\\frac{1}{2\\pi}\\int_0^{\\infty}' + , '\\newcommand\\smallskipamount' + , '\\end{eqnarray}']) + + + + From abc0536444284c5f24c0d6cab6d7902888aed413 Mon Sep 17 00:00:00 2001 From: Edward Bian Date: Fri, 14 Oct 2016 18:01:26 -0400 Subject: [PATCH 363/402] Fixes that go with unit tests --- KLSadd_insertion/src/updateChapters.py | 13 ++++--------- 1 file changed, 4 insertions(+), 9 deletions(-) diff --git a/KLSadd_insertion/src/updateChapters.py b/KLSadd_insertion/src/updateChapters.py index d0e162c..7360301 100644 --- a/KLSadd_insertion/src/updateChapters.py +++ b/KLSadd_insertion/src/updateChapters.py @@ -208,11 +208,6 @@ def fix_chapter_sort(kls, chap, word, sortloc, klsaddparas, sortmatch_2, tempref # print hyper_headers_chap # print word sortmatch_2.append(k_hyper_sub_chap) - if "generating function" in name_chap: - print "Stuff for checking" - print k_hyper_sub_chap - print khyper_header_chap - print hyper_headers_chap return chap @@ -278,7 +273,6 @@ def get_commands(kls, new_commands): new_commands.append(index - 1) if "mybibitem[1]" in word: new_commands.append(index) - print new_commands return kls[new_commands[0]:new_commands[1]] @@ -466,17 +460,19 @@ def fix_chapter(chap, references, paragraphs_to_be_added, kls, kls_list_all, cha reference_placer(chap, references, paragraphs_to_be_added, chapticker2) chap = prepare_for_pdf(chap) cms = get_commands(kls,new_commands) - print cms chap = insert_commands(kls, chap, cms) commentticker = 0 # These commands mess up PDF reading and mus tbe commented out + print"OUTPUT" for word in chap: word2 = chap[chap.index(word)-1] + print word2 if "\\newcommand{\qhypK}[5]{\,\mbox{}_{#1}\phi_{#2}\!\left(" not in word2: if "\\newcommand\\half{\\frac12}" in word: wordtoadd = "%" + word chap[commentticker] = wordtoadd + print("TRIGGERED") elif "\\newcommand{\\hyp}[5]{\\,\\mbox{}_{#1}F_{#2}\\!\\left(" in word: wordtoadd = "%" + word chap[commentticker] = wordtoadd @@ -487,14 +483,13 @@ def fix_chapter(chap, references, paragraphs_to_be_added, kls, kls_list_all, cha wordtoadd = "%" + word chap[commentticker] = wordtoadd commentticker += 1 - + print"END OUTPUT" ticker1 = 0 # Formatting to make the Latex file run while ticker1 < len(chap): if '\\myciteKLS' in chap[ticker1]: chap[ticker1] = chap[ticker1].replace('\\myciteKLS', '\\cite') ticker1 += 1 - print chap return chap From 5ba19031dc71d85c8facdafb2ab23504b1fa7fa9 Mon Sep 17 00:00:00 2001 From: Joon Bang Date: Wed, 20 Jul 2016 15:37:40 -0400 Subject: [PATCH 364/402] Fix issues with file paths --- maple2latex/src/main.py | 16 +++++---- maple2latex/src/translator.py | 2 +- maple2latex/test/test_translator.py | 53 ----------------------------- 3 files changed, 10 insertions(+), 61 deletions(-) delete mode 100644 maple2latex/test/test_translator.py diff --git a/maple2latex/src/main.py b/maple2latex/src/main.py index cbaadf9..c01657d 100644 --- a/maple2latex/src/main.py +++ b/maple2latex/src/main.py @@ -5,10 +5,10 @@ import os import copy -from src.translator import MapleEquation, make_equation +from translator import MapleEquation, make_equation TABLE = dict() # stores meaning of folder names -with open("data/section_names") as name_info: +with open("maple2latex/data/section_names") as name_info: for line in name_info.read().split("\n"): if line != "" and "%" not in line: key, value = line.split(" : ") @@ -28,7 +28,7 @@ "qhyper" ] -ROOT = "functions" # root directory +ROOT = "maple2latex/functions" # root directory class MapleFile(object): @@ -53,8 +53,9 @@ def translate_file(filename): # type: (str) """Translates all the formulae in a file.""" - with open("out/test.tex", "w") as test, open("out/primer") as primer: - test.write(primer.read() + MapleFile(filename).convert_formulae() + "\n\\end{document}") + with open("maple2latex/out/test.tex", "w") as test: + with open("maple2latex/out/primer") as primer: + test.write(primer.read() + MapleFile(filename).convert_formulae() + "\n\\end{document}") def translate_directories(): @@ -80,8 +81,9 @@ def translate_directories(): MapleFile(info[0] + "/" + file_name).convert_formulae() + "\n\n" # write output to file - with open("out/test.tex", "w") as test, open("out/primer") as primer: - test.write(primer.read() + result + "\n\\end{document}") + with open("maple2latex/out/test.tex", "w") as test: + with open("maple2latex/out/primer") as primer: + test.write(primer.read() + result + "\n\\end{document}") if __name__ == '__main__': translate_directories() diff --git a/maple2latex/src/translator.py b/maple2latex/src/translator.py index b1e8138..fad4e41 100644 --- a/maple2latex/src/translator.py +++ b/maple2latex/src/translator.py @@ -5,7 +5,7 @@ import copy import json -from src.maple_tokenize import tokenize +from maple_tokenize import tokenize INFO = json.loads(open("maple2latex/data/keys.json").read()) diff --git a/maple2latex/test/test_translator.py b/maple2latex/test/test_translator.py deleted file mode 100644 index 6c2914d..0000000 --- a/maple2latex/test/test_translator.py +++ /dev/null @@ -1,53 +0,0 @@ -#!/usr/bin/env python - -__author__ = "Joon Bang" -__status__ = "Development" - -import json -from unittest import TestCase -import src.translator as t - -TEST_CASES = json.loads(open("maple2latex/test/test_cases.json").read())["translator"] - -parse_brackets_test_cases = TEST_CASES["parse_brackets"] -trim_parens_test_cases = TEST_CASES["trim_parens"] -basic_translate_test_cases = TEST_CASES["basic_translate"] -get_arguments_test_cases = TEST_CASES["get_arguments"] -translate_test_cases = TEST_CASES["translate"] -make_equation_test_cases = TEST_CASES["make_equation"] - - -class TestParseBrackets(TestCase): - def test_parse_brackets(self): - for case in parse_brackets_test_cases: - self.assertEqual(t.parse_brackets(case["exp"]), case["res"]) - - -class TestTrimParens(TestCase): - def test_trim_parens(self): - for case in trim_parens_test_cases: - self.assertEqual(t.trim_parens(case["exp"]), case["res"]) - - -class TestBasicTranslate(TestCase): - def test_basic_translate(self): - for case in basic_translate_test_cases: - self.assertEqual(t.basic_translate(case["exp"]), case["res"]) - - -class TestGetArguments(TestCase): - def test_get_arguments(self): - for case in get_arguments_test_cases: - self.assertEqual(t.get_arguments(case["function"], case["arg_string"]), case["res"]) - - -class TestTranslate(TestCase): - def test_translate(self): - for case in translate_test_cases: - self.assertEqual(t.translate(case["exp"]), case["res"]) - - -class TestMakeEquation(TestCase): - def test_make_equation(self): - for case in make_equation_test_cases: - self.assertEqual(t.make_equation(t.MapleEquation(case["eq"]), case["view_metadata"]), case["res"]) From 084627f918abf91d08f2b352a9faaeb87fa87314 Mon Sep 17 00:00:00 2001 From: Joon Bang Date: Thu, 21 Jul 2016 11:27:20 -0400 Subject: [PATCH 365/402] Add LatexEquation class to support sorting --- maple2latex/src/main.py | 68 +++++----- maple2latex/src/translator.py | 193 +++++++++++++++------------- maple2latex/test/test_cases.json | 8 +- maple2latex/test/test_translator.py | 53 ++++++++ 4 files changed, 197 insertions(+), 125 deletions(-) create mode 100644 maple2latex/test/test_translator.py diff --git a/maple2latex/src/main.py b/maple2latex/src/main.py index c01657d..b88765a 100644 --- a/maple2latex/src/main.py +++ b/maple2latex/src/main.py @@ -4,8 +4,7 @@ __status__ = "Development" import os -import copy -from translator import MapleEquation, make_equation +from translator import MapleEquation, LatexEquation TABLE = dict() # stores meaning of folder names with open("maple2latex/data/section_names") as name_info: @@ -14,7 +13,7 @@ key, value = line.split(" : ") TABLE[key] = value -FILES = [ # files to be translated +FILES = [ "bessel", "modbessel", "confluent", "confluentlimit", "kummer", "parabolic", "whittaker", "apery", "archimedes", "catalan", "delian", "eulersconstant", "eulersnumber", "goldenratio", "gompertz", @@ -26,9 +25,7 @@ "binet", "incompletegamma", "polygamma", "tetragamma", "trigamma", "hypergeometric", "qhyper" -] - -ROOT = "maple2latex/functions" # root directory +] # files to be translated class MapleFile(object): @@ -42,43 +39,56 @@ def obtain_formulae(self): return [MapleEquation(piece.split("\n")) for piece in contents.split("create(") if "):" in piece or ");" in piece] - def convert_formulae(self): - return '\n\n'.join([make_equation(copy.copy(formula)) for formula in self.formulae]) - def __str__(self): return "MapleFile of " + self.filename -def translate_file(filename): - # type: (str) - """Translates all the formulae in a file.""" - - with open("maple2latex/out/test.tex", "w") as test: - with open("maple2latex/out/primer") as primer: - test.write(primer.read() + MapleFile(filename).convert_formulae() + "\n\\end{document}") - - -def translate_directories(): +def translate_files(root_directory): # type: () - """Generates and writes the results of traversing the root directory and translating all Maple files.""" + """Generates and writes the results of traversing the root directory, and translating all files in FILES.""" - root_depth = len(ROOT.split("/")) + root_depth = len(root_directory.split("/")) - dirs = [d for d in os.walk(ROOT) if d[0].count("/") == root_depth] + dirs = [d for d in os.walk(root_directory) if d[0].count("/") == root_depth] - result = "" + sections = dict() for info in dirs: # search through directories - depth = len(["" for ch in info[0] if ch == "/"]) + depth = info[0].count("/") if depth == root_depth: # section - result += "\n\\section{" + TABLE[info[0].split("/")[-1]] + "}\n" + section = TABLE[info[0].split("/")[-1]] + sections[section] = dict() for file_name in info[2]: folder = ''.join(file_name.split(".")[:-1]) - if folder in FILES: # subsection - result += "\\subsection{" + TABLE[folder] + "}\n" + \ - MapleFile(info[0] + "/" + file_name).convert_formulae() + "\n\n" + if folder in FILES: + # generate equation code, sorted by equation number + subsection = TABLE[file_name[:-4]] + formulae = map(LatexEquation.from_maple, MapleFile(info[0] + "/" + file_name).formulae) + formulae = sorted(formulae, key=LatexEquation.get_sortable_label) + + repr_label = formulae[len(formulae) / 2].label + sections[section][subsection] = [repr_label, "\n".join(map(str, formulae))] + + # generate subsection headers + for section, subsections in sections.iteritems(): + print section + + keys = sorted(subsections.keys(), key=lambda x: subsections[x][0]) + + text = "" + for subsection in keys: + print " " + subsection + text += "\n\\subsection{" + subsection + "}\n" + text += subsections[subsection][1] + + sections[section] = text + + # generate section headers + result = "" + for section, section_data in sections.iteritems(): + result += "\\section{" + section + "}\n" + section_data + "\n\n\n" # write output to file with open("maple2latex/out/test.tex", "w") as test: @@ -86,4 +96,4 @@ def translate_directories(): test.write(primer.read() + result + "\n\\end{document}") if __name__ == '__main__': - translate_directories() + translate_files("maple2latex/functions") diff --git a/maple2latex/src/translator.py b/maple2latex/src/translator.py index fad4e41..e529f37 100644 --- a/maple2latex/src/translator.py +++ b/maple2latex/src/translator.py @@ -5,6 +5,7 @@ import copy import json +from string import ascii_lowercase from maple_tokenize import tokenize INFO = json.loads(open("maple2latex/data/keys.json").read()) @@ -13,7 +14,8 @@ SYMBOLS = INFO["symbols"] CONSTRAINTS = INFO["constraints"] -SPECIAL = {"(": "\\left(", ")": "\\right)", "+-": "-", "\\subplus-": "-", "^{1}": "", "\\inNot": "\\notin"} +SPECIAL = {"(": "\\left(", ")": "\\right)", "+-": "-", "\\subplus-": "-", "^{1}": "", "\\inNot": "\\notin", + "\\imaginarynumber": "i"} class MapleEquation(object): @@ -39,7 +41,7 @@ def __init__(self, inp): line[1] = str(temp)[1:-1] elif line[0] == "booklabelv1" and line[1] == '"",': - line[1] = "No label" + line[1] = "x.x.x" elif line[0] in ["booklabelv1", "booklabelv2", "general", "constraints", "begin", "parameters"]: line[1] = line[1].strip()[1:-2].strip() @@ -68,6 +70,100 @@ def __init__(self, inp): self.general = [self.fields["even"], self.fields["odd"]] +class LatexEquation(object): + def __init__(self, label, equation, metadata): + self.label = label + self.equation = equation + self.metadata = {"constraint": metadata[0], "category": metadata[1]} + + @classmethod + def from_maple(cls, eq): + # modify fields + eq.lhs = translate(eq.lhs) + eq.factor = translate(eq.fields["factor"]) + eq.front = translate(eq.fields["front"]) + + equation = eq.lhs + "\n = " + + if eq.factor == "1": + eq.factor = "" + + # translates the Maple information (with spacing) + if eq.eq_type == "series": + eq.general = translate(eq.general[0]) + + if eq.factor != "": + equation += eq.factor + " " + + elif eq.front != "": + equation += eq.front + "+" + + equation += "\\sum_{k=0}^\\infty " + + if eq.category == "power series": + equation += eq.general + elif eq.category == "asymptotic series": # make sure to fix asymptotic series + equation += "(" + eq.general + ")" + + elif eq.eq_type == "contfrac": + pieces = parse_brackets(eq.general[0])[0] + + if eq.begin != "": + eq.begin = parse_brackets(eq.begin) + + start = 1 # in case the value of start isn't assigned + + # add terms before general + if eq.front != "": + equation += eq.front + "+" + start = 1 + + if eq.begin != "": + for piece in eq.begin: + equation += make_frac(piece) + " \\subplus " + start += 1 + + if eq.factor != "": + if eq.factor == "-1": + equation += "-" + else: + equation += eq.factor + " " + + # trim unnecessary parentheses + for i, element in enumerate(pieces): + pieces[i] = trim_parens(translate(element)) + + if pieces != ["0", "1"]: + equation += "\\CFK{m}{" + str(start) + "}{\\infty}@@{" + pieces[0] + "}{" + pieces[1] + "}" + else: + equation += "\\dots" + + return cls(eq.label, replace_strings(equation, SPECIAL), [translate(eq.constraints), eq.category]) + + @classmethod + def get_sortable_label(cls, equation): + if equation.label == "x.x.x": + return [100, 100, 100] # should be three very large numbers + + label = copy.copy(equation.label) + for i, ch in enumerate(ascii_lowercase): + if label[-1] == ch: + label = label[:-1] + "." + str(i + 1) + + if label[-1] == ".": + label = label[:-1] + + return map(int, label.split(".")) + + def __str__(self): + metadata = "" + for k, v in self.metadata.iteritems(): + metadata += " % \\" + k + "{" + v + "}\n" + + return "\\begin{equation*}\\tag{" + self.label + "}\n " + self.equation + "\n" + \ + metadata + "\\end{equation*}\n" + + def replace_strings(string, keys): # type: (str, dict) -> str """Replaces key strings with their mapped value.""" @@ -77,11 +173,11 @@ def replace_strings(string, keys): return string -def parse_brackets(exp): +def parse_brackets(string): # type: ((str, list)) -> list """Obtains the contents from data encapsulated in square brackets.""" - exp = exp[1:-1].split(",") + exp = string[1:-1].split(",") for i, e in enumerate(exp): exp[i] = replace_strings(e, {"[": "", "]": ""}).strip() @@ -135,14 +231,14 @@ def basic_translate(exp): """Translates basic mathematical operations.""" # translates operations in place - for order in range(3): # order of operations + for order in range(3): # order of operations i = 0 while i < len(exp): modified = False # the imaginary number if exp[i] == "I": - exp[i] = "i" + exp[i] = "\\imaginarynumber" # factorial elif exp[i] == "!" and order == 0: @@ -166,7 +262,8 @@ def basic_translate(exp): # division elif exp[i] == "/" and order == 1: - for index in [i - 1, i + 1]: # removes extra parentheses + # remove extra parentheses + for index in [i - 1, i + 1]: exp[index] = trim_parens(exp[index]) exp[i - 1] = "\\frac{" + exp[i - 1] + "}{" + exp.pop(i + 1) + "}" modified = True @@ -279,85 +376,3 @@ def translate(exp): i -= 1 return basic_translate(exp) - - -def make_equation(eq, view_metadata=False): - # type: (MapleEquation{, bool}) -> str - """Make a LaTeX equation based on a MapleEquation object.""" - - # modify fields - eq.lhs = translate(eq.lhs) - eq.factor = translate(eq.fields["factor"]) - eq.front = translate(eq.fields["front"]) - - if eq.eq_type == "series": - eq.general = translate(eq.general[0]) - - elif eq.eq_type == "contfrac": - eq.general = parse_brackets(eq.general[0])[0] - - if eq.begin != "": - eq.begin = parse_brackets(eq.begin) - - equation = "\\begin{equation*}\\tag{" + eq.label + "}\n " + eq.lhs + "\n = " - - if eq.factor == "1": - eq.factor = "" - - # translates the Maple information (with spacing) - if eq.eq_type == "series": - if eq.factor != "": - equation += eq.factor + " " - - elif eq.front != "": - equation += eq.front + "+" - - equation += "\\sum_{k=0}^\\infty " - - if eq.category == "power series": - equation += eq.general - elif eq.category == "asymptotic series": # make sure to fix asymptotic series - equation += "(" + eq.general + ")" - - elif eq.eq_type == "contfrac": - start = 1 # in case the value of start isn't assigned - - # add terms before general - if eq.front != "": - equation += eq.front + "+" - start = 1 - - if eq.begin != "": - for piece in eq.begin: - equation += make_frac(piece) + " \\subplus " - start += 1 - - if eq.factor != "": - if eq.factor == "-1": - equation += "-" - else: - equation += eq.factor + " " - - # trim unnecessary parentheses - for i, element in enumerate(eq.general): - eq.general[i] = trim_parens(translate(element)) - - if eq.general != ["0", "1"]: - equation += "\\CFK{m}{" + str(start) + "}{\\infty}@@{" + eq.general[0] + "}{" + eq.general[1] + "}" - else: - equation += "\\dots" - - # adds metadata - if view_metadata: - equation += "\n\\end{equation*}" - equation += "\n\\begin{center}" - equation += "\nParameters: $$" + eq.parameters + "$$" - equation += "\n$$" + translate(eq.constraints) + "$$" - equation += "\n" + eq.category - equation += "\n\\end{center}" - else: - equation += "\n % \\constraint{$" + translate(eq.constraints) + "$}" - equation += "\n % \\category{" + eq.category + "}" - equation += "\n\\end{equation*}" - - return replace_strings(equation, SPECIAL) diff --git a/maple2latex/test/test_cases.json b/maple2latex/test/test_cases.json index 58c79fb..01ed4c4 100644 --- a/maple2latex/test/test_cases.json +++ b/maple2latex/test/test_cases.json @@ -85,7 +85,7 @@ } ], - "make_equation": [ + "test_from_maple": [ { "eq": [ "create( 'series',", @@ -96,7 +96,6 @@ "general = [ 2 * k - 3 ],", "constraints = { k::integer }," ], - "view_metadata": false, "res": "\\begin{equation*}\\tag{0.0.0}\n n\n = \\frac{4}{5} \\sum_{k=0}^\\infty 2k-3\n % \\constraint{$k\\in\\mathbb{Z}$}\n % \\category{power series}\n\\end{equation*}" }, @@ -109,7 +108,6 @@ "front = 17,", "general = [ k / 2 ]," ], - "view_metadata": false, "res": "\\begin{equation*}\\tag{0.0.0}\n p\n = 17+\\sum_{k=0}^\\infty \\left(\\frac{k}{2}\\right)\n % \\constraint{$$}\n % \\category{asymptotic series}\n\\end{equation*}" }, @@ -122,7 +120,6 @@ "factor = 15,", "general = [ [3, 5] ]," ], - "view_metadata": false, "res": "\\begin{equation*}\\tag{0.0.0}\n \\zeta\n = 15 \\CFK{m}{1}{\\infty}@@{3}{5}\n % \\constraint{$$}\n % \\category{test_data}\n\\end{equation*}" }, @@ -136,7 +133,6 @@ "front = 3,", "general = [[4, 5]]," ], - "view_metadata": false, "res": "\\begin{equation*}\\tag{0.0.0}\n \\zeta\n = 3+\\frac{1}{2} \\subplus \\CFK{m}{2}{\\infty}@@{4}{5}\n % \\constraint{$$}\n % \\category{test_data}\n\\end{equation*}" }, @@ -151,7 +147,6 @@ "front = 4,", "general = [[0, 1]]," ], - "view_metadata": false, "res": "\\begin{equation*}\\tag{0.0.0}\n \\zeta\n = 4+\\frac{1}{2} \\subplus 3 \\dots\n % \\constraint{$$}\n % \\category{test_data}\n\\end{equation*}" }, @@ -167,7 +162,6 @@ "even = [[5, 6]],", "odd = [[7, 8]]," ], - "view_metadata": true, "res": "\\begin{equation*}\\tag{0.0.0}\n \\zeta\n = 4+\\frac{1}{2} \\subplus -\\CFK{m}{2}{\\infty}@@{5}{6}\n\\end{equation*}\n\\begin{center}\nParameters: $$$$\n$$$$\ntest_data\n\\end{center}" } ] diff --git a/maple2latex/test/test_translator.py b/maple2latex/test/test_translator.py new file mode 100644 index 0000000..30eeb22 --- /dev/null +++ b/maple2latex/test/test_translator.py @@ -0,0 +1,53 @@ +#!/usr/bin/env python + +__author__ = "Joon Bang" +__status__ = "Development" + +import json +from unittest import TestCase +import src.translator as t + +TEST_CASES = json.loads(open("maple2latex/test/test_cases.json").read())["translator"] + +parse_brackets_test_cases = TEST_CASES["parse_brackets"] +trim_parens_test_cases = TEST_CASES["trim_parens"] +basic_translate_test_cases = TEST_CASES["basic_translate"] +get_arguments_test_cases = TEST_CASES["get_arguments"] +translate_test_cases = TEST_CASES["translate"] +from_maple_test_cases = TEST_CASES["from_maple"] + + +class TestParseBrackets(TestCase): + def test_parse_brackets(self): + for case in parse_brackets_test_cases: + self.assertEqual(t.parse_brackets(case["exp"]), case["res"]) + + +class TestTrimParens(TestCase): + def test_trim_parens(self): + for case in trim_parens_test_cases: + self.assertEqual(t.trim_parens(case["exp"]), case["res"]) + + +class TestBasicTranslate(TestCase): + def test_basic_translate(self): + for case in basic_translate_test_cases: + self.assertEqual(t.basic_translate(case["exp"]), case["res"]) + + +class TestGetArguments(TestCase): + def test_get_arguments(self): + for case in get_arguments_test_cases: + self.assertEqual(t.get_arguments(case["function"], case["arg_string"]), case["res"]) + + +class TestTranslate(TestCase): + def test_translate(self): + for case in translate_test_cases: + self.assertEqual(t.translate(case["exp"]), case["res"]) + + +class TestFromMaple(TestCase): + def test_from_maple(self): + for case in from_maple_test_cases: + self.assertEqual(t.LatexEquation.from_maple(t.MapleEquation(case["eq"])), case["res"]) \ No newline at end of file From 46e903430621a9155f6cd3eefc8028d3de91011a Mon Sep 17 00:00:00 2001 From: Joon Bang Date: Thu, 21 Jul 2016 11:39:21 -0400 Subject: [PATCH 366/402] Fix test cases --- maple2latex/src/translator.py | 8 +++++--- maple2latex/test/test_cases.json | 32 ++++++++--------------------- maple2latex/test/test_translator.py | 2 +- 3 files changed, 15 insertions(+), 27 deletions(-) diff --git a/maple2latex/src/translator.py b/maple2latex/src/translator.py index e529f37..d284a54 100644 --- a/maple2latex/src/translator.py +++ b/maple2latex/src/translator.py @@ -157,9 +157,11 @@ def get_sortable_label(cls, equation): def __str__(self): metadata = "" - for k, v in self.metadata.iteritems(): - metadata += " % \\" + k + "{" + v + "}\n" - + for data_type, data in self.metadata.iteritems(): + if data_type in ["constraint"]: # mathmode + metadata += " % \\" + data_type + "{$" + data + "$}\n" + else: + metadata += " % \\" + data_type + "{" + data + "}\n" return "\\begin{equation*}\\tag{" + self.label + "}\n " + self.equation + "\n" + \ metadata + "\\end{equation*}\n" diff --git a/maple2latex/test/test_cases.json b/maple2latex/test/test_cases.json index 01ed4c4..0df535e 100644 --- a/maple2latex/test/test_cases.json +++ b/maple2latex/test/test_cases.json @@ -37,7 +37,7 @@ { "exp": ["\\pi", "*", "I"], - "res": "\\pi i" + "res": "\\pi\\imaginarynumber" } ], @@ -85,7 +85,7 @@ } ], - "test_from_maple": [ + "from_maple": [ { "eq": [ "create( 'series',", @@ -96,7 +96,7 @@ "general = [ 2 * k - 3 ],", "constraints = { k::integer }," ], - "res": "\\begin{equation*}\\tag{0.0.0}\n n\n = \\frac{4}{5} \\sum_{k=0}^\\infty 2k-3\n % \\constraint{$k\\in\\mathbb{Z}$}\n % \\category{power series}\n\\end{equation*}" + "res": "\\begin{equation*}\\tag{0.0.0}\n n\n = \\frac{4}{5} \\sum_{k=0}^\\infty 2k-3\n % \\category{power series}\n % \\constraint{$k\\in\\mathbb{Z}$}\n\\end{equation*}\n" }, { @@ -108,7 +108,7 @@ "front = 17,", "general = [ k / 2 ]," ], - "res": "\\begin{equation*}\\tag{0.0.0}\n p\n = 17+\\sum_{k=0}^\\infty \\left(\\frac{k}{2}\\right)\n % \\constraint{$$}\n % \\category{asymptotic series}\n\\end{equation*}" + "res": "\\begin{equation*}\\tag{0.0.0}\n p\n = 17+\\sum_{k=0}^\\infty \\left(\\frac{k}{2}\\right)\n % \\category{asymptotic series}\n % \\constraint{$$}\n\\end{equation*}\n" }, { @@ -120,7 +120,7 @@ "factor = 15,", "general = [ [3, 5] ]," ], - "res": "\\begin{equation*}\\tag{0.0.0}\n \\zeta\n = 15 \\CFK{m}{1}{\\infty}@@{3}{5}\n % \\constraint{$$}\n % \\category{test_data}\n\\end{equation*}" + "res": "\\begin{equation*}\\tag{0.0.0}\n \\zeta\n = 15 \\CFK{m}{1}{\\infty}@@{3}{5}\n % \\category{test_data}\n % \\constraint{$$}\n\\end{equation*}\n" }, { @@ -133,7 +133,7 @@ "front = 3,", "general = [[4, 5]]," ], - "res": "\\begin{equation*}\\tag{0.0.0}\n \\zeta\n = 3+\\frac{1}{2} \\subplus \\CFK{m}{2}{\\infty}@@{4}{5}\n % \\constraint{$$}\n % \\category{test_data}\n\\end{equation*}" + "res": "\\begin{equation*}\\tag{0.0.0}\n \\zeta\n = 3+\\frac{1}{2} \\subplus \\CFK{m}{2}{\\infty}@@{4}{5}\n % \\category{test_data}\n % \\constraint{$$}\n\\end{equation*}\n" }, { @@ -145,24 +145,10 @@ "begin = [[1,2]],", "factor = 3,", "front = 4,", - "general = [[0, 1]]," + "even = [[0, 1]],", + "odd = [[2, 3]]," ], - "res": "\\begin{equation*}\\tag{0.0.0}\n \\zeta\n = 4+\\frac{1}{2} \\subplus 3 \\dots\n % \\constraint{$$}\n % \\category{test_data}\n\\end{equation*}" - }, - - { - "eq": [ - "create( 'contfrac', ", - "booklabelv1 = \"0.0.0\",", - "category = \"test_data\"", - "lhs = zeta,", - "begin = [[1,2]],", - "factor = -1,", - "front = 4,", - "even = [[5, 6]],", - "odd = [[7, 8]]," - ], - "res": "\\begin{equation*}\\tag{0.0.0}\n \\zeta\n = 4+\\frac{1}{2} \\subplus -\\CFK{m}{2}{\\infty}@@{5}{6}\n\\end{equation*}\n\\begin{center}\nParameters: $$$$\n$$$$\ntest_data\n\\end{center}" + "res": "\\begin{equation*}\\tag{0.0.0}\n \\zeta\n = 4+\\frac{1}{2} \\subplus 3 \\dots\n % \\category{test_data}\n % \\constraint{$$}\n\\end{equation*}\n" } ] } diff --git a/maple2latex/test/test_translator.py b/maple2latex/test/test_translator.py index 30eeb22..512b174 100644 --- a/maple2latex/test/test_translator.py +++ b/maple2latex/test/test_translator.py @@ -50,4 +50,4 @@ def test_translate(self): class TestFromMaple(TestCase): def test_from_maple(self): for case in from_maple_test_cases: - self.assertEqual(t.LatexEquation.from_maple(t.MapleEquation(case["eq"])), case["res"]) \ No newline at end of file + self.assertEqual(str(t.LatexEquation.from_maple(t.MapleEquation(case["eq"]))), case["res"]) \ No newline at end of file From 562c177f83006e30d8386540b1a21d0a47e0001b Mon Sep 17 00:00:00 2001 From: Joon Bang Date: Fri, 22 Jul 2016 13:10:44 -0400 Subject: [PATCH 367/402] Add basic substitutions --- maple2latex/data/keys.json | 46 ++++++++++++++++++++++- maple2latex/src/main.py | 11 ++++-- maple2latex/src/translator.py | 63 ++++++++++++++++++++++++++++---- maple2latex/test/test_cases.json | 2 +- 4 files changed, 109 insertions(+), 13 deletions(-) diff --git a/maple2latex/data/keys.json b/maple2latex/data/keys.json index a7ea2f8..01e8c0d 100644 --- a/maple2latex/data/keys.json +++ b/maple2latex/data/keys.json @@ -38,7 +38,7 @@ "exp": [ { "args": 1, - "repr": ["e^{", "}"] + "repr": ["\\expe^{", "}"] } ], @@ -433,6 +433,48 @@ "args": 1, "repr": ["(", ")"] } + ], + + "regincBeta": [ + { + "args": 3, + "repr": ["\\IncI{", "}@{", "}{", "}"] + } + ], + + "Beta": [ + { + "args": 2, + "repr": ["\\EulerBeta@{", "}{", "}"] + } + ], + + "gammacdf": [ + { + "args": 3, + "repr": ["\\GammaP@{", "}{", "}{", "}"] + } + ], + + "cgammacdf": [ + { + "args": 3, + "repr": ["\\GammaQ@{", "}{", "}{", "}"] + } + ], + + "cnormalI": [ + { + "args": 2, + "repr": ["\\RepInterfc{", "}@{", "}"] + } + ], + + "Mills": [ + { + "args": 1, + "repr": ["\\Mills@{", "}"] + } ] }, @@ -461,7 +503,7 @@ "psi": "\\psi", "omega": "\\omega", "Pi": "\\pi", - "I": "i", + "I": "\\ImaginaryNumber", "Catalan": "\\CatalansConstant", "infinity": "\\infty", "Delian": "\\sqrt[\\leftroot{-2}\\uproot{2} 3]{2}", diff --git a/maple2latex/src/main.py b/maple2latex/src/main.py index b88765a..3fe555a 100644 --- a/maple2latex/src/main.py +++ b/maple2latex/src/main.py @@ -24,7 +24,8 @@ "expintegrals", "related", "binet", "incompletegamma", "polygamma", "tetragamma", "trigamma", "hypergeometric", - "qhyper" + "qhyper", + "beta_f_t", "gamma_chisquare", "normal", "repeated" ] # files to be translated @@ -64,12 +65,16 @@ def translate_files(root_directory): folder = ''.join(file_name.split(".")[:-1]) if folder in FILES: # generate equation code, sorted by equation number - subsection = TABLE[file_name[:-4]] + subsection = TABLE[folder] formulae = map(LatexEquation.from_maple, MapleFile(info[0] + "/" + file_name).formulae) formulae = sorted(formulae, key=LatexEquation.get_sortable_label) repr_label = formulae[len(formulae) / 2].label - sections[section][subsection] = [repr_label, "\n".join(map(str, formulae))] + + if subsection in sections[section]: + sections[section][subsection][1] += "\n".join(map(str, formulae)) + else: + sections[section][subsection] = [repr_label, "\n".join(map(str, formulae))] # generate subsection headers for section, subsections in sections.iteritems(): diff --git a/maple2latex/src/translator.py b/maple2latex/src/translator.py index d284a54..3788b1e 100644 --- a/maple2latex/src/translator.py +++ b/maple2latex/src/translator.py @@ -15,7 +15,10 @@ CONSTRAINTS = INFO["constraints"] SPECIAL = {"(": "\\left(", ")": "\\right)", "+-": "-", "\\subplus-": "-", "^{1}": "", "\\inNot": "\\notin", - "\\imaginarynumber": "i"} + "\\ImaginaryNumber": "i"} + +# TODO: gamma_chisquare.mpl (no macro.) +# TODO: normal.mpl (no macro.) class MapleEquation(object): @@ -74,7 +77,7 @@ class LatexEquation(object): def __init__(self, label, equation, metadata): self.label = label self.equation = equation - self.metadata = {"constraint": metadata[0], "category": metadata[1]} + self.metadata = metadata @classmethod def from_maple(cls, eq): @@ -88,6 +91,8 @@ def from_maple(cls, eq): if eq.factor == "1": eq.factor = "" + metadata = dict() + # translates the Maple information (with spacing) if eq.eq_type == "series": eq.general = translate(eq.general[0]) @@ -106,7 +111,44 @@ def from_maple(cls, eq): equation += "(" + eq.general + ")" elif eq.eq_type == "contfrac": - pieces = parse_brackets(eq.general[0])[0] + forms = list() + + if len(parse_brackets(eq.general[0])) > 1: + # print eq.label + forms = parse_brackets(eq.general[0]) + + if len(eq.general) == 2 or forms: + if not forms: # forms is empty + for form in eq.general: + forms += parse_brackets(form) + + replacements = list() + if len(eq.general) == 2: + replacements = ["2j", "2j+1"] + elif forms: + for i in range(len(forms)): + replacement = str(len(forms)) + "j" + if i < len(forms) - 1: + replacement += "-" + str(len(forms) - i - 1) + + replacements.append(replacement) + + for i, form in enumerate(forms): + for j, half in enumerate(form): + half = tokenize(half) + for k, ch in enumerate(half): + if ch == "m": + half[k] = "(" + replacements[i] + ")" + + form[j] = ' '.join(half) + + forms[i] = "s_{" + replacements[i] + "} = " + make_frac(form) + + metadata["substitution"] = ','.join(forms) + pieces = ["s_m", "1"] + + else: + pieces = parse_brackets(eq.general[0])[0] if eq.begin != "": eq.begin = parse_brackets(eq.begin) @@ -138,7 +180,10 @@ def from_maple(cls, eq): else: equation += "\\dots" - return cls(eq.label, replace_strings(equation, SPECIAL), [translate(eq.constraints), eq.category]) + metadata["constraint"] = translate(eq.constraints) + metadata["category"] = eq.category + + return cls(eq.label, replace_strings(equation, SPECIAL), metadata) @classmethod def get_sortable_label(cls, equation): @@ -158,10 +203,14 @@ def get_sortable_label(cls, equation): def __str__(self): metadata = "" for data_type, data in self.metadata.iteritems(): - if data_type in ["constraint"]: # mathmode - metadata += " % \\" + data_type + "{$" + data + "$}\n" + if data_type in ["constraint", "substitution"]: # mathmode + metadata += " % \\" + data_type + "{" + replace_strings(data, SPECIAL) + "}\n" else: metadata += " % \\" + data_type + "{" + data + "}\n" + + # if data_type in ["substitution"]: + # metadata += " " + replace_strings(data, SPECIAL) + "\n" + return "\\begin{equation*}\\tag{" + self.label + "}\n " + self.equation + "\n" + \ metadata + "\\end{equation*}\n" @@ -240,7 +289,7 @@ def basic_translate(exp): # the imaginary number if exp[i] == "I": - exp[i] = "\\imaginarynumber" + exp[i] = "\\ImaginaryNumber" # factorial elif exp[i] == "!" and order == 0: diff --git a/maple2latex/test/test_cases.json b/maple2latex/test/test_cases.json index 0df535e..3048031 100644 --- a/maple2latex/test/test_cases.json +++ b/maple2latex/test/test_cases.json @@ -37,7 +37,7 @@ { "exp": ["\\pi", "*", "I"], - "res": "\\pi\\imaginarynumber" + "res": "\\pi\\ImaginaryNumber" } ], From 1e66ce4a0cd1df40538ae81036c92e9050961818 Mon Sep 17 00:00:00 2001 From: Joon Bang Date: Mon, 1 Aug 2016 11:27:44 -0400 Subject: [PATCH 368/402] Update keys.json with new macros --- maple2latex/data/keys.json | 28 +++++++++++++++++++++------- 1 file changed, 21 insertions(+), 7 deletions(-) diff --git a/maple2latex/data/keys.json b/maple2latex/data/keys.json index 01e8c0d..63a40fc 100644 --- a/maple2latex/data/keys.json +++ b/maple2latex/data/keys.json @@ -449,31 +449,45 @@ } ], + "cnormalI": [ + { + "args": 2, + "repr": ["\\RepInterfc{", "}@{", "}"] + } + ], + + "Mills": [ + { + "args": 1, + "repr": ["\\Mills@{", "}"] + } + ], + "gammacdf": [ { "args": 3, - "repr": ["\\GammaP@{", "}{", "}{", "}"] + "repr": ["\\RegGammaP@{", "}{", "}{", "}"] } ], "cgammacdf": [ { "args": 3, - "repr": ["\\GammaQ@{", "}{", "}{", "}"] + "repr": ["\\RegGammaQ@{", "}{", "}{", "}"] } ], - "cnormalI": [ + "normaldcdf": [ { - "args": 2, - "repr": ["\\RepInterfc{", "}@{", "}"] + "args": 1, + "repr": ["\\GaussDistF@{", "}"] } ], - "Mills": [ + "cnormaldcdf": [ { "args": 1, - "repr": ["\\Mills@{", "}"] + "repr": ["\\GaussDistQ@{", "}"] } ] }, From 82100b75e0fcda9a537eaa679c7570e88748e019 Mon Sep 17 00:00:00 2001 From: Joon Bang Date: Thu, 11 Aug 2016 12:04:20 -0400 Subject: [PATCH 369/402] Fix tests --- maple2latex/src/translator.py | 5 +---- maple2latex/test/test_cases.json | 4 ++-- 2 files changed, 3 insertions(+), 6 deletions(-) diff --git a/maple2latex/src/translator.py b/maple2latex/src/translator.py index 3788b1e..972fe4d 100644 --- a/maple2latex/src/translator.py +++ b/maple2latex/src/translator.py @@ -204,7 +204,7 @@ def __str__(self): metadata = "" for data_type, data in self.metadata.iteritems(): if data_type in ["constraint", "substitution"]: # mathmode - metadata += " % \\" + data_type + "{" + replace_strings(data, SPECIAL) + "}\n" + metadata += " % \\" + data_type + "{$" + replace_strings(data, SPECIAL) + "$}\n" else: metadata += " % \\" + data_type + "{" + data + "}\n" @@ -262,9 +262,6 @@ def trim_parens(exp): else: s.pop() - if len(s) != 0: - return exp - return test return exp diff --git a/maple2latex/test/test_cases.json b/maple2latex/test/test_cases.json index 3048031..02a6885 100644 --- a/maple2latex/test/test_cases.json +++ b/maple2latex/test/test_cases.json @@ -139,7 +139,7 @@ { "eq": [ "create( 'contfrac', ", - "booklabelv1 = \"0.0.0\",", + "booklabelv1 = \"\",", "category = \"test_data\"", "lhs = zeta,", "begin = [[1,2]],", @@ -148,7 +148,7 @@ "even = [[0, 1]],", "odd = [[2, 3]]," ], - "res": "\\begin{equation*}\\tag{0.0.0}\n \\zeta\n = 4+\\frac{1}{2} \\subplus 3 \\dots\n % \\category{test_data}\n % \\constraint{$$}\n\\end{equation*}\n" + "res": "\\begin{equation*}\\tag{x.x.x}\n \\zeta\n = 4+\\frac{1}{2} \\subplus 3 \\CFK{m}{2}{\\infty}@@{s_m}{1}\n % \\category{test_data}\n % \\constraint{$$}\n % \\substitution{$s_{2j} = \\frac{0}{1},s_{2j+1} = \\frac{2}{3}$}\n\\end{equation*}\n" } ] } From e082f587e334c328c1e3331683f469ff02ac4324 Mon Sep 17 00:00:00 2001 From: Joon Bang Date: Thu, 11 Aug 2016 14:40:05 -0400 Subject: [PATCH 370/402] Fix constraints --- maple2latex/data/keys.json | 15 +++++++++++++-- maple2latex/src/translator.py | 22 ++++++++++++++++++---- 2 files changed, 31 insertions(+), 6 deletions(-) diff --git a/maple2latex/data/keys.json b/maple2latex/data/keys.json index 63a40fc..a3c75e7 100644 --- a/maple2latex/data/keys.json +++ b/maple2latex/data/keys.json @@ -489,6 +489,13 @@ "args": 1, "repr": ["\\GaussDistQ@{", "}"] } + ], + + "not": [ + { + "args": 1, + "repr": ["", ""] + } ] }, @@ -522,7 +529,10 @@ "infinity": "\\infty", "Delian": "\\sqrt[\\leftroot{-2}\\uproot{2} 3]{2}", "Theodorus": "\\sqrt{3}", - "integer": "\\mathbb{Z}" + "integer": "\\mathbb{Z}", + "or": "\\lor ", + "and": "\\wedge ", + "realcons": "\\RR" }, "constraints": { @@ -536,6 +546,7 @@ "::positive": "> 0", "::nonnegative": "\\geq 0", "::nonneg": "\\geq 0", - "::": "\\in " + "::": "\\in ", + "<>": "\\neq " } } \ No newline at end of file diff --git a/maple2latex/src/translator.py b/maple2latex/src/translator.py index 972fe4d..7dfb66a 100644 --- a/maple2latex/src/translator.py +++ b/maple2latex/src/translator.py @@ -17,9 +17,6 @@ SPECIAL = {"(": "\\left(", ")": "\\right)", "+-": "-", "\\subplus-": "-", "^{1}": "", "\\inNot": "\\notin", "\\ImaginaryNumber": "i"} -# TODO: gamma_chisquare.mpl (no macro.) -# TODO: normal.mpl (no macro.) - class MapleEquation(object): def __init__(self, inp): @@ -114,7 +111,6 @@ def from_maple(cls, eq): forms = list() if len(parse_brackets(eq.general[0])) > 1: - # print eq.label forms = parse_brackets(eq.general[0]) if len(eq.general) == 2 or forms: @@ -204,6 +200,8 @@ def __str__(self): metadata = "" for data_type, data in self.metadata.iteritems(): if data_type in ["constraint", "substitution"]: # mathmode + if data == "": + data = "Empty" metadata += " % \\" + data_type + "{$" + replace_strings(data, SPECIAL) + "$}\n" else: metadata += " % \\" + data_type + "{" + data + "}\n" @@ -334,6 +332,22 @@ def get_arguments(function, arg_string): if arg_string == ["(", ")"]: return [] + elif function == "not": + inversion = {"<": "\\geq ", ">": "\\leq ", "\\in": "\\notin "} + + for i, ch in enumerate(arg_string): + if ch in inversion: + arg_string[i] = inversion[ch] + + args = [basic_translate(arg_string[1:-1])] + + elif function == "RealRange": + for i, piece in enumerate(arg_string): + if trim_parens(piece) != piece: + arg_string[i] = trim_parens(piece) + + args = basic_translate(arg_string[1:-1]).split(",") + # handling for hypergeometric, q-hypergeometric functions elif function in ["hypergeom", "qhyper"]: args = list() From b5fbe242dfe73beb2c2fa9bd677d38c9013a893e Mon Sep 17 00:00:00 2001 From: Joon Bang Date: Fri, 12 Aug 2016 11:02:56 -0400 Subject: [PATCH 371/402] Add selective parenthese handling --- maple2latex/data/keys.json | 249 +++++++++++++++++----------------- maple2latex/src/translator.py | 63 +++++---- 2 files changed, 168 insertions(+), 144 deletions(-) diff --git a/maple2latex/data/keys.json b/maple2latex/data/keys.json index a3c75e7..51b9e5d 100644 --- a/maple2latex/data/keys.json +++ b/maple2latex/data/keys.json @@ -1,283 +1,311 @@ { "functions": { - "sqrt": [ + "sin": [ { "args": 1, - "repr": ["\\sqrt{", "}"] + "repr": ["\\sin@{", "}"] } ], - "sin": [ + "cos": [ { "args": 1, - "repr": ["\\sin(", ")"] + "repr": ["\\cos@{", "}"] } ], - "cos": [ + "tan": [ { "args": 1, - "repr": ["\\cos(", ")"] + "repr": ["\\tan@{", "}"] } ], - "tan": [ + "arccos": [ { "args": 1, - "repr": ["\\tan(", ")"] + "repr": ["\\acos@{", "}"] } ], - "abs": [ + "arccosh": [ { "args": 1, - "repr": ["\\abs{", "}"] + "repr": ["\\acosh@{", "}"] } ], - "exp": [ + "arcsin": [ { "args": 1, - "repr": ["\\expe^{", "}"] + "repr": ["\\asin@{", "}"] } ], - "log": [ + "arcsinh": [ { "args": 1, - "repr": ["\\log{(", ")}"] - }, + "repr": ["\\asinh@{", "}"] + } + ], + "arctan": [ { - "args": 2, - "repr": ["\\log_{", "} {(", ")}"] + "args": 1, + "repr": ["\\atan@{", "}"] } ], - "ln": [ + "arctanh": [ { "args": 1, - "repr": ["\\ln{(", ")}"] + "repr": ["\\atanh@{", "}"] } ], - "floor": [ + "sinh": [ { "args": 1, - "repr": ["\\floor{", "}"] + "repr": ["\\sinh@{", "}"] } ], - "sum": [ + "cosh": [ { - "args": 3, - "repr": ["\\sum_{", "}^{", "} (", ")"] + "args": 1, + "repr": ["\\cosh@{", "}"] } ], - "GAMMA": [ + "coth": [ { "args": 1, - "repr": ["\\EulerGamma@{", "}"] + "repr": ["\\coth@{", "}"] } ], - "bernoulli": [ + "tanh": [ { "args": 1, - "repr": ["\\BernoulliB{", "}"] + "repr": ["\\tanh@{", "}"] } ], - "BesselI": [ + "erfc": [ { - "args": 2, - "repr": ["\\BesselI{", "}@{", "}"] + "args": 1, + "repr": ["\\erfc@{", "}"] } ], - "BesselJ": [ + "erf": [ { - "args": 2, - "repr": ["\\BesselJ{", "}@{", "}"] + "args": 1, + "repr": ["\\erf@{", "}"] } ], - "BesselK": [ + "log": [ + { + "args": 1, + "repr": ["\\log@{", "}"] + }, + { "args": 2, - "repr": ["\\BesselK{", "}@{", "}"] + "repr": ["\\log_{", "}@{", "}"] } ], - "BesselY": [ + "ln": [ { - "args": 2, - "repr": ["\\BesselY{", "}@{", "}"] + "args": 1, + "repr": ["\\ln@{", "}"] } ], - "dawson": [ + "cerf": [ { "args": 1, - "repr": ["\\DawsonsInt@{", "}"] + "repr": ["\\erfw@{", "}"] } ], - "Ei": [ + "Ein": [ + { + "args": 1, + "repr": ["\\ExpIntEin@{", "}"] + } + ], + + "erfcI": [ { "args": 2, - "repr": ["\\ExpIntn{", "}@{", "}"] - }, + "repr": ["I^{", "} \\erfc@{", "}"] + } + ], + "sqrt": [ { "args": 1, - "repr": ["\\ExpInt@{", "}"] + "repr": ["\\sqrt{", "}"] } ], - "erfc": [ + "abs": [ { "args": 1, - "repr": ["\\erfc@@{(", ")}"] + "repr": ["\\abs{", "}"] } ], - "erf": [ + "exp": [ { "args": 1, - "repr": ["\\erf@@{(", ")}"] + "repr": ["\\expe^{", "}"] } ], - "FresnelC": [ + "floor": [ { "args": 1, - "repr": ["\\FresnelCos@{", "}"] + "repr": ["\\floor{", "}"] } ], - "FresnelS": [ + "sum": [ { - "args": 1, - "repr": ["\\FresnelSin@{", "}"] + "args": 3, + "repr": ["\\sum_{", "}^{", "} (", ")"] } ], - "HankelH1": [ + "GAMMA": [ { - "args": 2, - "repr": ["\\HankelHi{", "}@{", "}"] + "args": 1, + "repr": ["\\EulerGamma@{", "}"] } ], - "KummerU": [ + "bernoulli": [ { - "args": 3, - "repr": ["\\KummerU@{", "}{", "}{", "}"] + "args": 1, + "repr": ["\\BernoulliB{", "}"] } ], - "CylinderU": [ + "BesselI": [ { "args": 2, - "repr": ["\\ParabolicU@{", "}{", "}"] + "repr": ["\\BesselI{", "}@{", "}"] } ], - "Psi": [ + "BesselJ": [ { "args": 2, - "repr": ["\\polygamma{", "}@{", "}"] - }, + "repr": ["\\BesselJ{", "}@{", "}"] + } + ], + "BesselK": [ { - "args": 1, - "repr": ["\\digamma@{", "}"] + "args": 2, + "repr": ["\\BesselK{", "}@{", "}"] } ], - "WhittakerW": [ + "BesselY": [ { - "args": 3, - "repr": ["\\WhitW{", "}{", "}@{", "}"] + "args": 2, + "repr": ["\\BesselY{", "}@{", "}"] } ], - "Zeta": [ + "dawson": [ { "args": 1, - "repr": ["\\RiemannZeta@{", "}"] + "repr": ["\\DawsonsInt@{", "}"] } ], - "arccos": [ + "Ei": [ + { + "args": 2, + "repr": ["\\ExpIntn{", "}@{", "}"] + }, + { "args": 1, - "repr": ["\\Acos@@{(", ")}"] + "repr": ["\\ExpInt@{", "}"] } ], - "arccosh": [ + "FresnelC": [ { "args": 1, - "repr": ["\\Acosh@@{(", ")}"] + "repr": ["\\FresnelCos@{", "}"] } ], - "arcsin": [ + "FresnelS": [ { "args": 1, - "repr": ["\\Asin@@{(", ")}"] + "repr": ["\\FresnelSin@{", "}"] } ], - "arcsinh": [ + "HankelH1": [ { - "args": 1, - "repr": ["\\Asinh@@{(", ")}"] + "args": 2, + "repr": ["\\HankelHi{", "}@{", "}"] } ], - "arctan": [ + "KummerU": [ { - "args": 1, - "repr": ["\\Atan@@{(", ")}"] + "args": 3, + "repr": ["\\KummerU@{", "}{", "}{", "}"] } ], - "arctanh": [ + "CylinderU": [ { - "args": 1, - "repr": ["\\Atanh@@{(", ")}"] + "args": 2, + "repr": ["\\ParabolicU@{", "}{", "}"] } ], - "sinh": [ + "Psi": [ + { + "args": 2, + "repr": ["\\polygamma{", "}@{", "}"] + }, + { "args": 1, - "repr": ["\\sinh@@{(", ")}"] + "repr": ["\\digamma@{", "}"] } ], - "cosh": [ + "special-trigamma": [ { "args": 1, - "repr": ["\\cosh@@{(", ")}"] + "repr": ["\\digamma'@{", "}"] } ], - "coth": [ + "WhittakerW": [ { - "args": 1, - "repr": ["\\coth@@{(", ")}"] + "args": 3, + "repr": ["\\WhitW{", "}{", "}@{", "}"] } ], - "tanh": [ + "Zeta": [ { "args": 1, - "repr": ["\\tanh@@{(", ")}"] + "repr": ["\\RiemannZeta@{", "}"] } ], @@ -330,27 +358,6 @@ } ], - "cerf": [ - { - "args": 1, - "repr": ["\\erfw@@{(", ")}"] - } - ], - - "Ein": [ - { - "args": 1, - "repr": ["\\ExpIntEin@{", "}"] - } - ], - - "erfcI": [ - { - "args": 2, - "repr": ["I^{", "} \\erfc@@{(", ")}"] - } - ], - "hankelsymb": [ { "args": 2, @@ -417,7 +424,7 @@ "argument": [ { "args": 1, - "repr": ["\\arg{(", ")}"] + "repr": ["\\arg@{", "}"] } ], @@ -514,7 +521,7 @@ "mu": "\\mu", "nu": "\\nu", "xi": "\\xi", - "pi": "\\pi", + "pi": "\\cpi", "rho": "\\rho", "sigma": "\\sigma", "tau": "\\tau", @@ -524,7 +531,6 @@ "psi": "\\psi", "omega": "\\omega", "Pi": "\\pi", - "I": "\\ImaginaryNumber", "Catalan": "\\CatalansConstant", "infinity": "\\infty", "Delian": "\\sqrt[\\leftroot{-2}\\uproot{2} 3]{2}", @@ -532,7 +538,8 @@ "integer": "\\mathbb{Z}", "or": "\\lor ", "and": "\\wedge ", - "realcons": "\\RR" + "realcons": "\\RR", + "I": "\\iunit" }, "constraints": { diff --git a/maple2latex/src/translator.py b/maple2latex/src/translator.py index 7dfb66a..12ef620 100644 --- a/maple2latex/src/translator.py +++ b/maple2latex/src/translator.py @@ -14,8 +14,10 @@ SYMBOLS = INFO["symbols"] CONSTRAINTS = INFO["constraints"] -SPECIAL = {"(": "\\left(", ")": "\\right)", "+-": "-", "\\subplus-": "-", "^{1}": "", "\\inNot": "\\notin", - "\\ImaginaryNumber": "i"} +SPECIAL = {"(": "\\left(", ")": "\\right)", "+-": "-", "\\subplus-": "-", "^{1}": "", "\\inNot": "\\notin"} + +MULTI_ARGS = ["sin", "cos", "tan", "arccos", "arccosh", "arcsin", "arcsinh", "arctanh", "sinh", "cosh", "coth", "tanh", + "erfc", "erf", "log", "ln"] class MapleEquation(object): @@ -239,12 +241,15 @@ def parse_brackets(string): def trim_parens(exp): - # type: (str) -> str + # type: (str/list) -> str """Removes unnecessary parentheses.""" - if exp == "": + if type(exp) == str and exp == "": return "" + elif type(exp) == list and exp == []: + return [] + # checks whether the outer set of parentheses are necessary if exp[0] == "(" and exp[-1] == ")": test = exp[1:-1] @@ -282,12 +287,8 @@ def basic_translate(exp): while i < len(exp): modified = False - # the imaginary number - if exp[i] == "I": - exp[i] = "\\ImaginaryNumber" - # factorial - elif exp[i] == "!" and order == 0: + if exp[i] == "!" and order == 0: exp[i - 1] += "!" modified = True @@ -328,9 +329,11 @@ def get_arguments(function, arg_string): # type: (str, list) -> list """Obtains the arguments of a function.""" + parens_mod = False + # no arguments - if arg_string == ["(", ")"]: - return [] + if not arg_string: + args = [] elif function == "not": inversion = {"<": "\\geq ", ">": "\\leq ", "\\in": "\\notin "} @@ -339,19 +342,24 @@ def get_arguments(function, arg_string): if ch in inversion: arg_string[i] = inversion[ch] - args = [basic_translate(arg_string[1:-1])] + args = [basic_translate(arg_string)] elif function == "RealRange": for i, piece in enumerate(arg_string): if trim_parens(piece) != piece: arg_string[i] = trim_parens(piece) - args = basic_translate(arg_string[1:-1]).split(",") + args = basic_translate(arg_string).split(",") + + # handling for trigamma + elif function == "Psi" and arg_string[0] == "1": + function = "special-trigamma" + args = basic_translate(arg_string[2]) # handling for hypergeometric, q-hypergeometric functions elif function in ["hypergeom", "qhyper"]: args = list() - for s in ' '.join(arg_string[1:-1]).split("] , "): + for s in ' '.join(arg_string).split("] , "): args.append(basic_translate(replace_strings(s, {"[": "", "]": ""}).split())) if function == "qhyper": @@ -366,26 +374,35 @@ def get_arguments(function, arg_string): # handling for sums elif function == "sum": - args = basic_translate(arg_string[1:-1]).split(",") + args = basic_translate(arg_string).split(",") args = args.pop(1).split("..") + [args[0]] if args[1] == "infinity": args[1] = "\\infty" + elif function in MULTI_ARGS and len(arg_string) == 1: + parens_mod = True + args = basic_translate(arg_string).split(",") + else: - args = basic_translate(arg_string[1:-1]).split(",") + args = basic_translate(arg_string).split(",") - return args + result = list() + for n in FUNCTIONS: + for variant in FUNCTIONS[n]: + if function == n and len(args) == variant["args"]: + result = copy.copy(variant["repr"]) + + if parens_mod: + result[0] = result[0].replace("@", "@@") # make the macro without parens + + return [result, args] def generate_function(name, args): # type: (str, list) -> str """Generate a function with the provided function name and arguments.""" - result = list() - for n in FUNCTIONS: - for variant in FUNCTIONS[n]: - if name == n and len(args) == variant["args"]: - result = copy.copy(variant["repr"]) + result, args = get_arguments(name, args) # places arguments between shell of function for n in range(1, len(result)): @@ -423,7 +440,7 @@ def translate(exp): if exp[i - 1] in FUNCTIONS: # handling for functions i -= 1 - piece = generate_function(exp[i], get_arguments(exp[i], piece)) + piece = generate_function(exp[i], trim_parens(piece)) else: piece = basic_translate(piece) From ec3213b20b860ea522902199591c5b44cb0ce12a Mon Sep 17 00:00:00 2001 From: Joon Bang Date: Fri, 12 Aug 2016 11:11:02 -0400 Subject: [PATCH 372/402] Add Binet function --- maple2latex/data/keys.json | 7 +++++++ 1 file changed, 7 insertions(+) diff --git a/maple2latex/data/keys.json b/maple2latex/data/keys.json index 51b9e5d..06412fa 100644 --- a/maple2latex/data/keys.json +++ b/maple2latex/data/keys.json @@ -503,6 +503,13 @@ "args": 1, "repr": ["", ""] } + ], + + "Binet": [ + { + "args": 1, + "repr": ["\\Binet@{", "}"] + } ] }, From 68a151fb79294ea53812eb983cf0f01ba30f7ad6 Mon Sep 17 00:00:00 2001 From: Joon Bang Date: Fri, 12 Aug 2016 11:30:17 -0400 Subject: [PATCH 373/402] Fix unit tests --- maple2latex/src/translator.py | 6 ++---- maple2latex/test/test_cases.json | 28 ++++++++++++++-------------- 2 files changed, 16 insertions(+), 18 deletions(-) diff --git a/maple2latex/src/translator.py b/maple2latex/src/translator.py index 12ef620..89a5b81 100644 --- a/maple2latex/src/translator.py +++ b/maple2latex/src/translator.py @@ -202,8 +202,6 @@ def __str__(self): metadata = "" for data_type, data in self.metadata.iteritems(): if data_type in ["constraint", "substitution"]: # mathmode - if data == "": - data = "Empty" metadata += " % \\" + data_type + "{$" + replace_strings(data, SPECIAL) + "$}\n" else: metadata += " % \\" + data_type + "{" + data + "}\n" @@ -326,8 +324,8 @@ def basic_translate(exp): def get_arguments(function, arg_string): - # type: (str, list) -> list - """Obtains the arguments of a function.""" + # type: (str, list) -> (list, list) + """Generates the function pieces and the arguments.""" parens_mod = False diff --git a/maple2latex/test/test_cases.json b/maple2latex/test/test_cases.json index 02a6885..fe3b3d9 100644 --- a/maple2latex/test/test_cases.json +++ b/maple2latex/test/test_cases.json @@ -36,52 +36,52 @@ }, { - "exp": ["\\pi", "*", "I"], - "res": "\\pi\\ImaginaryNumber" + "exp": ["\\cpi", "*", "3"], + "res": "\\cpi 3" } ], "get_arguments": [ { "function": "rabbit", - "arg_string": ["(", ")"], - "res": [] + "arg_string": [], + "res": [["\\rabbitConstant"], []] }, { "function": "hypergeom", - "arg_string": ["(", "[1", ",", "2]", ",", "[]", ",", "4", ")"], - "res": ["2", "0", "1,2", "", "4"] + "arg_string": ["[1", ",", "2]", ",", "[]", ",", "4"], + "res": [["\\HyperpFq{", "}{", "}@@{", "}{", "}{", "}"], ["2", "0", "1,2", "", "4"]] }, { "function": "qhyper", - "arg_string": ["(", "[1", ",", "2", ",", "3]", ",", "[4", ",", "5]", ",", "6", ",", "7", ")"], - "res": ["3", "2", "1,2,3", "4,5", "6", "7"] + "arg_string": ["[1", ",", "2", ",", "3]", ",", "[4", ",", "5]", ",", "6", ",", "7"], + "res": [["\\qHyperrphis{", "}{", "}@@{", "}{", "}{", "}{", "}"], ["3", "2", "1,2,3", "4,5", "6", "7"]] }, { "function": "sum", - "arg_string": ["(", "m", ",", "1..infinity", ")"], - "res": ["1", "\\infty", "m"] + "arg_string": ["m", ",", "1..infinity"], + "res": [["\\sum_{", "}^{", "} (", ")"], ["1", "\\infty", "m"]] }, { "function": "BesselJ", - "arg_string": ["(", "\\nu", ",", "z", ")"], - "res": ["\\nu", "z"] + "arg_string": ["\\nu", ",", "z"], + "res": [["\\BesselJ{", "}@{", "}"], ["\\nu", "z"]] } ], "translate": [ { "exp": "sin(x)", - "res": "\\sin(x)" + "res": "\\sin@@{x}" }, { "exp": "log[beta](nu)", - "res": "\\log_{\\beta} {(\\nu)}" + "res": "\\log_{\\beta}@{\\nu}" } ], From 3eedb129ef467187893e4ac5b9efdb38ee4876af Mon Sep 17 00:00:00 2001 From: Joon Bang Date: Tue, 16 Aug 2016 20:16:44 -0400 Subject: [PATCH 374/402] Update README.md --- maple2latex/README.md | 19 +++++++++++++++++++ 1 file changed, 19 insertions(+) diff --git a/maple2latex/README.md b/maple2latex/README.md index e69de29..5fe9d94 100644 --- a/maple2latex/README.md +++ b/maple2latex/README.md @@ -0,0 +1,19 @@ +# Maple CFSF Seeding Project +This project uses Python to convert the continued fractions for special functions library, written in Maple, into LaTeX. + +## Prerequisites +This program requires Python 2.6+ to run. + +## Usage +To run the program, ensure that you are in the directory where maple2latex is located. Then, run the command `python maple2latex/src/main.py`. + +Currently, the program translates only the CFSF library of functions. In the future, support will be added for +translating individual .mpl files. + +## Code Explanation +There are three files, `main.py`, `maple_tokenize.py`, and `translator.py`. The bulk of the code is located in `translator.py.` + +The program works by parsing through "tokens," which are created by running the `tokenize` function on a string. +It then goes through the tokens, joining them when necessary. It initially joins terms grouped in parentheses, starting from +the innermost set of parentheses and working its way outwards. It has separate functions handling translation of mathematical +functions, and basic operations (addition, multiplication, division, etc.). From b4c335933bf5b8efc64c6293e4a5937279fe83cc Mon Sep 17 00:00:00 2001 From: Joon Bang Date: Tue, 16 Aug 2016 20:23:48 -0400 Subject: [PATCH 375/402] Add .gitignore --- main_page/.gitignore | 5 +++++ maple2latex/.gitignore | 11 +++++++++++ 2 files changed, 16 insertions(+) create mode 100644 main_page/.gitignore create mode 100644 maple2latex/.gitignore diff --git a/main_page/.gitignore b/main_page/.gitignore new file mode 100644 index 0000000..dbf9da8 --- /dev/null +++ b/main_page/.gitignore @@ -0,0 +1,5 @@ +.DS_Store +backups/ +../.DS_Store +*Glossary.csv* +!test/fake.Glossary.csv diff --git a/maple2latex/.gitignore b/maple2latex/.gitignore new file mode 100644 index 0000000..10d7823 --- /dev/null +++ b/maple2latex/.gitignore @@ -0,0 +1,11 @@ +functions/ +out/ +!out/test.pdf +notes.txt +resources/ +.DS_Store +../.DS_Store +*differences.py +../AzeemOldCode/ +../tex2wiki/ +../main_page/ From fa8c15bc468e553f103bdcc520499177f984e98a Mon Sep 17 00:00:00 2001 From: Joon Bang Date: Wed, 17 Aug 2016 10:07:12 -0400 Subject: [PATCH 376/402] Update .gitignore --- main_page/.gitignore | 1 - maple2latex/.gitignore | 6 ------ 2 files changed, 7 deletions(-) diff --git a/main_page/.gitignore b/main_page/.gitignore index dbf9da8..3caccc4 100644 --- a/main_page/.gitignore +++ b/main_page/.gitignore @@ -1,5 +1,4 @@ .DS_Store backups/ -../.DS_Store *Glossary.csv* !test/fake.Glossary.csv diff --git a/maple2latex/.gitignore b/maple2latex/.gitignore index 10d7823..1beb704 100644 --- a/maple2latex/.gitignore +++ b/maple2latex/.gitignore @@ -1,11 +1,5 @@ functions/ out/ -!out/test.pdf notes.txt resources/ .DS_Store -../.DS_Store -*differences.py -../AzeemOldCode/ -../tex2wiki/ -../main_page/ From 492943ce5d7f5d75d5317d56ef17a607ccc1f4a5 Mon Sep 17 00:00:00 2001 From: Joon Bang Date: Wed, 17 Aug 2016 10:37:46 -0400 Subject: [PATCH 377/402] Fix unit tests --- maple2latex/src/translator.py | 8 +-- maple2latex/test/test_cases.json | 93 ++++++++++++++++++++++++++++- maple2latex/test/test_translator.py | 9 ++- 3 files changed, 102 insertions(+), 8 deletions(-) diff --git a/maple2latex/src/translator.py b/maple2latex/src/translator.py index 89a5b81..be41033 100644 --- a/maple2latex/src/translator.py +++ b/maple2latex/src/translator.py @@ -183,8 +183,8 @@ def from_maple(cls, eq): return cls(eq.label, replace_strings(equation, SPECIAL), metadata) - @classmethod - def get_sortable_label(cls, equation): + @staticmethod + def get_sortable_label(equation): if equation.label == "x.x.x": return [100, 100, 100] # should be three very large numbers @@ -242,7 +242,7 @@ def trim_parens(exp): # type: (str/list) -> str """Removes unnecessary parentheses.""" - if type(exp) == str and exp == "": + if type(exp) in [unicode, str] and exp == "": return "" elif type(exp) == list and exp == []: @@ -352,7 +352,7 @@ def get_arguments(function, arg_string): # handling for trigamma elif function == "Psi" and arg_string[0] == "1": function = "special-trigamma" - args = basic_translate(arg_string[2]) + args = [basic_translate(arg_string[2])] # handling for hypergeometric, q-hypergeometric functions elif function in ["hypergeom", "qhyper"]: diff --git a/maple2latex/test/test_cases.json b/maple2latex/test/test_cases.json index fe3b3d9..e316eeb 100644 --- a/maple2latex/test/test_cases.json +++ b/maple2latex/test/test_cases.json @@ -26,6 +26,16 @@ { "exp": "((1 + 2))", "res": "(1 + 2)" + }, + + { + "exp": "", + "res": "" + }, + + { + "exp": [], + "res": [] } ], @@ -70,6 +80,24 @@ "function": "BesselJ", "arg_string": ["\\nu", ",", "z"], "res": [["\\BesselJ{", "}@{", "}"], ["\\nu", "z"]] + }, + + { + "function": "RealRange", + "arg_string": ["0", ",", "(1)"], + "res": [["(", ", ", ")"], ["0", "1"]] + }, + + { + "function": "Psi", + "arg_string": ["1", ",", "x"], + "res": [["\\digamma'@{", "}"], ["x"]] + }, + + { + "function": "not", + "arg_string": ["x", "<", "3"], + "res": [["", ""], ["x\\geq 3"]] } ], @@ -129,11 +157,15 @@ "booklabelv1 = \"0.0.0\",", "category = \"test_data\"", "lhs = zeta,", - "begin = [[1,2]],", + "begin = map(proc (x) [1, x] end proc,", + "[", + "1,2,3,4,5,6,7,8,9,10,11", + "]),", "front = 3,", - "general = [[4, 5]]," + "factor = 1,", + "general = [[0, 1]]," ], - "res": "\\begin{equation*}\\tag{0.0.0}\n \\zeta\n = 3+\\frac{1}{2} \\subplus \\CFK{m}{2}{\\infty}@@{4}{5}\n % \\category{test_data}\n % \\constraint{$$}\n\\end{equation*}\n" + "res": "\\begin{equation*}\\tag{0.0.0}\n \\zeta\n = 3+\\frac{1}{1} \\subplus \\frac{1}{2} \\subplus \\frac{1}{3} \\subplus \\frac{1}{4} \\subplus \\frac{1}{5} \\subplus \\frac{1}{6} \\subplus \\frac{1}{7} \\subplus \\frac{1}{8} \\subplus \\frac{1}{9} \\subplus \\frac{1}{10} \\subplus \\dots\n % \\category{test_data}\n % \\constraint{$$}\n\\end{equation*}\n" }, { @@ -149,6 +181,61 @@ "odd = [[2, 3]]," ], "res": "\\begin{equation*}\\tag{x.x.x}\n \\zeta\n = 4+\\frac{1}{2} \\subplus 3 \\CFK{m}{2}{\\infty}@@{s_m}{1}\n % \\category{test_data}\n % \\constraint{$$}\n % \\substitution{$s_{2j} = \\frac{0}{1},s_{2j+1} = \\frac{2}{3}$}\n\\end{equation*}\n" + }, + + { + "eq": [ + "create( 'contfrac', ", + "booklabelv1 = \"\",", + "category = \"test_data\"", + "lhs = zeta,", + "begin = [[1,2]],", + "factor = -1,", + "front = 4,", + "general = [[m, 2], [2, 3], [3, 4]]," + ], + "res": "\\begin{equation*}\\tag{x.x.x}\n \\zeta\n = 4+\\frac{1}{2} \\subplus -\\CFK{m}{2}{\\infty}@@{s_m}{1}\n % \\category{test_data}\n % \\constraint{$$}\n % \\substitution{$s_{3j-2} = \\frac{\\left(3j-2\\right)}{2},s_{3j-1} = \\frac{2}{3},s_{3j} = \\frac{3}{4}$}\n\\end{equation*}\n" + } + ], + + "get_sortable_label": [ + { + "eq": [ + "create( 'contfrac', ", + "booklabelv1 = \"\",", + "category = \"test_data\"", + "lhs = zeta,", + "begin = [[1,2]],", + "factor = 3,", + "front = 4,", + "even = [[0, 1]],", + "odd = [[2, 3]]," + ], + "res": [100, 100, 100] + }, + + { + "eq": [ + "create( 'contfrac', ", + "booklabelv1 = \"3.4.5a\",", + "category = \"test_data\"", + "lhs = zeta,", + "factor = 15,", + "general = [ [3, 5] ]," + ], + "res": [3, 4, 5, 1] + }, + + { + "eq": [ + "create( 'contfrac', ", + "booklabelv1 = \"3.4.5.\",", + "category = \"test_data\"", + "lhs = zeta,", + "factor = 15,", + "general = [ [3, 5] ]," + ], + "res": [3, 4, 5] } ] } diff --git a/maple2latex/test/test_translator.py b/maple2latex/test/test_translator.py index 512b174..c0a09c2 100644 --- a/maple2latex/test/test_translator.py +++ b/maple2latex/test/test_translator.py @@ -15,6 +15,7 @@ get_arguments_test_cases = TEST_CASES["get_arguments"] translate_test_cases = TEST_CASES["translate"] from_maple_test_cases = TEST_CASES["from_maple"] +get_sortable_label_test_cases = TEST_CASES["get_sortable_label"] class TestParseBrackets(TestCase): @@ -50,4 +51,10 @@ def test_translate(self): class TestFromMaple(TestCase): def test_from_maple(self): for case in from_maple_test_cases: - self.assertEqual(str(t.LatexEquation.from_maple(t.MapleEquation(case["eq"]))), case["res"]) \ No newline at end of file + self.assertEqual(str(t.LatexEquation.from_maple(t.MapleEquation(case["eq"]))), case["res"]) + + +class TestGetSortableLabel(TestCase): + def test_get_sortable_label(self): + for case in get_sortable_label_test_cases: + self.assertEqual(t.LatexEquation.get_sortable_label(t.MapleEquation(case["eq"])), case["res"]) From a6e7085ab15fa6f3b5343dc2b4a6039116ff8f32 Mon Sep 17 00:00:00 2001 From: Joon Bang Date: Wed, 17 Aug 2016 10:59:35 -0400 Subject: [PATCH 378/402] Fix landscape issues --- maple2latex/src/translator.py | 57 +++++++++++++++++++++-------------- 1 file changed, 34 insertions(+), 23 deletions(-) diff --git a/maple2latex/src/translator.py b/maple2latex/src/translator.py index be41033..d99b326 100644 --- a/maple2latex/src/translator.py +++ b/maple2latex/src/translator.py @@ -112,6 +112,7 @@ def from_maple(cls, eq): elif eq.eq_type == "contfrac": forms = list() + # substitution handling if len(parse_brackets(eq.general[0])) > 1: forms = parse_brackets(eq.general[0]) @@ -120,29 +121,7 @@ def from_maple(cls, eq): for form in eq.general: forms += parse_brackets(form) - replacements = list() - if len(eq.general) == 2: - replacements = ["2j", "2j+1"] - elif forms: - for i in range(len(forms)): - replacement = str(len(forms)) + "j" - if i < len(forms) - 1: - replacement += "-" + str(len(forms) - i - 1) - - replacements.append(replacement) - - for i, form in enumerate(forms): - for j, half in enumerate(form): - half = tokenize(half) - for k, ch in enumerate(half): - if ch == "m": - half[k] = "(" + replacements[i] + ")" - - form[j] = ' '.join(half) - - forms[i] = "s_{" + replacements[i] + "} = " + make_frac(form) - - metadata["substitution"] = ','.join(forms) + metadata["substitution"] = ','.join(perform_substitution(eq, forms)) pieces = ["s_m", "1"] else: @@ -323,6 +302,35 @@ def basic_translate(exp): return ''.join(exp) +def perform_substitution(eq, forms): + # (MapleEquation, list) -> list + """Performs variable substitutions on the strings in eq.general.""" + + replacements = list() + if len(eq.general) == 2: + replacements = ["2j", "2j+1"] + elif forms: + for i, _ in enumerate(forms): + replacement = str(len(forms)) + "j" + if i < len(forms) - 1: + replacement += "-" + str(len(forms) - i - 1) + + replacements.append(replacement) + + for i, form in enumerate(forms): + for j, half in enumerate(form): + half = tokenize(half) + for k, ch in enumerate(half): + if ch == "m": + half[k] = "(" + replacements[i] + ")" + + form[j] = ' '.join(half) + + forms[i] = "s_{" + replacements[i] + "} = " + make_frac(form) + + return forms + + def get_arguments(function, arg_string): # type: (str, list) -> (list, list) """Generates the function pieces and the arguments.""" @@ -333,6 +341,7 @@ def get_arguments(function, arg_string): if not arg_string: args = [] + # handling for not elif function == "not": inversion = {"<": "\\geq ", ">": "\\leq ", "\\in": "\\notin "} @@ -342,6 +351,7 @@ def get_arguments(function, arg_string): args = [basic_translate(arg_string)] + # handling for ranges (constraints) elif function == "RealRange": for i, piece in enumerate(arg_string): if trim_parens(piece) != piece: @@ -377,6 +387,7 @@ def get_arguments(function, arg_string): if args[1] == "infinity": args[1] = "\\infty" + # handling for function in case it has optional parentheses elif function in MULTI_ARGS and len(arg_string) == 1: parens_mod = True args = basic_translate(arg_string).split(",") From 622105e29410b5016b48880081ee5d7d8dbde9a5 Mon Sep 17 00:00:00 2001 From: Joon Bang Date: Thu, 18 Aug 2016 13:58:29 -0400 Subject: [PATCH 379/402] Improve code quality --- maple2latex/src/translator.py | 110 ++++++++++++++-------------------- 1 file changed, 46 insertions(+), 64 deletions(-) diff --git a/maple2latex/src/translator.py b/maple2latex/src/translator.py index d99b326..8c3d7ce 100644 --- a/maple2latex/src/translator.py +++ b/maple2latex/src/translator.py @@ -110,43 +110,58 @@ def from_maple(cls, eq): equation += "(" + eq.general + ")" elif eq.eq_type == "contfrac": - forms = list() + forms = parse_brackets(eq.general[0]) - # substitution handling - if len(parse_brackets(eq.general[0])) > 1: - forms = parse_brackets(eq.general[0]) - - if len(eq.general) == 2 or forms: - if not forms: # forms is empty + if len(eq.general) == 2 or len(forms) > 1: + if len(forms) == 1: + forms = list() for form in eq.general: forms += parse_brackets(form) - metadata["substitution"] = ','.join(perform_substitution(eq, forms)) + replacements = list() + if len(eq.general) == 2: + replacements = ["2j", "2j+1"] + elif forms: + for i in range(len(forms)): + replacement = str(len(forms)) + "j" + if i < len(forms) - 1: + replacement += "-" + str(len(forms) - i - 1) + + replacements.append(replacement) + + for i, form in enumerate(forms): + for j, half in enumerate(form): + half = tokenize(half) + for k, ch in enumerate(half): + if ch == "m": + half[k] = "(" + replacements[i] + ")" + + form[j] = ' '.join(half) + + forms[i] = "s_{" + replacements[i] + "} = " + make_frac(form) + + metadata["substitution"] = ','.join(forms) pieces = ["s_m", "1"] else: pieces = parse_brackets(eq.general[0])[0] - if eq.begin != "": - eq.begin = parse_brackets(eq.begin) - start = 1 # in case the value of start isn't assigned + eq.begin = parse_brackets(eq.begin) + for piece in eq.begin: + equation += make_frac(piece) + " \\subplus " + start += 1 + # add terms before general if eq.front != "": equation += eq.front + "+" start = 1 - if eq.begin != "": - for piece in eq.begin: - equation += make_frac(piece) + " \\subplus " - start += 1 - - if eq.factor != "": - if eq.factor == "-1": - equation += "-" - else: - equation += eq.factor + " " + if eq.factor == "-1": + equation += "-" + elif eq.factor != "": + equation += eq.factor + " " # trim unnecessary parentheses for i, element in enumerate(pieces): @@ -162,20 +177,17 @@ def from_maple(cls, eq): return cls(eq.label, replace_strings(equation, SPECIAL), metadata) - @staticmethod - def get_sortable_label(equation): + @classmethod + def get_sortable_label(cls, equation): if equation.label == "x.x.x": - return [100, 100, 100] # should be three very large numbers + return [float("inf"), float("inf"), float("inf")] # should be three very large numbers label = copy.copy(equation.label) for i, ch in enumerate(ascii_lowercase): if label[-1] == ch: label = label[:-1] + "." + str(i + 1) - if label[-1] == ".": - label = label[:-1] - - return map(int, label.split(".")) + return map(int, filter(lambda x: x != "", label.split("."))) def __str__(self): metadata = "" @@ -205,9 +217,11 @@ def parse_brackets(string): # type: ((str, list)) -> list """Obtains the contents from data encapsulated in square brackets.""" + if string == "": + return "" + + string = replace_strings(string, {"[": "", "]": ""}) exp = string[1:-1].split(",") - for i, e in enumerate(exp): - exp[i] = replace_strings(e, {"[": "", "]": ""}).strip() i = 0 while i + 1 < len(exp): @@ -221,7 +235,7 @@ def trim_parens(exp): # type: (str/list) -> str """Removes unnecessary parentheses.""" - if type(exp) in [unicode, str] and exp == "": + if type(exp) == str and exp == "": return "" elif type(exp) == list and exp == []: @@ -302,35 +316,6 @@ def basic_translate(exp): return ''.join(exp) -def perform_substitution(eq, forms): - # (MapleEquation, list) -> list - """Performs variable substitutions on the strings in eq.general.""" - - replacements = list() - if len(eq.general) == 2: - replacements = ["2j", "2j+1"] - elif forms: - for i, _ in enumerate(forms): - replacement = str(len(forms)) + "j" - if i < len(forms) - 1: - replacement += "-" + str(len(forms) - i - 1) - - replacements.append(replacement) - - for i, form in enumerate(forms): - for j, half in enumerate(form): - half = tokenize(half) - for k, ch in enumerate(half): - if ch == "m": - half[k] = "(" + replacements[i] + ")" - - form[j] = ' '.join(half) - - forms[i] = "s_{" + replacements[i] + "} = " + make_frac(form) - - return forms - - def get_arguments(function, arg_string): # type: (str, list) -> (list, list) """Generates the function pieces and the arguments.""" @@ -341,7 +326,6 @@ def get_arguments(function, arg_string): if not arg_string: args = [] - # handling for not elif function == "not": inversion = {"<": "\\geq ", ">": "\\leq ", "\\in": "\\notin "} @@ -351,7 +335,6 @@ def get_arguments(function, arg_string): args = [basic_translate(arg_string)] - # handling for ranges (constraints) elif function == "RealRange": for i, piece in enumerate(arg_string): if trim_parens(piece) != piece: @@ -362,7 +345,7 @@ def get_arguments(function, arg_string): # handling for trigamma elif function == "Psi" and arg_string[0] == "1": function = "special-trigamma" - args = [basic_translate(arg_string[2])] + args = basic_translate(arg_string[2]) # handling for hypergeometric, q-hypergeometric functions elif function in ["hypergeom", "qhyper"]: @@ -387,7 +370,6 @@ def get_arguments(function, arg_string): if args[1] == "infinity": args[1] = "\\infty" - # handling for function in case it has optional parentheses elif function in MULTI_ARGS and len(arg_string) == 1: parens_mod = True args = basic_translate(arg_string).split(",") From 8eaf3657c14da5cfe3e6611a11755fb56753e554 Mon Sep 17 00:00:00 2001 From: Joon Bang Date: Tue, 6 Sep 2016 14:03:30 -0400 Subject: [PATCH 380/402] Fix unit tests (again) --- .DS_Store | Bin 6148 -> 0 bytes maple2latex/src/main.py | 29 +++-- maple2latex/src/translator.py | 197 ++++++++++++++++--------------- maple2latex/test/test_cases.json | 14 +-- 4 files changed, 128 insertions(+), 112 deletions(-) delete mode 100644 .DS_Store diff --git a/.DS_Store b/.DS_Store deleted file mode 100644 index 13a31254b7e6c325c5ff2fb524c0a254b62b3d0b..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 6148 zcmeHKO>fgc5S?vPa8n^%0a7I{K7v$GQ_+uOnx>!%3Q<#9A&P=s8)D&lqr`1msz^C= z<-(co|AFAd4}hP-sk~Wlk)5{J3eb)<`^M{eJNE2kH%mmKvg4PCY$CFtjD;ea6NI0$ z9+M&6a{(yS7*%p9CP~k588I0!8Ti{Xz`xxhEl@-O?NR3UyBx_aFOnXf^WtzDdhJjN zETR!`UV;Md*a|=db%{}luGlxN#|g0esh+n0l}D}KtGDmofB5+6^OvtG0VJ>`D7mEZ2)@BMB|W?Kb{xt0IXswMRH;v> z36z+Ap1l2h;)%;GJ6#slP#7)vMygnXyuX^Ltp6yAVhP1uMWHO6RFcfdm86Bz3s8g! z-pEDk4n*J*%to-Ni#K@zJ<*>FE;l~tXCr!otg*8RXQG%4m<;?w8Q}H7fijjg_ES_# z2O4z*02a`#1UCO&V2-1)tg)XWo&?gk3 zqGNnVxsxcTXj+p2lYx8&hSOV{_y48S&;NOnxic9s8ThX dict + """Obtain data from the functions/ directory.""" sections = dict() for info in dirs: # search through directories @@ -76,15 +71,25 @@ def translate_files(root_directory): else: sections[section][subsection] = [repr_label, "\n".join(map(str, formulae))] + return sections + + +def translate_files(root_directory): + # type: () + """Generates and writes the results of traversing the root directory, and translating all files in FILES.""" + + root_depth = len(root_directory.split("/")) + + dirs = [d for d in os.walk(root_directory) if d[0].count("/") == root_depth] + + sections = get_sections_data(dirs, root_depth=root_depth) + # generate subsection headers for section, subsections in sections.iteritems(): - print section - keys = sorted(subsections.keys(), key=lambda x: subsections[x][0]) text = "" for subsection in keys: - print " " + subsection text += "\n\\subsection{" + subsection + "}\n" text += subsections[subsection][1] @@ -98,7 +103,7 @@ def translate_files(root_directory): # write output to file with open("maple2latex/out/test.tex", "w") as test: with open("maple2latex/out/primer") as primer: - test.write(primer.read() + result + "\n\\end{document}") + test.write(primer.read() + result + "\n\\end{document}\n") if __name__ == '__main__': translate_files("maple2latex/functions") diff --git a/maple2latex/src/translator.py b/maple2latex/src/translator.py index 8c3d7ce..8fa3e7f 100644 --- a/maple2latex/src/translator.py +++ b/maple2latex/src/translator.py @@ -5,6 +5,7 @@ import copy import json +import sys from string import ascii_lowercase from maple_tokenize import tokenize @@ -42,9 +43,6 @@ def __init__(self, inp): temp = temp[:10] line[1] = str(temp)[1:-1] - elif line[0] == "booklabelv1" and line[1] == '"",': - line[1] = "x.x.x" - elif line[0] in ["booklabelv1", "booklabelv2", "general", "constraints", "begin", "parameters"]: line[1] = line[1].strip()[1:-2].strip() @@ -67,7 +65,7 @@ def __init__(self, inp): # even-odd case handling if "general" in self.fields: - self.general = [self.fields["general"]] + self.general = self.fields["general"] elif "even" in self.fields and "odd" in self.fields: self.general = [self.fields["even"], self.fields["odd"]] @@ -80,88 +78,61 @@ def __init__(self, label, equation, metadata): @classmethod def from_maple(cls, eq): + # (List[MapleEquation]) -> LatexEquation # modify fields eq.lhs = translate(eq.lhs) eq.factor = translate(eq.fields["factor"]) eq.front = translate(eq.fields["front"]) - - equation = eq.lhs + "\n = " + eq.begin = parse_brackets(eq.begin) if eq.factor == "1": eq.factor = "" + equation = eq.lhs + "\n = " metadata = dict() # translates the Maple information (with spacing) if eq.eq_type == "series": - eq.general = translate(eq.general[0]) - - if eq.factor != "": - equation += eq.factor + " " - - elif eq.front != "": - equation += eq.front + "+" - + # add factor and front with accompanying symbols, if they exist + equation += evaluate(eq.factor + " ", eq.factor) + evaluate(eq.front + "+", eq.front) equation += "\\sum_{k=0}^\\infty " + eq.general = translate(eq.general) + if eq.category == "power series": equation += eq.general elif eq.category == "asymptotic series": # make sure to fix asymptotic series equation += "(" + eq.general + ")" elif eq.eq_type == "contfrac": - forms = parse_brackets(eq.general[0]) - - if len(eq.general) == 2 or len(forms) > 1: - if len(forms) == 1: - forms = list() - for form in eq.general: - forms += parse_brackets(form) - - replacements = list() - if len(eq.general) == 2: - replacements = ["2j", "2j+1"] - elif forms: - for i in range(len(forms)): - replacement = str(len(forms)) + "j" - if i < len(forms) - 1: - replacement += "-" + str(len(forms) - i - 1) - - replacements.append(replacement) - - for i, form in enumerate(forms): - for j, half in enumerate(form): - half = tokenize(half) - for k, ch in enumerate(half): - if ch == "m": - half[k] = "(" + replacements[i] + ")" - - form[j] = ' '.join(half) - - forms[i] = "s_{" + replacements[i] + "} = " + make_frac(form) - - metadata["substitution"] = ','.join(forms) + # obtain data from eq.general + try: + pieces = parse_brackets(eq.general) + except AttributeError: # in form of even, odd + pieces = list() + for piece in eq.general: + pieces += parse_brackets(piece) + + # if there are multiple forms (requires substitution) + if len(pieces) > 1: + metadata["substitution"] = ','.join(perform_substitution(eq, pieces)) pieces = ["s_m", "1"] - else: - pieces = parse_brackets(eq.general[0])[0] + pieces = pieces[0] start = 1 # in case the value of start isn't assigned - eq.begin = parse_brackets(eq.begin) + # add terms before general + equation += evaluate(eq.front + "+", eq.front) + for piece in eq.begin: equation += make_frac(piece) + " \\subplus " start += 1 - # add terms before general - if eq.front != "": - equation += eq.front + "+" - start = 1 - if eq.factor == "-1": equation += "-" - elif eq.factor != "": - equation += eq.factor + " " + else: + equation += evaluate(eq.factor + " ", eq.factor) # trim unnecessary parentheses for i, element in enumerate(pieces): @@ -174,13 +145,14 @@ def from_maple(cls, eq): metadata["constraint"] = translate(eq.constraints) metadata["category"] = eq.category + metadata["mapletag"] = eq.label return cls(eq.label, replace_strings(equation, SPECIAL), metadata) - @classmethod - def get_sortable_label(cls, equation): - if equation.label == "x.x.x": - return [float("inf"), float("inf"), float("inf")] # should be three very large numbers + @staticmethod + def get_sortable_label(equation): + if equation.label == "": + return [sys.maxsize, sys.maxsize, sys.maxsize] label = copy.copy(equation.label) for i, ch in enumerate(ascii_lowercase): @@ -192,16 +164,25 @@ def get_sortable_label(cls, equation): def __str__(self): metadata = "" for data_type, data in self.metadata.iteritems(): + if data == "": + continue + if data_type in ["constraint", "substitution"]: # mathmode metadata += " % \\" + data_type + "{$" + replace_strings(data, SPECIAL) + "$}\n" else: - metadata += " % \\" + data_type + "{" + data + "}\n" + metadata += " % \\" + data_type + "{" + data + "}\n"\ + + return "\\begin{equation}\n " + self.equation + "\n" + metadata + "\\end{equation}\n" - # if data_type in ["substitution"]: - # metadata += " " + replace_strings(data, SPECIAL) + "\n" - return "\\begin{equation*}\\tag{" + self.label + "}\n " + self.equation + "\n" + \ - metadata + "\\end{equation*}\n" +def evaluate(data, key): + # (Any, bool) -> Any + """ + Will return data as long as key is not False. + If key is False, returns 0 or blank string, depending on type(data). + Used as replacement for "if data != "": ...", and similar statements. + """ + return data * bool(key) def replace_strings(string, keys): @@ -214,14 +195,15 @@ def replace_strings(string, keys): def parse_brackets(string): - # type: ((str, list)) -> list + # type: (str) -> List[str] """Obtains the contents from data encapsulated in square brackets.""" - if string == "": + if not string: return "" - string = replace_strings(string, {"[": "", "]": ""}) exp = string[1:-1].split(",") + for i, e in enumerate(exp): + exp[i] = replace_strings(e, {"[": "", "]": ""}).strip() i = 0 while i + 1 < len(exp): @@ -232,12 +214,11 @@ def parse_brackets(string): def trim_parens(exp): - # type: (str/list) -> str + # type: (str/List[str]) -> str """Removes unnecessary parentheses.""" - if type(exp) == str and exp == "": + if type(exp) in [unicode, str] and exp == "": return "" - elif type(exp) == list and exp == []: return [] @@ -262,14 +243,14 @@ def trim_parens(exp): def make_frac(parts): - # type: (list) -> str + # type: (List[str]) -> str """Generate a LaTeX fraction from its numerator and denominator.""" return translate("(" + parts[0] + ") / (" + parts[1] + ")") def basic_translate(exp): - # type: (list) -> str + # type: (List[str]) -> str """Translates basic mathematical operations.""" # translates operations in place @@ -316,17 +297,47 @@ def basic_translate(exp): return ''.join(exp) -def get_arguments(function, arg_string): - # type: (str, list) -> (list, list) +def perform_substitution(eq, forms): + # (MapleEquation, List[str]) -> List[str] + """Performs variable substitutions on the strings in eq.general.""" + + replacements = list() + if len(eq.general) == 2: + replacements = ["2j", "2j+1"] + elif forms: + for i, _ in enumerate(forms): + replacement = str(len(forms)) + "j" + if i < len(forms) - 1: + replacement += "-" + str(len(forms) - i - 1) + + replacements.append(replacement) + + for i, form in enumerate(forms): + for j, half in enumerate(form): + half = tokenize(half) + for k, ch in enumerate(half): + if ch == "m": + half[k] = "(" + replacements[i] + ")" + + form[j] = ' '.join(half) + + forms[i] = "s_{" + replacements[i] + "} = " + make_frac(form) + + return forms + + +def get_arguments(function_name, arg_string): + # type: (str, List[str]) -> (List[str], List[str]) """Generates the function pieces and the arguments.""" parens_mod = False # no arguments - if not arg_string: + if not arg_string: # arg_string == [] args = [] - elif function == "not": + # handling for not + elif function_name == "not": inversion = {"<": "\\geq ", ">": "\\leq ", "\\in": "\\notin "} for i, ch in enumerate(arg_string): @@ -335,7 +346,8 @@ def get_arguments(function, arg_string): args = [basic_translate(arg_string)] - elif function == "RealRange": + # handling for ranges (constraints) + elif function_name == "RealRange": for i, piece in enumerate(arg_string): if trim_parens(piece) != piece: arg_string[i] = trim_parens(piece) @@ -343,34 +355,32 @@ def get_arguments(function, arg_string): args = basic_translate(arg_string).split(",") # handling for trigamma - elif function == "Psi" and arg_string[0] == "1": - function = "special-trigamma" - args = basic_translate(arg_string[2]) + elif function_name == "Psi" and arg_string[0] == "1": + function_name = "special-trigamma" + args = [basic_translate(arg_string[2])] # handling for hypergeometric, q-hypergeometric functions - elif function in ["hypergeom", "qhyper"]: + elif function_name in ["hypergeom", "qhyper"]: args = list() for s in ' '.join(arg_string).split("] , "): args.append(basic_translate(replace_strings(s, {"[": "", "]": ""}).split())) - if function == "qhyper": + if function_name == "qhyper": args += args.pop(2).split(",") for p, i in enumerate([1, 0]): - arg_count = 0 - if args[i + p] != "": - arg_count = args[i + p].count(",") + 1 - + arg_count = evaluate(args[i + p].count(",") + 1, args[i + p]) args.insert(0, str(arg_count)) # handling for sums - elif function == "sum": + elif function_name == "sum": args = basic_translate(arg_string).split(",") args = args.pop(1).split("..") + [args[0]] if args[1] == "infinity": args[1] = "\\infty" - elif function in MULTI_ARGS and len(arg_string) == 1: + # handling for function in case it has optional parentheses + elif function_name in MULTI_ARGS and len(arg_string) == 1: parens_mod = True args = basic_translate(arg_string).split(",") @@ -378,19 +388,20 @@ def get_arguments(function, arg_string): args = basic_translate(arg_string).split(",") result = list() - for n in FUNCTIONS: - for variant in FUNCTIONS[n]: - if function == n and len(args) == variant["args"]: + for function in FUNCTIONS: + for variant in FUNCTIONS[function]: + if function_name == function and len(args) == variant["args"]: result = copy.copy(variant["repr"]) + # modify macro to form without parentheses if parens_mod: - result[0] = result[0].replace("@", "@@") # make the macro without parens + result[0] = result[0].replace("@", "@@") return [result, args] def generate_function(name, args): - # type: (str, list) -> str + # type: (str, List[str]) -> str """Generate a function with the provided function name and arguments.""" result, args = get_arguments(name, args) diff --git a/maple2latex/test/test_cases.json b/maple2latex/test/test_cases.json index e316eeb..85881ee 100644 --- a/maple2latex/test/test_cases.json +++ b/maple2latex/test/test_cases.json @@ -124,7 +124,7 @@ "general = [ 2 * k - 3 ],", "constraints = { k::integer }," ], - "res": "\\begin{equation*}\\tag{0.0.0}\n n\n = \\frac{4}{5} \\sum_{k=0}^\\infty 2k-3\n % \\category{power series}\n % \\constraint{$k\\in\\mathbb{Z}$}\n\\end{equation*}\n" + "res": "\\begin{equation}\n n\n = \\frac{4}{5} \\sum_{k=0}^\\infty 2k-3\n % \\category{power series}\n % \\mapletag{0.0.0}\n % \\constraint{$k\\in\\mathbb{Z}$}\n\\end{equation}\n" }, { @@ -136,7 +136,7 @@ "front = 17,", "general = [ k / 2 ]," ], - "res": "\\begin{equation*}\\tag{0.0.0}\n p\n = 17+\\sum_{k=0}^\\infty \\left(\\frac{k}{2}\\right)\n % \\category{asymptotic series}\n % \\constraint{$$}\n\\end{equation*}\n" + "res": "\\begin{equation}\n p\n = 17+\\sum_{k=0}^\\infty \\left(\\frac{k}{2}\\right)\n % \\category{asymptotic series}\n % \\mapletag{0.0.0}\n\\end{equation}\n" }, { @@ -148,7 +148,7 @@ "factor = 15,", "general = [ [3, 5] ]," ], - "res": "\\begin{equation*}\\tag{0.0.0}\n \\zeta\n = 15 \\CFK{m}{1}{\\infty}@@{3}{5}\n % \\category{test_data}\n % \\constraint{$$}\n\\end{equation*}\n" + "res": "\\begin{equation}\n \\zeta\n = 15 \\CFK{m}{1}{\\infty}@@{3}{5}\n % \\category{test_data}\n % \\mapletag{0.0.0}\n\\end{equation}\n" }, { @@ -165,7 +165,7 @@ "factor = 1,", "general = [[0, 1]]," ], - "res": "\\begin{equation*}\\tag{0.0.0}\n \\zeta\n = 3+\\frac{1}{1} \\subplus \\frac{1}{2} \\subplus \\frac{1}{3} \\subplus \\frac{1}{4} \\subplus \\frac{1}{5} \\subplus \\frac{1}{6} \\subplus \\frac{1}{7} \\subplus \\frac{1}{8} \\subplus \\frac{1}{9} \\subplus \\frac{1}{10} \\subplus \\dots\n % \\category{test_data}\n % \\constraint{$$}\n\\end{equation*}\n" + "res": "\\begin{equation}\n \\zeta\n = 3+\\frac{1}{1} \\subplus \\frac{1}{2} \\subplus \\frac{1}{3} \\subplus \\frac{1}{4} \\subplus \\frac{1}{5} \\subplus \\frac{1}{6} \\subplus \\frac{1}{7} \\subplus \\frac{1}{8} \\subplus \\frac{1}{9} \\subplus \\frac{1}{10} \\subplus \\dots\n % \\category{test_data}\n % \\mapletag{0.0.0}\n\\end{equation}\n" }, { @@ -180,7 +180,7 @@ "even = [[0, 1]],", "odd = [[2, 3]]," ], - "res": "\\begin{equation*}\\tag{x.x.x}\n \\zeta\n = 4+\\frac{1}{2} \\subplus 3 \\CFK{m}{2}{\\infty}@@{s_m}{1}\n % \\category{test_data}\n % \\constraint{$$}\n % \\substitution{$s_{2j} = \\frac{0}{1},s_{2j+1} = \\frac{2}{3}$}\n\\end{equation*}\n" + "res": "\\begin{equation}\n \\zeta\n = 4+\\frac{1}{2} \\subplus 3 \\CFK{m}{2}{\\infty}@@{s_m}{1}\n % \\category{test_data}\n % \\substitution{$s_{2j} = \\frac{0}{1},s_{2j+1} = \\frac{2}{3}$}\n\\end{equation}\n" }, { @@ -194,7 +194,7 @@ "front = 4,", "general = [[m, 2], [2, 3], [3, 4]]," ], - "res": "\\begin{equation*}\\tag{x.x.x}\n \\zeta\n = 4+\\frac{1}{2} \\subplus -\\CFK{m}{2}{\\infty}@@{s_m}{1}\n % \\category{test_data}\n % \\constraint{$$}\n % \\substitution{$s_{3j-2} = \\frac{\\left(3j-2\\right)}{2},s_{3j-1} = \\frac{2}{3},s_{3j} = \\frac{3}{4}$}\n\\end{equation*}\n" + "res": "\\begin{equation}\n \\zeta\n = 4+\\frac{1}{2} \\subplus -\\CFK{m}{2}{\\infty}@@{s_m}{1}\n % \\category{test_data}\n % \\substitution{$s_{3j-2} = \\frac{\\left(3j-2\\right)}{2},s_{3j-1} = \\frac{2}{3},s_{3j} = \\frac{3}{4}$}\n\\end{equation}\n" } ], @@ -211,7 +211,7 @@ "even = [[0, 1]],", "odd = [[2, 3]]," ], - "res": [100, 100, 100] + "res": [9223372036854775807, 9223372036854775807, 9223372036854775807] }, { From 9144b1706c3c00a1a9a7e259efa943da4c30847c Mon Sep 17 00:00:00 2001 From: Joon Bang Date: Fri, 28 Oct 2016 17:19:11 -0400 Subject: [PATCH 381/402] Add option for output directory --- maple2latex/src/main.py | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/maple2latex/src/main.py b/maple2latex/src/main.py index 49de8cb..36fe1df 100644 --- a/maple2latex/src/main.py +++ b/maple2latex/src/main.py @@ -74,7 +74,7 @@ def get_sections_data(dirs, root_depth=0): return sections -def translate_files(root_directory): +def translate_files(root_directory, output_file): # type: () """Generates and writes the results of traversing the root directory, and translating all files in FILES.""" @@ -101,9 +101,9 @@ def translate_files(root_directory): result += "\\section{" + section + "}\n" + section_data + "\n\n\n" # write output to file - with open("maple2latex/out/test.tex", "w") as test: + with open(output_file, "w") as test: with open("maple2latex/out/primer") as primer: test.write(primer.read() + result + "\n\\end{document}\n") if __name__ == '__main__': - translate_files("maple2latex/functions") + translate_files("maple2latex/functions", "maple2latex/out/test.tex") From 084cef7e9f6a2d58b3df1228b7b0f66297ce43b2 Mon Sep 17 00:00:00 2001 From: Joon Date: Wed, 23 Nov 2016 20:49:49 -0500 Subject: [PATCH 382/402] Polish methods, fix bug with equation handling --- tex2wiki/src/tex2wiki.py | 96 ++++++++++++++++++++-------------------- 1 file changed, 47 insertions(+), 49 deletions(-) diff --git a/tex2wiki/src/tex2wiki.py b/tex2wiki/src/tex2wiki.py index 8aa4c5e..2d12bbe 100644 --- a/tex2wiki/src/tex2wiki.py +++ b/tex2wiki/src/tex2wiki.py @@ -1,9 +1,10 @@ __author__ = "Joon Bang" __status__ = "Prototype" -INPUT_FILE = "tex2wiki/data/01outb.tex" -OUTPUT_FILE = "tex2wiki/data/01outb.mmd" +INPUT_FILE = "tex2wiki/data/test.tex" +OUTPUT_FILE = "tex2wiki/data/test.mmd" GLOSSARY_LOCATION = "tex2wiki/data/new.Glossary.csv" +TITLE_STRING = "Orthogonal Polynomials" METADATA_TYPES = ["substitution", "constraint"] METADATA_MEANING = {"substitution": "Substitution(s)", "constraint": "Constraint(s)"} @@ -18,7 +19,7 @@ def __init__(self, label, equation, metadata): def generate_html(tag_name, text, options=None, spacing=2): - # (str, str(, dict, bool)) -> str + # type: (str, str(, dict, bool)) -> str """ Generates an html tag, with optional html parameters. When spacing = 0, there should be no spacing. @@ -42,9 +43,8 @@ def generate_html(tag_name, text, options=None, spacing=2): def generate_math_html(text, options=None, spacing=True): - """ - Special case of generate_html, where the tag is "math". - """ + # type: (str(, dict, bool)) -> str + """Special case of generate_html, where the tag is "math".""" if options is None: options = {} @@ -60,8 +60,8 @@ def generate_math_html(text, options=None, spacing=True): def generate_link(left, right=""): - """Generates a link thingie.""" - + # type: (str(, str)) -> str + """Generates a MediaWiki link.""" if right == "": return "[[" + left + "|" + left + "]]" @@ -69,7 +69,8 @@ def generate_link(left, right=""): def multi_split(s, seps): - """Copy pasted from internet!""" + # type: (str, list) -> list + """Splits a string on multiple characters.""" res = [s] for sep in seps: s, res = res, [] @@ -79,14 +80,16 @@ def multi_split(s, seps): def convert_dollar_signs(string): + # type: (str) -> str + """Converts dollar signs to html for math mode.""" count = 0 result = "" for i, ch in enumerate(string): - if ch == "$" and count % 2 == 0: - result += "{\\displaystyle " - count += 1 - elif ch == "$": - result += "}" + if ch == "$": + if count % 2 == 0: + result += "{\\displaystyle " + else: + result += "}" count += 1 else: result += ch @@ -94,24 +97,9 @@ def convert_dollar_signs(string): return result -def format_formula(formula): - if formula[0] == ":": - return formula[1:].zfill(2) - - formula = multi_split(formula.split("Formula:", 1)[1], [".", ":"]) - - for j in [-1, -2, -3]: - formula[j] = formula[j].zfill(2) - - if len(formula) == 3: - formula = formula[0] + "." + formula[1] + ":" + formula[2] - else: - formula = ":".join(formula[:-3]) + ":" + formula[-3] + "." + formula[-2] + ":" + formula[-1] - - return formula - - def format_metadata(string): + # type: (str) -> str + """Formats the metadata of an equation.""" if string == "": return "" @@ -134,20 +122,22 @@ def format_metadata(string): def extract_data(data): + # type: (list) -> list + """Extracts the equations and pertinent data from the tex file.""" result = list() + for section_data in data: equations = section_data[1][:-1] for i, equation in enumerate(equations): # formula stuff try: - formula = format_formula(get_data_str(equation, latex="\\formula")) + formula = get_data_str(equation, latex="\\mapletag") except IndexError: # there is no formula. break - equation = equation.split("\n")[1:] + equation = equation.split("\n") # get metadata - percent_list = list() raw_metadata = "" j = 0 @@ -161,8 +151,6 @@ def extract_data(data): for data_type in METADATA_TYPES: metadata[data_type] = format_metadata(get_data_str(raw_metadata, latex="\\"+data_type)) - # print metadata - equations[i] = LatexEquation(formula, '\n'.join(equation), metadata) result.append([section_data[0], equations]) @@ -171,6 +159,8 @@ def extract_data(data): def find_end(text, left_delimiter, right_delimiter, start=0): + # type: (str, str, str(, int)) -> int + """A .find that accounts for nested delimiters.""" net = 0 # left delimiters encountered - right delimiters encountered for i, ch in enumerate(text[start:]): if ch == left_delimiter: @@ -185,14 +175,19 @@ def find_end(text, left_delimiter, right_delimiter, start=0): def get_data_str(text, latex=""): - if latex + "{" not in text: + # type: (str(, str)) -> str + """Gets the string in between curly brackets.""" + start = text.find(latex + "{") + + if start == -1: return "" - start = text.find(latex + "{") return text[start + len(latex + "{"):find_end(text, "{", "}", start)] def generate_nav_bar(info): + # type: (list) -> list + """Generates the navigation bar code for a page.""" links = list() for link, text in info: link = link.replace("''", "") @@ -200,8 +195,8 @@ def generate_nav_bar(info): links.append(generate_link(link, text)) nav_section = generate_html("div", "<< " + links[0], options={"id": "alignleft"}, spacing=1) + "\n" + \ - generate_html("div", links[1], options={"id": "aligncenter"}, spacing=1) + "\n" + \ - generate_html("div", links[2] + " >>", options={"id": "alignright"}, spacing=1) + generate_html("div", links[1], options={"id": "aligncenter"}, spacing=1) + "\n" + \ + generate_html("div", links[2] + " >>", options={"id": "alignright"}, spacing=1) header = generate_html("div", nav_section, options={"id": "drmf_head"}) footer = generate_html("div", nav_section, options={"id": "drmf_foot"}) @@ -210,7 +205,7 @@ def generate_nav_bar(info): def get_macro_name(macro): - # (str) -> str + # type: (str) -> str """Obtains the macro name.""" macro_name = "" for ch in macro: @@ -223,7 +218,7 @@ def get_macro_name(macro): def find_all(pattern, string): - # (str, str) -> generator + # type: (str, str) -> generator """Finds all instances of pattern in string.""" i = string.find(pattern) @@ -233,7 +228,7 @@ def find_all(pattern, string): def get_symbols(text, glossary): - # (str, dict) -> str + # type: (str, dict) -> str """Generates span text based on symbols present in text.""" symbols = set() @@ -275,18 +270,20 @@ def get_symbols(text, glossary): def create_general_pages(data, title): + # type: (list, str) -> str + """Creates the 'index' pages for each section.""" ret = "" # get list of section names section_names = [d[0] for d in data] - section_names = ["Orthogonal Polynomials"] + section_names + ["Orthogonal Polynomials"] + section_names = [TITLE_STRING] + section_names + [TITLE_STRING] for i, section_data in enumerate(data): section_name = section_data[0] result = "drmf_bof\n'''" + section_name.replace("''", "") + "'''\n{{DISPLAYTITLE:" + section_name + "}}\n" # get header and footer - center_text = ("Orthogonal Polynomials" + "#Sections in " + title).replace(" ", "_") + center_text = (TITLE_STRING + "#Sections in " + title).replace(" ", "_") link_info = [[section_names[i], section_names[i]], [center_text, section_names[i + 1]], [section_names[i + 2], section_names[i + 2]]] header, footer = generate_nav_bar(link_info) @@ -295,9 +292,7 @@ def create_general_pages(data, title): text = "" for eq in section_data[1]: - # equation is of type LatexEquation - print eq - print + # equation should be of type LatexEquation text += generate_math_html(eq.equation, options={"id": eq.label}) metadata_exists = False @@ -318,6 +313,8 @@ def create_general_pages(data, title): def create_specific_pages(data, glossary): + # type: (list, dict) -> str + """Creates specific pages for each formula.""" formulae = list() for section_data in data: for equations in section_data[1:]: @@ -325,7 +322,7 @@ def create_specific_pages(data, glossary): formulae.append("Formula:" + eq.label) formulae.append(section_data[0]) - formulae = ["Orthogonal Polynomials"] + formulae[:-1] + ["Orthogonal Polynomials"] + formulae = [TITLE_STRING] + formulae[:-1] + [TITLE_STRING] i = 0 pages = list() @@ -404,6 +401,7 @@ def main(): data.append([section_name, equations]) data = extract_data(data) + output = create_general_pages(data, title) + create_specific_pages(data, glossary) with open(OUTPUT_FILE, "w") as output_file: From 3cbbac2905d59713d7ecf555e6a8867c0494970d Mon Sep 17 00:00:00 2001 From: Joon Date: Sun, 27 Nov 2016 19:18:48 -0500 Subject: [PATCH 383/402] Lay framework for better data storage/management --- tex2wiki/src/tex2wiki.py | 89 +++++++++++++++++++++++++++++++++++++++- 1 file changed, 88 insertions(+), 1 deletion(-) diff --git a/tex2wiki/src/tex2wiki.py b/tex2wiki/src/tex2wiki.py index 2d12bbe..5ce3a1c 100644 --- a/tex2wiki/src/tex2wiki.py +++ b/tex2wiki/src/tex2wiki.py @@ -4,7 +4,7 @@ INPUT_FILE = "tex2wiki/data/test.tex" OUTPUT_FILE = "tex2wiki/data/test.mmd" GLOSSARY_LOCATION = "tex2wiki/data/new.Glossary.csv" -TITLE_STRING = "Orthogonal Polynomials" +TITLE_STRING = "CFSF Dataset" METADATA_TYPES = ["substitution", "constraint"] METADATA_MEANING = {"substitution": "Substitution(s)", "constraint": "Constraint(s)"} @@ -17,6 +17,56 @@ def __init__(self, label, equation, metadata): self.equation = equation self.metadata = metadata + def __str__(self): + return self.label + "\n" + self.equation + "\n" + str(self.metadata) + + @staticmethod + def make_from_raw(raw): + equations = raw[:-1] + for i, equation in enumerate(equations): + # formula stuff + try: + formula = get_data_str(equation, latex="\\mapletag") + except IndexError: # there is no formula. + break + + equation = equation.split("\n") + + # get metadata + raw_metadata = "" + + j = 0 + while j < len(equation): + if "%" in equation[j]: + raw_metadata += equation.pop(j)[1:].strip().strip("\n") + "\n" + else: + j += 1 + + metadata = dict() + for data_type in METADATA_TYPES: + metadata[data_type] = format_metadata(get_data_str(raw_metadata, latex="\\" + data_type)) + + equations[i] = LatexEquation(formula, '\n'.join(equation), metadata) + + return equations + + +class DataUnit(object): + def __init__(self, title, subunits): + self.title = title + self.subunits = subunits + + def __str__(self): + result = self.title + "\n" + + if type(self.subunits) == list and type(self.subunits[0]) == DataUnit: + for sub in self.subunits: + result += str(sub) + "\n" + else: + result += str([str(eq) for eq in self.subunits]) + "\n" + + return result + def generate_html(tag_name, text, options=None, spacing=2): # type: (str, str(, dict, bool)) -> str @@ -376,6 +426,39 @@ def create_specific_pages(data, glossary): return "\n".join(pages) + "\n" +def section_split(string, sub=0): + # type: (str) -> DataUnit + """ + Split string into DataTree objects. + Will eventually become replacement for large part of main() + extract_data(), + and should theoretically make the create_general_pages and create_specific_pages methods simpler, + as well as more easily store all data from the .tex file. + """ + + string = string.split("\\" + sub * "sub" + "section") + + # base case; when depth too far + if len(string) == 1: + title = get_data_str(string[0]).replace("$", "''") + + equations = string[0].split("\\end{equation}") + for i, equation in enumerate(equations[:-1]): + equations[i] = equation.split("\\begin{equation}", 1)[1].strip() + + equations = LatexEquation.make_from_raw(equations) + + return DataUnit(title, equations) + + chunk_data = list() # list of DataSection + + for chunk in string[1:]: + chunk_data.append(section_split(chunk, sub + 1)) + + title = get_data_str(string[0]).replace("$", "''") + + return DataUnit(title, chunk_data) + + def main(): with open(INPUT_FILE) as input_file: text = input_file.read() @@ -387,6 +470,10 @@ def main(): glossary[get_macro_name(row[0])] = row text = text.split("\\begin{document}", 1)[1] + + info = section_split(text) # creates tree, split into section, subsection, subsubsection, etc. + # TODO: change create_general_pages & create_specific_pages to utilize DataUnit format rather than list format + text = text.split("\\section") title = get_data_str(text[0]) From eb001541991b4ea96e3b314d9935f73537ef285a Mon Sep 17 00:00:00 2001 From: Edward Bian Date: Fri, 9 Dec 2016 17:05:34 -0500 Subject: [PATCH 384/402] Updated README according to Github --- KLSadd_insertion/README.md | 37 ++++++++++++------------------------- 1 file changed, 12 insertions(+), 25 deletions(-) diff --git a/KLSadd_insertion/README.md b/KLSadd_insertion/README.md index 104f5d4..451afd9 100644 --- a/KLSadd_insertion/README.md +++ b/KLSadd_insertion/README.md @@ -1,8 +1,8 @@ ##KLS Addendum Insertion Project -This project uses Python and string manipulation to insert sections of the KLSadd.tex file into the appropriate DRMF chapter sections. The updateChapters.py program is the most recently updated and cleanest version, however it is not completed. Should be run with a simple call to +This project uses python and string manipulation to insert sections of the KLSadd.tex file into the appropriate DRMF chapter sections. The updateChapters.py program is the most recently updated and cleanest version, however it is not completed. Should be run with a simple call to ``` -Python updateChapters.py +python updateChapters.py ``` The program must have a tempchap9.tex and a tempchap14.tex as well as the KLSadd.tex file in the same directory! The linetest.py program is the original program file fully updated. It is much harder to read and should only be used as a reference to update the updateChapters.py program. @@ -25,17 +25,19 @@ First: the files - KLSadd.tex: This is the addendum we are working with. It has sections that correspond with sections in the chapter files, and it contains paragraphs that must be *inserted* into the chapter files. CAN BE FOUND ONINE IN PDF FORM: https://staff.fnwi.uva.nl/t.h.koornwinder/art/informal/KLSadd.pdf - - chap09.tex and chap14.tex: these are chapter files. These are LATEX documents that are chapters in a book. This is a book written by smart math people. But they did some things wrong, so another math person wants to fix them. That math person wrote an addendum file called KLSadd.tex and our job is to pull paragraphs out of KLSadd and insert them at the end of the relevant section in the chapter files. Every chapter is made up of sections and every section has subsections + - chap09.tex and chap14.tex: these are chapter files. These are LaTeX documents that are chapters in a book. This is a book written by smart math people. But they did some things wrong, so another math person wants to fix them. That math person wrote an addendum file called KLSadd.tex and our job is to pull paragraphs out of KLSadd and insert them at the end of the relevant section in the chapter files. Every chapter is made up of sections and every section has subsections - - updateChapters.py: this is our code. This document will be going over the variables and methods in this program. This program should: take paragraphs from every section in KLSadd and insert them into the relevant chapter file and into the relevant section within that chapter. + - updateChapters.py: This is thr code currently being worked on and sorting chapters. This document will be going over the variables and methods in this program. This program should: take paragraphs from every section in KLSadd and insert them into the relevant chapter file and into the relevant section within that chapter. - linetest.py: this abomination of code is the original program. It did the job, and it did it well. However, I realized that it was very very unreadable. This program is the child I never wanted. This program does 100% work and its outputs can be used to check against updateChapters.py's output. - - newtempchap09.tex and newtempchap14.tex these are output files of linetest.py and this is what the output to updateChapters.py should be like!!!!!!! + - newtempchap09.tex and newtempchap14.tex these are output files of linetest.py and this is what a standard output looks like. They are no longer up to date for updateChapters as it sorts by subsection, but still serve as good reference Next: the specifications -Toward the end of every section in the chapter file, there is a subsection called "References". It basically just contains a bunch of references to sources and other papers. Every paragraph in every subsection in KLSadd.tex must be put *before* the "References" subsection in the corresponding section in the chapter file. For example: +Each section has subsections like "Hypergeometric Representation", sections in KLSadd with "Hypergeometric Representation" should be seorted there. + +Toward the end of every section in the chapter file, there is a subsection called "References". It basically just contains a bunch of references to sources and other papers. This is where unsorted materials go For example: **These aren't real sections, obviously, just an example** in chap09.tex: @@ -56,28 +58,13 @@ The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetric in $a,b,c,d$. ``` -So in the chapter 9 file, section 15 "Dogs", before the References subsection, we need to add in the Beagles paragraph. HOWEVER, there are some changes that need to be made before we insert. The easiest change is adding a comment that tells us something was inserted. We just append something like "%RS insertion begin" and "%RS insertion complete" before and after, respectively, the Beagles paragraph when we insert. Another change is adding \large\bf before the paragraph name. This changes the size and format so it looks prettier. So the heading in this example would be: +So in the chapter 9 file, section 15 "Dogs", We change the formatting with "\bf KLS Addendum: ". This changes the format so it looks prettier and stands out. So the heading in this example would be: ```latex -\paragraph{\large\bf Beagles} +\paragraph{\bf KLS Addendum: Beagles} ``` --- -So after updateChapters.py is called, the chapter 9 file should look like this: - -```latex -\section{Dogs}\index{Dog polynomials} -\subsection*{Hypergeometric dogs} -blah blah blah -%RS insertion begin -\paragraph{\large\bf Beagles} -The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetric -in $a,b,c,d$. -%RS insertion complete -\subsection*{References} -\cite{Dogs101} -``` - The variables: These are the variables in the beginning @@ -88,13 +75,13 @@ These are the variables in the beginning -mathPeople: this holds the *name* of every section in KLSadd. It stores names like Wilson, Racah, Dual Hahn, etc. In our example with the Dogs above, it would hold "Dogs". This is useful to finding where to insert the correct paragraph --newCommands: this is a list that holds ints representing line numbers in KLSadd.tex that correspond to commands. There are a few special commands in the file that help turn LATEX files into PDF files and they need to be copied over into both chapter files +-newCommands: this is a list that holds ints representing line numbers in KLSadd.tex that correspond to commands. There are a few special commands in the file that help turn LaTeX files into PDF files and they need to be copied over into both chapter files -comms: holds the actual strings of the commands found from the line numbers stored in newCommands --- -The methods: +The functions: -prepareForPDF(chap): this method inserts some packages needed by LATEX files to be turned into pdf files. It takes a list called chap which is a String containing the contents of the chapter fie. It returns the chap String edited with the special packages in place. From 7938b41304030084e859ed32b8ec99e4b90fc784 Mon Sep 17 00:00:00 2001 From: Edward Bian Date: Fri, 9 Dec 2016 17:15:44 -0500 Subject: [PATCH 385/402] Updated comments according to Github --- KLSadd_insertion/src/linetest.py | 9 +++------ 1 file changed, 3 insertions(+), 6 deletions(-) diff --git a/KLSadd_insertion/src/linetest.py b/KLSadd_insertion/src/linetest.py index 19de78a..4d738a7 100644 --- a/KLSadd_insertion/src/linetest.py +++ b/KLSadd_insertion/src/linetest.py @@ -1,10 +1,7 @@ """ -TODO: Rewrite the whole thing so another human being can read it -jeez this is like my handwriting; it's nonsense to everyone but me -This started out as a test file but I just kept writing it -and writing it -and writing it -and now its the project +Written by Rahul Shah +Outdated project in terms of function, now that updateChapters does what it does and more + Kept for reference """ entireFile = "" From f0c5b3761cfeb437fd887ef813c4644c707a5941 Mon Sep 17 00:00:00 2001 From: Edward Bian Date: Tue, 13 Dec 2016 18:00:55 -0500 Subject: [PATCH 386/402] Comment changes --- KLSadd_insertion/src/updateChapters.py | 171 ++++++++++++++++--------- 1 file changed, 111 insertions(+), 60 deletions(-) diff --git a/KLSadd_insertion/src/updateChapters.py b/KLSadd_insertion/src/updateChapters.py index 7360301..749db96 100644 --- a/KLSadd_insertion/src/updateChapters.py +++ b/KLSadd_insertion/src/updateChapters.py @@ -13,23 +13,34 @@ # sorter_check = [[0 for _ in range(w)] for __ in range(h)] -def chap1placer(chap1, kls, klsaddparas): - chapterstart = True +def chap_1_placer(chap1, kls, klsaddparas): + """ + + :param chap1: Chapter 1 text file + :param kls: KLSadd, the addendum + :param klsaddparas: Paragraphs that the code has identified to be sorted + :return: Returns chapter 1 with the sections inserted into the right place from the introduction of KLSadd + """ + chapter_start = True index = 0 kls_sub_chap1 = [] kls_header_chap1 = [] for item in kls: index += 1 line = str(item) + print line if "\\subsection" in line: temp = line[line.find(" ", 12) + 1: line.find("}", 12)+1] # get just the name (like mathpeople) - if chapterstart and ("\\paragraph{" in line or "\\subsubsection*{" in line): + # The 12 is not arbitrary and comes from the number of characters in "//subsection{" + if chapter_start and ("\\paragraph{" in line or "\\subsubsection*{" in line): + print chapter_start + print line for item in klsaddparas: if index < item: klsloc = klsaddparas.index(item) break t = ''.join(kls[index-1: klsaddparas[klsloc]]) - kls_sub_chap1.append(t) # append the whole paragraph, pray every paragraph ends with a % comment + kls_sub_chap1.append(t) # append the whole paragraph, every paragraph should end with a % comment kls_header_chap1.append(temp) # append the name of subsection break intro_to_add = ''.join(kls_sub_chap1) @@ -39,14 +50,14 @@ def chap1placer(chap1, kls, klsaddparas): return total_chap1 -def extraneous_section_deleter(list): +def extraneous_section_deleter(entered_list): """ Removes sections that are irrelevant and will slow or confuse the program - :param list: + :param entered_list: List of paragraphs :return: Extraneous name free output """ - return [item for item in list if "reference" not in item.lower() and "limit relation" not in item.lower() + return [item for item in entered_list if "reference" not in item.lower() and "limit relation" not in item.lower() and "symmetry" not in item.lower() and " hypergeometric representation" not in item.lower() and "hypergeometric representation " not in item.lower()] @@ -94,9 +105,9 @@ def fix_chapter_sort(kls, chap, word, sortloc, klsaddparas, sortmatch_2, tempref hyper_headers_chap = [] hyper_subs_chap = [] + special_input = 0 name_chap = word.lower() - print name_chap if name_chap == "orthogonality": special_input = 1 elif name_chap == "special value": @@ -121,6 +132,10 @@ def fix_chapter_sort(kls, chap, word, sortloc, klsaddparas, sortmatch_2, tempref if special_input in (0, 2, 4) and name_chap in line.lower() and chapterstart and ("\\paragraph{" in line or "\\subsubsection*{" in line) or \ (special_input in (1, 3) and "orthogonality relation" not in line.lower() and name_chap in line.lower() and chapterstart and ("\\paragraph{" in line or "\\subsubsection*{" in line)): + testing = item + if "Basic hypergeometric representation" in testing: + print item + print "DETECTED" for item in klsaddparas: if index < item: klsloc = klsaddparas.index(item) @@ -128,7 +143,8 @@ def fix_chapter_sort(kls, chap, word, sortloc, klsaddparas, sortmatch_2, tempref t = ''.join(kls[index: klsaddparas[klsloc]]) k_hyper_sub_chap.append(t) # append the whole paragraph, pray every paragraph ends with a % comment khyper_header_chap.append(temp) # append the name of subsection - + if "Basic hypergeometric representation" in testing: + print t for item in khyper_header_chap: if item == "Pseudo Jacobi (or Routh-Romanovski)": khyper_header_chap[khyper_header_chap.index(item)] = "Pseudo Jacobi" @@ -140,25 +156,25 @@ def fix_chapter_sort(kls, chap, word, sortloc, klsaddparas, sortmatch_2, tempref try: if khyper_header_chap[item] == khyper_header_chap[item + 1] and "generating functions" in name_chap: k_hyper_sub_chap[item + 1] = k_hyper_sub_chap[item] + "\paragraph{\\bf KLS Addendum: Bilateral generating functions}" + k_hyper_sub_chap[item + 1] - khyper_header_chap[item] = "memes" - k_hyper_sub_chap[item] = "memes" + khyper_header_chap[item] = "/x00" + k_hyper_sub_chap[item] = "/x00" elif khyper_header_chap[item] == khyper_header_chap[item+1]: k_hyper_sub_chap[item + 1] = k_hyper_sub_chap[item] + "\paragraph{\\bf KLS Addendum: " + \ word + "}" + k_hyper_sub_chap[item + 1] - khyper_header_chap[item] = "memes" - k_hyper_sub_chap[item] = "memes" + khyper_header_chap[item] = "/x00" + k_hyper_sub_chap[item] = "/x00" else: item += 1 except IndexError: item += 1 - a = 0 - while a < len(khyper_header_chap): - if khyper_header_chap[a] == "memes": - del khyper_header_chap[a] - del k_hyper_sub_chap[a] + temp_counter = 0 + while temp_counter < len(khyper_header_chap): + if khyper_header_chap[temp_counter] == "/x00": + del khyper_header_chap[temp_counter] + del k_hyper_sub_chap[temp_counter] else: - a += 1 + temp_counter += 1 #global sorter_check chap9 = 0 @@ -174,8 +190,11 @@ def fix_chapter_sort(kls, chap, word, sortloc, klsaddparas, sortmatch_2, tempref if special_input in (2, 3): name_chap = 'hypergeometric representation' + # Uses of numbers like 12, or 9 are for specific lengths of strings (like \\subsection{) where the # section does not change + ''' + ''' for d in range(0, len(tempref)): # check every section and subsection line item = tempref[d] line = str(chap[item]) @@ -187,20 +206,41 @@ def fix_chapter_sort(kls, chap, word, sortloc, klsaddparas, sortmatch_2, tempref if name_chap in line.lower(): if special_input in (0, 2, 3, 4) or special_input == 1 and "orthogonality relation" not in line: + + hyper_subs_chap.append([tempref[d + 1]]) # appends the index for the line before following subsection hyper_headers_chap.append(temp) # appends the name of the section the hypergeo subsection is in - + ''' + Section below should + ''' + counteri = 0 + for i in khyper_header_chap: + i = i.replace(' I','~I') + khyper_header_chap[counteri] = i + if "Discrete $q$-Hermite II" in i: + print("FIX ACTIVATED") + #khyper_header_chap[counteri] = "Discrete $q$-Hermite~II" + counteri += 1; + + print khyper_header_chap if temp in khyper_header_chap: + print temp try: chap[tempref[d + 1] - 1] += "\paragraph{\\bf KLS Addendum: " + word + "}" + \ k_hyper_sub_chap[k_hyp_index_iii + offset] + if "\wt h_n(x;q)=x^n\,\qhyp21{q^{-n},q^{-n+1}}0{q^2,-q^2 x^{-2}}." in k_hyper_sub_chap[k_hyp_index_iii + offset]: + print k_hyper_sub_chap[k_hyp_index_iii + offset] + print "LOCATED" if chap9 == 1: sorter_check[sortloc][1] += 1 k_hyp_index_iii += 1 except IndexError: print("Warning! Code has found an error involving section finding for '" + name_chap + "'.") - - + ''' + ''' + if sortloc == 7: + with open(DATA_DIR + "testoutput.tex", "w") as thing: + thing.write(''.join(chap)) if len(hyper_headers_chap) != 0: # print "Stuff for checking" # print k_hyper_sub_chap @@ -222,23 +262,23 @@ def cut_words(word_to_find, word_to_search_in): :param word_to_search_in: The big word that is being searched :return: The big word without the word that was searched for """ - a = word_to_find - b = word_to_search_in + find_this_string = word_to_find + search_in_this_string = word_to_search_in precheck = 1 - if a in b: - if "\\paragraph{\\bf KLS Addendum: " in b or "\\subsubsection*{\\bf KLS Addendum: " in b: + if find_this_string in search_in_this_string: + if "\\paragraph{\\bf KLS Addendum: " in search_in_this_string or "\\subsubsection*{\\bf KLS Addendum: " in search_in_this_string: while True: - if "\\paragraph{\\bf KLS Addendum: " in b[b.find(a)-precheck:b.find(a)] or \ - "\\subsubsection*{\\bf KLS Addendum: " in b[b.find(a) - precheck:b.find(a)]: - return b[:b.find(a) - precheck] + b[b.find(a) + len(a):] + if "\\paragraph{\\bf KLS Addendum: " in search_in_this_string[search_in_this_string.find(find_this_string)-precheck:search_in_this_string.find(find_this_string)] or \ + "\\subsubsection*{\\bf KLS Addendum: " in search_in_this_string[search_in_this_string.find(find_this_string) - precheck:search_in_this_string.find(find_this_string)]: + return search_in_this_string[:search_in_this_string.find(find_this_string) - precheck] + search_in_this_string[search_in_this_string.find(find_this_string) + len(find_this_string):] else: precheck += 1 else: - cut = b[:b.find(a)] + b[b.find(a) + len(a):] + cut = search_in_this_string[:search_in_this_string.find(find_this_string)] + search_in_this_string[search_in_this_string.find(find_this_string) + len(find_this_string):] return cut else: - return b + return search_in_this_string def prepare_for_pdf(chap): @@ -296,11 +336,11 @@ def insert_commands(kls, chap, cms): index += 1 if "begin{document}" in word: begin_index += index - tempIndex = 0 + temp_index = 0 for i in cms: - chap.insert(begin_index + tempIndex, i) - tempIndex += 1 + chap.insert(begin_index + temp_index, i) + temp_index += 1 return chap @@ -325,15 +365,15 @@ def find_references(chapter, chapticker, math_people): elif chapticker == 1: chaptercheck = str(14) # canAdd tells the program whether the next section is a reference - canadd = False + add_next_section = False for word in chapter: index += 1 special_detector = 1 # check sections and subsections if("section{" in word or "subsection*{" in word) and ("subsubsection*{" not in word): - w = word[word.find("{")+1: word.find("}")] - ws = word[word.find("{")+1: word.find("~")] + processed_word = word[word.find("{")+1: word.find("}")] + specially_formatted_word = word[word.find("{")+1: word.find("~")] if "bessel" in word.lower() and chapticker == 0: ref9_3.append(index) if ("big $q$-legendre" in word.lower() or "little $q$-legendre" in word.lower() or "continuous $q$-legendre" in word.lower()) and chapticker == 1: @@ -341,22 +381,22 @@ def find_references(chapter, chapticker, math_people): for unit in math_people: subunit = unit[unit.find(" ")+1: unit.find("#")] # System of checks that verifies if section is in chapter - if (w in subunit or ws in subunit) and (chaptercheck in unit) and len(w) == len(subunit) or\ - ("Pseudo Jacobi" in w and "Pseudo Jacobi (or Routh-Romanovski)" in subunit): - canadd = True + if (processed_word in subunit or specially_formatted_word in subunit) and (chaptercheck in unit) and len(processed_word) == len(subunit) or\ + ("Pseudo Jacobi" in processed_word and "Pseudo Jacobi (or Routh-Romanovski)" in subunit): + add_next_section = True if chapticker == 0: ref9_3.append(index) elif chapticker == 1: ref14_3.append(index) special_detector = 0 - if "\\subsection*{References}" in word and canadd: + if "\\subsection*{References}" in word and add_next_section: # Appends valid locations references.append(index) if chapticker == 0: ref9_3.append(index) elif chapticker == 1: ref14_3.append(index) - canadd = False + add_next_section = False if "subsection*{" in word and "References" not in word: if chapticker == 0: ref9_3.append(index) @@ -384,7 +424,7 @@ def reference_placer(chap, references, p, chapticker2): :param references: List containing references :param p: List containing additions to be added. :param chapticker2: Which chapter is being searched (9 or 14). - :return: + :return: Returns the chapter, complete with additions """ # count is used to represent the values in count count = 0 @@ -429,7 +469,7 @@ def fix_chapter(chap, references, paragraphs_to_be_added, kls, kls_list_all, cha :param kls: The addendum that contains sections to be added (not processed). :param kls_list_all: List of all identified keywords, fed into the chapter sorter. :param chapticker2: Which chapter is being searched (9 or 14). - :return: + :return: entire chapter, with all other methods applied to it """ sort_location = 0 @@ -438,36 +478,41 @@ def fix_chapter(chap, references, paragraphs_to_be_added, kls, kls_list_all, cha fix_chapter_sort(kls, chap, i, sort_location, klsaddparas, sortmatch_2, tempref, sorter_check) sort_location += 1 - for a in range(len(paragraphs_to_be_added)-1): - for b in sortmatch_2: - for c in b: + print "SEARCH HERE FIRST" + print paragraphs_to_be_added + for paragraph in range(len(paragraphs_to_be_added)): + for subsection in sortmatch_2: + for lines_in_subsection in subsection: with open(DATA_DIR + "compare.tex", "a") as spook: - spook.write(paragraphs_to_be_added[a]) + spook.write(paragraphs_to_be_added[paragraph]) spook.write("NEXT: ") - if "%" == c[-2]: - c = c[:-3] + if "%" == lines_in_subsection[-2]: + lines_in_subsection = lines_in_subsection[:-3] #print "memes" #print c - elif "%" == c[-1]: - c = c[:-2] + elif "%" == lines_in_subsection[-1]: + lines_in_subsection = lines_in_subsection[:-2] # if "Formula (9.8.15) was first obtained by Brafman" in c and "Formula (9.8.15) was first obtained by Brafman"in paragraphs_to_be_added[a]: # print c # print "DING" # print paragraphs_to_be_added[a] - - paragraphs_to_be_added[a] = cut_words(c, paragraphs_to_be_added[a]) + if "\wt h_n(x;q)=x^n\,\qhyp21{q^{-n},q^{-n+1}}0{q^2,-q^2 x^{-2}}." in lines_in_subsection: + print "OUTPUTS" + print lines_in_subsection + print paragraphs_to_be_added[paragraph] + paragraphs_to_be_added[paragraph] = cut_words(lines_in_subsection, paragraphs_to_be_added[paragraph]) reference_placer(chap, references, paragraphs_to_be_added, chapticker2) + chap = prepare_for_pdf(chap) cms = get_commands(kls,new_commands) chap = insert_commands(kls, chap, cms) commentticker = 0 # These commands mess up PDF reading and mus tbe commented out - print"OUTPUT" + print "OUTPUT" for word in chap: word2 = chap[chap.index(word)-1] - print word2 if "\\newcommand{\qhypK}[5]{\,\mbox{}_{#1}\phi_{#2}\!\left(" not in word2: if "\\newcommand\\half{\\frac12}" in word: wordtoadd = "%" + word @@ -498,7 +543,6 @@ def main(): Runs all of the other functions and outputs their results into an output file as well as putting together the list of additions that is fed into fix_chapter. - :return: """ chap_nums = [] new_commands = [] # used to hold the indexes of the commands @@ -566,7 +610,9 @@ def main(): # now indexes holds all of the places there is a section # using these indexes, get all of the words in between and add that to the paras[] paras = [] + for i in range(len(indexes)-1): + box = ''.join(addendum[indexes[i]: indexes[i+1]-1]) paras.append(box) @@ -582,11 +628,14 @@ def main(): entire1 = ch1.readlines() # reads in as a list of strings with open(DATA_DIR + "updated1.tex", "w") as temp1: - temp1.write(chap1placer(entire1, addendum, klsaddparas)) + temp1.write(chap_1_placer(entire1, addendum, klsaddparas)) # chapter 9 with open(DATA_DIR + "chap09.tex", "r") as ch9: entire9 = ch9.readlines() # reads in as a list of strings + with open(DATA_DIR + "testoutput.tex", "w") as thing: + thing.write(''.join(entire9)) + # chapter 14 with open(DATA_DIR + "chap14.tex", "r") as ch14: entire14 = ch14.readlines() @@ -604,8 +653,9 @@ def main(): references14 = references14[0] - print ref14_3 + # call the fixChapter method to get a list with the addendum paragraphs added in + chapticker2 = 0 str9 = ''.join(fix_chapter(entire9, references9, paras, addendum, kls_list_all, chapticker2, new_commands, klsaddparas, sortmatch_2, ref9_3, sorter_check)) @@ -626,6 +676,7 @@ def main(): temp14.write(str14) print sorter_check - + print kls_list_all + print kls_list_all[7] if __name__ == '__main__': main() \ No newline at end of file From 2b9989710b38276a23865317c93734c3ec9daf16 Mon Sep 17 00:00:00 2001 From: Edward Bian Date: Fri, 16 Dec 2016 17:47:51 -0500 Subject: [PATCH 387/402] updateChapters.py with all the Oct. 5 fixes --- KLSadd_insertion/src/updateChapters.py | 136 ++++++++++--------------- 1 file changed, 55 insertions(+), 81 deletions(-) diff --git a/KLSadd_insertion/src/updateChapters.py b/KLSadd_insertion/src/updateChapters.py index 749db96..94e3c8c 100644 --- a/KLSadd_insertion/src/updateChapters.py +++ b/KLSadd_insertion/src/updateChapters.py @@ -28,13 +28,10 @@ def chap_1_placer(chap1, kls, klsaddparas): for item in kls: index += 1 line = str(item) - print line if "\\subsection" in line: temp = line[line.find(" ", 12) + 1: line.find("}", 12)+1] # get just the name (like mathpeople) # The 12 is not arbitrary and comes from the number of characters in "//subsection{" if chapter_start and ("\\paragraph{" in line or "\\subsubsection*{" in line): - print chapter_start - print line for item in klsaddparas: if index < item: klsloc = klsaddparas.index(item) @@ -68,7 +65,8 @@ def new_keywords(kls, kls_list): It then takes that list and removes duplicates, as well as printing the output to another file. :param kls: The addendum that provides the words. - :return: Inputs results into a global list. + :param kls_list: List of identified keywords. + :return: Outputs results as a list """ kls_list_chap = [] @@ -85,6 +83,7 @@ def new_keywords(kls, kls_list): kls_list_chap.append("Special value") w = 2 h = len(kls_list_chap)+1 + #sorter_check tells the program which chapter the code is on and how many sections under a keyword have been sorted. sorter_check = [[0 for _ in range(w)] for __ in range(h)] return kls_list_chap, sorter_check @@ -98,8 +97,8 @@ def fix_chapter_sort(kls, chap, word, sortloc, klsaddparas, sortmatch_2, tempref :param chap: The destination of inserted sections. :param word: The keyword that is being processed. :param sortloc: The location in the list of words, of the word being sorted. - :param klsaddparas: - :param sortmatch_2: + :param klsaddparas: Locations of possible sections + :param sortmatch_2: Subsections that the program finds a match for in the chapter (a list of those that will be sorted) :return: This function outputs the processed chapter into a larger function for further processing. """ hyper_headers_chap = [] @@ -133,9 +132,6 @@ def fix_chapter_sort(kls, chap, word, sortloc, klsaddparas, sortmatch_2, tempref (special_input in (1, 3) and "orthogonality relation" not in line.lower() and name_chap in line.lower() and chapterstart and ("\\paragraph{" in line or "\\subsubsection*{" in line)): testing = item - if "Basic hypergeometric representation" in testing: - print item - print "DETECTED" for item in klsaddparas: if index < item: klsloc = klsaddparas.index(item) @@ -143,8 +139,7 @@ def fix_chapter_sort(kls, chap, word, sortloc, klsaddparas, sortmatch_2, tempref t = ''.join(kls[index: klsaddparas[klsloc]]) k_hyper_sub_chap.append(t) # append the whole paragraph, pray every paragraph ends with a % comment khyper_header_chap.append(temp) # append the name of subsection - if "Basic hypergeometric representation" in testing: - print t + for item in khyper_header_chap: if item == "Pseudo Jacobi (or Routh-Romanovski)": khyper_header_chap[khyper_header_chap.index(item)] = "Pseudo Jacobi" @@ -176,7 +171,6 @@ def fix_chapter_sort(kls, chap, word, sortloc, klsaddparas, sortmatch_2, tempref else: temp_counter += 1 - #global sorter_check chap9 = 0 if sorter_check[sortloc][0] == 0: @@ -205,7 +199,7 @@ def fix_chapter_sort(kls, chap, word, sortloc, klsaddparas, sortmatch_2, tempref temp = line[9:line.find("}", 7)] if name_chap in line.lower(): - if special_input in (0, 2, 3, 4) or special_input == 1 and "orthogonality relation" not in line: + if special_input in (0, 2, 3, 4) or "orthogonality relation" not in line: hyper_subs_chap.append([tempref[d + 1]]) # appends the index for the line before following subsection @@ -217,20 +211,12 @@ def fix_chapter_sort(kls, chap, word, sortloc, klsaddparas, sortmatch_2, tempref for i in khyper_header_chap: i = i.replace(' I','~I') khyper_header_chap[counteri] = i - if "Discrete $q$-Hermite II" in i: - print("FIX ACTIVATED") - #khyper_header_chap[counteri] = "Discrete $q$-Hermite~II" counteri += 1; - print khyper_header_chap if temp in khyper_header_chap: - print temp try: chap[tempref[d + 1] - 1] += "\paragraph{\\bf KLS Addendum: " + word + "}" + \ k_hyper_sub_chap[k_hyp_index_iii + offset] - if "\wt h_n(x;q)=x^n\,\qhyp21{q^{-n},q^{-n+1}}0{q^2,-q^2 x^{-2}}." in k_hyper_sub_chap[k_hyp_index_iii + offset]: - print k_hyper_sub_chap[k_hyp_index_iii + offset] - print "LOCATED" if chap9 == 1: sorter_check[sortloc][1] += 1 k_hyp_index_iii += 1 @@ -242,11 +228,6 @@ def fix_chapter_sort(kls, chap, word, sortloc, klsaddparas, sortmatch_2, tempref with open(DATA_DIR + "testoutput.tex", "w") as thing: thing.write(''.join(chap)) if len(hyper_headers_chap) != 0: - # print "Stuff for checking" - # print k_hyper_sub_chap - # print khyper_header_chap - # print hyper_headers_chap - # print word sortmatch_2.append(k_hyper_sub_chap) return chap @@ -285,15 +266,22 @@ def prepare_for_pdf(chap): """ Edits the chapter string sent to include hyperref, xparse, and cite packages - :param chap: The chapter (9 or 14) that is being processed + :param chap: The chapter (9 or 14) that is being processed as a list of lines in each LaTeX chapter :return: The processed chapter, ready for additional processing """ + # (list) -> list foot_misc_index = 0 + for i, line in enumerate(chap): + if "footmisc" in line: + foot_misc_index += i + 1 + + ''' index = 0 - for word in chap: + for line in chap: index += 1 - if "footmisc" in word: + if "footmisc" in line: foot_misc_index += index + ''' # str[footmiscIndex] += "\\usepackage[pdftex]{hyperref} \n\\usepackage {xparse} \n\\usepackage{cite} \n" chap.insert(foot_misc_index, "\\usepackage[pdftex]{hyperref} \n\\usepackage {xparse} \n\\usepackage{cite} \n") return chap @@ -316,16 +304,17 @@ def get_commands(kls, new_commands): return kls[new_commands[0]:new_commands[1]] -# 2/18/16 this method addresses the goal of hardcoding in the necessary commands to let the chapter files run as pdf's. -# Currently only works with chapter 9 + def insert_commands(kls, chap, cms): """ Inserts commands identified in previous functions. + This method addresses the goal of hardcoding in the necessary commands to let the chapter files run as pdf's. + Currently only works with chapter 9 :param kls: Addendum to look through - :param chap: The chapter that is receiving commands (9 or 14) + :param chap: The chapter that is receiving commands (9 or 14) as a list of lines in each LaTeX chapter :param cms: Commands to be inserted - :return: + :return: chap, processed """ # reads in the newCommands[] and puts them in chap begin_index = -1 # the index of the "begin document" keyphrase, this is where the new commands need to be inserted. @@ -344,7 +333,6 @@ def insert_commands(kls, chap, cms): return chap -# method to find the indices of the reference paragraphs def find_references(chapter, chapticker, math_people): """ This function searches the chapters and locates potential destinations of additions from the addendum. @@ -352,7 +340,7 @@ def find_references(chapter, chapticker, math_people): :param chapter: The chapter being searched. :param chapticker: Which chapter is being searched (9 or 14). - :return: List of references. + :return: List of references, ref9_3 and ref14_3 are references for chapter 9 and 14 specifically """ ref9_3 = [] ref14_3 = [] @@ -360,9 +348,9 @@ def find_references(chapter, chapticker, math_people): index = -1 # chaptercheck designates which chapter is being searched for references chaptercheck = 0 - if chapticker == 0: + if chapticker == 9: chaptercheck = str(9) - elif chapticker == 1: + elif chapticker == 14: chaptercheck = str(14) # canAdd tells the program whether the next section is a reference add_next_section = False @@ -374,9 +362,9 @@ def find_references(chapter, chapticker, math_people): if("section{" in word or "subsection*{" in word) and ("subsubsection*{" not in word): processed_word = word[word.find("{")+1: word.find("}")] specially_formatted_word = word[word.find("{")+1: word.find("~")] - if "bessel" in word.lower() and chapticker == 0: + if "bessel" in word.lower() and chapticker == 9: ref9_3.append(index) - if ("big $q$-legendre" in word.lower() or "little $q$-legendre" in word.lower() or "continuous $q$-legendre" in word.lower()) and chapticker == 1: + if ("big $q$-legendre" in word.lower() or "little $q$-legendre" in word.lower() or "continuous $q$-legendre" in word.lower()) and chapticker == 14: ref14_3.append(index) for unit in math_people: subunit = unit[unit.find(" ")+1: unit.find("#")] @@ -384,44 +372,39 @@ def find_references(chapter, chapticker, math_people): if (processed_word in subunit or specially_formatted_word in subunit) and (chaptercheck in unit) and len(processed_word) == len(subunit) or\ ("Pseudo Jacobi" in processed_word and "Pseudo Jacobi (or Routh-Romanovski)" in subunit): add_next_section = True - if chapticker == 0: + if chapticker == 9: ref9_3.append(index) - elif chapticker == 1: + elif chapticker == 14: ref14_3.append(index) special_detector = 0 if "\\subsection*{References}" in word and add_next_section: # Appends valid locations references.append(index) - if chapticker == 0: + if chapticker == 9: ref9_3.append(index) - elif chapticker == 1: + elif chapticker == 14: ref14_3.append(index) add_next_section = False if "subsection*{" in word and "References" not in word: - if chapticker == 0: + if chapticker == 9: ref9_3.append(index) - elif chapticker == 1: + elif chapticker == 14: ref14_3.append(index) if "\\section{" in word and special_detector == 1 and chaptercheck == "14": w2 = word[word.find("{") + 1: word.find("}")] if "Bessel" not in w2: ref14_3.append(index) - bqlegendrecheck = 2 - - # Special lines that the program does not normally register - if bqlegendrecheck > 0: - pass - return references, ref9_3, ref14_3 def reference_placer(chap, references, p, chapticker2): """ - Places identified additions from the addendum into the chapter. + Places identified additions from the addendum list into the chapter list. The list returned will be turned into a + string and written into the new text document - :param chap: Chapter receiving the additions. - :param references: List containing references + :param chap: LaTeX Chapter receiving the additions, with each line an element of a list. + :param references: List containing references to where subsections begin/end :param p: List containing additions to be added. :param chapticker2: Which chapter is being searched (9 or 14). :return: Returns the chapter, complete with additions @@ -430,9 +413,9 @@ def reference_placer(chap, references, p, chapticker2): count = 0 # Tells which chapter it's on designator = 0 - if chapticker2 == 0: + if chapticker2 == 9: designator = "9." - elif chapticker2 == 1: + elif chapticker2 == 14: designator = "14." for i in references: @@ -456,9 +439,10 @@ def reference_placer(chap, references, p, chapticker2): return chap -# method to change file string(actually a list right now), returns string to be written to file -# If you write a method that changes something, it is preffered that you call the method in here + def fix_chapter(chap, references, paragraphs_to_be_added, kls, kls_list_all, chapticker2, new_commands, klsaddparas, sortmatch_2, tempref, sorter_check): + # method to change file string(actually a list right now), returns string to be written to file + # If you write a method that changes something, it is preferred that you call the method in here """ Removes specific lines stopping the latex file from converting into python, as well as running the functions responsible for sorting sections and placing the correct additions in the correct places @@ -474,12 +458,11 @@ def fix_chapter(chap, references, paragraphs_to_be_added, kls, kls_list_all, cha sort_location = 0 - for i in kls_list_all: - fix_chapter_sort(kls, chap, i, sort_location, klsaddparas, sortmatch_2, tempref, sorter_check) + for name in kls_list_all: + fix_chapter_sort(kls, chap, name, sort_location, klsaddparas, sortmatch_2, tempref, sorter_check) sort_location += 1 - print "SEARCH HERE FIRST" - print paragraphs_to_be_added + for paragraph in range(len(paragraphs_to_be_added)): for subsection in sortmatch_2: for lines_in_subsection in subsection: @@ -496,10 +479,7 @@ def fix_chapter(chap, references, paragraphs_to_be_added, kls, kls_list_all, cha # print c # print "DING" # print paragraphs_to_be_added[a] - if "\wt h_n(x;q)=x^n\,\qhyp21{q^{-n},q^{-n+1}}0{q^2,-q^2 x^{-2}}." in lines_in_subsection: - print "OUTPUTS" - print lines_in_subsection - print paragraphs_to_be_added[paragraph] + paragraphs_to_be_added[paragraph] = cut_words(lines_in_subsection, paragraphs_to_be_added[paragraph]) reference_placer(chap, references, paragraphs_to_be_added, chapticker2) @@ -510,14 +490,12 @@ def fix_chapter(chap, references, paragraphs_to_be_added, kls, kls_list_all, cha commentticker = 0 # These commands mess up PDF reading and mus tbe commented out - print "OUTPUT" for word in chap: word2 = chap[chap.index(word)-1] if "\\newcommand{\qhypK}[5]{\,\mbox{}_{#1}\phi_{#2}\!\left(" not in word2: if "\\newcommand\\half{\\frac12}" in word: wordtoadd = "%" + word chap[commentticker] = wordtoadd - print("TRIGGERED") elif "\\newcommand{\\hyp}[5]{\\,\\mbox{}_{#1}F_{#2}\\!\\left(" in word: wordtoadd = "%" + word chap[commentticker] = wordtoadd @@ -528,7 +506,6 @@ def fix_chapter(chap, references, paragraphs_to_be_added, kls, kls_list_all, cha wordtoadd = "%" + word chap[commentticker] = wordtoadd commentticker += 1 - print"END OUTPUT" ticker1 = 0 # Formatting to make the Latex file run while ticker1 < len(chap): @@ -580,8 +557,9 @@ def main(): indexes = [] # Designates sections that need stuff added - # get the index - startindex = 9999 + # gets the index + startindex = 99999 + #number must be larger than all the others, thus 99999 for word in addendum: index += 1 if "." in word and "\\subsection*{" in word: @@ -594,8 +572,8 @@ def main(): name = word[word.find("{") + 1: word.find("}")] math_people.append(name + "#") if name == "9.1 Wilson": + #9.1 Wilson is here as it is the start of chapter 9, and where the introduction ends startindex = addendum.index(word) - print startindex indexes.append(index-1) if "paragraph{" in word and index > startindex-1: klsaddparas.append(index-1) @@ -606,7 +584,7 @@ def main(): if "\\renewcommand{\\refname}{Standard references}" in word: klsaddparas.append(index-1) indexes.append(index - 1) - print(indexes) + print(klsaddparas) # now indexes holds all of the places there is a section # using these indexes, get all of the words in between and add that to the paras[] paras = [] @@ -621,7 +599,6 @@ def main(): # section(like Wilson, Racah, Hahn, etc.) so we use the mathPeople variable # we can use the section names to place the relevant paragraphs in the right place - # as of 2/8/16 the paragraphs will go before the References paragraph of the relevant section # parse both files 9 and 14 as strings with open(DATA_DIR + "chap01.tex", "r") as ch1: @@ -643,11 +620,11 @@ def main(): # call the findReferences method to find the index of the References paragraph in the two file strings # two variables for the references lists one for chapter 9 one for chapter 14 - chapticker = 0 + chapticker = 9 references9 = find_references(entire9, chapticker, math_people) ref9_3 = references9[1] references9 = references9[0] - chapticker += 1 + chapticker = 14 references14 = find_references(entire14, chapticker, math_people) ref14_3 = references14[2] references14 = references14[0] @@ -656,10 +633,10 @@ def main(): # call the fixChapter method to get a list with the addendum paragraphs added in - chapticker2 = 0 + chapticker2 = 9 str9 = ''.join(fix_chapter(entire9, references9, paras, addendum, kls_list_all, chapticker2, new_commands, klsaddparas, sortmatch_2, ref9_3, sorter_check)) - chapticker2 += 1 + chapticker2 = 14 str14 = ''.join(fix_chapter(entire14, references14, paras, addendum, kls_list_all, chapticker2, new_commands, klsaddparas, sortmatch_2, ref14_3, sorter_check)) # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ @@ -675,8 +652,5 @@ def main(): with open(DATA_DIR + "updated14.tex", "w") as temp14: temp14.write(str14) - print sorter_check - print kls_list_all - print kls_list_all[7] if __name__ == '__main__': main() \ No newline at end of file From af111853d318c589e0878fb1c3343255218890f8 Mon Sep 17 00:00:00 2001 From: Edward Bian Date: Tue, 20 Dec 2016 16:45:53 -0500 Subject: [PATCH 388/402] Refactored unit tests --- KLSadd_insertion/test/test_chap1_placer.py | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/KLSadd_insertion/test/test_chap1_placer.py b/KLSadd_insertion/test/test_chap1_placer.py index 121e02e..90aacfd 100644 --- a/KLSadd_insertion/test/test_chap1_placer.py +++ b/KLSadd_insertion/test/test_chap1_placer.py @@ -3,14 +3,14 @@ __status__ = 'Development' from unittest import TestCase -from updateChapters import chap1placer +from updateChapters import chap_1_placer class TestChap1Placer(TestCase): def test_chap1_placer(self): - self.assertEquals(chap1placer(['WordsAndSomeMoreWords','SampleEquation', '\\end{document}'] - , ['\\subsection*{Generalities}', '\\paragraph{MathFunction}', 'MathEquations', 'WordsAndStuff', + self.assertEquals(chap_1_placer(['WordsAndSomeMoreWords', 'SampleEquation', '\\end{document}'] + , ['\\subsection*{Generalities}', '\\paragraph{MathFunction}', 'MathEquations', 'WordsAndStuff', '\\subsection*{9.1 Wilson}', '\\paragraph{Symmetry}', 'WordsAndStuff'], [0,1,4,5]) - , 'WordsAndSomeMoreWordsSampleEquation\\paragraph{\\bf KLS Addendum: Generalities}\\paragraph{MathFunction}MathEquationsWordsAndStuff\\end{document}') + , 'WordsAndSomeMoreWordsSampleEquation\\paragraph{\\bf KLS Addendum: Generalities}\\paragraph{MathFunction}MathEquationsWordsAndStuff\\end{document}') From 18f01165a5a52ef67bf2c6e55dd4503716ff58b2 Mon Sep 17 00:00:00 2001 From: Joon Date: Mon, 2 Jan 2017 13:43:24 -0500 Subject: [PATCH 389/402] Change core methods to use DataUnit --- tex2wiki/.gitignore | 3 +- tex2wiki/src/tex2wiki.py | 104 ++++++++++++++++++++------------------- 2 files changed, 56 insertions(+), 51 deletions(-) diff --git a/tex2wiki/.gitignore b/tex2wiki/.gitignore index 1fa607a..1fea6f9 100644 --- a/tex2wiki/.gitignore +++ b/tex2wiki/.gitignore @@ -2,4 +2,5 @@ AzeemOldCode/ *Glossary.csv* *.mmd *.bak -*.tex \ No newline at end of file +*.tex +data/* \ No newline at end of file diff --git a/tex2wiki/src/tex2wiki.py b/tex2wiki/src/tex2wiki.py index 5ce3a1c..e702600 100644 --- a/tex2wiki/src/tex2wiki.py +++ b/tex2wiki/src/tex2wiki.py @@ -1,10 +1,10 @@ __author__ = "Joon Bang" __status__ = "Prototype" -INPUT_FILE = "tex2wiki/data/test.tex" -OUTPUT_FILE = "tex2wiki/data/test.mmd" +INPUT_FILE = "tex2wiki/data/01outb.tex" +OUTPUT_FILE = "tex2wiki/data/01outb.mmd" GLOSSARY_LOCATION = "tex2wiki/data/new.Glossary.csv" -TITLE_STRING = "CFSF Dataset" +TITLE_STRING = "Orthogonal Polynomials" METADATA_TYPES = ["substitution", "constraint"] METADATA_MEANING = {"substitution": "Substitution(s)", "constraint": "Constraint(s)"} @@ -26,7 +26,7 @@ def make_from_raw(raw): for i, equation in enumerate(equations): # formula stuff try: - formula = get_data_str(equation, latex="\\mapletag") + formula = format_formula(get_data_str(equation, latex="\\formula")) except IndexError: # there is no formula. break @@ -102,7 +102,7 @@ def generate_math_html(text, options=None, spacing=True): result = "" if spacing: - result += "{\\displaystyle\n" + text + "\n}\n" + result += "{\\displaystyle \n" + text + "\n}\n" else: result += "{\\displaystyle " + text + "}\n" @@ -147,6 +147,23 @@ def convert_dollar_signs(string): return result +def format_formula(formula): + if formula[0] == ":": + return formula[1:].zfill(2) + + formula = multi_split(formula.split("Formula:", 1)[1], [".", ":"]) + + for j in [-1, -2, -3]: + formula[j] = formula[j].zfill(2) + + if len(formula) == 3: + formula = formula[0] + "." + formula[1] + ":" + formula[2] + else: + formula = ":".join(formula[:-3]) + ":" + formula[-3] + "." + formula[-2] + ":" + formula[-1] + + return formula + + def format_metadata(string): # type: (str) -> str """Formats the metadata of an equation.""" @@ -181,7 +198,7 @@ def extract_data(data): for i, equation in enumerate(equations): # formula stuff try: - formula = get_data_str(equation, latex="\\mapletag") + formula = format_formula(get_data_str(equation, latex="\\formula")) except IndexError: # there is no formula. break @@ -319,29 +336,29 @@ def get_symbols(text, glossary): return span_text[:-7] # slice off the extra br and endline -def create_general_pages(data, title): - # type: (list, str) -> str - """Creates the 'index' pages for each section.""" +def create_general_pages(data): + # type: (DataUnit) -> str + """Creates the 'index' pages for each section. Corrected for use of DataUnit.""" ret = "" - # get list of section names - section_names = [d[0] for d in data] - section_names = [TITLE_STRING] + section_names + [TITLE_STRING] + section_names = [TITLE_STRING] + [unit.title for unit in data.subunits] + [TITLE_STRING] + + # deep down, subunits is a list of LatexEquation(s) - for i, section_data in enumerate(data): - section_name = section_data[0] - result = "drmf_bof\n'''" + section_name.replace("''", "") + "'''\n{{DISPLAYTITLE:" + section_name + "}}\n" + for i, section in enumerate(data.subunits): + result = "drmf_bof\n'''" + section.title.replace("''", "") + "'''\n{{DISPLAYTITLE:" + section.title + "}}\n" # get header and footer - center_text = (TITLE_STRING + "#Sections in " + title).replace(" ", "_") + center_text = (TITLE_STRING + "#").replace(" ", "_") + section.title link_info = [[section_names[i], section_names[i]], [center_text, section_names[i + 1]], [section_names[i + 2], section_names[i + 2]]] header, footer = generate_nav_bar(link_info) - result += header + "\n== " + section_name + " ==\n\n" + result += header + "\n== " + section.title + " ==\n\n" text = "" - for eq in section_data[1]: + metadata_exists = False + for eq in section.subunits: # equation should be of type LatexEquation text += generate_math_html(eq.equation, options={"id": eq.label}) @@ -355,33 +372,36 @@ def create_general_pages(data, title): if not metadata_exists: text = text[:-1] + "
\n" + if not metadata_exists: + text = text[:-7] + "\n" + result += text + footer + "\n" + "drmf_eof\n" ret += result return ret - def create_specific_pages(data, glossary): - # type: (list, dict) -> str - """Creates specific pages for each formula.""" + # type: (DataUnit, dict) -> str + """Creates specific pages for each formula. Corrected for use with DataUnit.""" + formulae = list() - for section_data in data: - for equations in section_data[1:]: - for eq in equations: - formulae.append("Formula:" + eq.label) - formulae.append(section_data[0]) + for unit in data.subunits: + for eq in unit.subunits: + formulae.append("Formula:" + eq.label) + formulae.append(unit.title) formulae = [TITLE_STRING] + formulae[:-1] + [TITLE_STRING] + print formulae + i = 0 pages = list() - for j, section_data in enumerate(data): - section_name = section_data[0] - for eq in section_data[1]: + for j, unit in enumerate(data.subunits): + for eq in unit.subunits: # get header and footer - center_text = (section_name + "#" + eq.label).replace(" ", "_") - middle = "formula in " + section_name + center_text = (unit.title + "#" + eq.label).replace(" ", "_") + middle = "formula in " + unit.title link_info = [[formulae[i].replace(" ", "_"), formulae[i]], [center_text, middle], [formulae[i + 2].replace(" ", "_"), formulae[i + 2]]] header, footer = generate_nav_bar(link_info) @@ -431,8 +451,8 @@ def section_split(string, sub=0): """ Split string into DataTree objects. Will eventually become replacement for large part of main() + extract_data(), - and should theoretically make the create_general_pages and create_specific_pages methods simpler, - as well as more easily store all data from the .tex file. + and should theoretically make the create_general_pages and create_specific_pages methods simpler, faster, + and able to more easily store all data from the .tex file. """ string = string.split("\\" + sub * "sub" + "section") @@ -472,24 +492,8 @@ def main(): text = text.split("\\begin{document}", 1)[1] info = section_split(text) # creates tree, split into section, subsection, subsubsection, etc. - # TODO: change create_general_pages & create_specific_pages to utilize DataUnit format rather than list format - - text = text.split("\\section") - - title = get_data_str(text[0]) - - data = list() - for section in text[1:]: - section_name = get_data_str(section).replace("$", "''") - equations = section.split("\\end{equation}") - for i, equation in enumerate(equations[:-1]): - equations[i] = equation.split("\\begin{equation}", 1)[1].strip() - - data.append([section_name, equations]) - - data = extract_data(data) - output = create_general_pages(data, title) + create_specific_pages(data, glossary) + output = create_general_pages(info) + create_specific_pages(info, glossary) with open(OUTPUT_FILE, "w") as output_file: output_file.write(output) From b442d3cbf63b945a70d8d79f3b9068708d1c7adf Mon Sep 17 00:00:00 2001 From: Edward Bian Date: Tue, 3 Jan 2017 17:52:18 -0500 Subject: [PATCH 390/402] Jagan Readme fix --- KLSadd_insertion/README.md | 22 +++++++++++----------- 1 file changed, 11 insertions(+), 11 deletions(-) diff --git a/KLSadd_insertion/README.md b/KLSadd_insertion/README.md index 451afd9..bfa1d4b 100644 --- a/KLSadd_insertion/README.md +++ b/KLSadd_insertion/README.md @@ -1,8 +1,8 @@ ##KLS Addendum Insertion Project -This project uses python and string manipulation to insert sections of the KLSadd.tex file into the appropriate DRMF chapter sections. The updateChapters.py program is the most recently updated and cleanest version, however it is not completed. Should be run with a simple call to +This project uses Python and string manipulation to insert sections of the KLSadd.tex file into the appropriate DRMF chapter sections. updateChapters.py is the most recently updated and cleanest version, however it is not completed. Should be run with a simple call to ``` -python updateChapters.py +Python updateChapters.py ``` The program must have a tempchap9.tex and a tempchap14.tex as well as the KLSadd.tex file in the same directory! The linetest.py program is the original program file fully updated. It is much harder to read and should only be used as a reference to update the updateChapters.py program. @@ -11,7 +11,7 @@ NOTE: Both the updateChapters.py and linetest.py lack a pdf version of their cha NOTE: The KLSadd.tex file only deals with chapters 9 and 14, as stated in the document itself. -NOTE: when you run updateChapters.py or linetest.py you need the KLSadd.tex file, tempchap9.tex, and tempchap14.tex in order to run the program. The program *generates* updated9.tex and updated14.tex files. +NOTE: when you run updateChapters.py or linetest.py you need the KLSadd.tex file, tempchap9.tex, and tempchap14.tex in order to run the program. The program generates updated9.tex and updated14.tex files. **DO NOT REMOVE INPUT FILES FROM .GITIGNORE, THEY ARE NOT PUBLIC** @@ -21,11 +21,11 @@ This is a roadmap of updateChapters.py. It will help explain every piece of code _______________________________________________________________________________ -First: the files +# First: The Files - - KLSadd.tex: This is the addendum we are working with. It has sections that correspond with sections in the chapter files, and it contains paragraphs that must be *inserted* into the chapter files. CAN BE FOUND ONINE IN PDF FORM: https://staff.fnwi.uva.nl/t.h.koornwinder/art/informal/KLSadd.pdf + - KLSadd.tex: This is the addendum we are working with. It has sections that correspond with sections in the chapter files, and it contains paragraphs that must be inserted into the chapter files. CAN BE FOUND ONINE IN PDF FORM: https://staff.fnwi.uva.nl/t.h.koornwinder/art/informal/KLSadd.pdf - - chap09.tex and chap14.tex: these are chapter files. These are LaTeX documents that are chapters in a book. This is a book written by smart math people. But they did some things wrong, so another math person wants to fix them. That math person wrote an addendum file called KLSadd.tex and our job is to pull paragraphs out of KLSadd and insert them at the end of the relevant section in the chapter files. Every chapter is made up of sections and every section has subsections + - chap09.tex and chap14.tex: these are chapter files. These are LaTeX documents that are chapters in a book. This is a book written by smart math people. But they did some things wrong, so another math person wants to fix them. That math person wrote an addendum file called KLSadd.tex and our job is to pull paragraphs out of KLSadd and insert them at the end of the relevant section in the chapter files. Every chapter is made up of sections and every section has subsections. - updateChapters.py: This is thr code currently being worked on and sorting chapters. This document will be going over the variables and methods in this program. This program should: take paragraphs from every section in KLSadd and insert them into the relevant chapter file and into the relevant section within that chapter. @@ -33,14 +33,14 @@ First: the files - newtempchap09.tex and newtempchap14.tex these are output files of linetest.py and this is what a standard output looks like. They are no longer up to date for updateChapters as it sorts by subsection, but still serve as good reference -Next: the specifications +# Next: the specifications Each section has subsections like "Hypergeometric Representation", sections in KLSadd with "Hypergeometric Representation" should be seorted there. Toward the end of every section in the chapter file, there is a subsection called "References". It basically just contains a bunch of references to sources and other papers. This is where unsorted materials go For example: -**These aren't real sections, obviously, just an example** +**These aren't real sections, just an example** -in chap09.tex: +In chap09.tex: ```latex \section{Dogs}\index{Dog polynomials} \subsection*{Hypergeometric dogs} @@ -49,7 +49,7 @@ blah blah blah \cite{Dogs101} ``` -in KLSadd.tex: +In KLSadd.tex: ```latex \subsection*{9.15 Dogs} @@ -67,7 +67,7 @@ So in the chapter 9 file, section 15 "Dogs", We change the formatting with "\bf The variables: -These are the variables in the beginning +These are the variables in the beginning: -chapNums: chapNums is a list, denoted by the [], that holds the chapter number (in this case it's either a 9 or a 14) that corresponds to where a section in KLSadd should go From 00227046b74fb1ca32d0590b150b6d78bb1394bb Mon Sep 17 00:00:00 2001 From: Edward Bian Date: Tue, 3 Jan 2017 17:53:20 -0500 Subject: [PATCH 391/402] Jagan linetest fix --- KLSadd_insertion/src/linetest.py | 3 +-- 1 file changed, 1 insertion(+), 2 deletions(-) diff --git a/KLSadd_insertion/src/linetest.py b/KLSadd_insertion/src/linetest.py index 4d738a7..b86133b 100644 --- a/KLSadd_insertion/src/linetest.py +++ b/KLSadd_insertion/src/linetest.py @@ -11,9 +11,8 @@ sections = [] mathPeople = [] newCommands = [] -#can you tell I like lists? name = "" -temp = "" +temp = "" with open("KLSadd.tex", "r") as file: entireFile = file.readlines() for word in entireFile: From 9d891ccff279234150e3c526cfc6e3db31dc0a82 Mon Sep 17 00:00:00 2001 From: Joon Date: Sat, 7 Jan 2017 21:49:23 -0500 Subject: [PATCH 392/402] Fix issues with
statements --- tex2wiki/src/compare_files.py | 12 +++ tex2wiki/src/tex2wiki.py | 189 +++++++++++++++++++++++++--------- 2 files changed, 155 insertions(+), 46 deletions(-) create mode 100644 tex2wiki/src/compare_files.py diff --git a/tex2wiki/src/compare_files.py b/tex2wiki/src/compare_files.py new file mode 100644 index 0000000..b1ec786 --- /dev/null +++ b/tex2wiki/src/compare_files.py @@ -0,0 +1,12 @@ +import os + + +def compare(output_file): + commands = [ + "TYPE %s.mmd | FIND /V \"\" > %s.frmt" % (output_file, output_file), + "TYPE %s.mmd.bak | FIND /V \"\" > %s.bak.frmt" % (output_file, output_file), + "fc /N %s.frmt %s.bak.frmt > tex2wiki\\data\\output.txt" % (output_file, output_file) + ] + + for command in commands: + os.system(command) diff --git a/tex2wiki/src/tex2wiki.py b/tex2wiki/src/tex2wiki.py index e702600..f35dd83 100644 --- a/tex2wiki/src/tex2wiki.py +++ b/tex2wiki/src/tex2wiki.py @@ -1,13 +1,15 @@ __author__ = "Joon Bang" __status__ = "Prototype" -INPUT_FILE = "tex2wiki/data/01outb.tex" -OUTPUT_FILE = "tex2wiki/data/01outb.mmd" +INPUT_FILE = "tex2wiki/data/09outb.tex" +OUTPUT_FILE = "tex2wiki/data/09outb.mmd" GLOSSARY_LOCATION = "tex2wiki/data/new.Glossary.csv" TITLE_STRING = "Orthogonal Polynomials" METADATA_TYPES = ["substitution", "constraint"] METADATA_MEANING = {"substitution": "Substitution(s)", "constraint": "Constraint(s)"} +import copy +import compare_files import csv @@ -44,13 +46,28 @@ def make_from_raw(raw): metadata = dict() for data_type in METADATA_TYPES: - metadata[data_type] = format_metadata(get_data_str(raw_metadata, latex="\\" + data_type)) + temp = format_metadata(get_data_str(raw_metadata, latex="\\" + data_type)).split("\n") + for k, line in enumerate(temp): + if line.rstrip().endswith("&"): + temp[k] = line.rstrip() + "
" + + metadata[data_type] = "\n".join(temp) equations[i] = LatexEquation(formula, '\n'.join(equation), metadata) return equations +class FormattedEquation(object): + def __init__(self, label, equation, metadata): + self.label = label + self.equation = equation + self.metadata = metadata + + def __str__(self): + return "Equation: " + self.equation + "\nMetadata: " + ("empty\n" if self.metadata == "" else self.metadata) + + class DataUnit(object): def __init__(self, title, subunits): self.title = title @@ -59,11 +76,11 @@ def __init__(self, title, subunits): def __str__(self): result = self.title + "\n" - if type(self.subunits) == list and type(self.subunits[0]) == DataUnit: + if type(self.subunits) == list and len(self.subunits) and type(self.subunits[0]) == DataUnit: for sub in self.subunits: result += str(sub) + "\n" else: - result += str([str(eq) for eq in self.subunits]) + "\n" + result += str("\n".join([str(eq) for eq in self.subunits])) + "\n" return result @@ -92,8 +109,8 @@ def generate_html(tag_name, text, options=None, spacing=2): return result -def generate_math_html(text, options=None, spacing=True): - # type: (str(, dict, bool)) -> str +def generate_math_html(text, options=None, spacing=2): + # type: (str(, dict, int)) -> str """Special case of generate_html, where the tag is "math".""" if options is None: options = {} @@ -101,10 +118,12 @@ def generate_math_html(text, options=None, spacing=True): option_text = [key + "=\"" + value + "\"" for key, value in options.iteritems()] result = "" - if spacing: + if spacing == 2: result += "{\\displaystyle \n" + text + "\n}\n" - else: + elif spacing == 1: result += "{\\displaystyle " + text + "}\n" + else: + result += "{\\displaystyle " + text + "}" return result @@ -296,7 +315,7 @@ def find_all(pattern, string): def get_symbols(text, glossary): # type: (str, dict) -> str - """Generates span text based on symbols present in text.""" + """Generates span text based on symbols present in text. Equivalent of old symbols_list module.""" symbols = set() for keyword in glossary: @@ -344,7 +363,6 @@ def create_general_pages(data): section_names = [TITLE_STRING] + [unit.title for unit in data.subunits] + [TITLE_STRING] # deep down, subunits is a list of LatexEquation(s) - for i, section in enumerate(data.subunits): result = "drmf_bof\n'''" + section.title.replace("''", "") + "'''\n{{DISPLAYTITLE:" + section.title + "}}\n" @@ -354,54 +372,113 @@ def create_general_pages(data): [section_names[i + 2], section_names[i + 2]]] header, footer = generate_nav_bar(link_info) - result += header + "\n== " + section.title + " ==\n\n" + result += header + "\n" + general_equation_format(section)[0] + "\n" + footer + "\n" + "drmf_eof\n" - text = "" - metadata_exists = False - for eq in section.subunits: - # equation should be of type LatexEquation - text += generate_math_html(eq.equation, options={"id": eq.label}) + ret += result - metadata_exists = False - for data_type in sorted(eq.metadata.keys()): - if eq.metadata[data_type] != "": - text += generate_html("div", METADATA_MEANING[data_type] + ": " + eq.metadata[data_type], - options={"align": "right"}, spacing=False) + "
\n" - metadata_exists = True + # post-processing + ret = ret.replace("\n\n\n", "\n") - if not metadata_exists: - text = text[:-1] + "
\n" + return ret - if not metadata_exists: - text = text[:-7] + "\n" - result += text + footer + "\n" + "drmf_eof\n" +def format_stuffs(data): + # type: (DataUnit) -> str + """Format the equations for the 'general' pages.""" + + result = copy.deepcopy(data) + + for i, unit in enumerate(result.subunits): + if type(unit) == LatexEquation: + equation = generate_math_html(unit.equation, options={"id": unit.label}) + metadata = list() + for data_type in sorted(unit.metadata.keys()): + if unit.metadata[data_type] != "": + metadata.append( + generate_html("div", METADATA_MEANING[data_type] + ": " + unit.metadata[data_type], + options={"align": "right"}, spacing=0) + ) + + text = equation.rstrip("\n") + "\n" * bool(len(metadata)) + "
\n".join(metadata) + "
\n" + result.subunits[i] = text + else: + result.subunits[i] = format_stuffs(result.subunits[i]) - ret += result + return result + + +def general_equation_format(data, depth=0): + # type: (DataUnit) -> str + """Format the equations for the 'general' pages.""" + + border = "=" * (2 + int(depth >= 2)) # '==' when depth < 2; '===' when depth >= 2 + text = "%s %s %s\n\n" % (border, data.title, border) + + contains_deeper_depth = False + metadata = list() + for i, unit in enumerate(data.subunits): + if type(unit) == LatexEquation: + equation = generate_math_html(unit.equation, options={"id": unit.label}) + metadata = list() + for data_type in sorted(unit.metadata.keys()): + if unit.metadata[data_type] != "": + metadata.append( + generate_html("div", METADATA_MEANING[data_type] + ": " + unit.metadata[data_type], + options={"align": "right"}, spacing=0) + ) + + text += equation.rstrip("\n") + "\n" * bool(len(metadata)) + "
\n".join(metadata) + "
\n" + else: + contains_deeper_depth = True + temp = general_equation_format(unit, depth + 1) + text = text.rstrip("\n") + "\n\n" + temp[0] + + if depth + 1 >= 2 and i == len(data.subunits) - 1 and temp[1]: + text = remove_break(text) + + if depth < 2 and metadata == list() and not contains_deeper_depth: + text = remove_break(text) + + return text, metadata == list() - return ret def create_specific_pages(data, glossary): # type: (DataUnit, dict) -> str """Creates specific pages for each formula. Corrected for use with DataUnit.""" + formulae = [TITLE_STRING] + make_formula_list(data)[:-1] + [TITLE_STRING] + + i = 0 + pages = list() + for j, unit in enumerate(data.subunits): + res, i = specific_page_format(unit, unit.title, formulae, glossary, j, i) + pages += res + i += 1 + + return "\n".join(pages) + "\n" + + +def make_formula_list(info): formulae = list() - for unit in data.subunits: - for eq in unit.subunits: + if len(info.subunits) and type(info.subunits[0]) != DataUnit: + for eq in info.subunits: formulae.append("Formula:" + eq.label) - formulae.append(unit.title) + formulae.append(info.title) + else: + for subunit in info.subunits: + formulae += make_formula_list(subunit) - formulae = [TITLE_STRING] + formulae[:-1] + [TITLE_STRING] + return formulae - print formulae - i = 0 +def specific_page_format(info, title, formulae, glossary, j, i=0): pages = list() - for j, unit in enumerate(data.subunits): - for eq in unit.subunits: + + if len(info.subunits) and type(info.subunits[0]) != DataUnit: + for eq in info.subunits: # get header and footer - center_text = (unit.title + "#" + eq.label).replace(" ", "_") - middle = "formula in " + unit.title + center_text = (title + "#" + eq.label).replace(" ", "_") + middle = "formula in " + title link_info = [[formulae[i].replace(" ", "_"), formulae[i]], [center_text, middle], [formulae[i + 2].replace(" ", "_"), formulae[i + 2]]] header, footer = generate_nav_bar(link_info) @@ -429,21 +506,38 @@ def create_specific_pages(data, glossary): result += get_symbols(result, glossary) + "\n
\n\n" # bibliography section - result += "== Bibliography ==\n\n" + result += "== Bibliography==\n\n" # TODO: Fix typo after feature parity has been met result += "[http://homepage.tudelft.nl/11r49/askey/contents.html " \ - "Equation in Section 1." + str(j + 1) + "] of [[Bibliography#KLS|'''KLS''']]." - - result += "\n\n== URL links ==\n\nWe ask users to provide relevant URL links in this space.\n\n" + "Equation in Section 1." + str(j + 1) + "] of [[Bibliography#KLS|'''KLS''']]." # TODO: FIX # end of page result += "
" + footer + "\ndrmf_eof" pages.append(result) i += 1 + else: + for subunit in info.subunits: + res = specific_page_format(subunit, title, formulae, glossary, j, i) + pages += res[0] + i = res[1] - i += 1 + return pages, i - return "\n".join(pages) + "\n" + +def remove_break(string): + while string.rstrip("\n").endswith("
"): + string = string.rstrip("\n")[:-6] + + return string + + +def rstrip(string, delimiter): + # type: (str, str) -> str + """A more intuitive version of rstrip, which simply removes delimiter if it is present.""" + while string.endswith(delimiter): + string = string[:(-1 * len(delimiter))] + + return string def section_split(string, sub=0): @@ -494,9 +588,12 @@ def main(): info = section_split(text) # creates tree, split into section, subsection, subsubsection, etc. output = create_general_pages(info) + create_specific_pages(info, glossary) + output = output.replace("

", "
") with open(OUTPUT_FILE, "w") as output_file: output_file.write(output) if __name__ == '__main__': main() + + compare_files.compare(OUTPUT_FILE.replace("/", "\\").replace(".mmd", "")) From 8287f6ce55f7c6374d560466ef418c6290387f7d Mon Sep 17 00:00:00 2001 From: Joon Date: Wed, 25 Jan 2017 23:04:57 -0500 Subject: [PATCH 393/402] Fix formula id, header bugs, add missing sections --- tex2wiki/src/tex2wiki.py | 141 ++++++++++++++++++++------------------- 1 file changed, 74 insertions(+), 67 deletions(-) diff --git a/tex2wiki/src/tex2wiki.py b/tex2wiki/src/tex2wiki.py index f35dd83..a43650b 100644 --- a/tex2wiki/src/tex2wiki.py +++ b/tex2wiki/src/tex2wiki.py @@ -12,10 +12,20 @@ import compare_files import csv +# TODO: Fix bugs with the special cases formula formatting; causes issues with math id="...", spantext +# TODO: Notes about bugs of old (Azeem) output: +# TODO: 09.25:27 (headers are bugged) +# TODO: 09.07:07 (misses \iunit) +# TODO: 09.07:19 (misses \cos@@) +# TODO: 09.08:19 (headers are bugged) +# Was Azeem's program run with an older version of Glossary.csv? It would explain some discrepancies. +# Header bugs are inexplicable. + class LatexEquation(object): - def __init__(self, label, equation, metadata): + def __init__(self, label, raw_label, equation, metadata): self.label = label + self.raw_label = raw_label self.equation = equation self.metadata = metadata @@ -28,7 +38,8 @@ def make_from_raw(raw): for i, equation in enumerate(equations): # formula stuff try: - formula = format_formula(get_data_str(equation, latex="\\formula")) + # print get_data_str(equation, latex="\\formula") + raw_formula, formula = format_formula(get_data_str(equation, latex="\\formula")) except IndexError: # there is no formula. break @@ -53,21 +64,11 @@ def make_from_raw(raw): metadata[data_type] = "\n".join(temp) - equations[i] = LatexEquation(formula, '\n'.join(equation), metadata) + equations[i] = LatexEquation(formula, raw_formula, '\n'.join(equation), metadata) return equations -class FormattedEquation(object): - def __init__(self, label, equation, metadata): - self.label = label - self.equation = equation - self.metadata = metadata - - def __str__(self): - return "Equation: " + self.equation + "\nMetadata: " + ("empty\n" if self.metadata == "" else self.metadata) - - class DataUnit(object): def __init__(self, title, subunits): self.title = title @@ -167,20 +168,32 @@ def convert_dollar_signs(string): def format_formula(formula): - if formula[0] == ":": - return formula[1:].zfill(2) + """Obtain the raw formula (integers), as well as the formatted version.""" + if formula[0] == ":": # handling for Jacobi special stuff + return [str(int(formula[1:]))], formula[1:].zfill(2) formula = multi_split(formula.split("Formula:", 1)[1], [".", ":"]) for j in [-1, -2, -3]: formula[j] = formula[j].zfill(2) + raw = copy.deepcopy(formula) + + # remove any text from the raw equation + i = 0 + while i < len(raw): + if not raw[i].isdigit(): + raw.pop(i) + else: + raw[i] = str(int(raw[i])) # removes any 0s that are present from left end. + i += 1 + if len(formula) == 3: formula = formula[0] + "." + formula[1] + ":" + formula[2] else: formula = ":".join(formula[:-3]) + ":" + formula[-3] + "." + formula[-2] + ":" + formula[-1] - return formula + return raw, formula def format_metadata(string): @@ -318,6 +331,11 @@ def get_symbols(text, glossary): """Generates span text based on symbols present in text. Equivalent of old symbols_list module.""" symbols = set() + special_cases = {"&": "& : logical and
"} + acknowledged = dict() + for case in special_cases: + acknowledged[case] = False + for keyword in glossary: for index in find_all(keyword, text): # if the macro is present in the text @@ -327,7 +345,15 @@ def get_symbols(text, glossary): symbols.add(keyword) span_text = "" - for symbol in sorted(symbols, key=str.lower): + + # code to handle special cases + for keyword in special_cases: + for index in find_all(keyword, text): + if index != -1 and not acknowledged[keyword]: + span_text += special_cases[keyword] + "\n" + acknowledged[keyword] = True # to prevent duplicates + + for symbol in sorted(symbols, key=text.index): links = list() for cell in glossary[symbol]: if "http://" in cell or "https://" in cell: @@ -352,7 +378,7 @@ def get_symbols(text, glossary): span_text += "[" + id_link + " {\\displaystyle " + appearance + \ "}] : " + ''.join(meaning) + " : " + " ".join(links) + "
\n" - return span_text[:-7] # slice off the extra br and endline + return remove_break(span_text) # slice off the extra br and endline def create_general_pages(data): @@ -372,7 +398,7 @@ def create_general_pages(data): [section_names[i + 2], section_names[i + 2]]] header, footer = generate_nav_bar(link_info) - result += header + "\n" + general_equation_format(section)[0] + "\n" + footer + "\n" + "drmf_eof\n" + result += header + "\n" + equation_list_format(section)[0].rstrip("\n") + "\n" + footer + "\n" + "drmf_eof\n" ret += result @@ -382,34 +408,9 @@ def create_general_pages(data): return ret -def format_stuffs(data): - # type: (DataUnit) -> str - """Format the equations for the 'general' pages.""" - - result = copy.deepcopy(data) - - for i, unit in enumerate(result.subunits): - if type(unit) == LatexEquation: - equation = generate_math_html(unit.equation, options={"id": unit.label}) - metadata = list() - for data_type in sorted(unit.metadata.keys()): - if unit.metadata[data_type] != "": - metadata.append( - generate_html("div", METADATA_MEANING[data_type] + ": " + unit.metadata[data_type], - options={"align": "right"}, spacing=0) - ) - - text = equation.rstrip("\n") + "\n" * bool(len(metadata)) + "
\n".join(metadata) + "
\n" - result.subunits[i] = text - else: - result.subunits[i] = format_stuffs(result.subunits[i]) - - return result - - -def general_equation_format(data, depth=0): +def equation_list_format(data, depth=0): # type: (DataUnit) -> str - """Format the equations for the 'general' pages.""" + """Format the equations in a section into a list style.""" border = "=" * (2 + int(depth >= 2)) # '==' when depth < 2; '===' when depth >= 2 text = "%s %s %s\n\n" % (border, data.title, border) @@ -430,7 +431,7 @@ def general_equation_format(data, depth=0): text += equation.rstrip("\n") + "\n" * bool(len(metadata)) + "
\n".join(metadata) + "
\n" else: contains_deeper_depth = True - temp = general_equation_format(unit, depth + 1) + temp = equation_list_format(unit, depth + 1) text = text.rstrip("\n") + "\n\n" + temp[0] if depth + 1 >= 2 and i == len(data.subunits) - 1 and temp[1]: @@ -446,32 +447,38 @@ def create_specific_pages(data, glossary): # type: (DataUnit, dict) -> str """Creates specific pages for each formula. Corrected for use with DataUnit.""" - formulae = [TITLE_STRING] + make_formula_list(data)[:-1] + [TITLE_STRING] + formulae = [TITLE_STRING] + make_formula_list(data)[:-1][0] + [TITLE_STRING] + + # print "\n".join(formulae) i = 0 pages = list() for j, unit in enumerate(data.subunits): - res, i = specific_page_format(unit, unit.title, formulae, glossary, j, i) + res, i = equation_page_format(unit, unit.title, formulae, glossary, i) pages += res i += 1 + open("tex2wiki/data/darn.txt", "w").write("\n".join(res)) + return "\n".join(pages) + "\n" -def make_formula_list(info): +def make_formula_list(info, depth=0): formulae = list() if len(info.subunits) and type(info.subunits[0]) != DataUnit: for eq in info.subunits: formulae.append("Formula:" + eq.label) - formulae.append(info.title) else: for subunit in info.subunits: - formulae += make_formula_list(subunit) + formulae += make_formula_list(subunit, depth + 1)[0] + + if depth < 2: + formulae.append(info.title) - return formulae + return formulae, depth -def specific_page_format(info, title, formulae, glossary, j, i=0): +def equation_page_format(info, title, formulae, glossary, i=0): pages = list() if len(info.subunits) and type(info.subunits[0]) != DataUnit: @@ -479,8 +486,14 @@ def specific_page_format(info, title, formulae, glossary, j, i=0): # get header and footer center_text = (title + "#" + eq.label).replace(" ", "_") middle = "formula in " + title + + last_index = i + 2 + + if "Formula" not in formulae[i + 2]: + last_index = i + 3 + link_info = [[formulae[i].replace(" ", "_"), formulae[i]], [center_text, middle], - [formulae[i + 2].replace(" ", "_"), formulae[i + 2]]] + [formulae[last_index].replace(" ", "_"), formulae[last_index]]] header, footer = generate_nav_bar(link_info) # add title of page, navigation headers @@ -505,10 +518,13 @@ def specific_page_format(info, title, formulae, glossary, j, i=0): result += "== Symbols List ==\n\n" result += get_symbols(result, glossary) + "\n
\n\n" - # bibliography section + # bibliography section TODO: fix links! result += "== Bibliography==\n\n" # TODO: Fix typo after feature parity has been met result += "[http://homepage.tudelft.nl/11r49/askey/contents.html " \ - "Equation in Section 1." + str(j + 1) + "] of [[Bibliography#KLS|'''KLS''']]." # TODO: FIX + "Equation in Section " + ".".join(eq.raw_label[:2]) + "] of [[Bibliography#KLS|'''KLS''']].\n\n" + + # url links placeholder + result += "== URL links ==\n\nWe ask users to provide relevant URL links in this space.\n\n" # end of page result += "
" + footer + "\ndrmf_eof" @@ -517,7 +533,7 @@ def specific_page_format(info, title, formulae, glossary, j, i=0): i += 1 else: for subunit in info.subunits: - res = specific_page_format(subunit, title, formulae, glossary, j, i) + res = equation_page_format(subunit, title, formulae, glossary, i) pages += res[0] i = res[1] @@ -531,15 +547,6 @@ def remove_break(string): return string -def rstrip(string, delimiter): - # type: (str, str) -> str - """A more intuitive version of rstrip, which simply removes delimiter if it is present.""" - while string.endswith(delimiter): - string = string[:(-1 * len(delimiter))] - - return string - - def section_split(string, sub=0): # type: (str) -> DataUnit """ From 0aee4ef3ff6b1ef9dfe6c80457bf0771be1e7cbb Mon Sep 17 00:00:00 2001 From: Joon Bang Date: Wed, 8 Feb 2017 11:14:29 -0500 Subject: [PATCH 394/402] Wrap up Orthogonal Polynomials testing --- tex2wiki/src/tex2wiki.py | 36 ++++++++++++++++++++++++------------ 1 file changed, 24 insertions(+), 12 deletions(-) diff --git a/tex2wiki/src/tex2wiki.py b/tex2wiki/src/tex2wiki.py index a43650b..51abced 100644 --- a/tex2wiki/src/tex2wiki.py +++ b/tex2wiki/src/tex2wiki.py @@ -1,8 +1,8 @@ __author__ = "Joon Bang" __status__ = "Prototype" -INPUT_FILE = "tex2wiki/data/09outb.tex" -OUTPUT_FILE = "tex2wiki/data/09outb.mmd" +INPUT_FILE = "tex2wiki/data/14outb.tex" +OUTPUT_FILE = "tex2wiki/data/14outb.mmd" GLOSSARY_LOCATION = "tex2wiki/data/new.Glossary.csv" TITLE_STRING = "Orthogonal Polynomials" METADATA_TYPES = ["substitution", "constraint"] @@ -18,7 +18,9 @@ # TODO: 09.07:07 (misses \iunit) # TODO: 09.07:19 (misses \cos@@) # TODO: 09.08:19 (headers are bugged) -# Was Azeem's program run with an older version of Glossary.csv? It would explain some discrepancies. +# TODO: 09.15:17 (misses \HyperpFq) +# TODO: 09.15:19 (misses \cos@@) +# Was Azeem's program run with a much older version of Glossary.csv? It could explain some discrepancies. # Header bugs are inexplicable. @@ -38,7 +40,6 @@ def make_from_raw(raw): for i, equation in enumerate(equations): # formula stuff try: - # print get_data_str(equation, latex="\\formula") raw_formula, formula = format_formula(get_data_str(equation, latex="\\formula")) except IndexError: # there is no formula. break @@ -447,7 +448,7 @@ def create_specific_pages(data, glossary): # type: (DataUnit, dict) -> str """Creates specific pages for each formula. Corrected for use with DataUnit.""" - formulae = [TITLE_STRING] + make_formula_list(data)[:-1][0] + [TITLE_STRING] + formulae = [TITLE_STRING] + make_formula_list(data)[0][:-1] + [TITLE_STRING] # print "\n".join(formulae) @@ -464,6 +465,9 @@ def create_specific_pages(data, glossary): def make_formula_list(info, depth=0): + # (str(, int)) -> str, int + """Generates the list of formulae contained in the file. Used for generating headers & footers.""" + formulae = list() if len(info.subunits) and type(info.subunits[0]) != DataUnit: for eq in info.subunits: @@ -479,6 +483,7 @@ def make_formula_list(info, depth=0): def equation_page_format(info, title, formulae, glossary, i=0): + """Formats equations into individual MediaWiki pages.""" pages = list() if len(info.subunits) and type(info.subunits[0]) != DataUnit: @@ -487,10 +492,7 @@ def equation_page_format(info, title, formulae, glossary, i=0): center_text = (title + "#" + eq.label).replace(" ", "_") middle = "formula in " + title - last_index = i + 2 - - if "Formula" not in formulae[i + 2]: - last_index = i + 3 + last_index = i + 2 + int("Formula" not in formulae[i + 2]) # sets the index for the "next" link link_info = [[formulae[i].replace(" ", "_"), formulae[i]], [center_text, middle], [formulae[last_index].replace(" ", "_"), formulae[last_index]]] @@ -518,10 +520,13 @@ def equation_page_format(info, title, formulae, glossary, i=0): result += "== Symbols List ==\n\n" result += get_symbols(result, glossary) + "\n
\n\n" - # bibliography section TODO: fix links! + # bibliography section result += "== Bibliography==\n\n" # TODO: Fix typo after feature parity has been met - result += "[http://homepage.tudelft.nl/11r49/askey/contents.html " \ - "Equation in Section " + ".".join(eq.raw_label[:2]) + "] of [[Bibliography#KLS|'''KLS''']].\n\n" + result += "" \ + "[http://homepage.tudelft.nl/11r49/askey/contents.html " \ + "Equation in Section " + \ + ".".join(eq.raw_label[:2]) + \ + "] of [[Bibliography#KLS|'''KLS''']].\n\n" # Meaning? Need to check validity of this. # url links placeholder result += "== URL links ==\n\nWe ask users to provide relevant URL links in this space.\n\n" @@ -541,6 +546,9 @@ def equation_page_format(info, title, formulae, glossary, i=0): def remove_break(string): + # (str) -> str + """Removes
from the end of a string.""" + while string.rstrip("\n").endswith("
"): string = string.rstrip("\n")[:-6] @@ -603,4 +611,8 @@ def main(): if __name__ == '__main__': main() + print "Complete. Now comparing files..." + compare_files.compare(OUTPUT_FILE.replace("/", "\\").replace(".mmd", "")) + + print "File comparison successful." From a1efca5acf689baf150d163067a890844dc2ecb8 Mon Sep 17 00:00:00 2001 From: Joon Bang Date: Sun, 12 Feb 2017 10:37:23 -0500 Subject: [PATCH 395/402] Fix bug with symbols list sorting --- tex2wiki/src/tex2wiki.py | 69 ++++++++-------------------------------- 1 file changed, 14 insertions(+), 55 deletions(-) diff --git a/tex2wiki/src/tex2wiki.py b/tex2wiki/src/tex2wiki.py index 51abced..7e16380 100644 --- a/tex2wiki/src/tex2wiki.py +++ b/tex2wiki/src/tex2wiki.py @@ -20,8 +20,6 @@ # TODO: 09.08:19 (headers are bugged) # TODO: 09.15:17 (misses \HyperpFq) # TODO: 09.15:19 (misses \cos@@) -# Was Azeem's program run with a much older version of Glossary.csv? It could explain some discrepancies. -# Header bugs are inexplicable. class LatexEquation(object): @@ -113,7 +111,7 @@ def generate_html(tag_name, text, options=None, spacing=2): def generate_math_html(text, options=None, spacing=2): # type: (str(, dict, int)) -> str - """Special case of generate_html, where the tag is "math".""" + """Special case of generate_html, where the tag is "".""" if options is None: options = {} @@ -141,12 +139,13 @@ def generate_link(left, right=""): def multi_split(s, seps): # type: (str, list) -> list - """Splits a string on multiple characters.""" + """Splits a string on multiple characters, specified in "seps".""" res = [s] for sep in seps: s, res = res, [] for seq in s: res += seq.split(sep) + return res @@ -169,7 +168,8 @@ def convert_dollar_signs(string): def format_formula(formula): - """Obtain the raw formula (integers), as well as the formatted version.""" + # (list) -> list + """Obtain the raw formula (the numerical values), as well as the formatted version.""" if formula[0] == ":": # handling for Jacobi special stuff return [str(int(formula[1:]))], formula[1:].zfill(2) @@ -221,43 +221,6 @@ def format_metadata(string): return result.strip() -def extract_data(data): - # type: (list) -> list - """Extracts the equations and pertinent data from the tex file.""" - result = list() - - for section_data in data: - equations = section_data[1][:-1] - for i, equation in enumerate(equations): - # formula stuff - try: - formula = format_formula(get_data_str(equation, latex="\\formula")) - except IndexError: # there is no formula. - break - - equation = equation.split("\n") - - # get metadata - raw_metadata = "" - - j = 0 - while j < len(equation): - if "%" in equation[j]: - raw_metadata += equation.pop(j)[1:].strip().strip("\n") + "\n" - else: - j += 1 - - metadata = dict() - for data_type in METADATA_TYPES: - metadata[data_type] = format_metadata(get_data_str(raw_metadata, latex="\\"+data_type)) - - equations[i] = LatexEquation(formula, '\n'.join(equation), metadata) - - result.append([section_data[0], equations]) - - return result - - def find_end(text, left_delimiter, right_delimiter, start=0): # type: (str, str, str(, int)) -> int """A .find that accounts for nested delimiters.""" @@ -330,7 +293,7 @@ def find_all(pattern, string): def get_symbols(text, glossary): # type: (str, dict) -> str """Generates span text based on symbols present in text. Equivalent of old symbols_list module.""" - symbols = set() + symbols = list() special_cases = {"&": "& : logical and
"} acknowledged = dict() @@ -343,7 +306,8 @@ def get_symbols(text, glossary): if index != -1: index += len(keyword) # now index of next character if index >= len(text) or not text[index].isalpha(): - symbols.add(keyword) + symbols.append([keyword, index]) + break span_text = "" @@ -354,7 +318,8 @@ def get_symbols(text, glossary): span_text += special_cases[keyword] + "\n" acknowledged[keyword] = True # to prevent duplicates - for symbol in sorted(symbols, key=text.index): + for symbol in sorted(symbols, key=lambda l: l[1]): # sort by list index. + symbol = symbol[0] links = list() for cell in glossary[symbol]: if "http://" in cell or "https://" in cell: @@ -459,13 +424,11 @@ def create_specific_pages(data, glossary): pages += res i += 1 - open("tex2wiki/data/darn.txt", "w").write("\n".join(res)) - return "\n".join(pages) + "\n" def make_formula_list(info, depth=0): - # (str(, int)) -> str, int + # (str(, int)) -> (str, int) """Generates the list of formulae contained in the file. Used for generating headers & footers.""" formulae = list() @@ -483,6 +446,7 @@ def make_formula_list(info, depth=0): def equation_page_format(info, title, formulae, glossary, i=0): + # (DataUnit, str, list, dict(, int)) -> (str, int) """Formats equations into individual MediaWiki pages.""" pages = list() @@ -557,12 +521,7 @@ def remove_break(string): def section_split(string, sub=0): # type: (str) -> DataUnit - """ - Split string into DataTree objects. - Will eventually become replacement for large part of main() + extract_data(), - and should theoretically make the create_general_pages and create_specific_pages methods simpler, faster, - and able to more easily store all data from the .tex file. - """ + """Split string into DataTree objects.""" string = string.split("\\" + sub * "sub" + "section") @@ -613,6 +572,6 @@ def main(): print "Complete. Now comparing files..." - compare_files.compare(OUTPUT_FILE.replace("/", "\\").replace(".mmd", "")) + compare_files.compare(OUTPUT_FILE.replace("/", "\\").replace(".mmd", "")) # only works for Windows print "File comparison successful." From e6cac8c7c68cd334127ad3ecdb567672804bf3ab Mon Sep 17 00:00:00 2001 From: Joon Bang Date: Sun, 12 Feb 2017 11:22:57 -0500 Subject: [PATCH 396/402] Improve main(), add usage() --- tex2wiki/src/compare_files.py | 4 +- tex2wiki/src/tex2wiki.py | 71 +++++++++++++++++++++-------------- 2 files changed, 45 insertions(+), 30 deletions(-) diff --git a/tex2wiki/src/compare_files.py b/tex2wiki/src/compare_files.py index b1ec786..982644f 100644 --- a/tex2wiki/src/compare_files.py +++ b/tex2wiki/src/compare_files.py @@ -1,11 +1,11 @@ import os -def compare(output_file): +def compare(output_file, number=0): commands = [ "TYPE %s.mmd | FIND /V \"\" > %s.frmt" % (output_file, output_file), "TYPE %s.mmd.bak | FIND /V \"\" > %s.bak.frmt" % (output_file, output_file), - "fc /N %s.frmt %s.bak.frmt > tex2wiki\\data\\output.txt" % (output_file, output_file) + "fc /N %s.frmt %s.bak.frmt > tex2wiki\\data\\output%s.txt" % (output_file, output_file, str(number)) ] for command in commands: diff --git a/tex2wiki/src/tex2wiki.py b/tex2wiki/src/tex2wiki.py index 7e16380..7fb06a3 100644 --- a/tex2wiki/src/tex2wiki.py +++ b/tex2wiki/src/tex2wiki.py @@ -1,16 +1,14 @@ __author__ = "Joon Bang" -__status__ = "Prototype" - -INPUT_FILE = "tex2wiki/data/14outb.tex" -OUTPUT_FILE = "tex2wiki/data/14outb.mmd" -GLOSSARY_LOCATION = "tex2wiki/data/new.Glossary.csv" -TITLE_STRING = "Orthogonal Polynomials" -METADATA_TYPES = ["substitution", "constraint"] -METADATA_MEANING = {"substitution": "Substitution(s)", "constraint": "Constraint(s)"} +__status__ = "Development" import copy import compare_files import csv +import sys + +GLOSSARY_LOCATION = "tex2wiki/data/new.Glossary.csv" +METADATA_TYPES = ["substitution", "constraint"] +METADATA_MEANING = {"substitution": "Substitution(s)", "constraint": "Constraint(s)"} # TODO: Fix bugs with the special cases formula formatting; causes issues with math id="...", spantext # TODO: Notes about bugs of old (Azeem) output: @@ -347,19 +345,19 @@ def get_symbols(text, glossary): return remove_break(span_text) # slice off the extra br and endline -def create_general_pages(data): - # type: (DataUnit) -> str +def create_general_pages(data, title_string=""): + # type: (DataUnit(, str)) -> str """Creates the 'index' pages for each section. Corrected for use of DataUnit.""" ret = "" - section_names = [TITLE_STRING] + [unit.title for unit in data.subunits] + [TITLE_STRING] + section_names = [title_string] + [unit.title for unit in data.subunits] + [title_string] # deep down, subunits is a list of LatexEquation(s) for i, section in enumerate(data.subunits): result = "drmf_bof\n'''" + section.title.replace("''", "") + "'''\n{{DISPLAYTITLE:" + section.title + "}}\n" # get header and footer - center_text = (TITLE_STRING + "#").replace(" ", "_") + section.title + center_text = (title_string + "#").replace(" ", "_") + section.title link_info = [[section_names[i], section_names[i]], [center_text, section_names[i + 1]], [section_names[i + 2], section_names[i + 2]]] header, footer = generate_nav_bar(link_info) @@ -368,14 +366,14 @@ def create_general_pages(data): ret += result - # post-processing + # some post-processing ret = ret.replace("\n\n\n", "\n") return ret def equation_list_format(data, depth=0): - # type: (DataUnit) -> str + # type: (DataUnit(, str)) -> str """Format the equations in a section into a list style.""" border = "=" * (2 + int(depth >= 2)) # '==' when depth < 2; '===' when depth >= 2 @@ -409,13 +407,11 @@ def equation_list_format(data, depth=0): return text, metadata == list() -def create_specific_pages(data, glossary): +def create_specific_pages(data, glossary, title_string): # type: (DataUnit, dict) -> str """Creates specific pages for each formula. Corrected for use with DataUnit.""" - formulae = [TITLE_STRING] + make_formula_list(data)[0][:-1] + [TITLE_STRING] - - # print "\n".join(formulae) + formulae = [title_string] + make_formula_list(data)[0][:-1] + [title_string] i = 0 pages = list() @@ -547,9 +543,19 @@ def section_split(string, sub=0): return DataUnit(title, chunk_data) -def main(): - with open(INPUT_FILE) as input_file: - text = input_file.read() +def usage(): + """Prints usage of program.""" + + print "Usage: python tex2wiki.py filename1 filename2 ...\nNote: filename(s) should NOT include the extension." + sys.exit(0) + + +def main(input_file, output_file, title): + # (str, str, str) -> None + """Converts a .tex file to MediaWiki page(s).""" + + with open(input_file) as f: + text = f.read() glossary = dict() with open(GLOSSARY_LOCATION, "rb") as csv_file: @@ -561,17 +567,26 @@ def main(): info = section_split(text) # creates tree, split into section, subsection, subsubsection, etc. - output = create_general_pages(info) + create_specific_pages(info, glossary) + output = create_general_pages(info, title) + create_specific_pages(info, glossary, title) output = output.replace("

", "
") - with open(OUTPUT_FILE, "w") as output_file: - output_file.write(output) + with open(output_file, "w") as f: + f.write(output) + if __name__ == '__main__': - main() + if len(sys.argv) < 2: + usage() + + for i, filename in enumerate(sys.argv[1:]): + main( + input_file=filename + ".tex", + output_file=filename + ".mmd", + title="Orthogonal Polynomials" + ) - print "Complete. Now comparing files..." + print "Complete. Now comparing files..." - compare_files.compare(OUTPUT_FILE.replace("/", "\\").replace(".mmd", "")) # only works for Windows + compare_files.compare(filename.replace("/", "\\"), i) # only works for Windows - print "File comparison successful." + print "File comparison successful." From ab1741287b67a3d95a63244236c8a8e73253ee89 Mon Sep 17 00:00:00 2001 From: Edward Bian Date: Fri, 3 Mar 2017 15:49:23 -0500 Subject: [PATCH 397/402] more Jagan fixes --- KLSadd_insertion/README.md | 56 +++++++++----- KLSadd_insertion/test/test_main.py | 115 +++++++++++++++++++++++++++++ 2 files changed, 151 insertions(+), 20 deletions(-) create mode 100644 KLSadd_insertion/test/test_main.py diff --git a/KLSadd_insertion/README.md b/KLSadd_insertion/README.md index bfa1d4b..02d624e 100644 --- a/KLSadd_insertion/README.md +++ b/KLSadd_insertion/README.md @@ -5,9 +5,10 @@ This project uses Python and string manipulation to insert sections of the KLSad Python updateChapters.py ``` The program must have a tempchap9.tex and a tempchap14.tex as well as the KLSadd.tex file in the same directory! -The linetest.py program is the original program file fully updated. It is much harder to read and should only be used as a reference to update the updateChapters.py program. -NOTE: Both the updateChapters.py and linetest.py lack a pdf version of their chapter 14 output. One must be generated to view results. +The linetest.py program is the original program file fully updated. It is much harder to read and should only be used as a reference to update updateChapters.py. + +NOTE: Both the updateChapters.py and linetest.py lack a .PDF version of their chapter 14 output. One must be generated to view results. NOTE: The KLSadd.tex file only deals with chapters 9 and 14, as stated in the document itself. @@ -23,15 +24,20 @@ _______________________________________________________________________________ # First: The Files - - KLSadd.tex: This is the addendum we are working with. It has sections that correspond with sections in the chapter files, and it contains paragraphs that must be inserted into the chapter files. CAN BE FOUND ONINE IN PDF FORM: https://staff.fnwi.uva.nl/t.h.koornwinder/art/informal/KLSadd.pdf + ## - KLSadd.tex: + This is the addendum we are working with. It has sections that correspond with sections in the chapter files, and it contains paragraphs that must be inserted into the chapter files. CAN BE FOUND ONINE IN .PDF FORM: https://staff.fnwi.uva.nl/t.h.koornwinder/art/informal/KLSadd.pdf - - chap09.tex and chap14.tex: these are chapter files. These are LaTeX documents that are chapters in a book. This is a book written by smart math people. But they did some things wrong, so another math person wants to fix them. That math person wrote an addendum file called KLSadd.tex and our job is to pull paragraphs out of KLSadd and insert them at the end of the relevant section in the chapter files. Every chapter is made up of sections and every section has subsections. + ## - chap09.tex and chap14.tex: + These are chapter files. These are LaTeX documents that are chapters in a book. This is a book written by smart math people. But they did some things wrong, so another math person wants to fix them. That math person wrote an addendum file called KLSadd.tex and our job is to pull paragraphs out of KLSadd and insert them at the end of the relevant section in the chapter files. Every chapter is made up of sections and every section has subsections. - - updateChapters.py: This is thr code currently being worked on and sorting chapters. This document will be going over the variables and methods in this program. This program should: take paragraphs from every section in KLSadd and insert them into the relevant chapter file and into the relevant section within that chapter. + ## - updateChapters.py: + This is the code currently being worked on and sorting chapters. This document will be going over the variables and methods in this program. This program should: take paragraphs from every section in KLSadd and insert them into the relevant chapter file and into the relevant section within that chapter. - - linetest.py: this abomination of code is the original program. It did the job, and it did it well. However, I realized that it was very very unreadable. This program is the child I never wanted. This program does 100% work and its outputs can be used to check against updateChapters.py's output. + ## - linetest.py: + This is the original program. It did the job, and it did it well. However, I realized that it was very very unreadable. This program is the child I never wanted. This program does 100% work and its outputs can be used to check against updateChapters.py's output. - - newtempchap09.tex and newtempchap14.tex these are output files of linetest.py and this is what a standard output looks like. They are no longer up to date for updateChapters as it sorts by subsection, but still serve as good reference + ## - newtempchap09.tex and newtempchap14.tex: + These are output files of linetest.py and this is what a standard output looks like. They are no longer up to date for updateChapters as it sorts by subsection, but still serve as good reference # Next: the specifications @@ -65,33 +71,43 @@ So in the chapter 9 file, section 15 "Dogs", We change the formatting with "\bf ``` --- -The variables: +# The variables: These are the variables in the beginning: --chapNums: chapNums is a list, denoted by the [], that holds the chapter number (in this case it's either a 9 or a 14) that corresponds to where a section in KLSadd should go + ## -chapNums: + chapNums is a list, denoted by the [], that holds the chapter number (in this case it's either a 9 or a 14) that corresponds to where a section in KLSadd should go --paras: holds all of the text in the paragraphs of KLSadd.tex. This is the list that gets read when its time to insert paragraphs into the chapter file. Every element is a loooong string + ## -paras: + Holds all of the text in the paragraphs of KLSadd.tex. This is the list that gets read when its time to insert paragraphs into the chapter file. Every element is a loooong string --mathPeople: this holds the *name* of every section in KLSadd. It stores names like Wilson, Racah, Dual Hahn, etc. In our example with the Dogs above, it would hold "Dogs". This is useful to finding where to insert the correct paragraph + ## -mathPeople: T + This holds the *name* of every section in KLSadd. It stores names like Wilson, Racah, Dual Hahn, etc. In our example with the Dogs above, it would hold "Dogs". This is useful to finding where to insert the correct paragraph --newCommands: this is a list that holds ints representing line numbers in KLSadd.tex that correspond to commands. There are a few special commands in the file that help turn LaTeX files into PDF files and they need to be copied over into both chapter files + ## -newCommands: + This is a list that holds ints representing line numbers in KLSadd.tex that correspond to commands. There are a few special commands in the file that help turn LaTeX files into .PDF files and they need to be copied over into both chapter files --comms: holds the actual strings of the commands found from the line numbers stored in newCommands + ## -comms: + Holds the actual strings of the commands found from the line numbers stored in newCommands --- -The functions: +# The functions: --prepareForPDF(chap): this method inserts some packages needed by LATEX files to be turned into pdf files. It takes a list called chap which is a String containing the contents of the chapter fie. It returns the chap String edited with the special packages in place. + ## -prepareForPDF(chap): + This method inserts some packages needed by LATEX files to be turned into .PDF files. It takes a list called chap which is a String containing the contents of the chapter fie. It returns the chap String edited with the special packages in place. --getCommands(kls): takes the KLSadd.tex as a string as a parameter and stores the special commands in the comms variable mentioned earlier + ## -getCommands(kls): + Takes the KLSadd.tex as a string as a parameter and stores the special commands in the comms variable mentioned earlier --insertCommands(kls, chap, cms): kls is KLSadd.tex, chap is a chapter file string, cms is a list of commands, it is the comms variable. This method returns the chap with the special commands inserted in place + ## -insertCommands(kls, chap, cms): + kls is KLSadd.tex, chap is a chapter file string, cms is a list of commands, it is the comms variable. This method returns the chap with the special commands inserted in place --findReferences(str): takes a string representation of a chapter file. Returns a list of the line numbers of the "References" subsections. This references variable is used as an index as to where the paragraphs in paras should be inserted + ## -findReferences(str): + Takes a string representation of a chapter file. Returns a list of the line numbers of the "References" subsections. This references variable is used as an index as to where the paragraphs in paras should be inserted --fixChapter(chap, references, p, kls): takes a chapter file string, a references list, the paras variable, and the KLSadd file string. This method basically just calls all of the methods above and adds all of the extra stuff like the commands and packages. + ## -fixChapter(chap, references, p, kls): + Takes a chapter file string, a references list, the paras variable, and the KLSadd file string. This method basically just calls all of the methods above and adds all of the extra stuff like the commands and packages. @@ -104,6 +120,6 @@ Then the strings are written into seperate updated chapter files. --- -At this point in time updateChapters.py is mostly functional (there are still a few areas that do not work correctly). Currently, unit tests and additional functions are being added. +At this point in time updateChapters.py is functional (there are still a few areas that are ill formatted or inefficient). Currently, unit tests and additional functions are being added. diff --git a/KLSadd_insertion/test/test_main.py b/KLSadd_insertion/test/test_main.py new file mode 100644 index 0000000..91bfd31 --- /dev/null +++ b/KLSadd_insertion/test/test_main.py @@ -0,0 +1,115 @@ + +__author__ = 'Edward Bian' +__status__ = 'Development' + +from unittest import TestCase +from updateChapters import fix_chapter + + +class TestFixChapter(TestCase): + def test_fix_chapter(self): + +#fix_chapter(chap, references , paragraphs_to_be_added , kls, kls_list_all (basically word), chapticker2, new_commands, klsaddparas, sortmatch_2, tempref, sorter_check): +#fix_chapter_sort(chap, [5] , [\\subsection*{9.1 Wilson}\n\\label{sec9.1}\n%\n, kls, word, sortloc(should be 0) , 0, , blank?, , klsaddparas, sortmatch_2, tempref, sorter_check): +# fix_chapter_sort( kls, chap, word, sortloc , klsaddparas, sortmatch_2, tempref, sorter_check): +# fix_chapter( chap, references, paragraphs_to_be_added, kls, kls_list_all, chapticker2, new_commands , klsaddparas, sortmatch_2, tempref, sorter_check): + self.assertEquals(fix_chapter(['\\newcommand{\qhypK}[5]{\,\mbox{}_{#1}\phi_{#2}\!\left(' + , '\\newcommand\half{\\frac50}' + , '\\newcommand\half{\\frac12}' + , '\\newcommand{\\hyp}[5]{\\,\\mbox{}_{#1}F_{#2}\\!\\left(' + , '\\genfrac{}{}{0pt}{}{#3}{#4};#5\\right)}' + , '\\newcommand{\\qhyp}[5]{\\,\\mbox{}_{#1}\\phi_{#2}\\!\\left(' + , '\\section{Wilson}\\index{Wilson polynomials}' + , '\\subsection*{Hypergeometric representation}' + , '\\begin{eqnarray}' + , '& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' + , '\\end{eqnarray}' + , '\\subsection*{Orthogonality relation}' + , '\\begin{eqnarray}' + , '\\label{OrthIWilson}' + , '\\myciteKLS' + , '\\end{eqnarray}'] + ,[5],['\\subsection*{9.1 Wilson}\\n\\label{sec9.1}\\n%\\n'] + , ['\\newcommand\\smallskipamount' + , '\\newcommand\\mybibitem[1]' + ,'%' + , '\\subsection*{9.1 Wilson}' + , '\\paragraph{Hypergeometric Representation}' + , 'The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetric' + , 'in $a,b,c,d$.' + , '\\renewcommand{\\refname}{Standard references}'] + , ['Hypergeometric Representation'] + , 9, [], [3, 4, 7], [['%ab']], [6, 7, 12, 15], [[0, 0], [0, 0]]) + + # Expected output: + + , ['\\newcommand\\mybibitem[1]' + , '\\usepackage[pdftex]{hyperref} \n\\usepackage {xparse} \n\\usepackage{cite} \n' + , '\\newcommand{\\qhypK}[5]{\\,\\mbox{}_{#1}\\phi_{#2}\\!\\left(' + , '\\newcommand\half{\\frac50}' + , '%\\newcommand\\half{\\frac12}' + , '%\\newcommand{\\hyp}[5]{\\,\\mbox{}_{#1}F_{#2}\\!\\left(%Begin KLS Addendum additions\\subsection*{9.1 Wilson}\\n\\label{sec9.1}\\n%\\n%End of KLS Addendum additions' + , '%\\genfrac{}{}{0pt}{}{#3}{#4};#5\\right)}' + , '%\\newcommand{\\qhyp}[5]{\\,\\mbox{}_{#1}\\phi_{#2}\\!\\left(' + , '\\section{Wilson}\\index{Wilson polynomials}' + , '\\subsection*{Hypergeometric representation}' + , '\\begin{eqnarray}' + , '& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' + , '\\end{eqnarray}' + , '\\subsection*{Orthogonality relation}\\paragraph{\\bf KLS Addendum: Hypergeometric Representation}The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetricin $a,b,c,d$.' + , '\\begin{eqnarray}' + , '\\label{OrthIWilson}' + , '\\cite' + , '\\newcommand\\smallskipamount' + , '\\end{eqnarray}']) + self.assertEquals(fix_chapter(['\\newcommand{\qhypK}[5]{\,\mbox{}_{#1}\phi_{#2}\!\left(' + , '\\newcommand\half{\\frac50}' + , '\\newcommand\half{\\frac12}' + , '\\newcommand{\\hyp}[5]{\\,\\mbox{}_{#1}F_{#2}\\!\\left(' + , '\\genfrac{}{}{0pt}{}{#3}{#4};#5\\right)}' + , '\\newcommand{\\qhyp}[5]{\\,\\mbox{}_{#1}\\phi_{#2}\\!\\left(' + , '\\section{Wilson}\\index{Wilson polynomials}' + , '\\subsection*{Hypergeometric representation}' + , '\\begin{eqnarray}' + , '& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' + , '\\end{eqnarray}' + , '\\subsection*{Orthogonality relation}' + , '\\begin{eqnarray}' + , '\\label{OrthIWilson}' + , '\\myciteKLS' + , '\\end{eqnarray}'] + ,[5],['\\subsection*{9.1 Wilson}\\n\\label{sec9.1}\\n%\\n'] + , ['\\newcommand\\smallskipamount' + , '\\newcommand\\mybibitem[1]' + ,'%' + , '\\subsection*{9.1 Wilson}' + , '\\paragraph{Hypergeometric Representation}' + , 'The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetric' + , 'in $a,b,c,d$.' + , '\\renewcommand{\\refname}{Standard references}'] + , ['Hypergeometric Representation'] + , 9, [], [3, 4, 7], [['%a']], [6, 7, 12, 15], [[0, 0], [0, 0]]) + + # Expected output: + + , ['\\newcommand\\mybibitem[1]' + , '\\usepackage[pdftex]{hyperref} \n\\usepackage {xparse} \n\\usepackage{cite} \n' + , '\\newcommand{\\qhypK}[5]{\\,\\mbox{}_{#1}\\phi_{#2}\\!\\left(' + , '\\newcommand\half{\\frac50}' + , '%\\newcommand\\half{\\frac12}' + , '%\\newcommand{\\hyp}[5]{\\,\\mbox{}_{#1}F_{#2}\\!\\left(%Begin KLS Addendum additions\\subsection*{9.1 Wilson}\\n\\label{sec9.1}\\n%\\n%End of KLS Addendum additions' + , '%\\genfrac{}{}{0pt}{}{#3}{#4};#5\\right)}' + , '%\\newcommand{\\qhyp}[5]{\\,\\mbox{}_{#1}\\phi_{#2}\\!\\left(' + , '\\section{Wilson}\\index{Wilson polynomials}' + , '\\subsection*{Hypergeometric representation}' + , '\\begin{eqnarray}' + , '& &\\frac{W_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}\\nonumber\\' + , '\\end{eqnarray}' + , '\\subsection*{Orthogonality relation}\\paragraph{\\bf KLS Addendum: Hypergeometric Representation}The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetricin $a,b,c,d$.' + , '\\begin{eqnarray}' + , '\\label{OrthIWilson}' + , '\\cite' + , '\\newcommand\\smallskipamount' + , '\\end{eqnarray}']) + + From 6dbdfbec2dd5b277de7199f9b3e9c7033363c90f Mon Sep 17 00:00:00 2001 From: Moritz Schubotz Date: Mon, 13 Mar 2017 14:47:52 +0100 Subject: [PATCH 398/402] Update README.md Add link to the DRMF project. --- README.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/README.md b/README.md index b5b6afb..f6d8acf 100644 --- a/README.md +++ b/README.md @@ -6,7 +6,7 @@ [![Code Health](https://landscape.io/github/DRMF/DRMF-Seeding-Project/master/landscape.svg?style=flat)](https://landscape.io/github/DRMF/DRMF-Seeding-Project/master) This project is meant to convert different designated source formats to semantic LaTeX for the -DRMF project. One major aspect of this conversion is the inclusion, insertion, and replacement +[DRMF project](https://drmf.wmflabs.org). One major aspect of this conversion is the inclusion, insertion, and replacement of LaTeX semantic macros. ## MediaWiki Wikitext Generation From bcbd077dd9e2009c9a2ec6c81370b8e5ecfb8d0a Mon Sep 17 00:00:00 2001 From: Edward Bian Date: Fri, 21 Apr 2017 17:55:13 -0400 Subject: [PATCH 399/402] Minor pdf generation fixes --- KLSadd_insertion/src/updateChapters.py | 213 +++++++++++++++++-------- 1 file changed, 150 insertions(+), 63 deletions(-) diff --git a/KLSadd_insertion/src/updateChapters.py b/KLSadd_insertion/src/updateChapters.py index 94e3c8c..7f3a985 100644 --- a/KLSadd_insertion/src/updateChapters.py +++ b/KLSadd_insertion/src/updateChapters.py @@ -12,8 +12,41 @@ # w, h = 2, 100 # sorter_check = [[0 for _ in range(w)] for __ in range(h)] +def bracket_finder(word_to_search): + """ + + :param word_to_search: + :return: The word between the first set of brackets + """ + first_bracket = True; + first_bracket_location = 0; + last_bracket_location = 0; + layers = 0; + counter = 0; + first_char = 0; + for character_examined in word_to_search: + if character_examined == "{" and first_bracket == True: + first_bracket_location = word_to_search.index(character_examined) + first_bracket = False; + elif character_examined == "{" and first_bracket != True: + layers += 1; + elif character_examined == "}": + if layers == 0: + last_bracket_location = counter + #word_to_search.index(character_examined) + else: + layers -= 1 + counter += 1 + word_to_search2 = word_to_search[first_bracket_location + 1:last_bracket_location] + for character_examined in word_to_search2: + if character_examined.isalpha(): + first_char = word_to_search2.index(character_examined) + return word_to_search2[first_char:] + return word_to_search[first_bracket_location+1:last_bracket_location] + + -def chap_1_placer(chap1, kls, klsaddparas): +def chap_1_placer(chap1, kls, klsaddparas, cp1commands): """ :param chap1: Chapter 1 text file @@ -21,6 +54,8 @@ def chap_1_placer(chap1, kls, klsaddparas): :param klsaddparas: Paragraphs that the code has identified to be sorted :return: Returns chapter 1 with the sections inserted into the right place from the introduction of KLSadd """ + + chapter_start = True index = 0 kls_sub_chap1 = [] @@ -29,7 +64,8 @@ def chap_1_placer(chap1, kls, klsaddparas): index += 1 line = str(item) if "\\subsection" in line: - temp = line[line.find(" ", 12) + 1: line.find("}", 12)+1] # get just the name (like mathpeople) + temp = bracket_finder(line) + # temp = line[line.find(" ", 12) + 1: line.find("}", 12)+1] # get just the name (like mathpeople) (edit this) and line 128 # The 12 is not arbitrary and comes from the number of characters in "//subsection{" if chapter_start and ("\\paragraph{" in line or "\\subsubsection*{" in line): for item in klsaddparas: @@ -39,11 +75,57 @@ def chap_1_placer(chap1, kls, klsaddparas): t = ''.join(kls[index-1: klsaddparas[klsloc]]) kls_sub_chap1.append(t) # append the whole paragraph, every paragraph should end with a % comment kls_header_chap1.append(temp) # append the name of subsection + break intro_to_add = ''.join(kls_sub_chap1) chap1 = chap1[:len(chap1)-1] + commentticker = 0 + + # These commands mess up PDF reading and mus tbe commented out + for line in cp1commands: + prev_line = cp1commands[cp1commands.index(line) - 1] + if "\\newcommand{\qhypK}[5]{\,\mbox{}_{#1}\phi_{#2}\!\left(" not in prev_line: + if "\\newcommand\\half{\\frac12}" in line: + linetoadd = "%" + line + cp1commands[commentticker] = linetoadd + + elif "\\newcommand{\\hyp}[5]{\\,\\mbox{}_{#1}F_{#2}\\!\\left(" in line: + linetoadd = "%" + line + cp1commands[commentticker] = linetoadd + + elif "\\genfrac{}{}{0pt}{}{#3}{#4};#5\\right)}" in line: + linetoadd = "%" + line + cp1commands[commentticker] = linetoadd + + elif "\\newcommand{\\qhyp}[5]{\\,\\mbox{}_{#1}\\phi_{#2}\\!\\left(" in line: + linetoadd = "%" + line + cp1commands[commentticker] = linetoadd + + commentticker += 1 + if "\\newpage" in line and "\hbox{}" not in line: + chap1[chap1.index(line)] = "" + if '\myciteKLS' in line: + linetoadd = line.replace('\myciteKLS', '\cite') + cp1commands[commentticker] = linetoadd + + cp1commandsstring = ''.join(cp1commands) + index = 0 + for word in chap1: + index += 1 + if "\\newcommand" in chap1[index]: + chap1[index - 1] = cp1commandsstring + print"ding" + break + ticker1 = 0 + # Formatting to make the Latex file run + while ticker1 < len(chap1): + if '\\myciteKLS' in chap1[ticker1]: + chap1[ticker1] = chap1[ticker1].replace('\\myciteKLS', '\\cite') + ticker1 += 1 + chap1 = ''.join(chap1) total_chap1 = chap1 + "\\paragraph{\\bf KLS Addendum: Generalities}" + intro_to_add + "\\end{document}" + return total_chap1 @@ -125,13 +207,15 @@ def fix_chapter_sort(kls, chap, word, sortloc, klsaddparas, sortmatch_2, tempref index += 1 line = str(item) if "\\subsection" in line: - temp = line[line.find(" ", 12)+1: line.find("}", 12)] # get just the name (like mathpeople) + temp = bracket_finder(line) + #line[line.find(" ", 12)+1: line.find("}", 12)] + # get just the name (like mathpeople) if temp.lower() == 'wilson': chapterstart = True if special_input in (0, 2, 4) and name_chap in line.lower() and chapterstart and ("\\paragraph{" in line or "\\subsubsection*{" in line) or \ (special_input in (1, 3) and "orthogonality relation" not in line.lower() and name_chap in line.lower() and chapterstart and ("\\paragraph{" in line or "\\subsubsection*{" in line)): - testing = item + testing = item #CLEAN UP(?) for item in klsaddparas: if index < item: klsloc = klsaddparas.index(item) @@ -194,9 +278,9 @@ def fix_chapter_sort(kls, chap, word, sortloc, klsaddparas, sortmatch_2, tempref line = str(chap[item]) if "\\section{" in line or "\\subsection{" in line: if "\\subsection{" in line: - temp = line[12:line.find("}", 7)] + temp = bracket_finder(line) else: - temp = line[9:line.find("}", 7)] + temp = bracket_finder(line) if name_chap in line.lower(): if special_input in (0, 2, 3, 4) or "orthogonality relation" not in line: @@ -220,13 +304,15 @@ def fix_chapter_sort(kls, chap, word, sortloc, klsaddparas, sortmatch_2, tempref if chap9 == 1: sorter_check[sortloc][1] += 1 k_hyp_index_iii += 1 + with open(DATA_DIR + "shiftedsections.tex", "a") as shifted: + shifted.write("KLS Addendum: " + word + " to " + temp + "\n") except IndexError: + print chap[tempref[d+1] - 1] + print print("Warning! Code has found an error involving section finding for '" + name_chap + "'.") ''' ''' - if sortloc == 7: - with open(DATA_DIR + "testoutput.tex", "w") as thing: - thing.write(''.join(chap)) + if len(hyper_headers_chap) != 0: sortmatch_2.append(k_hyper_sub_chap) return chap @@ -243,23 +329,23 @@ def cut_words(word_to_find, word_to_search_in): :param word_to_search_in: The big word that is being searched :return: The big word without the word that was searched for """ - find_this_string = word_to_find - search_in_this_string = word_to_search_in + find_string = word_to_find + search_string = word_to_search_in precheck = 1 - if find_this_string in search_in_this_string: - if "\\paragraph{\\bf KLS Addendum: " in search_in_this_string or "\\subsubsection*{\\bf KLS Addendum: " in search_in_this_string: + if find_string in search_string: + if "\\paragraph{\\bf KLS Addendum: " in search_string or "\\subsubsection*{\\bf KLS Addendum: " in search_string: while True: - if "\\paragraph{\\bf KLS Addendum: " in search_in_this_string[search_in_this_string.find(find_this_string)-precheck:search_in_this_string.find(find_this_string)] or \ - "\\subsubsection*{\\bf KLS Addendum: " in search_in_this_string[search_in_this_string.find(find_this_string) - precheck:search_in_this_string.find(find_this_string)]: - return search_in_this_string[:search_in_this_string.find(find_this_string) - precheck] + search_in_this_string[search_in_this_string.find(find_this_string) + len(find_this_string):] + if "\\paragraph{\\bf KLS Addendum: " in search_string[search_string.find(find_string)-precheck:search_string.find(find_string)] or \ + "\\subsubsection*{\\bf KLS Addendum: " in search_string[search_string.find(find_string) - precheck:search_string.find(find_string)]: + return search_string[:search_string.find(find_string) - precheck] + search_string[search_string.find(find_string) + len(find_string):] else: precheck += 1 else: - cut = search_in_this_string[:search_in_this_string.find(find_this_string)] + search_in_this_string[search_in_this_string.find(find_this_string) + len(find_this_string):] + cut = search_string[:search_string.find(find_string)] + search_string[search_string.find(find_string) + len(find_string):] return cut else: - return search_in_this_string + return search_string def prepare_for_pdf(chap): @@ -269,20 +355,13 @@ def prepare_for_pdf(chap): :param chap: The chapter (9 or 14) that is being processed as a list of lines in each LaTeX chapter :return: The processed chapter, ready for additional processing """ + # (list) -> list foot_misc_index = 0 for i, line in enumerate(chap): if "footmisc" in line: foot_misc_index += i + 1 - ''' - index = 0 - for line in chap: - index += 1 - if "footmisc" in line: - foot_misc_index += index - ''' - # str[footmiscIndex] += "\\usepackage[pdftex]{hyperref} \n\\usepackage {xparse} \n\\usepackage{cite} \n" chap.insert(foot_misc_index, "\\usepackage[pdftex]{hyperref} \n\\usepackage {xparse} \n\\usepackage{cite} \n") return chap @@ -305,15 +384,15 @@ def get_commands(kls, new_commands): -def insert_commands(kls, chap, cms): +def insert_commands(kls, chap, commands): """ Inserts commands identified in previous functions. - This method addresses the goal of hardcoding in the necessary commands to let the chapter files run as pdf's. + This method addresses the goal of hardcoding in the necessary comma vnds to let the chapter files run as pdf's. Currently only works with chapter 9 :param kls: Addendum to look through :param chap: The chapter that is receiving commands (9 or 14) as a list of lines in each LaTeX chapter - :param cms: Commands to be inserted + :param commands: Commands to be inserted :return: chap, processed """ # reads in the newCommands[] and puts them in chap @@ -327,7 +406,7 @@ def insert_commands(kls, chap, cms): begin_index += index temp_index = 0 - for i in cms: + for i in commands: chap.insert(begin_index + temp_index, i) temp_index += 1 return chap @@ -423,8 +502,11 @@ def reference_placer(chap, references, p, chapticker2): word1 = str(p[count]) if designator in word1[word1.find("\\subsection*{") + 1: word1.find("}")]: chap[i - 2] += "%Begin KLS Addendum additions" + if "In addition to the Chebyshev poly" in p[count]: + chap[i - 2] += "\paragraph{\\bf KLS Addendum Addition}" chap[i - 2] += p[count] chap[i - 2] += "%End of KLS Addendum additions" + count += 1 else: while designator not in word1[word1.find("\\subsection*{") + 1: word1.find("}")]: @@ -436,6 +518,7 @@ def reference_placer(chap, references, p, chapticker2): count += 1 else: count += 1 + return chap @@ -457,7 +540,7 @@ def fix_chapter(chap, references, paragraphs_to_be_added, kls, kls_list_all, cha """ sort_location = 0 - + print chapticker2 for name in kls_list_all: fix_chapter_sort(kls, chap, name, sort_location, klsaddparas, sortmatch_2, tempref, sorter_check) sort_location += 1 @@ -466,9 +549,6 @@ def fix_chapter(chap, references, paragraphs_to_be_added, kls, kls_list_all, cha for paragraph in range(len(paragraphs_to_be_added)): for subsection in sortmatch_2: for lines_in_subsection in subsection: - with open(DATA_DIR + "compare.tex", "a") as spook: - spook.write(paragraphs_to_be_added[paragraph]) - spook.write("NEXT: ") if "%" == lines_in_subsection[-2]: lines_in_subsection = lines_in_subsection[:-3] #print "memes" @@ -486,26 +566,29 @@ def fix_chapter(chap, references, paragraphs_to_be_added, kls, kls_list_all, cha chap = prepare_for_pdf(chap) cms = get_commands(kls,new_commands) + print cms chap = insert_commands(kls, chap, cms) commentticker = 0 # These commands mess up PDF reading and mus tbe commented out - for word in chap: - word2 = chap[chap.index(word)-1] - if "\\newcommand{\qhypK}[5]{\,\mbox{}_{#1}\phi_{#2}\!\left(" not in word2: - if "\\newcommand\\half{\\frac12}" in word: - wordtoadd = "%" + word - chap[commentticker] = wordtoadd - elif "\\newcommand{\\hyp}[5]{\\,\\mbox{}_{#1}F_{#2}\\!\\left(" in word: - wordtoadd = "%" + word - chap[commentticker] = wordtoadd - elif "\\genfrac{}{}{0pt}{}{#3}{#4};#5\\right)}" in word: - wordtoadd = "%" + word - chap[commentticker] = wordtoadd - elif "\\newcommand{\\qhyp}[5]{\\,\\mbox{}_{#1}\\phi_{#2}\\!\\left(" in word: - wordtoadd = "%" + word - chap[commentticker] = wordtoadd + for line in chap: + prev_line = chap[chap.index(line)-1] + if "\\newcommand{\qhypK}[5]{\,\mbox{}_{#1}\phi_{#2}\!\left(" not in prev_line: + if "\\newcommand\\half{\\frac12}" in line: + linetoadd = "%" + line + chap[commentticker] = linetoadd + elif "\\newcommand{\\hyp}[5]{\\,\\mbox{}_{#1}F_{#2}\\!\\left(" in line: + linetoadd = "%" + line + chap[commentticker] = linetoadd + elif "\\genfrac{}{}{0pt}{}{#3}{#4};#5\\right)}" in line: + linetoadd = "%" + line + chap[commentticker] = linetoadd + elif "\\newcommand{\\qhyp}[5]{\\,\\mbox{}_{#1}\\phi_{#2}\\!\\left(" in line: + linetoadd = "%" + line + chap[commentticker] = linetoadd commentticker += 1 + if "\\newpage" in line and "\hbox{}" not in line: + chap[chap.index(line)] = "" ticker1 = 0 # Formatting to make the Latex file run while ticker1 < len(chap): @@ -515,7 +598,7 @@ def fix_chapter(chap, references, paragraphs_to_be_added, kls, kls_list_all, cha return chap -def main(): +def main(klsfile, klswrite9, klswrite14, klswrite1, klsread9, klsread14, klsread1): """ Runs all of the other functions and outputs their results into an output file as well as putting together the list of additions that is fed into fix_chapter. @@ -534,7 +617,7 @@ def main(): sortmatch_2 = [] # open the KLSadd file to do things with - with open(DATA_DIR + "KLSaddII.tex", "r") as add: + with open(DATA_DIR + klsfile, "r") as add: # store the file as a string addendum = add.readlines() kls_list_all = new_keywords(addendum, kls_list_full)[0] @@ -584,7 +667,7 @@ def main(): if "\\renewcommand{\\refname}{Standard references}" in word: klsaddparas.append(index-1) indexes.append(index - 1) - print(klsaddparas) + # now indexes holds all of the places there is a section # using these indexes, get all of the words in between and add that to the paras[] paras = [] @@ -601,20 +684,21 @@ def main(): # parse both files 9 and 14 as strings - with open(DATA_DIR + "chap01.tex", "r") as ch1: + with open(DATA_DIR + klsread1, "r") as ch1: entire1 = ch1.readlines() # reads in as a list of strings - with open(DATA_DIR + "updated1.tex", "w") as temp1: - temp1.write(chap_1_placer(entire1, addendum, klsaddparas)) + cp1commands = get_commands(addendum, new_commands) + print cp1commands + with open(DATA_DIR + klswrite1, "w") as temp1: + temp1.write(chap_1_placer(entire1, addendum, klsaddparas, cp1commands)) + + # chapter 9 - with open(DATA_DIR + "chap09.tex", "r") as ch9: + with open(DATA_DIR + klsread9, "r") as ch9: entire9 = ch9.readlines() # reads in as a list of strings - with open(DATA_DIR + "testoutput.tex", "w") as thing: - thing.write(''.join(entire9)) - # chapter 14 - with open(DATA_DIR + "chap14.tex", "r") as ch14: + with open(DATA_DIR + klsread14, "r") as ch14: entire14 = ch14.readlines() # call the findReferences method to find the index of the References paragraph in the two file strings @@ -639,6 +723,8 @@ def main(): chapticker2 = 14 str14 = ''.join(fix_chapter(entire14, references14, paras, addendum, kls_list_all, chapticker2, new_commands, klsaddparas, sortmatch_2, ref14_3, sorter_check)) + + # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ # If you are writing something that will make a change to the chapter files, write it BEFORE this line, this part # is where the lists representing the words/strings in the chapter are joined together and updated as a string! @@ -646,11 +732,12 @@ def main(): # write to files # new output files for safety - with open(DATA_DIR + "updated9.tex", "w") as temp9: + with open(DATA_DIR + klswrite9, "w") as temp9: temp9.write(str9) - with open(DATA_DIR + "updated14.tex", "w") as temp14: + with open(DATA_DIR + klswrite14, "w") as temp14: temp14.write(str14) + print(bracket_finder("words{19.4 morewords{}}ess")) if __name__ == '__main__': - main() \ No newline at end of file + main("KLSaddII.tex", "updated9.tex", "updated14.tex", "updated1.tex", "chap09.tex", "chap14.tex", "chap01.tex") \ No newline at end of file From 1325134a90d6423b01e05fa7d5955a2a0c5a5826 Mon Sep 17 00:00:00 2001 From: Jagan Prem Date: Tue, 25 Apr 2017 16:56:00 -0400 Subject: [PATCH 400/402] Fix \pi replacement --- DLMF_preprocessing/src/replace_special.py | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/DLMF_preprocessing/src/replace_special.py b/DLMF_preprocessing/src/replace_special.py index 09d39c3..624fa67 100644 --- a/DLMF_preprocessing/src/replace_special.py +++ b/DLMF_preprocessing/src/replace_special.py @@ -79,7 +79,7 @@ def remove_special(content): in_eq = True - pi_pat = re.compile(r'(\s*)\\pi(\s*\b|[aeiou])') + pi_pat = re.compile(r'(\s*)\\pi(?![a-zA-Z])') expe_pat = re.compile(r'\b([^\\]?\W*)\s*e\s*\^') spaces_pat = re.compile(r' {2,}') From 0eae492a6e650f158446d6782968c8d1ef25e8ac Mon Sep 17 00:00:00 2001 From: Jagan Prem Date: Tue, 25 Apr 2017 17:03:24 -0400 Subject: [PATCH 401/402] Remove excessive group --- DLMF_preprocessing/src/replace_special.py | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/DLMF_preprocessing/src/replace_special.py b/DLMF_preprocessing/src/replace_special.py index 624fa67..a13ffe9 100644 --- a/DLMF_preprocessing/src/replace_special.py +++ b/DLMF_preprocessing/src/replace_special.py @@ -100,7 +100,7 @@ def remove_special(content): is_comment = line.lstrip().startswith("%") - line = pi_pat.sub(r'\1\\cpi\2', line) + line = pi_pat.sub(r'\1\\cpi', line) line = expe_pat.sub(r'\1\\expe^', line) # only check for flags if the line is comment From 1c45e94f144c8a364f7a0ec9f98e17b739fdb8a7 Mon Sep 17 00:00:00 2001 From: Edward Bian Date: Fri, 9 Jun 2017 17:51:13 -0400 Subject: [PATCH 402/402] Commented version --- KLSadd_insertion/src/updateChapters.py | 52 +++++++++++++------------- 1 file changed, 25 insertions(+), 27 deletions(-) diff --git a/KLSadd_insertion/src/updateChapters.py b/KLSadd_insertion/src/updateChapters.py index 7f3a985..22c6a85 100644 --- a/KLSadd_insertion/src/updateChapters.py +++ b/KLSadd_insertion/src/updateChapters.py @@ -12,18 +12,18 @@ # w, h = 2, 100 # sorter_check = [[0 for _ in range(w)] for __ in range(h)] + def bracket_finder(word_to_search): """ - :param word_to_search: - :return: The word between the first set of brackets + :return: The string between the first set of brackets """ + # This function is mainly for convenience and layered brackets first_bracket = True; first_bracket_location = 0; last_bracket_location = 0; layers = 0; counter = 0; - first_char = 0; for character_examined in word_to_search: if character_examined == "{" and first_bracket == True: first_bracket_location = word_to_search.index(character_examined) @@ -33,26 +33,26 @@ def bracket_finder(word_to_search): elif character_examined == "}": if layers == 0: last_bracket_location = counter - #word_to_search.index(character_examined) else: layers -= 1 counter += 1 + # Goes through every character while recording layer to not extract the wrong string word_to_search2 = word_to_search[first_bracket_location + 1:last_bracket_location] for character_examined in word_to_search2: if character_examined.isalpha(): first_char = word_to_search2.index(character_examined) return word_to_search2[first_char:] + # This loop is for chapter sections that may start with numbers or other symbols, it makes the output start on the section name return word_to_search[first_bracket_location+1:last_bracket_location] - def chap_1_placer(chap1, kls, klsaddparas, cp1commands): """ - :param chap1: Chapter 1 text file :param kls: KLSadd, the addendum :param klsaddparas: Paragraphs that the code has identified to be sorted - :return: Returns chapter 1 with the sections inserted into the right place from the introduction of KLSadd + :return: Returns chapter 1 with the sections inserted into the right places from the introduction of KLSadd + It's bascially a modified version of fix_chapter_sort """ @@ -65,8 +65,6 @@ def chap_1_placer(chap1, kls, klsaddparas, cp1commands): line = str(item) if "\\subsection" in line: temp = bracket_finder(line) - # temp = line[line.find(" ", 12) + 1: line.find("}", 12)+1] # get just the name (like mathpeople) (edit this) and line 128 - # The 12 is not arbitrary and comes from the number of characters in "//subsection{" if chapter_start and ("\\paragraph{" in line or "\\subsubsection*{" in line): for item in klsaddparas: if index < item: @@ -114,7 +112,6 @@ def chap_1_placer(chap1, kls, klsaddparas, cp1commands): index += 1 if "\\newcommand" in chap1[index]: chap1[index - 1] = cp1commandsstring - print"ding" break ticker1 = 0 # Formatting to make the Latex file run @@ -136,9 +133,9 @@ def extraneous_section_deleter(entered_list): :param entered_list: List of paragraphs :return: Extraneous name free output """ + # Returns the sections with unique names (per section) (as the following are often duplicated) return [item for item in entered_list if "reference" not in item.lower() and "limit relation" not in item.lower() - and "symmetry" not in item.lower() and " hypergeometric representation" not in item.lower() - and "hypergeometric representation " not in item.lower()] + and "symmetry" not in item.lower() and " hypergeometric representation" not in item.lower()] def new_keywords(kls, kls_list): @@ -161,6 +158,7 @@ def new_keywords(kls, kls_list): for item in kls_list: if item not in kls_list_chap: kls_list_chap.append(item) + #Preserves original kls_list, while allowing editting (special value is classed under symmetry) kls_list_chap[kls_list_chap.index("Special value")] = "Symmetry" kls_list_chap.append("Special value") w = 2 @@ -197,6 +195,7 @@ def fix_chapter_sort(kls, chap, word, sortloc, klsaddparas, sortmatch_2, tempref special_input = 3 elif name_chap == "bilateral generating function": special_input = 4 + #These sections either share names with others or are categorized under others, so they must be processed specially khyper_header_chap = [] k_hyper_sub_chap = [] @@ -208,14 +207,11 @@ def fix_chapter_sort(kls, chap, word, sortloc, klsaddparas, sortmatch_2, tempref line = str(item) if "\\subsection" in line: temp = bracket_finder(line) - #line[line.find(" ", 12)+1: line.find("}", 12)] - # get just the name (like mathpeople) if temp.lower() == 'wilson': chapterstart = True if special_input in (0, 2, 4) and name_chap in line.lower() and chapterstart and ("\\paragraph{" in line or "\\subsubsection*{" in line) or \ (special_input in (1, 3) and "orthogonality relation" not in line.lower() and name_chap in line.lower() and chapterstart and ("\\paragraph{" in line or "\\subsubsection*{" in line)): - testing = item #CLEAN UP(?) for item in klsaddparas: if index < item: klsloc = klsaddparas.index(item) @@ -237,6 +233,7 @@ def fix_chapter_sort(kls, chap, word, sortloc, klsaddparas, sortmatch_2, tempref k_hyper_sub_chap[item + 1] = k_hyper_sub_chap[item] + "\paragraph{\\bf KLS Addendum: Bilateral generating functions}" + k_hyper_sub_chap[item + 1] khyper_header_chap[item] = "/x00" k_hyper_sub_chap[item] = "/x00" + #/x00 is a filler character that must be unique, it can be changed, as long as the replacement isn't ever going to be an actual section elif khyper_header_chap[item] == khyper_header_chap[item+1]: k_hyper_sub_chap[item + 1] = k_hyper_sub_chap[item] + "\paragraph{\\bf KLS Addendum: " + \ word + "}" + k_hyper_sub_chap[item + 1] @@ -284,8 +281,6 @@ def fix_chapter_sort(kls, chap, word, sortloc, klsaddparas, sortmatch_2, tempref if name_chap in line.lower(): if special_input in (0, 2, 3, 4) or "orthogonality relation" not in line: - - hyper_subs_chap.append([tempref[d + 1]]) # appends the index for the line before following subsection hyper_headers_chap.append(temp) # appends the name of the section the hypergeo subsection is in ''' @@ -310,8 +305,6 @@ def fix_chapter_sort(kls, chap, word, sortloc, klsaddparas, sortmatch_2, tempref print chap[tempref[d+1] - 1] print print("Warning! Code has found an error involving section finding for '" + name_chap + "'.") - ''' - ''' if len(hyper_headers_chap) != 0: sortmatch_2.append(k_hyper_sub_chap) @@ -323,8 +316,6 @@ def cut_words(word_to_find, word_to_search_in): This function checks through the outputs of later sections and removes duplicates, so that the output file does not have duplicates. - IN DEVELOPMENT - :param word_to_find: Word that is being found :param word_to_search_in: The big word that is being searched :return: The big word without the word that was searched for @@ -363,6 +354,7 @@ def prepare_for_pdf(chap): foot_misc_index += i + 1 chap.insert(foot_misc_index, "\\usepackage[pdftex]{hyperref} \n\\usepackage {xparse} \n\\usepackage{cite} \n") + #This is so things generate correctly return chap @@ -381,6 +373,7 @@ def get_commands(kls, new_commands): if "mybibitem[1]" in word: new_commands.append(index) return kls[new_commands[0]:new_commands[1]] + #Addendum needs these to generate correctly @@ -410,6 +403,7 @@ def insert_commands(kls, chap, commands): chap.insert(begin_index + temp_index, i) temp_index += 1 return chap + #Puts the commands in the chapter so they do something def find_references(chapter, chapticker, math_people): @@ -464,6 +458,7 @@ def find_references(chapter, chapticker, math_people): elif chapticker == 14: ref14_3.append(index) add_next_section = False + #The if statement differentiates chapters 9 and 14 if "subsection*{" in word and "References" not in word: if chapticker == 9: ref9_3.append(index) @@ -540,12 +535,11 @@ def fix_chapter(chap, references, paragraphs_to_be_added, kls, kls_list_all, cha """ sort_location = 0 - print chapticker2 + for name in kls_list_all: fix_chapter_sort(kls, chap, name, sort_location, klsaddparas, sortmatch_2, tempref, sorter_check) sort_location += 1 - for paragraph in range(len(paragraphs_to_be_added)): for subsection in sortmatch_2: for lines_in_subsection in subsection: @@ -563,11 +557,13 @@ def fix_chapter(chap, references, paragraphs_to_be_added, kls, kls_list_all, cha paragraphs_to_be_added[paragraph] = cut_words(lines_in_subsection, paragraphs_to_be_added[paragraph]) reference_placer(chap, references, paragraphs_to_be_added, chapticker2) - + # Reference_placer is the one that does all the work + # If anything needs to be done before commands are put in, do it here chap = prepare_for_pdf(chap) cms = get_commands(kls,new_commands) - print cms chap = insert_commands(kls, chap, cms) + # The above 3 lines insert comments into the new revised chapter + # The final output (addendum + chapter) is made here commentticker = 0 # These commands mess up PDF reading and mus tbe commented out @@ -668,6 +664,7 @@ def main(klsfile, klswrite9, klswrite14, klswrite1, klsread9, klsread14, klsread klsaddparas.append(index-1) indexes.append(index - 1) + # Looks for sections to sort # now indexes holds all of the places there is a section # using these indexes, get all of the words in between and add that to the paras[] paras = [] @@ -688,7 +685,6 @@ def main(klsfile, klswrite9, klswrite14, klswrite1, klsread9, klsread14, klsread entire1 = ch1.readlines() # reads in as a list of strings cp1commands = get_commands(addendum, new_commands) - print cp1commands with open(DATA_DIR + klswrite1, "w") as temp1: temp1.write(chap_1_placer(entire1, addendum, klsaddparas, cp1commands)) @@ -737,7 +733,9 @@ def main(klsfile, klswrite9, klswrite14, klswrite1, klsread9, klsread14, klsread with open(DATA_DIR + klswrite14, "w") as temp14: temp14.write(str14) - print(bracket_finder("words{19.4 morewords{}}ess")) + + print "Files sorted" + # Will print when code done if __name__ == '__main__': main("KLSaddII.tex", "updated9.tex", "updated14.tex", "updated1.tex", "chap09.tex", "chap14.tex", "chap01.tex") \ No newline at end of file
\n") - wiki.write("
<< [["+KLS.secLabel(sections[secCounter-1][0])+"|"+KLS.secLabel(sections[secCounter-1][0])+"]]
\n") - wiki.write("
[[Zeta_and_Related_Functions#"+ + KLS.append_text("drmf_bof\n") + KLS.append_text("\'\'\'"+KLS.secLabel(KLS.getString(line))+"\'\'\'\n") + KLS.append_text("{{DISPLAYTITLE:"+(sections[secCounter][0])+"}}\n") + KLS.append_text("
\n") + KLS.append_text("
<< [["+KLS.secLabel(sections[secCounter-1][0])+"|"+KLS.secLabel(sections[secCounter-1][0])+"]]
\n") + KLS.append_text("
[[Zeta_and_Related_Functions#"+ "Sections_in_"+chapter.replace(" ","_")+"|"+KLS.secLabel(sections[secCounter][0])+"]]
\n") - wiki.write("
[["+KLS.secLabel(sections[(secCounter+1)%len(sections)][0])+ + KLS.append_text("
[["+KLS.secLabel(sections[(secCounter+1)%len(sections)][0])+ "|"+KLS.secLabel(sections[(secCounter+1)%len(sections)][0])+"]] >>
\n
\n\n") head=True - wiki.write("== "+KLS.getString(line)+" ==\n") + KLS.append_text("== "+KLS.getString(line)+" ==\n") elif ("\\section" in lines[(i+1)%len(lines)] or "\\end{document}" in lines[(i+1)%len(lines)]) and parse: - wiki.write("
\n") - wiki.write("
<< [["+KLS.secLabel(sections[secCounter-1][0])+"|"+KLS.secLabel(sections[secCounter-1][0])+"]]
\n") - wiki.write("
[[Zeta_and_Related_Functions#"+"Sections_in_" + KLS.append_text("
\n") + KLS.append_text("
<< [["+KLS.secLabel(sections[secCounter-1][0])+"|"+KLS.secLabel(sections[secCounter-1][0])+"]]
\n") + KLS.append_text("
[[Zeta_and_Related_Functions#"+"Sections_in_" ""+chapter.replace(" ","_")+"|"+KLS.secLabel(sections[secCounter][0])+"]]
\n") - wiki.write("
[["+KLS.secLabel(sections[(secCounter+1)%len(sections)][0])+ + KLS.append_text("
[["+KLS.secLabel(sections[(secCounter+1)%len(sections)][0])+ "|"+KLS.secLabel(sections[(secCounter+1)%len(sections)][0])+"]] >>
\n
\n\n") - wiki.write("drmf_eof\n") + KLS.append_text("drmf_eof\n") sections[secCounter].append(eqCounter) eqCounter=0 elif "\\subsection" in line and parse: - wiki.write("\n== "+KLS.getString(line)+" ==\n") + KLS.append_text("\n== "+KLS.getString(line)+" ==\n") head=True elif "\\paragraph" in line and parse: - wiki.write("\n=== "+KLS.getString(line)+" ===\n") + KLS.append_text("\n=== "+KLS.getString(line)+" ===\n") head=True elif "\\subsubsection" in line and parse: - wiki.write("\n=== "+KLS.getString(line)+" ===\n") + KLS.append_text("\n=== "+KLS.getString(line)+" ===\n") head=True elif "\\begin{equation}" in line and parse: # symLine="" if head: - wiki.write("\n") + KLS.append_text("\n") head=False sLabel=line.find("\\label{")+7 eLabel=line.find("}",sLabel) @@ -207,8 +208,8 @@ def DLMF(n): label=KLS.modLabel(rlabel) labels.append("Formula:"+rlabel) eqs.append("") - #wiki.write("\n\n") - wiki.write("{\displaystyle \n") + #KLS.append_text("\n\n") + KLS.append_text("{\displaystyle \n") math=True elif "\\begin{equation}" in line and not parse: sLabel=line.find("\\label{")+7 @@ -228,13 +229,13 @@ def DLMF(n): substitution=True math=False subLine="" - #wiki.write("
Substitution(s): "+KLS.getEq(line)+"

\n") + #KLS.append_text("
Substitution(s): "+KLS.getEq(line)+"

\n") elif "\\proof" in line and parse: math=False elif "\\drmfn" in line and parse: math=False if "\\drmfname" in line and parse: - wiki.write("
This formula has the name: "+KLS.getString(line)+"

\n") + KLS.append_text("
This formula has the name: "+KLS.getString(line)+"

\n") elif math and parse: flagM=True eqs[len(eqs)-1]+=line @@ -251,19 +252,19 @@ def DLMF(n): flagM2=True if not(flagM2): - wiki.write(line.rstrip("\n")) - wiki.write("\n}
\n") + KLS.append_text(line.rstrip("\n")) + KLS.append_text("\n}
\n") else: - wiki.write(line.rstrip("\n")) - wiki.write("\n}\n") + KLS.append_text(line.rstrip("\n")) + KLS.append_text("\n}\n") elif "\\end{equation}" in lines[i+1]: - wiki.write(line.rstrip("\n")) - wiki.write("\n}\n") + KLS.append_text(line.rstrip("\n")) + KLS.append_text("\n}\n") elif "\\constraint" in lines[i+1] or "\\substitution" in lines[i+1] or "\\drmfn" in lines[i+1]: - wiki.write(line.rstrip("\n")) - wiki.write("\n}\n") + KLS.append_text(line.rstrip("\n")) + KLS.append_text("\n}\n") else: - wiki.write(line) + KLS.append_text(line) elif math and not parse: eqs[len(eqs)-1]+=line if "\\end{equation}" in lines[i+1] or "\\constraint" in lines[i+1] or "\\substitution" in lines[i+1] or "\\drmfn" in lines[i+1]: @@ -273,14 +274,14 @@ def DLMF(n): if "\\end{equation}" in lines[i+1] or "\\substitution" in lines[i+1]\ or "\\constraint" in lines[i+1] or "\\drmfn" in lines[i+1] or "\\proof" in lines[i+1]: substitution=False - wiki.write("
Substitution(s): "+KLS.getEq(subLine)+"

\n") + KLS.append_text("
Substitution(s): "+KLS.getEq(subLine)+"

\n") if constraint and parse: conLine=conLine+line.replace("&","&
") if "\\end{equation}" in lines[i+1] or "\\substitution" \ in lines[i+1] or "\\constraint" in lines[i+1] or "\\drmfn" in lines[i+1] or "\\proof" in lines[i+1]: constraint=False - wiki.write("
Constraint(s): "+KLS.getEq(conLine)+"

\n") + KLS.append_text("
Constraint(s): "+KLS.getEq(conLine)+"

\n") eqCounter=n endNum=len(labels)-1 @@ -319,58 +320,58 @@ def DLMF(n): parse=True symbols=[] eqCounter+=1 - wiki.write("drmf_bof\n") + KLS.append_text("drmf_bof\n") label=labels[eqCounter] #if not "DLMF" in label:eqCounter+=1 label=labels[eqCounter] - wiki.write("\'\'\'"+KLS.secLabel(label)+"\'\'\'\n") - wiki.write("{{DISPLAYTITLE:"+(labels[eqCounter])+"}}\n") + KLS.append_text("\'\'\'"+KLS.secLabel(label)+"\'\'\'\n") + KLS.append_text("{{DISPLAYTITLE:"+(labels[eqCounter])+"}}\n") if eqCounter==len(labels)-1: break if eqCounter\n") + KLS.append_text("
\n") if newSec: - wiki.write("
" + KLS.append_text("
" "<< [["+KLS.secLabel(sections[secCount][0]).replace(" ","_")+"|"+KLS.secLabel(sections[secCount][0])+"]]
\n") else: - wiki.write("
" + KLS.append_text("
" "<< [["+KLS.secLabel(labels[eqCounter-1]).replace(" ","_")+"|"+KLS.secLabel(labels[eqCounter-1])+"]]
\n") - wiki.write("
[["+KLS.secLabel(sections[secCount+1][0]).replace(" ","_")+"#"+ + KLS.append_text("
[["+KLS.secLabel(sections[secCount+1][0]).replace(" ","_")+"#"+ KLS.secLabel(labels[eqCounter][len("Formula:"):])+"|formula in "+KLS.secLabel(sections[secCount+1][0])+"]]
\n") #if eqS==sections[secCount][1]: if True: - wiki.write("
[["+KLS.secLabel(labels[(eqCounter+1)%(endNum+1)]).replace(" ","_")+ + KLS.append_text("
[["+KLS.secLabel(labels[(eqCounter+1)%(endNum+1)]).replace(" ","_")+ "|"+KLS.secLabel(labels[(eqCounter+1)%(endNum+1)])+"]] >>
\n") - wiki.write("
\n\n") + KLS.append_text("
\n\n") elif eqCounter==endNum: - wiki.write("
\n") + KLS.append_text("
\n") if newSec: newSec=False - wiki.write("
<< [["+KLS.secLabel(sections[secCount][0]).replace(" ","_")+"|"+ + KLS.append_text("
<< [["+KLS.secLabel(sections[secCount][0]).replace(" ","_")+"|"+ KLS.secLabel(sections[secCount][0])+"]]
\n") else: - wiki.write("
" + KLS.append_text("
" "<< [["+KLS.secLabel(labels[eqCounter-1]).replace(" ","_")+"|"+KLS.secLabel(labels[eqCounter-1])+"]]
\n") - wiki.write("
[["+ + KLS.append_text("
[["+ KLS.secLabel(sections[secCount+1][0]).replace(" ","_")+ "#"+KLS.secLabel(labels[eqCounter][len("Formula:"):])+ "|formula in "+KLS.secLabel(sections[secCount+1][0])+ "]]
\n") - wiki.write("
[["+KLS.secLabel(labels[(eqCounter+1)%(endNum+1)].replace(" ","_"))+ + KLS.append_text("
[["+KLS.secLabel(labels[(eqCounter+1)%(endNum+1)].replace(" ","_"))+ "|"+KLS.secLabel(labels[(eqCounter+1)%(endNum+1)])+"]]
\n") - wiki.write("
\n\n") + KLS.append_text("
\n\n") - wiki.write("
{\displaystyle \n") + KLS.append_text("
{\displaystyle \n") math=True elif "\\end{equation}" in line: - wiki.write(comToWrite) + KLS.append_text(comToWrite) parse=False math=False if hProof: - wiki.write("\n== Proof ==\n\nWe ask users to provide proof(s), \ + KLS.append_text("\n== Proof ==\n\nWe ask users to provide proof(s), \ reference(s) to proof(s), or further clarification on the proof(s) in this space.\n") - wiki.write("\n== Symbols List ==\n\n") + KLS.append_text("\n== Symbols List ==\n\n") newSym=[] #if "09.07:04" in label: for x in symbols: @@ -550,21 +551,21 @@ def DLMF(n): gFlag=True #finSym.reverse() if ampFlag: - wiki.write("& : logical and") + KLS.append_text("& : logical and") gFlag=False for y in finSym: if y=="& : logical and": pass elif gFlag: gFlag=False - wiki.write("["+y) + KLS.append_text("["+y) else: - wiki.write("
\n["+y) + KLS.append_text("
\n["+y) - wiki.write("\n
\n") + KLS.append_text("\n
\n") - wiki.write("\n== Bibliography==\n\n")#should there be a space between bibliography and ==? + KLS.append_text("\n== Bibliography==\n\n")#should there be a space between bibliography and ==? r=KLS.unmodlabel(labels[eqCounter]) q=r.find("DLMF:")+5 p=r.find(":",q) @@ -575,33 +576,33 @@ def DLMF(n): equation=equation[0:equation.find(":")] if is_number(section) == False: return eqCounter - wiki.write("[HTTP://DLMF.NIST.GOV/"+ + KLS.append_text("[HTTP://DLMF.NIST.GOV/"+ section+"#"+equation+" Equation ("+equation[1:]+"), " "Section "+section+"] of [[Bibliography#DLMF|'''DLMF''']].\n\n") - wiki.write("== URL links ==\n\nWe ask users to provide relevant URL links in this space.\n\n") + KLS.append_text("== URL links ==\n\nWe ask users to provide relevant URL links in this space.\n\n") if eqCounter
\n") + KLS.append_text("
\n") if newSec: newSec=False - wiki.write("
<< [["+KLS.secLabel(sections[secCount][0]).replace(" ","_")+ + KLS.append_text("
<< [["+KLS.secLabel(sections[secCount][0]).replace(" ","_")+ "|"+KLS.secLabel(sections[secCount][0])+"]]
\n") else: - wiki.write("
<< [["+KLS.secLabel(labels[eqCounter-1]).replace(" ","_")+"|"+ + KLS.append_text("
<< [["+KLS.secLabel(labels[eqCounter-1]).replace(" ","_")+"|"+ KLS.secLabel(labels[eqCounter-1])+"]]
\n") - wiki.write("
[["+KLS.secLabel(sections[secCount+1][0]).replace(" ","_")+"#"+ + KLS.append_text("
[["+KLS.secLabel(sections[secCount+1][0]).replace(" ","_")+"#"+ KLS.secLabel(labels[eqCounter][len("Formula:"):])+"|formula in "+KLS.secLabel(sections[secCount+1][0])+"]]
\n") #if eqS==sections[secCount][1]: #else: if True: - wiki.write("
[["+KLS.secLabel(labels[(eqCounter+1)% + KLS.append_text("
[["+KLS.secLabel(labels[(eqCounter+1)% (endNum+1)]).replace(" ","_")+"|"+KLS.secLabel(labels[(eqCounter+1)%(endNum+1)])+"]] >>
\n") - wiki.write("
\n\ndrmf_eof\n") + KLS.append_text("
\n\ndrmf_eof\n") else: #FOR EXTRA EQUATIONS - wiki.write("
\n") - wiki.write("
<< [["+labels[endNum-1].replace(" ","_")+"|"+labels[endNum-1]+"]]
\n") - wiki.write("
[["+labels[0].replace(" ","_")+"#"+labels[endNum][8:]+"|formula in "+labels[0]+"]]
\n") - wiki.write("
[["+labels[(0)%endNum].replace(" ","_")+"|"+labels[0%endNum]+"]]
\n") - wiki.write("
\n\ndrmf_eof\n") + KLS.append_text("
\n") + KLS.append_text("
<< [["+labels[endNum-1].replace(" ","_")+"|"+labels[endNum-1]+"]]
\n") + KLS.append_text("
[["+labels[0].replace(" ","_")+"#"+labels[endNum][8:]+"|formula in "+labels[0]+"]]
\n") + KLS.append_text("
[["+labels[(0)%endNum].replace(" ","_")+"|"+labels[0%endNum]+"]]
\n") + KLS.append_text("
\n\ndrmf_eof\n") @@ -614,14 +615,14 @@ def DLMF(n): constraint=True math=False conLine="" - #wiki.write("
"+KLS.getEq(line)+"

\n") + #KLS.append_text("
"+KLS.getEq(line)+"

\n") elif "\\substitution" in line and parse: #symbols=symbols+KLS.getSym(line) symLine=line.strip("\n") if hSub: comToWrite=comToWrite+"\n== Substitution(s) ==\n\n" hSub=False - #wiki.write("
"+KLS.getEq(line)+"

\n") + #KLS.append_text("
"+KLS.getEq(line)+"

\n") substitution=True math=False subLine="" @@ -687,12 +688,12 @@ def DLMF(n): if "\\end{equation}" in lines[i+1]: proof=False #symLine+=line.strip("\n") - wiki.write(comToWrite+KLS.getEqP(proofLine)+"
\n
\n") + KLS.append_text(comToWrite+KLS.getEqP(proofLine)+"
\n
\n") comToWrite="" symbols=symbols+KLS.getSym(symLine) symLine="" - #wiki.write(line) + #KLS.append_text(line) elif proof: symLine+=line.strip("\n") @@ -733,7 +734,7 @@ def DLMF(n): if "\\end{equation}" in lines[i+1]: proof=False #symLine+=line.strip("\n") - wiki.write(comToWrite+KLS.getEqP(proofLine).rstrip("\n")+"
\n
\n") + KLS.append_text(comToWrite+KLS.getEqP(proofLine).rstrip("\n")+"
\n
\n") comToWrite="" symbols=symbols+KLS.getSym(symLine) symLine="" @@ -741,14 +742,14 @@ def DLMF(n): elif math: if "\\end{equation}" in lines[i+1] or "\\constraint" in lines[i+1] \ or "\\substitution" in lines[i+1] or "\\proof" in lines[i+1] or "\\drmfnote" in lines[i+1] or "\\drmfname" in lines[i+1]: - wiki.write(line.rstrip("\n")) + KLS.append_text(line.rstrip("\n")) symLine+=line.strip("\n") symbols=symbols+KLS.getSym(symLine) symLine="" - wiki.write("\n}
\n") + KLS.append_text("\n}
\n") else: symLine+=line.strip("\n") - wiki.write(line) + KLS.append_text(line) if note and parse: noteLine=noteLine+line symbols=symbols+KLS.getSym(line) @@ -775,7 +776,7 @@ def DLMF(n): constraint=False symbols=symbols+KLS.getSym(symLine) symLine="" - wiki.write(comToWrite+"
"+KLS.getEq(conLine)+"

\n") + KLS.append_text(comToWrite+"
"+KLS.getEq(conLine)+"

\n") comToWrite="" if substitution and parse: subLine=subLine+line.replace("&","&
") @@ -787,7 +788,7 @@ def DLMF(n): substitution=False symbols=symbols+KLS.getSym(symLine) symLine="" - wiki.write(comToWrite+"
"+KLS.getEq(subLine)+"

\n") + KLS.append_text(comToWrite+"
"+KLS.getEq(subLine)+"

\n") comToWrite="" if __name__ == "__main__": From ee71e2c5af3b0c3b4a24d74feffd643f4c6f9a45 Mon Sep 17 00:00:00 2001 From: joonbang Date: Thu, 19 May 2016 18:28:37 -0400 Subject: [PATCH 093/402] Complete section/subsection generation code (#40) * Improve code for section/subsection generation * Fix issues with category generator code * Update .gitignore --- Joon/.gitignore | 4 +- Joon/category_generator.py | 86 ++++++------------ Joon/converter.py | 2 +- Joon/keys/section_names | 69 ++++++++++++++ Joon/tests/primer | 50 ---------- Joon/tests/test.pdf | Bin 97230 -> 56226 bytes Joon/tests/test.tex | 182 +++++++++++++++++-------------------- 7 files changed, 184 insertions(+), 209 deletions(-) create mode 100644 Joon/keys/section_names delete mode 100644 Joon/tests/primer diff --git a/Joon/.gitignore b/Joon/.gitignore index 4321898..2133789 100644 --- a/Joon/.gitignore +++ b/Joon/.gitignore @@ -1,3 +1,5 @@ +Glossary.csv tests/ +functions/ !tests/test.tex -!tests/test.pdf \ No newline at end of file +!tests/test.pdf diff --git a/Joon/category_generator.py b/Joon/category_generator.py index 1a3dbdb..35a5317 100644 --- a/Joon/category_generator.py +++ b/Joon/category_generator.py @@ -2,84 +2,50 @@ import os names = ["BS", "CH", "CN", "EF", "ER", "EX", "GA", "HY", "QH", "SM"] -name_dictionary = {'naturallogarithm': 'naturallogarithm', 'archimedes': 'archimedes', 'arcsinh': 'arcsinh', - 'pow': 'pow', 'fresnel': 'fresnel', 'modbessel': 'Modified Bessel functions', 'apery': 'apery', - 'arccos': 'arccos', 'HY': 'HY', 'BS': 'Bessel', 'tan': 'tan', 'GA': 'GA', 'gamma_chisquare': - 'gamma_chisquare', 'powerandroot': 'powerandroot', 'error': 'error', 'gompertz': 'gompertz', - 'ln': 'ln', 'kummer': 'kummer', 'whittaker': 'whittaker', 'arctan': 'arctan', 'trigamma': 'trigamma', - 'eulersconstant': 'eulersconstant', 'binet': 'binet', 'parabolic': 'parabolic', 'repint': 'repint', - 'bessel': 'Bessel functions', 'sin': 'sin', 'incompletegamma': 'incompletegamma', 'CH': 'CH', - 'pythagoras': 'pythagoras', 'CN': 'CN', 'normal': 'normal', 'arcsin': 'arcsin', - 'confluentlimit': 'confluentlimit', 'confluent': 'confluent', 'eulersnumber': 'eulersnumber', - 'catalan': 'catalan', 'repeated': 'repeated', 'arctanh': 'arctanh', 'EX': 'EX', 'rabbit': 'rabbit', - 'QH': 'QH', 'beta_f_t': 'beta_f_t', 'cosh': 'cosh', 'coth': 'coth', 'ER': 'ER', 'sinh': 'sinh', - 'cos': 'cos', 'SM': 'SM', 'EF': 'EF', 'tanh': 'tanh', 'expintegrals': 'expintegrals', 'polygamma': - 'polygamma', 'exp': 'exp', 'goldenratio': 'goldenratio', 'comperror': 'comperror', 'arccosh': - 'arccosh', 'tetragamma': 'tetragamma', 'related': 'related', "": ""} +translate = dict(tuple(line.split(" : ")) for line in open("keys/section_names").read().split("\n") + if line != "" and "%" not in line) + +print translate def generate_categories(): """ Generates section, subsection """ - root_directory = "/home/jnb8/CFSF/functions" + root_directory = "functions" depth = len(root_directory.split("/")) - - dirs = [x for x in os.walk(root_directory)] - - sections_created = list() - # current_section = "" + dirs = [d for d in os.walk(root_directory)] text = "" - - # print name_dictionary - - other_text = "" subsections = {"no-subsection": list()} - for i in range(len(dirs)+1): - if i < len(dirs): - d = dirs[i] - else: - d = [root_directory, [], []] - - if "/functions/" in d[0]: # only subdirectories of functions/ - print str(d) - other_text += d[0].split("/")[-1]+"\n" - # so the first one will be the list of "names" - section = d[0].split("/")[depth] # should be in names - if len(d[0].split("/")) == depth+1 or i == len(dirs): # switched to a new section - # print "Switched to new section" - for qq in subsections: - thing = subsections[qq] - if not thing and qq == "no-subsection": - pass - else: - text += " \\subsection{"+qq+"}\n" - for j in thing: - text += " "+j+"\n" - text += "\n" + for info in dirs: + print info - subsections = {"no-subsection": list()} + directory = info[0].split("/") - if section not in sections_created: - text += "\\section{"+name_dictionary[section]+"}\n" - sections_created.append(section) + if len(directory) == depth+1: + if text: + for subsection in subsections: + text += " \\subsection{"+subsection+"}\n" + for file_name in subsections[subsection]: + text += " "+file_name+"\n" + text += "\n" - for thing in d[1]: - subsections[name_dictionary[thing]] = list() + text += " \\section{"+translate[directory[depth]]+"}\n" + subsections = {"no-subsection": list()} - for thing in d[2]: - if len(d[0].split("/")) == depth+1: - subsections["no-subsection"].append(thing) - else: - subsections[name_dictionary[d[0].split("/")[depth+1]]].append(thing) + elif len(directory) == depth+2 and translate[directory[depth+1]] not in subsections: + if "no-subsection" in subsections: + subsections.pop("no-subsection") - # maybe write some better code. - print '\n'.join(text.split("\n")[:-2]).replace(" \subsection{no-subsection}\n ", "") + subsections[translate[directory[depth+1]]] = list() - print other_text + print text + with open("tests/test.tex", "w") as f: + text = open("tests/primers/primer").read() + text + "\\end{document}" + f.write(text) def main(): generate_categories() diff --git a/Joon/converter.py b/Joon/converter.py index 48c7dc0..d840c1d 100644 --- a/Joon/converter.py +++ b/Joon/converter.py @@ -261,7 +261,7 @@ def main(): print text with open("tests/test.tex", "w") as f: - text = open("tests/primer").read() + text + "\\end{document}" + text = open("tests/primers/primer").read() + text + "\\end{document}" f.write(text) if __name__ == '__main__': diff --git a/Joon/keys/section_names b/Joon/keys/section_names new file mode 100644 index 0000000..3486534 --- /dev/null +++ b/Joon/keys/section_names @@ -0,0 +1,69 @@ +% section names of folder in functions/ directory, and actual meaning +BS : Bessel functions +bessel : Bessel functions +modbessel : Modified Bessel functions + +CH : Confluent hypergeometric functions +confluent : Confluent hypergeometric series $_2F_0$ +confluentlimit : Confluent hypergeometric limit function +kummer : Kummer functions +parabolic : Parabolic cylinder functions +whittaker : Whittaker functions + +CN : Mathematical constants +apery : Ap\'{e}ry's constant$\zeta$(3) +archimedes : Archimedes' constant, symbol $\pi$ +catalan : Catalan's constant, symbol $C$ +eulersconstant : Euler's constant, symbol $\gamma$ +eulersnumber : Euler's number, base of the natural logarithm +goldenratio : Golden ratio, symbol $\phi$ +gompertz : Gompertz' constant, symbol $G$ +naturallogarithm : The natural logarithm, $\ln(2)$ +powerandroot : Regular continued fractions +pythagoras : Pythagoras' constant, the square root of two +rabbit : The rabbit constant, symbol $\rho$ + +EF : Elementary functions +arccos : Inverse trigonometric functions +arccosh : Inverse hyperbolic functions +arcsin : Inverse trigonometric functions +arcsinh : Inverse hyperbolic functions +arctan : Inverse trigonometric functions +arctanh : Inverse hyperbolic functions +cos : Trigonometric functions +cosh : Hyperbolic functions +coth : Hyperbolic functions +exp : The exponential function +ln : The natural logarithm +pow : The power function +sin : Trigonometric functions +sinh : Hyperbolic functions +tan : Trigonometric functions +tanh : Hyperbolic functions + +ER : Error function and related integrals +error : Error function and Dawson's integral +comperror : Complementary and complex error function +repint : Repeated integrals +fresnel : Fresnel integrals + +EX : Exponential integrals and related functions +expintegrals : Exponential integrals +related : Related functions + +GA : Gamma function and related functions +binet : Binet function +incompletegamma : Incomplete gamma functions +polygamma : Polygamma functions +tetragamma : Tetragamma function +trigamma : Trigamma function + +HY : Hypergeometric functions + +QH : q-Hypergeometric function + +SM : Probability functions +beta_f_t : Beta, F- and Student's $t$-distributions +gamma_chisquare : Gamma and chi-square distribution +normal : Normal and log-normal distributions +repeated : Repeated integrals diff --git a/Joon/tests/primer b/Joon/tests/primer deleted file mode 100644 index 66d090c..0000000 --- a/Joon/tests/primer +++ /dev/null @@ -1,50 +0,0 @@ -\documentclass[11pt]{article} -\usepackage[pdftex]{hyperref} -\usepackage{amsmath} -\usepackage{amsfonts} -\usepackage{DLMFmath} -\usepackage{DRMFfcns} -\usepackage{amssymb} -\usepackage{xparse} -\usepackage{cite} -\usepackage{forloop} - -\setlength{\textwidth}{16cm} -\setlength{\textheight}{21cm} -\setlength{\topmargin}{0cm} -\setlength{\oddsidemargin}{0.2mm} -\setlength{\evensidemargin}{0.2mm} - -\newcommand\sa{\smallskipamount} -\newcommand\sLP{\\[\sa]} -\newcommand\sPP{\\[\sa]\indent} -\newcommand\ba{\bigskipamount} -\newcommand\bLP{\\[\ba]} -\newcommand\CC{\mathbb{C}} -\newcommand\RR{\mathbb{R}} -\newcommand\ZZ{\mathbb{Z}} -\newcommand\al\alpha -\newcommand\be\beta -\newcommand\ga\gamma 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zq+I`j`~TUIBsCEH0^Jf)ViFS4+)`{(V&WXE?9x0u++vdK(qCAdQ<_tl^#306Rf-}2 zX!X?{VB_ZfFI-FdAJ}W}`h~w)r0u^zz9hih(E{)v(Cp&+)r)X|_gtl#=0gVqLk-z^ z99&MViw2wvXo1HE(Uvo)${)gl$AiF^P+>!mw^7yMj1{_G{!|22L;r|{;3ehpD5Et5 z&xwwK4R5!ONGS^sJdVgYj*PJmYsQVJ4GSv^3vZPTE0PUw9EOb@Ha!2y$mXMW5O&Z< zuW~-A4Yhl}#emMUG)j}xr4{b!_ Date: Sat, 28 May 2016 00:03:25 +0200 Subject: [PATCH 094/402] Homogenize DLMFtex2Wiki.py and tex2Wik (#48) * Reformat file and import append_text * Homogenize DLMFtex2Wiki.py and tex2Wiki * remove {\displaystyle ... } wrapper * make DLMF (in DLMFtex2Wiki) accept 3 arguments like readin (in tex2Wiki) i.e., ofname,glossary,mmd * remove all commented code such as #tester=open("testData.txt",'w') * convert bof eof to append_revision * Fix code style warnings --- Azeem/src/DLMFtex2Wiki.py | 1265 ++++++++++++++++++------------------- 1 file changed, 632 insertions(+), 633 deletions(-) diff --git a/Azeem/src/DLMFtex2Wiki.py b/Azeem/src/DLMFtex2Wiki.py index f78f42f..d8e2814 100644 --- a/Azeem/src/DLMFtex2Wiki.py +++ b/Azeem/src/DLMFtex2Wiki.py @@ -1,795 +1,794 @@ -#Version 13: Minor Fixes -#Convert tex to wikiText -import csv #imported for using csv format -import sys #imported for getting args -import os #imported for copying file -import tex2Wiki as KLS - #Change Accordingly based on project directory. -def is_number(char): - try: - "".join(["0"+str(float(char)) if float(char) < 10 else float(char)]) - return True - except ValueError: - return False +# Version 15: Minor Fixes +# Convert tex to wikiText +import csv # imported for using csv format +import sys # imported for getting args +from shutil import copyfile +from tex2Wiki import append_text, append_revision, getString,\ + getSym, getEq, secLabel, getEqP, unmodLabel, isnumber + + + + def modLabel(label): - #label.replace("Formula:KLS:","KLS;") - isNumer=False - newlabel="" - num="" - for i in range(0,len(label)): - if isNumer and not is_number(label[i]): - if len(num)>1: - newlabel+=num - isNumer=False - num="" + isNumer = False + newlabel = "" + num = "" + for i in range(0, len(label)): + if isNumer and not isnumber(label[i]): + if len(num) > 1: + newlabel += num + num = "" else: - newlabel+="0"+str(num) - num="" - isNumer=False - if is_number(label[i]): - isNumer=True - num+=str(label[i]) + newlabel += "0" + str(num) + num = "" + if isnumber(label[i]): + isNumer = True + num += str(label[i]) else: - isNumer=False - newlabel+=label[i] - if len(num)>1: - newlabel+=num - elif len(num)==1: - newlabel+="0"+num - return(newlabel) - - -def DLMF(n): - for iterations in range(0,1): - try: - tex=open(sys.argv[1],'r') - wiki = open(sys.argv[2], 'w') - glossary = open(sys.argv[3],"r") - main=open(sys.argv[4],"r") - lLinks = open(sys.argv[5], 'r') - except (IOError,IndexError): - tex = open("../../data/ZE.3.tex","r") - wiki = open("../../data/ZE.4.xml","w") - glossary = open("../../data/new.Glossary.csv","r") - main = open("../../data/OrthogonalPolynomials.mmd","r") - lLinks = open("../../data/BruceLabelLinks","r") - mainText=main.read() - mainPrepend="" - mainWrite=open("ZetaFunctions.mmd.new","w") - #tester=open("testData.txt",'w') - #glossary=open('Glossary', 'r') - try: - gCSV=csv.reader(glossary, delimiter=',', quotechar='\"') - except: - raise - lLink=lLinks.readlines() - math=False - constraint=False - substitution=False - symbols=[] - lines=tex.readlines() - refLines=[] - sections=[] - labels=[] - eqs=[] - refEqs=[] - parse=False - head=False + isNumer = False + newlabel += label[i] + if len(num) > 1: + newlabel += num + elif len(num) == 1: + newlabel += "0" + num + return newlabel + + +def DLMF(ofname,mmd,llinks,n): + for iterations in range(0, 1): + tex = open(ofname, 'r') + main_file = open(mmd, "r") + lLinks = open(llinks, 'r') + mainPrepend = "" + mainWrite = open("ZetaFunctions.mmd.new", "w") + lLink = lLinks.readlines() + math = False + constraint = False + substitution = False + symbols = [] + lines = tex.readlines() + refLines = [] + sections = [] + labels = [] + eqs = [] + refEqs = [] + parse = False + head = False try: - chapRef=[("GA",open("../../data/GA.tex",'r')),("ZE",open("../../data/ZE.3.tex",'r'))] + chapRef = [("GA", open("../../data/GA.tex", 'r')), ("ZE", open("../../data/ZE.3.tex", 'r'))] except IOError: - chapRef=[("GA",open("GA.tex",'r')),("ZE",open("ZE.3.tex",'r'))] - refLabels=[] + chapRef = [("GA", open("GA.tex", 'r')), ("ZE", open("ZE.3.tex", 'r'))] + refLabels = [] for c in chapRef: - refLines=refLines+(c[1].readlines()) + refLines = refLines + (c[1].readlines()) c[1].close() - for i in range(0,len(refLines)): - line=refLines[i] + for i in range(0, len(refLines)): + line = refLines[i] if "\\begin{equation}" in line: - sLabel=line.find("\\label{")+7 - eLabel=line.find("}",sLabel) - label=line[sLabel:eLabel] + sLabel = line.find("\\label{") + 7 + eLabel = line.find("}", sLabel) + label = line[sLabel:eLabel] for l in lLink: - if l.find(label)!=-1 and l[len(label)+1]=="=": - rlabel=l[l.find("=>")+3:l.find("\\n")] - rlabel=rlabel.replace("/","") - rlabel=rlabel.replace("#",":") + if l.find(label) != -1 and l[len(label) + 1] == "=": + rlabel = l[l.find("=>") + 3:l.find("\\n")] + rlabel = rlabel.replace("/", "") + rlabel = rlabel.replace("#", ":") break - label=rlabel + label = rlabel refLabels.append(label) refEqs.append("") - math=True + math = True elif math: - refEqs[len(refEqs)-1]+=line - if "\\end{equation}" in refLines[i+1] or "\\constraint" in refLines[i+1] or "\\substitution" in refLines[i+1] or "\\drmfn" in refLines[i+1]: - math=False - - startFlag=True - for i in range(0,len(lines)): - line=lines[i] + refEqs[len(refEqs) - 1] += line + if "\\end{equation}" in refLines[i + 1] or "\\constraint" in refLines[i + 1] or "\\substitution" in \ + refLines[i + 1] or "\\drmfn" in refLines[i + 1]: + math = False + + startFlag = True + for i in range(0, len(lines)): + line = lines[i] if "\\begin{document}" in line: - #KLS.append_text("drmf_bof\n") - parse=True + parse = True elif "\\end{document}" in line: - #KLS.append_text("
\n") - mainPrepend+="
\n" - mainText="" - mainText=mainPrepend+mainText - mainText=mainText.replace("drmf_bof\n","") - mainText=mainText.replace("drmf_eof\n","") - mainText=mainText.replace("\'\'\'Zeta and Related Functions\'\'\'\n","") - mainText=mainText.replace("{{#set:Section=0}}\n","") - mainText="drmf_bof\n\'\'\'Zeta and Related Functions\'\'\'\n{{#set:Section=0}}"+mainText+"\ndrmf_eof\n" + mainPrepend += "
\n" + mainText = mainPrepend + mainText = mainText.replace("drmf_bof\n", "") + mainText = mainText.replace("drmf_eof\n", "") + mainText = mainText.replace("\'\'\'Zeta and Related Functions\'\'\'\n", "") + mainText = mainText.replace("{{#set:Section=0}}\n", "") + append_revision('Zeta and Related Functions'); + mainText = "{{#set:Section=0}}\n" + mainText mainWrite.write(mainText) mainWrite.close() - main.close() - os.system("cp -f ZetaFunctions.mmd.new ZetaFunctions.mmd") - #KLS.append_text("\ndrmf_eof\n") - parse=False + main_file.close() + copyfile(mmd, 'ZetaFunctions.mmd.new') + parse = False elif "\\title" in line and parse: labels.append("Zeta and Related Functions") - sections.append(["Zeta and Related Functions",0]) + sections.append(["Zeta and Related Functions", 0]) elif "\\part" in line: - if KLS.getString(line)=="BOF": - parse=False - elif KLS.getString(line)=="EOF": - parse=True + if getString(line) == "BOF": + parse = False + elif getString(line) == "EOF": + parse = True elif parse: - stringWrite="\'\'\'" - stringWrite+=KLS.getString(line)+"\'\'\'\n" - chapter=KLS.getString(line) + stringWrite = "\'\'\'" + stringWrite += getString(line) + "\'\'\'\n" + chapter = getString(line) if startFlag: - mainPrepend+=("\n== Sections in "+chapter+" ==\n\n
\n") - startFlag=False + mainPrepend += ( + "\n== Sections in " + chapter + " ==\n\n
\n") + startFlag = False else: - mainPrepend+=("
\n\n== Sections in "+chapter+" ==\n\n
\n") - head=True + mainPrepend += ( + "
\n\n== Sections in " + chapter + " ==\n\n
\n") + head = True elif "\\section" in line: - mainPrepend+=("* [["+KLS.secLabel(KLS.getString(line))+"|"+KLS.getString(line)+"]]\n") - sections.append([KLS.getString(line)]) + mainPrepend += ("* [[" + secLabel(getString(line)) + "|" + getString(line) + "]]\n") + sections.append([getString(line)]) - secCounter=0 - eqCounter=0 - for i in range(0,len(lines)): - line=lines[i] + secCounter = 0 + eqCounter = 0 + for i in range(0, len(lines)): + line = lines[i] if "\\section" in line: - parse=True - secCounter+=1 - KLS.append_text("drmf_bof\n") - KLS.append_text("\'\'\'"+KLS.secLabel(KLS.getString(line))+"\'\'\'\n") - KLS.append_text("{{DISPLAYTITLE:"+(sections[secCounter][0])+"}}\n") - KLS.append_text("
\n") - KLS.append_text("
<< [["+KLS.secLabel(sections[secCounter-1][0])+"|"+KLS.secLabel(sections[secCounter-1][0])+"]]
\n") - KLS.append_text("
[[Zeta_and_Related_Functions#"+ - "Sections_in_"+chapter.replace(" ","_")+"|"+KLS.secLabel(sections[secCounter][0])+"]]
\n") - KLS.append_text("
[["+KLS.secLabel(sections[(secCounter+1)%len(sections)][0])+ - "|"+KLS.secLabel(sections[(secCounter+1)%len(sections)][0])+"]] >>
\n
\n\n") - head=True - KLS.append_text("== "+KLS.getString(line)+" ==\n") - elif ("\\section" in lines[(i+1)%len(lines)] or "\\end{document}" in lines[(i+1)%len(lines)]) and parse: - KLS.append_text("
\n") - KLS.append_text("
<< [["+KLS.secLabel(sections[secCounter-1][0])+"|"+KLS.secLabel(sections[secCounter-1][0])+"]]
\n") - KLS.append_text("
[[Zeta_and_Related_Functions#"+"Sections_in_" - ""+chapter.replace(" ","_")+"|"+KLS.secLabel(sections[secCounter][0])+"]]
\n") - KLS.append_text("
[["+KLS.secLabel(sections[(secCounter+1)%len(sections)][0])+ - "|"+KLS.secLabel(sections[(secCounter+1)%len(sections)][0])+"]] >>
\n
\n\n") - KLS.append_text("drmf_eof\n") + parse = True + secCounter += 1 + append_revision(secLabel(getString(line))) + append_text("{{DISPLAYTITLE:" + (sections[secCounter][0]) + "}}\n") + append_text("
\n") + append_text( + "
<< [[" + secLabel(sections[secCounter - 1][0]) + "|" + secLabel( + sections[secCounter - 1][0]) + "]]
\n") + append_text("
[[Zeta_and_Related_Functions#" + + "Sections_in_" + chapter.replace(" ", "_") + "|" + secLabel( + sections[secCounter][0]) + "]]
\n") + append_text( + "
[[" + secLabel(sections[(secCounter + 1) % len(sections)][0]) + + "|" + secLabel(sections[(secCounter + 1) % len(sections)][0]) + "]] >>
\n
\n\n") + head = True + append_text("== " + getString(line) + " ==\n") + elif ("\\section" in lines[(i + 1) % len(lines)] or "\\end{document}" in lines[ + (i + 1) % len(lines)]) and parse: + append_text("
\n") + append_text( + "
<< [[" + secLabel(sections[secCounter - 1][0]) + "|" + secLabel( + sections[secCounter - 1][0]) + "]]
\n") + append_text("
[[Zeta_and_Related_Functions#" + "Sections_in_" + "" + chapter.replace(" ", + "_") + "|" + secLabel( + sections[secCounter][0]) + "]]
\n") + append_text( + "
[[" + secLabel(sections[(secCounter + 1) % len(sections)][0]) + + "|" + secLabel(sections[(secCounter + 1) % len(sections)][0]) + "]] >>
\n
\n\n") + append_text("drmf_eof\n") sections[secCounter].append(eqCounter) - eqCounter=0 + eqCounter = 0 elif "\\subsection" in line and parse: - KLS.append_text("\n== "+KLS.getString(line)+" ==\n") - head=True + append_text("\n== " + getString(line) + " ==\n") + head = True elif "\\paragraph" in line and parse: - KLS.append_text("\n=== "+KLS.getString(line)+" ===\n") - head=True + append_text("\n=== " + getString(line) + " ===\n") + head = True elif "\\subsubsection" in line and parse: - KLS.append_text("\n=== "+KLS.getString(line)+" ===\n") - head=True + append_text("\n=== " + getString(line) + " ===\n") + head = True elif "\\begin{equation}" in line and parse: - # symLine="" if head: - KLS.append_text("\n") - head=False - sLabel=line.find("\\label{")+7 - eLabel=line.find("}",sLabel) - label=(line[sLabel:eLabel]) - eqCounter+=1 + append_text("\n") + head = False + sLabel = line.find("\\label{") + 7 + eLabel = line.find("}", sLabel) + label = (line[sLabel:eLabel]) + eqCounter += 1 for l in lLink: - if label==l[0:l.find("=")-1]: - rlabel=l[l.find("=>")+3:l.find("\\n")] - rlabel=rlabel.replace("/","") - rlabel=rlabel.replace("#",":") - rlabel=rlabel.replace("!",":") + if label == l[0:l.find("=") - 1]: + rlabel = l[l.find("=>") + 3:l.find("\\n")] + rlabel = rlabel.replace("/", "") + rlabel = rlabel.replace("#", ":") + rlabel = rlabel.replace("!", ":") break - label=KLS.modLabel(rlabel) - labels.append("Formula:"+rlabel) + label = modLabel(rlabel) + labels.append("Formula:" + rlabel) eqs.append("") - #KLS.append_text("\n\n") - KLS.append_text("{\displaystyle \n") - math=True + append_text("\n") + math = True elif "\\begin{equation}" in line and not parse: - sLabel=line.find("\\label{")+7 - eLabel=line.find("}",sLabel) - label=KLS.modLabel(line[sLabel:eLabel]) - labels.append("*"+label) #special marker + sLabel = line.find("\\label{") + 7 + eLabel = line.find("}", sLabel) + label = modLabel(line[sLabel:eLabel]) + labels.append("*" + label) # special marker eqs.append("") - math=True + math = True elif "\\end{equation}" in line: - math=False + math = False elif "\\constraint" in line and parse: - constraint=True - math=False - conLine="" + constraint = True + math = False + conLine = "" elif "\\substitution" in line and parse: - substitution=True - math=False - subLine="" - #KLS.append_text("
Substitution(s): "+KLS.getEq(line)+"

\n") + substitution = True + math = False + subLine = "" elif "\\proof" in line and parse: - math=False + math = False elif "\\drmfn" in line and parse: - math=False + math = False if "\\drmfname" in line and parse: - KLS.append_text("
This formula has the name: "+KLS.getString(line)+"

\n") + append_text("
This formula has the name: " + getString(line) + "

\n") elif math and parse: - flagM=True - eqs[len(eqs)-1]+=line + flagM = True + eqs[len(eqs) - 1] += line - if "\\end{equation}" in lines[i+1] and not "\\subsection" in lines[i+3] and not "\\section" in lines[i+3] and not "\\part" in lines[i+3]: - u=i - flagM2=False + if not ((not ("\\end{equation}" in lines[i + 1]) or "\\subsection" in lines[i + 3]) or "\\section" in \ + lines[i + 3]) and not "\\part" in lines[i + 3]: + u = i + flagM2 = False while flagM: - u+=1 - if "\\begin{equation}"in lines[u] in lines[u]: - flagM=False - if "\\section" in lines[u] or "\\subsection" in lines[i] or "\\part" in lines[u] or "\\end{document}" in lines[u]: - flagM=False - flagM2=True - if not(flagM2): - - KLS.append_text(line.rstrip("\n")) - KLS.append_text("\n}
\n") + u += 1 + if "\\begin{equation}" in lines[u] in lines[u]: + flagM = False + if "\\section" in lines[u] or "\\subsection" in lines[i] or "\\part" in lines[ + u] or "\\end{document}" in lines[u]: + flagM = False + flagM2 = True + if not flagM2: + + append_text(line.rstrip("\n")) + append_text("\n
\n") else: - KLS.append_text(line.rstrip("\n")) - KLS.append_text("\n}\n") - elif "\\end{equation}" in lines[i+1]: - KLS.append_text(line.rstrip("\n")) - KLS.append_text("\n}\n") - elif "\\constraint" in lines[i+1] or "\\substitution" in lines[i+1] or "\\drmfn" in lines[i+1]: - KLS.append_text(line.rstrip("\n")) - KLS.append_text("\n}\n") + append_text(line.rstrip("\n")) + append_text("\n\n") + elif "\\end{equation}" in lines[i + 1]: + append_text(line.rstrip("\n")) + append_text("\n\n") + elif "\\constraint" in lines[i + 1] or "\\substitution" in lines[i + 1] or "\\drmfn" in lines[i + 1]: + append_text(line.rstrip("\n")) + append_text("\n\n") else: - KLS.append_text(line) + append_text(line) elif math and not parse: - eqs[len(eqs)-1]+=line - if "\\end{equation}" in lines[i+1] or "\\constraint" in lines[i+1] or "\\substitution" in lines[i+1] or "\\drmfn" in lines[i+1]: - math=False + eqs[len(eqs) - 1] += line + if "\\end{equation}" in lines[i + 1] or "\\constraint" in lines[i + 1] or "\\substitution" in lines[ + i + 1] or "\\drmfn" in lines[i + 1]: + math = False if substitution and parse: - subLine=subLine+line.replace("&","&
") - if "\\end{equation}" in lines[i+1] or "\\substitution" in lines[i+1]\ - or "\\constraint" in lines[i+1] or "\\drmfn" in lines[i+1] or "\\proof" in lines[i+1]: - substitution=False - KLS.append_text("
Substitution(s): "+KLS.getEq(subLine)+"

\n") + subLine += line.replace("&", "&
") + if "\\end{equation}" in lines[i + 1] or "\\substitution" in lines[i + 1] \ + or "\\constraint" in lines[i + 1] or "\\drmfn" in lines[i + 1] or "\\proof" in lines[i + 1]: + substitution = False + append_text("
Substitution(s): " + getEq(subLine) + "

\n") if constraint and parse: - conLine=conLine+line.replace("&","&
") - if "\\end{equation}" in lines[i+1] or "\\substitution" \ - in lines[i+1] or "\\constraint" in lines[i+1] or "\\drmfn" in lines[i+1] or "\\proof" in lines[i+1]: - constraint=False - KLS.append_text("
Constraint(s): "+KLS.getEq(conLine)+"

\n") - - eqCounter=n - endNum=len(labels)-1 - parse=False - constraint=False - substitution=False - note=False - hCon=True - hSub=True - hNote=True - hProof=True - proof=False - comToWrite="" - secCount=-1 - newSec=False - for i in range(0,len(lines)): - line=lines[i] + conLine += line.replace("&", "&
") + if "\\end{equation}" in lines[i + 1] or "\\substitution" \ + in lines[i + 1] or "\\constraint" in lines[i + 1] or "\\drmfn" in lines[i + 1] or "\\proof" in \ + lines[i + 1]: + constraint = False + append_text("
Constraint(s): " + getEq(conLine) + "

\n") + + eqCounter = n + endNum = len(labels) - 1 + parse = False + constraint = False + substitution = False + note = False + hCon = True + hSub = True + hNote = True + hProof = True + proof = False + comToWrite = "" + secCount = -1 + newSec = False + for i in range(0, len(lines)): + line = lines[i] if "\\section" in line: - secCount+=1 - newSec=True - eqS=0 + secCount += 1 + newSec = True + eqS = 0 if "\\begin{equation}" in line: - symLine=line.strip("\n") - eqS+=1 - constraint=False - substitution=False - note=False - comToWrite="" - hCon=True - hSub=True - hNote=True - hProof=True - proof=False - parse=True - symbols=[] - eqCounter+=1 - KLS.append_text("drmf_bof\n") - label=labels[eqCounter] - #if not "DLMF" in label:eqCounter+=1 - label=labels[eqCounter] - KLS.append_text("\'\'\'"+KLS.secLabel(label)+"\'\'\'\n") - KLS.append_text("{{DISPLAYTITLE:"+(labels[eqCounter])+"}}\n") - if eqCounter==len(labels)-1: + symLine = line.strip("\n") + eqS += 1 + constraint = False + substitution = False + note = False + comToWrite = "" + hCon = True + hSub = True + hNote = True + hProof = True + proof = False + parse = True + symbols = [] + eqCounter += 1 + label = labels[eqCounter] + append_revision(secLabel(label)) + append_text("{{DISPLAYTITLE:" + (labels[eqCounter]) + "}}\n") + if eqCounter == len(labels) - 1: break - if eqCounter\n") + if eqCounter < endNum: # FOR ANYTHING THAT IS NOT THE EXTRA EQUATIONS + append_text("
\n") if newSec: - KLS.append_text("
" - "<< [["+KLS.secLabel(sections[secCount][0]).replace(" ","_")+"|"+KLS.secLabel(sections[secCount][0])+"]]
\n") + append_text("
" + "<< [[" + secLabel(sections[secCount][0]).replace(" ", + "_") + "|" + secLabel( + sections[secCount][0]) + "]]
\n") else: - KLS.append_text("
" - "<< [["+KLS.secLabel(labels[eqCounter-1]).replace(" ","_")+"|"+KLS.secLabel(labels[eqCounter-1])+"]]
\n") - KLS.append_text("
[["+KLS.secLabel(sections[secCount+1][0]).replace(" ","_")+"#"+ - KLS.secLabel(labels[eqCounter][len("Formula:"):])+"|formula in "+KLS.secLabel(sections[secCount+1][0])+"]]
\n") - #if eqS==sections[secCount][1]: + append_text("
" + "<< [[" + secLabel(labels[eqCounter - 1]).replace(" ", + "_") + "|" + secLabel( + labels[eqCounter - 1]) + "]]
\n") + append_text("
[[" + secLabel(sections[secCount + 1][0]).replace(" ", + "_") + "#" + + secLabel(labels[eqCounter][len("Formula:"):]) + "|formula in " + secLabel( + sections[secCount + 1][0]) + "]]
\n") if True: - KLS.append_text("
[["+KLS.secLabel(labels[(eqCounter+1)%(endNum+1)]).replace(" ","_")+ - "|"+KLS.secLabel(labels[(eqCounter+1)%(endNum+1)])+"]] >>
\n") - KLS.append_text("
\n\n") - elif eqCounter==endNum: - KLS.append_text("
\n") + append_text( + "
[[" + secLabel(labels[(eqCounter + 1) % (endNum + 1)]).replace( + " ", "_") + + "|" + secLabel(labels[(eqCounter + 1) % (endNum + 1)]) + "]] >>
\n") + append_text("
\n\n") + elif eqCounter == endNum: + append_text("
\n") if newSec: - newSec=False - KLS.append_text("
<< [["+KLS.secLabel(sections[secCount][0]).replace(" ","_")+"|"+ - KLS.secLabel(sections[secCount][0])+"]]
\n") + newSec = False + append_text( + "
<< [[" + secLabel(sections[secCount][0]).replace(" ", + "_") + "|" + + secLabel(sections[secCount][0]) + "]]
\n") else: - KLS.append_text("
" - "<< [["+KLS.secLabel(labels[eqCounter-1]).replace(" ","_")+"|"+KLS.secLabel(labels[eqCounter-1])+"]]
\n") - KLS.append_text("
[["+ - KLS.secLabel(sections[secCount+1][0]).replace(" ","_")+ - "#"+KLS.secLabel(labels[eqCounter][len("Formula:"):])+ - "|formula in "+KLS.secLabel(sections[secCount+1][0])+ - "]]
\n") - KLS.append_text("
[["+KLS.secLabel(labels[(eqCounter+1)%(endNum+1)].replace(" ","_"))+ - "|"+KLS.secLabel(labels[(eqCounter+1)%(endNum+1)])+"]]
\n") - KLS.append_text("
\n\n") - - KLS.append_text("
{\displaystyle \n") - math=True + append_text("
" + "<< [[" + secLabel(labels[eqCounter - 1]).replace(" ", + "_") + "|" + secLabel( + labels[eqCounter - 1]) + "]]
\n") + append_text("
[[" + + secLabel(sections[secCount + 1][0]).replace(" ", "_") + + "#" + secLabel(labels[eqCounter][len("Formula:"):]) + + "|formula in " + secLabel(sections[secCount + 1][0]) + + "]]
\n") + append_text("
[[" + secLabel( + labels[(eqCounter + 1) % (endNum + 1)].replace(" ", "_")) + + "|" + secLabel(labels[(eqCounter + 1) % (endNum + 1)]) + "]]
\n") + append_text("
\n\n") + + append_text("
\n") + math = True elif "\\end{equation}" in line: - KLS.append_text(comToWrite) - parse=False - math=False + append_text(comToWrite) + parse = False + math = False if hProof: - KLS.append_text("\n== Proof ==\n\nWe ask users to provide proof(s), \ + append_text("\n== Proof ==\n\nWe ask users to provide proof(s), \ reference(s) to proof(s), or further clarification on the proof(s) in this space.\n") - KLS.append_text("\n== Symbols List ==\n\n") - newSym=[] - #if "09.07:04" in label: + append_text("\n== Symbols List ==\n\n") + newSym = [] + # if "09.07:04" in label: for x in symbols: - flagA=True - #if x not in newSym: - cN=0 - cC=0 - flag=True - ArgCx=0 + flagA = True + # if x not in newSym: + cN = 0 + cC = 0 + flag = True + ArgCx = 0 for z in x: - if z.isalpha() and flag or z=="&": - cN+=1 + if z.isalpha() and flag or z == "&": + cN += 1 else: - flag=False - if z=="{" or z=="[": - cC+=1 - if z=="}" or z=="]": - cC-=1 - if cC==0: - ArgCx+=1 - noA=x[:cN] + flag = False + if z == "{" or z == "[": + cC += 1 + if z == "}" or z == "]": + cC -= 1 + if cC == 0: + ArgCx += 1 + noA = x[:cN] for y in newSym: - cN=0 - cC=0 - ArgC=0 - flag=True + cN = 0 + cC = 0 + ArgC = 0 + flag = True for z in y: - if z.isalpha() and flag or z=="&": - cN+=1 + if z.isalpha() and flag or z == "&": + cN += 1 else: - flag=False - if z=="{" or z=="[": - cC+=1 - if z=="}" or z=="]": - cC-=1 - if cC==0: - ArgC+=1 - - - if y[:cN]==noA: #and ArgC==ArgCx: - flagA=False + flag = False + if z == "{" or z == "[": + cC += 1 + if z == "}" or z == "]": + cC -= 1 + if cC == 0: + ArgC += 1 + + if y[:cN] == noA: # and ArgC==ArgCx: + flagA = False break if flagA: newSym.append(x) newSym.reverse() - ampFlag=False - finSym=[] - for s in range(len(newSym)-1,-1,-1): - symbolPar="\\"+newSym[s] - ArgCx=0 - parCx=0 - parFlag=False - cC=0 + ampFlag = False + finSym = [] + for s in range(len(newSym) - 1, -1, -1): + symbolPar = "\\" + newSym[s] + ArgCx = 0 + parCx = 0 + parFlag = False + cC = 0 for z in symbolPar: - if z=="@": - parFlag=True - elif z.isalpha() or z=="&": - cN+=1 + if z == "@": + parFlag = True + elif z.isalpha() or z == "&": + cN += 1 else: - if z=="{" or z=="[": - cC+=1 - if z=="}" or z=="]": - cC-=1 - if cC==0: + if z == "{" or z == "[": + cC += 1 + if z == "}" or z == "]": + cC -= 1 + if cC == 0: if parFlag: - parCx+=1 + parCx += 1 else: - ArgCx+=1 + ArgCx += 1 - if symbolPar.find("{")!=-1 or symbolPar.find("[")!=-1: - if symbolPar.find("[")==-1: - symbol=symbolPar[0:symbolPar.find("{")] - elif symbolPar.find("{")==-1 or symbolPar.find("[")len(symbol) and (Q[len(symbol)]=="{" or Q[len(symbol)]=="["): - ap="" - for o in range(len(symbol),len(Q)): - if Q[o]=="{" or z=="[": + Q = symbolPar + listArgs = [] + if len(Q) > len(symbol) and (Q[len(symbol)] == "{" or Q[len(symbol)] == "["): + ap = "" + for o in range(len(symbol), len(Q)): + if Q[o] == "{" or z == "[": pass - elif Q[o]=="}" or z=="]": + elif Q[o] == "}" or z == "]": listArgs.append(ap) - ap="" + ap = "" else: - ap+=Q[o] - #websiteF=G[4].strip("\n") - websiteF="" - web1=G[5] - for t in range(5,len(G)): - if G[t]!="": - websiteF=websiteF+" ["+G[t]+" "+G[t]+"]" - - #p1=Q - #if Q.find("@")!=-1: - #p1=Q[:Q.find("@")] - p1=G[4].strip("$") - p1="{\\displaystyle "+p1+"}" - #if checkFlag: - new2="" - pause=False - mathF=True - p2=G[1] - for k in range(0,len(p2)): - if p2[k]=="$": + ap += Q[o] + websiteF = "" + web1 = G[5] + for t in range(5, len(G)): + if G[t] != "": + websiteF = websiteF + " [" + G[t] + " " + G[t] + "]" + p1 = G[4].strip("$") + p1 = "" + p1 + "" + # if checkFlag: + new2 = "" + pause = False + mathF = True + p2 = G[1] + for k in range(0, len(p2)): + if p2[k] == "$": if mathF: - new2+="{\\displaystyle " + new2 += " " else: - new2+="}" - mathF=not mathF + new2 += "" + mathF = not mathF else: - new2+=p2[k] - p2=new2 - finSym.append(web1+" "+p1+"] : "+p2+" :"+websiteF) + new2 += p2[k] + p2 = new2 + finSym.append(web1 + " " + p1 + "] : " + p2 + " :" + websiteF) break - - #preG=S if not gFlag: del newSym[s] - gFlag=True - #finSym.reverse() + gFlag = True if ampFlag: - KLS.append_text("& : logical and") - gFlag=False + append_text("& : logical and") + gFlag = False for y in finSym: - if y=="& : logical and": + if y == "& : logical and": pass elif gFlag: - gFlag=False - KLS.append_text("["+y) + gFlag = False + append_text("[" + y) else: - KLS.append_text("
\n["+y) - - - KLS.append_text("\n
\n") - - KLS.append_text("\n== Bibliography==\n\n")#should there be a space between bibliography and ==? - r=KLS.unmodlabel(labels[eqCounter]) - q=r.find("DLMF:")+5 - p=r.find(":",q) - section=r[q:p] - equation=r[p+1:] - #print("Section", section, r, p, q,) - if equation.find(":")!=-1: - equation=equation[0:equation.find(":")] - if is_number(section) == False: + append_text("
\n[" + y) + + append_text("\n
\n") + + append_text("\n== Bibliography==\n\n") # should there be a space between bibliography and ==? + r = unmodLabel(labels[eqCounter]) + q = r.find("DLMF:") + 5 + p = r.find(":", q) + section = r[q:p] + equation = r[p + 1:] + if equation.find(":") != -1: + equation = equation[0:equation.find(":")] + if isnumber(section) == False: return eqCounter - KLS.append_text("[HTTP://DLMF.NIST.GOV/"+ - section+"#"+equation+" Equation ("+equation[1:]+"), " - "Section "+section+"] of [[Bibliography#DLMF|'''DLMF''']].\n\n") - KLS.append_text("== URL links ==\n\nWe ask users to provide relevant URL links in this space.\n\n") - if eqCounter
\n") + append_text("[HTTP://DLMF.NIST.GOV/" + + section + "#" + equation + " Equation (" + equation[1:] + "), Section " + section + + "] of [[Bibliography#DLMF|'''DLMF''']].\n\n") + append_text("== URL links ==\n\nWe ask users to provide relevant URL links in this space.\n\n") + if eqCounter < endNum: + append_text("
\n") if newSec: - newSec=False - KLS.append_text("
<< [["+KLS.secLabel(sections[secCount][0]).replace(" ","_")+ - "|"+KLS.secLabel(sections[secCount][0])+"]]
\n") + newSec = False + append_text( + "
<< [[" + secLabel(sections[secCount][0]).replace(" ", "_") + + "|" + secLabel(sections[secCount][0]) + "]]
\n") else: - KLS.append_text("
<< [["+KLS.secLabel(labels[eqCounter-1]).replace(" ","_")+"|"+ - KLS.secLabel(labels[eqCounter-1])+"]]
\n") - KLS.append_text("
[["+KLS.secLabel(sections[secCount+1][0]).replace(" ","_")+"#"+ - KLS.secLabel(labels[eqCounter][len("Formula:"):])+"|formula in "+KLS.secLabel(sections[secCount+1][0])+"]]
\n") - #if eqS==sections[secCount][1]: - #else: + append_text( + "
<< [[" + secLabel(labels[eqCounter - 1]).replace(" ", + "_") + "|" + + secLabel(labels[eqCounter - 1]) + "]]
\n") + append_text("
[[" + secLabel(sections[secCount + 1][0]).replace(" ", + "_") + "#" + + secLabel(labels[eqCounter][len("Formula:"):]) + "|formula in " + secLabel( + sections[secCount + 1][0]) + "]]
\n") if True: - KLS.append_text("
[["+KLS.secLabel(labels[(eqCounter+1)% - (endNum+1)]).replace(" ","_")+"|"+KLS.secLabel(labels[(eqCounter+1)%(endNum+1)])+"]] >>
\n") - KLS.append_text("
\n\ndrmf_eof\n") - else: #FOR EXTRA EQUATIONS - KLS.append_text("
\n") - KLS.append_text("
<< [["+labels[endNum-1].replace(" ","_")+"|"+labels[endNum-1]+"]]
\n") - KLS.append_text("
[["+labels[0].replace(" ","_")+"#"+labels[endNum][8:]+"|formula in "+labels[0]+"]]
\n") - KLS.append_text("
[["+labels[(0)%endNum].replace(" ","_")+"|"+labels[0%endNum]+"]]
\n") - KLS.append_text("
\n\ndrmf_eof\n") + append_text("
[[" + secLabel(labels[(eqCounter + 1) % + (endNum + 1)]).replace(" ", + "_") + "|" + secLabel( + labels[(eqCounter + 1) % (endNum + 1)]) + "]] >>
\n") + append_text("
\n\ndrmf_eof\n") + else: # FOR EXTRA EQUATIONS + append_text("
\n") + append_text( + "
<< [[" + labels[endNum - 1].replace(" ", "_") + "|" + labels[ + endNum - 1] + "]]
\n") + append_text("
[[" + labels[0].replace(" ", "_") + "#" + labels[endNum][ + 8:] + "|formula in " + + labels[0] + "]]
\n") + append_text( + "
[[" + labels[(0) % endNum].replace(" ", "_") + "|" + labels[ + 0 % endNum] + "]]
\n") + append_text("
\n\ndrmf_eof\n") elif "\\constraint" in line and parse: - #symbols=symbols+KLS.getSym(line) - symLine=line.strip("\n") + symLine = line.strip("\n") if hCon: - comToWrite=comToWrite+"\n== Constraint(s) ==\n\n" - hCon=False - constraint=True - math=False - conLine="" - #KLS.append_text("
"+KLS.getEq(line)+"

\n") + comToWrite += "\n== Constraint(s) ==\n\n" + hCon = False + constraint = True + math = False + conLine = "" elif "\\substitution" in line and parse: - #symbols=symbols+KLS.getSym(line) - symLine=line.strip("\n") + symLine = line.strip("\n") if hSub: - comToWrite=comToWrite+"\n== Substitution(s) ==\n\n" - hSub=False - #KLS.append_text("
"+KLS.getEq(line)+"

\n") - substitution=True - math=False - subLine="" + comToWrite += "\n== Substitution(s) ==\n\n" + hSub = False + substitution = True + math = False + subLine = "" elif "\\drmfname" in line and parse: - math=False - comToWrite="\n== Name ==\n\n
"+KLS.getString(line)+"

\n"+comToWrite + math = False + comToWrite = "\n== Name ==\n\n
" + getString( + line) + "

\n" + comToWrite elif "\\drmfnote" in line and parse: - symbols=symbols+KLS.getSym(line) + symbols = symbols + getSym(line) if hNote: - comToWrite=comToWrite+"\n== Note(s) ==\n\n" - hNote=False - note=True - math=False - noteLine="" + comToWrite += "\n== Note(s) ==\n\n" + hNote = False + note = True + math = False + noteLine = "" elif "\\proof" in line and parse: - #symbols=symbols+KLS.getSym(line) - symLine=line.strip("\n") + symLine = line.strip("\n") if hProof: - hProof=False - comToWrite=comToWrite+"\n== Proof ==\n\nWe ask users to provide proof(s), reference(s) to proof(s), " \ - "or further clarification on the proof(s) in this space. \n

\n
" - proof=True - proofLine="" - pause=False - pauseP=False - for ind in range(0,len(line)): - if line[ind:ind+7]=="\\eqref{": - pause=True - eqR=line[ind:line.find("}",ind)+1] - rLab=KLS.getString(eqR) + hProof = False + comToWrite += "\n== Proof ==\n\nWe ask users to provide proof(s), reference(s) to proof(s), " \ + "or further clarification on the proof(s) in this space. \n

\n
" + proof = True + proofLine = "" + pause = False + pauseP = False + for ind in range(0, len(line)): + if line[ind:ind + 7] == "\\eqref{": + pause = True + eqR = line[ind:line.find("}", ind) + 1] + rLab = getString(eqR) for l in lLink: - if rLab==l[0:l.find("=")-1]: - rlabel=l[l.find("=>")+3:l.find("\\n")] - rlabel=rlabel.replace("/","") - rlabel=rlabel.replace("#",":") - rlabel=rlabel.replace("!",":") + if rLab == l[0:l.find("=") - 1]: + rlabel = l[l.find("=>") + 3:l.find("\\n")] + rlabel = rlabel.replace("/", "") + rlabel = rlabel.replace("#", ":") + rlabel = rlabel.replace("!", ":") break - eInd=refLabels.index(""+rlabel) - z=line[line.find("}",ind+7)+1] - if z=="." or z==",": - pauseP=True - proofLine+=("
\n{\displaystyle \n"+refEqs[eInd]+"}"+z+"
\n") + eInd = refLabels.index("" + rlabel) + z = line[line.find("}", ind + 7) + 1] + if z == "." or z == ",": + pauseP = True + proofLine += ("
\n\n" + refEqs[ + eInd] + "" + z + "
\n") else: - if z=="}": - proofLine+=("
\n{\displaystyle \n"+refEqs[eInd]+"}
") + if z == "}": + proofLine += ( + "
\n\n" + refEqs[ + eInd] + "
") else: - proofLine+=("
\n{\displaystyle \n"+refEqs[eInd]+"}
\n") - - + proofLine += ("
\n\n" + refEqs[ + eInd] + "
\n") else: if pause: - if line[ind]=="}": - pause=False + if line[ind] == "}": + pause = False elif pauseP: - pauseP=False - elif line[ind]=="\n" and "\\end{equation}" in lines[i+1]: + pauseP = False + elif line[ind] == "\n" and "\\end{equation}" in lines[i + 1]: pass else: - proofLine+=(line[ind]) - if "\\end{equation}" in lines[i+1]: - proof=False - #symLine+=line.strip("\n") - KLS.append_text(comToWrite+KLS.getEqP(proofLine)+"
\n
\n") - comToWrite="" - symbols=symbols+KLS.getSym(symLine) - symLine="" - - #KLS.append_text(line) - + proofLine += (line[ind]) + if "\\end{equation}" in lines[i + 1]: + proof = False + append_text(comToWrite + getEqP(proofLine) + "
\n
\n") + comToWrite = "" + symbols = symbols + getSym(symLine) + symLine = "" elif proof: - symLine+=line.strip("\n") - pauseP=False - #symbols=symbols+KLS.getSym(line) - for ind in range(0,len(line)): - if line[ind:ind+7]=="\\eqref{": - pause=True - eqR=line[ind:line.find("}",ind)+1] - rLab=KLS.getString(eqR) + symLine += line.strip("\n") + pauseP = False + for ind in range(0, len(line)): + if line[ind:ind + 7] == "\\eqref{": + pause = True + eqR = line[ind:line.find("}", ind) + 1] + rLab = getString(eqR) for l in lLink: - if rLab==l[0:l.find("=")-1]: - rlabel=l[l.find("=>")+3:l.find("\\n")] - rlabel=rlabel.replace("/","") - rlabel=rlabel.replace("#",":") - rlabel=rlabel.replace("!",":") + if rLab == l[0:l.find("=") - 1]: + rlabel = l[l.find("=>") + 3:l.find("\\n")] + rlabel = rlabel.replace("/", "") + rlabel = rlabel.replace("#", ":") + rlabel = rlabel.replace("!", ":") break - eInd=refLabels.index(""+rlabel.lstrip("Formula:")) - z=line[line.find("}",ind+7)+1] - if z=="." or z==",": - pauseP=True - proofLine+=("
\n{\displaystyle \n"+refEqs[eInd]+"}"+z+"
\n") + eInd = refLabels.index("" + rlabel.lstrip("Formula:")) + z = line[line.find("}", ind + 7) + 1] + if z == "." or z == ",": + pauseP = True + proofLine += ("
\n\n" + refEqs[ + eInd] + "" + z + "
\n") else: - proofLine+=("
\n{\displaystyle \n"+refEqs[eInd]+"}
\n") + proofLine += ("
\n\n" + refEqs[ + eInd] + "
\n") else: if pause: - if line[ind]=="}": - pause=False + if line[ind] == "}": + pause = False elif pauseP: - pauseP=False - elif line[ind]=="\n" and "\\end{equation}" in lines[i+1]: + pauseP = False + elif line[ind] == "\n" and "\\end{equation}" in lines[i + 1]: pass else: - proofLine+=(line[ind]) - if "\\end{equation}" in lines[i+1]: - proof=False - #symLine+=line.strip("\n") - KLS.append_text(comToWrite+KLS.getEqP(proofLine).rstrip("\n")+"
\n
\n") - comToWrite="" - symbols=symbols+KLS.getSym(symLine) - symLine="" + proofLine += (line[ind]) + if "\\end{equation}" in lines[i + 1]: + proof = False + append_text(comToWrite + getEqP(proofLine).rstrip("\n") + "
\n
\n") + comToWrite = "" + symbols = symbols + getSym(symLine) + symLine = "" elif math: - if "\\end{equation}" in lines[i+1] or "\\constraint" in lines[i+1] \ - or "\\substitution" in lines[i+1] or "\\proof" in lines[i+1] or "\\drmfnote" in lines[i+1] or "\\drmfname" in lines[i+1]: - KLS.append_text(line.rstrip("\n")) - symLine+=line.strip("\n") - symbols=symbols+KLS.getSym(symLine) - symLine="" - KLS.append_text("\n}
\n") + if "\\end{equation}" in lines[i + 1] or "\\constraint" in lines[i + 1] \ + or "\\substitution" in lines[i + 1] or "\\proof" in lines[i + 1] or "\\drmfnote" in lines[ + i + 1] or "\\drmfname" in lines[i + 1]: + append_text(line.rstrip("\n")) + symLine += line.strip("\n") + symbols = symbols + getSym(symLine) + symLine = "" + append_text("\n
\n") else: - symLine+=line.strip("\n") - KLS.append_text(line) + symLine += line.strip("\n") + append_text(line) if note and parse: - noteLine=noteLine+line - symbols=symbols+KLS.getSym(line) - if "\\end{equation}" in lines[i+1] or "\\drmfn" in lines[i+1] \ - or "\\constraint" in lines[i+1] \ - or "\\substitution" in lines[i+1] \ - or "\\proof" in lines[i+1]: - note=False + noteLine = noteLine + line + symbols = symbols + getSym(line) + if "\\end{equation}" in lines[i + 1] or "\\drmfn" in lines[i + 1] \ + or "\\constraint" in lines[i + 1] \ + or "\\substitution" in lines[i + 1] \ + or "\\proof" in lines[i + 1]: + note = False if "\\emph" in noteLine: - noteLine=noteLine[0:noteLine.find("\\emph{")]+"\'\'"+noteLine[noteLine.find("\\emph{")+len("\\emph{"): - noteLine.find("}",noteLine.find("\\emph{")+len("\\emph{"))]+"\'\'"+\ - noteLine[noteLine.find("}",noteLine.find("\\emph{")+len("\\emph{"))+1:] - comToWrite=comToWrite+"
"+KLS.getEq(noteLine)+"

\n" + noteLine = noteLine[0:noteLine.find("\\emph{")] + "\'\'" + noteLine[ + noteLine.find("\\emph{") + len( + "\\emph{"): + noteLine.find("}", noteLine.find( + "\\emph{") + len( + "\\emph{"))] + "\'\'" + \ + noteLine[noteLine.find("}", noteLine.find("\\emph{") + len("\\emph{")) + 1:] + comToWrite = comToWrite + "
" + getEq(noteLine) + "

\n" if constraint and parse: - conLine=conLine+line.replace("&","&
") - - symLine+=line.strip("\n") - #symbols=symbols+KLS.getSym(line) - if "\\end{equation}" in lines[i+1] or "\\drmfn" in lines[i+1] \ - or "\\constraint" in lines[i+1]\ - or "\\substitution" in lines[i+1] \ - or "\\proof" in lines[i+1]: - constraint=False - symbols=symbols+KLS.getSym(symLine) - symLine="" - KLS.append_text(comToWrite+"
"+KLS.getEq(conLine)+"

\n") - comToWrite="" + conLine += line.replace("&", "&
") + + symLine += line.strip("\n") + # symbols=symbols+getSym(line) + if "\\end{equation}" in lines[i + 1] or "\\drmfn" in lines[i + 1] \ + or "\\constraint" in lines[i + 1] \ + or "\\substitution" in lines[i + 1] \ + or "\\proof" in lines[i + 1]: + constraint = False + symbols = symbols + getSym(symLine) + symLine = "" + append_text(comToWrite + "
" + getEq(conLine) + "

\n") + comToWrite = "" if substitution and parse: - subLine=subLine+line.replace("&","&
") - - symLine+=line.strip("\n") - #symbols=symbols+KLS.getSym(line) - if "\\end{equation}" in lines[i+1] or "\\drmfn" in lines[i+1] or \ - "\\substitution" in lines[i+1] or "\\constraint" in lines[i+1] or "\\proof" in lines[i+1]: - substitution=False - symbols=symbols+KLS.getSym(symLine) - symLine="" - KLS.append_text(comToWrite+"
"+KLS.getEq(subLine)+"

\n") - comToWrite="" + subLine = subLine + line.replace("&", "&
") + + symLine += line.strip("\n") + # symbols=symbols+getSym(line) + if "\\end{equation}" in lines[i + 1] or "\\drmfn" in lines[i + 1] or \ + "\\substitution" in lines[i + 1] or "\\constraint" in lines[i + 1] or "\\proof" in \ + lines[i + 1]: + substitution = False + symbols = symbols + getSym(symLine) + symLine = "" + append_text(comToWrite + "
" + getEq(subLine) + "

\n") + comToWrite = "" + if __name__ == "__main__": - DLMF(0); \ No newline at end of file + tex = "../../data/ZE.3.tex" + mainFile = "../../data/OrthogonalPolynomials.mmd" + lLinks = "../../data/BruceLabelLinks" + DLMF(tex, mainFile, lLinks, 0) From 0cf12eae70d89b7bb56f11023d0f050bc654f228 Mon Sep 17 00:00:00 2001 From: joonbang Date: Fri, 10 Jun 2016 08:28:18 -0400 Subject: [PATCH 095/402] Extend compatibility to confluent, confluent limit functions; organize code into classes (#49) * Extend compatibility to confluent, confluent limit functions * Reorganize output files * Reorganize files * Add folder for unit tests * Change to object-oriented style of programming * Remove wildcard imports * Fix file directory issues --- Joon/.gitignore | 5 - Joon/Cuytbook.pdf | Bin 2744504 -> 0 bytes Joon/Glossary.csv | 691 --------------------------- Joon/converter.py | 268 ----------- Joon/keys/functions | 17 - Joon/keys/section_names | 69 --- Joon/keys/spacing | 11 - Joon/keys/special | 5 - Joon/keys/symbols | 35 -- Joon/mpl/bessel.mpl | 169 ------- Joon/mpl/confluent.mpl | 13 - Joon/mpl/modbessel.mpl | 156 ------ Joon/src/MapleFormula.py | 120 +++++ Joon/src/TranslationMethods.py | 128 +++++ Joon/{ => src}/category_generator.py | 6 +- Joon/src/main.py | 42 ++ Joon/test/README.md | 1 + Joon/tests/test.pdf | Bin 56226 -> 0 bytes Joon/tests/test.tex | 137 ------ Joon/todo.txt | 2 - 20 files changed, 295 insertions(+), 1580 deletions(-) delete mode 100644 Joon/.gitignore delete mode 100644 Joon/Cuytbook.pdf delete mode 100644 Joon/Glossary.csv delete mode 100644 Joon/converter.py delete mode 100644 Joon/keys/functions delete mode 100644 Joon/keys/section_names delete mode 100644 Joon/keys/spacing delete mode 100644 Joon/keys/special delete mode 100644 Joon/keys/symbols delete mode 100644 Joon/mpl/bessel.mpl delete mode 100644 Joon/mpl/confluent.mpl delete mode 100644 Joon/mpl/modbessel.mpl create mode 100644 Joon/src/MapleFormula.py create mode 100644 Joon/src/TranslationMethods.py rename Joon/{ => src}/category_generator.py (88%) create mode 100644 Joon/src/main.py create mode 100644 Joon/test/README.md delete mode 100644 Joon/tests/test.pdf delete mode 100644 Joon/tests/test.tex delete mode 100644 Joon/todo.txt diff --git a/Joon/.gitignore b/Joon/.gitignore deleted file mode 100644 index 2133789..0000000 --- a/Joon/.gitignore +++ /dev/null @@ -1,5 +0,0 @@ -Glossary.csv -tests/ -functions/ -!tests/test.tex -!tests/test.pdf diff --git a/Joon/Cuytbook.pdf b/Joon/Cuytbook.pdf deleted file mode 100644 index 914a06ba364ee74724703159776c03a78f5095df..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 2744504 zcma&NV~}XgwzXNdZQHhO+qUhhUAAr8wr%XPZQK2xj*j!4?tA;j{gaXF&%zTc=Nvia zOi~3AFz7lo0pdNbb-AKhlAQU<`)&ut=52w#Weta822(mGqw< zEhy=<3%FA}-1AQaPiD#?BS4d~+*N@zX^k4x53!^B{yis9ux6Z~&LJN<5c-i=ft_uG z#$QJ&IL7yd<0D!U{T1kHmco2w1sO`yeA;+gYKU!M>joPOUdCI0+IkZqA?g#2W}qhs 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\n
\n") comToWrite = "" symbols = symbols + getSym(symLine) symLine = "" @@ -707,11 +1019,19 @@ def DLMF(ofname,mmd,llinks,n): z = line[line.find("}", ind + 7) + 1] if z == "." or z == ",": pauseP = True - proofLine += ("
\n\n" + refEqs[ - eInd] + "" + z + "
\n") + proofLine += ("
\n\n" + + refEqs[eInd] + + "" + + z + + "
\n") else: - proofLine += ("
\n\n" + refEqs[ - eInd] + "
\n") + proofLine += ("
\n\n" + + refEqs[eInd] + + "
\n") else: if pause: @@ -726,15 +1046,28 @@ def DLMF(ofname,mmd,llinks,n): proofLine += (line[ind]) if "\\end{equation}" in lines[i + 1]: proof = False - append_text(comToWrite + getEqP(proofLine).rstrip("\n") + "
\n
\n") + append_text( + comToWrite + + getEqP(proofLine).rstrip("\n") + + "
\n
\n") comToWrite = "" symbols = symbols + getSym(symLine) symLine = "" elif math: - if "\\end{equation}" in lines[i + 1] or "\\constraint" in lines[i + 1] \ - or "\\substitution" in lines[i + 1] or "\\proof" in lines[i + 1] or "\\drmfnote" in lines[ - i + 1] or "\\drmfname" in lines[i + 1]: + if "\\end{equation}" in lines[ + i + + 1] or "\\constraint" in lines[ + i + + 1] or "\\substitution" in lines[ + i + + 1] or "\\proof" in lines[ + i + + 1] or "\\drmfnote" in lines[ + i + + 1] or "\\drmfname" in lines[ + i + + 1]: append_text(line.rstrip("\n")) symLine += line.strip("\n") symbols = symbols + getSym(symLine) @@ -746,47 +1079,77 @@ def DLMF(ofname,mmd,llinks,n): if note and parse: noteLine = noteLine + line symbols = symbols + getSym(line) - if "\\end{equation}" in lines[i + 1] or "\\drmfn" in lines[i + 1] \ - or "\\constraint" in lines[i + 1] \ - or "\\substitution" in lines[i + 1] \ - or "\\proof" in lines[i + 1]: + if "\\end{equation}" in lines[ + i + + 1] or "\\drmfn" in lines[ + i + + 1] or "\\constraint" in lines[ + i + + 1] or "\\substitution" in lines[ + i + + 1] or "\\proof" in lines[ + i + + 1]: note = False if "\\emph" in noteLine: - noteLine = noteLine[0:noteLine.find("\\emph{")] + "\'\'" + noteLine[ - noteLine.find("\\emph{") + len( - "\\emph{"): - noteLine.find("}", noteLine.find( - "\\emph{") + len( - "\\emph{"))] + "\'\'" + \ - noteLine[noteLine.find("}", noteLine.find("\\emph{") + len("\\emph{")) + 1:] - comToWrite = comToWrite + "
" + getEq(noteLine) + "

\n" + noteLine = noteLine[ + 0:noteLine.find("\\emph{")] + "\'\'" + noteLine[ + noteLine.find("\\emph{") + len("\\emph{"): noteLine.find( + "}", noteLine.find("\\emph{") + len("\\emph{"))] + "\'\'" + noteLine[ + noteLine.find( + "}", noteLine.find("\\emph{") + len("\\emph{")) + 1:] + comToWrite = comToWrite + "
" + \ + getEq(noteLine) + "

\n" if constraint and parse: conLine += line.replace("&", "&
") symLine += line.strip("\n") # symbols=symbols+getSym(line) - if "\\end{equation}" in lines[i + 1] or "\\drmfn" in lines[i + 1] \ - or "\\constraint" in lines[i + 1] \ - or "\\substitution" in lines[i + 1] \ - or "\\proof" in lines[i + 1]: + if "\\end{equation}" in lines[ + i + + 1] or "\\drmfn" in lines[ + i + + 1] or "\\constraint" in lines[ + i + + 1] or "\\substitution" in lines[ + i + + 1] or "\\proof" in lines[ + i + + 1]: constraint = False symbols = symbols + getSym(symLine) symLine = "" - append_text(comToWrite + "
" + getEq(conLine) + "

\n") + append_text( + comToWrite + + "
" + + getEq(conLine) + + "

\n") comToWrite = "" if substitution and parse: subLine = subLine + line.replace("&", "&
") symLine += line.strip("\n") # symbols=symbols+getSym(line) - if "\\end{equation}" in lines[i + 1] or "\\drmfn" in lines[i + 1] or \ - "\\substitution" in lines[i + 1] or "\\constraint" in lines[i + 1] or "\\proof" in \ - lines[i + 1]: + if "\\end{equation}" in lines[ + i + + 1] or "\\drmfn" in lines[ + i + + 1] or "\\substitution" in lines[ + i + + 1] or "\\constraint" in lines[ + i + + 1] or "\\proof" in lines[ + i + + 1]: substitution = False symbols = symbols + getSym(symLine) symLine = "" - append_text(comToWrite + "
" + getEq(subLine) + "

\n") + append_text( + comToWrite + + "
" + + getEq(subLine) + + "

\n") comToWrite = "" diff --git a/tex2Wiki/src/tex2Wiki.py b/tex2Wiki/src/tex2Wiki.py index 16ef557..5bb0c24 100644 --- a/tex2Wiki/src/tex2Wiki.py +++ b/tex2Wiki/src/tex2Wiki.py @@ -32,13 +32,15 @@ def getString(line): # Gets all data within curly braces on a line if c == "}": # no more info needed break elif getStr: # if within curly braces - if not (pW == c and c == "-"): # if there is no double dash (makes single dash) + if not ( + pW == c and c == "-"): # if there is no double dash (makes single dash) if c != "$": # if not a $ sign stringWrite += c # this character is part of the data else: # replace $ with '' stringWrite += "\'\'" # add double ' pW = c # change last character for finding double dashes - return stringWrite.rstrip('\n').lstrip() # return the data without newlines and without leading spaces + # return the data without newlines and without leading spaces + return stringWrite.rstrip('\n').lstrip() def getG(line): # gets equation for symbols list @@ -85,7 +87,9 @@ def getEq(line): # Gets all data within constraints,substitutions elif c == "\n" and per == 0: # if newline stringWrite = stringWrite.strip() # remove all leading and trailing whitespace - # should above be rstrip?<-------------------------------------------------------------------CHECK THIS + # should above be + # rstrip?<-------------------------------------------------------------------CHECK + # THIS stringWrite += c # add the newline character per += 1 # watch for % signs @@ -102,7 +106,8 @@ def getEqP(line): # Gets all data within proofs count = 0 length = 0 for c in line: - if fEq and c != " " and c != "$" and c != "{" and c != "}" and not c.isalpha(): + if fEq and c != " " and c != "$" and c != "{" and c != "}" and not c.isalpha( + ): length += 1 if count >= 0 and per != 0: per += 1 @@ -146,7 +151,9 @@ def getSym(line): # Gets all symbols on a line for symbols list argFlag = False cC = 0 for i in range(0, len(line)): - # if "MeixnerPollaczek{\\lambda}{n}@{x}{\\phi}=\frac{\\pochhammer{2\\lambda}{n}}" in line: + # if + # "MeixnerPollaczek{\\lambda}{n}@{x}{\\phi}=\frac{\\pochhammer{2\\lambda}{n}}" + # in line: c = line[i] if symFlag: if c == "{" or c == "[": @@ -170,7 +177,9 @@ def getSym(line): # Gets all symbols on a line for symbols list p = line[i + 1] if cC <= 0 and p != "{" and p != "[" and p != "@": symFlag = False - # if "monicAskeyWilson{n+1}@{x}{a}{b}{c}{d}{q}+\\frac{1}{2}" in line: + # if + # "monicAskeyWilson{n+1}@{x}{a}{b}{c}{d}{q}+\\frac{1}{2}" + # in line: symList.append(symbol) symList += (getSym(symbol)) argFlag = False @@ -181,7 +190,9 @@ def getSym(line): # Gets all symbols on a line for symbols list elif c == "&" and not (i > line.find("\\begin{array}") and i < line.find("\\end{array}")): symList.append("&") - # if "MeixnerPollaczek{\\lambda}{n}@{x}{\\phi}=\frac{\\pochhammer{2\\lambda}{n}}" in line: + # if + # "MeixnerPollaczek{\\lambda}{n}@{x}{\\phi}=\frac{\\pochhammer{2\\lambda}{n}}" + # in line: symList.append(symbol) symList += getSym(symbol) return symList @@ -275,8 +286,8 @@ def main(): fname = "../../data/ZE.3.tex" ofname = "../../data/ZE.4.xml" lname = "../../data/BruceLabelLinks" - glossary = "../../data/new.Glossary.csv" - mmd = "../../data/OrthogonalPolynomials.mmd" + glossary = "../../data/new.Glossary.csv" + mmd = "../../data/OrthogonalPolynomials.mmd" else: fname = sys.argv[1] @@ -285,7 +296,7 @@ def main(): glossary = sys.argv[4] mmd = sys.argv[5] setup_label_links(lname) - readin(fname,glossary,mmd) + readin(fname, glossary, mmd) writeout(ofname) @@ -294,7 +305,7 @@ def setup_label_links(ofname): lLink = open(ofname, "r").readlines() -def readin(ofname,glossary,mmd): +def readin(ofname, glossary, mmd): # try: for iterations in range(0, 1): tex = open(ofname, 'r') @@ -326,8 +337,15 @@ def readin(ofname,glossary,mmd): math = True elif math: refEqs[len(refEqs) - 1] += line - if "\\end{equation}" in refLines[i + 1] or "\\constraint" in refLines[i + 1] or "\\substitution" in \ - refLines[i + 1] or "\\drmfn" in refLines[i + 1]: + if "\\end{equation}" in refLines[ + i + + 1] or "\\constraint" in refLines[ + i + + 1] or "\\substitution" in refLines[ + i + + 1] or "\\drmfn" in refLines[ + i + + 1]: math = False for i in range(0, len(lines)): @@ -337,7 +355,8 @@ def readin(ofname,glossary,mmd): elif "\\end{document}" in line and parse: mainPrepend += "
\n" mainText = mainPrepend + mainText - mainText = mainText.replace("\'\'\'Orthogonal Polynomials\'\'\'\n", "") + mainText = mainText.replace( + "\'\'\'Orthogonal Polynomials\'\'\'\n", "") mainText = mainText.replace("{{#set:Section=0}}\n", "") mainText = mainText[0:mainText.rfind("== Sections ")] append_revision('Orthogonal Polynomials') @@ -354,7 +373,9 @@ def readin(ofname,glossary,mmd): sections.append(["Orthogonal Polynomials", 0]) chapter = getString(line) mainPrepend += ( - "\n== Sections in " + chapter + " ==\n\n
\n") + "\n== Sections in " + + chapter + + " ==\n\n
\n") elif "\\part" in line: if getString(line) == "BOF": parse = False @@ -364,7 +385,11 @@ def readin(ofname,glossary,mmd): mainPrepend += ("\n
\n= " + getString(line) + " =\n") head = True elif "\\section" in line: - mainPrepend += ("* [[" + secLabel(getString(line)) + "|" + getString(line) + "]]\n") + mainPrepend += ("* [[" + + secLabel(getString(line)) + + "|" + + getString(line) + + "]]\n") sections.append([getString(line)]) secCounter = 0 @@ -377,7 +402,8 @@ def readin(ofname,glossary,mmd): parse = True secCounter += 1 append_revision(secLabel(getString(line))) - append_text("{{DISPLAYTITLE:" + (sections[secCounter][0]) + "}}\n") + append_text( + "{{DISPLAYTITLE:" + (sections[secCounter][0]) + "}}\n") append_text("{{#set:Chapter=" + chapter + "}}\n") append_text("{{#set:Section=" + str(secCounter) + "}}\n") append_text("{{headSection}}\n") @@ -430,21 +456,31 @@ def readin(ofname,glossary,mmd): elif "\\drmfn" in line and parse: math = False if "\\drmfname" in line and parse: - append_text("
This formula has the name: " + getString(line) + "

\n") + append_text( + "
This formula has the name: " + + getString(line) + + "

\n") elif math and parse: flagM = True eqs[len(eqs) - 1] += line - if "\\end{equation}" in lines[i + 1] and "\\subsection" not in lines[i + 3] and "\\section" not in \ - lines[i + 3] and "\\part" not in lines[i + 3]: + if "\\end{equation}" in lines[ + i + + 1] and "\\subsection" not in lines[ + i + + 3] and "\\section" not in lines[ + i + + 3] and "\\part" not in lines[ + i + + 3]: u = i flagM2 = False while flagM: u += 1 if "\\begin{equation}" in lines[u] in lines[u]: flagM = False - if "\\section" in lines[u] or "\\subsection" in lines[i] or "\\part" in lines[u] \ - or "\\end{document}" in lines[u]: + if "\\section" in lines[u] or "\\subsection" in lines[ + i] or "\\part" in lines[u] or "\\end{document}" in lines[u]: flagM = False flagM2 = True if not flagM2: @@ -464,13 +500,29 @@ def readin(ofname,glossary,mmd): append_text(line) elif math and not parse: eqs[len(eqs) - 1] += line - if "\\end{equation}" in lines[i + 1] or "\\constraint" in lines[i + 1] or "\\substitution" in lines[ - i + 1] or "\\drmfn" in lines[i + 1]: + if "\\end{equation}" in lines[ + i + + 1] or "\\constraint" in lines[ + i + + 1] or "\\substitution" in lines[ + i + + 1] or "\\drmfn" in lines[ + i + + 1]: math = False if substitution and parse: subLine += line.replace("&", "&
") - if "\\end{equation}" in lines[i + 1] or "\\substitution" in lines[i + 1] or "\\constraint" in lines[ - i + 1] or "\\drmfn" in lines[i + 1] or "\\proof" in lines[i + 1]: + if "\\end{equation}" in lines[ + i + + 1] or "\\substitution" in lines[ + i + + 1] or "\\constraint" in lines[ + i + + 1] or "\\drmfn" in lines[ + i + + 1] or "\\proof" in lines[ + i + + 1]: lineR = "" for i in range(0, len(subLine)): if subLine[i] == "&" and not ( @@ -479,14 +531,29 @@ def readin(ofname,glossary,mmd): else: lineR += subLine[i] substitution = False - append_text("
Substitution(s): " + getEq(subLine) + "

\n") + append_text( + "
Substitution(s): " + + getEq(subLine) + + "

\n") if constraint and parse: conLine += line.replace("&", "&
") - if "\\end{equation}" in lines[i + 1] or "\\substitution" in lines[i + 1] or "\\constraint" in lines[ - i + 1] or "\\drmfn" in lines[i + 1] or "\\proof" in lines[i + 1]: + if "\\end{equation}" in lines[ + i + + 1] or "\\substitution" in lines[ + i + + 1] or "\\constraint" in lines[ + i + + 1] or "\\drmfn" in lines[ + i + + 1] or "\\proof" in lines[ + i + + 1]: constraint = False - append_text("
Constraint(s): " + getEq(conLine) + "

\n") + append_text( + "
Constraint(s): " + + getEq(conLine) + + "

\n") eqCounter = 0 endNum = len(labels) - 1 @@ -536,40 +603,151 @@ def readin(ofname,glossary,mmd): if eqCounter < endNum: # FOR ANYTHING THAT IS NOT THE EXTRA EQUATIONS append_text("
\n") if newSec: - append_text("
<< [[" + secLabel(sections[secCount][0]).replace(" ", "_") + - "|" + secLabel(sections[secCount][0]) + "]]
\n") + append_text( + "
<< [[" + + secLabel( + sections[secCount][0]).replace( + " ", + "_") + + "|" + + secLabel( + sections[secCount][0]) + + "]]
\n") else: - append_text("
<< [[" + secLabel(labels[eqCounter - 1]).replace(" ", "_") + - "|" + secLabel(labels[eqCounter - 1]) + "]]
\n") - append_text("
[[" + secLabel(sections[secCount + 1][0]).replace(" ", "_") + - "#" + secLabel(labels[eqCounter][len("Formula:"):]) + "|formula in " + - secLabel(sections[secCount + 1][0]) + "]]
\n") + append_text( + "
<< [[" + + secLabel( + labels[ + eqCounter - + 1]).replace( + " ", + "_") + + "|" + + secLabel( + labels[ + eqCounter - + 1]) + + "]]
\n") + append_text( + "
[[" + + secLabel( + sections[ + secCount + + 1][0]).replace( + " ", + "_") + + "#" + + secLabel( + labels[eqCounter][ + len("Formula:"):]) + + "|formula in " + + secLabel( + sections[ + secCount + + 1][0]) + + "]]
\n") if eqS == sections[secCount][1]: append_text( - "
[[" + secLabel( - sections[(secCount + 1) % len(sections)][0]).replace( - " ", "_") + "|" + secLabel( - sections[(secCount + 1) % len(sections)][0]) + "]] >>
\n") + "
[[" + + secLabel( + sections[ + (secCount + + 1) % + len(sections)][0]).replace( + " ", + "_") + + "|" + + secLabel( + sections[ + (secCount + + 1) % + len(sections)][0]) + + "]] >>
\n") else: - append_text("
[[" + - secLabel(labels[(eqCounter + 1) % (endNum + 1)]).replace(" ", "_") + - "|" + secLabel(labels[(eqCounter + 1) % (endNum + 1)]) + "]] >>
\n") + append_text( + "
[[" + + secLabel( + labels[ + (eqCounter + + 1) % + (endNum + + 1)]).replace( + " ", + "_") + + "|" + + secLabel( + labels[ + (eqCounter + + 1) % + (endNum + + 1)]) + + "]] >>
\n") append_text("
\n\n") elif eqCounter == endNum: append_text("
\n") if newSec: newSec = False - append_text("
<< [[" + secLabel(sections[secCount][0]).replace(" ", "_") + - "|" + secLabel(sections[secCount][0]) + "]]
\n") + append_text( + "
<< [[" + + secLabel( + sections[secCount][0]).replace( + " ", + "_") + + "|" + + secLabel( + sections[secCount][0]) + + "]]
\n") else: - append_text("
<< [[" + secLabel(labels[eqCounter - 1]).replace(" ", "_") + - "|" + secLabel(labels[eqCounter - 1]) + "]]
\n") - append_text("
[[" + secLabel(sections[secCount + 1][0]).replace(" ", "_") + - "#" + secLabel(labels[eqCounter][len("Formula:"):]) + "|formula in " + - secLabel(sections[secCount + 1][0]) + "]]
\n") - append_text("
[[" + secLabel( - labels[(eqCounter + 1) % (endNum + 1)].replace(" ", "_")) + "|" + secLabel( - labels[(eqCounter + 1) % (endNum + 1)]) + "]]
\n") + append_text( + "
<< [[" + + secLabel( + labels[ + eqCounter - + 1]).replace( + " ", + "_") + + "|" + + secLabel( + labels[ + eqCounter - + 1]) + + "]]
\n") + append_text( + "
[[" + + secLabel( + sections[ + secCount + + 1][0]).replace( + " ", + "_") + + "#" + + secLabel( + labels[eqCounter][ + len("Formula:"):]) + + "|formula in " + + secLabel( + sections[ + secCount + + 1][0]) + + "]]
\n") + append_text( + "
[[" + + secLabel( + labels[ + (eqCounter + + 1) % + (endNum + + 1)].replace( + " ", + "_")) + + "|" + + secLabel( + labels[ + (eqCounter + + 1) % + (endNum + + 1)]) + + "]]
\n") append_text("
\n\n") append_text("
\n") @@ -579,8 +757,9 @@ def readin(ofname,glossary,mmd): parse = False math = False if hProof: - append_text("\n== Proof ==\n\nWe ask users to provide proof(s), reference(s) to proof(s), or" - " further clarification on the proof(s) in this space.\n") + append_text( + "\n== Proof ==\n\nWe ask users to provide proof(s), reference(s) to proof(s), or" + " further clarification on the proof(s) in this space.\n") append_text("\n== Symbols List ==\n\n") newSym = [] # if "09.07:04" in label: @@ -663,7 +842,12 @@ def readin(ofname,glossary,mmd): gFlag = False checkFlag = False get = False - gCSV = csv.reader(open(glossary, 'rb'), delimiter=',', quotechar='\"') + gCSV = csv.reader( + open( + glossary, + 'rb'), + delimiter=',', + quotechar='\"') preG = "" if symbol == "\\&": ampFlag = True @@ -691,7 +875,8 @@ def readin(ofname,glossary,mmd): parCx += 1 else: ArgCx += 1 - if G[0].find(symbol) == 0 and (len(G[0]) == len(symbol) or not G[0][len(symbol)].isalpha()): + if G[0].find(symbol) == 0 and (len(G[0]) == len( + symbol) or not G[0][len(symbol)].isalpha()): checkFlag = True get = True preG = S @@ -707,7 +892,8 @@ def readin(ofname,glossary,mmd): else: Q = symbolPar listArgs = [] - if len(Q) > len(symbol) and (Q[len(symbol)] == "{" or Q[len(symbol)] == "["): + if len(Q) > len(symbol) and (Q[ + len(symbol)] == "{" or Q[len(symbol)] == "["): ap = "" for o in range(len(symbol), len(Q)): if Q[o] == "}" or z == "]": @@ -719,7 +905,8 @@ def readin(ofname,glossary,mmd): web1 = G[5] for t in range(5, len(G)): if G[t] != "": - websiteF = websiteF + " [" + G[t] + " " + G[t] + "]" + websiteF = websiteF + \ + " [" + G[t] + " " + G[t] + "]" p1 = G[4].strip("$") p1 = "" + p1 + "" # if checkFlag: @@ -737,7 +924,8 @@ def readin(ofname,glossary,mmd): else: new2 += p2[k] p2 = new2 - finSym.append(web1 + " " + p1 + "] : " + p2 + " :" + websiteF) + finSym.append( + web1 + " " + p1 + "] : " + p2 + " :" + websiteF) break if not gFlag: del newSym[s] @@ -757,7 +945,8 @@ def readin(ofname,glossary,mmd): append_text("\n
\n") - append_text("\n== Bibliography==\n\n") # should there be a space between bibliography and ==? + # should there be a space between bibliography and ==? + append_text("\n== Bibliography==\n\n") r = unmodLabel(labels[eqCounter]) q = r.find("KLS:") + 4 p = r.find(":", q) @@ -765,38 +954,132 @@ def readin(ofname,glossary,mmd): equation = r[p + 1:] if equation.find(":") != -1: equation = equation[0:equation.find(":")] - append_text("[http://homepage.tudelft.nl/11r49/askey/contents.html " - "Equation in Section " + section + "] of [[Bibliography#KLS|'''KLS''']].\n\n") - append_text("== URL links ==\n\nWe ask users to provide relevant URL links in this space.\n\n") + append_text( + "[http://homepage.tudelft.nl/11r49/askey/contents.html " + "Equation in Section " + + section + + "] of [[Bibliography#KLS|'''KLS''']].\n\n") + append_text( + "== URL links ==\n\nWe ask users to provide relevant URL links in this space.\n\n") if eqCounter < endNum: append_text("
\n") if newSec: newSec = False - append_text("
<< [[" + secLabel(sections[secCount][0]).replace(" ", "_") + - "|" + secLabel(sections[secCount][0]) + "]]
\n") + append_text( + "
<< [[" + + secLabel( + sections[secCount][0]).replace( + " ", + "_") + + "|" + + secLabel( + sections[secCount][0]) + + "]]
\n") else: - append_text("
<< [[" + secLabel(labels[eqCounter - 1]).replace(" ", "_") + - "|" + secLabel(labels[eqCounter - 1]) + "]]
\n") - append_text("
[[" + secLabel(sections[secCount + 1][0]).replace(" ", "_") + - "#" + secLabel(labels[eqCounter][len("Formula:"):]) + "|formula in " + - secLabel(sections[secCount + 1][0]) + "]]
\n") + append_text( + "
<< [[" + + secLabel( + labels[ + eqCounter - + 1]).replace( + " ", + "_") + + "|" + + secLabel( + labels[ + eqCounter - + 1]) + + "]]
\n") + append_text( + "
[[" + + secLabel( + sections[ + secCount + + 1][0]).replace( + " ", + "_") + + "#" + + secLabel( + labels[eqCounter][ + len("Formula:"):]) + + "|formula in " + + secLabel( + sections[ + secCount + + 1][0]) + + "]]
\n") if eqS == sections[secCount][1]: - append_text("
[[" + - sections[(secCount + 1) % len(sections)][0].replace(" ", "_") + "|" + - sections[(secCount + 1) % len(sections)][0] + "]] >>
\n") + append_text( + "
[[" + + sections[ + (secCount + + 1) % + len(sections)][0].replace( + " ", + "_") + + "|" + + sections[ + (secCount + + 1) % + len(sections)][0] + + "]] >>
\n") else: - append_text("
[[" + - secLabel(labels[(eqCounter + 1) % (endNum + 1)]).replace(" ", "_") + "|" + - secLabel(labels[(eqCounter + 1) % (endNum + 1)]) + "]] >>
\n") + append_text( + "
[[" + + secLabel( + labels[ + (eqCounter + + 1) % + (endNum + + 1)]).replace( + " ", + "_") + + "|" + + secLabel( + labels[ + (eqCounter + + 1) % + (endNum + + 1)]) + + "]] >>
\n") append_text("
\n") else: # FOR EXTRA EQUATIONS append_text("
\n") - append_text("
<< [[" + labels[endNum - 1].replace(" ", "_") + "|" + labels[ - endNum - 1] + "]]
\n") - append_text("
[[" + labels[0].replace(" ", "_") + "#" + labels[endNum][8:] + - "|formula in " + labels[0] + "]]
\n") - append_text("
[[" + labels[0 % endNum].replace(" ", "_") + "|" + labels[ - 0 % endNum] + "]]
\n") + append_text( + "
<< [[" + + labels[ + endNum - + 1].replace( + " ", + "_") + + "|" + + labels[ + endNum - + 1] + + "]]
\n") + append_text( + "
[[" + + labels[0].replace( + " ", + "_") + + "#" + + labels[endNum][ + 8:] + + "|formula in " + + labels[0] + + "]]
\n") + append_text( + "
[[" + + labels[ + 0 % + endNum].replace( + " ", + "_") + + "|" + + labels[ + 0 % + endNum] + + "]]
\n") append_text("
\n") elif "\\constraint" in line and parse: # symbols=symbols+getSym(line) @@ -818,7 +1101,8 @@ def readin(ofname,glossary,mmd): subLine = "" elif "\\drmfname" in line and parse: math = False - comToWrite = "\n== Name ==\n\n
" + getString(line) + "

\n" + comToWrite + comToWrite = "\n== Name ==\n\n
" + \ + getString(line) + "

\n" + comToWrite elif "\\drmfnote" in line and parse: symbols = symbols + getSym(line) @@ -846,21 +1130,29 @@ def readin(ofname,glossary,mmd): # TODO: figure out how eqR is defined # rLab = getString(eqR) pause = True - eInd = refLabels.index("" + label) # This should be rLab + eInd = refLabels.index( + "" + label) # This should be rLab z = line[line.find("}", ind + 7) + 1] if z == "." or z == ",": pauseP = True - proofLine += ("
\n" + refEqs[ - eInd] + "" + z + "
\n") + proofLine += ("
\n" + + refEqs[eInd] + + "" + + z + + "
\n") else: if z == "}": proofLine += ( "
\n" + refEqs[ eInd] + "
") else: - proofLine += ( - "
\n \n" + refEqs[ - eInd] + "
\n") + proofLine += ("
\n \n" + + refEqs[eInd] + + "
\n") else: if pause: if line[ind] == "}": @@ -873,7 +1165,10 @@ def readin(ofname,glossary,mmd): proofLine += (line[ind]) if "\\end{equation}" in lines[i + 1]: proof = False - append_text(comToWrite + getEqP(proofLine) + "
\n
\n") + append_text( + comToWrite + + getEqP(proofLine) + + "
\n
\n") comToWrite = "" symbols = symbols + getSym(symLine) symLine = "" @@ -891,12 +1186,19 @@ def readin(ofname,glossary,mmd): z = line[line.find("}", ind + 7) + 1] if z == "." or z == ",": pauseP = True - proofLine += ("
\n" + refEqs[ - eInd] + "" + z + "
\n") + proofLine += ("
\n" + + refEqs[eInd] + + "" + + z + + "
\n") else: - proofLine += ( - "
\n" + refEqs[ - eInd] + "
\n") + proofLine += ("
\n" + + refEqs[eInd] + + "
\n") else: if pause: @@ -911,16 +1213,28 @@ def readin(ofname,glossary,mmd): proofLine += (line[ind]) if "\\end{equation}" in lines[i + 1]: proof = False - append_text(comToWrite + getEqP(proofLine).rstrip("\n") + "
\n
\n") + append_text( + comToWrite + + getEqP(proofLine).rstrip("\n") + + "
\n
\n") comToWrite = "" symbols = symbols + getSym(symLine) symLine = "" elif math: - if "\\end{equation}" in lines[i + 1] or "\\constraint" in lines[i + 1] or "\\substitution" in lines[ - i + 1] or "\\proof" in lines[i + 1] or "\\drmfnote" in lines[i + 1] or "\\drmfname" in \ - lines[ - i + 1]: + if "\\end{equation}" in lines[ + i + + 1] or "\\constraint" in lines[ + i + + 1] or "\\substitution" in lines[ + i + + 1] or "\\proof" in lines[ + i + + 1] or "\\drmfnote" in lines[ + i + + 1] or "\\drmfname" in lines[ + i + + 1]: append_text(line.rstrip("\n")) symLine += line.strip("\n") symbols = symbols + getSym(symLine) @@ -932,37 +1246,69 @@ def readin(ofname,glossary,mmd): if note and parse: noteLine += line symbols = symbols + getSym(line) - if "\\end{equation}" in lines[i + 1] or "\\drmfn" in lines[i + 1] or "\\constraint" in lines[ - i + 1] or "\\substitution" in lines[i + 1] or "\\proof" in lines[i + 1]: + if "\\end{equation}" in lines[ + i + + 1] or "\\drmfn" in lines[ + i + + 1] or "\\constraint" in lines[ + i + + 1] or "\\substitution" in lines[ + i + + 1] or "\\proof" in lines[ + i + + 1]: note = False if "\\emph" in noteLine: - noteLine = noteLine[0:noteLine.find("\\emph{")] + "\'\'" + noteLine[noteLine.find("\\emph{") + len( - "\\emph{"):noteLine.find("}", noteLine.find("\\emph{") + len("\\emph{"))] + "\'\'" + noteLine[ - noteLine.find( - "}", - noteLine.find( - "\\emph{") + len( - "\\emph{")) + 1:] - comToWrite = comToWrite + "
" + getEq(noteLine) + "

\n" + noteLine = noteLine[ + 0:noteLine.find("\\emph{")] + "\'\'" + noteLine[ + noteLine.find("\\emph{") + len("\\emph{"):noteLine.find( + "}", noteLine.find("\\emph{") + len("\\emph{"))] + "\'\'" + noteLine[ + noteLine.find( + "}", noteLine.find("\\emph{") + len("\\emph{")) + 1:] + comToWrite = comToWrite + "
" + \ + getEq(noteLine) + "

\n" if constraint and parse: conLine += line.replace("&", "&
") symLine += line.strip("\n") # symbols=symbols+getSym(line) - if "\\end{equation}" in lines[i + 1] or "\\drmfn" in lines[i + 1] or "\\constraint" in lines[ - i + 1] or "\\substitution" in lines[i + 1] or "\\proof" in lines[i + 1]: + if "\\end{equation}" in lines[ + i + + 1] or "\\drmfn" in lines[ + i + + 1] or "\\constraint" in lines[ + i + + 1] or "\\substitution" in lines[ + i + + 1] or "\\proof" in lines[ + i + + 1]: constraint = False symbols = symbols + getSym(symLine) symLine = "" - append_text(comToWrite + "
" + getEq(conLine) + "

\n") + append_text( + comToWrite + + "
" + + getEq(conLine) + + "

\n") comToWrite = "" if substitution and parse: - subLine += line.replace("&", "&
") #TODO: Figure out if .replace is needed + # TODO: Figure out if .replace is needed + subLine += line.replace("&", "&
") symLine += line.strip("\n") - if "\\end{equation}" in lines[i + 1] or "\\drmfn" in lines[i + 1] or "\\substitution" in lines[ - i + 1] or "\\constraint" in lines[i + 1] or "\\proof" in lines[i + 1]: + if "\\end{equation}" in lines[ + i + + 1] or "\\drmfn" in lines[ + i + + 1] or "\\substitution" in lines[ + i + + 1] or "\\constraint" in lines[ + i + + 1] or "\\proof" in lines[ + i + + 1]: substitution = False symbols = symbols + getSym(symLine) symLine = "" @@ -973,7 +1319,11 @@ def readin(ofname,glossary,mmd): lineR += "&
" else: lineR += subLine[i] - append_text(comToWrite + "
" + getEq(subLine) + "

\n") + append_text( + comToWrite + + "
" + + getEq(subLine) + + "

\n") comToWrite = "" From 9e82fb6c19b280175f76ea986f9641edc2e1379d Mon Sep 17 00:00:00 2001 From: Parth Oza Date: Tue, 19 Jul 2016 10:55:59 -0400 Subject: [PATCH 161/402] Resolve some QuantifyCode issues --- DLMF_preprocessing/src/conversion_ecf.py | 64 ++++++++++++------------ macro_replacement/src/drmfconfig.py | 1 - macro_replacement/src/monics.py | 1 - 3 files changed, 32 insertions(+), 34 deletions(-) diff --git a/DLMF_preprocessing/src/conversion_ecf.py b/DLMF_preprocessing/src/conversion_ecf.py index c108fc1..724d786 100644 --- a/DLMF_preprocessing/src/conversion_ecf.py +++ b/DLMF_preprocessing/src/conversion_ecf.py @@ -4,7 +4,7 @@ import re # existing file named text.txt -file = open('test1.txt', 'r').read() +existing_file = open('test1.txt', 'r').read() # output written to newText.txt newFile = open('newIdentities.txt', 'w') @@ -568,40 +568,40 @@ def equationSetUp(s): return (newFile + "\\end {equation} \n\n") -file = removeInactive(file) -file = replaceGreekVars(file) -file = replaceSqrt(file) -file = replacePolygamma(file) -file = replaceCos(file) -file = replaceSin(file) -file = replaceTan(file) -file = replaceCsc(file) -file = replaceSec(file) -file = replaceCot(file) -file = replaceArcCos(file) -file = replaceArcSin(file) -file = replaceArcTan(file) -file = replaceArcCsc(file) -file = replaceArcSec(file) -file = replaceArcCot(file) -file = replaceCosh(file) -file = replaceSinh(file) -file = replaceTanh(file) -file = replaceCsch(file) -file = replaceSech(file) -file = replaceCoth(file) -file = replaceInfinity(file) -file = ContinuedFractionK(file) -file = replaceAbs(file) -file = comparativeRelators(file) +existing_file = removeInactive(existing_file) +existing_file = replaceGreekVars(existing_file) +existing_file = replaceSqrt(existing_file) +existing_file = replacePolygamma(existing_file) +existing_file = replaceCos(existing_file) +existing_file = replaceSin(existing_file) +existing_file = replaceTan(existing_file) +existing_file = replaceCsc(existing_file) +existing_file = replaceSec(existing_file) +existing_file = replaceCot(existing_file) +existing_file = replaceArcCos(existing_file) +existing_file = replaceArcSin(existing_file) +existing_file = replaceArcTan(existing_file) +existing_file = replaceArcCsc(existing_file) +existing_file = replaceArcSec(existing_file) +existing_file = replaceArcCot(existing_file) +existing_file = replaceCosh(existing_file) +existing_file = replaceSinh(existing_file) +existing_file = replaceTanh(existing_file) +existing_file = replaceCsch(existing_file) +existing_file = replaceSech(existing_file) +existing_file = replaceCoth(existing_file) +existing_file = replaceInfinity(existing_file) +existing_file = ContinuedFractionK(existing_file) +existing_file = replaceAbs(existing_file) +existing_file = comparativeRelators(existing_file) count = 0 position = 0 yorn = False -for index in range(len(file)): - if file[index: index + 2] == "(*": +for index in range(len(existing_file)): + if existing_file[index: index + 2] == "(*": count = index - position = file.find("\n", index) - position = file.find("\n", position + 1) - newFile.write(equationSetUp(file[index: position]).replace("*", " ")) + position = existing_file.find("\n", index) + position = existing_file.find("\n", position + 1) + newFile.write(equationSetUp(existing_file[index: position]).replace("*", " ")) newFile.close() diff --git a/macro_replacement/src/drmfconfig.py b/macro_replacement/src/drmfconfig.py index 4670d48..c0b09a2 100644 --- a/macro_replacement/src/drmfconfig.py +++ b/macro_replacement/src/drmfconfig.py @@ -3,7 +3,6 @@ import os import sys -from snippets import * from function import Function diff --git a/macro_replacement/src/monics.py b/macro_replacement/src/monics.py index a919e5f..80cb3cb 100644 --- a/macro_replacement/src/monics.py +++ b/macro_replacement/src/monics.py @@ -4,7 +4,6 @@ import re import sys -global fullmacro fullmacro = '' From e3cd35bddc56cfe17204b1f72ea1669681228dd5 Mon Sep 17 00:00:00 2001 From: Parth Oza Date: Tue, 19 Jul 2016 13:01:16 -0400 Subject: [PATCH 162/402] Remove wildcard imports (which apparently weren't used anyway) --- KLSadd_insertion/src/FileChangerLatex.py | 6 +++--- macro_replacement/src/drmfconfig.py | 3 +++ macro_replacement/src/monics.py | 3 +++ macro_replacement/src/normalized.py | 3 ++- macro_replacement/src/replace.py | 5 ++++- 5 files changed, 15 insertions(+), 5 deletions(-) diff --git a/KLSadd_insertion/src/FileChangerLatex.py b/KLSadd_insertion/src/FileChangerLatex.py index f0ed347..6fe754e 100644 --- a/KLSadd_insertion/src/FileChangerLatex.py +++ b/KLSadd_insertion/src/FileChangerLatex.py @@ -15,7 +15,7 @@ subsection = "" a = -1 chapter = "" -list = [] +chapters = [] with open("KLSadd.tex", "rb") as file: for line in file: # finds all mentions of subsections in KLSadd @@ -37,7 +37,7 @@ chapter += "." if chapter is not "": - list.append(chapter) + chapters.append(chapter) # END OF LOOP why can't I have nice brackets like java :( """ list contains all of the subsections. Begin chapter file searching @@ -49,7 +49,7 @@ print(words) numFile = "" fileList = [] - for s in list: + for s in chapters: numFile = "" if ("." in s[0:2]): numFile = s[0] diff --git a/macro_replacement/src/drmfconfig.py b/macro_replacement/src/drmfconfig.py index c0b09a2..fd9d6c7 100644 --- a/macro_replacement/src/drmfconfig.py +++ b/macro_replacement/src/drmfconfig.py @@ -5,6 +5,9 @@ import sys from function import Function +# If there are errors it probably has to do with the missing import below +# from snippets import * + class ConfigFile(object): """Represents a configuration file (containing regular expressions) used for seeding the DRMF""" diff --git a/macro_replacement/src/monics.py b/macro_replacement/src/monics.py index 80cb3cb..d7f57e2 100644 --- a/macro_replacement/src/monics.py +++ b/macro_replacement/src/monics.py @@ -4,6 +4,9 @@ import re import sys +# If there are errors it probably has to do with the missing import below +# from snippets import * + fullmacro = '' diff --git a/macro_replacement/src/normalized.py b/macro_replacement/src/normalized.py index 38a82bc..45ee7ad 100644 --- a/macro_replacement/src/normalized.py +++ b/macro_replacement/src/normalized.py @@ -3,7 +3,8 @@ import re import sys -from snippets import * +# If there are errors it probably has to do with the missing import below +# from snippets import * def main(): diff --git a/macro_replacement/src/replace.py b/macro_replacement/src/replace.py index 8831861..c53a64a 100644 --- a/macro_replacement/src/replace.py +++ b/macro_replacement/src/replace.py @@ -8,7 +8,10 @@ import sys import re import os -from snippets import * + +# If there are errors it probably has to do with the missing import below +# from snippets import * + from function import Function from replace_special import remove_special from monics import replace_monics From c79f80343d37d99af6bbb31ac99113e8101e2021 Mon Sep 17 00:00:00 2001 From: Parth Oza Date: Tue, 19 Jul 2016 13:10:11 -0400 Subject: [PATCH 163/402] Fix some more QuantifyCode issues --- DLMF_preprocessing/src/conversion_ecf.py | 82 ++++++++++++------------ tex2Wiki/src/DLMFtex2Wiki.py | 6 +- tex2Wiki/src/tex2Wiki.py | 6 +- 3 files changed, 46 insertions(+), 48 deletions(-) diff --git a/DLMF_preprocessing/src/conversion_ecf.py b/DLMF_preprocessing/src/conversion_ecf.py index 724d786..753a7a3 100644 --- a/DLMF_preprocessing/src/conversion_ecf.py +++ b/DLMF_preprocessing/src/conversion_ecf.py @@ -137,7 +137,7 @@ def findArgs(s, functionName): def replaceSqrt(s): arguments = findArgs(s, "Sqrt") for i in arguments: - s = s.replace("Sqrt" + i, "\\sqrt{" + i[1:len(i) - 1] + "}") + s = s.replace("Sqrt" + i, "\\sqrt{" + i[1:-1] + "}") return (s) @@ -150,20 +150,18 @@ def replacePolygamma(s): i, "\\polygamma{" + i[1:commaLoc] + "}@{" + - i[commaLoc + - 1:len(i) - - 1] + + i[commaLoc + 1:-1] + "}") elif i[1:2] == "0": s = s.replace("Polygamma" + i, - "\\digamma@{" + i[3:len(i) - 1] + "}") + "\\digamma@{" + i[3:-1] + "}") return s def removeInactive(s): arguments = findArgs(s, "Inactive") for i in arguments: - s = s.replace("Inactive" + i, i[1:len(i) - 1]) + s = s.replace("Inactive" + i, i[1:-1]) return s @@ -171,9 +169,9 @@ def replaceCos(s): arguments = findArgs(s, "Cos") for i in arguments: if len(i) == 3: - s = s.replace("Cos" + i, "\\cos@@{" + i[1:len(i) - 1] + "}") + s = s.replace("Cos" + i, "\\cos@@{" + i[1:-1] + "}") else: - s = s.replace("Cos" + i, "\\cos@{" + i[1:len(i) - 1] + "}") + s = s.replace("Cos" + i, "\\cos@{" + i[1:-1] + "}") return s @@ -181,9 +179,9 @@ def replaceSin(s): arguments = findArgs(s, "Sin") for i in arguments: if len(i) == 3: - s = s.replace("Sin" + i, "\\sin@@{" + i[1:len(i) - 1] + "}") + s = s.replace("Sin" + i, "\\sin@@{" + i[1:-1] + "}") else: - s = s.replace("Sin" + i, "\\sin@{" + i[1:len(i) - 1] + "}") + s = s.replace("Sin" + i, "\\sin@{" + i[1:-1] + "}") return s @@ -191,9 +189,9 @@ def replaceTan(s): arguments = findArgs(s, "Tan") for i in arguments: if len(i) == 3: - s = s.replace("Tan" + i, "\\tan@@{" + i[1:len(i) - 1] + "}") + s = s.replace("Tan" + i, "\\tan@@{" + i[1:-1] + "}") else: - s = s.replace("Tan" + i, "\\tan@{" + i[1:len(i) - 1] + "}") + s = s.replace("Tan" + i, "\\tan@{" + i[1:-1] + "}") return s @@ -201,9 +199,9 @@ def replaceCsc(s): arguments = findArgs(s, "Csc") for i in arguments: if len(i) == 3: - s = s.replace("Csc" + i, "\\csc@@{" + i[1:len(i) - 1] + "}") + s = s.replace("Csc" + i, "\\csc@@{" + i[1:-1] + "}") else: - s = s.replace("Csc" + i, "\\csc@{" + i[1:len(i) - 1] + "}") + s = s.replace("Csc" + i, "\\csc@{" + i[1:-1] + "}") return s @@ -211,9 +209,9 @@ def replaceSec(s): arguments = findArgs(s, "Sec") for i in arguments: if len(i) == 3: - s = s.replace("Sec" + i, "\\sec@@{" + i[1:len(i) - 1] + "}") + s = s.replace("Sec" + i, "\\sec@@{" + i[1:-1] + "}") else: - s = s.replace("Sec" + i, "\\sec@{" + i[1:len(i) - 1] + "}") + s = s.replace("Sec" + i, "\\sec@{" + i[1:-1] + "}") return s @@ -221,9 +219,9 @@ def replaceCot(s): arguments = findArgs(s, "Cot") for i in arguments: if len(i) == 3: - s = s.replace("Cot" + i, "\\cot@@{" + i[1:len(i) - 1] + "}") + s = s.replace("Cot" + i, "\\cot@@{" + i[1:-1] + "}") else: - s = s.replace("Cot" + i, "\\cot@{" + i[1:len(i) - 1] + "}") + s = s.replace("Cot" + i, "\\cot@{" + i[1:-1] + "}") return s @@ -231,9 +229,9 @@ def replaceCosh(s): arguments = findArgs(s, "Cosh") for i in arguments: if len(i) == 3: - s = s.replace("Cosh" + i, "\\cosh@@{" + i[1:len(i) - 1] + "}") + s = s.replace("Cosh" + i, "\\cosh@@{" + i[1:-1] + "}") else: - s = s.replace("Cosh" + i, "\\cosh@{" + i[1:len(i) - 1] + "}") + s = s.replace("Cosh" + i, "\\cosh@{" + i[1:-1] + "}") return s @@ -241,9 +239,9 @@ def replaceSinh(s): arguments = findArgs(s, "Sinh") for i in arguments: if len(i) == 3: - s = s.replace("Sinh" + i, "\\sinh@@{" + i[1:len(i) - 1] + "}") + s = s.replace("Sinh" + i, "\\sinh@@{" + i[1:-1] + "}") else: - s = s.replace("Sinh" + i, "\\sinh@{" + i[1:len(i) - 1] + "}") + s = s.replace("Sinh" + i, "\\sinh@{" + i[1:-1] + "}") return s @@ -251,9 +249,9 @@ def replaceTanh(s): arguments = findArgs(s, "Tanh") for i in arguments: if len(i) == 3: - s = s.replace("Tanh" + i, "\\tanh@@{" + i[1:len(i) - 1] + "}") + s = s.replace("Tanh" + i, "\\tanh@@{" + i[1:-1] + "}") else: - s = s.replace("Tanh" + i, "\\tanh@{" + i[1:len(i) - 1] + "}") + s = s.replace("Tanh" + i, "\\tanh@{" + i[1:-1] + "}") return s @@ -261,9 +259,9 @@ def replaceCsch(s): arguments = findArgs(s, "Csch") for i in arguments: if len(i) == 3: - s = s.replace("Csch" + i, "\\csch@@{" + i[1:len(i) - 1] + "}") + s = s.replace("Csch" + i, "\\csch@@{" + i[1:-1] + "}") else: - s = s.replace("Csch" + i, "\\csch@{" + i[1:len(i) - 1] + "}") + s = s.replace("Csch" + i, "\\csch@{" + i[1:-1] + "}") return s @@ -271,9 +269,9 @@ def replaceSech(s): arguments = findArgs(s, "Sech") for i in arguments: if len(i) == 3: - s = s.replace("Sech" + i, "\\sech@@{" + i[1:len(i) - 1] + "}") + s = s.replace("Sech" + i, "\\sech@@{" + i[1:-1] + "}") else: - s = s.replace("Sech" + i, "\\sech@{" + i[1:len(i) - 1] + "}") + s = s.replace("Sech" + i, "\\sech@{" + i[1:-1] + "}") return s @@ -281,9 +279,9 @@ def replaceCoth(s): arguments = findArgs(s, "Coth") for i in arguments: if len(i) == 3: - s = s.replace("Coth" + i, "\\coth@@{" + i[1:len(i) - 1] + "}") + s = s.replace("Coth" + i, "\\coth@@{" + i[1:-1] + "}") else: - s = s.replace("Coth" + i, "\\coth@{" + i[1:len(i) - 1] + "}") + s = s.replace("Coth" + i, "\\coth@{" + i[1:-1] + "}") return s @@ -291,9 +289,9 @@ def replaceArcCos(s): arguments = findArgs(s, "ArcCos") for i in arguments: if len(i) == 3: - s = s.replace("ArcCos" + i, "\\acos@@{" + i[1:len(i) - 1] + "}") + s = s.replace("ArcCos" + i, "\\acos@@{" + i[1:-1] + "}") else: - s = s.replace("ArcCos" + i, "\\acos@{" + i[1:len(i) - 1] + "}") + s = s.replace("ArcCos" + i, "\\acos@{" + i[1:-1] + "}") return s @@ -301,9 +299,9 @@ def replaceArcSin(s): arguments = findArgs(s, "ArcSin") for i in arguments: if len(i) == 3: - s = s.replace("ArcSin" + i, "\\asin@@{" + i[1:len(i) - 1] + "}") + s = s.replace("ArcSin" + i, "\\asin@@{" + i[1:-1] + "}") else: - s = s.replace("ArcSin" + i, "\\asin@{" + i[1:len(i) - 1] + "}") + s = s.replace("ArcSin" + i, "\\asin@{" + i[1:-1] + "}") return s @@ -311,9 +309,9 @@ def replaceArcTan(s): arguments = findArgs(s, "ArcTan") for i in arguments: if len(i) == 3: - s = s.replace("ArcTan" + i, "\\atan@@{" + i[1:len(i) - 1] + "}") + s = s.replace("ArcTan" + i, "\\atan@@{" + i[1:-1] + "}") else: - s = s.replace("ArcTan" + i, "\\atan@{" + i[1:len(i) - 1] + "}") + s = s.replace("ArcTan" + i, "\\atan@{" + i[1:-1] + "}") return s @@ -321,9 +319,9 @@ def replaceArcCsc(s): arguments = findArgs(s, "ArcCsc") for i in arguments: if len(i) == 3: - s = s.replace("ArcCsc" + i, "\\acsc@@{" + i[1:len(i) - 1] + "}") + s = s.replace("ArcCsc" + i, "\\acsc@@{" + i[1:-1] + "}") else: - s = s.replace("ArcCsc" + i, "\\acsc@{" + i[1:len(i) - 1] + "}") + s = s.replace("ArcCsc" + i, "\\acsc@{" + i[1:-1] + "}") return s @@ -331,9 +329,9 @@ def replaceArcSec(s): arguments = findArgs(s, "ArcSec") for i in arguments: if len(i) == 3: - s = s.replace("ArcSec" + i, "\\asec@@{" + i[1:len(i) - 1] + "}") + s = s.replace("ArcSec" + i, "\\asec@@{" + i[1:-1] + "}") else: - s = s.replace("ArcSec" + i, "\\asec@{" + i[1:len(i) - 1] + "}") + s = s.replace("ArcSec" + i, "\\asec@{" + i[1:-1] + "}") return s @@ -341,9 +339,9 @@ def replaceArcCot(s): arguments = findArgs(s, "ArcCot") for i in arguments: if len(i) == 3: - s = s.replace("ArcCot" + i, "\\acot@@{" + i[1:len(i) - 1] + "}") + s = s.replace("ArcCot" + i, "\\acot@@{" + i[1:-1] + "}") else: - s = s.replace("ArcCot" + i, "\\acot@{" + i[1:len(i) - 1] + "}") + s = s.replace("ArcCot" + i, "\\acot@{" + i[1:-1] + "}") return s diff --git a/tex2Wiki/src/DLMFtex2Wiki.py b/tex2Wiki/src/DLMFtex2Wiki.py index 9b2577c..f53aa29 100644 --- a/tex2Wiki/src/DLMFtex2Wiki.py +++ b/tex2Wiki/src/DLMFtex2Wiki.py @@ -84,7 +84,7 @@ def DLMF(ofname, mmd, llinks, n): refEqs.append("") math = True elif math: - refEqs[len(refEqs) - 1] += line + refEqs[-1] += line if "\\end{equation}" in refLines[ i + 1] or "\\constraint" in refLines[ @@ -303,7 +303,7 @@ def DLMF(ofname, mmd, llinks, n): "

\n") elif math and parse: flagM = True - eqs[len(eqs) - 1] += line + eqs[-1] += line if not ( (not ( @@ -342,7 +342,7 @@ def DLMF(ofname, mmd, llinks, n): else: append_text(line) elif math and not parse: - eqs[len(eqs) - 1] += line + eqs[-1] += line if "\\end{equation}" in lines[ i + 1] or "\\constraint" in lines[ diff --git a/tex2Wiki/src/tex2Wiki.py b/tex2Wiki/src/tex2Wiki.py index 5bb0c24..c5e70a2 100644 --- a/tex2Wiki/src/tex2Wiki.py +++ b/tex2Wiki/src/tex2Wiki.py @@ -336,7 +336,7 @@ def readin(ofname, glossary, mmd): refEqs.append("") math = True elif math: - refEqs[len(refEqs) - 1] += line + refEqs[-1] += line if "\\end{equation}" in refLines[ i + 1] or "\\constraint" in refLines[ @@ -462,7 +462,7 @@ def readin(ofname, glossary, mmd): "

\n") elif math and parse: flagM = True - eqs[len(eqs) - 1] += line + eqs[-1] += line if "\\end{equation}" in lines[ i + @@ -499,7 +499,7 @@ def readin(ofname, glossary, mmd): else: append_text(line) elif math and not parse: - eqs[len(eqs) - 1] += line + eqs[-1] += line if "\\end{equation}" in lines[ i + 1] or "\\constraint" in lines[ From 250a46b8aa69922daf9406f04a69916da4dfac64 Mon Sep 17 00:00:00 2001 From: Joon Bang Date: Tue, 19 Jul 2016 14:02:49 -0400 Subject: [PATCH 164/402] Add unit tests --- maple2latex/README.md | 0 maple2latex/data/keys.json | 966 +++++++++--------- maple2latex/src/__init__.py | 0 maple2latex/src/classes.py | 79 -- maple2latex/src/main.py | 30 +- maple2latex/src/maple_tokenize.py | 2 + .../{translation_methods.py => translator.py} | 119 ++- 7 files changed, 595 insertions(+), 601 deletions(-) create mode 100644 maple2latex/README.md create mode 100644 maple2latex/src/__init__.py delete mode 100644 maple2latex/src/classes.py rename maple2latex/src/{translation_methods.py => translator.py} (68%) diff --git a/maple2latex/README.md b/maple2latex/README.md new file mode 100644 index 0000000..e69de29 diff --git a/maple2latex/data/keys.json b/maple2latex/data/keys.json index f0c8da9..a7ea2f8 100644 --- a/maple2latex/data/keys.json +++ b/maple2latex/data/keys.json @@ -1,485 +1,485 @@ { - "functions": { - "sqrt": [ - { - "args": 1, - "repr": ["\\sqrt{", "}"] - } - ], - - "sin": [ - { - "args": 1, - "repr": ["\\sin(", ")"] - } - ], - - "cos": [ - { - "args": 1, - "repr": ["\\cos(", ")"] - } - ], - - "tan": [ - { - "args": 1, - "repr": ["\\tan(", ")"] - } - ], - - "abs": [ - { - "args": 1, - "repr": ["\\abs{", "}"] - } - ], - - "exp": [ - { - "args": 1, - "repr": ["e^{", "}"] - } - ], - - "log": [ - { - "args": 1, - "repr": ["\\log{(", ")}"] - }, - - { - "args": 2, - "repr": ["\\log_{", "} (", ")"] - } - ], - - "ln": [ - { - "args": 1, - "repr": ["\\ln{(", ")}"] - } - ], - - "floor": [ - { - "args": 1, - "repr": ["\\floor{", "}"] - } - ], - - "sum": [ - { - "args": 3, - "repr": ["\\sum_{", "}^{", "} (", ")"] - } - ], - - "GAMMA": [ - { - "args": 1, - "repr": ["\\EulerGamma@{", "}"] - } - ], - - "bernoulli": [ - { - "args": 1, - "repr": ["\\BernoulliB{", "}"] - } - ], - - "BesselI": [ - { - "args": 2, - "repr": ["\\BesselI{", "}@{", "}"] - } - ], - - "BesselJ": [ - { - "args": 2, - "repr": ["\\BesselJ{", "}@{", "}"] - } - ], - - "BesselK": [ - { - "args": 2, - "repr": ["\\BesselK{", "}@{", "}"] - } - ], - - "BesselY": [ - { - "args": 2, - "repr": ["\\BesselY{", "}@{", "}"] - } - ], - - "dawson": [ - { - "args": 1, - "repr": ["\\DawsonsInt@{", "}"] - } - ], - - "Ei": [ - { - "args": 2, - "repr": ["\\ExpIntn{", "}@{", "}"] - }, - - { - "args": 1, - "repr": ["\\ExpInt@{", "}"] - } - ], - - "erfc": [ - { - "args": 1, - "repr": ["\\erfc@@{(", ")}"] - } - ], - - "erf": [ - { - "args": 1, - "repr": ["\\erf@@{(", ")}"] - } - ], - - "FresnelC": [ - { - "args": 1, - "repr": ["\\FresnelCos@{", "}"] - } - ], - - "FresnelS": [ - { - "args": 1, - "repr": ["\\FresnelSin@{", "}"] - } - ], - - "HankelH1": [ - { - "args": 2, - "repr": ["\\HankelHi{", "}@{", "}"] - } - ], - - "KummerU": [ - { - "args": 3, - "repr": ["\\KummerU@{", "}{", "}{", "}"] - } - ], - - "CylinderU": [ - { - "args": 2, - "repr": ["\\ParabolicU@{", "}{", "}"] - } - ], - - "Psi": [ - { - "args": 2, - "repr": ["\\polygamma{", "}@{", "}"] - }, - - { - "args": 1, - "repr": ["\\digamma@{", "}"] - } - ], - - "WhittakerW": [ - { - "args": 3, - "repr": ["\\WhitW{", "}{", "}@{", "}"] - } - ], - - "Zeta": [ - { - "args": 1, - "repr": ["\\RiemannZeta@{", "}"] - } - ], - - "arccos": [ - { - "args": 1, - "repr": ["\\Acos@@{(", ")}"] - } - ], - - "arccosh": [ - { - "args": 1, - "repr": ["\\Acosh@@{(", ")}"] - } - ], - - "arcsin": [ - { - "args": 1, - "repr": ["\\Asin@@{(", ")}"] - } - ], - - "arcsinh": [ - { - "args": 1, - "repr": ["\\Asinh@@{(", ")}"] - } - ], - - "arctan": [ - { - "args": 1, - "repr": ["\\Atan@@{(", ")}"] - } - ], - - "arctanh": [ - { - "args": 1, - "repr": ["\\Atanh@@{(", ")}"] - } - ], - - "sinh": [ - { - "args": 1, - "repr": ["\\sinh@@{(", ")}"] - } - ], - - "cosh": [ - { - "args": 1, - "repr": ["\\cosh@@{(", ")}"] - } - ], - - "coth": [ - { - "args": 1, - "repr": ["\\coth@@{(", ")}"] - } - ], - - "tanh": [ - { - "args": 1, - "repr": ["\\tanh@@{(", ")}"] - } - ], - - "Re": [ - { - "args": 1, - "repr": ["\\Re(", ")"] - } - ], - - "Im": [ - { - "args": 1, - "repr": ["\\Im(", ")"] - } - ], - - "doublefactorial": [ - { - "args": 1, - "repr": ["(", ")!!"] - } - ], - - "hypergeom": [ - { - "args": 5, - "repr": ["\\HyperpFq{", "}{", "}@@{", "}{", "}{", "}"] - } - ], - - "pochhammer": [ - { - "args": 2, - "repr": ["\\pochhammer{", "}{", "}"] - } - ], - - "Besseli": [ - { - "args": 2, - "repr": ["\\SphBesselIi{", "}@{", "}"] - } - ], - - "Besselj": [ - { - "args": 2, - "repr": ["\\SphBesselJ{", "}@{", "}"] - } - ], - - "cerf": [ - { - "args": 1, - "repr": ["\\erfw@@{(", ")}"] - } - ], - - "Ein": [ - { - "args": 1, - "repr": ["\\ExpIntEin@{", "}"] - } - ], - - "erfcI": [ - { - "args": 2, - "repr": ["I^{", "} \\erfc@@{(", ")}"] - } - ], - - "hankelsymb": [ - { - "args": 2, - "repr": ["\\HankelSym{", "}{", "}"] - } - ], - - "lincgamma": [ - { - "args": 2, - "repr": ["\\incgamma@{", "}{", "}"] - } - ], - - "qhyper": [ - { - "args": 6, - "repr": ["\\qHyperrphis{", "}{", "}@@{", "}{", "}{", "}{", "}"] - } - ], - - "uincgamma": [ - { - "args": 2, - "repr": ["\\IncGamma@{", "}{", "}"] - } - ], - - "WhittakerPsi": [ - { - "args": 3, - "repr": ["\\WhitPsi{", "}{", "}@{", "}"] - } - ], - - "goldenratio": [ - { - "args": 0, - "repr": ["\\GoldenRatio"] - } - ], - - "Gompertz": [ - { - "args": 0, - "repr": ["\\GompertzConstant"] - } - ], - - "rabbit": [ - { - "args": 0, - "repr": ["\\rabbitConstant"] - } - ], - - "combinatfibonacci": [ - { - "args": 1, - "repr": ["\\FibonacciNumber{", "}"] - } - ], - - "argument": [ - { - "args": 1, - "repr": ["\\arg{(", ")}"] - } - ], - - "RealRange": [ - { - "args": 2, - "repr": ["(", ", ", ")"] - } - ], - - "Open": [ - { - "args": 1, - "repr": ["(", ")"] - } - ] - }, - - "symbols": { - "alpha": "n", - "beta": "\\beta", - "gamma": "\\gamma", - "delta": "\\delta", - "epsilon": "\\epsilon", - "zeta": "\\zeta", - "eta": "\\eta", - "theta": "\\theta", - "iota": "\\iota", - "kappa": "\\kappa", - "lambda": "\\lambda", - "mu": "\\mu", - "nu": "\\nu", - "xi": "\\xi", - "pi": "\\pi", - "rho": "\\rho", - "sigma": "\\sigma", - "tau": "\\tau", - "upsilon": "\\upsilon", - "phi": "\\phi", - "chi": "\\chi", - "psi": "\\psi", - "omega": "\\omega", - "Pi": "\\pi", - "I": "i", - "Catalan": "\\CatalansConstant", - "infinity": "\\infty", - "Delian": "\\sqrt[\\leftroot{-2}\\uproot{2} 3]{2}", - "Theodorus": "\\sqrt{3}", - "integer": "\\mathbb{Z}" - }, - - "constraints": { - "Not(negint)": "\\CC\\backslash\\ZZ^-", - "Not(nonposint)": "\\CC\\backslash\\ZZ_0^-", - "Not(posint)": "\\CC\\backslash\\ZZ^+", - "::Not(positive)": "< 0", - "Not(integer)": "\\CC\\backslash\\ZZ", - "posint": "\\mathbb{Z^+}", - "nonnegint": "\\ZZ\\backslash\\ZZ^-", - "::positive": "> 0", - "::nonnegative": "\\geq 0", - "::nonneg": "\\geq 0", - "::": "\\in " - } + "functions": { + "sqrt": [ + { + "args": 1, + "repr": ["\\sqrt{", "}"] + } + ], + + "sin": [ + { + "args": 1, + "repr": ["\\sin(", ")"] + } + ], + + "cos": [ + { + "args": 1, + "repr": ["\\cos(", ")"] + } + ], + + "tan": [ + { + "args": 1, + "repr": ["\\tan(", ")"] + } + ], + + "abs": [ + { + "args": 1, + "repr": ["\\abs{", "}"] + } + ], + + "exp": [ + { + "args": 1, + "repr": ["e^{", "}"] + } + ], + + "log": [ + { + "args": 1, + "repr": ["\\log{(", ")}"] + }, + + { + "args": 2, + "repr": ["\\log_{", "} {(", ")}"] + } + ], + + "ln": [ + { + "args": 1, + "repr": ["\\ln{(", ")}"] + } + ], + + "floor": [ + { + "args": 1, + "repr": ["\\floor{", "}"] + } + ], + + "sum": [ + { + "args": 3, + "repr": ["\\sum_{", "}^{", "} (", ")"] + } + ], + + "GAMMA": [ + { + "args": 1, + "repr": ["\\EulerGamma@{", "}"] + } + ], + + "bernoulli": [ + { + "args": 1, + "repr": ["\\BernoulliB{", "}"] + } + ], + + "BesselI": [ + { + "args": 2, + "repr": ["\\BesselI{", "}@{", "}"] + } + ], + + "BesselJ": [ + { + "args": 2, + "repr": ["\\BesselJ{", "}@{", "}"] + } + ], + + "BesselK": [ + { + "args": 2, + "repr": ["\\BesselK{", "}@{", "}"] + } + ], + + "BesselY": [ + { + "args": 2, + "repr": ["\\BesselY{", "}@{", "}"] + } + ], + + "dawson": [ + { + "args": 1, + "repr": ["\\DawsonsInt@{", "}"] + } + ], + + "Ei": [ + { + "args": 2, + "repr": ["\\ExpIntn{", "}@{", "}"] + }, + + { + "args": 1, + "repr": ["\\ExpInt@{", "}"] + } + ], + + "erfc": [ + { + "args": 1, + "repr": ["\\erfc@@{(", ")}"] + } + ], + + "erf": [ + { + "args": 1, + "repr": ["\\erf@@{(", ")}"] + } + ], + + "FresnelC": [ + { + "args": 1, + "repr": ["\\FresnelCos@{", "}"] + } + ], + + "FresnelS": [ + { + "args": 1, + "repr": ["\\FresnelSin@{", "}"] + } + ], + + "HankelH1": [ + { + "args": 2, + "repr": ["\\HankelHi{", "}@{", "}"] + } + ], + + "KummerU": [ + { + "args": 3, + "repr": ["\\KummerU@{", "}{", "}{", "}"] + } + ], + + "CylinderU": [ + { + "args": 2, + "repr": ["\\ParabolicU@{", "}{", "}"] + } + ], + + "Psi": [ + { + "args": 2, + "repr": ["\\polygamma{", "}@{", "}"] + }, + + { + "args": 1, + "repr": ["\\digamma@{", "}"] + } + ], + + "WhittakerW": [ + { + "args": 3, + "repr": ["\\WhitW{", "}{", "}@{", "}"] + } + ], + + "Zeta": [ + { + "args": 1, + "repr": ["\\RiemannZeta@{", "}"] + } + ], + + "arccos": [ + { + "args": 1, + "repr": ["\\Acos@@{(", ")}"] + } + ], + + "arccosh": [ + { + "args": 1, + "repr": ["\\Acosh@@{(", ")}"] + } + ], + + "arcsin": [ + { + "args": 1, + "repr": ["\\Asin@@{(", ")}"] + } + ], + + "arcsinh": [ + { + "args": 1, + "repr": ["\\Asinh@@{(", ")}"] + } + ], + + "arctan": [ + { + "args": 1, + "repr": ["\\Atan@@{(", ")}"] + } + ], + + "arctanh": [ + { + "args": 1, + "repr": ["\\Atanh@@{(", ")}"] + } + ], + + "sinh": [ + { + "args": 1, + "repr": ["\\sinh@@{(", ")}"] + } + ], + + "cosh": [ + { + "args": 1, + "repr": ["\\cosh@@{(", ")}"] + } + ], + + "coth": [ + { + "args": 1, + "repr": ["\\coth@@{(", ")}"] + } + ], + + "tanh": [ + { + "args": 1, + "repr": ["\\tanh@@{(", ")}"] + } + ], + + "Re": [ + { + "args": 1, + "repr": ["\\Re(", ")"] + } + ], + + "Im": [ + { + "args": 1, + "repr": ["\\Im(", ")"] + } + ], + + "doublefactorial": [ + { + "args": 1, + "repr": ["(", ")!!"] + } + ], + + "hypergeom": [ + { + "args": 5, + "repr": ["\\HyperpFq{", "}{", "}@@{", "}{", "}{", "}"] + } + ], + + "pochhammer": [ + { + "args": 2, + "repr": ["\\pochhammer{", "}{", "}"] + } + ], + + "Besseli": [ + { + "args": 2, + "repr": ["\\SphBesselIi{", "}@{", "}"] + } + ], + + "Besselj": [ + { + "args": 2, + "repr": ["\\SphBesselJ{", "}@{", "}"] + } + ], + + "cerf": [ + { + "args": 1, + "repr": ["\\erfw@@{(", ")}"] + } + ], + + "Ein": [ + { + "args": 1, + "repr": ["\\ExpIntEin@{", "}"] + } + ], + + "erfcI": [ + { + "args": 2, + "repr": ["I^{", "} \\erfc@@{(", ")}"] + } + ], + + "hankelsymb": [ + { + "args": 2, + "repr": ["\\HankelSym{", "}{", "}"] + } + ], + + "lincgamma": [ + { + "args": 2, + "repr": ["\\incgamma@{", "}{", "}"] + } + ], + + "qhyper": [ + { + "args": 6, + "repr": ["\\qHyperrphis{", "}{", "}@@{", "}{", "}{", "}{", "}"] + } + ], + + "uincgamma": [ + { + "args": 2, + "repr": ["\\IncGamma@{", "}{", "}"] + } + ], + + "WhittakerPsi": [ + { + "args": 3, + "repr": ["\\WhitPsi{", "}{", "}@{", "}"] + } + ], + + "goldenratio": [ + { + "args": 0, + "repr": ["\\GoldenRatio"] + } + ], + + "Gompertz": [ + { + "args": 0, + "repr": ["\\GompertzConstant"] + } + ], + + "rabbit": [ + { + "args": 0, + "repr": ["\\rabbitConstant"] + } + ], + + "combinatfibonacci": [ + { + "args": 1, + "repr": ["\\FibonacciNumber{", "}"] + } + ], + + "argument": [ + { + "args": 1, + "repr": ["\\arg{(", ")}"] + } + ], + + "RealRange": [ + { + "args": 2, + "repr": ["(", ", ", ")"] + } + ], + + "Open": [ + { + "args": 1, + "repr": ["(", ")"] + } + ] + }, + + "symbols": { + "alpha": "n", + "beta": "\\beta", + "gamma": "\\gamma", + "delta": "\\delta", + "epsilon": "\\epsilon", + "zeta": "\\zeta", + "eta": "\\eta", + "theta": "\\theta", + "iota": "\\iota", + "kappa": "\\kappa", + "lambda": "\\lambda", + "mu": "\\mu", + "nu": "\\nu", + "xi": "\\xi", + "pi": "\\pi", + "rho": "\\rho", + "sigma": "\\sigma", + "tau": "\\tau", + "upsilon": "\\upsilon", + "phi": "\\phi", + "chi": "\\chi", + "psi": "\\psi", + "omega": "\\omega", + "Pi": "\\pi", + "I": "i", + "Catalan": "\\CatalansConstant", + "infinity": "\\infty", + "Delian": "\\sqrt[\\leftroot{-2}\\uproot{2} 3]{2}", + "Theodorus": "\\sqrt{3}", + "integer": "\\mathbb{Z}" + }, + + "constraints": { + "Not(negint)": "\\CC\\backslash\\ZZ^-", + "Not(nonposint)": "\\CC\\backslash\\ZZ_0^-", + "Not(posint)": "\\CC\\backslash\\ZZ^+", + "::Not(positive)": "< 0", + "Not(integer)": "\\CC\\backslash\\ZZ", + "posint": "\\mathbb{Z^+}", + "nonnegint": "\\ZZ\\backslash\\ZZ^-", + "::positive": "> 0", + "::nonnegative": "\\geq 0", + "::nonneg": "\\geq 0", + "::": "\\in " + } } \ No newline at end of file diff --git a/maple2latex/src/__init__.py b/maple2latex/src/__init__.py new file mode 100644 index 0000000..e69de29 diff --git a/maple2latex/src/classes.py b/maple2latex/src/classes.py deleted file mode 100644 index 3709e92..0000000 --- a/maple2latex/src/classes.py +++ /dev/null @@ -1,79 +0,0 @@ -#!/user/bin/env python - -__author__ = "Joon Bang" -__status__ = "Development" - -from translation_methods import make_equation -from copy import copy - - -class MapleFile(object): - def __init__(self, filename): - self.filename = filename - self.formulae = self.obtain_formulae() - - def obtain_formulae(self): - contents = open(self.filename).read() - - return [MapleEquation(piece.split("\n")) for piece in contents.split("create(") - if "):" in piece or ");" in piece] - - def convert_formulae(self): - return '\n\n'.join([make_equation(copy(formula)) for formula in self.formulae]) - - def __str__(self): - return "MapleFile of " + self.filename - - -class MapleEquation(object): - def __init__(self, inp): - # creates a dictionary called "fields", containing all the Maple fields - self.fields = {"category": "", "constraints": "", "begin": "", "factor": "", "front": "", "parameters": "", - "type": inp.pop(0).split("'")[1]} - - for i, line in enumerate(inp): - line = line.split(" = ", 1) - - if len(line) > 1: - line[0] = line[0].strip() - - if line[0] in ["category"]: - line[1] = line[1].strip()[1:-1].strip() - - elif line[0] == "begin" and "proc (x) [1, x] end proc" in line[1]: - temp = [[1, int(d)] for d in inp[i + 2].split(",")] - if len(temp) > 10: - temp = temp[:10] - line[1] = str(temp)[1:-1] - - elif line[0] == "booklabelv1" and line[1] == '"",': - line[1] = "No label" - - elif line[0] in ["booklabelv1", "booklabelv2", "general", "constraints", "begin", "parameters"]: - line[1] = line[1].strip()[1:-2].strip() - - elif line[0] in ["lhs", "factor", "front", "even", "odd"]: - if line[1].strip()[-1] == ",": - line[1] = line[1].strip()[:-1] - line[1] = line[1].strip() - self.fields[line[0]] = line[1] - - # assign fields containing information about the equation - self.eq_type = self.fields["type"] - self.category = self.fields["category"] - self.label = self.fields["booklabelv1"] - self.lhs = self.fields["lhs"] - self.factor = self.fields["factor"] - self.front = self.fields["front"] - self.begin = self.fields["begin"] - self.constraints = self.fields["constraints"] - self.parameters = self.fields["parameters"] - - # even-odd case handling - if "general" in self.fields: - self.general = [self.fields["general"]] - elif "even" in self.fields and "odd" in self.fields: - self.general = [self.fields["even"], self.fields["odd"]] - - def __str__(self): - return '\n'.join([s + ": " + self.fields[s] for s in self.fields]) diff --git a/maple2latex/src/main.py b/maple2latex/src/main.py index 14a7af4..fd8ae06 100644 --- a/maple2latex/src/main.py +++ b/maple2latex/src/main.py @@ -4,10 +4,11 @@ __status__ = "Development" import os -from classes import MapleFile +import copy +from src.translator import MapleEquation, make_equation TABLE = dict() -with open("info/section_names") as name_info: +with open("data/section_names") as name_info: for line in name_info.read().split("\n"): if line != "" and "%" not in line: key, value = line.split(" : ") @@ -30,7 +31,26 @@ ROOT = "functions" -def translate_file(filename: str): +class MapleFile(object): + def __init__(self, filename): + self.filename = filename + self.formulae = self.obtain_formulae() + + def obtain_formulae(self): + contents = open(self.filename).read() + + return [MapleEquation(piece.split("\n")) for piece in contents.split("create(") + if "):" in piece or ");" in piece] + + def convert_formulae(self): + return '\n\n'.join([make_equation(copy.copy(formula)) for formula in self.formulae]) + + def __str__(self): + return "MapleFile of " + self.filename + + +def translate_file(filename): + # type: (str) """Translates all the formulae in a file.""" with open("out/test.tex", "w") as test, open("out/primer") as primer: @@ -38,6 +58,7 @@ def translate_file(filename: str): def translate_directories(): + # type: () """Generates and writes the results of traversing the root directory and translating all Maple files.""" root_depth = len(ROOT.split("/")) @@ -59,7 +80,8 @@ def translate_directories(): MapleFile(info[0] + "/" + file_name).convert_formulae() + "\n\n" with open("out/test.tex", "w") as test, open("out/primer") as primer: - test.write(primer.read() + result + "\n\\end{document}") + result = primer.read() + result + "\n\\end{document}" + test.write(result) if __name__ == '__main__': translate_directories() diff --git a/maple2latex/src/maple_tokenize.py b/maple2latex/src/maple_tokenize.py index 588784b..7561fce 100644 --- a/maple2latex/src/maple_tokenize.py +++ b/maple2latex/src/maple_tokenize.py @@ -21,6 +21,8 @@ ")", "{", "}", + "[", + "]", "\'", "\"", "|", diff --git a/maple2latex/src/translation_methods.py b/maple2latex/src/translator.py similarity index 68% rename from maple2latex/src/translation_methods.py rename to maple2latex/src/translator.py index c5013a5..933659b 100644 --- a/maple2latex/src/translation_methods.py +++ b/maple2latex/src/translator.py @@ -3,9 +3,9 @@ __author__ = "Joon Bang" __status__ = "Development" +import copy import json -from maple_tokenize import tokenize -from copy import copy +from src.maple_tokenize import tokenize INFO = json.loads(open("data/keys.json").read()) @@ -16,19 +16,68 @@ SPECIAL = {"(": "\\left(", ")": "\\right)", "+-": "-", "\\subplus-": "-", "^{1}": "", "\\inNot": "\\notin"} -def replace_strings(string: str, keys: dict) -> str: - """ - Replaces multiple strings with multiple other strings, stored in lists of length two in li. - The first element is the string to find, and the second element is the replacement string. - A dictionary can also be given. - """ +class MapleEquation(object): + def __init__(self, inp): + # creates a dictionary called "fields", containing all the Maple fields + self.fields = {"category": "", "constraints": "", "begin": "", "factor": "", "front": "", "parameters": "", + "type": inp.pop(0).split("'")[1]} + + for i, line in enumerate(inp): + line = line.split(" = ", 1) + + if len(line) > 1: + line[0] = line[0].strip() + + if line[0] in ["category"]: + line[1] = line[1].strip()[1:-1].strip() + + elif line[0] == "begin" and "proc (x) [1, x] end proc" in line[1]: + temp = [[1, int(d)] for d in inp[i + 2].split(",")] + if len(temp) > 10: + temp = temp[:10] + line[1] = str(temp)[1:-1] + + elif line[0] == "booklabelv1" and line[1] == '"",': + line[1] = "No label" + + elif line[0] in ["booklabelv1", "booklabelv2", "general", "constraints", "begin", "parameters"]: + line[1] = line[1].strip()[1:-2].strip() + + elif line[0] in ["lhs", "factor", "front", "even", "odd"]: + if line[1].strip()[-1] == ",": + line[1] = line[1].strip()[:-1] + line[1] = line[1].strip() + self.fields[line[0]] = line[1] + + # assign fields containing information about the equation + self.eq_type = self.fields["type"] + self.category = self.fields["category"] + self.label = self.fields["booklabelv1"] + self.lhs = self.fields["lhs"] + self.factor = self.fields["factor"] + self.front = self.fields["front"] + self.begin = self.fields["begin"] + self.constraints = self.fields["constraints"] + self.parameters = self.fields["parameters"] + + # even-odd case handling + if "general" in self.fields: + self.general = [self.fields["general"]] + elif "even" in self.fields and "odd" in self.fields: + self.general = [self.fields["even"], self.fields["odd"]] + + +def replace_strings(string, keys): + # type: (str, dict) -> str + """Replaces key strings with their mapped value.""" for key in list(keys): string = string.replace(key, keys[key]) return string -def parse_brackets(exp: (str, list)) -> list: +def parse_brackets(exp): + # type: ((str, list)) -> list """Obtains the contents from data encapsulated in square brackets.""" exp = exp[1:-1].split(",") @@ -43,7 +92,8 @@ def parse_brackets(exp: (str, list)) -> list: return exp -def trim_parens(exp: str) -> str: +def trim_parens(exp): + # type: (str) -> str """Removes unnecessary parentheses.""" if exp == "": @@ -72,13 +122,15 @@ def trim_parens(exp: str) -> str: return exp -def make_frac(parts: list): +def make_frac(parts): + # type: (list) -> str """Generate a LaTeX fraction from its numerator and denominator.""" return translate("(" + parts[0] + ") / (" + parts[1] + ")") -def basic_translate(exp: list) -> str: +def basic_translate(exp): + # type: (list) -> str """Translates basic mathematical operations.""" for order in range(3): @@ -95,11 +147,11 @@ def basic_translate(exp: list) -> str: elif exp[i] == "^" and order == 0: power = trim_parens(exp.pop(i + 1)) - exp[i - 1] += "^{%s}" % power + exp[i - 1] += "^{" + power + "}" modified = True elif exp[i] == "*" and order == 1: - if "\\" in exp[i - 1] and exp[i + 1][0] != "\\": + if exp[i - 1][-1] not in "}]" and "\\" in exp[i - 1] and exp[i + 1][0] != "\\": exp[i - 1] += " " exp[i - 1] += exp.pop(i + 1) @@ -120,15 +172,8 @@ def basic_translate(exp: list) -> str: return ''.join(exp) -def count_args(string: str, pattern: str) -> int: - """Counts the arguments contained in a string, based on a delimiter pattern.""" - if string == "": - return 0 - - return string.count(pattern) + 1 - - -def get_arguments(function: str, arg_string: str) -> list: +def get_arguments(function, arg_string): + # type: (str, list) -> list """Obtains the arguments of a function.""" if arg_string == ["(", ")"]: @@ -142,10 +187,10 @@ def get_arguments(function: str, arg_string: str) -> list: if function == "qhyper": args += args.pop(2).split(",") - for i in [1, 0]: + for p, i in enumerate([1, 0]): arg_count = 0 - if args[i] != "": - arg_count = arg_string.count(",") + 1 + if args[i + p] != "": + arg_count = args[i + p].count(",") + 1 args.insert(0, str(arg_count)) @@ -161,14 +206,15 @@ def get_arguments(function: str, arg_string: str) -> list: return args -def generate_function(name: str, args: list) -> str: +def generate_function(name, args): + # type: (str, list) -> str """Generate a function with the provided function name and arguments.""" result = list() for n in FUNCTIONS: for variant in FUNCTIONS[n]: if name == n and len(args) == variant["args"]: - result = copy(variant["repr"]) + result = copy.copy(variant["repr"]) for n in range(1, len(result)): result.insert(2 * n - 1, args[n - 1]) @@ -176,7 +222,8 @@ def generate_function(name: str, args: list) -> str: return ''.join(result) -def translate(exp: str) -> str: +def translate(exp): + # type: (str) -> str """Format Maple code as LaTeX.""" if exp == "": @@ -191,7 +238,8 @@ def translate(exp: str) -> str: if exp[i] == s: exp[i] = SYMBOLS[s] - for i in range(len(exp)-1, -1, -1): + i = len(exp) - 1 + while i >= 0: if exp[i] == "(": r = i + exp[i:].index(")") piece = exp[i:r + 1] @@ -214,10 +262,13 @@ def translate(exp: str) -> str: exp = exp[0:i] + [piece] + exp[r + 1:] + i -= 1 + return basic_translate(exp) -def make_equation(eq: "MapleEquation") -> str: +def make_equation(eq, view_metadata=False): + # type: (MapleEquation{, bool}) -> str """Make a LaTeX equation based on a MapleEquation object.""" eq.lhs = translate(eq.lhs) @@ -233,7 +284,7 @@ def make_equation(eq: "MapleEquation") -> str: if eq.begin != "": eq.begin = parse_brackets(eq.begin) - equation = "\\begin{equation*}\\tag{%s}\n %s\n = " % (eq.label, eq.lhs) + equation = "\\begin{equation*}\\tag{" + eq.label + "}\n " + eq.lhs + "\n = " if eq.factor == "1": eq.factor = "" @@ -280,14 +331,12 @@ def make_equation(eq: "MapleEquation") -> str: equation += "\\dots" # adds metadata - view_metadata = False - if view_metadata: equation += "\n\\end{equation*}" equation += "\n\\begin{center}" equation += "\nParameters: $$" + eq.parameters + "$$" equation += "\n$$" + translate(eq.constraints) + "$$" - equation += eq.category + equation += "\n" + eq.category equation += "\n\\end{center}" else: equation += "\n % \\constraint{$" + translate(eq.constraints) + "$}" From 1c977d73a06947ad334f0deab738fe3c04499925 Mon Sep 17 00:00:00 2001 From: Joon Bang Date: Tue, 19 Jul 2016 14:05:24 -0400 Subject: [PATCH 165/402] Fix issues with for loop --- maple2latex/src/translator.py | 5 ++--- 1 file changed, 2 insertions(+), 3 deletions(-) diff --git a/maple2latex/src/translator.py b/maple2latex/src/translator.py index 933659b..eda0864 100644 --- a/maple2latex/src/translator.py +++ b/maple2latex/src/translator.py @@ -234,9 +234,8 @@ def translate(exp): exp = tokenize(exp) for i in range(len(exp)): - for s in SYMBOLS: - if exp[i] == s: - exp[i] = SYMBOLS[s] + if exp[i] in SYMBOLS: + exp[i] = SYMBOLS[exp[i]] i = len(exp) - 1 while i >= 0: From 4b6f4b08bcc15618caecf3a87689aa7b2b882f5d Mon Sep 17 00:00:00 2001 From: Joon Bang Date: Tue, 19 Jul 2016 14:08:49 -0400 Subject: [PATCH 166/402] Fix more issues --- maple2latex/src/translator.py | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/maple2latex/src/translator.py b/maple2latex/src/translator.py index eda0864..1c07228 100644 --- a/maple2latex/src/translator.py +++ b/maple2latex/src/translator.py @@ -233,9 +233,9 @@ def translate(exp): exp = replace_strings(exp, {"functions:-": ""}) exp = tokenize(exp) - for i in range(len(exp)): + for i, s in enumerate(exp): if exp[i] in SYMBOLS: - exp[i] = SYMBOLS[exp[i]] + exp[i] = SYMBOLS[s] i = len(exp) - 1 while i >= 0: From c5c16aca0077abe2d8970a2b81cde82aaec68332 Mon Sep 17 00:00:00 2001 From: joonbang Date: Tue, 19 Jul 2016 14:39:52 -0400 Subject: [PATCH 167/402] Update CONTRIBUTING.md --- CONTRIBUTING.md | 54 ++++++++++++++++++++++++++++++++++++++++--------- 1 file changed, 45 insertions(+), 9 deletions(-) diff --git a/CONTRIBUTING.md b/CONTRIBUTING.md index d834a55..258684e 100644 --- a/CONTRIBUTING.md +++ b/CONTRIBUTING.md @@ -1,13 +1,49 @@ -Plese use pull request to contribute to the project. -Check out our guide at http://drmf.wmflabs.org/wiki/GitHub how to use git. +# Guidelines to Contributing -## Example -See the following pull request: +## Making Changes to the Repository +Please use pull requests to contribute to the project. -https://github.com/DRMF/DRMF-Seeding-Project/pull/19 +If you are unfamiliar with GitHub, read the [DRMF guide](http://drmf.wmflabs.org/wiki/GitHub). + +### Sample Pull Request +[This](https://github.com/DRMF/DRMF-Seeding-Project/pull/19) is an example of a good pull request. This pull request satisfies the following properties: -1. It is small (incremental) -2. It improves the readability -3. It adds more tests -4. It demonstrates how to add issues for aspects that can be fixed later + +1. It is small/incremental. + +2. It improves the readability of the code. + +3. It adds more tests. + +4. It demonstrates how to add issues for aspects that can be fixed later. + +## Recommended IDEs +PyCharm is recommended for Python development. For other languages, their respective JetBrains IDEs should be used. + +## Coding Style + +All Python code should be written for Python 2.7, and should follow [PEP-8 conventions](https://www.python.org/dev/peps/pep-0008/). In PyCharm, code can be reformatted to follow some PEP-8 conventions, using Code -> Reformat Code. Be warned, Reformat Code will *not* fix all issues. In addition, PyCharm will warn you about code that violates PEP-8 standards. + +Code in other languages should follow standard conventions for that language. + +## Unit Tests + +All important code meant for use should have unit tests written for them. In Python, this can be achieved through the built-in [```unittest```](https://docs.python.org/2.7/library/unittest.html) module. + +Ideally, unit tests should cover every single possibility, every input and outcome, every line of code. A more realistic goal is "90%" coverage. Coverage can be checked through Coveralls. The idea is that, by the time code gets to production, there should be no possibility of unexpected output, since all possible types of inputs should have been tested. + +Unit tests should be stored in a separate "test" folder. + +## Best Practices + +Please follow the instructions [here](http://chris.beams.io/posts/git-commit/) on how to write git commit messages. + +PyCharm can help resolve merge conflicts. To resolve conflicts in your project, navigate to VCS -> Git -> Resolve Conflicts. Merge conflicts can also be resolved manually, through the steps in [this guide](https://confluence.atlassian.com/bitbucket/resolve-merge-conflicts-704414003.html). + +Code should be stored in a "src" folder. + +All subprojects must have README.md files. A guide may be found [here](https://gist.github.com/PurpleBooth/109311bb0361f32d87a2). Note that some of these requirements may not make sense to put in your README.md. + +Make good use of .gitignore. Anything that has been generated by code in the repository (compiled code, test files, test output, etc.) should be in your .gitignore. The easiest way to do this is to move these files into a folder, and gitignore that folder. + From 4d1cd2955239c842aa197f0e7bc14917bd682ac1 Mon Sep 17 00:00:00 2001 From: Oksana Tkach Date: Tue, 19 Jul 2016 14:48:18 -0400 Subject: [PATCH 168/402] Fix crash error on test_replace_i.py --- AlexDanoff/src/math_function.py | 72 +++++++++++++++++++++++------- AlexDanoff/src/replace_special.py | 73 ++++++++++++++----------------- AlexDanoff/test/test_replace_i.py | 4 +- 3 files changed, 90 insertions(+), 59 deletions(-) diff --git a/AlexDanoff/src/math_function.py b/AlexDanoff/src/math_function.py index 9536ffc..33ae8de 100644 --- a/AlexDanoff/src/math_function.py +++ b/AlexDanoff/src/math_function.py @@ -1,7 +1,5 @@ import math_mode import string -from utilities import readin -from utilities import writeout import re def math_string(file): @@ -9,6 +7,7 @@ def math_string(file): string = open(file).read() output = [] ranges = math_mode.find_math_ranges(string) + #print ranges for i in ranges: new = string[i[0]:i[1]] output.append(new) @@ -27,24 +26,68 @@ def change_original(o_file, changed_math_string): return edited -def index_spacing(file, out_file): - used = open(file).read() +def formatting(file_str): + # Proper spacing for the indexes and gets rid of all non begin{eq end{eq text + + commands = r'\\[a-zA-Z]+' updated = [] - IND_START = r'\index{' + IND_START = r'\index\{' ind_str = "" in_ind = False previous = "" - lines = used.split("\n") + + ranges = math_mode.find_math_ranges(file_str) + + lines = file_str.split('\n') + in_eq = False + been_in_eq = False + + + section = r'\\[a-zA-Z]*section' + + re.compile(commands) + re.compile(section) + past_last = 0 for line in lines: - # if this line is an index start storing it, or write it if we're done with the indexes + past_last += len(line) + + if '\\begin{equation}' in line: + in_eq = True + been_in_eq = False + if '\\end{equation}' in line: + updated.append(line) + in_eq = False + continue + + if in_eq: + updated.append(line) + + else: + # not in eq + print line + if not been_in_eq: + # if text line != command + if re.match(commands, line) or line == '': + updated.append(line) + else: + # not in eq but already been in eq + if '\\index' in line or re.match(section,line) or line == '': + updated.append(line) + + if past_last > ranges[-1][1]: + # In section after last eq + if re.match(commands,line) or line == '': + updated.append(line) + + # if this line is an index start storing it,or write it if we're done with the indexes if IND_START in line: in_ind = True ind_str += line + "\n" - continue - elif in_ind: + + if in_ind: in_ind = False # add a preceding newline if one is not already present @@ -52,21 +95,18 @@ def index_spacing(file, out_file): ind_str = "\n" + ind_str fullsplit = ind_str.split("\n") - updated.extend(fullsplit) ind_str = "" + previous = line - updated.append(line) wrote = "\n".join(updated) + wrote = wrote # remove consecutive blank lines and blank lines between \index groups spaces_pat = re.compile(r'\n{2,}[ ]?\n+') wrote = spaces_pat.sub('\n\n', wrote) wrote = re.sub(r'\\index{(.*?)}\n\n\\index{(.*?)}', r'\\index{\1}\n\\index{\2}', wrote) - out = open(out_file, 'w') - out.write(wrote) - out.close() - return out -# 4/26, look at chapter 16 and prevent converting the \ApellFiii into \ApellFii\iunit + return wrote +#formatting(open('/home/ont1/DLMF/25.ZE/newMoritz').read()) \ No newline at end of file diff --git a/AlexDanoff/src/replace_special.py b/AlexDanoff/src/replace_special.py index eca08a9..930d3f2 100644 --- a/AlexDanoff/src/replace_special.py +++ b/AlexDanoff/src/replace_special.py @@ -10,6 +10,7 @@ except ImportError: izip = zip +import math_mode import math_function import parentheses from utilities import writeout @@ -46,18 +47,24 @@ def main(): # Below: index_str writes to output file, math_string takes output file as input, and change_original # writes to the output based on the previous output file - index_str = math_function.index_spacing(fname, ofname) - my_string = math_function.math_string(ofname) + # print math_mode.find_math_ranges(open(fname).read()) - math_string = remove_special(my_string) + unchanged_math = math_function.math_string(fname) + math_string = remove_special(unchanged_math) + changed_math = math_function.change_original(fname,math_string) + formatted = math_function.formatting(changed_math) - writeout(ofname, math_function.change_original(ofname, math_string)) + writeout(ofname, formatted) def remove_special(content): - """Removes the excess pieces from the given content and returns the updated version as a string.""" - print(content) + """Removes the excess pieces from the given content and returns the updated version as a string. + IN -> list + OUT -> string""" + if isinstance(content, str): + content = content.split('\n') counter = 0 + for function in content: lines = function.split('\n') @@ -84,37 +91,12 @@ def remove_special(content): dollar_pat = re.compile(r'(? 1: + continue surrounding = "" @@ -265,13 +251,18 @@ def _replace_i(words): if surrounding[0] != "\\": replacement = r'\iunit ' - else: # neither of the characters surrounding the "i" are alphabetic, replace - + else: + # neither of the characters surrounding the "i" are alphabetic, replace replacement = r'\iunit' + + # print("Surrounding: {0} - replacement made: {1}".format(surrounding, replacement != "i")) - words = words[:iloc] + replacement + words[iloc + 1:] + if iloc != -1: + words = words[:iloc] + replacement + words[iloc + 1:] + iloc = words.find("i", iloc + len(replacement)) + else: + words = words[:iloc] + replacement - iloc = words.find("i", iloc + len(replacement)) return words diff --git a/AlexDanoff/test/test_replace_i.py b/AlexDanoff/test/test_replace_i.py index 236d70c..725d657 100644 --- a/AlexDanoff/test/test_replace_i.py +++ b/AlexDanoff/test/test_replace_i.py @@ -4,6 +4,6 @@ class TestReplaceSpecial(TestCase): def test__replace_special(self): - tex = 'z^a = \\underbrace{z \\cdot z \\cdots z}_{n \\text{ times}} = 1 / z^{-a}.' - result = replace_special(tex) + tex = "z^a = \\underbrace{z \\cdot z \\cdots z}_{n \n \\text{ times}} = 1 / z^{-a}" + result = remove_special(tex) self.assertEqual(tex, result) From 3337906a5f7ead8b3336f4768fd1b738e5491ee7 Mon Sep 17 00:00:00 2001 From: Joon Bang Date: Tue, 19 Jul 2016 14:52:20 -0400 Subject: [PATCH 169/402] Add comments --- maple2latex/src/main.py | 14 +++++++------- maple2latex/src/translator.py | 28 +++++++++++++++++++++++----- 2 files changed, 30 insertions(+), 12 deletions(-) diff --git a/maple2latex/src/main.py b/maple2latex/src/main.py index fd8ae06..cbaadf9 100644 --- a/maple2latex/src/main.py +++ b/maple2latex/src/main.py @@ -7,14 +7,14 @@ import copy from src.translator import MapleEquation, make_equation -TABLE = dict() +TABLE = dict() # stores meaning of folder names with open("data/section_names") as name_info: for line in name_info.read().split("\n"): if line != "" and "%" not in line: key, value = line.split(" : ") TABLE[key] = value -FILES = [ +FILES = [ # files to be translated "bessel", "modbessel", "confluent", "confluentlimit", "kummer", "parabolic", "whittaker", "apery", "archimedes", "catalan", "delian", "eulersconstant", "eulersnumber", "goldenratio", "gompertz", @@ -28,7 +28,7 @@ "qhyper" ] -ROOT = "functions" +ROOT = "functions" # root directory class MapleFile(object): @@ -67,7 +67,7 @@ def translate_directories(): result = "" - for info in dirs: + for info in dirs: # search through directories depth = len(["" for ch in info[0] if ch == "/"]) if depth == root_depth: # section @@ -75,13 +75,13 @@ def translate_directories(): for file_name in info[2]: folder = ''.join(file_name.split(".")[:-1]) - if folder in FILES: + if folder in FILES: # subsection result += "\\subsection{" + TABLE[folder] + "}\n" + \ MapleFile(info[0] + "/" + file_name).convert_formulae() + "\n\n" + # write output to file with open("out/test.tex", "w") as test, open("out/primer") as primer: - result = primer.read() + result + "\n\\end{document}" - test.write(result) + test.write(primer.read() + result + "\n\\end{document}") if __name__ == '__main__': translate_directories() diff --git a/maple2latex/src/translator.py b/maple2latex/src/translator.py index 1c07228..f3f50f1 100644 --- a/maple2latex/src/translator.py +++ b/maple2latex/src/translator.py @@ -22,6 +22,7 @@ def __init__(self, inp): self.fields = {"category": "", "constraints": "", "begin": "", "factor": "", "front": "", "parameters": "", "type": inp.pop(0).split("'")[1]} + # read data from the Maple create() statement for i, line in enumerate(inp): line = line.split(" = ", 1) @@ -133,36 +134,44 @@ def basic_translate(exp): # type: (list) -> str """Translates basic mathematical operations.""" - for order in range(3): + # translates operations in place + for order in range(3): # order of operations i = 0 while i < len(exp): modified = False + # the imaginary number if exp[i] == "I": exp[i] = "i" + # factorial elif exp[i] == "!" and order == 0: exp[i - 1] += "!" modified = True + # power elif exp[i] == "^" and order == 0: power = trim_parens(exp.pop(i + 1)) exp[i - 1] += "^{" + power + "}" modified = True + # multiplication elif exp[i] == "*" and order == 1: + # adds spacing when necessary if exp[i - 1][-1] not in "}]" and "\\" in exp[i - 1] and exp[i + 1][0] != "\\": exp[i - 1] += " " exp[i - 1] += exp.pop(i + 1) modified = True + # division elif exp[i] == "/" and order == 1: - for index in [i - 1, i + 1]: + for index in [i - 1, i + 1]: # removes extra parentheses exp[index] = trim_parens(exp[index]) exp[i - 1] = "\\frac{" + exp[i - 1] + "}{" + exp.pop(i + 1) + "}" modified = True + # removes extra characters if modified: exp.pop(i) i -= 1 @@ -176,9 +185,11 @@ def get_arguments(function, arg_string): # type: (str, list) -> list """Obtains the arguments of a function.""" + # no arguments if arg_string == ["(", ")"]: return [] + # handling for hypergeometric, q-hypergeometric functions elif function in ["hypergeom", "qhyper"]: args = list() for s in ' '.join(arg_string[1:-1]).split("] , "): @@ -194,6 +205,7 @@ def get_arguments(function, arg_string): args.insert(0, str(arg_count)) + # handling for sums elif function == "sum": args = basic_translate(arg_string[1:-1]).split(",") args = args.pop(1).split("..") + [args[0]] @@ -216,6 +228,7 @@ def generate_function(name, args): if name == n and len(args) == variant["args"]: result = copy.copy(variant["repr"]) + # places arguments between shell of function for n in range(1, len(result)): result.insert(2 * n - 1, args[n - 1]) @@ -229,6 +242,7 @@ def translate(exp): if exp == "": return "" + # initial formatting of input string exp = replace_strings(exp.strip(), CONSTRAINTS) exp = replace_strings(exp, {"functions:-": ""}) exp = tokenize(exp) @@ -239,22 +253,23 @@ def translate(exp): i = len(exp) - 1 while i >= 0: - if exp[i] == "(": + if exp[i] == "(": # handle contents between parentheses r = i + exp[i:].index(")") piece = exp[i:r + 1] - if exp[i - 1] == "]": + if exp[i - 1] == "]": # for logarithm function sq = i - 2 - exp[:i - 1][::-1].index("[") piece = [piece[0]] + exp[sq + 1:i - 1] + [","] + piece[1:] i = sq - if exp[i - 1] in FUNCTIONS: + if exp[i - 1] in FUNCTIONS: # handling for functions i -= 1 piece = generate_function(exp[i], get_arguments(exp[i], piece)) else: piece = basic_translate(piece) + # remove unnecessary parentheses around fractions if "frac" in replace_strings(piece, {"(": ""})[:6] and (r + 2 > len(exp) or exp[r + 1] != "^"): while piece != trim_parens(piece): piece = trim_parens(piece) @@ -270,6 +285,7 @@ def make_equation(eq, view_metadata=False): # type: (MapleEquation{, bool}) -> str """Make a LaTeX equation based on a MapleEquation object.""" + # modify fields eq.lhs = translate(eq.lhs) eq.factor = translate(eq.fields["factor"]) eq.front = translate(eq.fields["front"]) @@ -306,6 +322,7 @@ def make_equation(eq, view_metadata=False): elif eq.eq_type == "contfrac": start = 1 # in case the value of start isn't assigned + # add terms before general if eq.front != "": equation += eq.front + "+" start = 1 @@ -321,6 +338,7 @@ def make_equation(eq, view_metadata=False): else: equation += eq.factor + " " + # trim unnecessary parentheses for i, element in enumerate(eq.general): eq.general[i] = trim_parens(translate(element)) From 0f44033c6fad60221d980d87a96e0ea60e940b3d Mon Sep 17 00:00:00 2001 From: Joon Bang Date: Tue, 19 Jul 2016 15:00:29 -0400 Subject: [PATCH 170/402] Add unit tests (for real) --- maple2latex/test/__init__.py | 0 maple2latex/test/test_cases.json | 175 ++++++++++++++++++++++++++++ maple2latex/test/test_translator.py | 53 +++++++++ 3 files changed, 228 insertions(+) create mode 100644 maple2latex/test/__init__.py create mode 100644 maple2latex/test/test_cases.json create mode 100644 maple2latex/test/test_translator.py diff --git a/maple2latex/test/__init__.py b/maple2latex/test/__init__.py new file mode 100644 index 0000000..e69de29 diff --git a/maple2latex/test/test_cases.json b/maple2latex/test/test_cases.json new file mode 100644 index 0000000..58c79fb --- /dev/null +++ b/maple2latex/test/test_cases.json @@ -0,0 +1,175 @@ +{ + "translator": { + "parse_brackets": [ + { + "exp": "[1,2, 3]", + "res": [["1", "2"], "3"] + }, + + { + "exp": "[1]", + "res": ["1"] + } + ], + + "trim_parens": [ + { + "exp": "(1 + 2)", + "res": "1 + 2" + }, + + { + "exp": "(3)(4)", + "res": "(3)(4)" + }, + + { + "exp": "((1 + 2))", + "res": "(1 + 2)" + } + ], + + "basic_translate": [ + { + "exp": ["1", "+", "2", "/", "(3)", "*", "4", "-", "5", "!", "^", "-6"], + "res": "1+\\frac{2}{3}4-5!^{-6}" + }, + + { + "exp": ["\\pi", "*", "I"], + "res": "\\pi i" + } + ], + + "get_arguments": [ + { + "function": "rabbit", + "arg_string": ["(", ")"], + "res": [] + }, + + { + "function": "hypergeom", + "arg_string": ["(", "[1", ",", "2]", ",", "[]", ",", "4", ")"], + "res": ["2", "0", "1,2", "", "4"] + }, + + { + "function": "qhyper", + "arg_string": ["(", "[1", ",", "2", ",", "3]", ",", "[4", ",", "5]", ",", "6", ",", "7", ")"], + "res": ["3", "2", "1,2,3", "4,5", "6", "7"] + }, + + { + "function": "sum", + "arg_string": ["(", "m", ",", "1..infinity", ")"], + "res": ["1", "\\infty", "m"] + }, + + { + "function": "BesselJ", + "arg_string": ["(", "\\nu", ",", "z", ")"], + "res": ["\\nu", "z"] + } + ], + + "translate": [ + { + "exp": "sin(x)", + "res": "\\sin(x)" + }, + + { + "exp": "log[beta](nu)", + "res": "\\log_{\\beta} {(\\nu)}" + } + ], + + "make_equation": [ + { + "eq": [ + "create( 'series',", + "booklabelv1 = \"0.0.0\",", + "category = \"power series\"", + "lhs = n,", + "factor = 4/5,", + "general = [ 2 * k - 3 ],", + "constraints = { k::integer }," + ], + "view_metadata": false, + "res": "\\begin{equation*}\\tag{0.0.0}\n n\n = \\frac{4}{5} \\sum_{k=0}^\\infty 2k-3\n % \\constraint{$k\\in\\mathbb{Z}$}\n % \\category{power series}\n\\end{equation*}" + }, + + { + "eq": [ + "create( 'series',", + "booklabelv1 = \"0.0.0\",", + "category = \"asymptotic series\"", + "lhs = p,", + "front = 17,", + "general = [ k / 2 ]," + ], + "view_metadata": false, + "res": "\\begin{equation*}\\tag{0.0.0}\n p\n = 17+\\sum_{k=0}^\\infty \\left(\\frac{k}{2}\\right)\n % \\constraint{$$}\n % \\category{asymptotic series}\n\\end{equation*}" + }, + + { + "eq": [ + "create( 'contfrac', ", + "booklabelv1 = \"0.0.0\",", + "category = \"test_data\"", + "lhs = zeta,", + "factor = 15,", + "general = [ [3, 5] ]," + ], + "view_metadata": false, + "res": "\\begin{equation*}\\tag{0.0.0}\n \\zeta\n = 15 \\CFK{m}{1}{\\infty}@@{3}{5}\n % \\constraint{$$}\n % \\category{test_data}\n\\end{equation*}" + }, + + { + "eq": [ + "create( 'contfrac', ", + "booklabelv1 = \"0.0.0\",", + "category = \"test_data\"", + "lhs = zeta,", + "begin = [[1,2]],", + "front = 3,", + "general = [[4, 5]]," + ], + "view_metadata": false, + "res": "\\begin{equation*}\\tag{0.0.0}\n \\zeta\n = 3+\\frac{1}{2} \\subplus \\CFK{m}{2}{\\infty}@@{4}{5}\n % \\constraint{$$}\n % \\category{test_data}\n\\end{equation*}" + }, + + { + "eq": [ + "create( 'contfrac', ", + "booklabelv1 = \"0.0.0\",", + "category = \"test_data\"", + "lhs = zeta,", + "begin = [[1,2]],", + "factor = 3,", + "front = 4,", + "general = [[0, 1]]," + ], + "view_metadata": false, + "res": "\\begin{equation*}\\tag{0.0.0}\n \\zeta\n = 4+\\frac{1}{2} \\subplus 3 \\dots\n % \\constraint{$$}\n % \\category{test_data}\n\\end{equation*}" + }, + + { + "eq": [ + "create( 'contfrac', ", + "booklabelv1 = \"0.0.0\",", + "category = \"test_data\"", + "lhs = zeta,", + "begin = [[1,2]],", + "factor = -1,", + "front = 4,", + "even = [[5, 6]],", + "odd = [[7, 8]]," + ], + "view_metadata": true, + "res": "\\begin{equation*}\\tag{0.0.0}\n \\zeta\n = 4+\\frac{1}{2} \\subplus -\\CFK{m}{2}{\\infty}@@{5}{6}\n\\end{equation*}\n\\begin{center}\nParameters: $$$$\n$$$$\ntest_data\n\\end{center}" + } + ] + } +} \ No newline at end of file diff --git a/maple2latex/test/test_translator.py b/maple2latex/test/test_translator.py new file mode 100644 index 0000000..b9b3bfb --- /dev/null +++ b/maple2latex/test/test_translator.py @@ -0,0 +1,53 @@ +#!/usr/bin/env python + +__author__ = "Joon Bang" +__status__ = "Development" + +import json +from unittest import TestCase +import src.translator as t + +TEST_CASES = json.loads(open("test/test_cases.json").read())["translator"] + +parse_brackets_test_cases = TEST_CASES["parse_brackets"] +trim_parens_test_cases = TEST_CASES["trim_parens"] +basic_translate_test_cases = TEST_CASES["basic_translate"] +get_arguments_test_cases = TEST_CASES["get_arguments"] +translate_test_cases = TEST_CASES["translate"] +make_equation_test_cases = TEST_CASES["make_equation"] + + +class TestParseBrackets(TestCase): + def test_parse_brackets(self): + for case in parse_brackets_test_cases: + self.assertEqual(t.parse_brackets(case["exp"]), case["res"]) + + +class TestTrimParens(TestCase): + def test_trim_parens(self): + for case in trim_parens_test_cases: + self.assertEqual(t.trim_parens(case["exp"]), case["res"]) + + +class TestBasicTranslate(TestCase): + def test_basic_translate(self): + for case in basic_translate_test_cases: + self.assertEqual(t.basic_translate(case["exp"]), case["res"]) + + +class TestGetArguments(TestCase): + def test_get_arguments(self): + for case in get_arguments_test_cases: + self.assertEqual(t.get_arguments(case["function"], case["arg_string"]), case["res"]) + + +class TestTranslate(TestCase): + def test_translate(self): + for case in translate_test_cases: + self.assertEqual(t.translate(case["exp"]), case["res"]) + + +class TestMakeEquation(TestCase): + def test_make_equation(self): + for case in make_equation_test_cases: + self.assertEqual(t.make_equation(t.MapleEquation(case["eq"]), case["view_metadata"]), case["res"]) From c1520ba5fab0864d5eb70821d05536bb0c484a03 Mon Sep 17 00:00:00 2001 From: Joon Bang Date: Tue, 19 Jul 2016 15:00:29 -0400 Subject: [PATCH 171/402] Add unit tests (for real) --- maple2latex/test/__init__.py | 0 maple2latex/test/test_cases.json | 175 ++++++++++++++++++++++++++++ maple2latex/test/test_translator.py | 53 +++++++++ 3 files changed, 228 insertions(+) create mode 100644 maple2latex/test/__init__.py create mode 100644 maple2latex/test/test_cases.json create mode 100644 maple2latex/test/test_translator.py diff --git a/maple2latex/test/__init__.py b/maple2latex/test/__init__.py new file mode 100644 index 0000000..e69de29 diff --git a/maple2latex/test/test_cases.json b/maple2latex/test/test_cases.json new file mode 100644 index 0000000..58c79fb --- /dev/null +++ b/maple2latex/test/test_cases.json @@ -0,0 +1,175 @@ +{ + "translator": { + "parse_brackets": [ + { + "exp": "[1,2, 3]", + "res": [["1", "2"], "3"] + }, + + { + "exp": "[1]", + "res": ["1"] + } + ], + + "trim_parens": [ + { + "exp": "(1 + 2)", + "res": "1 + 2" + }, + + { + "exp": "(3)(4)", + "res": "(3)(4)" + }, + + { + "exp": "((1 + 2))", + "res": "(1 + 2)" + } + ], + + "basic_translate": [ + { + "exp": ["1", "+", "2", "/", "(3)", "*", "4", "-", "5", "!", "^", "-6"], + "res": "1+\\frac{2}{3}4-5!^{-6}" + }, + + { + "exp": ["\\pi", "*", "I"], + "res": "\\pi i" + } + ], + + "get_arguments": [ + { + "function": "rabbit", + "arg_string": ["(", ")"], + "res": [] + }, + + { + "function": "hypergeom", + "arg_string": ["(", "[1", ",", "2]", ",", "[]", ",", "4", ")"], + "res": ["2", "0", "1,2", "", "4"] + }, + + { + "function": "qhyper", + "arg_string": ["(", "[1", ",", "2", ",", "3]", ",", "[4", ",", "5]", ",", "6", ",", "7", ")"], + "res": ["3", "2", "1,2,3", "4,5", "6", "7"] + }, + + { + "function": "sum", + "arg_string": ["(", "m", ",", "1..infinity", ")"], + "res": ["1", "\\infty", "m"] + }, + + { + "function": "BesselJ", + "arg_string": ["(", "\\nu", ",", "z", ")"], + "res": ["\\nu", "z"] + } + ], + + "translate": [ + { + "exp": "sin(x)", + "res": "\\sin(x)" + }, + + { + "exp": "log[beta](nu)", + "res": "\\log_{\\beta} {(\\nu)}" + } + ], + + "make_equation": [ + { + "eq": [ + "create( 'series',", + "booklabelv1 = \"0.0.0\",", + "category = \"power series\"", + "lhs = n,", + "factor = 4/5,", + "general = [ 2 * k - 3 ],", + "constraints = { k::integer }," + ], + "view_metadata": false, + "res": "\\begin{equation*}\\tag{0.0.0}\n n\n = \\frac{4}{5} \\sum_{k=0}^\\infty 2k-3\n % \\constraint{$k\\in\\mathbb{Z}$}\n % \\category{power series}\n\\end{equation*}" + }, + + { + "eq": [ + "create( 'series',", + "booklabelv1 = \"0.0.0\",", + "category = \"asymptotic series\"", + "lhs = p,", + "front = 17,", + "general = [ k / 2 ]," + ], + "view_metadata": false, + "res": "\\begin{equation*}\\tag{0.0.0}\n p\n = 17+\\sum_{k=0}^\\infty \\left(\\frac{k}{2}\\right)\n % \\constraint{$$}\n % \\category{asymptotic series}\n\\end{equation*}" + }, + + { + "eq": [ + "create( 'contfrac', ", + "booklabelv1 = \"0.0.0\",", + "category = \"test_data\"", + "lhs = zeta,", + "factor = 15,", + "general = [ [3, 5] ]," + ], + "view_metadata": false, + "res": "\\begin{equation*}\\tag{0.0.0}\n \\zeta\n = 15 \\CFK{m}{1}{\\infty}@@{3}{5}\n % \\constraint{$$}\n % \\category{test_data}\n\\end{equation*}" + }, + + { + "eq": [ + "create( 'contfrac', ", + "booklabelv1 = \"0.0.0\",", + "category = \"test_data\"", + "lhs = zeta,", + "begin = [[1,2]],", + "front = 3,", + "general = [[4, 5]]," + ], + "view_metadata": false, + "res": "\\begin{equation*}\\tag{0.0.0}\n \\zeta\n = 3+\\frac{1}{2} \\subplus \\CFK{m}{2}{\\infty}@@{4}{5}\n % \\constraint{$$}\n % \\category{test_data}\n\\end{equation*}" + }, + + { + "eq": [ + "create( 'contfrac', ", + "booklabelv1 = \"0.0.0\",", + "category = \"test_data\"", + "lhs = zeta,", + "begin = [[1,2]],", + "factor = 3,", + "front = 4,", + "general = [[0, 1]]," + ], + "view_metadata": false, + "res": "\\begin{equation*}\\tag{0.0.0}\n \\zeta\n = 4+\\frac{1}{2} \\subplus 3 \\dots\n % \\constraint{$$}\n % \\category{test_data}\n\\end{equation*}" + }, + + { + "eq": [ + "create( 'contfrac', ", + "booklabelv1 = \"0.0.0\",", + "category = \"test_data\"", + "lhs = zeta,", + "begin = [[1,2]],", + "factor = -1,", + "front = 4,", + "even = [[5, 6]],", + "odd = [[7, 8]]," + ], + "view_metadata": true, + "res": "\\begin{equation*}\\tag{0.0.0}\n \\zeta\n = 4+\\frac{1}{2} \\subplus -\\CFK{m}{2}{\\infty}@@{5}{6}\n\\end{equation*}\n\\begin{center}\nParameters: $$$$\n$$$$\ntest_data\n\\end{center}" + } + ] + } +} \ No newline at end of file diff --git a/maple2latex/test/test_translator.py b/maple2latex/test/test_translator.py new file mode 100644 index 0000000..b9b3bfb --- /dev/null +++ b/maple2latex/test/test_translator.py @@ -0,0 +1,53 @@ +#!/usr/bin/env python + +__author__ = "Joon Bang" +__status__ = "Development" + +import json +from unittest import TestCase +import src.translator as t + +TEST_CASES = json.loads(open("test/test_cases.json").read())["translator"] + +parse_brackets_test_cases = TEST_CASES["parse_brackets"] +trim_parens_test_cases = TEST_CASES["trim_parens"] +basic_translate_test_cases = TEST_CASES["basic_translate"] +get_arguments_test_cases = TEST_CASES["get_arguments"] +translate_test_cases = TEST_CASES["translate"] +make_equation_test_cases = TEST_CASES["make_equation"] + + +class TestParseBrackets(TestCase): + def test_parse_brackets(self): + for case in parse_brackets_test_cases: + self.assertEqual(t.parse_brackets(case["exp"]), case["res"]) + + +class TestTrimParens(TestCase): + def test_trim_parens(self): + for case in trim_parens_test_cases: + self.assertEqual(t.trim_parens(case["exp"]), case["res"]) + + +class TestBasicTranslate(TestCase): + def test_basic_translate(self): + for case in basic_translate_test_cases: + self.assertEqual(t.basic_translate(case["exp"]), case["res"]) + + +class TestGetArguments(TestCase): + def test_get_arguments(self): + for case in get_arguments_test_cases: + self.assertEqual(t.get_arguments(case["function"], case["arg_string"]), case["res"]) + + +class TestTranslate(TestCase): + def test_translate(self): + for case in translate_test_cases: + self.assertEqual(t.translate(case["exp"]), case["res"]) + + +class TestMakeEquation(TestCase): + def test_make_equation(self): + for case in make_equation_test_cases: + self.assertEqual(t.make_equation(t.MapleEquation(case["eq"]), case["view_metadata"]), case["res"]) From 96b7f647ca4e36a3c05356dac0da8d1454aa1e6e Mon Sep 17 00:00:00 2001 From: Jagan Prem Date: Tue, 19 Jul 2016 15:10:23 -0400 Subject: [PATCH 172/402] Add tests for mathmode.py and re-add missing incrementation. --- mathmode/mathmode.py | 15 +++-- mathmode/mathmodetest.py | 115 +++++++++++++++++++++++++++++++++++++++ 2 files changed, 122 insertions(+), 8 deletions(-) create mode 100644 mathmode/mathmodetest.py diff --git a/mathmode/mathmode.py b/mathmode/mathmode.py index 5091dbf..7927bc0 100644 --- a/mathmode/mathmode.py +++ b/mathmode/mathmode.py @@ -1,6 +1,4 @@ -""" -Dictionary containing math mode delimiters and their respective endpoints. -""" +# Dictionary containing math mode delimiters and their respective endpoints. MATH_START = {"\\[": "\\]", "\\(": "\\)", "$$": "$$", @@ -12,9 +10,7 @@ "\\begin{multline}": "\\end{multline}", "\\begin{multline*}": "\\end{multline*}"} -""" -List of delimiters that exit math mode. -""" +# List of delimiters that exit math mode. MATH_END = ["\\hbox{", "\\mbox{", "\\text{"] @@ -106,7 +102,7 @@ def parse_math(string, start, ranges): ranges.append((begin, start + i)) return i + len(MATH_START[delim]) - 1 i += 1 - return i + raise SyntaxError("missing " + MATH_START[delim]) def parse_non_math(string, start, ranges): @@ -136,11 +132,14 @@ def parse_non_math(string, start, ranges): level += 1 elif string[i] == "}": if level == 0 and delim != "": + i += 1 return i else: level -= 1 i += 1 - return i + if delim == "" and level == 0: + return i + raise SyntaxError("missing end bracket") def find_math_ranges(string): diff --git a/mathmode/mathmodetest.py b/mathmode/mathmodetest.py new file mode 100644 index 0000000..86118e6 --- /dev/null +++ b/mathmode/mathmodetest.py @@ -0,0 +1,115 @@ +import unittest +import mathmode + +DELIM_TESTS = [ + { + "args": ["$test$"], + "output": "$" + }, + { + "args": ["$$test$$"], + "output": "$$" + }, + { + "args": [" \\begin{equation}test\\end{equation} "], + "output": "\\begin{equation}" + }, + { + "args": ["\\hbox{}", False], + "output": "\\hbox{" + } +] + +MATH_TESTS = [ + { + "string": "$\\frac 1 2$ test", + "start": 0, + "output": 10 + }, + { + "string": "\\[2\\hbox{Test}\\]", + "start": 22, + "output": 15 + }, + { + "string": "$\\$$", + "start": 12, + "output": 3 + }, + [ + (1, 10), + (24, 25), + (13, 15) + ] +] + +NON_MATH_TESTS = [ + { + "string": "\\mbox{Test\\(test\\)} ", + "start": 14, + "output": 19 + }, + { + "string": "test \\\\(\\begin{align*}\\end{align*}\\\\[", + "start": 56, + "output": 37 + }, + { + "string": "\\begin{multline}\n\\left(2\\right)\n\\end{multline}", + "start": 3, + "output": 46 + }, + [ + (26, 30), + (19, 35) + ] +] + +RANGE_TESTS = [ + { + "string": "test \\begin{multline*}\ntest \\hbox{Test}\\end{multline*}", + "output": [ + (22, 28) + ] + }, + { + "string": "$2$ \\begin{align}Test\\end{align}", + "output": [ + (1, 2), + (17, 21) + ] + }, + { + "string": "test", + "output": [] + } +] + + +class TestMathMode(unittest.TestCase): + + def test_first_delim(self): + for test in DELIM_TESTS: + self.assertEqual(mathmode.first_delim(*test["args"]), test["output"]) + + def test_parse_math(self): + ranges = [] + for test in MATH_TESTS[:-1]: + self.assertEqual(mathmode.parse_math(test["string"], test["start"], ranges), test["output"]) + self.assertEqual(ranges, MATH_TESTS[-1]) + self.assertRaises(SyntaxError, mathmode.parse_math, "\\begin{equation*}", 23, []) + + def test_parse_non_math(self): + ranges = [] + for test in NON_MATH_TESTS[:-1]: + self.assertEqual(mathmode.parse_non_math(test["string"], test["start"], ranges), test["output"]) + self.assertEqual(ranges, NON_MATH_TESTS[-1]) + self.assertRaises(SyntaxError, mathmode.parse_non_math, "\\text{", 9, [(1, 4)]) + + def test_find_math_ranges(self): + for test in RANGE_TESTS: + self.assertEqual(mathmode.find_math_ranges(test["string"]), test["output"]) + self.assertRaises(SyntaxError, mathmode.find_math_ranges("{$$}$$")) + +if __name__ == "__main__": + unittest.main() From ae47f06d9274aca9aed4dc262849532000f52816 Mon Sep 17 00:00:00 2001 From: Jagan Prem Date: Tue, 19 Jul 2016 15:25:01 -0400 Subject: [PATCH 173/402] Moved math mode code to AlexDanoff and updated test format. --- AlexDanoff/src/math_mode.py | 57 +++---- AlexDanoff/test/test_math_mode.py | 115 +++++++++++++ .../test/test_math_mode.py~ | 0 mathmode/mathmode.py | 154 ------------------ 4 files changed, 141 insertions(+), 185 deletions(-) create mode 100644 AlexDanoff/test/test_math_mode.py rename mathmode/mathmodetest.py => AlexDanoff/test/test_math_mode.py~ (100%) delete mode 100644 mathmode/mathmode.py diff --git a/AlexDanoff/src/math_mode.py b/AlexDanoff/src/math_mode.py index 16954f8..7927bc0 100644 --- a/AlexDanoff/src/math_mode.py +++ b/AlexDanoff/src/math_mode.py @@ -1,17 +1,19 @@ -math_start = {"\\[": "\\]", - "\\(": "\\)", - "$$": "$$", - "$": "$", - "\\begin{equation}": "\\end{equation}", - "\\begin{equation*}": "\\end{equation*}", - "\\begin{align}": "\\end{align}", - "\\begin{align*}": "\\end{align*}", - "\\begin{multline}": "\\end{multline}", - "\\begin{multline*}": "\\end{multline*}"} +# Dictionary containing math mode delimiters and their respective endpoints. +MATH_START = {"\\[": "\\]", + "\\(": "\\)", + "$$": "$$", + "$": "$", + "\\begin{equation}": "\\end{equation}", + "\\begin{equation*}": "\\end{equation*}", + "\\begin{align}": "\\end{align}", + "\\begin{align*}": "\\end{align*}", + "\\begin{multline}": "\\end{multline}", + "\\begin{multline*}": "\\end{multline*}"} -math_end = ["\\hbox{", - "\\mbox{", - "\\text{"] +# List of delimiters that exit math mode. +MATH_END = ["\\hbox{", + "\\mbox{", + "\\text{"] def find_first(string, delim, start=0): @@ -41,7 +43,7 @@ def first_delim(string, enter=True): :return: The delimiter that first appears in the string. """ minm = ["\\hbox{", "\\]"][enter] - for delim in [math_end, math_start][enter]: + for delim in [MATH_END, MATH_START][enter]: i = find_first(string, delim) if i != -1 and i < find_first(string, minm) or minm not in string: minm = delim @@ -57,7 +59,7 @@ def does_exit(string): :param string: The string to check. :return: Whether the string starts with a text mode delimiter. """ - return any(string.startswith(delim) for delim in math_end) + return any(string.startswith(delim) for delim in MATH_END) def does_enter(string): @@ -67,7 +69,7 @@ def does_enter(string): :param string: The string to check. :return: Whether the string starts with a math mode delimiter. """ - return any(string.startswith(delim) for delim in math_start) + return any(string.startswith(delim) for delim in MATH_START) def parse_math(string, start, ranges): @@ -95,12 +97,12 @@ def parse_math(string, start, ranges): ranges.append((begin, start + i)) i += parse_non_math(string[i:], start + i, ranges) begin = start + i - if string[i:].startswith(math_start[delim]): + if string[i:].startswith(MATH_START[delim]): if begin != start + i: ranges.append((begin, start + i)) - return i + len(math_start[delim]) - 1 + return i + len(MATH_START[delim]) - 1 i += 1 - return i + raise SyntaxError("missing " + MATH_START[delim]) def parse_non_math(string, start, ranges): @@ -135,25 +137,18 @@ def parse_non_math(string, start, ranges): else: level -= 1 i += 1 - return i + if delim == "" and level == 0: + return i + raise SyntaxError("missing end bracket") -def find_math_ranges(string,drmf): +def find_math_ranges(string): # type: (str) -> list """ Returns a list of tuples, each tuple denoting a range of math mode. :param string: The string to search within. :return: A list of tuples denoting math mode ranges. """ - global math_end - if drmf == True: - string = string.replace("\\drmfnote","\\drmfname") - math_end.extend(["\\constraint{", - "\\substitution" - "\\drmfnote{", - "\\drmfname{", - "\\proof{", - "\\label{"]) ranges = [] parse_non_math(string, 0, ranges) - return ranges[:] \ No newline at end of file + return ranges[:] diff --git a/AlexDanoff/test/test_math_mode.py b/AlexDanoff/test/test_math_mode.py new file mode 100644 index 0000000..3008742 --- /dev/null +++ b/AlexDanoff/test/test_math_mode.py @@ -0,0 +1,115 @@ +import unittest +import math_mode + +DELIM_TESTS = [ + { + "args": ["$test$"], + "output": "$" + }, + { + "args": ["$$test$$"], + "output": "$$" + }, + { + "args": [" \\begin{equation}test\\end{equation} "], + "output": "\\begin{equation}" + }, + { + "args": ["\\hbox{}", False], + "output": "\\hbox{" + } +] + +MATH_TESTS = [ + { + "string": "$\\frac 1 2$ test", + "start": 0, + "output": 10 + }, + { + "string": "\\[2\\hbox{Test}\\]", + "start": 22, + "output": 15 + }, + { + "string": "$\\$$", + "start": 12, + "output": 3 + }, + [ + (1, 10), + (24, 25), + (13, 15) + ] +] + +NON_MATH_TESTS = [ + { + "string": "\\mbox{Test\\(test\\)} ", + "start": 14, + "output": 19 + }, + { + "string": "test \\\\(\\begin{align*}\\end{align*}\\\\[", + "start": 56, + "output": 37 + }, + { + "string": "\\begin{multline}\n\\left(2\\right)\n\\end{multline}", + "start": 3, + "output": 46 + }, + [ + (26, 30), + (19, 35) + ] +] + +RANGE_TESTS = [ + { + "string": "test \\begin{multline*}\ntest \\hbox{Test}\\end{multline*}", + "output": [ + (22, 28) + ] + }, + { + "string": "$2$ \\begin{align}Test\\end{align}", + "output": [ + (1, 2), + (17, 21) + ] + }, + { + "string": "test", + "output": [] + } +] + + +class Testmath_mode(unittest.TestCase): + + def test_first_delim(self): + for test in DELIM_TESTS: + self.assertEqual(math_mode.first_delim(*test["args"]), test["output"]) + + def test_parse_math(self): + ranges = [] + for test in MATH_TESTS[:-1]: + self.assertEqual(math_mode.parse_math(test["string"], test["start"], ranges), test["output"]) + self.assertEqual(ranges, MATH_TESTS[-1]) + self.assertRaises(SyntaxError, math_mode.parse_math, "\\begin{equation*}", 23, []) + + def test_parse_non_math(self): + ranges = [] + for test in NON_MATH_TESTS[:-1]: + self.assertEqual(math_mode.parse_non_math(test["string"], test["start"], ranges), test["output"]) + self.assertEqual(ranges, NON_MATH_TESTS[-1]) + self.assertRaises(SyntaxError, math_mode.parse_non_math, "\\text{", 9, [(1, 4)]) + + def test_find_math_ranges(self): + for test in RANGE_TESTS: + self.assertEqual(math_mode.find_math_ranges(test["string"]), test["output"]) + self.assertRaises(SyntaxError, math_mode.find_math_ranges, "{$$}$$") + +if __name__ == "__main__": + unittest.main() diff --git a/mathmode/mathmodetest.py b/AlexDanoff/test/test_math_mode.py~ similarity index 100% rename from mathmode/mathmodetest.py rename to AlexDanoff/test/test_math_mode.py~ diff --git a/mathmode/mathmode.py b/mathmode/mathmode.py deleted file mode 100644 index 7927bc0..0000000 --- a/mathmode/mathmode.py +++ /dev/null @@ -1,154 +0,0 @@ -# Dictionary containing math mode delimiters and their respective endpoints. -MATH_START = {"\\[": "\\]", - "\\(": "\\)", - "$$": "$$", - "$": "$", - "\\begin{equation}": "\\end{equation}", - "\\begin{equation*}": "\\end{equation*}", - "\\begin{align}": "\\end{align}", - "\\begin{align*}": "\\end{align*}", - "\\begin{multline}": "\\end{multline}", - "\\begin{multline*}": "\\end{multline*}"} - -# List of delimiters that exit math mode. -MATH_END = ["\\hbox{", - "\\mbox{", - "\\text{"] - - -def find_first(string, delim, start=0): - # type: (str, str, int) -> int - """ - Finds the first occurrence of a delimiter within a string. - :param string: The string to search within. - :param delim: The delimiter to search for. - :param start: The starting index in the string. - :return: The index of the first occurrence. - """ - i = string.find(delim, start) - if delim in ["$", "\\[", "\\("]: - if i == -1: - return -1 - elif i != 0 and string[i - 1] == "\\": - return find_first(string, delim, i + len(delim)) - return i - - -def first_delim(string, enter=True): - # type: (str, bool) -> str - """ - Finds the first math or text mode delimiter within a string. - :param string: The string to search within. - :param enter: Whether to find a math mode or a text mode delimiter. - :return: The delimiter that first appears in the string. - """ - minm = ["\\hbox{", "\\]"][enter] - for delim in [MATH_END, MATH_START][enter]: - i = find_first(string, delim) - if i != -1 and i < find_first(string, minm) or minm not in string: - minm = delim - if minm == "$" and string.find("$$") == string.find("$"): - minm = "$$" - return minm - - -def does_exit(string): - # type: (str) -> bool - """ - Returns whether a string starts with a text mode delimiter. - :param string: The string to check. - :return: Whether the string starts with a text mode delimiter. - """ - return any(string.startswith(delim) for delim in MATH_END) - - -def does_enter(string): - # type: (str) -> bool - """ - Returns whether a string starts with a math mode delimiter. - :param string: The string to check. - :return: Whether the string starts with a math mode delimiter. - """ - return any(string.startswith(delim) for delim in MATH_START) - - -def parse_math(string, start, ranges): - # type: (str) -> str, int - """ - Finds the ranges of the current math mode segment. - :param string: The string to parse. - :param start: The starting index in the original string. - :param ranges: The ranges of all math mode segments. - :return i: The length of the math mode segment. - """ - delim = first_delim(string) - i = len(delim) - begin = start + i - while i < len(string): - if string[i:].startswith("\\$"): - i += 1 - elif string[i:].startswith("\\\\]"): - i += 2 - elif string[i:].startswith("\\\\)"): - i += 2 - else: - if does_exit(string[i:]): - if begin != start + i: - ranges.append((begin, start + i)) - i += parse_non_math(string[i:], start + i, ranges) - begin = start + i - if string[i:].startswith(MATH_START[delim]): - if begin != start + i: - ranges.append((begin, start + i)) - return i + len(MATH_START[delim]) - 1 - i += 1 - raise SyntaxError("missing " + MATH_START[delim]) - - -def parse_non_math(string, start, ranges): - # type: (str) -> str, int - """ - Finds occurrences of math mode within a segment of text mode. - :param string: The string to parse. - :param start: The starting index in the original string. - :param ranges: The ranges of all math mode segments. - :return i: The length of the text mode segment. - """ - delim = first_delim(string, False) - if not string.startswith(delim): - delim = "" - level = 0 - i = len(delim) - while i < len(string): - if string[i:].startswith("\\$"): - i += 1 - elif string[i:].startswith("\\\\["): - i += 2 - elif string[i:].startswith("\\\\("): - i += 2 - elif does_enter(string[i:]): - i += parse_math(string[i:], start + i, ranges) - elif string[i] == "{": - level += 1 - elif string[i] == "}": - if level == 0 and delim != "": - i += 1 - return i - else: - level -= 1 - i += 1 - if delim == "" and level == 0: - return i - raise SyntaxError("missing end bracket") - - -def find_math_ranges(string): - # type: (str) -> list - """ - Returns a list of tuples, each tuple denoting a range of math mode. - :param string: The string to search within. - :return: A list of tuples denoting math mode ranges. - """ - ranges = [] - parse_non_math(string, 0, ranges) - return ranges[:] From 038ced202820844f6b967b6f5b630cf70ea1a380 Mon Sep 17 00:00:00 2001 From: Parth Oza Date: Tue, 19 Jul 2016 16:01:31 -0400 Subject: [PATCH 174/402] Added symono.cls back --- KLSadd_insertion/src/svmono.cls | 1690 +++++++++++++++++++++++++++++++ 1 file changed, 1690 insertions(+) create mode 100644 KLSadd_insertion/src/svmono.cls diff --git a/KLSadd_insertion/src/svmono.cls b/KLSadd_insertion/src/svmono.cls new file mode 100644 index 0000000..2341bee --- /dev/null +++ b/KLSadd_insertion/src/svmono.cls @@ -0,0 +1,1690 @@ +% SVMONO DOCUMENT CLASS -- version 3.9 (23-Jul-03) +% Springer Verlag global LaTeX2e support for monographs +%% +%% +%% \CharacterTable +%% {Upper-case \A\B\C\D\E\F\G\H\I\J\K\L\M\N\O\P\Q\R\S\T\U\V\W\X\Y\Z +%% Lower-case \a\b\c\d\e\f\g\h\i\j\k\l\m\n\o\p\q\r\s\t\u\v\w\x\y\z +%% Digits \0\1\2\3\4\5\6\7\8\9 +%% Exclamation \! Double quote \" Hash (number) \# +%% Dollar \$ Percent \% Ampersand \& +%% Acute accent \' Left paren \( Right paren \) +%% Asterisk \* Plus \+ Comma \, +%% Minus \- Point \. Solidus \/ +%% Colon \: Semicolon \; Less than \< +%% Equals \= Greater than \> Question mark \? +%% Commercial at \@ Left bracket \[ Backslash \\ +%% Right bracket \] Circumflex \^ Underscore \_ +%% Grave accent \` Left brace \{ Vertical bar \| +%% Right brace \} Tilde \~} +%% +\NeedsTeXFormat{LaTeX2e}[1995/12/01] +\ProvidesClass{svmono}[2003/07/23 3.9 +^^JSpringer Verlag global LaTeX document class for monographs] + +% Options +% citations +\DeclareOption{natbib}{\ExecuteOptions{oribibl}% +\AtEndOfClass{% Loading package 'NATBIB' +\RequirePackage{natbib} +% Changing some parameters of NATBIB +\setlength{\bibhang}{\parindent} +%\setlength{\bibsep}{0mm} +\let\bibfont=\small +\def\@biblabel#1{#1.} +\newcommand{\etal}{\textit{et al}.} +%\bibpunct[,]{(}{)}{;}{a}{}{,}}} +}} +% Springer environment +\let\if@spthms\iftrue +\DeclareOption{nospthms}{\let\if@spthms\iffalse} +% +\let\envankh\@empty % no anchor for "theorems" +% +\let\if@envcntreset\iffalse % environment counter is not reset +\let\if@envcntresetsect=\iffalse % reset each section? +\DeclareOption{envcountresetchap}{\let\if@envcntreset\iftrue} +\DeclareOption{envcountresetsect}{\let\if@envcntreset\iftrue +\let\if@envcntresetsect=\iftrue} +% +\let\if@envcntsame\iffalse % NOT all environments work like "Theorem", + % each using its own counter +\DeclareOption{envcountsame}{\let\if@envcntsame\iftrue} +% +\let\if@envcntshowhiercnt=\iffalse % do not show hierarchy counter at all +% +% enhance theorem counter +\DeclareOption{envcountchap}{\def\envankh{chapter}% show \thechapter along with theorem number +\let\if@envcntshowhiercnt=\iftrue +\ExecuteOptions{envcountreset}} +% +\DeclareOption{envcountsect}{\def\envankh{section}% show \thesection along with theorem number +\let\if@envcntshowhiercnt=\iftrue +\ExecuteOptions{envcountreset}} +% +% languages +\let\switcht@@therlang\relax +\let\svlanginfo\relax +\def\ds@deutsch{\def\switcht@@therlang{\switcht@deutsch}% +\gdef\svlanginfo{\typeout{Man spricht deutsch.}\global\let\svlanginfo\relax}} +\def\ds@francais{\def\switcht@@therlang{\switcht@francais}% +\gdef\svlanginfo{\typeout{On parle francais.}\global\let\svlanginfo\relax}} +% +\AtBeginDocument{\@ifpackageloaded{babel}{% +\@ifundefined{extrasenglish}{}{\addto\extrasenglish{\switcht@albion}}% +\@ifundefined{extrasUKenglish}{}{\addto\extrasUKenglish{\switcht@albion}}% +\@ifundefined{extrasfrenchb}{}{\addto\extrasfrenchb{\switcht@francais}}% +\@ifundefined{extrasgerman}{}{\addto\extrasgerman{\switcht@deutsch}}% +}{\switcht@@therlang}% +} +% numbering style of floats, equations +\newif\if@numart \@numartfalse +\DeclareOption{numart}{\@numarttrue} +\def\set@numbering{\if@numart\else\num@book\fi} +\AtEndOfClass{\set@numbering} +% style for vectors +\DeclareOption{vecphys}{\def\vec@style{phys}} +\DeclareOption{vecarrow}{\def\vec@style{arrow}} +% running heads +\let\if@runhead\iftrue +\DeclareOption{norunningheads}{\let\if@runhead\iffalse} +% referee option +\let\if@referee\iffalse +\def\makereferee{\def\baselinestretch{2}\selectfont +\newbox\refereebox +\setbox\refereebox=\vbox to\z@{\vskip0.5cm% + \hbox to\textwidth{\normalsize\tt\hrulefill\lower0.5ex + \hbox{\kern5\p@ referee's copy\kern5\p@}\hrulefill}\vss}% +\def\@oddfoot{\copy\refereebox}\let\@evenfoot=\@oddfoot} +\DeclareOption{referee}{\let\if@referee\iftrue +\AtBeginDocument{\makereferee\small\normalsize}} +% modification of thebibliography +\let\if@openbib\iffalse +\DeclareOption{openbib}{\let\if@openbib\iftrue} +% LaTeX standard, sectionwise references +\DeclareOption{oribibl}{\let\oribibl=Y} +\DeclareOption{sectrefs}{\let\secbibl=Y} +% +% footinfo option (provides an informatory line on every page) +\def\SpringerMacroPackageNameA{svmono.cls} +% \thetime, \thedate and \timstamp are macros to include +% time, date (or both) of the TeX run in the document +\def\maketimestamp{\count255=\time +\divide\count255 by 60\relax +\edef\thetime{\the\count255:}% +\multiply\count255 by-60\relax +\advance\count255 by\time +\edef\thetime{\thetime\ifnum\count255<10 0\fi\the\count255} +\edef\thedate{\number\day-\ifcase\month\or Jan\or Feb\or Mar\or + Apr\or May\or Jun\or Jul\or Aug\or Sep\or Oct\or + Nov\or Dec\fi-\number\year} +\def\timstamp{\hbox to\hsize{\tt\hfil\thedate\hfil\thetime\hfil}}} +\maketimestamp +% +% \footinfo generates a info footline on every page containing +% pagenumber, jobname, macroname, and timestamp +\DeclareOption{footinfo}{\AtBeginDocument{\maketimestamp + \def\@oddfoot{\footnotesize\tt Page: \thepage\hfil job: \jobname\hfil + macro: \SpringerMacroPackageNameA\hfil + date/time: \thedate/\thetime}% + \let\@evenfoot=\@oddfoot}} +% +% start new chapter on any page +\newif\if@openright \@openrighttrue +\DeclareOption{openany}{\@openrightfalse} +% +% no size changing allowed +\DeclareOption{11pt}{\OptionNotUsed} +\DeclareOption{12pt}{\OptionNotUsed} +% options for the article class +\def\@rticle@options{10pt,twoside} +% fleqn +\DeclareOption{fleqn}{\def\@rticle@options{10pt,twoside,fleqn}% +\AtEndOfClass{\let\leftlegendglue\relax}% +\AtBeginDocument{\mathindent\parindent}} +% hanging sectioning titles +\let\if@sechang\iffalse +\DeclareOption{sechang}{\let\if@sechang\iftrue} +\def\ClassInfoNoLine#1#2{% + \ClassInfo{#1}{#2\@gobble}% +} +\let\SVMonoOpt\@empty +\DeclareOption*{\InputIfFileExists{sv\CurrentOption.clo}{% +\global\let\SVMonoOpt\CurrentOption}{% +\ClassWarning{Springer-SVMono}{Specified option or subpackage +"\CurrentOption" \MessageBreak not found +passing it to article class \MessageBreak +-}\PassOptionsToClass{\CurrentOption}{article}% +}} +\ProcessOptions\relax +\ifx\SVMonoOpt\@empty\relax +\ClassInfoNoLine{Springer-SVMono}{extra/valid Springer sub-package +\MessageBreak not found in option list - using "global" style}{} +\fi +\LoadClass[\@rticle@options]{article} +\raggedbottom + +% various sizes and settings for monographs + +%\setlength{\textwidth}{28pc} % 11.8cm +\setlength{\textwidth}{43pc} % 11.8cm +%\setlength{\textheight}{12pt}\multiply\textheight by 45\relax +%\setlength{\textheight}{540\p@} +\setlength{\textheight}{720\p@} +%\setlength{\topmargin}{0cm} +\setlength{\topmargin}{-2cm} +%\setlength\oddsidemargin {63\p@} +%\setlength\evensidemargin {63\p@} +\setlength\oddsidemargin {-28\p@} +\setlength\evensidemargin {-28\p@} +\setlength\marginparwidth{90\p@} +\setlength\headsep {12\p@} + +\setlength{\parindent}{15\p@} +\setlength{\parskip}{\z@ \@plus \p@} +\setlength{\hfuzz}{2\p@} +\setlength{\arraycolsep}{1.5\p@} + +\frenchspacing + +\tolerance=500 + +\predisplaypenalty=0 +\clubpenalty=10000 +\widowpenalty=10000 + +\setlength\footnotesep{7.7\p@} + +\newdimen\betweenumberspace % dimension for space between +\betweenumberspace=5\p@ % number and text of titles +\newdimen\headlineindent % dimension for space of +\headlineindent=2.5cc % number and gap of running heads + +% fonts, sizes, and the like +\renewcommand\small{% + \@setfontsize\small\@ixpt{11}% + \abovedisplayskip 8.5\p@ \@plus3\p@ \@minus4\p@ + \abovedisplayshortskip \z@ \@plus2\p@ + \belowdisplayshortskip 4\p@ \@plus2\p@ \@minus2\p@ + \def\@listi{\leftmargin\leftmargini + \parsep \z@ \@plus\p@ \@minus\p@ + \topsep 6\p@ \@plus2\p@ \@minus4\p@ + \itemsep\z@}% + \belowdisplayskip \abovedisplayskip +} +% +\let\footnotesize=\small +% +\newenvironment{petit}{\par\addvspace{6\p@}\small}{\par\addvspace{6\p@}} +% + +% modification of automatic positioning of floating objects +\setlength\@fptop{\z@ } +\setlength\@fpsep{12\p@ } +\setlength\@fpbot{\z@ \@plus 1fil } +\def\textfraction{.01} +\def\floatpagefraction{.8} +\setlength{\intextsep}{20\p@ \@plus 2\p@ \@minus 2\p@} +\setcounter{topnumber}{4} +\def\topfraction{.9} +\setcounter{bottomnumber}{2} +\def\bottomfraction{.7} +\setcounter{totalnumber}{6} +% +% size and style of headings +\newcommand{\partsize}{\Large} +\newcommand{\partstyle}{\bfseries\boldmath} +\newcommand{\chapsize}{\Large} +\newcommand{\chapstyle}{\bfseries\boldmath} +\newcommand{\secsize}{\large} +\newcommand{\secstyle}{\bfseries\boldmath} +\newcommand{\subsecsize}{\normalsize} +\newcommand{\subsecstyle}{\bfseries\boldmath} + +\def\newendpage {\par + \ifdim \pagetotal>\topskip + \else\thispagestyle{empty} + \fi + \vfil\penalty -\@M} +\def\clearpage{% + \ifvmode + \ifnum \@dbltopnum =\m@ne + \ifdim \pagetotal <\topskip + \hbox{} + \fi + \fi + \fi + \newendpage + \write\m@ne{} + \vbox{} + \penalty -\@Mi +} +\def\cleardoublepage{\clearpage\if@twoside \ifodd\c@page\else + \hbox{}\newendpage\if@twocolumn\hbox{}\newpage\fi\fi\fi} + +\newcommand{\clearemptydoublepage}{% + \newpage{\pagestyle{empty}\cleardoublepage}} +\newcommand{\startnewpage}{\if@openright\clearemptydoublepage\else\clearpage\fi} + +% redefinition of \part +\renewcommand\part{\clearemptydoublepage + \thispagestyle{empty} + \if@twocolumn + \onecolumn + \@tempswatrue + \else + \@tempswafalse + \fi + \@ifundefined{thispagecropped}{}{\thispagecropped} + \secdef\@part\@spart} + +\def\@part[#1]#2{\ifnum \c@secnumdepth >-2\relax + \refstepcounter{part} + \addcontentsline{toc}{part}{\partname\ + \thepart\thechapterend\hskip\betweenumberspace + #1}\else + \addcontentsline{toc}{part}{#1}\fi + \markboth{}{} + {\raggedleft + \ifnum \c@secnumdepth >-2\relax + \normalfont\partstyle\partsize\vrule height 34pt width 0pt depth 0pt% + \partname\ \thepart\llap{\smash{\lower 5pt\hbox to\textwidth{\hrulefill}}} + \par + \vskip 128.3\p@ \fi + #2\par}\@endpart} +% +% \@endpart finishes the part page +% +\def\@endpart{\vfil\newpage + \if@twoside + \hbox{} + \thispagestyle{empty} + \newpage + \fi + \if@tempswa + \twocolumn + \fi} +% +\def\@spart#1{{\raggedleft + \normalfont\partsize\partstyle + #1\par}\@endpart} +% +% (re)define sectioning +\setcounter{secnumdepth}{2} + +\def\seccounterend{} +\def\seccountergap{\hskip\betweenumberspace} +\def\@seccntformat#1{\csname the#1\endcsname\seccounterend\seccountergap\ignorespaces} +% +\let\firstmark=\botmark +% +\@ifundefined{thechapterend}{\def\thechapterend{}}{} +% +\if@sechang + \def\sec@hangfrom#1{\setbox\@tempboxa\hbox{#1}% + \hangindent\wd\@tempboxa\noindent\box\@tempboxa} +\else + \def\sec@hangfrom#1{\setbox\@tempboxa\hbox{#1}% + \hangindent\z@\noindent\box\@tempboxa} +\fi + +\def\chap@hangfrom#1{\noindent\vrule height 34pt width 0pt depth 0pt +\rlap{\smash{\lower 5pt\hbox to\textwidth{\hrulefill}}}\hbox{#1} +\vskip10pt} +\def\schap@hangfrom{\chap@hangfrom{}} + +\newcounter{chapter} +% +\@addtoreset{section}{chapter} +\@addtoreset{footnote}{chapter} + +\newif\if@mainmatter \@mainmattertrue +\newcommand\frontmatter{\startnewpage + \@mainmatterfalse\pagenumbering{Roman} + \setcounter{page}{5}} +% +\newcommand\mainmatter{\clearemptydoublepage + \@mainmattertrue\pagenumbering{arabic}} +% +\newcommand\backmatter{\clearemptydoublepage\@mainmatterfalse} + +\def\@chapapp{\chaptername} + +\newdimen\chapstarthookwidth +\newcommand\chapstarthook[2][0.66\textwidth]{% +\setlength{\chapstarthookwidth}{#1}% +\gdef\chapst@rthook{#2}} + +\newcommand{\processchapstarthook}{\@ifundefined{chapst@rthook}{}{% + \setbox0=\hbox{\vbox{ + \begin{flushright} + \begin{minipage}{\chapstarthookwidth} + \vrule\@width\z@\@height21\p@\@depth\z@ + \normalfont\small\itshape\chapst@rthook + \end{minipage} + \end{flushright}}}% + \@tempdima=\pagetotal + \advance\@tempdima by\ht0 + \ifdim\@tempdima<106\p@ + \multiply\@tempdima by-1 + \advance\@tempdima by106\p@ + \vskip\@tempdima + \fi + \box0\par + \global\let\chapst@rthook=\undefined}} + +\newcommand\chapter{\startnewpage + \@ifundefined{thispagecropped}{}{\thispagecropped} + \thispagestyle{empty}% + \global\@topnum\z@ + \@afterindentfalse + \secdef\@chapter\@schapter} + +\def\@chapter[#1]#2{\ifnum \c@secnumdepth >\m@ne + \refstepcounter{chapter}% + \if@mainmatter + \typeout{\@chapapp\space\thechapter.}% + \addcontentsline{toc}{chapter}{\protect + \numberline{\thechapter\thechapterend}#1}% + \else + \addcontentsline{toc}{chapter}{#1}% + \fi + \else + \addcontentsline{toc}{chapter}{#1}% + \fi + \chaptermark{#1}% + \addtocontents{lof}{\protect\addvspace{10\p@}}% + \addtocontents{lot}{\protect\addvspace{10\p@}}% + \if@twocolumn + \@topnewpage[\@makechapterhead{#2}]% + \else + \@makechapterhead{#2}% + \@afterheading + \fi} + +\def\@schapter#1{\if@twocolumn + \@topnewpage[\@makeschapterhead{#1}]% + \else + \@makeschapterhead{#1}% + \@afterheading + \fi} + +%%changes position and layout of numbered chapter headings +\def\@makechapterhead#1{{\parindent\z@\raggedright\normalfont + \hyphenpenalty \@M + \interlinepenalty\@M + \chapsize\chapstyle + \chap@hangfrom{\thechapter\thechapterend\hskip\betweenumberspace}%!!! + \ignorespaces#1\par\nobreak + \processchapstarthook + \ifdim\pagetotal>157\p@ + \vskip 11\p@ + \else + \@tempdima=168\p@\advance\@tempdima by-\pagetotal + \vskip\@tempdima + \fi}} + +%%changes position and layout of unnumbered chapter headings +\def\@makeschapterhead#1{{\parindent \z@ \raggedright\normalfont + \hyphenpenalty \@M + \interlinepenalty\@M + \chapsize\chapstyle + \schap@hangfrom + \ignorespaces#1\par\nobreak + \processchapstarthook + \ifdim\pagetotal>157\p@ + \vskip 11\p@ + \else + \@tempdima=168\p@\advance\@tempdima by-\pagetotal + \vskip\@tempdima + \fi}} + +% predefined unnumbered headings +\newcommand{\preface}[1][\prefacename]{\chapter*{#1}\markboth{#1}{#1}} + +% measures and setting of sections +\renewcommand\section{\@startsection{section}{1}{\z@}% + {-24\p@ \@plus -4\p@ \@minus -4\p@}% + {12\p@ \@plus 4\p@ \@minus 4\p@}% + {\normalfont\secsize\secstyle + \rightskip=\z@ \@plus 8em\pretolerance=10000 }} +\renewcommand\subsection{\@startsection{subsection}{2}{\z@}% + {-17\p@ \@plus -4\p@ \@minus -4\p@}% + {10\p@ \@plus 4\p@ \@minus 4\p@}% + {\normalfont\subsecsize\subsecstyle + \rightskip=\z@ \@plus 8em\pretolerance=10000 }} +\renewcommand\subsubsection{\@startsection{subsubsection}{3}{\z@}% + {-17\p@ \@plus -4\p@ \@minus -4\p@}% + {10\p@ \@plus 4\p@ \@minus 4\p@}% + {\normalfont\normalsize\subsecstyle + \rightskip=\z@ \@plus 8em\pretolerance=10000 }} +\renewcommand\paragraph{\@startsection{paragraph}{4}{\z@}% + {-10\p@ \@plus -4\p@ \@minus -4\p@}% + {10\p@ \@plus 4\p@ \@minus 4\p@}% + {\normalfont\normalsize\itshape + \rightskip=\z@ \@plus 8em\pretolerance=10000 }} +\def\subparagraph{\@startsection{subparagraph}{5}{\z@}% + {-5.388\p@ \@plus-4\p@ \@minus-4\p@}{-5\p@}{\normalfont\normalsize\itshape}} + +% Appendix +\renewcommand\appendix{\par + \stepcounter{chapter} + \setcounter{chapter}{0} + \stepcounter{section} + \setcounter{section}{0} + \setcounter{equation}{0} + \setcounter{figure}{0} + \setcounter{table}{0} + \setcounter{footnote}{0} + \def\@chapapp{\appendixname}% + \renewcommand\thechapter{\@Alph\c@chapter}} + +% definition of sections +% \hyphenpenalty and \raggedright added, so that there is no +% hyphenation and the text is set ragged-right in sectioning + +\def\runinsep{} +\def\aftertext{\unskip\runinsep} +% +\def\thesection{\thechapter.\arabic{section}} +\def\thesubsection{\thesection.\arabic{subsection}} +\def\thesubsubsection{\thesubsection.\arabic{subsubsection}} +\def\theparagraph{\thesubsubsection.\arabic{paragraph}} +\def\thesubparagraph{\theparagraph.\arabic{subparagraph}} +\def\chaptermark#1{} +% +\def\@ssect#1#2#3#4#5{% + \@tempskipa #3\relax + \ifdim \@tempskipa>\z@ + \begingroup + #4{% + \@hangfrom{\hskip #1}% + \raggedright + \hyphenpenalty \@M + \interlinepenalty \@M #5\@@par}% + \endgroup + \else + \def\@svsechd{#4{\hskip #1\relax #5}}% + \fi + \@xsect{#3}} +% +\def\@sect#1#2#3#4#5#6[#7]#8{% + \ifnum #2>\c@secnumdepth + \let\@svsec\@empty + \else + \refstepcounter{#1}% + \protected@edef\@svsec{\@seccntformat{#1}\relax}% + \fi + \@tempskipa #5\relax + \ifdim \@tempskipa>\z@ + \begingroup #6\relax + \sec@hangfrom{\hskip #3\relax\@svsec}% + {\raggedright + \hyphenpenalty \@M + \interlinepenalty \@M #8\@@par}% + \endgroup + \csname #1mark\endcsname{#7\seccounterend}% + \addcontentsline{toc}{#1}{\ifnum #2>\c@secnumdepth + \else + \protect\numberline{\csname the#1\endcsname\seccounterend}% + \fi + #7}% + \else + \def\@svsechd{% + #6\hskip #3\relax + \@svsec #8\aftertext\ignorespaces + \csname #1mark\endcsname{#7}% + \addcontentsline{toc}{#1}{% + \ifnum #2>\c@secnumdepth \else + \protect\numberline{\csname the#1\endcsname\seccounterend}% + \fi + #7}}% + \fi + \@xsect{#5}} + +% figures and tables are processed in small print +\def \@floatboxreset {% + \reset@font + \small + \@setnobreak + \@setminipage +} +\def\fps@figure{htbp} +\def\fps@table{htbp} + +% Frame for paste-in figures or tables +\def\mpicplace#1#2{% #1 =width #2 =height +\vbox{\hbox to #1{\vrule\@width \fboxrule \@height #2\hfill}}} + +% labels of enumerate +\renewcommand\labelenumii{\theenumii)} +\renewcommand\theenumii{\@alph\c@enumii} + +% labels of itemize +\renewcommand\labelitemi{\textbullet} +\renewcommand\labelitemii{\textendash} +\let\labelitemiii=\labelitemiv + +% labels of description +\renewcommand*\descriptionlabel[1]{\hspace\labelsep #1\hfil} + +% fixed indentation for standard itemize-environment +\newdimen\svitemindent \setlength{\svitemindent}{\parindent} + + +% make indentations changeable + +\def\setitemindent#1{\settowidth{\labelwidth}{#1}% + \let\setit@m=Y% + \leftmargini\labelwidth + \advance\leftmargini\labelsep + \def\@listi{\leftmargin\leftmargini + \labelwidth\leftmargini\advance\labelwidth by -\labelsep + \parsep=\parskip + \topsep=\medskipamount + \itemsep=\parskip \advance\itemsep by -\parsep}} +\def\setitemitemindent#1{\settowidth{\labelwidth}{#1}% + \let\setit@m=Y% + \leftmarginii\labelwidth + \advance\leftmarginii\labelsep +\def\@listii{\leftmargin\leftmarginii + \labelwidth\leftmarginii\advance\labelwidth by -\labelsep + \parsep=\parskip + \topsep=\z@ + \itemsep=\parskip \advance\itemsep by -\parsep}} +% +% adjusted environment "description" +% if an optional parameter (at the first two levels of lists) +% is present, its width is considered to be the widest mark +% throughout the current list. +\def\description{\@ifnextchar[{\@describe}{\list{}{\labelwidth\z@ + \itemindent-\leftmargin \let\makelabel\descriptionlabel}}} +% +\def\describelabel#1{#1\hfil} +\def\@describe[#1]{\relax\ifnum\@listdepth=0 +\setitemindent{#1}\else\ifnum\@listdepth=1 +\setitemitemindent{#1}\fi\fi +\list{--}{\let\makelabel\describelabel}} +% +\def\itemize{% + \ifnum \@itemdepth >\thr@@\@toodeep\else + \advance\@itemdepth\@ne + \ifx\setit@m\undefined + \ifnum \@itemdepth=1 \leftmargini=\svitemindent + \labelwidth\leftmargini\advance\labelwidth-\labelsep + \leftmarginii=\leftmargini \leftmarginiii=\leftmargini + \fi + \fi + \edef\@itemitem{labelitem\romannumeral\the\@itemdepth}% + \expandafter\list + \csname\@itemitem\endcsname + {\def\makelabel##1{\rlap{##1}\hss}}% + \fi} +% +\newdimen\verbatimindent \verbatimindent\parindent +\def\verbatim{\advance\@totalleftmargin by\verbatimindent +\@verbatim \frenchspacing\@vobeyspaces \@xverbatim} + +% +% special signs and characters +\newcommand{\D}{\mathrm{d}} +\newcommand{\E}{\mathrm{e}} +\let\eul=\E +\newcommand{\I}{{\rm i}} +\let\imag=\I +% +% the definition of uppercase Greek characters +% Springer likes them as italics to depict variables +\DeclareMathSymbol{\Gamma}{\mathalpha}{letters}{"00} +\DeclareMathSymbol{\Delta}{\mathalpha}{letters}{"01} +\DeclareMathSymbol{\Theta}{\mathalpha}{letters}{"02} +\DeclareMathSymbol{\Lambda}{\mathalpha}{letters}{"03} +\DeclareMathSymbol{\Xi}{\mathalpha}{letters}{"04} +\DeclareMathSymbol{\Pi}{\mathalpha}{letters}{"05} +\DeclareMathSymbol{\Sigma}{\mathalpha}{letters}{"06} +\DeclareMathSymbol{\Upsilon}{\mathalpha}{letters}{"07} +\DeclareMathSymbol{\Phi}{\mathalpha}{letters}{"08} +\DeclareMathSymbol{\Psi}{\mathalpha}{letters}{"09} +\DeclareMathSymbol{\Omega}{\mathalpha}{letters}{"0A} +% the upright forms are defined here as \var +\DeclareMathSymbol{\varGamma}{\mathalpha}{operators}{"00} +\DeclareMathSymbol{\varDelta}{\mathalpha}{operators}{"01} +\DeclareMathSymbol{\varTheta}{\mathalpha}{operators}{"02} +\DeclareMathSymbol{\varLambda}{\mathalpha}{operators}{"03} +\DeclareMathSymbol{\varXi}{\mathalpha}{operators}{"04} +\DeclareMathSymbol{\varPi}{\mathalpha}{operators}{"05} +\DeclareMathSymbol{\varSigma}{\mathalpha}{operators}{"06} +\DeclareMathSymbol{\varUpsilon}{\mathalpha}{operators}{"07} +\DeclareMathSymbol{\varPhi}{\mathalpha}{operators}{"08} +\DeclareMathSymbol{\varPsi}{\mathalpha}{operators}{"09} +\DeclareMathSymbol{\varOmega}{\mathalpha}{operators}{"0A} +% Upright Lower Case Greek letters without using a new MathAlphabet +\newcommand{\greeksym}[1]{\usefont{U}{psy}{m}{n}#1} +\newcommand{\greeksymbold}[1]{{\usefont{U}{psy}{b}{n}#1}} +\newcommand{\allmodesymb}[2]{\relax\ifmmode{\mathchoice +{\mbox{\fontsize{\tf@size}{\tf@size}#1{#2}}} +{\mbox{\fontsize{\tf@size}{\tf@size}#1{#2}}} +{\mbox{\fontsize{\sf@size}{\sf@size}#1{#2}}} +{\mbox{\fontsize{\ssf@size}{\ssf@size}#1{#2}}}} +\else +\mbox{#1{#2}}\fi} +% Definition of lower case Greek letters +\newcommand{\ualpha}{\allmodesymb{\greeksym}{a}} +\newcommand{\ubeta}{\allmodesymb{\greeksym}{b}} +\newcommand{\uchi}{\allmodesymb{\greeksym}{c}} +\newcommand{\udelta}{\allmodesymb{\greeksym}{d}} +\newcommand{\ugamma}{\allmodesymb{\greeksym}{g}} +\newcommand{\umu}{\allmodesymb{\greeksym}{m}} +\newcommand{\unu}{\allmodesymb{\greeksym}{n}} +\newcommand{\upi}{\allmodesymb{\greeksym}{p}} +\newcommand{\utau}{\allmodesymb{\greeksym}{t}} +% redefines the \vec accent to a bold character - if desired +\def\fig@type{arrow}% temporarily abused +\ifx\vec@style\fig@type\else +\@ifundefined{vec@style}{% + \def\vec#1{\ensuremath{\mathchoice + {\mbox{\boldmath$\displaystyle\mathbf{#1}$}} + {\mbox{\boldmath$\textstyle\mathbf{#1}$}} + {\mbox{\boldmath$\scriptstyle\mathbf{#1}$}} + {\mbox{\boldmath$\scriptscriptstyle\mathbf{#1}$}}}}% +} +{\def\vec#1{\ensuremath{\mathchoice + {\mbox{\boldmath$\displaystyle#1$}} + {\mbox{\boldmath$\textstyle#1$}} + {\mbox{\boldmath$\scriptstyle#1$}} + {\mbox{\boldmath$\scriptscriptstyle#1$}}}}% +} +\fi +% tensor +\def\tens#1{\relax\ifmmode\mathsf{#1}\else\textsf{#1}\fi} + +% end of proof symbol +\newcommand\qedsymbol{\hbox{\rlap{$\sqcap$}$\sqcup$}} +\newcommand\qed{\relax\ifmmode\else\unskip\quad\fi\qedsymbol} +\newcommand\smartqed{\renewcommand\qed{\relax\ifmmode\qedsymbol\else + {\unskip\nobreak\hfil\penalty50\hskip1em\null\nobreak\hfil\qedsymbol + \parfillskip=\z@\finalhyphendemerits=0\endgraf}\fi}} +% +\def\num@book{% +\renewcommand\thesection{\thechapter.\@arabic\c@section}% +\renewcommand\thesubsection{\thesection.\@arabic\c@subsection}% +\renewcommand\theequation{\thechapter.\@arabic\c@equation}% +\renewcommand\thefigure{\thechapter.\@arabic\c@figure}% +\renewcommand\thetable{\thechapter.\@arabic\c@table}% +\@addtoreset{section}{chapter}% +\@addtoreset{figure}{chapter}% +\@addtoreset{table}{chapter}% +\@addtoreset{equation}{chapter}} +% +% Ragged bottom for the actual page +\def\thisbottomragged{\def\@textbottom{\vskip\z@ \@plus.0001fil +\global\let\@textbottom\relax}} + +% This is texte.tex +% it defines various texts and their translations +% called up with documentstyle options +\def\switcht@albion{% +\def\abstractname{Summary.}% +\def\ackname{Acknowledgement.}% +\def\andname{and}% +\def\bibname{References}% +\def\lastandname{, and}% +\def\appendixname{Appendix}% +\def\chaptername{Chapter}% +\def\claimname{Claim}% +\def\conjecturename{Conjecture}% +\def\contentsname{Contents}% +\def\corollaryname{Corollary}% +\def\definitionname{Definition}% +\def\examplename{Example}% +\def\exercisename{Exercise}% +\def\figurename{Fig.}% +\def\keywordname{{\bf Key words:}}% +\def\indexname{Index}% +\def\lemmaname{Lemma}% +\def\contriblistname{List of Contributors}% +\def\listfigurename{List of Figures}% +\def\listtablename{List of Tables}% +\def\mailname{{\it Correspondence to\/}:}% +\def\noteaddname{Note added in proof}% +\def\notename{Note}% +\def\partname{Part}% +\def\prefacename{Preface}% +\def\problemname{Problem}% +\def\proofname{Proof}% +\def\propertyname{Property}% +\def\propositionname{Proposition}% +\def\questionname{Question}% +\def\refname{References}% +\def\remarkname{Remark}% +\def\seename{see}% +\def\solutionname{Solution}% +\def\subclassname{{\it Subject Classifications\/}:}% +\def\tablename{Table}% +\def\theoremname{Theorem}} +\switcht@albion +% Names of theorem like environments are already defined +% but must be translated if another language is chosen +% +% French section +\def\switcht@francais{\svlanginfo + \def\abstractname{R\'esum\'e.}% + \def\ackname{Remerciements.}% + \def\andname{et}% + \def\lastandname{ et}% + \def\appendixname{Appendice}% + \def\bibname{Bibliographie}% + \def\chaptername{Chapitre}% + \def\claimname{Pr\'etention}% + \def\conjecturename{Hypoth\`ese}% + \def\contentsname{Table des mati\`eres}% + \def\corollaryname{Corollaire}% + \def\definitionname{D\'efinition}% + \def\examplename{Exemple}% + \def\exercisename{Exercice}% + \def\figurename{Fig.}% + \def\keywordname{{\bf Mots-cl\'e:}}% + \def\indexname{Index}% + \def\lemmaname{Lemme}% + \def\contriblistname{Liste des contributeurs}% + \def\listfigurename{Liste des figures}% + \def\listtablename{Liste des tables}% + \def\mailname{{\it Correspondence to\/}:}% + \def\noteaddname{Note ajout\'ee \`a l'\'epreuve}% + \def\notename{Remarque}% + \def\partname{Partie}% + \def\prefacename{Avant-propos}% ou Pr\'eface + \def\problemname{Probl\`eme}% + \def\proofname{Preuve}% + \def\propertyname{Caract\'eristique}% +%\def\propositionname{Proposition}% + \def\questionname{Question}% + \def\refname{Lit\'erature}% + \def\remarkname{Remarque}% + \def\seename{voir}% + \def\solutionname{Solution}% + \def\subclassname{{\it Subject Classifications\/}:}% + \def\tablename{Tableau}% + \def\theoremname{Th\'eor\`eme}% +} +% +% German section +\def\switcht@deutsch{\svlanginfo + \def\abstractname{Zusammenfassung.}% + \def\ackname{Danksagung.}% + \def\andname{und}% + \def\lastandname{ und}% + \def\appendixname{Anhang}% + \def\bibname{Literaturverzeichnis}% + \def\chaptername{Kapitel}% + \def\claimname{Behauptung}% + \def\conjecturename{Hypothese}% + \def\contentsname{Inhaltsverzeichnis}% + \def\corollaryname{Korollar}% +%\def\definitionname{Definition}% + \def\examplename{Beispiel}% + \def\exercisename{\"Ubung}% + \def\figurename{Abb.}% + \def\keywordname{{\bf Schl\"usselw\"orter:}}% + \def\indexname{Sachverzeichnis}% +%\def\lemmaname{Lemma}% + \def\contriblistname{Mitarbeiter}% + \def\listfigurename{Abbildungsverzeichnis}% + \def\listtablename{Tabellenverzeichnis}% + \def\mailname{{\it Correspondence to\/}:}% + \def\noteaddname{Nachtrag}% + \def\notename{Anmerkung}% + \def\partname{Teil}% + \def\prefacename{Vorwort}% +%\def\problemname{Problem}% + \def\proofname{Beweis}% + \def\propertyname{Eigenschaft}% +%\def\propositionname{Proposition}% + \def\questionname{Frage}% + \def\refname{Literaturverzeichnis}% + \def\remarkname{Anmerkung}% + \def\seename{siehe}% + \def\solutionname{L\"osung}% + \def\subclassname{{\it Subject Classifications\/}:}% + \def\tablename{Tabelle}% +%\def\theoremname{Theorem}% +} + +\def\getsto{\mathrel{\mathchoice {\vcenter{\offinterlineskip +\halign{\hfil +$\displaystyle##$\hfil\cr\gets\cr\to\cr}}} +{\vcenter{\offinterlineskip\halign{\hfil$\textstyle##$\hfil\cr\gets +\cr\to\cr}}} +{\vcenter{\offinterlineskip\halign{\hfil$\scriptstyle##$\hfil\cr\gets +\cr\to\cr}}} +{\vcenter{\offinterlineskip\halign{\hfil$\scriptscriptstyle##$\hfil\cr +\gets\cr\to\cr}}}}} +\def\lid{\mathrel{\mathchoice {\vcenter{\offinterlineskip\halign{\hfil +$\displaystyle##$\hfil\cr<\cr\noalign{\vskip1.2\p@}=\cr}}} +{\vcenter{\offinterlineskip\halign{\hfil$\textstyle##$\hfil\cr<\cr +\noalign{\vskip1.2\p@}=\cr}}} +{\vcenter{\offinterlineskip\halign{\hfil$\scriptstyle##$\hfil\cr<\cr +\noalign{\vskip\p@}=\cr}}} +{\vcenter{\offinterlineskip\halign{\hfil$\scriptscriptstyle##$\hfil\cr +<\cr +\noalign{\vskip0.9\p@}=\cr}}}}} +\def\gid{\mathrel{\mathchoice {\vcenter{\offinterlineskip\halign{\hfil +$\displaystyle##$\hfil\cr>\cr\noalign{\vskip1.2\p@}=\cr}}} +{\vcenter{\offinterlineskip\halign{\hfil$\textstyle##$\hfil\cr>\cr +\noalign{\vskip1.2\p@}=\cr}}} +{\vcenter{\offinterlineskip\halign{\hfil$\scriptstyle##$\hfil\cr>\cr +\noalign{\vskip\p@}=\cr}}} +{\vcenter{\offinterlineskip\halign{\hfil$\scriptscriptstyle##$\hfil\cr +>\cr +\noalign{\vskip0.9\p@}=\cr}}}}} +\def\grole{\mathrel{\mathchoice {\vcenter{\offinterlineskip +\halign{\hfil +$\displaystyle##$\hfil\cr>\cr\noalign{\vskip-\p@}<\cr}}} +{\vcenter{\offinterlineskip\halign{\hfil$\textstyle##$\hfil\cr +>\cr\noalign{\vskip-\p@}<\cr}}} +{\vcenter{\offinterlineskip\halign{\hfil$\scriptstyle##$\hfil\cr +>\cr\noalign{\vskip-0.8\p@}<\cr}}} +{\vcenter{\offinterlineskip\halign{\hfil$\scriptscriptstyle##$\hfil\cr +>\cr\noalign{\vskip-0.3\p@}<\cr}}}}} +\def\bbbr{{\rm I\!R}} %reelle Zahlen +\def\bbbm{{\rm I\!M}} +\def\bbbn{{\rm I\!N}} %natuerliche Zahlen +\def\bbbf{{\rm I\!F}} +\def\bbbh{{\rm I\!H}} +\def\bbbk{{\rm I\!K}} +\def\bbbp{{\rm I\!P}} +\def\bbbone{{\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l} +{\rm 1\mskip-4.5mu l} {\rm 1\mskip-5mu l}}} +\def\bbbc{{\mathchoice {\setbox0=\hbox{$\displaystyle\rm C$}\hbox{\hbox +to\z@{\kern0.4\wd0\vrule\@height0.9\ht0\hss}\box0}} +{\setbox0=\hbox{$\textstyle\rm C$}\hbox{\hbox +to\z@{\kern0.4\wd0\vrule\@height0.9\ht0\hss}\box0}} +{\setbox0=\hbox{$\scriptstyle\rm C$}\hbox{\hbox +to\z@{\kern0.4\wd0\vrule\@height0.9\ht0\hss}\box0}} +{\setbox0=\hbox{$\scriptscriptstyle\rm C$}\hbox{\hbox +to\z@{\kern0.4\wd0\vrule\@height0.9\ht0\hss}\box0}}}} +\def\bbbq{{\mathchoice {\setbox0=\hbox{$\displaystyle\rm +Q$}\hbox{\raise +0.15\ht0\hbox to\z@{\kern0.4\wd0\vrule\@height0.8\ht0\hss}\box0}} +{\setbox0=\hbox{$\textstyle\rm Q$}\hbox{\raise +0.15\ht0\hbox to\z@{\kern0.4\wd0\vrule\@height0.8\ht0\hss}\box0}} +{\setbox0=\hbox{$\scriptstyle\rm Q$}\hbox{\raise +0.15\ht0\hbox to\z@{\kern0.4\wd0\vrule\@height0.7\ht0\hss}\box0}} +{\setbox0=\hbox{$\scriptscriptstyle\rm Q$}\hbox{\raise +0.15\ht0\hbox to\z@{\kern0.4\wd0\vrule\@height0.7\ht0\hss}\box0}}}} +\def\bbbt{{\mathchoice {\setbox0=\hbox{$\displaystyle\rm +T$}\hbox{\hbox to\z@{\kern0.3\wd0\vrule\@height0.9\ht0\hss}\box0}} +{\setbox0=\hbox{$\textstyle\rm T$}\hbox{\hbox +to\z@{\kern0.3\wd0\vrule\@height0.9\ht0\hss}\box0}} +{\setbox0=\hbox{$\scriptstyle\rm T$}\hbox{\hbox +to\z@{\kern0.3\wd0\vrule\@height0.9\ht0\hss}\box0}} +{\setbox0=\hbox{$\scriptscriptstyle\rm T$}\hbox{\hbox +to\z@{\kern0.3\wd0\vrule\@height0.9\ht0\hss}\box0}}}} +\def\bbbs{{\mathchoice +{\setbox0=\hbox{$\displaystyle \rm S$}\hbox{\raise0.5\ht0\hbox +to\z@{\kern0.35\wd0\vrule\@height0.45\ht0\hss}\hbox +to\z@{\kern0.55\wd0\vrule\@height0.5\ht0\hss}\box0}} +{\setbox0=\hbox{$\textstyle \rm S$}\hbox{\raise0.5\ht0\hbox +to\z@{\kern0.35\wd0\vrule\@height0.45\ht0\hss}\hbox +to\z@{\kern0.55\wd0\vrule\@height0.5\ht0\hss}\box0}} +{\setbox0=\hbox{$\scriptstyle \rm S$}\hbox{\raise0.5\ht0\hbox +to\z@{\kern0.35\wd0\vrule\@height0.45\ht0\hss}\raise0.05\ht0\hbox +to\z@{\kern0.5\wd0\vrule\@height0.45\ht0\hss}\box0}} +{\setbox0=\hbox{$\scriptscriptstyle\rm S$}\hbox{\raise0.5\ht0\hbox +to\z@{\kern0.4\wd0\vrule\@height0.45\ht0\hss}\raise0.05\ht0\hbox +to\z@{\kern0.55\wd0\vrule\@height0.45\ht0\hss}\box0}}}} +\def\bbbz{{\mathchoice {\hbox{$\textstyle\sf Z\kern-0.4em Z$}} +{\hbox{$\textstyle\sf Z\kern-0.4em Z$}} +{\hbox{$\scriptstyle\sf Z\kern-0.3em Z$}} +{\hbox{$\scriptscriptstyle\sf Z\kern-0.2em Z$}}}} + +\let\ts\, + +\setlength \labelsep {5\p@} +\setlength\leftmargini {17\p@} +\setlength\leftmargin {\leftmargini} +\setlength\leftmarginii {\leftmargini} +\setlength\leftmarginiii {\leftmargini} +\setlength\leftmarginiv {\leftmargini} +\setlength\labelwidth {\leftmargini} +\addtolength\labelwidth{-\labelsep} + +\def\@listI{\leftmargin\leftmargini + \parsep=\parskip + \topsep=\medskipamount + \itemsep=\parskip \advance\itemsep by -\parsep} +\let\@listi\@listI +\@listi + +\def\@listii{\leftmargin\leftmarginii + \labelwidth\leftmarginii + \advance\labelwidth by -\labelsep + \parsep=\parskip + \topsep=\z@ + \itemsep=\parskip + \advance\itemsep by -\parsep} + +\def\@listiii{\leftmargin\leftmarginiii + \labelwidth\leftmarginiii\advance\labelwidth by -\labelsep + \parsep=\parskip + \topsep=\z@ + \itemsep=\parskip + \advance\itemsep by -\parsep + \partopsep=\topsep} + +\setlength\arraycolsep{1.5\p@} +\setlength\tabcolsep{1.5\p@} + +\def\tableofcontents{\@restonecolfalse\if@twocolumn\@restonecoltrue\onecolumn + \fi\chapter*{\contentsname \@mkboth{{\contentsname}}{{\contentsname}}} + \@starttoc{toc}\if@restonecol\twocolumn\fi} + +\setcounter{tocdepth}{2} + +\def\l@part#1#2{\addpenalty{\@secpenalty}% + \addvspace{2em \@plus\p@}% + \begingroup + \parindent \z@ + \rightskip \z@ \@plus 5em + \hrule\vskip5\p@ + \bfseries\boldmath + \leavevmode + #1\par + \vskip5\p@ + \hrule + \vskip\p@ + \nobreak + \endgroup} + +\def\@dotsep{2} + +\def\addnumcontentsmark#1#2#3{% +\addtocontents{#1}{\protect\contentsline{#2}{\protect\numberline + {\thechapter}#3}{\thepage}}} +\def\addcontentsmark#1#2#3{% +\addtocontents{#1}{\protect\contentsline{#2}{#3}{\thepage}}} +\def\addcontentsmarkwop#1#2#3{% +\addtocontents{#1}{\protect\contentsline{#2}{#3}{0}}} + +\def\@adcmk[#1]{\ifcase #1 \or +\def\@gtempa{\addnumcontentsmark}% + \or \def\@gtempa{\addcontentsmark}% + \or \def\@gtempa{\addcontentsmarkwop}% + \fi\@gtempa{toc}{chapter}} +\def\addtocmark{\@ifnextchar[{\@adcmk}{\@adcmk[3]}} + +\def\l@chapter#1#2{\par\addpenalty{-\@highpenalty} + \addvspace{1.0em \@plus \p@} + \@tempdima \tocchpnum \begingroup + \parindent \z@ \rightskip \@tocrmarg + \advance\rightskip by \z@ \@plus 2cm + \parfillskip -\rightskip \pretolerance=10000 + \leavevmode \advance\leftskip\@tempdima \hskip -\leftskip + {\bfseries\boldmath#1}\ifx0#2\hfil\null + \else + \nobreak + \leaders\hbox{$\m@th \mkern \@dotsep mu.\mkern + \@dotsep mu$}\hfill + \nobreak\hbox to\@pnumwidth{\hss #2}% + \fi\par + \penalty\@highpenalty \endgroup} + +\newdimen\tocchpnum +\newdimen\tocsecnum +\newdimen\tocsectotal +\newdimen\tocsubsecnum +\newdimen\tocsubsectotal +\newdimen\tocsubsubsecnum +\newdimen\tocsubsubsectotal +\newdimen\tocparanum +\newdimen\tocparatotal +\newdimen\tocsubparanum +\tocchpnum=20\p@ % chapter {\bf 88.} \@plus 5.3\p@ +\tocsecnum=22.5\p@ % section 88.8. plus 4.722\p@ +\tocsubsecnum=30.5\p@ % subsection 88.8.8 plus 4.944\p@ +\tocsubsubsecnum=38\p@ % subsubsection 88.8.8.8 plus 4.666\p@ +\tocparanum=45\p@ % paragraph 88.8.8.8.8 plus 3.888\p@ +\tocsubparanum=53\p@ % subparagraph 88.8.8.8.8.8 plus 4.11\p@ +\def\calctocindent{% +\tocsectotal=\tocchpnum +\advance\tocsectotal by\tocsecnum +\tocsubsectotal=\tocsectotal +\advance\tocsubsectotal by\tocsubsecnum +\tocsubsubsectotal=\tocsubsectotal +\advance\tocsubsubsectotal by\tocsubsubsecnum +\tocparatotal=\tocsubsubsectotal +\advance\tocparatotal by\tocparanum} +\calctocindent + +\def\@dottedtocline#1#2#3#4#5{% + \ifnum #1>\c@tocdepth \else + \vskip \z@ \@plus.2\p@ + {\leftskip #2\relax \rightskip \@tocrmarg \advance\rightskip by \z@ \@plus 2cm + \parfillskip -\rightskip \pretolerance=10000 + \parindent #2\relax\@afterindenttrue + \interlinepenalty\@M + \leavevmode + \@tempdima #3\relax + \advance\leftskip \@tempdima \null\nobreak\hskip -\leftskip + {#4}\nobreak + \leaders\hbox{$\m@th + \mkern \@dotsep mu\hbox{.}\mkern \@dotsep + mu$}\hfill + \nobreak + \hb@xt@\@pnumwidth{\hfil\normalfont \normalcolor #5}% + \par}% + \fi} +% +\def\l@section{\@dottedtocline{1}{\tocchpnum}{\tocsecnum}} +\def\l@subsection{\@dottedtocline{2}{\tocsectotal}{\tocsubsecnum}} +\def\l@subsubsection{\@dottedtocline{3}{\tocsubsectotal}{\tocsubsubsecnum}} +\def\l@paragraph{\@dottedtocline{4}{\tocsubsubsectotal}{\tocparanum}} +\def\l@subparagraph{\@dottedtocline{5}{\tocparatotal}{\tocsubparanum}} + +\renewcommand\listoffigures{% + \chapter*{\listfigurename + \@mkboth{\listfigurename}{\listfigurename}}% + \@starttoc{lof}% + } + +\renewcommand\listoftables{% + \chapter*{\listtablename + \@mkboth{\listtablename}{\listtablename}}% + \@starttoc{lot}% + } + +\renewcommand\footnoterule{% + \kern-3\p@ + \hrule\@width 50\p@ + \kern2.6\p@} + +\newdimen\foot@parindent +\foot@parindent 10.83\p@ + +\long\def\@makefntext#1{\@setpar{\@@par\@tempdima \hsize + \advance\@tempdima-\foot@parindent\parshape\@ne\foot@parindent + \@tempdima}\par + \parindent \foot@parindent\noindent \hbox to \z@{% + \hss\hss$^{\@thefnmark}$ }#1} + +\if@spthms +% Definition of the "\spnewtheorem" command. +% +% Usage: +% +% \spnewtheorem{env_nam}{caption}[within]{cap_font}{body_font} +% or \spnewtheorem{env_nam}[numbered_like]{caption}{cap_font}{body_font} +% or \spnewtheorem*{env_nam}{caption}{cap_font}{body_font} +% +% New is "cap_font" and "body_font". It stands for +% fontdefinition of the caption and the text itself. +% +% "\spnewtheorem*" gives a theorem without number. +% +% A defined spnewthoerem environment is used as described +% by Lamport. +% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\def\@thmcountersep{.} +\def\@thmcounterend{.} +\newcommand\nocaption{\noexpand\@gobble} +\newdimen\spthmsep \spthmsep=3pt + +\def\spnewtheorem{\@ifstar{\@sthm}{\@Sthm}} + +% definition of \spnewtheorem with number + +\def\@spnthm#1#2{% + \@ifnextchar[{\@spxnthm{#1}{#2}}{\@spynthm{#1}{#2}}} +\def\@Sthm#1{\@ifnextchar[{\@spothm{#1}}{\@spnthm{#1}}} + +\def\@spxnthm#1#2[#3]#4#5{\expandafter\@ifdefinable\csname #1\endcsname + {\@definecounter{#1}\@addtoreset{#1}{#3}% + \expandafter\xdef\csname the#1\endcsname{\expandafter\noexpand + \csname the#3\endcsname \noexpand\@thmcountersep \@thmcounter{#1}}% + \expandafter\xdef\csname #1name\endcsname{#2}% + \global\@namedef{#1}{\@spthm{#1}{\csname #1name\endcsname}{#4}{#5}}% + \global\@namedef{end#1}{\@endtheorem}}} + +\def\@spynthm#1#2#3#4{\expandafter\@ifdefinable\csname #1\endcsname + {\@definecounter{#1}% + \expandafter\xdef\csname the#1\endcsname{\@thmcounter{#1}}% + \expandafter\xdef\csname #1name\endcsname{#2}% + \global\@namedef{#1}{\@spthm{#1}{\csname #1name\endcsname}{#3}{#4}}% + \global\@namedef{end#1}{\@endtheorem}}} + +\def\@spothm#1[#2]#3#4#5{% + \@ifundefined{c@#2}{\@latexerr{No theorem environment `#2' defined}\@eha}% + {\expandafter\@ifdefinable\csname #1\endcsname + {\global\@namedef{the#1}{\@nameuse{the#2}}% + \expandafter\xdef\csname #1name\endcsname{#3}% + \global\@namedef{#1}{\@spthm{#2}{\csname #1name\endcsname}{#4}{#5}}% + \global\@namedef{end#1}{\@endtheorem}}}} + +\def\@spthm#1#2#3#4{\topsep 7\p@ \@plus2\p@ \@minus4\p@ +\labelsep=\spthmsep\refstepcounter{#1}% +\@ifnextchar[{\@spythm{#1}{#2}{#3}{#4}}{\@spxthm{#1}{#2}{#3}{#4}}} + +\def\@spxthm#1#2#3#4{\@spbegintheorem{#2}{\csname the#1\endcsname}{#3}{#4}% + \ignorespaces} + +\def\@spythm#1#2#3#4[#5]{\@spopargbegintheorem{#2}{\csname + the#1\endcsname}{#5}{#3}{#4}\ignorespaces} + +\def\normalthmheadings{\def\@spbegintheorem##1##2##3##4{\trivlist + \item[\hskip\labelsep{##3##1\ ##2\@thmcounterend}]##4} +\def\@spopargbegintheorem##1##2##3##4##5{\trivlist + \item[\hskip\labelsep{##4##1\ ##2}]{##4(##3)\@thmcounterend\ }##5}} +\normalthmheadings + +\def\reversethmheadings{\def\@spbegintheorem##1##2##3##4{\trivlist + \item[\hskip\labelsep{##3##2\ ##1\@thmcounterend}]##4} +\def\@spopargbegintheorem##1##2##3##4##5{\trivlist + \item[\hskip\labelsep{##4##2\ ##1}]{##4(##3)\@thmcounterend\ }##5}} + +% definition of \spnewtheorem* without number + +\def\@sthm#1#2{\@Ynthm{#1}{#2}} + +\def\@Ynthm#1#2#3#4{\expandafter\@ifdefinable\csname #1\endcsname + {\global\@namedef{#1}{\@Thm{\csname #1name\endcsname}{#3}{#4}}% + \expandafter\xdef\csname #1name\endcsname{#2}% + \global\@namedef{end#1}{\@endtheorem}}} + +\def\@Thm#1#2#3{\topsep 7\p@ \@plus2\p@ \@minus4\p@ +\@ifnextchar[{\@Ythm{#1}{#2}{#3}}{\@Xthm{#1}{#2}{#3}}} + +\def\@Xthm#1#2#3{\@Begintheorem{#1}{#2}{#3}\ignorespaces} + +\def\@Ythm#1#2#3[#4]{\@Opargbegintheorem{#1} + {#4}{#2}{#3}\ignorespaces} + +\def\@Begintheorem#1#2#3{#3\trivlist + \item[\hskip\labelsep{#2#1\@thmcounterend}]} + +\def\@Opargbegintheorem#1#2#3#4{#4\trivlist + \item[\hskip\labelsep{#3#1}]{#3(#2)\@thmcounterend\ }} + +% initialize theorem environment + +\if@envcntshowhiercnt % show hierarchy counter + \def\@thmcountersep{.} + \spnewtheorem{theorem}{Theorem}[\envankh]{\bfseries}{\itshape} + \@addtoreset{theorem}{chapter} +\else % theorem counter only + \spnewtheorem{theorem}{Theorem}{\bfseries}{\itshape} + \if@envcntreset + \@addtoreset{theorem}{chapter} + \if@envcntresetsect + \@addtoreset{theorem}{section} + \fi + \fi +\fi + +%definition of divers theorem environments +\spnewtheorem*{claim}{Claim}{\itshape}{\rmfamily} +\spnewtheorem*{proof}{Proof}{\itshape}{\rmfamily} +% +\if@envcntsame % all environments like "Theorem" - using its counter + \def\spn@wtheorem#1#2#3#4{\@spothm{#1}[theorem]{#2}{#3}{#4}} +\else % all environments with their own counter + \if@envcntshowhiercnt % show hierarchy counter + \def\spn@wtheorem#1#2#3#4{\@spxnthm{#1}{#2}[\envankh]{#3}{#4}} + \else % environment counter only + \if@envcntreset % environment counter is reset each section + \if@envcntresetsect + \def\spn@wtheorem#1#2#3#4{\@spynthm{#1}{#2}{#3}{#4} + \@addtoreset{#1}{chapter}\@addtoreset{#1}{section}} + \else + \def\spn@wtheorem#1#2#3#4{\@spynthm{#1}{#2}{#3}{#4} + \@addtoreset{#1}{chapter}} + \fi + \else + \let\spn@wtheorem=\@spynthm + \fi + \fi +\fi +% +\let\spdefaulttheorem=\spn@wtheorem +% +\spn@wtheorem{case}{Case}{\itshape}{\rmfamily} +\spn@wtheorem{conjecture}{Conjecture}{\itshape}{\rmfamily} +\spn@wtheorem{corollary}{Corollary}{\bfseries}{\itshape} +\spn@wtheorem{definition}{Definition}{\bfseries}{\itshape} +\spn@wtheorem{example}{Example}{\itshape}{\rmfamily} +\spn@wtheorem{exercise}{Exercise}{\bfseries}{\rmfamily} +\spn@wtheorem{lemma}{Lemma}{\bfseries}{\itshape} +\spn@wtheorem{note}{Note}{\itshape}{\rmfamily} +\spn@wtheorem{problem}{Problem}{\bfseries}{\rmfamily} +\spn@wtheorem{property}{Property}{\itshape}{\rmfamily} +\spn@wtheorem{proposition}{Proposition}{\bfseries}{\itshape} +\spn@wtheorem{question}{Question}{\itshape}{\rmfamily} +\spn@wtheorem{solution}{Solution}{\bfseries}{\rmfamily} +\spn@wtheorem{remark}{Remark}{\itshape}{\rmfamily} +% +\newenvironment{theopargself} + {\def\@spopargbegintheorem##1##2##3##4##5{\trivlist + \item[\hskip\labelsep{##4##1\ ##2}]{##4##3\@thmcounterend\ }##5} + \def\@Opargbegintheorem##1##2##3##4{##4\trivlist + \item[\hskip\labelsep{##3##1}]{##3##2\@thmcounterend\ }}}{} +\newenvironment{theopargself*} + {\def\@spopargbegintheorem##1##2##3##4##5{\trivlist + \item[\hskip\labelsep{##4##1\ ##2}]{\hspace*{-\labelsep}##4##3\@thmcounterend}##5} + \def\@Opargbegintheorem##1##2##3##4{##4\trivlist + \item[\hskip\labelsep{##3##1}]{\hspace*{-\labelsep}##3##2\@thmcounterend}}}{} +\fi + +\def\@takefromreset#1#2{% + \def\@tempa{#1}% + \let\@tempd\@elt + \def\@elt##1{% + \def\@tempb{##1}% + \ifx\@tempa\@tempb\else + \@addtoreset{##1}{#2}% + \fi}% + \expandafter\expandafter\let\expandafter\@tempc\csname cl@#2\endcsname + \expandafter\def\csname cl@#2\endcsname{}% + \@tempc + \let\@elt\@tempd} + +% redefininition of the captions for "figure" and "table" environments +% +\@ifundefined{floatlegendstyle}{\def\floatlegendstyle{\bfseries}}{} +\def\floatcounterend{.\ } +\def\capstrut{\vrule\@width\z@\@height\topskip} +\@ifundefined{captionstyle}{\def\captionstyle{\normalfont\small}}{} +\@ifundefined{instindent}{\newdimen\instindent}{} + +\long\def\@caption#1[#2]#3{\par\addcontentsline{\csname + ext@#1\endcsname}{#1}{\protect\numberline{\csname + the#1\endcsname}{\ignorespaces #2}}\begingroup + \@parboxrestore\if@minipage\@setminipage\fi + \@makecaption{\csname fnum@#1\endcsname}{\ignorespaces #3}\par + \endgroup} + +\def\twocaptionwidth#1#2{\def\first@capwidth{#1}\def\second@capwidth{#2}} +% Default: .46\textwidth +\twocaptionwidth{.46\textwidth}{.46\textwidth} + +\def\leftcaption{\refstepcounter\@captype\@dblarg% + {\@leftcaption\@captype}} + +\def\rightcaption{\refstepcounter\@captype\@dblarg% + {\@rightcaption\@captype}} + +\long\def\@leftcaption#1[#2]#3{\addcontentsline{\csname + ext@#1\endcsname}{#1}{\protect\numberline{\csname + the#1\endcsname}{\ignorespaces #2}}\begingroup + \@parboxrestore + \vskip\figcapgap + \@maketwocaptions{\csname fnum@#1\endcsname}{\ignorespaces #3}% + {\first@capwidth}\ignorespaces\hspace{.073\textwidth}\hfill% + \endgroup} + +\long\def\@rightcaption#1[#2]#3{\addcontentsline{\csname + ext@#1\endcsname}{#1}{\protect\numberline{\csname + the#1\endcsname}{\ignorespaces #2}}\begingroup + \@parboxrestore + \@maketwocaptions{\csname fnum@#1\endcsname}{\ignorespaces #3}% + {\second@capwidth}\par + \endgroup} + +\long\def\@maketwocaptions#1#2#3{% + \parbox[t]{#3}{{\floatlegendstyle #1\floatcounterend}#2}} + +\def\fig@pos{l} +\newcommand{\leftfigure}[2][\fig@pos]{\makebox[.4635\textwidth][#1]{#2}} +\let\rightfigure\leftfigure + +\newdimen\figgap\figgap=0.5cm % hgap between figure and sidecaption +% +\long\def\@makesidecaption#1#2{% + \setbox0=\vbox{\hsize=\@tempdima + \captionstyle{\floatlegendstyle + #1\floatcounterend}#2}% + \ifdim\instindent<\z@ + \ifdim\ht0>-\instindent + \advance\instindent by\ht0 + \typeout{^^JClass-Warning: Legend of \string\sidecaption\space for + \@captype\space\csname the\@captype\endcsname + ^^Jis \the\instindent\space taller than the corresponding float - + ^^Jyou'd better switch the environment. }% + \instindent\z@ + \fi + \else + \ifdim\ht0<\instindent + \advance\instindent by-\ht0 + \advance\instindent by-\dp0\relax + \advance\instindent by\topskip + \advance\instindent by-11\p@ + \else + \advance\instindent by-\ht0 + \instindent=-\instindent + \typeout{^^JClass-Warning: Legend of \string\sidecaption\space for + \@captype\space\csname the\@captype\endcsname + ^^Jis \the\instindent\space taller than the corresponding float - + ^^Jyou'd better switch the environment. }% + \instindent\z@ + \fi + \fi + \parbox[b]{\@tempdima}{\captionstyle{\floatlegendstyle + #1\floatcounterend}#2% + \ifdim\instindent>\z@ \\ + \vrule\@width\z@\@height\instindent + \@depth\z@ + \fi}} +\def\sidecaption{\@ifnextchar[\sidec@ption{\sidec@ption[b]}} +\def\sidec@ption[#1]#2\caption{% +\setbox\@tempboxa=\hbox{\ignorespaces#2\unskip}% +\if@twocolumn + \ifdim\hsize<\textwidth\else + \ifdim\wd\@tempboxa<\columnwidth + \typeout{Double column float fits into single column - + ^^Jyou'd better switch the environment. }% + \fi + \fi +\fi + \instindent=\ht\@tempboxa + \advance\instindent by\dp\@tempboxa +\if t#1 +\else + \instindent=-\instindent +\fi +\@tempdima=\hsize +\advance\@tempdima by-\figgap +\advance\@tempdima by-\wd\@tempboxa +\ifdim\@tempdima<3cm + \ClassWarning{SVMono}{\string\sidecaption: No sufficient room for the legend; + ^^Jusing normal \string\caption}% + \unhbox\@tempboxa + \let\@capcommand=\@caption +\else + \ifdim\@tempdima<4.5cm + \ClassWarning{SVMono}{\string\sidecaption: Room for the legend very narrow; + ^^Jusing \string\raggedright}% + \toks@\expandafter{\captionstyle\sloppy + \rightskip=\z@\@plus6mm\relax}% + \def\captionstyle{\the\toks@}% + \fi + \let\@capcommand=\@sidecaption + \leavevmode + \unhbox\@tempboxa + \hfill +\fi +\refstepcounter\@captype +\@dblarg{\@capcommand\@captype}} +\long\def\@sidecaption#1[#2]#3{\addcontentsline{\csname + ext@#1\endcsname}{#1}{\protect\numberline{\csname + the#1\endcsname}{\ignorespaces #2}}\begingroup + \@parboxrestore + \@makesidecaption{\csname fnum@#1\endcsname}{\ignorespaces #3}\par + \endgroup} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\def\fig@type{figure} + +\def\leftlegendglue{\hfil} +\newdimen\figcapgap\figcapgap=5\p@ % vgap between figure and caption +\newdimen\tabcapgap\tabcapgap=5.5\p@ % vgap between caption and table + +\long\def\@makecaption#1#2{% + \captionstyle + \ifx\@captype\fig@type + \vskip\figcapgap + \fi + \setbox\@tempboxa\hbox{{\floatlegendstyle #1\floatcounterend}% + \capstrut #2}% + \ifdim \wd\@tempboxa >\hsize + {\floatlegendstyle #1\floatcounterend}\capstrut #2\par + \else + \hbox to\hsize{\leftlegendglue\unhbox\@tempboxa\hfil}% + \fi + \ifx\@captype\fig@type\else + \vskip\tabcapgap + \fi} + +\newcounter{merk} + +\def\endfigure{\resetsubfig\end@float} + +\@namedef{endfigure*}{\resetsubfig\end@dblfloat} + +\def\resetsubfig{\global\let\last@subfig=\undefined} + +\def\r@setsubfig{\xdef\last@subfig{\number\value{figure}}% +\setcounter{figure}{\value{merk}}% +\setcounter{merk}{0}} + +\def\subfigures{\refstepcounter{figure}% + \@tempcnta=\value{merk}% + \setcounter{merk}{\value{figure}}% + \setcounter{figure}{\the\@tempcnta}% + \def\thefigure{\if@numart\else\thechapter.\fi + \@arabic\c@merk\alph{figure}}% + \let\resetsubfig=\r@setsubfig} + +\def\samenumber{\addtocounter{\@captype}{-1}% +\@ifundefined{last@subfig}{}{\setcounter{merk}{\last@subfig}}} + +% redefinition of the "bibliography" environment +% +\def\biblstarthook#1{\gdef\biblst@rthook{#1}} +% +\AtBeginDocument{% +\ifx\secbibl\undefined + \def\bibsection{\chapter*{\refname}\markboth{\refname}{\refname}% + \addcontentsline{toc}{chapter}{\refname}% + \csname biblst@rthook\endcsname} +\else + \def\bibsection{\section*{\refname}\markright{\refname}% + \addcontentsline{toc}{section}{\refname}% + \csname biblst@rthook\endcsname} +\fi} +\ifx\oribibl\undefined % Springer way of life + \renewenvironment{thebibliography}[1]{\bibsection + \global\let\biblst@rthook=\undefined + \def\@biblabel##1{##1.} + \small + \list{\@biblabel{\@arabic\c@enumiv}}% + {\settowidth\labelwidth{\@biblabel{#1}}% + \leftmargin\labelwidth + \advance\leftmargin\labelsep + \if@openbib + \advance\leftmargin\bibindent + \itemindent -\bibindent + \listparindent \itemindent + \parsep \z@ + \fi + \usecounter{enumiv}% + \let\p@enumiv\@empty + \renewcommand\theenumiv{\@arabic\c@enumiv}}% + \if@openbib + \renewcommand\newblock{\par}% + \else + \renewcommand\newblock{\hskip .11em \@plus.33em \@minus.07em}% + \fi + \sloppy\clubpenalty4000\widowpenalty4000% + \sfcode`\.=\@m} + {\def\@noitemerr + {\@latex@warning{Empty `thebibliography' environment}}% + \endlist} + \def\@lbibitem[#1]#2{\item[{[#1]}\hfill]\if@filesw + {\let\protect\noexpand\immediate + \write\@auxout{\string\bibcite{#2}{#1}}}\fi\ignorespaces} +\else % original bibliography is required + \let\bibname=\refname + \renewenvironment{thebibliography}[1] + {\chapter*{\bibname + \@mkboth{\bibname}{\bibname}}% + \list{\@biblabel{\@arabic\c@enumiv}}% + {\settowidth\labelwidth{\@biblabel{#1}}% + \leftmargin\labelwidth + \advance\leftmargin\labelsep + \@openbib@code + \usecounter{enumiv}% + \let\p@enumiv\@empty + \renewcommand\theenumiv{\@arabic\c@enumiv}}% + \sloppy + \clubpenalty4000 + \@clubpenalty \clubpenalty + \widowpenalty4000% + \sfcode`\.\@m} + {\def\@noitemerr + {\@latex@warning{Empty `thebibliography' environment}}% + \endlist} +\fi + +\let\if@threecolind\iffalse +\def\threecolindex{\let\if@threecolind\iftrue} +\def\indexstarthook#1{\gdef\indexst@rthook{#1}} +\renewenvironment{theindex} + {\if@twocolumn + \@restonecolfalse + \else + \@restonecoltrue + \fi + \columnseprule \z@ + \columnsep 1cc + \@nobreaktrue + \if@threecolind + \begin{multicols}{3}[\chapter*{\indexname}% + \else + \begin{multicols}{2}[\chapter*{\indexname}% + \fi + {\csname indexst@rthook\endcsname}]% + \global\let\indexst@rthook=\undefined + \markboth{\indexname}{\indexname}% + \addcontentsline{toc}{chapter}{\indexname}% + \flushbottom + \parindent\z@ + \rightskip\z@ \@plus 40\p@ + \parskip\z@ \@plus .3\p@\relax + \flushbottom + \let\item\@idxitem + \def\,{\relax\ifmmode\mskip\thinmuskip + \else\hskip0.2em\ignorespaces\fi}% + \normalfont\small} + {\end{multicols} + \global\let\if@threecolind\iffalse + \if@restonecol\onecolumn\else\clearpage\fi} + +\def\idxquad{\hskip 10\p@}% space that divides entry from number + +\def\@idxitem{\par\setbox0=\hbox{--\,--\,--\enspace}% + \hangindent\wd0\relax} + +\def\subitem{\par\noindent\setbox0=\hbox{--\enspace}% second order + \kern\wd0\setbox0=\hbox{--\,--\,--\enspace}% + \hangindent\wd0\relax}% indexentry + +\def\subsubitem{\par\noindent\setbox0=\hbox{--\,--\enspace}% third order + \kern\wd0\setbox0=\hbox{--\,--\,--\enspace}% + \hangindent\wd0\relax}% indexentry + +\def\indexspace{\par \vskip 10\p@ \@plus5\p@ \@minus3\p@\relax} + +\def\subtitle#1{\gdef\@subtitle{#1}} +\def\@subtitle{} + +\def\maketitle{\par + \begingroup + \def\thefootnote{\fnsymbol{footnote}}% + \def\@makefnmark{\hbox + to\z@{$\m@th^{\@thefnmark}$\hss}}% + \if@twocolumn + \twocolumn[\@maketitle]% + \else \newpage + \global\@topnum\z@ % Prevents figures from going at top of page. + \@maketitle \fi\thispagestyle{empty}\@thanks + \par\penalty -\@M + \endgroup + \setcounter{footnote}{0}% + \let\maketitle\relax + \let\@maketitle\relax + \gdef\@thanks{}\gdef\@author{}\gdef\@title{}\let\thanks\relax} + +\def\@maketitle{\newpage + \null + \vskip 2em % Vertical space above title. +\begingroup + \def\and{\unskip, } + \parindent=\z@ + \pretolerance=10000 + \rightskip=\z@ \@plus 3cm + {\LARGE % each author set in \LARGE + \lineskip .5em + \@author + \par}% + \vskip 2cm % Vertical space after author. + {\Huge \@title \par}% % Title set in \Huge size. + \vskip 1cm % Vertical space after title. + \if!\@subtitle!\else + {\LARGE\ignorespaces\@subtitle \par} + \vskip 1cm % Vertical space after subtitle. + \fi + \if!\@date!\else + {\large \@date}% % Date set in \large size. + \par + \vskip 1.5em % Vertical space after date. + \fi + \vfill + {\Large Springer\par} + \vskip 5\p@ + \large + Berlin\enspace Heidelberg\enspace New\kern0.1em York\\ + Hong\thinspace Kong\enspace London\\ + Milan\enspace Paris\enspace Tokyo\par +\endgroup} + +% Useful environments +\newenvironment{acknowledgement}{\par\addvspace{17\p@}\small\rm +\trivlist\item[\hskip\labelsep{\it\ackname}]} +{\endtrivlist\addvspace{6\p@}} +% +\newenvironment{noteadd}{\par\addvspace{17\p@}\small\rm +\trivlist\item[\hskip\labelsep{\it\noteaddname}]} +{\endtrivlist\addvspace{6\p@}} +% +\renewenvironment{abstract}{% + \advance\topsep by0.35cm\relax\small + \labelwidth=\z@ + \listparindent=\z@ + \itemindent\listparindent + \trivlist\item[\hskip\labelsep\bfseries\abstractname]% + \if!\abstractname!\hskip-\labelsep\fi + } + {\endtrivlist} + +% define the running headings of a twoside text +\def\runheadsize{\small} +\def\runheadstyle{\rmfamily\upshape} +\def\customizhead{\hspace{\headlineindent}} + +\def\ps@headings{\let\@mkboth\markboth + \let\@oddfoot\@empty\let\@evenfoot\@empty + \def\@evenhead{\runheadsize\runheadstyle\rlap{\thepage}\customizhead + \leftmark\hfil} + \def\@oddhead{\hfil\runheadsize\runheadstyle\rightmark\customizhead + \llap{\thepage}} + \def\chaptermark##1{\markboth{{\ifnum\c@secnumdepth>\m@ne + \thechapter\thechapterend\hskip\betweenumberspace\fi ##1}}{{\ifnum %!!! + \c@secnumdepth>\m@ne\thechapter\thechapterend\hskip\betweenumberspace\fi ##1}}}%!!! + \def\sectionmark##1{\markright{{\ifnum\c@secnumdepth>\z@ + \thesection\seccounterend\hskip\betweenumberspace\fi ##1}}}} + +\def\ps@myheadings{\let\@mkboth\@gobbletwo + \let\@oddfoot\@empty\let\@evenfoot\@empty + \def\@evenhead{\runheadsize\runheadstyle\rlap{\thepage}\customizhead + \leftmark\hfil} + \def\@oddhead{\hfil\runheadsize\runheadstyle\rightmark\customizhead + \llap{\thepage}} + \let\chaptermark\@gobble + \let\sectionmark\@gobble + \let\subsectionmark\@gobble} + + +\ps@headings + +\endinput +%end of file svmono.cls From feff28f33e91aca5bd187cac6cdfc54b5ea79c1e Mon Sep 17 00:00:00 2001 From: OksanaTk Date: Wed, 20 Jul 2016 08:49:53 -0400 Subject: [PATCH 175/402] follow proper Python format --- AlexDanoff/src/math_function.py | 4 ---- 1 file changed, 4 deletions(-) diff --git a/AlexDanoff/src/math_function.py b/AlexDanoff/src/math_function.py index 33ae8de..1704546 100644 --- a/AlexDanoff/src/math_function.py +++ b/AlexDanoff/src/math_function.py @@ -7,7 +7,6 @@ def math_string(file): string = open(file).read() output = [] ranges = math_mode.find_math_ranges(string) - #print ranges for i in ranges: new = string[i[0]:i[1]] output.append(new) @@ -100,7 +99,6 @@ def formatting(file_str): previous = line wrote = "\n".join(updated) - wrote = wrote # remove consecutive blank lines and blank lines between \index groups spaces_pat = re.compile(r'\n{2,}[ ]?\n+') @@ -108,5 +106,3 @@ def formatting(file_str): wrote = re.sub(r'\\index{(.*?)}\n\n\\index{(.*?)}', r'\\index{\1}\n\\index{\2}', wrote) return wrote - -#formatting(open('/home/ont1/DLMF/25.ZE/newMoritz').read()) \ No newline at end of file From 148bdbf8d7532f66e0bcb24ed0be3d01ee09fa71 Mon Sep 17 00:00:00 2001 From: OksanaTk Date: Wed, 20 Jul 2016 08:54:56 -0400 Subject: [PATCH 176/402] remove unnecessary comments --- AlexDanoff/src/replace_special.py | 7 ------- 1 file changed, 7 deletions(-) diff --git a/AlexDanoff/src/replace_special.py b/AlexDanoff/src/replace_special.py index 930d3f2..5123917 100644 --- a/AlexDanoff/src/replace_special.py +++ b/AlexDanoff/src/replace_special.py @@ -47,7 +47,6 @@ def main(): # Below: index_str writes to output file, math_string takes output file as input, and change_original # writes to the output based on the previous output file - # print math_mode.find_math_ranges(open(fname).read()) unchanged_math = math_function.math_string(fname) math_string = remove_special(unchanged_math) @@ -192,7 +191,6 @@ def remove_special(content): function = re.sub(r'\\index{(.*?)}\n\n\\index{(.*?)}', r'\\index{\1}\n\\index{\2}', function) content[counter] = function - # print 'content[counter]', content[counter] counter += 1 return "\n".join(content) @@ -239,11 +237,6 @@ def _replace_i(words): # one (but not both) of the surrounding characters IS alphabetic, may need to replace if any(s.isalpha() for s in surrounding): - """"# character before is alphabetic - if surrounding[0].isalpha(): - print('caught') # below was replacing i in ch 16 commands, shraeya wrote unnecessary if char before is vowel - # if surrounding[0] in "aeiou": - # replacement = r'\iunit'""" if surrounding[1].isalpha(): # character after is alphabetic From f5c3f14b00b67901e3aa3d6e7b0645edfdbbb27b Mon Sep 17 00:00:00 2001 From: Oksana Tkach Date: Wed, 20 Jul 2016 10:01:14 -0400 Subject: [PATCH 177/402] Temporary commit --- AlexDanoff/src/math_function.py | 18 ++++++++--------- AlexDanoff/test/test_math_function.py | 28 +++++++++++++++++++++++++++ 2 files changed, 37 insertions(+), 9 deletions(-) create mode 100644 AlexDanoff/test/test_math_function.py diff --git a/AlexDanoff/src/math_function.py b/AlexDanoff/src/math_function.py index 33ae8de..2d8d08f 100644 --- a/AlexDanoff/src/math_function.py +++ b/AlexDanoff/src/math_function.py @@ -2,12 +2,14 @@ import string import re -def math_string(file): - # Takes input file and returns list of strings when in math mode - string = open(file).read() +def math_string(in_file): + # Takes input file or string and returns list of strings when in math mode + try: #test if file or string + string = open(in_file).read() + except: + string = in_file output = [] - ranges = math_mode.find_math_ranges(string) - #print ranges + ranges = math_mode.find_math_ranges(string, drmf=True) for i in ranges: new = string[i[0]:i[1]] output.append(new) @@ -17,7 +19,7 @@ def math_string(file): def change_original(o_file, changed_math_string): # Places changed string from math mode back into place in the original function o_string = open(o_file).read() - ranges = math_mode.find_math_ranges(o_string) + ranges = math_mode.find_math_ranges(o_string, drmf=True) num = 0 edited = o_string for i in ranges: @@ -37,7 +39,7 @@ def formatting(file_str): in_ind = False previous = "" - ranges = math_mode.find_math_ranges(file_str) + ranges = math_mode.find_math_ranges(file_str, drmf=True) lines = file_str.split('\n') in_eq = False @@ -100,7 +102,6 @@ def formatting(file_str): previous = line wrote = "\n".join(updated) - wrote = wrote # remove consecutive blank lines and blank lines between \index groups spaces_pat = re.compile(r'\n{2,}[ ]?\n+') @@ -109,4 +110,3 @@ def formatting(file_str): return wrote -#formatting(open('/home/ont1/DLMF/25.ZE/newMoritz').read()) \ No newline at end of file diff --git a/AlexDanoff/test/test_math_function.py b/AlexDanoff/test/test_math_function.py new file mode 100644 index 0000000..59d024e --- /dev/null +++ b/AlexDanoff/test/test_math_function.py @@ -0,0 +1,28 @@ +from math_function import math_string +from unittest import TestCase + +class TestMathString(TestCase): + def test_math_string(self): + before = """ + \paragraph{Arithmetic Progression} + \index{arithmetic progression} + + \begin{equation}\label{eq:AL.ES.AR} + a + (a + d) + (a + 2d) + \dots + (a + (n-1)d) + = na + \tfrac{1}{2} n(n-1) d + = \tfrac{1}{2} n (a + \ell) + \end{equation} + + where $\ell$ = last term of the series = $a + (n-1)d$. + + \paragraph{Geometric Progression} + \index{geometric progression (or series)}""" + + after = ["""a + (a + d) + (a + 2d) + \dots + (a + (n-1)d) + = na + \tfrac{1}{2} n(n-1) d + = \tfrac{1}{2} n (a + \ell)""", "\ell", "a + (n-1)d"] + + math = math_string(before) + print 'math', math + print 'after', after + self.assertEqual(after, math) From bc486b97abcdba8248fa1935bd27966f891afadb Mon Sep 17 00:00:00 2001 From: Jagan Prem Date: Wed, 20 Jul 2016 10:57:52 -0400 Subject: [PATCH 178/402] Complete test cases and add formatting for author and status. --- DLMF_preprocessing/src/math_mode.py | 3 +++ DLMF_preprocessing/test/test_math_mode.py | 10 +++++----- 2 files changed, 8 insertions(+), 5 deletions(-) diff --git a/DLMF_preprocessing/src/math_mode.py b/DLMF_preprocessing/src/math_mode.py index 7927bc0..a68a847 100644 --- a/DLMF_preprocessing/src/math_mode.py +++ b/DLMF_preprocessing/src/math_mode.py @@ -1,3 +1,6 @@ +__author__ = "Jagan Prem" +__status__ = "Production" + # Dictionary containing math mode delimiters and their respective endpoints. MATH_START = {"\\[": "\\]", "\\(": "\\)", diff --git a/DLMF_preprocessing/test/test_math_mode.py b/DLMF_preprocessing/test/test_math_mode.py index 3008742..67b1d16 100644 --- a/DLMF_preprocessing/test/test_math_mode.py +++ b/DLMF_preprocessing/test/test_math_mode.py @@ -45,14 +45,14 @@ NON_MATH_TESTS = [ { - "string": "\\mbox{Test\\(test\\)} ", + "string": "\\mbox{\\$Test\\(test\\)} ", "start": 14, - "output": 19 + "output": 21 }, { - "string": "test \\\\(\\begin{align*}\\end{align*}\\\\[", + "string": "test{} \\\\(\\begin{align*}\\end{align*}\\\\[", "start": 56, - "output": 37 + "output": 39 }, { "string": "\\begin{multline}\n\\left(2\\right)\n\\end{multline}", @@ -60,7 +60,7 @@ "output": 46 }, [ - (26, 30), + (28, 32), (19, 35) ] ] From 0bd62baa6819461897aff6b6a879f352c85acdac Mon Sep 17 00:00:00 2001 From: Parth Oza Date: Wed, 20 Jul 2016 11:00:06 -0400 Subject: [PATCH 179/402] Update CONTRIBUTING.md for status things --- CONTRIBUTING.md | 2 ++ 1 file changed, 2 insertions(+) diff --git a/CONTRIBUTING.md b/CONTRIBUTING.md index 258684e..4e75335 100644 --- a/CONTRIBUTING.md +++ b/CONTRIBUTING.md @@ -43,6 +43,8 @@ PyCharm can help resolve merge conflicts. To resolve conflicts in your project, Code should be stored in a "src" folder. +All Python source files should have `__author__` and `__status__` set to the author's name and development status - either "Prototype", for the most basic, incomplete code, "Development", for code that is currently being worked on, and "Production" for code that is ready to be used without any changes. If there were multiple people working on the code, also set `__credits__` to a Python list of people who have worked on the code. + All subprojects must have README.md files. A guide may be found [here](https://gist.github.com/PurpleBooth/109311bb0361f32d87a2). Note that some of these requirements may not make sense to put in your README.md. Make good use of .gitignore. Anything that has been generated by code in the repository (compiled code, test files, test output, etc.) should be in your .gitignore. The easiest way to do this is to move these files into a folder, and gitignore that folder. From 638255c159194304a030a867d42919f04be39d03 Mon Sep 17 00:00:00 2001 From: Oksana Tkach Date: Wed, 20 Jul 2016 11:06:14 -0400 Subject: [PATCH 180/402] -Run unittests for file math_function.py -Fix text removal error in math_function.py -Add \label to list of commands bringing out of math mode in math_mode.py --- AlexDanoff/test/test_math_function.py | 28 ----------- DLMF_preprocessing/src/math_function.py | 21 +++++--- DLMF_preprocessing/src/math_mode.py | 3 +- DLMF_preprocessing/test/test_math_function.py | 49 +++++++++++++++++++ 4 files changed, 64 insertions(+), 37 deletions(-) delete mode 100644 AlexDanoff/test/test_math_function.py create mode 100644 DLMF_preprocessing/test/test_math_function.py diff --git a/AlexDanoff/test/test_math_function.py b/AlexDanoff/test/test_math_function.py deleted file mode 100644 index 59d024e..0000000 --- a/AlexDanoff/test/test_math_function.py +++ /dev/null @@ -1,28 +0,0 @@ -from math_function import math_string -from unittest import TestCase - -class TestMathString(TestCase): - def test_math_string(self): - before = """ - \paragraph{Arithmetic Progression} - \index{arithmetic progression} - - \begin{equation}\label{eq:AL.ES.AR} - a + (a + d) + (a + 2d) + \dots + (a + (n-1)d) - = na + \tfrac{1}{2} n(n-1) d - = \tfrac{1}{2} n (a + \ell) - \end{equation} - - where $\ell$ = last term of the series = $a + (n-1)d$. - - \paragraph{Geometric Progression} - \index{geometric progression (or series)}""" - - after = ["""a + (a + d) + (a + 2d) + \dots + (a + (n-1)d) - = na + \tfrac{1}{2} n(n-1) d - = \tfrac{1}{2} n (a + \ell)""", "\ell", "a + (n-1)d"] - - math = math_string(before) - print 'math', math - print 'after', after - self.assertEqual(after, math) diff --git a/DLMF_preprocessing/src/math_function.py b/DLMF_preprocessing/src/math_function.py index 2d8d08f..3cdfe88 100644 --- a/DLMF_preprocessing/src/math_function.py +++ b/DLMF_preprocessing/src/math_function.py @@ -2,14 +2,16 @@ import string import re + def math_string(in_file): # Takes input file or string and returns list of strings when in math mode - try: #test if file or string + try: # test if file or string string = open(in_file).read() except: string = in_file output = [] - ranges = math_mode.find_math_ranges(string, drmf=True) + ranges = math_mode.find_math_ranges(string) + print ranges for i in ranges: new = string[i[0]:i[1]] output.append(new) @@ -18,8 +20,11 @@ def math_string(in_file): def change_original(o_file, changed_math_string): # Places changed string from math mode back into place in the original function - o_string = open(o_file).read() - ranges = math_mode.find_math_ranges(o_string, drmf=True) + try: + o_string = open(o_file).read() + except: + o_string = o_file + ranges = math_mode.find_math_ranges(o_string) num = 0 edited = o_string for i in ranges: @@ -39,7 +44,7 @@ def formatting(file_str): in_ind = False previous = "" - ranges = math_mode.find_math_ranges(file_str, drmf=True) + ranges = math_mode.find_math_ranges(file_str) lines = file_str.split('\n') in_eq = False @@ -68,10 +73,10 @@ def formatting(file_str): else: # not in eq - print line if not been_in_eq: # if text line != command if re.match(commands, line) or line == '': + if '\\paragraph' not in line and '\\acknowledgements' not in line: updated.append(line) else: # not in eq but already been in eq @@ -81,7 +86,8 @@ def formatting(file_str): if past_last > ranges[-1][1]: # In section after last eq if re.match(commands,line) or line == '': - updated.append(line) + if '\\paragraph' not in line: + updated.append(line) # if this line is an index start storing it,or write it if we're done with the indexes if IND_START in line: @@ -96,7 +102,6 @@ def formatting(file_str): if previous.strip() != "": ind_str = "\n" + ind_str - fullsplit = ind_str.split("\n") ind_str = "" previous = line diff --git a/DLMF_preprocessing/src/math_mode.py b/DLMF_preprocessing/src/math_mode.py index 7927bc0..ab8d1c6 100644 --- a/DLMF_preprocessing/src/math_mode.py +++ b/DLMF_preprocessing/src/math_mode.py @@ -13,7 +13,8 @@ # List of delimiters that exit math mode. MATH_END = ["\\hbox{", "\\mbox{", - "\\text{"] + "\\text{", + "\\label{"] def find_first(string, delim, start=0): diff --git a/DLMF_preprocessing/test/test_math_function.py b/DLMF_preprocessing/test/test_math_function.py new file mode 100644 index 0000000..9a3c2de --- /dev/null +++ b/DLMF_preprocessing/test/test_math_function.py @@ -0,0 +1,49 @@ +import math_function +from unittest import TestCase + + +before = """ +\\paragraph{Arithmetic Progression} +\\index{arithmetic progression} + +\\begin{equation}\\label{eq:AL.ES.AR} + a + (a + d) + (a + 2d) + \\dots + (a + (n-1)d) + = na + \\tfrac{1}{2} n(n-1) d + = \\tfrac{1}{2} n (a + \\ell) +\\end{equation} + +where $\\ell$ = last term of the series = $a + (n-1)d$. + +\\paragraph{Geometric Progression} +\\index{geometric progression (or series)}""" + +just_math = [""" + a + (a + d) + (a + 2d) + \\dots + (a + (n-1)d) + = na + \\tfrac{1}{2} n(n-1) d + = \\tfrac{1}{2} n (a + \\ell)\n""", "\\ell", "a + (n-1)d"] + +no_text = """ +\\index{arithmetic progression} + +\\begin{equation}\\label{eq:AL.ES.AR} + a + (a + d) + (a + 2d) + \\dots + (a + (n-1)d) + = na + \\tfrac{1}{2} n(n-1) d + = \\tfrac{1}{2} n (a + \\ell) +\\end{equation} + +\\index{geometric progression (or series)}""" + + + +class TestMathString(TestCase): + def test_math_string(self): + math = math_function.math_string(before) + self.assertEqual(just_math, math) + + def test_change_original(self): + c_o = math_function.change_original(before, just_math) + self.assertEqual(before, c_o) + + def test_formatting(self): + formatted = math_function.formatting(before) + self.assertEqual(no_text, formatted) From 8c0f1802df42f624da0f1120ef3b16c22f11884c Mon Sep 17 00:00:00 2001 From: Jagan Prem Date: Wed, 20 Jul 2016 11:20:19 -0400 Subject: [PATCH 181/402] Add to test case and made small fix in source. --- DLMF_preprocessing/src/math_mode.py | 1 + DLMF_preprocessing/test/test_math_mode.py | 11 ++++++----- 2 files changed, 7 insertions(+), 5 deletions(-) diff --git a/DLMF_preprocessing/src/math_mode.py b/DLMF_preprocessing/src/math_mode.py index a68a847..5017006 100644 --- a/DLMF_preprocessing/src/math_mode.py +++ b/DLMF_preprocessing/src/math_mode.py @@ -100,6 +100,7 @@ def parse_math(string, start, ranges): ranges.append((begin, start + i)) i += parse_non_math(string[i:], start + i, ranges) begin = start + i + i -= 1 if string[i:].startswith(MATH_START[delim]): if begin != start + i: ranges.append((begin, start + i)) diff --git a/DLMF_preprocessing/test/test_math_mode.py b/DLMF_preprocessing/test/test_math_mode.py index 67b1d16..2308e51 100644 --- a/DLMF_preprocessing/test/test_math_mode.py +++ b/DLMF_preprocessing/test/test_math_mode.py @@ -27,9 +27,9 @@ "output": 10 }, { - "string": "\\[2\\hbox{Test}\\]", + "string": "\\[2\\hbox{Test}\\\\]\\]", "start": 22, - "output": 15 + "output": 18 }, { "string": "$\\$$", @@ -39,15 +39,16 @@ [ (1, 10), (24, 25), + (36, 39), (13, 15) ] ] NON_MATH_TESTS = [ { - "string": "\\mbox{\\$Test\\(test\\)} ", + "string": "\\mbox{\\$Test\\(test\\\\)\\)} ", "start": 14, - "output": 21 + "output": 24 }, { "string": "test{} \\\\(\\begin{align*}\\end{align*}\\\\[", @@ -60,7 +61,7 @@ "output": 46 }, [ - (28, 32), + (28, 35), (19, 35) ] ] From 9e709a76fe766ada719b2e520e7f1ea04f9c45b9 Mon Sep 17 00:00:00 2001 From: Oksana Tkach Date: Wed, 20 Jul 2016 11:27:59 -0400 Subject: [PATCH 182/402] remove unnecessary code --- DLMF_preprocessing/src/math_function.py | 4 ---- 1 file changed, 4 deletions(-) diff --git a/DLMF_preprocessing/src/math_function.py b/DLMF_preprocessing/src/math_function.py index 7809196..2fa1dea 100644 --- a/DLMF_preprocessing/src/math_function.py +++ b/DLMF_preprocessing/src/math_function.py @@ -11,10 +11,6 @@ def math_string(in_file): string = in_file output = [] ranges = math_mode.find_math_ranges(string) -<<<<<<< HEAD:DLMF_preprocessing/src/math_function.py - print ranges -======= ->>>>>>> 148bdbf8d7532f66e0bcb24ed0be3d01ee09fa71:AlexDanoff/src/math_function.py for i in ranges: new = string[i[0]:i[1]] output.append(new) From cfac6aed300aab96b4d8564537348f80f587167c Mon Sep 17 00:00:00 2001 From: Joon Bang Date: Thu, 7 Jul 2016 10:09:25 -0400 Subject: [PATCH 183/402] Create modded Azeem code to update only Main Page --- KLS-main-page/KLS_main_page.py | 14 ++ KLS-main-page/headers.py | 55 +++++ KLS-main-page/list.py | 86 ++++++++ KLS-main-page/modMain.py | 117 +++++++++++ KLS-main-page/symbolsList2.py | 373 +++++++++++++++++++++++++++++++++ 5 files changed, 645 insertions(+) create mode 100644 KLS-main-page/KLS_main_page.py create mode 100644 KLS-main-page/headers.py create mode 100644 KLS-main-page/list.py create mode 100644 KLS-main-page/modMain.py create mode 100644 KLS-main-page/symbolsList2.py diff --git a/KLS-main-page/KLS_main_page.py b/KLS-main-page/KLS_main_page.py new file mode 100644 index 0000000..fc1722a --- /dev/null +++ b/KLS-main-page/KLS_main_page.py @@ -0,0 +1,14 @@ +import os + +inp = raw_input("Update Main Page? (y or n)") +if inp != "n": + print("List") + os.system("python list.py") + print("Symbols List") + os.system("python symbolsList2.py") + print("Headers") + os.system("python headers.py") + print("List Ways") + os.system("python modMain.py") + +print("Done") diff --git a/KLS-main-page/headers.py b/KLS-main-page/headers.py new file mode 100644 index 0000000..bdaac81 --- /dev/null +++ b/KLS-main-page/headers.py @@ -0,0 +1,55 @@ +file = open("main_page.mmd", "r") +lines = file.read() +listStart = lines.find("= Definition Pages =") +listStart = lines.find("* [[Definition:", listStart) +listEnd = lines.find("" in h: + h = h[h.find(">>
\n

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